
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00032912/00001
Material Information
 Title:
 Time, energy and spacedependent neutron thermalization theory
 Creator:
 Kunaisch, Hassan Houseinn, 1933
 Publication Date:
 1964
 Language:
 English
 Physical Description:
 vii, 162 leaves : illustrations ; 28 cm
Subjects
 Subjects / Keywords:
 Adjoints ( jstor )
Approximation ( jstor ) Eigenvalues ( jstor ) Graphite ( jstor ) Matrices ( jstor ) Neutron thermalization ( jstor ) Neutrons ( jstor ) Polynomials ( jstor ) Subroutines ( jstor ) Time dependence ( jstor ) Neutrons ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaves 157161).
 General Note:
 Manuscript copy.
 General Note:
 Vita.
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 University of Florida
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 University of Florida
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 Classification:
 539.7213 K96t ( ddc )

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Full Text 
TIME, ENERGY AND SPACEDEPENDENT NEUTRON THERMALIZATION THEORY
By
HASSAN H. KUNAISH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1964
ACKNOWLEDGMENTS
The author wishes to acknowledge the guidance of the members of his supervising committee. He gratefully acknowledges the continuous guidance and invaluable advice of his committee chairman, Dr. R. B. Perez, without whose aid this work would have been impossible.
The author is indebted to the American Friends of the Middle East, Inc., and the Syrian Government for a scholarship during his graduate study toward the Ph.D. degree. He wishes to acknowledge a partial financial support by the University of Florida Computing Center for the computations in this work. He also wishes to thank Mr. R. Booth for permitting him to use his experimental results.
Special thanks are due Mrs. Gail Gyles and
Miss Barbara Gyles for their patience and enthusiasm during the typing of this manuscript.
Last, but not least, the author would like to express his deep appreciation for the encouragement and understanding of his wife, Subhieh, throughout the progress of this dissertation.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ........ ....................... ii
LIST OF FIGURES ........ .................... iv
ABSTRACT ............................. . . . v
Chapter
I. INTRODUCTION ...... ................ 1
II. THE THERMALIZATION MODEL OF THE CONSISTENT
P1 APPROXIMATION OF THE NEUTRON TRANSPORT
EQUATION ....... .................. 5
III. NEUTRON WAVES IN MODERATING SYSTEMS ..... 23
IV. PULSED NEUTRONS IN MODERATING SYSTEMS ..60 V. RESULTS AND CONCLUSIONS ... ........... 87
APPENDIXES
A OPERATORS IN GENERAL FORMS ....
B HEAVY GAS 1/v MODEL ... ........
C CALCULATION OF THE MATRICES
D COMPUTING CODES ..............
LIST OF REFERENCES ..............
BIOGRAPHICAL SKETCH ..... .............
102 109
* . . I . 118
* . . . . 126 . . .. . 157
* . . . . 162
iii
ï¿½ ï¿½ i Q ï¿½
LIST OF FIGURES
Figure Page
1. The Steady State Inverse Relaxation
Length vs. Number of Laguerre
Polynomials ..... ............... . 90
2. Components of the Complex Inverse Relaxation Length of the Fundamental Mode vs.
Source Frequency..... . . . . . . . 91
3. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 100 cps . . . . . 93
4. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 500 cps ..... .. 94
5. Phase Shift of Neutron Density vs.
Distance from the Source ........ ..96
6. Neutron Spectra at Various Positions
along the Central Axis of the Graphite
Assembly . .................. 97
7. Decay Constant, X , of the Fundamental
Mode of the Flux in the Neutronpulsed
Graphite System ................ 98
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
TIME, ENERGY AND SPACEDEPENDENT NEUTRON THERMALIZATION THEORY
By
Hassan H. Kunaish
December, 1964
Chairman: Dr. Rafael B. Perez Major Department: Nuclear Engineering
The time, energy and spacedependent neutron thermalization theory in moderating assemblies is developed in the consistent P1 approximation of the Boltzmann equation of neutron transport. In order to evaluate the scattering integral involved in this equation the neutron flux is analyzed into two components according to the expression
(rE, t) = m(E) 0 (r,E, t).
The Maxwellian function, m(E), represents the energy spectrum of neutrons in a nonabsorbing infinite medium. In a finite medium with a small absorption cross section the energy spectrum of the neutron flux resembles a Maxwellian distribution. Therefore, the nonMaxwellian
component, *(r,E,t), is expected to be a smooth function of energy, E, and hence its value at a different energy, E', can be well approximated by a finite Taylor series expansion about R. This analysis and the detailed balance principle are used in order to evaluate the scattering integral. The resulting form of the transport equation is a fourth order differential equation with respect to time, energy and space.
This theory is used in two applications of
practical interest, the neutron waves and the pulsed neutrons experiments. The use of the cosine functions in expressing the spacial dependence of the neutron flux leads, in both cases, to the modal equation of the flux as a function of time and energy. The energy dependence of the flux modes is then developed in terms of the associated Laguerre polynomials. This procedure leads to two similar complicated eigenvalue problems.
The source condition in the neutron waves problem is established from the P1 approximation and is used to determine the modal constants. In order to account for transient behavior of the pulsed systems the principle of neutron conservation is used and the final solution of the time dependent flux is obtained by the technique of Laplace transformation.
Numerical computations are made for both
applications using a heavy gas scattering kernel and a 1/v absorption cross section in graphite in order to test the applicability of this model and its compatibility with the P1 approximation. The comparison of these results with the experimental ones confirms that this model is conveniently accurate in describing the pulsed neutron experiment in relatively large systems of buckling up to 3x103 cm2. This model is also proven to provide accurate description of the neutron waves experiment with source frequencies up to 500 cycles per second which are equivalent to transients of periods down to about 2 milliseconds.
vii
CHAPTER I
INTRODUCTION
The theory of neutron thermalization has been given much emphasis in recent years. The interest of investigators has been focused on two objectives. The first objective is the theoretical and experimental understanding of the scattering law of thermal neutrons in various materials, while the second objective is to use the best available information about the scattering law in order to study the details of the neutron thermalization process and the neutron spectra which are important in determining the behavior and properties of nuclear systems. The mathematical framework of the latter is the development of an adequate solution for the Boltzmann equation describing the neutron transport phenomenon. The work presented in this dissertation can be classified in the second category.
In solving this equation investigators followed several approaches and based their solutions on certain assumptions in order to reduce the complexity of the mathematics involved in the problem. Effort was made,
1
2
however, in order for the problem to retain certain aspects of its physics depending on the goal of the investigator.
Wigner and Wilkins (l, 2)* established the monoatomic gas model of the scattering kernel on which investigators L3  17), and many others, based their works later on. Using this model, Barnard et al. (16) studied the time dependent neutron spectra in graphite varying the moderator to neutron mass ratio. The analytically calculated spectra for a fictitious mass ratio of 33 were in good agreement with the experimental results while for the mass ratio of 12 this agreement disappeared at times less than 300 u sec after the pulse and for energies below 0.1 ev.
Corngold et a () and Shapiro (14) showed
that, in the analysis of the time dependent thermalization problem in the neutron pulse technique, the consideration of a discrete set of eigenvalues is not adequate and continuous eigenvalues should also be considered for complete analysis. However, the question of how important these continuous eigenvalues and
* Underlined numbers will be used to indicate
references cited in the List of References of this
dissertation.
3
the corresponding eigenfunctions are has not been answered yet. Shapiro also found that the velocity associated Laguerre polynomials give faster convergence than the energy polynomials.
The technique of pulsed neutrons used by many investigators (Q  35) and others, has been reviewed by Beckurts (al'). The oscillatory neutron source method was first developed by Raievski and Horowitz
( 6). Using this technique Uhrig (37) studied the nuclear properties of subcritical assemblies while Perez and Uhrig (384 investigated the thermalization theory and calculated the thermalization parameters. They also showed the similarity between this technique and the pulsed neutrons technique from the analytical point of view. Experimental investigations have been carried out by Perez et al (39, 40) and Booth (41). According to Hetrick and Seale in a more recent publication (42} it is possible to calculate the specific heat integral and the zeropoint mean square displacement of the scattering atoms in terms of the diffusion coefficient and the second energy transfer moment by using this technique with low temperature experiments.
4
The objectives of this work are to formulate the neutron thermalization theory in the consistent P1 approximation and then utilize this formalism in both neutron waves and pulsed neutrons techniques. In Chapter II the thermalization model of the PN approximation is derived in general operational form and the P1 approximation is then considered in detail. The resulting form of the Boltzmann equation is a fourth order differential equation in space, energy and time.
The application of this equation to the neutron wave problem in Chapter III and to pulsed systems in Chapter IV using the Laguerre polynomials leads to rather complicated eigenvalue problems. In both cases the general solution is obtained with no specific assumptions about the scattering kernel. The results presented in Chapter V, however, are obtained for graphite in the heavy gas approximation and 1/v absorption.
The operators involved in the theory are
derived and discussed in Appendix A and calculated explicitly in Appendix B for the heavy gas scattering kernel. Finally, the computational scheme of the matrices involved in both applications is shown in Appendix C, while the IBM709 codes are listed in Appendix D.
CHAPTER II
THE THERMALIZATION MODEL OF THE CONSISTENT
P1 APPROXIMATION OF THE NEUTRON TRANSPORT EQUATION
General Theory
The space, direction, energy and time dependent Boltzmann equation describing the neutron flux in a homogeneous isotropic medium is
1
v at t
S(r, ,R,t) + f dE' f dQ' Zs(E'4E, ' ) (r,f',E',t) (2.1)
where r, , E, v and t are the neutron position vector, direction, energy, velocity and time, respectively. Primed symbols indicate the corresponding variables after a scattering collision. The terms appearing in Eq. (2.1) are defined in the following
1 a.
 *(rQE,t) d'dgdE the time rate of change of v at 1ha no+.nna Whieh nrnc
the space dr dE dQ,
the loss ratedue to the net transport of neutrons through the boundaries of
dil,
d' didE
E t(E) f('r, , E, t) dr d~dE
S (Ii,E,t) drdildE
f dE'f dQï¿½ Zs(E'EQ') E'l
x 0 (r.,' E ,t) drd~dE
the loss rate due to absorption, scattering out and slowing down,
rate of contribution from all sources in the space dr dQ dE,
contribution into dr dQ dE due to scattering from all directions n' and all energies E'.
The total space, energy and time dependent scalar flux is obtained by integrating the directional flux over all directions
J',Et)= f *('r,'a,E,t) dQ
(2.2)
In order to eliminate the angular (direction) dependence from Eq. (2.1) the flux, the source, and the scattering kernel are expanded in terms of the spherical harmonics
4.m n n
S(r, ,E,t) = #m(rEt) Y (a)
m=o n=m
S(r, QE,t)= Sn (r,E,t) y('n m m
mino nmr
(2.3)
(2.4)
S2m+l
2 E M m(E')E)P M()) m0m
m=o
 m n(4. n' 25
= r m(E' E)Ym (a)Y (a (2.5)
m=o n=m
where
o Cos eo (2.6)
is the cosine of the scattering angle in the laboratory system.
The spherical harmonics, yn (t), the Legendre
m
polynomials, Pm(vo), and their associated functions,
n
Pm(p), satisfy the following relations, see references (43) and (44).
ni n H n n eJnq (2.7) m 01) ym (I , %ï¿½) _ mPm (p)e
Wt+ n* n n jny n 4a) PYm (Pm()e = (nyn (') (2.8)
f y* 6 (2.9)
m p mp nq
Hn [(2m+l) (mn) 1/4w (m+n)!] /2 (2.10)
m
Hm = ( (m+n) ! / (mn) I] iHm
dn
sinn P(U)
(2.13)
M (mn) ! n , Jn(TY') Imr+n) ! (V PM(" ( )e
QI *z
and where 8, f, 8' and T' are, respectively, the azimuthal and longitudinal angles of the directions Q and n'.
The moments of the flux, the source and the scattering kernel are given by
n4
n (r,E,t) = f Ym(fl)S(r,',E,t)d m n
Sm(rE,t) . f Ym(Q)S(r,fl,,E,t)dQ
(2.17) (2.18)
pn Pm
(2.11)
(2.12)
Pm (o0) =
where
(2.14)
1 o= cos z
UI = COS 8' =
(2.15)
(2.16)
pn(,) =(ln (mn) I/ (r+n) I!]Pm(n
1 P+, E m(E I +E) f p )1 E  ,a')d
(2.19)
1
The use of Eqs. (2.3)(2.5) and the properties of the spherical harmonics expressed by Eqs. (2.7)(2.14) along with
07 = cos + sine cos  + sine sin
az ax By (2.20)
leads to the formulation of a set of coupled integrodifferential equations relating the moments of the flux up to any order of approximation, i.e.,
_[i _!+Et(E)],E t)+SmnrE t)+Inlr,E,t) v at m
(m+ln) (r+l+n) A~z) (2m+3) (2m+l)
n ( ) (mn) (m4+n) h n1 m+l. rI (2m+l)(2m ) Y m1 t
1 [m+n+2 (m+n+1) n+ + Et) _ (mn) (mn1) n+ l
(2m+3) (2m+l) m+l (2m+l) (2ml)m E
(m+n)_(m+n) 1 (mn+2) (mn+l) )5n
2, [# f ,(2m+l) (2 .., rEt) 1. (2m+3) ( 2Rm+l)\ +lCLDF t
(2.21)
where
In(rEt) = E L,(EI +E)n( E't) 'm (2.22)
Et
A(Z) = .(2.23)
3z
B(xy) (2.24)
ax
B*(xy) a (2.25)
ax a
In an infinite moderating medium with no
absorption the energy spectrum of the neutron flux is given by the Maxwellian function
m(E) = (E/T2)eE/T (2.26) where
T = the energy corresponding to the
most probable speed,
 Boltzmann constant x Kelvin
temperature (in energy units).
On the other hand, the energy spectrum in a finite medium with a small absorption cross section, compared with the scattering cross section, is not Maxwellian but similar to it in the general shape.
If the Maxwellian function is factored out from the flux moments, i.e.,
*n(r,E,t)  m(E)*n(r,E,t) (2.27) m m
n
the component, #m M(r,E,T is expected to be relatively smooth.
Similarly, the source moments are analyzed:
n4 n 4 Sm(r,,Et) = m(E)xm(r,E,t) (2.28)
n
To evaluate the integrals,Im(rE,t), the flux components**M(rE'It) , are expanded in Taylor series about E.
n k knr
*m(r,E',t)  EE) D MrEt) (2.29)
k=o
where
k n n D m(r,E,t)   m(r,Et)
WEk IEI=E
Using Eqs. (2.27) and (2.29) in Eq. (2.22) the integrals become I (r,Et) = f Em(E'.E)m(E') [  (EEk
El k=o
(2.30)
Dktn( ,Et)dE(
m
12
To put the integrals I n into a more workable form, the
m
detailed balance principle will be used. This principle states that an equilibrium is established for the product of the Maxwellian distribution and the scattering kernel between any two energy intervals, i.e.,
m(E')Es(E'.E) = m(E)E (EE') (2.31)
Notice that the relation (2.31) is completely independent of the amount of absorption in the medium and the size of the medium. Eq. (2.31) can be generalized for the moments of the kernel and is then stated as
m(E')E (E')E) = m(E): (EE') (2.32)
li m
Eq. (2.30) can be combined with Eq. (2.32) to yield
In( ,E,t) = m(E) f E (E+E') I 1 m E m (E'E) k m k=o k!
x k~n(rEt) (2.33) or
n kn+ Im(rmEt) = r(E) M (E) D 'n(rEt) (2.34)
k=o
where the energy moments are
Mk 1 = 1 1 km m (E) ï¿½m(E)A E(E) = k f[ f (EOE) E(EE )dE
m Id
E l _ (2.35)
Furthermore, the integrals of Eq. (2.34) can be written
in operator form as
In(r,E,t)
= m(E)Qm(E) m'r, E,t)
(2.36)
where the operators Qm(E) are defined as
Qm (E)
 k k = [ 1M(E) D
k=o
(2.37)
The use of Eqs. (2.22), (2.27), (2.281 and (2.36) in Eq. (2.21) yields:
Rm(Et)n(r,E,t) Xn(r,E,t)
(m+1In) tM++ln)
A((z+ (2 1e n +
L (2m+3) (2m+J.) rEt
(mn) (m+n) (2m+l) (2n+l)
n I rnMI (r,E,t)
1 [{(m+n+2) (m+n+l) B(x ,y) I (2m+3) (2m+l) 44Mi rErt)
(n n) (rnni)'
'2l (2m1)
I .
n+l ,). X. *mIrrp
(m+n) (m+n+l) n1 1 (mn+2) (mn+l) .;I + (2+l)(2m (rEt) (2m+3)(2m+l)
ni l
X *m+l(r,Et) (2.38) where
RM(Et) = QM(E)  ï¿½t(E) v at (2.39)
Hence, we have obtained the general form of the thermalization model of the Boltzmann equation in its spherical harmonics formulation, where its validity is only limited by the approximation involved in the Taylor expansion of ,(r,E',,t) about E. The operators Qm(E) and Rm(Eit) are discussed and calculated in Appendixes A and B.
P1 Approximation
When the index, m, in Eq. (2.38) is given the
values 0 and 1, the index n takes the values il, 0, and 1. Then, the infinite set of Eq. (2.38) is reduced to the following four equations which couple the moments of the flux in the socalled P1 approximation.
R Ej)O~jEt+Xo(r,E,t)_(jo ' (Z) Et)
R(E,t).0E o
+ ( ) B(x,y)*,(rEt)(). B*(xy)*(r,E,t) (2.40)
0 (
I(Ept)*j (rVEjt) +X (rE,t)( A(z)*o (rEpt) (2.41)
1.1 1 . 1l 0Rl(~ t~ l~rE~)+ l(rE,t)=( B* (x"y) *ï¿½(r,E,t) ( .2 R (Et)B (rEt)+ (xy) (rEt)
R1(L~t) 6(2.42)
R1(Et) i ( +t)+X
(2.43)
The scalar flux is obtained from Eq. (2.2)
f
G m
n n
a m=o n=m
0 0 0 0
0 (rE,t)IHo = W o(',E,t)
i=4 m() ï¿½(r,E t)
reE 0 (2.44)
where Eq.s (2.2), (2.3), (2.7), (2.10), (2.12) and (2.27) were used.
Similarly,
S(r,E,t) = /W m(E) Xï¿½(rE,t) (2.45)
Since the scalar flux is easily obtained from *ï¿½(,E,t) according to Eq. (2.44) the set of Eqs. (2.402.43) will be solved for *8(f,E,t). Multiplying Eq. (2.40)
by R1(E,t), Eq. (2.41) by A(z)/Vfi, Eq. (2.42) by x,y)/T6, Eq. (2.43) by B*(x,y)/f6 and adding up the resulting set yields
R1 (E,t) R0 (E,t)
1
+  clï¿½ ï¿½ t
v2] *ï¿½(r,Et) + R1(E,t) Xo(rE,t)
1 1
+ B (x,y)x1 (r,E,t)
1 1
 B*(xy) x, (r,E,t)
/ " (2.46)
1(Et) 1I + X1. , 0
(2.47)
R 1(]Eft) i(rEt) + X1(rEt)  B*(x,y) * (r,,,t) (2.48)
R (Eft) *r,E,t) + X1(rEpt) = 1 0
1 1 7~ (x,y) j(rE)
(2.49)
If the source is assumed to be isotropic, the only nonzero moment of the source will be So(B,E,t) or Xo(_r,E,t) . Then the set of Eqs. (2.46)(2.49) is
0
reduced to
(RpEEt)R0 (Et)  V2] (rEt) + RI(E,t)
0 ~3 0
x xo(r,E,t) = 0
(2.50)
+1 +
Rl(E,t) *p(r,E,t)  A(z) *ï¿½(r,E,t)
11 V30
(2.51)
1 0 1o .
R1 (E't) E(rEt)   B*(x,y)*o(rE,t) (2.52)
RI(E,t)*1(r,E,t) = B(x,y)*o(r,Ept) (2.53)
Eq. (2.50) is then the sought for thermalization model of the neutron transport equation in the consistent P1 approximation with space, energy, and time dependence. This equation is in a general operator form. To rewrite it in a more explicit form we make use of Eqs. (A.6), (A.7), (A.ll), (A.12) and (A.13) which give
[H(E) + 1 F(E) +  V2]2
v(E) at V2(E) at2 3
v(E) t + G(E)] X(r,Et) (2.54)
along with the definitions
O0
*(r,E,t) = /7"f *o(r,Et) (2.55)
X(r,E,t) = /4w XO(r,Et) (2.56)
k
H(E) = I hk(E) a (2.57)
k
k
F(E) = I fk(E) k (2.58)
k 3Ek
a k
G(E)  [ gk(E)  t2.59)
k aEk
where fk(E), hk(E) and gk(E), which are algebraic functions of the energy and the macroscopic properties of the medium, are given by Eqs. (A.14)  (A.22) and where the derivative operators, ak/aEk, arise from the Taylor expansion of the flux.
In Eqs. (2.57)  (2.59) all the energy
derivatives are kept and the truncation of the expansion depends on the model of the scattering kernel used and on the energy moments, M(E), which appear as coefficients of the derivatives, ak/aEk, in the Taylor expansion. See Eqs. (2.34) and (2.37). In the heavy gas model scattering kernel, which is applied in this work for numerical computation, the moments, M (EY, are zero for k greater than 2. Therefore, only the first and second derivatives are used.
It is very convenient to transform Eq. (2.54) from the energy domain, E, into the domain of the dimensionless variable, c , given by
= E/T (2.60)
This transformation leads to
1 a 1 2 1 1 .
[H(0  F(t)  +   V2] *(rEt)
at v2eC at2 3
1 a *
[ [G(E) +  ] X(rvt) (2.61)
V/ at
H(C) = [ hi(c)  (2.62) aei.
F(C) = fi(c) A (2.63)
i aE1
ac
G(C) = [ gi(c)  (2.64)
1ac '
See. Eqs. (A.271  (A.35). The functions fi' hi and gi are calculated in terms of E and ï¿½ for the heavy gas model in Appendix B.
Eqs. (2.50), (2.54) and (2.61) are all
equivalent and the last form will be used throughout the remainder of this work. Specifically, Eq. (2.61) will be used in Chapters III and IV to investigate the neutron waves and the pulsed neutrons techniques in moderating media.
EnerqvIndependent Cases
It is interesting to conclude this chapter by the examination of the energyindependent case of Eq. (2.50). In this case
P. 0
(r,E,t) = O(rit)
Dk *(r,E,t) = 0
(2.65)
k> 0
It is easily seen that the operators, Qm(E), become energy independent and are reduced to the moments of the scattering kernel, i.e.,
k Dk = MmD
Qm(E) = QM
=% = Em
(2.67)
where
Em = Pm (Io ) E (so )dO
1
(2.68)
In the P1 approximation m takes the values 0 and 1 which give
1
E0 f PoUo ) (dUo6
E s(Vo)duo E s (2.69)
(2.66w
f jl 1(v 5(vo)do 0
1
1
f U" o s(vo )do0
= EsJo
(2.70)
Then the operators, Ro(E,t) and RI(E,t), take the energyindependent forms
R (t) = = +  + (271)
0 s v at a v at R(t) = 0  3D  t ) (2.72) where
D = l/3(zt  Es)
(2.73)
is the diffusion coefficient. The energyindependent flux equation is obtained by the substitution of Eqs.
(2.71) and (2.72) in Eq. (2.50).
3D 2 1 2+D)
2 +t I (.1+3ZaD) + E DV2 *Cr,t)
3 3ot2 = 1+  x (r Pt)
v at]
(274)
22
This equation is easily recognized as the
telegrapher equation which includes the transport correction on the source. This correction is given by the term (3D/V) i(r,t) It should be noticed that this correction term is neglected in the telegrapher equation given by Meghreblian and Holmes (AV and by Weinberg and Wigner C.k). The neglection of this term is possible in a steady or quasi steady state operation where the source variation with time is very slow. On the contrary, the source derivative term is very important in time dependent kinetic problems which are sensitive to time variations of the source. In studying the two problems of neutron waves and pulsed neutrons, which are of most interest to this work, the source transport correction is specifically important when sharp neutron pulses are fed into the system and the transient flux is investigated, or when the source frequency becomes large. However, these two subjects will not be studied through the telegrapher equation but through the general space, energy and timedependent formulation given by Eq. (2.61).
CHAPTER III
NEUTRON WAVES IN MODERATING SYSTEMS
Analytical Formulation of the Neutron Wave Problem
Various theoretical and experimental
investigations of the neutron wave problem have been reviewed in Chapter I. This chapter is devoted to the investigation of the neutron wave problem through the time, energyand spacedependent flux equation, Eq. (2.61), derived from the consistent P1 approximation which gives rise to the thermalization and transport effects. As a convenient reference, Eq. (2.61) is restated here.
1 a 1 a2 1
(H(c) + L F(c)  +  a2  V2] 4(r,c,t)
Vo at V02C at2 3
1 a
 [G(E) + 7 ] X(r,E,t) (3.1)
Vo/ E at
Since the source does not exist inside the
system, the source term is dropped from Eq. (3.1) and a suitable source condition will be established later in this chapter. Thus, the neutron waves will be studied through the homogeneous equation 23
1 a 11
[H() +  F(C)  +
V0 att V(3.2)
where, as noted before, *(rc,,t) is the nonMaxwellian component of the flux
O(r,e,t) = m(e) *(r,c,t) (3.3)
The corresponding component of the source is X(r,Et) where
S(r,c,t) = m(c) x(r,e,t) (3.4)
When the source is sinusoidal in time its component X(r,c,t) is also sinusoidal and can be expressed by
4. jWt
x(r,,t) = S0(,e) + Re[S(r,)e t (3.5)
where so0e,) and s( ,ï¿½)jwt are, respectively, the timeindependent and dependent components of X(r,E,t). Similarly, the nonMaxwellian component of the flux has the same time behavior with the same angular frequency, f, or angular velocity, w = 2wf, but with some angular phase shift. This phase shift (lag), which is
25
a function of the source frequency and the position at which the flux is measured, will be determined later. Therefore, the time dependence of the flux can be given by the expression
*(r,c,t) = *o(re) + Re[(rc)ejwt] (3.6)
The combination of Eqs. (3.2) and (3.6), and then the separation of the steady state part of the resulting equation gives
2
[H(e) + F(e) +   V2] *(r,e)=O (3.7)
and
[H(E)  v2] jo(rE ) = 0 13..8)
3 0
Since the steady state solution can be easily found by replacing w by o in the time dependent solution, there will be no need to treat Eq. (3.8) separately but it suffices to solve Eq. (3.7).
The boundary conditions associated with Eq. (3.7) are
*(X, y, Z) = *(X, y, Z) ,
(3.9)
*(x,y,z) = *(x,y,z) (3.10)
yFZ) = 0 (3.11)
*(X,,z) = 0 (3.12)
=(xfy,6) = 0 (3.13)
where 21, 2E, and Z are the extrapolated dimensions of the nonmultiplicative assembly in the x, y and zdirections, respectively, and where the center of the xyface is taken as the center of the coordinate system. The source is located at the xyface with supposedly known spacial and energy distributions and will be specified later when the source condition is established.
The solution of Eq. (3.7) is developed in cosine functions for the x and ydependence, exponential functions for the zdependence and normalized associated Laguerre polynomial for the energy dependence, i.e.,
*(x,y,z,C) = I cm,nXm(x)Yn(Y)Zm,n(z)Emn() (3.14Y
m,n
where
2m 1
Xm (x = Cos a wIrx (3.15),
Yn(Y)  Cos \ 1Y (3.16
2b;
Z m,n(z) = exp(pm,nz) [1exp{2Pm,n(cz) }]
exp(Pm,nz) (3.17)
E (c) [A LL ( ) (3.18Y m,n m,n L
L
The term in brackets is the end effect correction due to the finite size of the system in the zdirection and p is the inverse relaxation length of the (m,n) spacial mode. The energy dependence of the (m,n) mode, E m,n is expanded in the normalized associated Laguerre polynomials of the first order, L () These polynomials and some of their properties are discussed in Appendix C.
The substitution of the solution given by Eq. (3.14) in Eq. (3.7) gives the equation
E H(c) + AF(e) + IV2 1 (p2 B2 m 3 mI,n mn
cm,nXm(x)Yn(y)Zmen(z)Em,n() = 0 (3.191
where
jW
v = 3.20)
and
B2 = +2n 1 (321) BIm,n 2a 26 f3.21)
is the transverse buckling of the (mn) spacial mode. Eq. t3.19), which contains all the spacial modes, can be separated into a decoupled set of equations each of which involves one mode only. This decoupling is achieved by operating on Eq. (3.19) by the special integral operator
P f a dx f6 dy Xm,(x)Yn,(y) (3.22)
1 5
This operation and the recognition of the orthogonality properties,
X Xm (X)X (x)dx = atm m (3.23)
Yn' (y)Yn(y)dy = b6n', (3.24)
leads to the modal equation
[H(F.)+AF(e)+ 1 &2 (P21() 0( .5
   (P2 B2 )] E (c)=0 (3.25)
S 3 mrn lm,n mn mn = 12,3e...
where we divided through by the constant Cm,n.
In order to determine the coefficients, Amno we will operate on Eq. (3.25Y by the energy integral operator
0 = de m(c) L(I) (c) (3.26)
0
where m(e) is the Maxwellian function previously defined by Eq. (2.26). This operation leads to the following set of algebraic linear homogeneous equations.
[ + B A + Y A2  r] At ( t ,L Ol,, L,, Loot m,n 0 (3.27)
30
where the matrices involved in this equation are defined by the following:
fc* m(c) L ()H() LM (M)dc (3.28)
0
B fJ m (E) L(I) (e)dE C3.29) ,'It 0t
(1) 1 L(1)
em() L (C)  (E)de (3.30) t', ï¿½ ' ï¿½"
Sf M(Ec) L ()(e) (c)dc = 1 6 (3.31) , LI 3 0 t. 3 Lift
and where
r = p2  B2
m,n 1m,n (3.32)
If the Laguerre expansion of Eq. (3.18> contains L+l polynomials, i.e., if t takes the values 0, 1, 2, S.., L, then, the set of Eqs. (3.27) can be rewritten as
L
[I+ ,+2 [a]A n=0 (3.33) L=o
31
with t' = 0, 1, 2, ..., L. This represents a set of L+l homogeneous equations with L+l unknowns, Arun' t = 0, 1, ..., L.
In order for this set to have a solution, its compatibility conditions
Dj, (Ar) = IL,, 8LLII + YL , 2  = 0 (3.34)
, . = 0, 1, 2, ..., L must be satisfied. This equation will be referred to as the eigenvalue equation and r will be referred to as the eigenvalue. From this equation we recognize that r is completely "spaceindependent" because it is a function of A and the matrices , %, 7ï¿½, and all of which are "spaceindependent." Consequently, and for similar reason, the coefficients, At , are "spaceindependent,"
m, n
as is seen from Eq. (3.33). This means that all the spacial modes have the same eigenvalues and the same coefficients which lead to the formulation of the eigenfunctions as we will see later. On this basis the indices m and n, which indicate the (m,n) spacial mode, can be omitted from Eq. (3.34) which is rewritten as
L
[off + + AY&2  0A (3.3 LUne
I = 0,1,2,**,L
32
However, it is important to realize that the complex inverse relaxation length,Pm,n, is "spacedependent" and is computed from the relationship
P2 =r + B (3.36)
m,n Im,n
To avoid confusion we must point out that the terms "spacedependence" and "spaceindependence" are used here to indicate, respectively, whether or not a certain quantity is a function of the spacial modes, or the spacial mode indices m and n. To solve the frequency dependent set of Eqs. (3.35) two alternatives can be used: the exact method and the perturbation method. In the following two sections both methods will be discussed.
Solution by the Exact Method
Due to the fact that the expansion of the
determinant in Eq. (3.34) gives a polynomial in F with a maximum power of L+l (the size of the determinant) the solution of this equation yields L+l values for r. These values will be distinguished by the subscripted variablerk, where k = 0, 1, 2, ..., L. It is then obvious from Eq. (3.35) that to each rk there corresponds a different set of coefficients, A ,k' which
33
leads to the formation of the energy eigenfunction or energy mode, E (c), through Eq. (3.18) which becomes
m, n
L
Ek (c) = Ek(c.) = I A L(1) (C) (3.37) Mtn L= tk L
Each set, A LkI is the solution of Eq. (3.35Y using the value rk for r, i.e.,
L
[ + A 2 A + rkJA (3 38) L=o
or
L
I DL,,t(A,rk)Atk = 0 (3.39) 1=0
This homogeneous set of L+l equations with L+l unknowns can be solved for L unknowns in terms of the other. If Eq. (3.39) is divided by Ao,k' it can be rearranged in the form
L
[ D,,L(A,rk)RLk = D,O(A,rk) (3.40)
with L' = 0, 1, 2, ..., LI.
34
The new unknowns appearing in this equation are the ratios
Rt,k = At,k/Ao,k (3.41)
where
Rok = Ao,k/Aok = 1 (3.42)
Notice that the equation corresponding to L' = L was omitted from the set represented by Eq. (3.40) in order to equalize the number of variables to the number of inhomogeneous equations, i.e., to remove the degeneracy of the solution. The set of Eqs. (3.40) can be solved for all the ratios, R ,k and the energy modes, Ek(c), cap be formed by recombining Eqs. (3.37) and (3.41).
L
Ek A A ~ l C
o,k L,k L(3.43)
Z=0
The exact method that has been discussed yields the correct solution as a function of A or w. The whole computational process must be carried out in complex algebra due to the fact that A and r are complex.
35
Furthermore, the whole solution must be repeated for every value of w since we are interested in the solution
for a wide range of source frequencies. Solution by the Perturbation Technique
This technique is based on Feenberg's
perturbation method discussed in (#I). A modified version of this technique was developed and used by Perez and Uhrig (38) to solve a similar problem. We will use this technique here to solve Eq. (3.35).
First, Eq. (3.35) is rewritten as
L
[a , L + age A + Y &2 0R (344) L', ,L L=0
where
R = At/AO. (3.45) Then, it is solved for the case of A = 0 by the exact method, i.e.,
L (o) (0) I[a L'OJ r ]A 0 0 (3.46)
with the compatibility condition
I L,,L  nj,, r(O)I 0 (3.47)
36
Equation (3.47 gives all the unperturbed or steady state eigenvalues, r(o) the use of each of which in
k
Eq. (3.46) yields the corresponding unperturbed ratios, R(o). Next the kth eigenvalue, rk, and the correstrk
ponding ratios, Rtk, for A y 0 are obtained from the equation
L
I + 8 A + Y A2  n rk]R =0 Left L to it k Lek (3.48) L=O
by the perturbation method.
In this method the eigenvalues, r and the ratios, Rzek, are expanded in power series in A.
r r (o) + r(1) A + r (2) A2/21 + ... k= k k k (3.49)
R = R (o) + (I) +(2) R R9 + R~ + R (2)A2/21 +  (3.50) ilk ilk ik ik
where
(V) av
k  rk (3.51)
k AI 1A=
R( =  R (3.52) ik aAv ik 1A0
This expansion is restricted by the condition
[ol(rad/cm) < 1 (3.53)
Once we find r(V) and R(v) the solution is obtained L,k
from Eqs. (3.49) and 3.50) by the substitution of the numerical value of A.
It remains now to determine the derivatives
r(v) and R(V). This determination is done step by step k L,k
from Eq. (3.48). First, the first derivative of the set (3.48) with respect to A at A = 0 is equated to
zero and solved for r(1) and R(1) in terms of r(o) and k Zk k R(o) which have been determined from Eqs. (3.46) and
L,k
(3.471. Then the same process is used for the second derivative of Eq. (3.48 to solve for r(2) and R(2) in k t,k
terms of r(o), r(l),R(O) and R(1) and so on. Suppose
k k Lk L ,k
now that we have found r(P) and R(P) for j = 0, 1, 2, k 1,k
..., v1 and want to find r(V) and k Applying the rule
3VV a) (vU)
 (A(x)B(x)= ( )A (x)B (x). (3.54) 3xv
I0=O
to Eq. (3.48), where
vI
wI o bt a
we obtain
V L
I ( ) a +
. R (V )
Raik At0 Realizing that
 (An)I dxI A0
1,' I
(,i)
r] i2,L k
= 0
nI
= 6
where 6 is the Kronecker delta function, Eq. (3.56) is rewritten as
L
I ( ( a 6 [ a y L 0 P I 114
(.)
+2Y, 6 niI r I
2 0& , U ' PI k
R(vP) = 0 R0k
(3.58)
Noticing that Ro,k = 1 for all values of k and
(n) (V) R o= for all values of k, and separating rk
o,k =o~n
and R(v) (z / 0), one can rearrange Eq. (3.58) in
t,k
the form
(3.55)
(3.56)
(3.57)
V Im
ak r(v) + I ak R(v) = bk V) ',o k t1 L )k where
a
ak
to O0
DL
L=
R(o) n Zek ,Itt
= r(0)  o
L',L k i',
t > 0
k L
bk (v) = L [R(o) {8 6 to LIL k i 'L I tv
Vi ) R{ V=il L,k
+2y 6
L',L 2,v
6 +2y 6 to ft } L, to o 2
} + u(v2)
n r , L 1', k
where the Heaviside unit step function
U (v2) = 1
0
for v > 2 for v < 2
indicates that the term following it exists only for v >, 2. In explicit matrix notations Eq. (3.59) takes the form
(3.59)
. (3.60)
(3.61)
(3.62)
(3.63)
ak ak ak . akr bkV 0,0 O,1 0,2 o,L k 01 k ak k k (V) k,v a a a a..Ra b l,o 1,1 1,2 1 ,L. 1, 1 ak ak ak ... ak R i (3.64) 2,0 2,1 2,2 2,L. 2,=k 2 SI
k ak ak ak R(V) L,o L,1 L,2 L,L 1 L, k L
These two equivalent sets, Eqs. (3.59) and (3.64), are linear inhomogeneous and consist of L+l equations with L+1 unknowns, r(v) and R(v) (t = 1, 2, ... L) which
k tfk
are completely determined by the solution of the set. It should be pointed out that the matrix, a ,o , does not depend on the order of derivative, v, and it is the same for all values of v. On the other hand, does depend on v and has to be calculated for each value of v. See Eqs. (3.60), (3.61) and (3.62). However, both ak and bk v depend on the eigenvalue
Pltt t0
used, as indicated by the superscript k. Both Bqs. (3.59) and (3.64) will be referred to as the perturbation equation.
It is very helpful to summarize the various
steps of this technique in the sequence used in actual computation.
41
1. The unperturbed equation, Eq. (3.46), and
its compatibility condition, Eq. (3.47),
are solved and the unperturbed quantities
r(o) and R(0) are determined.
k t,k
2. For each value of k, the perturbation
equation, (3.59) or (3.64), is solved for all values of v in increasing order up to
the desired maximum value. Each time
rv) and are calculated in terms of
r(P) and R(P) 0, 1, 2, v1.
k L,k
3. rk and R ,k are calculated from Eqs.(3.50) and
(3.50) using the desired value for A or w.
4. Finally, the energydependent flux modes,
or eigenfunctions, Ek{c), are formed
L
E (c) = AR LR,kLl) (C) (3.65) Z=O
It should be emphasized that this perturbation technique has been used here only in the energydependent equation (3.44), leading to the determination of the energy modes,
Ek(c)
Before we move on to further developments we
ought to compare the two methods. The exact method has
the advantage of yielding the functions Rk() without any approximation at any desired frequency and without any restriction on the maximum value of w that can be used. The disadvantage of this method is that all the computational processes must be performed in complex algebra and all the computational steps must be repeated for each value of w. On the other hand, the perturbation technique introduces some error in the solution due to the approximation hidden in Eqs. (3.49) and (3.50). The value of this error diminishes rapidly as the number of terms kept in these equations increases. The other disadvantage of this method is the restriction given by Eq. (3.53), namely
W < V (3.66)
It is obvious, of course, that the solution is improved with decreasing values of w. The advantages of the perturbation technique are that the algebra involved in it is easier to handle in actual computations and that the computation does not have to be repeated for each frequency. The computational scheme used in this work utilizes the perturbation technique.
43
In both methods the energy eigenfunctions take the same form, as shown by Eqs. (3.43) and (3.65). Having determined the energy dependence of * (x, y, z, e) by either one of these methods we can easily rewrite Eq. (3.14) in the form k Ek
(xfyvzr)= Ck X (X)y (Y)Zk (z) ( .67)
m,n m Mn(3.67
m,n k
where we redefine
Ck = c A (3.68)
m,n m,n o,k
Xm(x) cos (2m1._ Nx) (3.69)
2i
Yn(y) = COs 2n y) (3.70)
2b
zk (z) = exp ( k Z) (3.71) m,n Mn
kk
Ek(ï¿½) = [ ()()(3.72)
L,k ï¿½ s:
It remains now to determine the constants, Cm, n This will be done later from the source condition developed in the following section.
44
Neutron Net Current and Source Condition
Before we establish the source condition the
neutron net current has to be developed in terms of the neutron flux. To do so we first define the neutron net current. Let dA be an infinitesimal area whose norm is the vector N located at r and consider the directional flux, *(ra,E,t) where the vector Q makes with N the angleve given by
cos e = fï¿½N . (3.73) The net current can then be defined by the following relation
Jn{r,E,t) dA the net number of neutrons with energy E which pass through the area dA in all possible directions.
This definition is equivalent to the analytical statement
J (r,E,t) dA  dA f (r,E , t)( Nda 13.74)
n
where dA(Q  N is the apparent area seen by the neutrons traveling in the direction, Q, or
J n(r,E,t) = f *(r., E,t) (aeN)d (3.75)
40.
where J n(r,E,t) is then the net current through a unit area whose norm is N. Eq. (3.75) can be rewritten as
Jn (r,E,t) = N f f *(r,2,E,t)f d
= N * J(r,E,t) (3.76)
which means that the net current, Jn(r,Et), is the projection on N of the vector net current
J(r,Et) = f *(r,QE,t)sl da (3.77)
.
The flux and the vector, fl , are given by
S n r n)( ~) (3.78)
m n
= sine cos y u + sine siny v + cos8 w (3.79) where u, v and w are the unit vectors along the x, yand zaxes, respectively, and ' and e are the polar and azimuthal angles of a in spherical geometry. From Eqs. (2.7) and (2.12) it can be shown that cose = Y1ï¿½(')/HO . (3.80)
sine cos f  [Yl(Q)1 ()]/2H1 sine sin Y = [Y l(,)+Y ('9)]/2jH1
(3.81)
(3.82)
The combination of Eqs. (3.77)(3.82) and the use of the orthonormal property of Ym(Q) expressed in Eq. (2.7) we arrive to the expression J(r,E,t) = u1(r,Et)l(rE,,t)]/2H 1
JEt)= [ ' i ,Et)]/2jH
0 0
(3.83)
The moments, n( ,Et) satisfy the relations
*m(rEt)  m(E)m(r.,E,t)
mj 0
(r, Et) )j  R1 (EOt) 0(rE,t) 1' ax ay
(r,E,t)
1 a a 01
( +j ) R, (Et) 0 (rEt
ax ay
1 a R1 0 4
(rEt) 1 (Et)* (r,E,t) 13 3 Z
(3.84) (3.85)
(3.86)
(3.87)
47
and the scalar flux is obtained by
*(r,E,t) = m(E) (rEt)/H0
0 0
= m(E)*(r,Et) (3.88)
Eqs. 3.83)(3.88) yield
+1 4 4 a ) 1 a
J(rE,t) = m(E)[u Tx +vy +wTz] (E,t) (r,E,t)
_ m(E) VR1(E,t)*(r,E,t) (3.89)
3 1
In the case of N = w the net current through a unit area perpendicular to the zaxis at (x,y,z = 0) is given by
1 1 a.. l I
J (r,E,t) = ro rE) R(E,t) (r,E,t)
z=O 3 z 1z=O (3.90)
Having found the net current, the source
condition can be easily established. Let S(xy,E,t) be a plane source located at z = 0 evaluated in neutrons/cm2 sec emitted in all inward directions. Equating the neutron source strength to the net current we obtain the source condition
1 a 1 1
 m(E) az R1 (E,t)*(r,E,t) I = S(x,y,E,t)
(3.91)
By factoring out the Maxwellian component of the source and dividing Eq. (3.91) by m(E) it becomes
3 z R (Eft) V(r,E,t) z=O X(xyEt)
(3.92)
Finally, if we operate on Eq. (3.92) by R1(E,t) we obtain the final form of the source condition z (r,E~t) I == 3RI(Ept) X (xyEt) (.3
where the operator R1(E,t) is given by Eq. 2.36). The transformation of Eq. (3.93) into the domain of the dimensionless energy variable, E, yields
7z (rct) = 3R1(,t)X(x,y,C,t)
Iz=o 13.94)
or, by using Eqs. (A.7) and (A.29),
 ,(r ict) i" 3 [  + G(E)]X(x,y,ct) z a
(3.95)
This condition introduces two corrections on the source. The first one is the transport correction given by the
49
derivative with respect to time and the second one is the energy correction hidden in the energy dependence of the operator G(e) and the energy derivatives involved in it. When the transport correction is neglected and a Maxwellian source is used the nonMaxwellian component of the source becomes independent of energy. Then, the operator G(E) is reduced to
G(c) = [Et(c) sE (E) ] =
t 3D(c) (3.96)
and the condition of Eq. (3.95) becomes
 D ( ) a * ( r , E , t ) l z = = X ( x j 'y , C , t ) ( . 7
or
z *(r ,EZ,t) I S(x,y, ,t)
 Z Iz=O (3.98)
which is essentially the condition used by Perez and Uhrig (_8).
Before we use the source condition in Eq. (3.95) to determine the so far unknown constants, Ck two m ,n'
remarks should be made. The transport correction in this condition has a strong effect when the source frequency is high while its effect is small with low
50
frequencies. The energy correction is always persistent except when a pure Maxwellian source is used, in which case X(r,c,t) is constant and its derivatives are zero. Determination of the Constants, Ck n.m
The final step in the solution of the neutron
wave problem is to determine the constants, Ck n which appear in the flux expression of Eq. (3.67). This determination is carried out by using the source condition given by Eq. (3.95). The substitution of *(r, ,t[ from Eq. (3.67) into Eq. (3.95) gives
k k zk (zEk (c)e jWt
CnXm(X)Yn(Y)pmnZm,n(z) mn k
= 3[  + G(e)]x(x,yet) (3.99)
v 04Vi at
If we assume the separability of space, energy and time dependence of x(x,y,e,t), i.e., if we express it by
X(x#y,c t) = Xs(x)Ys(y)Es(c)eJWt (3.100)
Eq. (3.99) becomes
k k (x) k
Cm,n m,nXm Y(Y)Ek(')
3[T 7t + G(c)IXs(x)Ys(y) (c) (3.101)
51
where the subscript s is used to indicate the source and has no numerical values. To utilize this equation in determining the unknown constants, Cmn we first mulmmul
tiply it by Xm,(x) Yn' (y~dx dy and integrate it over the whole extrapolated xand ydimensions. This operation, along with the orthogonal properties
X , (x) Xm (x)dx = a 6m' m
 a m (3.102)
f~ Y n'(y) Yn(y)dy = 6 no,,n
b (3.103)
yields the equation
I C p Ek(') = a + G(e)]E (() k m,n m,n m;n 7 s (3.104)
which is valid for all values of m and n, where we have defined
1 a
a mn= f dx f dy Xm(x)Xs(x)Yn(x)Ys(x)
mmn bs (3.105)
a b
52
For given values of m and n Eq. (3.104) couples all the unknowns, Ck . corresponding to all the values of k. This equation can be reduced to a coupled set of equations by the multiplication by m(c)L(l)(elde and integration over e from 0 to .
k k
SCm,nPmndkIk 'za,n(A k' + (3.106)
k
where
d = f )L(1)() Ek(c)dc k',k o k' (3.107)
fk mle) L(I() _ Es(e)lde
ki () E / C (3.108)
0
f m(e)' (c) G(C)E (E)dc
k' s (3.109)
Using the expression for Ek(E) given by Eq. (3.72) in Eq. .!,U07) and considering the orthonormality property of the Laguerre polynomials, one obtains
d , f () L(1)k, (c) Rk LI (1)dc
= Rk 6k', R kk (3.110)
2.
Then, if we define
Dk',k Pk d k Rk m,n m,n k',k m,n k',k (3.111) k'
Bm,n amn (Ak' + ek') (3.112)
Eq. (3.106) takes the simple form
D k k Bk' .113)
m,n m,n m~n (3.113)
k
A more explicit form is
Dï¿½ o Dï¿½, 1 . . Dï¿½,L C0 Bo m,n m,n m,n m,n m,n
DIO DI ... DL C1 B1 m,n m,n m,n m,n m,n
This linear inhomogeneous set applies for all values of m and n and contains (L+1) equations with (L+l) unknowns, Ck which are completely determined by its solution.
At this stage we see that the solution for the flux has been completely found, as given by Eq. (3.67). All the constants have been determined. All the space
54
and energy eigenfunctions are known and given by Eqs. (3.691(3.77). The various inverse relaxation lengths, Pnk .are computed from the eigenvalues,rk , according to Eq. (3.36). It is important to recognize that the actual computational process must be carried out in complex arithmetic because, in general, all the quantities involved in this computation are complex.
The actual scalar flux, as a function of space, energy and time, is now given by
,t) =
Re [ C k X (x)Y (y)Zk (z)Ek (E)m()eJWt]
m n k n mn 0.115)
In order to compare Eq. (3.115) with the actual experimental results, the neutron density has to be computed. This arises from the fact that the experimental setup used by Booth (41) uses a 1/v detector and, hence, it gives the energy integrated, or total neutron density, as the output. The neutron density, which is related to the flux by the relation
4 1
N(r,,t) =  (r,,t)(3.116)
Vo(3/16
can be given by
N(r,c,t) =
k k k1 ~
Re [ Ck,nX m (Yn) z (y) Zk , i Ek(c)et
Re n Xkx v;n E e (3.117)
The total neutron flux and total neutron density are obtained simply by integrating Eqs. (3.115) and (3.117) over energy.
It was mentioned at the beginning of this
chapter that the flux, or hence, the neutron density, has the same time behavior as the source with a phase shift which depends on the detector location and the source frequency. Having found the flux and the density we can now determine the amplitudes and phase shifts for both of them using Eqs. (3.115) and (3.117) which are rewritten as
*(r,c,t) =
k
Re[j I k E(e)JAmkn(r)+jBk (r) }e (wtbm~nZ)
m n k (3.118)
N(r,c,t)=
k
Re[m I I Ek(E)JAk (+)+JBk ( (wtbm,nz) (3.119)
m n k d m,n m,n reI
where
k ak + Jbk
m,n m,n m,n (3.120)
k
Ak ( ) + jBk ( C) = Ck X (x)Y (y) eamnz m,n mn m,n m n
Ek(C) m(E)Ek(C). (3.122)
f
Ek(C)  m(C) Ek (e) (3.123) d V
The indices f and d have been used to indicate the neutron flux and neutron density, respectively. If we write
*(r,ct) = Af (r,) cos[wt  e f(r,)] (3.124)
N(r,c,t) = Ad(r,c) cos [wt  ed(r,E)] (3.125)
then Af(rs), ef(r,e), Ad(r,c) and 8d(rc) signify, respectively, the amplitude and phase shift of the neutron flux and density. If the righthand sides of Eqs. (3.118) and (3.124) are equated and expanded, the following expressions are obtained for the amplitude and the phase shift:
A (r)= I JA (r)+B (r)2 ()
(rr IE'(.) 2]
nk mn ,n f 3.126)
ef (r ,) =
k k k 1k ( Ak on (r) sin bmon zBmon (;*) cos bkmonz]E f[
tan k kk k (C)(3.127)
[mAnZ B cosin b,nz E
Similarly, we obtain the amplitude and phase shift for the neutron density. These are found to be
1
A d(re) = Af(r, e) (3.128)
e d(ro) = 0 f(rc) (3.129)
For the case of the total neutron flux and
density, Eqs. 13.118) and (3.119) are first integrated over energy and then the same procedure discussed above is used. This yields
2
f k m1 n Mn f (3.130)
f (r) =
f
mn k
Ak (r) sin bk z  Bk (r) COB bk Ek] m,n m,n m,n m,n J
tan1r k k
I I I [mA (r) Cos bnz+ B (r)
m n k m,n m,n m,n
Ad() [II jAk (rf+Bk ()2Ek2
in bk
men
ZIE k
(3.131)
(3.132)
ed (r)
mn k
[Ank (r) si bk 4  Bk (,) cos bk zEk C m,n m,n m,n M,n Id
kn CO bk n + Bk (') sin bk Z] Ek
Mn(3M,nz M3n rMn Ed
(3.133)
where
=k f E (e) dc
0
E k Ek(edc
0
(3.134)
(3.135)
tan1
rnk
59
Thus we have developed the neutron flux and density in all desired forms as functions of space, energy and time. Furthermore, we have seen the neutron traveling waves in the medium through the expressions of the amplitudes and phase shifts of various quantities of interest. In Chapter V we will consider and discuss the results obtained from the numerical computations using a heavy gas scattering kernel. We will also compare these results with the preliminary experimental results obtained by Booth
(41).
CHAPTER IV
PULSED NEUTRONS IN MODERATING SYSTEMS
The analytical treatment of the pulsed neutron problem is similar to that of the neutron waves. A major difference, however, arises in connection with the extra multiplicity introduced by the second order time derivative which introduces a quadratic term in the time eigenvalue. For this reason and in order to obtain a consistent set of conditions sufficient to determine all the modal amplitudes, we use the invariance of the expectation value of neutron population, (.* 1 0 , as our starting point. This involves the
v
knowledge of the adjoint flux which increases the complexity of the problem.
The pulsed neutrons problem will be studied here starting with Eq. (2.61). This linear inhomogeneous equation is rewritten here for convenience.
[H(e) + 1 F(1) _ + __ 2 &2) *] ,c,t)
VO at VO2ï¿½ et2 3
* [Ccz) + 1 a I X(IEt) (4.1)
Vo 2 t
61
where it should be recalled again that the functions
*re,t} and x(z,c,t) are the nonMaxwellian components of the flux and the source, respectively. The system configuration considered here consists of a rectangular parallelopiped block of moderator whose dimensions are 2a, 2b and 2c. The pulsed neutron source is located at the center of this system, i.e., at x=O, y=O and z=O. The boundary conditions associated with this problem are:
(1) Symmetry of flux.
(2) Flux vanishes at the extrapolated
boundaries.
(3) Finite flux everywhere in the system.
Since these conditions must hold regardless of the source energy distribution, it follows that they must hold true for the nonMaxwellian component, (rc,t), which contains the spacial and time dependence of the flux. Therefore, the above conditions can be given by the following analytical expressions.
*(x,y,z,ic,t) *(xyzqt) (4.2)
*(xy,zCt) = *(x,y,z,et)
(4.4)
0(Ia,yzE,t) = 0 (4.5)
(x,i,z,,,t) = 0 (4.6)
*(xay, 6,C t) 0 (4.7 Y
*(x,y,z,c,t) = finite (4.8)
where 2i, 2A and 2Z are the extrapolated dimensions of the system.
We will first solve the homogeneous part of Eq. (4.1), i.e.,
[H(c) + _L F(e) _L   CvI2O *( ,ct) 0
Vo at V02c at2 3 (4.9)
The solution of this equation is developed in cosine functions for the spacial dependence, exponential functions for the time dependence and Laguerre polynomials for the energy dependence, i.e.,
(*re,t)=uM C X (x)Y (y)Z (z)T (t) E (c) (4.10)
S m mnP n p m n p mnsp m.n~p
Considering the involved in thi X (x) = cos /2m Y (y) = cos j Zp(Z) = cos T (t) exp(x m,n,p
63
boundary conditions the functions
3 equation are given by
1
r Y) W Z)
m,np
E (e) = A' L(1)(e) (4.15 m,np ï¿½ m,nlp j
where c and AR are constants to be determined
m,n,p m,n,p
and m, n and p are positive integers from 1 to some desired maxima.
The combination of Eqs. (4.9)(4.15) gives the equation
I C
mnp mn,p
[H()  A
m,n,p
Fc) + A2
mnp
+ 1 B2
3 mn,p
X(X) Y (y) Z (z) T (t) E (E) 0 (
P mnp mmnp
(4.11) (4.12)
(4.13) (4.14)
)
(4.16)
where
AMnp m n p / vï¿½ (4.17)
B ml I2n1 I2 I2p1 1 2
B22 m1 . . (4.18)
,nop1 2& 1 26 I 2c1
Eq. (4.16) contains all the spacial modes corresponding to all the combinations of m, n and p values. To make this equation more useful we can separate it into a set of uncoupled equations by operating on it with the integral operator
0 = j dy dz Xm, (x) Yn' (y) (z) (4.19)
and using the orthogonality property
coS . U Cos U du = d 6 (4.20)
d2d / 2d q, q
which applies to each of Xm(x), Yn(y) and Zp(z). This leadsto the modal equation
Cm n~[H (c) AnF(E}+A 2 C1+ 1 a2 ]TmWpE ) = c [EcA m,np 3 m, n,p IM Snp
(4.21)
65
Since the operators in brackets do not depend on time, and hence do not operate on T) , this equation can m,n,P
be divided through by c Tit) The following m,n,p m,n,p
equation follows
[H(e)  A NOs) + A? EI + L B2 ] E (e) 0
m,np mnp 3 mnp mnp
(4.22)
This equation applies for all the spacial modes, i.e., for all values of m, n, and p. We call this equation "the modal equation" because its solution for given m,
n, and p determines E(E) , A and hence T(t) m,n,p m,n,p m,n,p and therefore, completes the solution of the (m,n,p) mode. To solve this modal equation, we first substitute for E(E) from Eq. (4.15) and then operate on
m,n,p
the resulting equation with the integral operator
w de m(e) Ll)(c) (4.23) it
0
This leads to the equation
[la  8 A + , A2 + B2 IAL :0 L "L t m,n,,p m,np Mnp ,p
(4.24)
66
where V' = 0, 1, 2, ..., and
aï¿½,ï¿½ = f m(e) L(1)(c) H(c) L (1)() dc
0 I
a 8 f m(c) L(1)(c)
0t f
0
0
FL) L(1)(c) dc
L(1 ()  L(1)(e) dc it, I
L() L , (g)
L ( ) de = 1l 61 t
Cc) dc 3
Notice that the matrices a, t,, 11, and 11Z are exactly the same as those for the neutron waves. See Eqs. (3.28)(3.31) of Chapter III.
If the maximum value of the index, i, is L, then Eq. (4.24) represents a set of (L+1) linear homogeneous equations in (L+l) unknowns, Am,n,p, where ï¿½ , ï¿½ ' = 0, 1, 2, ..., L. The compatibility condition of Eq. (4.24) is that its determinant be zero.
_ , Amnp + , ,
,B2 Z 0
1n,Z m,n,p I
11'A = 0, 1, 2, ..., L
(4.25)
(4.26)
(4.27) (4.28)
(4.29)
67
This equation will be called the eigenvalue equation since its solution determines the time eigenvalues, Am, ,1 which lead to the eigenfunctions, T(t)
n, m,n,p
according to Eqs. (4.14) and (4.17). Each element in the determinant of Eq. (4.29) is, in general, quadratic in A . Therefore, the expansion of this
m,n,p
determinant is a polynomial in A with a maximum m,n,p
power equal to 2(L+1). Hence, the solution of Eq. (4.29) yields 2(L+l) eigenvalues, A , which lead m,n,p'
to 2(L+1) time eigenfunctions
Tk(t) = exp(Xk t) (4.30)
mnp mn~p
where
Ak = v Ak (4.31)
mnp 0 mtn,p
and where k = 1, 2, ..., 2(L+1). The multiplicity in the time eigenvalues leads to the multiplicity not only in the time eigenfunction but also in the energy modes, since, for each eigenvalue, Ak , there corresponds a I k m,n,p
set of constants, A mn,p' as a solution of Eq. (4.24). Each set of constants leads to the formation of an energy mode given by
Em() k (k m L(1)() (432)
m ,n tp k m pn p I
68
k
For any eigenvalue, A , Eq. (4.24) can be solved for A (L=l,2,...,L) in terms of Aok or we can solve m,n,p m,n,p for the ratios
L k Ak Ak
R = AL~k / Aok L=l,2,...,L (4.33) m,n,p m,n,p m,n,p
from the set
L
8[91  Ak + Ak2 + B2 R ,k
L1'L m,n,p Zmnp Mnp mnp
 A k +y Ak2 + n B2
0mn,p Lo m,np 11P mtnp
.' 0,1,2,...,L1 (4.34)
where the equation corresponding to t' = L w~s neglected in order to make the set, Zq. (4.34), with L inhomogeneous equations and L unknowns,R I # k, 2 ..., L).
m,n,p
A FORTRAN subroutine, DETEX, was developed to
expand a determinant, the general element of which is a polynomial in an unknown, say x, of any size. The result of this expansion is a polynomial in x with a maximum power of IJwhere I is the size of the determinant and J is the size of its general element, i.e., the maximum power of x in this element. This subroutine is listed in Appendix D. The use of DETUX for Eq. (4.29) leads to an algebraic equation with a maximum power of 2(L+1). Then a polynomial solver, POLY, 48) is used to find all the roots of this equation,
69
i.e., Ak (k = 1, 2, ..., 2L+2). Having determined
m ,n,p
these eigenvalues the set of Eqs. (4.34) is solved for the ratios, RLk , using another subroutine, ELM,
m,n,p
which was developed for this purpose utilizing the Gauss elimination method discussed in (49). This subroutine is also listed in Appendix D.
The whole computational process discussed
above is then applied for all the spacial modes, i.e., for all the values of m, n and p. Finally, the flux is recombined using Eqs. (4.10)(4.151, (4.17), (4.30)(4.33), along with the relation
#(f'e't) = m(C) r(oest) (4.35)
This combination leads to the total flux expression
S X (x)Yn(Y)Z(z)Tk(t) Ek(t) r(C)
m~n.,p mqnqp m n T mqntp mntp
(4.36)
where we redefine
Ck Z c AOk (4.37) mnp m,np mnp
2m 1
Xm(x) a cos *x (4.38)
m 2a
2n 1
Yn(y) cos (Y. (439)
f2p 1
Zp (Z) = COs NZj (4.40) 2c
Tk(t) = exp( n t) (4.41) m,.n~p m,njP 'k () I RLRAk (1)
=,n,p m,n,p L (4.42)
Notice that all these functions have been determined except Ck which will be found later from the prinm,n,p
ciple of neutron conservation.
In the neutron waves case we saw that the source condition, Eq. (3.95), leads to the set of Eqs. (3.114) which are enough to determine all the constants, Ck,n (k = 1, 2, ..., L+l), for each (m,n) spacial mode. In the pulsed neutron case such a condition is not enough to determine all Ck,np not because of the additional space index, p, but because of the greater multiplicity in the energy modes, i.e., due to the fact that k takes twice as many values as in the wave problem, k = 1, 2, ï¿½.., 2(L+l). A more general condition based on the neutron conservation principle will be used in this case. The use of this principle requires the knowledge of the
71
adjoint equation and the adjoint flux which are studied in the following section.
The Adioint Flux Equation
The differential equation of the nonMaxwellian component of the flux is given by Eq. (4.1). Substituting for the operators H(c), F(c) and G(e) from Eqs. (A.30)(A.32), this equation becomes
[h0 (c) + hl(c)  + h2(C)  + l fo(t)+ fl(e)i B E 3 2 vo 0 B
a2 1 2 1
f ()  )  +  (re,t)
2 ae2 at v 2c at2 3
0
a a2 1 3
[go(,)*g(,) +g (C)  + i]x(r, e,t) (4.43) g0 (Be+1c)3 2 aC2 v /Cat
where hife), fi(c)and gi(c) are algebraic functions of E. To find the adjoint flux equation the following rule is used (IQ). This rule states that the adjoint operator of the differential operator
a(x,y,....z) am a a ap axm By n azp
is given by
m+n+. ..p
(1)
m n
m n ax ay
0..  a(x,y,...,z)
ap )z~
and its application to Eq. (4.43) gives the adjoint equation
[H+(c) 1 i_ F+(U)  + 1 a2 + ! A2]**(p,e ,t) = 0
at 02 3at2
(4.44)
with the adjoint operators defined as
H+() = h0 (c)   hl(c) +  h2(C)
a a2
F+(ï¿½) fo (C)   fl(c) +  f2(ï¿½)
3Cï¿½ a C2
t4.45) (4.46)
In solving Eq. (4.441 for the adjoint flux the same technique used in solving Eq. (4.9) for the ordinary flux is followed and we obtain the solution
I c + Xm(X)Yn(Y)Zp(Z)T+(t) E+(c) m(e) m~n,p mnp mnp m,np
(4.47)
where c+ are constant, Xm(x), Yn(y) and Z (z) are
m,n,p P those given by Eqs. (4.1l)(4.13) and
T+(t) exp(+ mn,p mnp
E+(E) A m + L (E) mn,p mnp
(4.48)
(4.49)
The use of Eqs. (4.11)(4.13), (4.47), (4.48) in Eq. (4.44) and then the application of the operator
O' f de L (1)()
0 to
(4.50)
lead to the adjoint modal equation
[H+()  A+
m,np
F+'(c) +  A+
C mnp
+ 1 B7
3 m,n,p
m,np
m(e) = 0
(4.52)
A+
m~n.,p
+
= n
m,n ,p
where
(4.51)
/ vo0
74.
The adjoint eigenvalue equation is then obtained by using Eqs. (4.49) and (4.23) in Eq. (4.51).
+ A+
mMn,p
+ A +2
msn,p
+ n+ B 2 A0
Lit
mtnp msntp
(4.53)
and the compatibility condition is
 8+ A+
ï¿½l m,n,p
2
+ y+ ^+
Aï¿½, m,n,p
+ TI + 2 tit Bm,np I*
(4.54)
+ z f L, (c) H+(O) mCc) L1)(t) de
0
f L(1)(O) F+CE) M(e) L(1)(c) de
0 t ï¿½
o. U(1) (e) 1 m(c) L(1)(c) de y
0 ZIA
1
3 0
f L(l)() m(c) L(1)(c) de a n
It ï¿½ lï¿½
(4.55)
(4.56) (4.57) (4.58)
We can easily show that these adjoint matrices are the rotational matrices of a * B y and nr Combining Eqs. (4.45) and (4.55) one can calculate the matrix a+
+
where
+
+ TI
n+,,
9
+
L 0
L ()[h( (c)  ' h (C) + 2 h()]
(1)(L) de
ï¿½m(c) L g d
4.59)
Two successive integrations by parts lead to the expression t
L *h2(C) 3C
z f
0
(1) H() L ( )
'2 ]L(1)()d DC2 11
C4.60)
Thus
a

(4.61)
Similarly,
2. ~ 2.
(4.62)
9,2.'
Due to the symmetry of Y
L',
and .
we can write
= Y
LgL L,,L'
(4.63)
(4.64)
f m(c) L(1)(0)[ho(c)+hl(C)
11 0
76
Hence, Eqs.,14.53) and (4.54) become
~ AA+2 B
[a a LIVh + A +2 + n B2 P ' ,L' mn,p L, m,np ,' mnp
0 0 (4.65) m,n ,p
and
Qq 8 A+2 +11 B2 =0 (4.66) i nmn,p ï¿½L, m,n,p t,t m,n,p
A simple comparison between Eqs. (4.29) and (4.66), keeping in mind that the rotation of a determinant about its main diagonal does not change its roots, we arrive at the conclusion that both the flux and its adjoint have the same time eigenvalues, i.e.,
A+ A (4.67)
m,n,p m,n,p
The essential difference is that the time eigenfunctions of the flux are decaying exponentials, while those of the adjoint are rising exponentials, as seen from Eqs. (4.14 and (4.48). This result is expected because the adjoint flux is the importance function and must rise as the neutron flux decays where the remaining neutrons become more important.
77
Thus, Eq. X4.66) does not have to be solved since its roots are given by Eq. (4.67). Defining the ratios
R+k A +1,k / +ok m,n,p mnp m,np
(4.68)
and solving Eq. (4.65) using all the eigenvalues one by one, we can finally recombine the adjoint flux.
**('r,c,t)=
I C+k X (X) Y (y) Z (Z) mnpkmgngp m n p
T+k(t) E+k(C) m(e)
mgntp mnp
where we redefine
C+k c + A+ ok m,np mtnp mn,p
T+k(t)= mn ,p
E+k(e)
(4.69)
(4.70)
(4.71)
exp( Xk
m,n,p
R+t'k L (1)() m,np LI
(4.72)
In this section we have derived the adjoint
flux equation, proved that it has the same time eigenvalues, and finally, solved the adjoint equation except
for the constants, C+k the determination of which m,n,p
is not necessary as we will see later.
Neutron Conservation
In order to complete the solution for the
neutron flux the quantities, Cknp must be determined. Before we do so it is helpful to discuss qualitatively the time behavior of the neutron population in the system. As the source pulse has a finite width there is some time interval during which neutrons are continuously fed into the system. Consequently, the neutron population at a certain position and a certain energy rises to reach a maximum and then decays. The time at which this maximum is reached depends on the amplitude, the time dependence and the duration of the pulse. Furthermore, the neutron density may reach an intermediate steady state depending on the pulse shape, as in the case of a broad rectangular pulse, or a pulse with a broad flat portion. For these reasons, and since the negative time exponentials alone can describe the time behavior of the neutron population only in the asymptotic region but not in the transient region, the quantities, Ck , must be functions of
t , n,p
time. Hence, the flux expression of Eq. (4.36) becomes
~Crct) 
XM(x) Yn(y) Zp(Z) Jk (t) Ek ()
m.,n p m,.n,p
(4.73)
where
Jk(t)
m,n,p
= ckt) x T(t)
m,n,p m,n,p
(4.74)
Consider the flux and the adjoint flux equations, Eqs. (4.1) and C4.44Y. Operating on the first by the operator
+ f dr f de ft dt **('rst)
r e 0
(4.75)
and on the second by the operator
Q =.f dr f dc f dt(1',ct)
r C o
(4.76)
and then subtracting the resulting equations, we obtain the integrodifferential equation
m,n,p,k
[Q+H* QH+*] kQ  _ QI+ _ ]
o at at
1 + a2 1 a2
+ [Q Q Q*J. (2 __._v at2 2.at2 Q
V0 Q+ _ a. Q +GX (4.77)
Vo / at X+ G
where all the arguments have been dropped for simplicity. Introducing Eqs. (4.45) and 44.46) into Eq. (4.77) and integrating by parts we can prove that
Q+H*  x 0 (4.78)
Q+F . *QF+  ** :f dr f [**Fï¿½] de (4.79)
at at. r t
Q a *Q a2 # Q*
c at2 v at2
= f dr C  ,ï¿½ ; ,] dE (4.80)
y ï¿½
Q+V2*  QV2l* = 0 (4.81)
where in this proof we have used the condition that the flux is zero at t=o and utilized the invariance of the
expectation of the flux, i.e.,
_ f dr f de **(P,,t) *(itt) = 0 (4.82) at~ ï¿½
at any time. Eq. (4.77) takes the form
2 f dr f [d*   4   c* dc+ 1 fdrf ,*F~de V0 C C at C at Vo rC
 + X + Q+GX (4.83)
V0 a
Next, we will assume for the source the expression
X(E,t) = (c) Ts (t) 6(x) 6(y) 6(z) (4.84)
i.e., we assume a point source at the origin with separable energy and time dependence. At this stage we introduce Eqs. (4.47), (4.73), (4.36) and (4.84) into Eq. (4.83). Performing the spacial and energy integrals in the resulting equation and considering the orthogonal property of the spacial modes we arrive at the equation
C~k' +k' F k9k
ST (t) 1 akik  __ k
nn,p k~k m,n,p m,n,p0
1 bktk ik(t) Vo mono mflnvp
PII
1 +k' k' akdk =n c+k' f T (t) tcm  + dpT(t)dt ,15c m np m,n 0 mnp minip9t ren (4.85)
where
jk(t) = ck(t) Tk(t) . (4.86)
mnp monop monop
m
a kk f E k() m(e) i Ek(,) de (4.87) mrnp 0 mnp C mnp
kk kk bk0 =Of E (c) m(e) F) E k() de (4.88) monop M np monop
m'f E+c) m(C) 1 (c) de {4.89)
0 mnp v S
k' +k[
D : f E Cc) m(c) G() E (c) de (4.90) mnp 0 monop
Equation 14.85) must be satisfied at any time regardless of what the constants, C+k , are. Therefore, the
m,n,p
coefficients of C+k' in both sides of Eq. (4.85) must
m,n,p
be equal. Hence, the following set is obtained.
ST+k(1) [1ak 1a  x+ bk1k ]Jk(t) k mnp 2 m,n,p at mgnp v0 mnp mbnp
1 t Tk() { ck A (t) dt
:  I f T+  +}I dt
ac k o m,n,p Vo m,np at m,n,p  a (4.91)
This equation holds true for all the values of m,n,p and k'. Hence it represents (M.N.P.K) differential equations in (M.N.P.K) unknowns, jknp, where MN,P and K are the total numbers assigned to m,np and k tor k), respectively. Rearranging Eq. (4.91) and integrating the righthand side by parts it becomes
1 k k'k a 1
T (t)[ a  +{bksk  Ak' ak'k}] jk(t) k V 2 at Vo
t
1[_ ck T+k(t) T8 (t) + {dk  Ak ck}f Tk(t)Ts(t)dt]
B k Vo 0
(4.92)
84
where the indices, m,n,p, have been dropped for simplicity, If Ts(t) is a known function the integral in Eq. (4.92) can be evaluated and this set of equations can be solved for the unknown functions, jk(t).
Formally, this completes the solution for the flux in the pulsed neutrons problem. To go into more detail in the solution a specific case has to be treated. Assume that the pulse takes a rectangular shape as a function of time. Analytically, this pulse can be given by
T5(t) = u(t)  u(t  '0, (4.93) where T is the width of the pulse and u(t) is the heaviside unit step function. With the aid of Eq. (4.93) it can be seen that the righthand side of Eq. (4.92) is a function of time in the range 0 < t T T, while it becomes a constant in therange t _ v. Therefore, this set must be solved in both regions, where it takes the two forms
+k' _ akpk .+ .Lbk'k Akjk(t)
k v. 2 at V0
dk T+k(t) for t < T (4.94)
(dk  Akck) T+k(t)/Xk for t (4.95
k'
Multiplying these two equations by v02 and dividing them by T+k' (t) and then using the value of the latter they can be reduced to the forms [Ak'k I + Bk'k]Jk(t)
2 pk exp[_(Xk + Xk')t] t < T (4.96)
k
SQ. exp( )k't ï¿½ (4.97)
where
Ak',k = ak,k (4.98) Bk',k vo(bk'kA k'a (4.99)
 c VO (dk/Ak) (4o00)
Qk  v (dk/kck) exp(X ) . (4.101) Taking the Laplace transform of Eqs. (4.96) and (4.97) one obtains
I [Ak1k s + Bksk]Jk(s)
k
p k (s  Xk') t < T (4.102)
 Qk / cs  xk') t >, ' Q k(4.103)
If these algebraic equations are solved for k() then the inverse Laplace transform
JTk(t) = k (t) (4.104)
gives the sought for functions which complete the solution.
In Chapter V we will discuss some of the numerical results obtained for graphite and compare these results with those obtained experimentally by Starr and Price
(33).
CHAPTER V
RESULTS AND CONCLUSIONS
The theoretical analysis of the theory of neutron thermalization in moderating media in the consistent P1 approximation was developed in Chapter II. This analysis is tested in two applications of practical interest, the neutron waves and pulsed neutron experiments as presented in Chapters III and IV. In both applications the heavy gas scattering kernel and the 1/v absorption cross sections are used, and the first order associated Laguerre polynomials are utilized to express the energy dependence of the flux. All the results presented in this chapter are obtained for AGOT type graphite with density equal to 1.67 gm/cm3.
Application to Neutron Waves Technicue
The experimental arrangement under consideration consists of a parallelopiped block of graphite with a sinusoidally modulated plane neutron source at its xy , face. This source is assumed to have a Maxwellian distribution in energy and a cosine shape in the xand ydirections. A 1/v neutron detector is used to 87
88
measure the neutron density at various distances from the source along the central zaxis of the graphite assembly. The experimental technique and the data analysis are fully discussed by Booth (41) who obtained the amplitude and phase shift of the neutron density at various positions in the graphite assembly for several source frequencies. The errors in all Booth's preliminary results used here are typically of the order of 3 per cent.
A code, NWP, has been developed for the IBM709 to calculate all the quantities involved in this discussion. A listing of this code and the subroutines associated with it is found in Appendix D. In all the computations the fundamental spacial mode alone was considered. Therefore, the spacial indices are dropped from all the quantities involved in this presentation. The computational scheme follows the analytical steps of Chapter III and the perturbation technique is used to solve the eigenvalue equation, Eq. (3.35).
The use of Laguerre polynomials in the expansion of the energy flux modes, k (E), leads to a number of eigenvalues equal to the number of polynomials in this expansion. Each eigenvalue, rk, is related to the inverse relaxation length, Pk , through Eq. (3.36).
010
.89 k
These eigenvalues were calculated using numbers of polynomials ranging from 1 to 10. First one polynomial is used and the fundamental eigenvalue, r0, is obtained. Then the number of polynomials is increased by 1 in each step and the next higher order eigenvalue arises. It was found that the use of L+1 polynomials does not change the L unperturbed eigenvalues obtained using L polynomials, but merely gives rise to the L+Ith eigenvalue which is greater than all the previous ones. Furthermore, the difference between two eigenvalues of successive orders decreases with increasing orders as seen in Fig. 1. Therefore, it is expected that the highest ordered eigenvalues tend to be so closely spaced that their distribution may approach a continuum.
The frequencydependent complex inverse
relaxation length, P0o of the fundamental energy mode is plotted vs. source frequency in Fig. 2 and is compared with the experimental values obtained by Booth. Based on the analytical development of the neutron waves problem in the diffusion equation by Perez and Uhriq (38) calculation was made in the heavy gas and heavy crystal models. The matrix elements for the later model were obtained from Razminas (51) and the results are also presented in Fig. 2. A very reasonable agreement
 1ï¿½ 0 2 O7 Nme o 0 m L
I IIII I I I I I
1 2 3 4 5 6 7 8 9 10
Number of Laguerre Polynomials, L
Fig. 1. The Steady State Inverse Relaxation
vs. Number of Laguerre Polynomials
1.0
0
M0
.t
J
(d
X Q) Q)
01.1
0.05
Length
0 100
200 300 400 500 600 700 800
900 1000
Source Frequency (cps)
Fig. 2. Components of the Complex Inverse Relaxation Length of the
Fundamental Mode vs. Source Frequency.
0 .0
0 C0
.08
.06
.04 .02 .01
92
between the theoretical results of the P1 approximation and the experiment is obtained for frequencies up to 500 cps. For higher frequencies the theoretical curves diverge from the experimental values giving a smaller real component and a greater imaginary component of P0 leading to less attenuation and greater phase shift, respectively. k
A more appropriate comparison may be made between the calculated and the experimental total neutron densities. In Fig. 3 and Fig. 4 the amplitude of the neutron density is plotted as a function of position for source frequencies of 100 and 500 cycles per second. In both cases the theoretical results agree very well with the experimental values up to about 25 cm from the source. The disagreement beyond this position becomes more pronounced where the effect of the higher modes disappears, due to their fast attenuation, and the fundamental mode dominates. This disagreement may be reduced by including more modes in the combined solution since the higher modes tend to raise the amplitude near the source and hence increase the slope of the theoretical curve toward that of the experimentally measured one.
The theoretical phase shift of the neutron density is in a much better agreement with the
1.0
)
0.01 :
00
0.00511
0 10 20 30 40 50 60 70 Distance from the Source (cm) Fig. 3. Amplitude of the Combined Neutron Density ys. Position along the zaxis for Source Frequency of 100 cps.,

Full Text 
77
Thus, Eq. {4.66} does not have to be solved
since its roots are given by Eq. (4.67). Defining
the ratios
R* = / A*0*
rnjn jp mynyp iti ^ n ^ p
(4.68)
and solving Eq. (4.65) using all the eigenvalues one
by one, we can finally recombine the adjoint flux.
^*(r,e,t) =
l C+k X (x)
m,n,p,kmn*P m
Yn(y)
Zp(z)
. T+k(t) E+k(e) m(e)
mfn fp m jri jp
(4.69)
where we redefine
C+k c+ a+ o,k
m,n,p m,n,p m,n,p
(4.70)
T+k(t)
m,n,p
exp(Xk t)
m,n,p
(4.71)
E+k(e)
l R+l'k L1)(G)
m,n,p l
(4.72)
In this section we have derived the adjoint
flux equation, proved that it has the same time eigen
values, and finally, solved the adjoint equation except
22
This equation is easily recognized as the
telegrapher equation which includes the transport cor
rection on the source. This correction is given by
g
the term (3D/v) x(r,t) It should be noticed that
3t
this correction term is neglected in the telegrapher
equation given by Meghreblian and Holmes (45) and by
Weinberg and Wigner (46). The neglection of this
term is possible in a steady or quasi steady state
operation where the source variation with time is very
slow. On the contrary, the source derivative term is
very important in time dependent kinetic problems
which are sensitive to time variations of the source.
In studying the two problems of neutron waves and
pulsed neutrons, which are of most interest to this work,
the source transport correction is specifically impor
tant when sharp neutron pulses are fed into the system
and the transient flux is investigated, or when the
source frequency becomes large. However, these two
subjects will not be studied through the telegrapher
equation but through the general space, energy and time
dependent formulation given by Eq. (2.61).
V
Amplitude
Fig. 3. Amplitude of the Combined Neutron Density vs. Position along the
zaxis for Source Frequency of 100 cps.
139
306 CONTINUE
DU 407 J 7=1,LE
ND=7
N7 = J 7 1
JP=1+N1+N2+N3+N4+N5+N6+N7
IF(LOND) 307,107,307
107 CONTINUE
L)U 07 M= 1 LU
CD(M,1)=CC(M, 1,J 1 )
CD(M,2)CC(M,2,J2)
CD(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4)
CD(M,5)=CC(M,5J5)
CD(M,6)=CC(M,6,J6)
CD(M,7)=CC(M,7,J7)
207 CONTINUE
CALL EVAUET(LD,CD,DET)
B(JP)=B(JPJ+DET
307 CONTINUE
L'O 408 JS=i,LE
NQ = 8
N8=J81
JP=1+N1+N2+N3+N4+N5+N6+N7+N8
IF(LDNU) 308,108,308
108 CONTINUE
DO 208 M=1,LD
CD(M,1)=CC(M,1,J1 )
CD(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3,J3)
CD(M,4)=CC(M,4,J4)
CD(M,3)=CC(M,6,J3)
CD(M,6)=CC(M,6,J6)
CD(M,7)=CC(N,7,J7)
CD(M,8)=CC{M,8,J8)
208 CONTINUE
CALL EVADET(LD,CD,DET )
til JP)=B( JPJ + DET
308 CUNTINUE
BO 409 J9=1,LE
N D = 9
N 9 = J 9 1
JP=i+NHN2+N3+N4+N5+N6+N7+N8+N9
IF(LDND) 309,109,309
109 CONTINUc
DU 209 M=1,LD
CU(M,1)=CCIM,1,J1)
CD(M,2)=CC(M,2,J2)
C D { M 3 ) C C ( M 3 J 3 )
CD(M,4)=CC(M,4,J4)
CD(M,9)=CC(M,5,J5)
CD(M,6)=CC(M,6,J6)
100
\(r,E',t) can be well approximated by a Taylor series
expansion about E. This assumption seems reasonable
because the flux energy distribution in moderating
assemblies is very similar in shape to the Maxwellian
distribution and hence the component, iii(Â£,E,t)^ is not
expected to have sharp variation in E. This assump
tion was essential in order to evaluate the scattering
integral in the Boltzmann equation.
This theory was applied to describe the neutron
waves and the pulsed neutrons experiments. In both
applications the theoretical analysis was general from
the standpoint that it made no specific reference to
what types of scattering kernel and absorption cross
section should be used. The analysis in both cases
used the associated Laguerre polynomials to formulate
the energy dependence of the flux where the solutions
led essentially to eigenvalue problems.
The validity of the heavy gas model and its
compatibility with the approximation of the Boltz
mann equation were tested in the numerical solutions
obtained for these two applications. From the results
that have been presented here it can be concluded that
the heavy gas model in the P^ approximation can be used
in studying the pulsed neutrons experiment in rela
tively large moderating assemblies with bucklings up
145
z4=fe2*fei6
Z5=Z3Z4
Z6=XCUN7*Z5
TE 1 = Z1Z2Z6
Z l = XC0N6 *TE13
Z2 = Z 1 *TE 14
Z3 = TÂ£ 12* FE 15
7.4=FE11*FE16
Z5 = Z 3 + Z4
Z6=XC0N 7*Z 5
TE2=Z2Z6
Z1=FE1*FE1
Z2= FE2*FE2
Z3=Z1+Z2
rEM=SQRTF(Z3)
IF(TE1)113,113,112
113 Zi=FEMFEl
72 = XCUN8 Z1
FE4 = SQR F F{Z2)
ZL=FE2/Ft4
TE 3 = XCONb Z 1
GQ FU 111
112 Zl=rCM+Tri
Z2=XCUN3*Z1
TE3 = SG,'RFF( Z2 )
I F r E2 ) 1 10,200,200
110 T E 3 = T E 3
200 Z1=TE2/Tc3
TE4=XC0N8*Z1
111 TE7=rE13+TE3
rE8=TE14+TE4
T E9= TE 13TE 3
FE 10F E14F E4
FE 1 = XC0W6*T E15
TE? = XC0i*6*TE16
Z1=FE7*F87
Z2= Ed*FE8
Z3=FÂ£9*FE9
Z4=F E10*TE10
IF ( Z 1 + Z2Z3Z4)204,204,205
204 E 7 = TF9
FE8=FE10
205 Zl=fE7*FE7
Z2=FE8*FE8
EMZ1+Z2
Zl=rEl*FÂ£7
Z2 = TE2* FE8
Z3 = Z 1 + Z2
IE3=Z3/FEM
Z 1 = FE2* E7
Z2 = T E1 F Â£8
Z3=Z1Z2
IÂ£4=Z3/TEM
74
The adjoint eigenvalue equation is then obtained by
using Eqs. (4.49) and (4.23) in Eq. (4.51).
u
Â£
A+ +^;tA+2 ]a+
m,n,p m,n,p m,n,p m,n,p
(4.53)
and the compatibility condition is
Â£(Â£
+ Y+i A+ + R2
Â£,Â£ m,n,p Â£;Â£ m,n,p Â£jÂ£ Dm,n,p
e + a+
where
l, = / H+(c) m(e) L^^(c) de
B+ = / L<1>(e) F+(e) m(e) ^^(t) de
If i It Â£
% w
T+, / L(^)(e) Im(e) L^i;(e) de = y ,
Â£',Â£ o' Â£' e Â£ Â£ Â£
(1)
n+ = / L^l)(e) m(e) L.(D(e) de = n
0 0 1 0 0 0
Â£Â£ 3 0' Â£
Â£; Â£
a 0
(4.54)
(4.55)
(4.56)
(4.57)
(4.58)
We can easily show that these adjoint matrices are the
rotational matrices of a g v and n .
Combining Eqs. (4.45) and (4.55) one can calculate the
matrix a+,
Â£ y Â£
on non
131
UNIT NO. 4
PERTURBATION TECHNIQUE
DO 200 K= 1 L
DO 25 M= 1 L
DO 24 N=1,L
C TR ( M, N)=ALPHA(M,N)ETA(M,N)*XS(K)
24 CTI(M,N)=ETA(M,N)*YS(K)
25 CONTINUE
DU 27 M=1,LM1
ML=M+LM1
DU 26 N=1 LMl
NL=N+LM L
C(M,N)=CTR(M,N+1)
C ( ML f N ) =CT I ( M N + I )
C(M,NL)=C(ML,N)
26 C ( ML ,NL ) =C ( M, N )
C(M,L2M1)=CTR(M,1)
27 C(ML,L2 M1)= C TI(M,l)
CALL CLEM(L2M2,C,DET)
RSR(K, 1) = 1.0
RSI(K,1)=0.0
DO 26 J = 2 L
R$R(K,J)=C(Jl,L2MI )
1=J+L1
28 RSI(K,J)=C(I1,L2M1)
CALCULATION CF THE HOMOGENEOUS MATRIX
WHICH IS THE SAME FUR ALL DERIVATIVES
00 31 M =1 L
ML=M+L
U 29 N = i L
NL=N+L
A(MfN)=CTR(M,N)
A(ML,N)=CTI(M,N)
A(M f NL)=A(ML N)
2 9 A(ML NL)=A(M,N)
AIM,1)=0.0
A(M,LP1)=0.0
DO 30 N = 1 L
A(M,1)=A(M,L)ETA(M,N)*RSR(K,N)
30 A(ML 1)=AIML,1JETA(M,N)*RSI(K,N)
A(M,LP1)=A(ML,1)
31 A(ML,LP1)=A(M,1)
C CALCULATION OF THE DERIVATIVES OF
C THE EIGENVALUES AND THE RATIOS
DU 33 M1,10
DO 32 N=l,10
32 G{M,N)= 0.0
33 G(M, 1) = 1.0
G ( 1,2 ) = 1.0
DO 35 M=2, 10
DU 34 N=2,10
34 G(M,N}=C(Ml,N)+G(MlfNl)
35 CONTINUE
CHAPTER III
NEUTRON WAVES IN MODERATING SYSTEMS
Analytical Formulation of the Neutron
Wave Problem
Various theoretical and experimental
investigations of the neutron wave problem have been
reviewed in Chapter I. This chapter is devoted to
the investigation of the neutron wave problem through
the time, energyand spacedependent flux equation,
Eq. (2.61), derived from the consistent P^ approxima
tion which gives rise to the thermalization and
transport effects. As a convenient reference, Eq.
(2.61) is restated here.
(H(e) + F ( e)
(3.1)
Since the source does not exist inside the
system, the source term is dropped from Eq. (3.1) and
a suitable source condition will be established later
in this chapter. Thus, the neutron waves will be
studied through the homogeneous equation
23
20
i> (r ,E,t)
'I'(r,t)
(2.65)
Dki>(r#E,t) = 0 k>0 2.66T
It is easily seen that the operators, Qm(E), become
energy independent and are reduced to the moments of
the scattering kernel, i.e.,
Qm(E) = Qm = [ Mm D Mm Em (2.67)
k
where
Em = / pm(uo} EsUo)dyo (2.68)
l
In the P^ approximation m takes the values 0 and 1
which give
Zo = I P0(uQ) zs(vj0)d'j0
1
(nQ)dM0
1
(2.69)
3
the corresponding eigenfunctions are has not been
answered yet. Shapiro also found that the velocity
associated Laguerre polynomials give faster con
vergence than the energy polynomials.
The technique of pulsed neutrons used by many
investigators (8 3_5) and others, has been reviewed
by Beckurts (21). The oscillatory neutron source
method was first developed by Raievski and Horowitz
(36). Using this technique Uhrig (37) studied the
nuclear properties of subcritical assemblies while
Perez and Uhrig (38) investigated the thermalization
theory and calculated the thermalization parameters.
They also showed the similarity between this tech
nique and the pulsed neutrons technique from the
analytical point of view. Experimental investiga
tions have been carried out by Perez et alt (39, 40)
and Booth (41). According to Hetrick and Seale in a
more recent publication (421 it is possible to cal
culate the specific heat integral and the zeropoint
mean square displacement of the scattering atoms in
terms of the diffusion coefficient and the second
energy transfer moment by using this technique with
low temperature experiments.
105
ho(E) E 2(E) + CM(B) M(E)] S (E)
o a. oia
i d o d
 M(E) S (E) M 2(E) E (E)
1 dE a L 21
(A. 14)
h, (E) = C M^(E) + m1(E)] Z (E) 2M 2 (E) Z (E)
x u 1 a 1 dE a
+ M^(E) [M(E) M^E)] + M}(E) M (E)
1 1 dE
+ M12(E) 2 *Â£<*)
(A.15)
h2(E) = [ Mq2(E) + M12(E)3 T a (E) + mJ(E) mÂ£(E)
+ Mq2(E) r mJ(E) M(E) ] + Mj(E) M02(E)
O d 1 n d2 O
+ 2M 2(E) M"" (E) + M 2(E) M 2 (E) (A. 16)
1 dE o 1 dE2
f 0 (E) = 2la(E) + M (E) M(E) + ~ m];(E) M^CE)
X Je* x
(A. 17)
f1(E) = M(E> M*(E) + ^ M12(E)
(A.18)
CHAPTER II
THE THERMALIZATION MODEL OF THE CONSISTENT
P, APPROXIMATION OF THE NEUTRON
TRANSPORT EQUATION
General Theory
The space, direction, energy and time dependent
Boltzmann equation describing the neutron flux in a
homogeneous isotropic medium is
i *(r,3,E,t)+E (E)
V 91 t
S(r,ft,R,t) + / dE'
E'
/ da' zg (E'E,ft'*ft) (r,ft',E',t)
a'
(2.1)
where r, ft E, v and t are the neutron position vector,
direction, energy, velocity and time, respectively.
Primed symbols indicate the corresponding variables
after a scattering collision. The terms appearing in
Eq. (2.1) are defined in the following
13 .> + +
$(r,QfE,t) drdadE = the time rate of change of
v the neutrons which occupy
the space dr dE d ft,
ft*V$(r,ft,E,t) drdftdE = the loss rate due to the
net transport of neutrons
through the boundaries of
dt,
5
140
CD(M,7)=CC(M,7,J7)
CD(M,8)=CC(M,8,J8)
CD(M,9)=CC(M,9J9)
209 CONTINUE
CALL EVADETILD,CD,DET)
B(JP)=B(JP)+DET
309 CONTINUE
U 410 J 10=l LE
ND= 1 0
N10 = N10 1
JP=l+N1+N2+N3+N4+N5+N6+N7+N8+N9+N10
IF(LDND) 310.110,310
110 CONTINUE
DO 210 M=1,LD
CO(M,1)=CC(M,1,J 1 )
C0(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3J3)
CD(M.4)=CC(M,4,J4)
CD{M,5)=CC(M,5,J5)
CD(M,6)=CC(M,6,J6)
CO(M,7)=CC(M,7,J 7)
CD(M,8)=CC(M,8,J8)
CD(M,9)=CC(M9,J9)
CD(M,10)=CC(M,10,J10)
210 CONTINUE
CALL EVADETILO,CD,OET)
{JP) = 8(J P)+DE T
310 CONIINUE
410 CONTINUE
409 CONTINUE
408 CONTINUE
407 CONTINUE
406 CONTINUE
405 CONTINUE
404 CONTINUE
403 CONTINUc
402 CONTINUE
401 CONTINUE
DO 50 J=1,JMAX
50 A(J)=B(J)
RE TURN
END
50
frequencies. The energy correction is always persistent
except when a pure Maxwellian source is used, in which
case x(r,e,tj is constant and its derivatives are zero.
Determination of the Constants,
:n.m
The final step in the solution of the neutron
wave problem is to determine the constants, n, which
appear in the flux expression of Eq. (3.67). This
determination is carried out by using the source condi
tion given by Eq. (3.95). The substitution of ^(r,e,tr
from Eq. (3.67) into Eq. (3.95) gives
l l I Cm,nVx>Vy>0m,nzm,n!z>Ek<=>e:'t
m n k
= 3 [~ + G(e) ] X(x,y,e,t) <3.99)
v /e 3t
o
If we assume the separability of space, energy and time
dependence of x(x,y,Â£,t), i.e., if we express it by
X(x,y,e,t) = Xg(x) Yg(y)Eg(e)e^wt
(3.100)
Eq. (3.99) becomes
l l l Ck o* X (x)Y (y)Ek(e)
** L u m,n m,n m n 1
m n k
 G(e)]Xs(x)Ys(y)Es(c)
(3.101)
UNIVERSITY
OF FLORIDA
LIBRARIES
ENGINEERING AND PHYSICS
LI BRARY
ACKNOWLEDGMENTS
The author wishes to acknowledge the guidance
of the members of his supervising committee. He
gratefully acknowledges the continuous guidance and
invaluable advice of his committee chairman, Dr. R.
B. Perez, without whose aid this work would have been
impossible.
The author is indebted to the American Friends
of the Middle East, Inc., and the Syrian Government
for a scholarship during his graduate study toward the
Ph.D. degree. He wishes to acknowledge a partial
financial support by the University of Florida Com
puting Center for the computations in this work. He
also wishes to thank Mr. R. Booth for permittinghim
to use his experimental results.
Special thanks are due Mrs. Gail Gyles and
Miss Barbara Gyles for their patience and enthusiasm
during the typing of this manuscript.
Last, but not least, the author would like to
express his deep appreciation for the encouragement
and understanding of his wife, Subhieh, throughout
the progress of this dissertation.
ii
59
Thus we have developed the neutron flux and
density in all desired forms as functions of space,
energy and time. Furthermore, we have seen the
neutron traveling waves in the medium through the
expressions of the amplitudes and phase shifts of
various quantities of interest. In Chapter V we will
consider and discuss the results obtained from the
numerical computations using a heavy gas scattering
kernel. We will also compare these results with the
preliminary experimental results obtained by Booth
(41)
Numerical computations are made for both
applications using a heavy gas scattering kernel and a
1/v absorption cross section in graphite in order to
test the applicability of this model and its compati
bility with the P^ approximation. The comparison of
these results with the experimental ones confirms that
this model is conveniently accurate in describing the
pulsed neutron experiment in relatively large systems
of buckling up to 3x10* cm2. This model is also
proven to provide accurate description of the neutron
waves experiment with source frequencies up to 500
cycles per second which are equivalent to transients of
periods down to about 2 milliseconds.
12 5
Having calculated VJ and
l\l
the
matrices, ljl# 6*,*, yl)lt V,i' and
follow immediately from the general form, Eq. (C.4), using
the proper functions, a Appendix D contains a list
i # J
ing of the FORTRAN Program, HGM, developed for all these
calculations.
i
non
135
UNIT NU. 7
SPACE DEPENDENT COMPUTATIONS
************
WRITE OUTPUT TAPE 6,366,L,OMEGA
366 FORMAT(1H1///,5X,34HAMPLITUDES AND PHASE SHIFTS OF THE/
1 5X,34HTOTAL NEUTRON DENSITY AND THE FLUX/
2 5X,34HENERGY DISTRIBUTION VS. POSITION,1/
3 5X,18HF0R THE CASE UF L= 12/
4 5X,16HAND OMEGA(CPS)= E16.8/5X,34(1H*)//)
IFIZDIC) 401,401,367
367 CONTINUE
DO 400 J = i,NZ
Z=ZPUINT(J)
ASIAU=0.0
SUDAU=0.0
U 381 K=1,L
SC=CUSF(RHUI(K)*Z)
SS = SlNF(RH1IK)*Z)
EX=EXPF(RHOR(K)*Z)
QDR^O.O
CD I = 0.0
DU 369 M=1 L
0DR=QDR+ZETA{M)*AR(K,M)
369 GDI = QDI+ZE TAlM)*A11K,M)
DAMP(K)=EX*SORTF(QDR**2+QD1**2)/220000.0
DTI=QDR*SSUDI*SC
DTR=QDR*SC+ODI*SS
UPHASE(K)=ATANF(DTI/DTR)
IF(DTR) 370,371,371
370 UPHASE( K)=DPHASE(K)+3.14159
371 CONTINUE
ASIAD=ASIAU+DAMP(K)*COSF(DPHASE(K))
SUDAD= SUDAD+DAMP{K)*SINF(DPHASE(K))
IF(EINT) 385,385,375
375 CONTINUE
DU 380 N=1,NE
C'FR = 0.0
WFI=0.0
DO 372 M= 1 ,L
QFR=QFR+ARIK,M)* E(M,N)
372 QFI=QFI+AI(K,M)E(M,N)
FAMP(K,N)=EX*SQRTF(QFR**2+QF1**2)
FTI=QFRSSOFI*SC
FTR=OFR*SC+OFI*SS
FPHASE(K,N)=ATANF(FTI/FTR)
IFIFTR) 373,374,374
373 FPHASEU,N)=FPHASE(K,N)+3.14159
374 CUNTINUE
380 CONTINUE
385 CONTINUE
381 CONTINUE '
TAMP=SQRTF(ASIAD**2+SUDAD**2)
TPHASE = ATANF(SUDAD/AS I AD)
26
4>(x,y,z) = \p (x,y ,z) (3.10)
D(a,y,z) = 0 <3.11)
*(x,E,z) = 0 (3.12)
where 2a, 2B, and c are the extrapolated dimensions of
the nonmultiplicative assembly in the x, y and z
directions, respectively, and where the center of the
xyface is taken as the center of the coordinate system.
The source is located at the xyface with supposedly
known spacial and energy distributions and will be
specified later when the source condition is estab
lished .
The solution of Eq. (3.7) is developed in cosine
functions for the x and ydependence, exponential
functions for the zdependence and normalized associated
Laguerre polynomial for the energy dependence, i.e.,
ip(xfy,z,c) = l cmfTlXm(x) Yn(y) Zm^n(z) Em,n(e) (3.14y
m,n
where
151
23 FORMAT(/ ,15X,2HN= *12)
25 WRITE OUTPUT TAPE 6,27,(U(M,N,I),I = 1.10)
27 FORMAT(5X,5F20.8)
30 CONTINUE
00 3 5 M= i L
DO 34 N = 2 L
NMI=N1
X=FLOATF(N)
XN=SQRTF( (X 1.0)/X)
DO 33 1 = 1, 10
V(M,N,I)=XN*(V(M,NMl,I)U(M,NMI,I))
33 W(M,N,I)=XN*(W(M,NMi,I)V(M,NMI,I))
34 CONIINUE
35 CONTINUE
DU 4 0 M =1,L
WRITE OUTPUT TAPE 6,36,M
36 FORMAT(1HI /////,52X,2HV(,I 2,1OH,N, I = 1, 10) ///)
DO 39 N=1 L
WRITE OUTPUT TAPE 6,23,N
39 WRITE OUTPUT TAPE 6,27,(V(M,N,I),1=110)
40 CONTINUE
DO 4 5 M =I.L
WRITE OUTPUT TAPE 6,42,M
42 F0RMATUH1 /////,52X,2HW(,12,10HtNf1=1,10) ///)
DO 44 N=I,L
WRITE OUTPUT TAPE 6,23,N
44 WRITE OUTPUT TAPE 6,27, (W(M,N.I)I = 1.10)
45 CONTINUE
RETURN
END
LIST OF FIGURES
Figure Page
1. The Steady State Inverse Relaxation
Length vs. Number of Laguerre
Polynomials 90
2. Components of the Complex Inverse Relax
ation Length of the Fundamental Mode vs.
Source Frequency 91
3. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 100 cps 93
4. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 500 cps 94
5. Phase Shift of Neutron Density vs.
Distance from the Source 96
6. Neutron Spectra at Various Positions
along the Central Axis of the Graphite
Assembly 97
7. Decay Constant, X of the Fundamental
Mode of the Flux in the Neutronpulsed
Graphite System 98
iv
l
4
The objectives of this work are to formulate
the neutron thermalization theory in the consistent
pl approximation and then utilize this formalism in
both neutron waves and pulsed neutrons techniques.
In Chapter II the thermalization model of the PN
approximation is derived in general operational form
and the P^ approximation is then considered in detail.
The resulting form of the Boltzmann equation is a
fourth order differential equation in space, energy
and time.
The application of this equation to the neutron
wave problem in Chapter III and to pulsed systems in
Chapter IV using the Laguerre polynomials leads to
rather complicated eigenvalue problems. In both cases
the general solution is obtained with no specific
assumptions about the scattering kernel. The results
presented in Chapter V, however, are obtained for
graphite in the heavy gas approximation and 1/v
absorption.
The operators involved in the theory are
derived and discussed in Appendix A and calculated ex
plicitly in Appendix B for the heavy gas scattering
kernel. Finally, the computational scheme of the
matrices involved in both applications is shown in
Appendix C, while the IBM709 codes are listed in
Appendix D.
101
3 2
to about 3x10 era In the neutron waves study the
heavy gas model in the P^ approximation can describe
the experiment with source frequencies up to about
500 cycles per second, or equivalently, with time
variations of periods down to about 2 milliseconds.
These results have interesting practical
repercussions in that if one is only interested in
the slow changing regimes of nuclear systems, the P^
heavy gas kernel approximation is proven to be fully
satisfactory. Then it does not pay off to go to more
sophisticated kernels unless the emphasis lies in
fast transients, in which case the flux distributions
become sensitive to the detailed behavior of the
scattering kernel according to Ohanian (32) .
This conclusion both confirms and extends the
results of de Sobrino (17) who concluded that in the
computation of the steady state regime of nuclear
reactors the heavy gas kernel approximation (Wilkins
Equation) was conveniently accurate.
Finally, it would be of much interest to
investigate the theory developed in this work using a
more sophisticated scattering kernel in order to
enlarge the applicability of this theory to fast
transients. Steps in this direction are being taken
by the author.
104
where
H(E) = ^(E) + QjlEJQqE) 0L(E)Zt(E) Zt(E) Q0(E)
(Pi. 8)
F(E) = 2Et(E) vEJQj^E) v1(E) Qq(E)
(A.9)
G^E) = Zt(E) QX(E)
(A. 10)
Notice that all the operators do not commute due to ,the
energy derivatives involved in them. See Eq. (A.2).
Using Eq. (A.2) in Eqs.(A.8), (A.9) and (A. 10), and
grouping the coefficients of the energy derivatives, we
can express the operators H(E), F(E) and G(E) in the
following equations:
H(Ei = h (E) + hx(E) 5 + h2{E) (A. 11)
3E 3 E2
3E 3 E2
(A.12)
(A.13)
where the functions h^(E), f^E) and g^fE) are found to be
53
Then, if we define
Dk'k=pk dv,
m,n v
m,n k' k P m, n Rk' k
Bm,n = m,n k* + V*
(3.111)
(3.112)
Eq. (3.106) takes the simple form
y Dk'tk ck = ak*
L m,n m,n m,n (3.113)
k
A more explicit form is

D'
m, n
D' 1
m, n
... D'L
m, n
C
m, n
B
m, n
D1,0
m, n
D1'1
m, n
... dx'l
m, n
C1
m, n
=
B1
m, n
*
dl'
m, n
m, n
... dl'l
m, n
c1
m, n
;
m, n
_ _
(3.114)
This linear inhomogeneous set applies for all values of
m and n and contains (L+1) equations with (L+1) unknowns,
C* which are completely determined by its solution,
m, n
At this stage we see that the solution for the
flux has been completely found, as given by Eq. (3.67).
All the constants have been determined. All the space
61
where it should be recalled again that the functions
4>(r, e,t) and x(r,e,t) are the nonMaxweIlian com
ponents of the flux and the source, respectively.
The system configuration considered here consists of
a rectangular parallelopiped block of moderator whose
dimensions are 2a, 2b and 2c. The pulsed neutron
source is located at the center of this system, i.e.,
at x=0, y=0 and z=0. The boundary conditions associ
ated with this problem are:
(1) Symmetry of flux.
(2) Flux vanishes at the extrapolated
boundaries.
(3) Finite flux everywhere in the system.
Since these conditions must hold regardless of
the source energy distribution, it follows that they
must hold true for the nonMaxwellian component,
'l>(r,e,t), which contains the spacial and time depend
ence of the flux. Therefore, the above conditions can
be given by the following analytical expressions.
= ii(x,y ,z,e ,t)
(4.2)
s 4i(x,y ,z,e ,t)
(4.3)
i*(x,y ,z,e ,t)
= ifi(x,y ,z,e ,t)
(4.4)
124
Upon the substitution from Eq. (C.24) in Eqs.
(C.6) and (C.7) we find the recursion relations
VJ
Â£ J Â£
wJ
Â£ J Â£
() u (Â£1) IV UJ 1
^tl^ [ Â£ J Â£1 Â£ J Â£lJ
(Vs u (Â£1) iwJ VJ ]
li+lj L 4}tl Â£ J Â£1J
(C.25)
(C.26)
Thus the computation of VJ and VT1 becomes very
Â£ Â£ Â£ Â£
straightforward once we compute the matrix, To
Xr
calculate this matrix we will write the polynomials in
the form
L(1: (e)
Â£
r yi i
i=o *
(C.27)
Then,
U
Â£ J Â£
l l b^ /* Â£i' + t+(j4)/2m(E)d(
i'=o i=o
(C.28)
or
u
Â£ J Â£
i' i j+2i'+ 2i4
I I b > r( )
4C.29)
i'=o i=o
where r(x) is the wellknown gamma function
APPENDIX B
HEAVY GAS 1/v MODEL
In this appendix the heavy gas model is adopted
for the scattering kernel and the absorption cross
section is considered to be of the 1/v type On this
basis, then, the energy moments, m(E)^, and the
functions h^tE) f^CE) and g^E) will be calculated.
Heavy Gas Scattering Kernels
In,this rather idealized model the moderator is
treated as a monatomic gas whose atoms are heavy com
pared to neutrons. The advantages of using this model
are that it offers an analytical expression for the
scattering kernel and "it bypasses the complex con
siderations of neutron interactions with chemically
bound atoms, but retains much of the essential physics
of the thermalization process," according to Hurwitz,
Nelkin and Habetler (15). "It does not include,
however, the effect of 'jumping' across a narrow
resonance."
Based on the treatment of this model in (15)
the scattering kernels for the Pn approximations can
be given by
109
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INGEST IEID EE5RPNBMP_3JXFJB INGEST_TIME 20151028T20:15:09Z PACKAGE AA00032912_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
36
Equation (3.47)^ gives all the unperturbed or steady
state eigenvalues, the use of each of which in
Eq. (3.46) yields the corresponding unperturbed ratios,
. Next the kth eigenvalue, rk, and the corres
ponding ratios, k, for A / 0 are obtained from the
equation
l t
n r, ] r =0
t k (3.48)
fc=0
by the perturbation method.
In this method the eigenvalues, rk, and the
ratios, R k, are expanded in power series in
rk= r^0) + rk1} + rk2) A2/2i +
(3.49)
R = R(o) + R(1) + R<2> A 2/2! +
Â£,k Â£,k Â£,k Â£,k
(3.50)
where
.(v)
a'
9 A
v rk
A=0
(3.51)
3 v
9 A v RÂ£k
A=0
i ,k
(3.52)
38
to Eq. (3.48), where
v!
U) 
y! (vy) 1
(3.55)
we obtain
v k (y)
y y ( [a +e a+y A2n r ]
y=0 Â£=0 l' z' Â£ Â£''Â£ Â£,'Â£ k
(vy)
Â£, k
Realizing that
= 0
(3.56)
A=0
d*
(An)
dxy
ni
A=*0
(ny) n'p
(3.57)
where 6 is the Kronecker delta function, Eq. (3.56)
n,y n
is rewritten as
v L (y) 1
y y ( M (a 6 +3 6 +2Y 6 n T J
_ n ly J V,Â£ o,y Â£,* i,y 'I', l 2,y 1*, k
yj 0 i>"0
= 0
Â£,k
(3.58)
Noticing that R =1 for all values of k and
O / K
(n) (v)
Rq k = 60 n for ald values of and separating
and R^ ( l / 0) one can rearrange Eq. (3.58J in
p K
the form
LIST OF REFERENCES
1. Wigner, E. P. and Wilkins, J. E., Jr., "Effect of the
Temperature of the Moderator on the Velocity Distri
bution of Neutrons with Numerical Calculation for H
as a Moderator," AECD2275 (1948).
2. Wilkins, J. E., Jr., CP2481 (1944).
3. Weiss, Z., "Neutron Spectrum Temperature Distribution
in Heterogeneous Media with Energy Dependent Absorp
tion," Nukleonika, Vol. VI, No. 78 (1961).
4. Weiss, Z., "The Transport Theory of Thermal Neutrons
in Heavy Gas Moderators," Nukleonika, Vol. VI, No.
11 11961).
5. Weiss, Z., "The PNLj Approximation of Neutron Thermal
ization in Heavy Gas Moderator," Nukleonika. Vol. VI,
No. 11 <1961).
6. de Sobrino, L. and Clark, M., Jr., "A Study of Wilkins'
Equation," Nuclear Science and Engineering 10, 388399
(1961).
7. Corngold, N., Michael, P. and Wallman, W., "The Time
Decay Constant in Neutron Thermalization," Nuclear
Science and Engineering 15, 1319 <1963).
8. Williams, M. M. R., "SpaceEnergy Separability in
Pulsed Neutron Systems," BNL 719 (C32), Vol. IV,
13311359 (1962).
9. Zelazny, R. S., "Transport Theory of Neutrons in Heavy
Gas in Plane Geometry," BNL 719 (C32), Vol. IV,
13601374 (1962).
10. Lindenmeier, C. W., "The Modified Heavy Gas Model and
Two Thermal Group Rethermalization Theory," BNL 719
(C32), Vol. IV, 12721298 (1962).
11. Katsuragi, S., "The Thermalization and Diffusion of
Neutrons in Heavy Gaseous Moderators," Nuclear Science
and Engineering 13, 215229 (1962).
157
25
a function of the source frequency and the position at
which the flux is measured, will be determined later.
Therefore, the tim dependence of the flux can be
given by the expression
*(r,Â£,t) = ipQ(r ,e) + Re (r ,e) e^)t] ^
The combination of Eqs. (3.2) and (3.6), and then the
separation of the steady state part of the resulting
equation gives
2
Ij U) \ 1 j j ) \ 1 +
) F(e) +  J J 72] i{i(r,E)=0 (3.7)
and
[H (e) \ V2] <Mr,e) = 0 (3..8)
Since the steady state solution can be easily found
by replacing u by o in the time dependent solution,
there will be no need to treat Eq. (3.8h separately
but it suffices to solve Eq. (3.7).
The boundary conditions associated with Eq.
(3.7) are
t>(X, y, z)
ii(x, y, z) ,
(3.9)
19
1
[H(e)+ F(Â£)
Vq
9t v02e 9t2
*(r,E,t)
1 9
= [G(E) + ~ ] x(r,e,t) (2.61)
v /T 9t
H(e) =
1 hi
i
91
9ei
(2.62)
F(e) =
I fi(Â£)
JL
9E1
(2.63)
i
0(E) =
9j
i
9
9ci
(2.64)
See. Eqs.
(A. 27)
 (A.35).
The functions f^, h^ and
g^ are calculated in terms of E and e for the heavy
gas model in Appendix B.
Eqs. (2.50), (2.54) and (2.61) are all
equivalent and the last form will be used throughout
the remainder of this work. Specifically, Eq. (2.61)
will be used in Chapters III and IV to investigate the
neutron waves and the pulsed neutrons techniques in
moderating media.
EnergyIndependent Cases
It is interesting to conclude this chapter by
the examination of the energyindependent case of
Eq. (2.50). In this case
78
+k
for the constants, C the determination of which
m,n,p
is not necessary as we will see later.
Neutron Conservation
In order to complete the solution for the
neutron flux the quantities, must be deter
m,n,p
mined. Before we do so it is helpful to discuss
qualitatively the time behavior of the neutron popula
tion in the system. As the source pulse has a finite
width there is some time interval during which neutrons
are continuously fed into the system. Consequently,
the neutron population at a certain position and a
certain energy rises to reach a maximum and then decays.
The time at which this maximum is reached depends on
the amplitude, the time dependence and the duration of
the pulse. Furthermore, the neutron density may reach
an intermediate steady state depending on the pulse
shape, as in the case of a broad rectangular pulse, or
a pulse with a broad flat portion. For these reasons,
and since the negative time exponentials alone can
describe the time behavior of the neutron population
only in the asymptotic region but not in the transient
region, the quantities, n must be functions of
time. Hence, the flux expression of Eq. (4.36) becomes
155
SUBROUTINE ABGE TZ
SUBROUTINE ABGETZ(L,NS,AA,BB,TT,U,V,W,
1 ALPHA,BET A,GAMMA,ETA,THETA,ZETA)
DIMENSION ALPHA!1G, 10),BETA(LG,10),GAMMA(10,10),
1 ETA!10,10),THETA!10),ZETA(10),DELTA( 10,10),
2 AllO,10),A A(3,10),B B(3,10),TT(10)*U(10,10,10) ,
3 V(10,10,10),W(10,10,10),C(50),G{50)
CUMMON ALPHA.BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB,TT,U,VW,C,G
DO 5 M= 1 L
DO 4 N= 1, L
ALPHA(M,N)=0.0
BE T A(M,N)=0.0
GAMMA(M,N)=0.0
4 ETA(M,N)=0.0
THE T A(M)=0.0
5 ZE T A(M)=0.0
DO 7 M= 1 L
DO N=i,L
6 DELTA!M,N)=0.0
7 DE LT A ( M M) = 1.0
NSP2=NS+2
NSP3=NS+3
NSP4=NS+4
DO 11 M=1,L
DO 9 N= 1 L
DU 8 J=l, 10
ALPHA!M,N)=ALPHA(M,N)+AA(1,J)*U(M,N,J)
1 + A A { 2 J ) V ( M, N J ) + AA { 3, J ) W { M M J )
BETA!M,N)=BETA(M,N)+BB(1,J)*U(M,N,J)
1 +BB!2,J)*V(M,N,J)+BB!3,J)*W(M,N,J)
8 CONTINUE
GAMMA{M,N) = U!M,N,NS P2)
9 CTA(M,N)=U(M,N,NSP4)/3.0
ZETA(M)=U(M,1.NSP3)
DO 10 J=l,10
10 THETA(M)=TH2TA(M)+TT(J)*U(M,1, J)
11 CONTINUE
WRITE OUTPUT TAPE 6,12
12 FORMAT(1H1 /////, 52X, 10HALPHA{M,N) )
DO 13 M=1,L
13 WRITE OUTPUT TAPE 6,14, (ALPHA(M,N),N=1,L)
14 FORMAT!// 5X,5E20.3/5X,5E20.8)
WRITE OUTPUT TAPE 6,15
15 FORMAT(1H1 /////,52X,10H BETAlM,N) )
DO 17 M=1,L
17 WRITE OUTPUT TAPE 6,14,( BETA(M,N),N=1,L)
WRITE OUTPUT TAPE 6,18
18 FORMAT!1H1 /////,52X,10HGAMMAlM,N))
DU 20 M=1,L
20 WRITE UUTPUT TAPE 6,14, (GAMMA(M,N),N = 1,L)
WRITE OUTPUT TAPE 6,21
32
However, it is important to realize that the complex
inverse relaxation length, Pm^n, is "spacedependent"
and is computed from the relationship
P2 = r + B (3.36)
m,n Jn,n
To avoid confusion we must point out that the terms
"spacedependence" and "spaceindependence" are used
here to indicate, respectively, whether or not a cer
tain quantity is a function of the spacial modes, or
the spacial mode indices m and n. To solve the
frequency dependent set of Eqs. (3.35) two alternatives
can be used: the exact method and the perturbation
method. In the following two sections both methods
will be discussed.
Solution by the Exact Method
Due to the fact that the expansion of the
determinant in Eq. (3.34) gives a polynomial in r with
a maximum power of L+l (the size of the determinant)
the solution of this equation yields b+1 values for r.
These values will be distinguished by the subscripted
variable,^, where k = 0, 1, 2, ..., L. It is then
obvious from Eq. (3.35} that to each there corres
ponds a different set of coefficients, A
which
112
These can be obtained simply by integrating Eqs. (B.6)
and (B.7) with respect to E', using the relations
dE' = dA
(B.9)
/ f (w)
o
6^n^tw)dw = (l)n
w=o
(B.10)
The integration leads to
0O(E) = <1 2y + yT / 2B)oq
(B.ll)
(^(E) = (2y/3) aQ
(B.12)
These are
kernel.
obtained
by N, the
the microscopic moments of the scattering
The corresponding macroscopic moments are
simply by multiplying Eq. {B.ll) and Eq. (B.12)
number of atoms in one cubic centimeter.
Z0(E) = (1 2y + yT /2E)Eq
(B.13)
E1 (E) = = (2V3)Eq
(B.14)
l
n o o
129
MAIN PRUGRAM NWP (NEUTRUN WAVES PROROGATION)
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DI ME NS I ON
DIMENSION
DIMENSION
DIMENSION
ALPHA! 10, 10),BETA! 10,10).GAMMA 110,10)
DELTA!10,10),XS(20),YS(20),CC(10,10,3)
C!20,21),CTR( 10,10),CTI( 10,10)
RSK!10,10),RS1(10,10),3UE(21)
A!20,21),G! 10,10), RR110,10,10)
RI(10,10,10),X(10,10) ,Y( 10,10)
XEV!10),YEV!10),XR(10,10),YR{10,10)
RHR!10),RHUI!10),THETA{ 10),ZETA(10)
ARI10,10),AIl10,10),BR(10),6 Il10)
DAMP! 10),DPHASE{ 10),FAMP( 10,51)
EPHASE!10,51),S(10,10),6(10,51)
SAMP(51)SPHASE!51),ETA(10,10)
RIURI 10) R2M 12 ( 10 ) ZPO IN T! 2 1) AMEG l 50)
UNIT NO. 1
INPUT DATA
1 FORMAT(1013)
2
FORMAT(5E14
.6)
READ
INPUT
TAPE
5,1 ,L
READ
INPUT
TAPE
5,1 NUM
READ
INPUT
TAPE
5,2,SUCK
READ
INPUT
TAPE
5,1,NE
READ
INPUT
TAPE
5,1,ND
READ
INPUT
TAPE
5,2,EINT
READ
INPUT
TAPE
5,2,ZDIC
READ
INPUT
TAPE
5,1,NZ
READ
INPUT
TAPE
5,2,(ZPOINT!J),J=1,NZ)
DO 3
M=l, 10
3
READ
INPUT
TAPE
5,2,(ALPHA!M,N),N=1,10)
DO 4
M=l, 10
4
READ
I NPUT
TAPE
5,2,!BETA!M,N),N=1,10)
DO 5
M= 1,10
5
READ
INPUT
TAPE
5,2,(GAMMA!M,N),N=1,10)
DO 2 3
M=l, 10
23
READ
INPUT
TAPE
5,2,(ETA(M,N),N=1,10)
DO 22
M= 1,10
22
READ
INPUT
TAPE
5,2,(S(M,N),N=1,10)
READ
INPUT
TAPE
5,2,(THETA(M),M=1,10)
READ
INPUT
TAPE
5,2,(ZETA(M),M=1,10)
WRITE OUTPUT TAPE 6,6
6 FORMAT!1H1////.52X, 10HALPHA!M,N)III)
DU 7 M=1, 1 0
7 RITE OUTPUT TAPE 6,8,(ALPHA(M,N),N=1,10)
8 FORMAT!// 5X.5E20.8/5X,5E20.8)
WRITE OUTPUT TAPE 6,9
9 FORMAT(1H1 ////,52X,10H BETA(M,N) ///)
DO 10 M=l,10
10 WRITE OUTPUT TAPE 6,8,!BETA!M,N),N=1 10)
WRITE OUTPUT TAPE 6,11
11 FORMAT( 1H1 ////,52X,10HGAMMA(M,N) ///)
132
DO 100 NU=1,NUM
NU1=NU~1
RR!K,1,NU)=0.0
RI( K 1,NU)=0.0
DO 40 M= 1, L
ML=M+L
A(M,L2P1)=0.0
A{ML,L2P1)=0.0
DO 49 N= 11 L
AT=8ETA(M,N)DELTA(NU,1)+2.0*GAMMA(M,N)*DELTA{NU,2)
A(M,L2P1)=A(M,L2P1)RSR(K,N)*AT
A(MLL2P1)=A{ML,L2P1)RSI!K,N)*AT
IF(NUl) 38,38,36
36 CONTINUE
DO 47 MU=i,NU 1
NM=NU~MU
AT=(3ETA!M,N)DELTAlMU,1)+2.0*GAMMAlM,N)DELTA(MU,2)
1 ETA(M,N)X(K,MU))*G(NU,MU)
T = ETA t M,N)*YlK,MU)*G(NU,MU)
A(ML2P1)=A!M,L2P1)AT*RR(K,N,NM)+BTRI(K N NM)
A!ML,L2P1)=A{ML,L2P1)AT*RI(K,N,NM)BT*RRK,N,NM)
37 CONTINUE
38 CONTINUL
49 CUNIINUt
40 CUNINUL
CALL ELcM(L2,A,DET)
X(K,NU)=A(1.L2P1)
Y(K,NU)=A(LP1 L2P1)
DO 41 M= 2,L
RR(K,M,NU)=A(M,L2P1)
M L = M + L
41 RI(K,M,NU)=A(ML,L2P1)
100 CONTINUE
200 CONTINUE
WRITE OUTPUT TAPE 6,201,L,NUM
201 FORMAT!1H1///5X,27HTHE TIME DEPENDENT SOLUTION /
1 5X,27( 1H*)/>X,22HNU OF LAGUERRE POLYN =13/
2 5X,22UMAX DERIVATIVE ORDER =13 //)
WRITE OUTPUT TAPE 6,202,(M,M=1,7)
202 FORMAT(9X,7HX(K,NU),1X,7(4X,5HRR!K,I1,4H,NU)))
WRITE OUTPUT TAPE 6,203,(M,M=1,7)
203 FORMAT(9X,7HY(K,NU) IX,7(4X,5HRI!K,11,4H,NU) ))
DO 208 K=1,L
WRITE OUTPUT TAPE 6,204,K
204 FORMAT!// 5X,2HK= 12)
WRITE OUTPUT TAPE 6,205,XS(K),(RSR(K,M),M=1,L)
WRITE OUTPUT TAPE 6,205,YS(K),{RSI(K,M),M=1,L)
205 FUKMATIE17.6,7E14.6)
DU 206 NU=1,NUM
WRITE OUTPUT TAPE 6,207 X(K,NU),(RR(K,M,NU),M=1,L)
206 WRITE OUTPUT TAPE 6,205,Y!K,NU),(RI!K,M,NU),M=1,L)
207 FURMAT!/El 7.6.7E14.6)
208 CONTINUE
JQM=0
40
1
P
0
o
0,1
CN
ak 
CL
O, L
r(v)
rk
bk,v
o
k
ai,o
k
ai,l
k
ai,2
k
a
1,L
R(V)
1, k
, k,v
b
1
i
; aK
2,0
J .
a^
2,1
ak
2,2
... ak
2, L
R(v)
2 ,k
23
bk,v
;
i ak
L, o
ak ,
L,1
ak
L, 2
ak
** aL,L
' L,*
i
!b?'v 
L
These two equivalent sets, Eqs. (3.59) and (3.64), are
linear inhomogeneous and consist of l&l equations with
L+l unknowns, and R^v) (i = 1, 2, ..., L) which
k % ,k
are completely determined by the solution of the set.
It should be pointed out that the matrix, a^, does
not depend on the order of derivative, v, and it is
V \i
the same for all values of v. On the other hand, b '
l1
does depend on v and has to be calculated for each
value of v. See Eqs. (3.60), (3.61) and (3.62). How
ever, both akt and bki'v depend on the eigenvalue
SL f i
used, as indicated by the superscript k. Both Eqs.
(3.59) and (3.64) will be referred to as the perturba
tion equation.
It is very helpful to summarize the various
steps of this technique in the sequence used in actual
computation.
UNIVERSITY
OF FLORIDA
LIBRARIES
ENGINEERING AND PHYSICS
LI BRARY
TIME, ENERGY AND SPACEDEPENDENT
NEUTRON THERMALIZATION THEORY
By
HASSAN H. KUNAISH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1964
ACKNOWLEDGMENTS
The author wishes to acknowledge the guidance
of the members of his supervising committee. He
gratefully acknowledges the continuous guidance and
invaluable advice of his committee chairman, Dr. R.
B. Perez, without whose aid this work would have been
impossible.
The author is indebted to the American Friends
of the Middle East, Inc., and the Syrian Government
for a scholarship during his graduate study toward the
Ph.D. degree. He wishes to acknowledge a partial
financial support by the University of Florida Com
puting Center for the computations in this work. He
also wishes to thank Mr. R. Booth for permittinghim
to use his experimental results.
Special thanks are due Mrs. Gail Gyles and
Miss Barbara Gyles for their patience and enthusiasm
during the typing of this manuscript.
Last, but not least, the author would like to
express his deep appreciation for the encouragement
and understanding of his wife, Subhieh, throughout
the progress of this dissertation.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF FIGURES iv
ABSTRACT V
Chapter
I. INTRODUCTION 1
II.THE THERMALIZATION MODEL OF THE CONSISTENT
PL APPROXIMATION OF THE NEUTRON TRANSPORT
EQUATION 5
III. NEUTRON WAVES IN MODERATING SYSTEMS 23
IV. PULSED NEUTRONS IN MODERATING SYSTEMS .... 60
V. RESULTS AND CONCLUSIONS 87
APPENDIXES
A OPERATORS IN GENERAL FORMS 102
B HEAVY GAS 1/v MODEL 109
C CALCULATION OF THE MATRICES
Bn', l' yi\l . i 118
D COMPUTING CODES 126
LIST OF REFERENCES 157
BIOGRAPHICAL SKETCH 162
iii
LIST OF FIGURES
Figure Page
1. The Steady State Inverse Relaxation
Length vs. Number of Laguerre
Polynomials 90
2. Components of the Complex Inverse Relax
ation Length of the Fundamental Mode vs.
Source Frequency 91
3. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 100 cps 93
4. Amplitude of the Combined Neutron
Density vs. Position along the zaxis
for Source Frequency of 500 cps 94
5. Phase Shift of Neutron Density vs.
Distance from the Source 96
6. Neutron Spectra at Various Positions
along the Central Axis of the Graphite
Assembly 97
7. Decay Constant, X of the Fundamental
Mode of the Flux in the Neutronpulsed
Graphite System 98
iv
l
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
TIME, ENERGY AND SPACEDEPENDENT NEUTRON
THERMALIZATION THEORY
By
Hassan H. Kunaish
December, 1964
Chairman: Dr. Rafael B. Perez
Major Department: Nuclear Engineering
The time, energy and spacedependent neutron
thermalization theory in moderating assemblies is
developed in the consistent P^ approximation of the
Boltzmann equation of neutron transport. In order to
evaluate the scattering integral involved in this
equation the neutron flux is analyzed into two com
ponents according to the expression
$ (r, E, t) = m(E) {Â¡i (r, E, t) .
The Maxwellian function, m(E), represents the energy
spectrum of neutrons in a nonabsorbing infinite medium.
In a finite medium with a small absorption cross section
the energy spectrum of the neutron flux resembles a
Maxwellian distribution. Therefore, the nonMaxwellian
v
component, */(r,E,t), is expected to be a smooth function
of energy, E, and hence its value at a different energy,
E1, can be well approximated by a finite Taylor series
expansion about E. This analysis and the detailed
balance principle are used in order to evaluate the
scattering integral. The resulting form of the trans
port equation is a fourth order differential equation
with respect to time, energy and space.
This theory is used in two applications of
practical interest, the neutron waves and the pulsed
neutrons experiments. The use of the cosine functions
in expressing the spacial dependence of the neutron flux
leads, in both cases, to the modal equation of the flux
as a function of time and energy. The energy dependence
of the flux modes is then developed in terms of the
associated Laguerre polynomials. This procedure leads
to two similar complicated eigenvalue problems.
The source condition in the neutron waves problem
is established from the P^ approximation and is used to
determine the modal constants. In order to account for
transient behavior of the pulsed systems the principle
of neutron conservation is used and the final solution
of the time dependent flux is obtained by the technique
of Laplace transformation.
i
vi
Numerical computations are made for both
applications using a heavy gas scattering kernel and a
1/v absorption cross section in graphite in order to
test the applicability of this model and its compati
bility with the P^ approximation. The comparison of
these results with the experimental ones confirms that
this model is conveniently accurate in describing the
pulsed neutron experiment in relatively large systems
of buckling up to 3x10* cm2. This model is also
proven to provide accurate description of the neutron
waves experiment with source frequencies up to 500
cycles per second which are equivalent to transients of
periods down to about 2 milliseconds.
CHAPTER I
INTRODUCTION
The theory of neutron thermalization has been
given much emphasis in recent years. The interest of
investigators has been focused on two objectives.
The first objective is the theoretical and experi
mental understanding of the scattering law of thermal
neutrons in various materials, while the second
objective is to use the best available information
about the scattering law in order to study the details
of the neutron thermalization process and the neutron
spectra which are important in determining the
behavior and properties of nuclear systems. The
mathematical framework of the latter is the develop
ment of an adequate solution for the Boltzmann
equation describing the neutron transport phenomenon.
The work presented in this dissertation can be
classified in the second category.
In solving this equation investigators followed
several approaches and based their solutions on certain
assumptions in order to reduce the complexity of the
mathematics involved in the problem. Effort was made,
1
2
however, in order for the problem to retain certain
aspects of its physics depending on the goal of the
investigator.
'ft
Wigner and Wilkins (A/ 2.) established the
monoatomic gas model of the scattering kernel on
which investigators (3. 17} and many others, based
their works later on. Using this model, Barnard et
aJL. (16) studied the time dependent neutron spectra
in graphite varying the moderator to neutron mass
ratio. The analytically calculated spectra for a
fictitious mass ratio of 33 were in good agreement
with the experimental results while for the mass ratio
of 12 this agreement disappeared at times less than
300 y sec after the pulse and for energies below 0.1
ev.
Corngold et cil, {!_) and Shapiro (14) showed
that, in the analysis of the time dependent thermaliza
tion problem in the neutron pulse technique, the
consideration of a discrete set of eigenvalues is not
adequate and continuous eigenvalues should also be
considered for complete analysis. However, the ques
tion of how important these continuous eigenvalues and
* Underlined numbers will be used to indicate
references cited in the List of References of this
dissertation.
3
the corresponding eigenfunctions are has not been
answered yet. Shapiro also found that the velocity
associated Laguerre polynomials give faster con
vergence than the energy polynomials.
The technique of pulsed neutrons used by many
investigators (8 3_5) and others, has been reviewed
by Beckurts (21). The oscillatory neutron source
method was first developed by Raievski and Horowitz
(36). Using this technique Uhrig (37) studied the
nuclear properties of subcritical assemblies while
Perez and Uhrig (38) investigated the thermalization
theory and calculated the thermalization parameters.
They also showed the similarity between this tech
nique and the pulsed neutrons technique from the
analytical point of view. Experimental investiga
tions have been carried out by Perez et alt (39, 40)
and Booth (41). According to Hetrick and Seale in a
more recent publication (421 it is possible to cal
culate the specific heat integral and the zeropoint
mean square displacement of the scattering atoms in
terms of the diffusion coefficient and the second
energy transfer moment by using this technique with
low temperature experiments.
4
The objectives of this work are to formulate
the neutron thermalization theory in the consistent
pl approximation and then utilize this formalism in
both neutron waves and pulsed neutrons techniques.
In Chapter II the thermalization model of the PN
approximation is derived in general operational form
and the P^ approximation is then considered in detail.
The resulting form of the Boltzmann equation is a
fourth order differential equation in space, energy
and time.
The application of this equation to the neutron
wave problem in Chapter III and to pulsed systems in
Chapter IV using the Laguerre polynomials leads to
rather complicated eigenvalue problems. In both cases
the general solution is obtained with no specific
assumptions about the scattering kernel. The results
presented in Chapter V, however, are obtained for
graphite in the heavy gas approximation and 1/v
absorption.
The operators involved in the theory are
derived and discussed in Appendix A and calculated ex
plicitly in Appendix B for the heavy gas scattering
kernel. Finally, the computational scheme of the
matrices involved in both applications is shown in
Appendix C, while the IBM709 codes are listed in
Appendix D.
CHAPTER II
THE THERMALIZATION MODEL OF THE CONSISTENT
P, APPROXIMATION OF THE NEUTRON
TRANSPORT EQUATION
General Theory
The space, direction, energy and time dependent
Boltzmann equation describing the neutron flux in a
homogeneous isotropic medium is
i *(r,3,E,t)+E (E)
V 91 t
S(r,ft,R,t) + / dE'
E'
/ da' zg (E'E,ft'*ft) (r,ft',E',t)
a'
(2.1)
where r, ft E, v and t are the neutron position vector,
direction, energy, velocity and time, respectively.
Primed symbols indicate the corresponding variables
after a scattering collision. The terms appearing in
Eq. (2.1) are defined in the following
13 .> + +
$(r,QfE,t) drdadE = the time rate of change of
v the neutrons which occupy
the space dr dE d ft,
ft*V$(r,ft,E,t) drdftdE = the loss rate due to the
net transport of neutrons
through the boundaries of
dt,
5
6
E (E) $ (r ,E, t) dr dadE =
S(r,S,E,t) drdadE
/ dE'J da' es(e'*e,')
e' a'
x ,E' ,t) drdadE
the loss rate due to
absorption, scattering
out and slowing down,
rate of contribution
from all sources in the
space dr da dE,
contribution into dr da
dE due to scattering
from all directions a'
and all energies E'.
The total space, energy and time dependent scalar flux
is obtained by integrating the directional flux over
all directions
(Hr,E,t) = / (r,a,E,t) da
a
(2.2)
In order to eliminate the angular (direction)
dependence from Eq. (2.1) the flux, the source, and
the scattering kernel are expanded in terms of the
spherical harmonics
4> (r ,a,E,t)
m
 I l Or,E,t> Y"(fl)
(2.3)
ra=o n=m
s(r,a,E,t) = l l s"(r,E,t) y"()
(2.4)
m=o n=m
I
7
v
Es (E'+Ejft+ft)
Z3(E'E,w0)
" 2m+l
= l ~ EmE,,E>VJ)
m=o
ra + +
 I I VE"E>Ym
m=o n=m
(2.5)
where
w0 = cos 0O = ft*ft* (2.6)
is the cosine of the scattering angle in the labora
tory system.
The spherical harmonics, ^(3), the Legendre
m
polynomials, pm(y0)< and their associated functions,
P^(u)/ satisfy the following relations, see refer
ences (43) and (44) .
YmW 2 s HmP>>ejlW
(2.7)
(V* (2*8)
f n + n Â¥ Â¥
f (ii)yJ(n)dQ
m p
ft
6 6
mp nq
H
n
m
1/2
(2.9)
[ (2m+l) (mn) /4ir (m+n) 1 ]
(2.10)
8
H~n 3 [(m+n)l/(mn)!] (2.11)
dn
Pra(w) 5 sinne Pm(M) (2.12)
m dyn m
P^n(u) = (l)n[(mn)i/(m+n)l]PÂ¡J(y) t (2.13)
Pm(^(
(mn) 1 n infipvM
.> I 7=TT7 V*>*Â£<>o3
nm
(2.14)
where
(2.15)
(2.16)
y = COS 9 = Si z
y' = COS 01 = SI1 Z
and where e,
azimuthal and longitudinal angles of the directions
si and si'.
The moments of the flux, the source and the
scattering kernel are given by
n(i,E,t) = / ^(n) (r,n,E,t)dn (2.17)
m m
SI
sÂ£(i,E,t) = / Ym(8)S(r,n,E,t)dn
SI
(2.18)
+ a
I
9
Zm(E'E) = /1Pm(y0) Es(EUE,n.n)du (2.19)
I
The use of Eqs. (2.3)(2.5) and the properties of
the spherical harmonics expressed by Eqs. (2.7)(2.14)
along with
3 3 8
7 = cose + sine cos V + sine sin y
3z 3x 3y (2.20)
leads to the formulation of a set of coupled integro
differential equations relating the moments of the flux
up to any order of approximation, i.e.,
I ~nt(E) ]*Â£(?,E,t)+s5J(r,E,t)+lÂ£(r,E,t) =
V 31
A(Z)
(m+ln) (m+l+n) ** (mn)(m+n) *5 + 1
) .+ E#t)+ } (r,E,t)
(2m+3)(2m+l) m+1 (2m+l)(2ml) m"1 J
+7 B(x,y)
(2m+3)(2m+l) w+1 (2m+l)(2ml) m1 J
+ie*(x,y) j
(m+n)(m+n1) h n1 ^ ^ r(ran+2)(mn+1) h>n]_ +
(2ra+l)(2ml)
HT* Ciir'E.t)!
(2m+3)(2m+l)
m+ltr'E't)]
(2.21)
where
10
ln(,E,t) = / I (E'+E) n(,E',t)dE' (2.22)
m m m
E*
A(z) =
3
(2.23)
3 z
{
B (x ,y)
3 3
3x +3 3y
(2.24)
B*(x,y)
3 _3_
3x 3y
(2.25)
In an infinite moderating medium with no
absorption the energy spectrum of the neutron flux is
given by the Maxwellian function
m(E) = (E/T2)e"E//T (2.26)
where
T = the energy corresponding to the
most probable speed,
= Boltzmann constant x Kelvin
temperature (in energy units).
On the other hand, the energy spectrum in a finite
medium with a small absorption cross section, com
pared with the scattering cross section, is not
Maxwellian but similar to it in the general shape.
11
If the Maxwellian function is factored out from the
flux moments, i. e.,
$n(r,E,t) = m(E) ij>n (r,E,t) (2.27)
m ra
the component, ij>n(r,E,T) is expected to be relatively
m #
smooth.
Similarly, the source moments are analyzed:
SÂ¡Â¡(r,E,t) = m(E) x^(r,E,t)
(2.28)
To evaluate the integrals,, the flux
components,^(r,E',t) are expanded in Taylor series
about E.
00 1 k k
^{r,E,,t) = l \E'E) d nÂ£(,E,t)
kiC i
=o
(2.29)
where
k n,* n *
D *m(r,E,t) r *m(r,E',t)
9E'K
E'=E
Using Eqs. (2.27) and (2.29) in Eq. (2.22) the
integrals become
" I E (E'E)m
E k=SQ k!
Dkipn (r,E,t) dE'
(2.30)
12
To put the integrals In into a more workable form, the
m
detailed balance principle will be used. This prin
ciple states that an equilibrium is established for
the product of the Maxwellian distribution and the
scattering kernel between any two energy intervals,
i.e. ,
mE'jZgtE'fE) = m (E) Eg (E*E 1 ) (2.31)
Notice that the relation (2.31) is completely inde
pendent of the amount of absorption in the medium and
the size of the medium. Eq. (2.31) can be general
ized for the moments of the kernel and is then stated
as
ra(E')E (E+E) = m(E)Â£ (E+E') (2.32)
mm v
Eq. (2.30) can be combined with Eq. (2.32) to yield
I^(r,E,t) = m(E) / Â£ (EE1) l L (EE)k
m E' m k=o k!
x DktÂ£(r,E,t) (2.33)
m
or
I^(,E,t) = ra(E) l
k=o
Mn
k,n,> .
D *m(r,E,t)
(2.34)
13
where the energy moments are
Mk (E)
m
;rr!:m(E)ivE>= in 1 (e'_e) zm(E*E,)dE'
E'
(2.35)
Furthermore, the integrals of Eq. (2.34) can be written
in operator form as
l(,E,t) = mtEjQ^Ej^r^t)
n
(2.36)
where the operators Q^E) are defined as
Qm = I 'VDk
k=o
(2.37)
The use of Eqs. (2.22), (2.27), (2.28) and
(2.36) in Eq. (2.21) yields:
RmiEft) ij;^(r,E,t) + *
f (m+lrn) (m+l+n) ^ n + (mn) (m+n) **
A(z)j^{ _} t;m(r,E,t) + { }
(2m+3)(2m+l)
(2m+l)(2n+l)
+ B(x,y)
2
x 'P^_1 (r ,E,t)
(m+n+2)(m+n+1) ^ n+1
(2m+3)(2m+l)
1 V+l (r,E,t) {
(mn)(mn1) %
) >
(2m+l) (2ml)
n+l,> ^ ..X
* *ml(r'E't)
14
(ran+2)(mn+1)M
n (m+n) (m+n+1) > n_!  *
+ B*(x,v) I } ij t (r,E,t) ( j
2 [ (2m+l)(2ml) m1 (2m+3)(2ra+l)
x C<'E't)
(2.38)
where
1 3
R (E,t) = Q (E) Z. (E) ~
m m t v
31
(2.39)
Hence, we have obtained the general form of the thermali
zation model of the Boltzmann equation in its spherical
harmonics formulation, where its validity is only limited
by the approximation involved in the Taylor expansion of
^(r,E',t) about E. The operators (^(E) and Rm{E,t)
are discussed and calculated in Appendixes A and B.
PjL Approximation
When the index, m, in Eq. (2.38) is given the
values 0 and 1, the index n takes the values il, 0, and 1.
Then, the infinite set of Eq. (2.38) is reduced to the
following four equations which couple the moments of
the flux in the socalled P^ approximation.
Ro(E't)'t,o(^E,t)+xo(^,E,t)4)^ A(Z)4'l(^E't>
+ (i)*5 B(x,y)^(,E,t)()^ B*(x,y)^(irE,t) (2<40)
(2.41)
15
Ri (E;t) \j)i" (r,E,t) +x^(r/Ert) = (i)^ B* (x,y) < (r,E,t)
6 (2.42)
R, (E,t)iA1(r,E,t)+x71(r,E,t)=(i)2 B(x,y) i>(r,E,t)
1 60 (2.43)
The scalar flux is obtained from Eq. (2.2)
$ (r ,E, t) = /
Q
m
 / I I Oi'E't>YS<)d(!
m=o n=m
= 0 (r ,E,t)  Hq Air 4>o (r fE,t)
= AT m(E) *g(r,E,t) (2.44)
where Eq.s (2.2), (2.3), (2.7), (2.10), (2.12) and
(2.27) were used.
Similarly,
S(r,E,t) = ATT xn(E) xÂ§(r,E,t) (2.45)
Since the scalar flux is easily obtained from ^g(r,EA)
according to Eq. (2.44) the set of Eqs. (2.402.43)
will be solved for ^g(i,E#t). Multiplying Eq. (2.40)
16
by Rj^t), Eq. (2.41) by A(z}/V3, Eq. (2.42) by
B
the resulting set yields
[R1(E,t)Ro(E/t)  V2]
1 1 1
+ yj A(z) x?(r,E,t) + B(x,y)x., (r,E,t)
v 1 yr 1
/6
1
r
B*(x,y) x^1(r,E,t)
(2.46)
R]_(E,t) *(r,E,t) + X(r,E,t)
1 0 +
: A(z) (r,E,t)
y 3
(2.47)
Rx(E,t) (r,E,t) + x];(r,E,t) = B* (x ,y) *(,E,t)
R_^(E,t) tj;^l(r,E,t) + x^(r,E,t) g(x,y) tJ>(r,E,t)
(2.49)
(2.48)
l
If the source is assumed to be isotropic, the
o .>
only nonzero moment of the source will be S0(r,E,t)
or Xo(r>E,t). Then the set of Eqs. (2.46)(2.49) is
reduced to
(R1(E,t)R0(E,t) V2]
x x0(r,E,t) = 0
(2.50)
R,(E,t) *?(r,E,t) ~ A(z) ii(r ,E,t)
11 /T
(2.51)
17
(2.52)
(2.53)
Eq. (2.50) is then the sought for thermalization model
of the neutron transport equation in the consistent
approximation with space, energy, and time dependence.
This equation is in a general operator form. To re
write it in a more explicit form we make use of Eqs.
(A.6), (A.7), (A.11), (A.12) and (A.13) which give
1 ala2
[H (E) + F (E)  + ~~
v(E) 3t v2(E) 3t2
a v
+ G(E) ] X(r,E,t)
(2.54)
along with the definitions
(2.55)
X (r ,E,t) = AT xg(r,E,t)
(2.56)
H
(2.57)
k
k
F(E) = l fk(E) 2
(2.58)
18
3k
G(E) = l gk(E) (2.59)
k SEk
where fk(E), hk'{E) and gk(E), which are algebraic
functions of the energy and the macroscopic properties
of the medium, are given by Eqs. (A.14} (A.22) and
where the derivative operators, ak/3Ek / arise from the
Taylor expansion of the flux.
In Eqs. (2.57) (2.59) all the energy
derivatives are kept and the truncation of the expansion
depends on the model of the scattering kernel used and
on the energy moments, M^(E), which appear as coeffici
ents of the derivatives, 3k/3Ek, in the Taylor expan
sion. See Eqs. (2.34) and (2.37). In the heavy gas
model scattering kernel, which is applied in this work
for numerical computation, the moments, Mk(E}, are
zero for k greater than 2. Therefore, only the first
and second derivatives are used.
It is very convenient to transform Eq. (2.54)
from the energy domain, E, into the domain of the
dimensionless variable, e given by
e = E/T (2.60)
This transformation leads to
19
1
[H(e)+ F(Â£)
Vq
9t v02e 9t2
*(r,E,t)
1 9
= [G(E) + ~ ] x(r,e,t) (2.61)
v /T 9t
H(e) =
1 hi
i
91
9ei
(2.62)
F(e) =
I fi(Â£)
JL
9E1
(2.63)
i
0(E) =
9j
i
9
9ci
(2.64)
See. Eqs.
(A. 27)
 (A.35).
The functions f^, h^ and
g^ are calculated in terms of E and e for the heavy
gas model in Appendix B.
Eqs. (2.50), (2.54) and (2.61) are all
equivalent and the last form will be used throughout
the remainder of this work. Specifically, Eq. (2.61)
will be used in Chapters III and IV to investigate the
neutron waves and the pulsed neutrons techniques in
moderating media.
EnergyIndependent Cases
It is interesting to conclude this chapter by
the examination of the energyindependent case of
Eq. (2.50). In this case
20
i> (r ,E,t)
'I'(r,t)
(2.65)
Dki>(r#E,t) = 0 k>0 2.66T
It is easily seen that the operators, Qm(E), become
energy independent and are reduced to the moments of
the scattering kernel, i.e.,
Qm(E) = Qm = [ Mm D Mm Em (2.67)
k
where
Em = / pm(uo} EsUo)dyo (2.68)
l
In the P^ approximation m takes the values 0 and 1
which give
Zo = I P0(uQ) zs(vj0)d'j0
1
(nQ)dM0
1
(2.69)
21
/ P]>0) "s^o)dlJo
= / Es(vJo)d;jo = Es*o
(2.70)
1
Then the operators, RQ(E,t) and Ri(E,t), take the
energyindependent forms
1 3
R (t) = l E =
o s t v at
1 8 v

R (t) qEs Et v 3t
 (x + k *)
'3D v 31'
where
(2.71)
(2.72)
D = 1/3(Zfc p0Eg)
(2.73)
is the diffusion coefficient. The energyindependent
flux equation is obtained by the substitution of Eqs.
(2.71) and (2.72) in Eq. (2.50).
3D a2 1 3
v2 3t2 + v <1+3EaD)at +
*(rrt)
1 +
3D 3 ~
v at
x(rt)
(2.74)
22
This equation is easily recognized as the
telegrapher equation which includes the transport cor
rection on the source. This correction is given by
g
the term (3D/v) x(r,t) It should be noticed that
3t
this correction term is neglected in the telegrapher
equation given by Meghreblian and Holmes (45) and by
Weinberg and Wigner (46). The neglection of this
term is possible in a steady or quasi steady state
operation where the source variation with time is very
slow. On the contrary, the source derivative term is
very important in time dependent kinetic problems
which are sensitive to time variations of the source.
In studying the two problems of neutron waves and
pulsed neutrons, which are of most interest to this work,
the source transport correction is specifically impor
tant when sharp neutron pulses are fed into the system
and the transient flux is investigated, or when the
source frequency becomes large. However, these two
subjects will not be studied through the telegrapher
equation but through the general space, energy and time
dependent formulation given by Eq. (2.61).
V
CHAPTER III
NEUTRON WAVES IN MODERATING SYSTEMS
Analytical Formulation of the Neutron
Wave Problem
Various theoretical and experimental
investigations of the neutron wave problem have been
reviewed in Chapter I. This chapter is devoted to
the investigation of the neutron wave problem through
the time, energyand spacedependent flux equation,
Eq. (2.61), derived from the consistent P^ approxima
tion which gives rise to the thermalization and
transport effects. As a convenient reference, Eq.
(2.61) is restated here.
(H(e) + F ( e)
(3.1)
Since the source does not exist inside the
system, the source term is dropped from Eq. (3.1) and
a suitable source condition will be established later
in this chapter. Thus, the neutron waves will be
studied through the homogeneous equation
23
24
[H (e)
3 1 32 1
31 + VQ2e 3t2 3
*(r,e,t)=0
(3.2)
where, as noted before, ij>(r,e,t) is the nonMaxweIlian
component of the flux
4>(r,e,t) = m(e)
The corresponding component of the source is x(r,Â£,t)
where
> >
S(r,e,t) = ra(e) x(r,e,t) (3.4)
When the source is sinusoidal in time its
component x(r,e,t) is also sinusoidal and can be
expressed by
x(r,E,t) = SQ(r,e) + Re[S(r,e)e^wt] ^
where sQVr,Â£) and s(r,E)^ut are, respectively, the
timeindependent and dependent components of x(r,e,t) .
Similarly, the nonMaxweIlian component of the flux
has the same time behavior with the same angular fre
quency, f, or angular velocity, w = 2Ttf, but with some
angular phase shift. This phase shift (lag), which is
25
a function of the source frequency and the position at
which the flux is measured, will be determined later.
Therefore, the tim dependence of the flux can be
given by the expression
*(r,Â£,t) = ipQ(r ,e) + Re (r ,e) e^)t] ^
The combination of Eqs. (3.2) and (3.6), and then the
separation of the steady state part of the resulting
equation gives
2
Ij U) \ 1 j j ) \ 1 +
) F(e) +  J J 72] i{i(r,E)=0 (3.7)
and
[H (e) \ V2] <Mr,e) = 0 (3..8)
Since the steady state solution can be easily found
by replacing u by o in the time dependent solution,
there will be no need to treat Eq. (3.8h separately
but it suffices to solve Eq. (3.7).
The boundary conditions associated with Eq.
(3.7) are
t>(X, y, z)
ii(x, y, z) ,
(3.9)
26
4>(x,y,z) = \p (x,y ,z) (3.10)
D(a,y,z) = 0 <3.11)
*(x,E,z) = 0 (3.12)
where 2a, 2B, and c are the extrapolated dimensions of
the nonmultiplicative assembly in the x, y and z
directions, respectively, and where the center of the
xyface is taken as the center of the coordinate system.
The source is located at the xyface with supposedly
known spacial and energy distributions and will be
specified later when the source condition is estab
lished .
The solution of Eq. (3.7) is developed in cosine
functions for the x and ydependence, exponential
functions for the zdependence and normalized associated
Laguerre polynomial for the energy dependence, i.e.,
ip(xfy,z,c) = l cmfTlXm(x) Yn(y) Zm^n(z) Em,n(e) (3.14y
m,n
where
27
2m~l
Xj^x) = cos TIX
2 I
(3.15)
Yn(y) cos
2nl
*y
2b
(3.16>
Zm,nU) exP<pm,nz> Uexp<2pm(n(Sz) )]
exp(~pm#nz)
(3.17)
E
m,n
(e)
I
l
a* t*1*
A Lo
m,n 1
U)
(3.18^
The terra in brackets is the end effect correction due
to the finite size of the system in the zdirection and
p is the inverse relaxation length of the (m,n)
ran
spacial mode. The energy dependence of the (m,n) mode',
Em n^e^ exPanc^ed in the normalized associated
Laguerre polynomials of the first order, (e) .
These polynomials and some of their properties are
discussed in Appendix C.
The substitution of the solution given by Eq.
(3.14) in Eq. (3.7) gives the equation
28
I [H(e) + AF (e) + 7 V2 (P2 _b2 )]
e 3 m,n Jm n'
m,n
, .cm,Xm(x)Yn(y)Zm(n(^)Em(n(c) O (3.19J
where
A
and
B?
im,n
(3.20)
(3.21)
is the transverse buckling of the (m,n) spacial mode.
Eq. (3.19), which contains all the spacial modes, can
be separated into a decoupled set of equations each of
which involves one mode only. This decoupling is
achieved by operating on Eq. (3.19) by the special
integral operator
P
as 
Ja dx / dy (x) Yn, (y)
a 6
(3.22)
This operation and the recognition of the orthogonality
properties,
29
/a X (x) X (x) dx = a
i m' m m',m
a
(3.23)
b
/ Yn'(y)Yn(^)dy = b6nl,n (3.24)
b
leads to the modal equation
[H(e)+4F(e)+ i 42 (P/5tfln>n)] Em,n(e)=0 (3.25)
ra,n = 1,2,3,
where we divided through by the constant C .
Ill /XI
In order to determine the coefficients, A ,
m, n
we will operate on Eq. (3.25^ by the energy integral
operator
0 = / de m(e) (e)
Xr
o
(3.26)
where m(e) is the Maxwellian function previously
defined by Eq. (2.26K* This operation leads to the
following set of algebraic linear homogeneous equations.
I [a + 0 A + y A2
Z l' ,1 l\i f ,1
n r]
Â£' ,i
m,n
= 0
(3.27)
30
where the matrices involved in this equation are defined
by the following:
/ m(e) L(*} (e)H(e) LU) (e)de
o Â£ Â£
(3.23)
Â£' ,Â£
r m(e) (e)F(e) (E)de
(3.29)
Â£' ,Â£
f m(e) L(1) (e) L(1) (e)de
Â£1 e Â£
(3.30)
Â£ Â£ 3
= 1 / m(e) L(1) (e) L(1)(e)de = 6 (3.31)
Â£ i jj, 3 Â£ Â£
and where
r
p2 B2
m ,n im, n
(3.32)
If the Laguerre expansion of Eq. (3.18h contains
L+l polynomials, i.e., if Â£ takes the values 0, 1, 2,
..., L, then, the set of Eqs. (3.27) can be rewritten as
l i
Â£=0
Â£' ,Â£
+ 3
Â£' ,Â£
A + y
2 _
V Â£
r]Am,n= (333)
I
31
with i' = 0, 1, 2, ..., L. This represents a set of
Â£
L+l homogeneous equations with L+l unknowns, n,
Â£ = 0, 1, L.
In order for this set to have a solution, its compati
bility conditions
A + y
(3.34)
l', Â£=0, 1, 2, ..., L must be satisfied. This
equation will be referred to as the eigenvalue equation
and r will be referred to as the eigenvalue. From this
equation we recognize that r is completely "space
independent" because it is a function of A and the
matrices a,, B , v c, and n ., all of which
are "spaceindependent." Consequently, and for similar
reason, the coefficients, are "spaceindependent,M
m, n
as is seen from Eq. (3.33). This means that all the
spacial modes have the same eigenvalues and the same
coefficients which lead to the formulation of the
eigenfunctions as we will see later. On this basis the
indices m and n, which indicate the (m,n) spacial mode,
can be omitted from Eq. (3.34) which is rewritten as
L
l
Â£=0
[V,* +
B A
Â£ ,Â£
2 _
V,*n\
(3.35)
Â£' = 0,1,2, ,L
32
However, it is important to realize that the complex
inverse relaxation length, Pm^n, is "spacedependent"
and is computed from the relationship
P2 = r + B (3.36)
m,n Jn,n
To avoid confusion we must point out that the terms
"spacedependence" and "spaceindependence" are used
here to indicate, respectively, whether or not a cer
tain quantity is a function of the spacial modes, or
the spacial mode indices m and n. To solve the
frequency dependent set of Eqs. (3.35) two alternatives
can be used: the exact method and the perturbation
method. In the following two sections both methods
will be discussed.
Solution by the Exact Method
Due to the fact that the expansion of the
determinant in Eq. (3.34) gives a polynomial in r with
a maximum power of L+l (the size of the determinant)
the solution of this equation yields b+1 values for r.
These values will be distinguished by the subscripted
variable,^, where k = 0, 1, 2, ..., L. It is then
obvious from Eq. (3.35} that to each there corres
ponds a different set of coefficients, A
which
33
leads to the formation of the energy eigenfunction or
energy mode, E (e), through Eq. (3.18) which becomes
m, n
L
Ek (e) = Ek(e) = l A L(1) (e) (3.37)
m,n 4=o Â£'k l
Each set, A is the solution of Eq. (3.35) using
l ,k
the value for F, i.e.,
L
1 [aA + rk^A4,k=0 (3*38)
l=o
or
L
Â£ V ,4 (A'Wk = 0 {3.39)
1=0
This homogeneous set of L+l equations with L+l unknowns
can be solved for L unknowns in terms of the other. If
Eq. (3.39) is divided by A it can be rearranged in
O / JC
the form
L
E 4* ,4(A,rk)R4,k = D4,,o(Ark) (3.40)
4=1
with i
= 0, 1, 2,
L1.
34
The new unknowns appearing in this equation are the
ratios
Ri,k Aa,k//Ao,k
where
Ro,k = Ao,k/Ao,k = 1
(3.41)
(3.42)
Notice that the equation corresponding to i = L was
omitted from the set represented by Eq. (3.40) in
order to equalize the number of variables to the number
of inhomogeneous equations, i.e., to remove the degen
eracy of the solution. The set of Eqs. (3.40) can be
solved for all the ratios, R and the energy modes,
X, f ix
Ek(e), can be formed by recombining Eqs. (3.37) and
(3.41).
Ek (e)
L
l
1=0
(3.43)
The exact method that has been discussed yields the
correct solution as a function of A or w. The whole
computational process must be carried out in complex
algebra due to the fact that A and r are complex.
35
Furthermore, the whole solution must be repeated for
every value of w since we are interested in the solution
for a wide range of source frequencies.
Solution by the Perturbation Technique
This technique is based on Feenberg's
perturbation method discussed in (47) A modified
version of this technique was developed and used by
Perez and Uhrig (38) to solve a similar problem.
We will use this technique here to solve Eq. (3.35).
First, Eq. (3.35) is rewritten as
L
l
1=0
+ 8
i a
A + y
0
where
(3.44)
R* = At/Ao. (3.45)
Then, it is solved for the case of A = 0 by the exact
method, i.e.,
L
1
1=0
[a
,' Z
 n
i' i
r(o)]A(o)
l
0
(3.46)
with the compatibility condition
V ,i ~ V ,i
\
r(o)  o
(3.47)
36
Equation (3.47)^ gives all the unperturbed or steady
state eigenvalues, the use of each of which in
Eq. (3.46) yields the corresponding unperturbed ratios,
. Next the kth eigenvalue, rk, and the corres
ponding ratios, k, for A / 0 are obtained from the
equation
l t
n r, ] r =0
t k (3.48)
fc=0
by the perturbation method.
In this method the eigenvalues, rk, and the
ratios, R k, are expanded in power series in
rk= r^0) + rk1} + rk2) A2/2i +
(3.49)
R = R(o) + R(1) + R<2> A 2/2! +
Â£,k Â£,k Â£,k Â£,k
(3.50)
where
.(v)
a'
9 A
v rk
A=0
(3.51)
3 v
9 A v RÂ£k
A=0
i ,k
(3.52)
37
This expansion is restricted by the condition
U)
v0 (rad/cm) < 1
(3.53)
( V ) ( V )
Once we find r and R the solution is obtained
l ,k
from Eqs. (3.49) and (3.50) by the substitution of the
numerical value of a.
It remains now to determine the derivatives
r(v) and R^v This determination is done step by step
from Eq. (3.48). First, the first derivative of the
set (3.48) with respect to A at A = 0 is equated to
zero and solved for r,^ and R^ in terms of and
k A,k k
, (o)
'a ,k
which have been determined from Eqs. (3.46) and
(3.47). Then the same process is used for the second
derivative of Eq. (3.48)^ to solve for I*/2) and R^2^ in
k i ,k
terms of r^^, and R^, and so on. Suppose
k k i,k Â£,k
now that we have found r ^p) and R ^ ^ for y = 0, 1, 2,
k , ,k
..., v1 and want to find and R^v,^. Applying the
k l ,k
rule
9x'
[A(x) B (x) ]
l {i)AU)(x, B(VB,(X,
y=o
(3.54)
38
to Eq. (3.48), where
v!
U) 
y! (vy) 1
(3.55)
we obtain
v k (y)
y y ( [a +e a+y A2n r ]
y=0 Â£=0 l' z' Â£ Â£''Â£ Â£,'Â£ k
(vy)
Â£, k
Realizing that
= 0
(3.56)
A=0
d*
(An)
dxy
ni
A=*0
(ny) n'p
(3.57)
where 6 is the Kronecker delta function, Eq. (3.56)
n,y n
is rewritten as
v L (y) 1
y y ( M (a 6 +3 6 +2Y 6 n T J
_ n ly J V,Â£ o,y Â£,* i,y 'I', l 2,y 1*, k
yj 0 i>"0
= 0
Â£,k
(3.58)
Noticing that R =1 for all values of k and
O / K
(n) (v)
Rq k = 60 n for ald values of and separating
and R^ ( l / 0) one can rearrange Eq. (3.58J in
p K
the form
39
ak r(v* + l k r(v) = bk (v)
O I A L O I 0 0 \r O V
i ,ok
Â£=1
Â£' Â£ Â£ ,k Â£'
(3.59)
where
Â£' ,o
L
I R
Â£=0
(o)
Â£ ,k nl1 Â£
(3.60)
= n
(o) _
Â£ ,Â£ Â£ ,Â£ k Â£ ,Â£
Â£ > 0
(3.61)
bK,(v) = l [R*0}(8 ,6 +2y 6 } + u(v2)
Â£1 Â£,k Â£' fSL 1 ,v Â£',Â£ 2 v
Â£=0
v1
l ( Â¡Â¡ ) R(v"y) (8 6 +2y 5 n r(y)
p = ]_ Â£^k Z 11 1 y Â£fÂ£ 2, V Â£'fÂ£ k
where the Heaviside unit step function
U (v2) = 1 for v i 2
}]
(3.62)
for v < 2
(3.63)
indicates that the term following it exists only for
v > 2 In explicit matrix notations Eq. (3.59) takes
the form
40
1
P
0
o
0,1
CN
ak 
CL
O, L
r(v)
rk
bk,v
o
k
ai,o
k
ai,l
k
ai,2
k
a
1,L
R(V)
1, k
, k,v
b
1
i
; aK
2,0
J .
a^
2,1
ak
2,2
... ak
2, L
R(v)
2 ,k
23
bk,v
;
i ak
L, o
ak ,
L,1
ak
L, 2
ak
** aL,L
' L,*
i
!b?'v 
L
These two equivalent sets, Eqs. (3.59) and (3.64), are
linear inhomogeneous and consist of l&l equations with
L+l unknowns, and R^v) (i = 1, 2, ..., L) which
k % ,k
are completely determined by the solution of the set.
It should be pointed out that the matrix, a^, does
not depend on the order of derivative, v, and it is
V \i
the same for all values of v. On the other hand, b '
l1
does depend on v and has to be calculated for each
value of v. See Eqs. (3.60), (3.61) and (3.62). How
ever, both akt and bki'v depend on the eigenvalue
SL f i
used, as indicated by the superscript k. Both Eqs.
(3.59) and (3.64) will be referred to as the perturba
tion equation.
It is very helpful to summarize the various
steps of this technique in the sequence used in actual
computation.
41
1. The unperturbed equation, Eq. (3.46), and
its compatibility condition, Eq. (3.47),
are solved and the unperturbed quantities
rand r() are determined,
k H,k
2. For each value of k, the perturbation
equation, (3.59) or (3.64), is solved for
all values of v in increasing order up to
the desired maximum value. Each time
v) and are calculated in terms of
k L ,k
r(vi) and y = 0, 1, 2 v1.
k i. ,k
3. r and R are calculated from Eqs.(3.50) and
k l ,k
(3.50) using the desired value for A or w.
4. Finally, the energydependent flux modes,
or eigenfunctions, Ek(e), are formed
Ek(Â£) Ao,k R1(kLi1>U) <365>
l0
It should be emphasized that this perturbation technique
has been used here only in the energydependent equation
(3.44), leading to the determination of the energy modes,
Ek(Â£) .
Before we move on to further developments we
ought to compare the two methods. The exact method has
42
the advantage of yielding the functions E (e without
any approximation at any desired frequency and without
any restriction on the maximum value of ui that can be
used. The disadvantage of this method is that all the
computational processes must be performed in complex
algebra and all the computational steps must be
repeated for each value of m. On the other hand, the
perturbation technique introduces some error in the
solution due to the approximation hidden in Eqs. (3.49)
and (3.50). The value of this error diminishes rapidly
as the number of terms kept in these equations in
creases. The other disadvantage of this method is the
restriction given by Eq. (3.53), namely
< VQ (3.66)
It is obvious, of course, that the solution is improved
with decreasing values of u. The advantages of the
perturbation technique are that the algebra involved in
it is easier to handle in actual computations and that
the computation does not have to be repeated for each
frequency. The computational scheme used in this work
utilizes the perturbation technique.
43
In both methods the energy eigenfunctions take
the same form, as shown by Eqs. (3.43) and (3.65).
Having determined the energy dependence of >J>(x,y,z,e) by
either one of these methods we can easily rewrite Eq.
(3.14) in the form
4> (x,y ,z,
o l
l Ck X (x) Y (y)Zk (z)Ek(e)
L m,n m n m*n
(3.67)
m,
n k
where we
redefine
ck
m,n
c A
m,n o
(3.68)
n
'x
X
cos (
2m1 \
TTX
2a
(3.69)
Yn(y) =
2nl ,
in 7 n \
tUo t
try J
2b
\ J / U j
Zk (z)
m,n
= exp (
k
"PmrnZJ
(3.71)
Ek(e) =
l R
; *
k Â£ '
(3.72)
It remains now
V
to determine the constants, CTO _.
ill / XI
This
will be done later from the source condition developed
in the following section.
44
Neutron Net Current and Source Condition
Before we establish the source condition the
neutron net current has to be developed in terms of the
neutron flux. To do so we first define the neutron net
current. Let dA be an infinitesimal area whose norm is
the vector N located at r and consider the directional
>  *
flux, $(r,n,E,t) where the vector Q makes with N the
anglevQ given by
cos 6 = Q*N (3.73)
The net current can then be defined by the following
relation
Jn(r,E,t) dA = the net number of neutrons
with energy E which pass
through the area dA in all
possible directions.
This definition is equivalent to the analytical
statement
J (r,E,t)dA = dA / $ (rf,E,t) (lN)d2 (3.74)
.
where dA(n N) is the apparent area seen by the
neutrons traveling in the direction, ii, or
Jn(r,E,t) = / 4 (r,n,E,t) (nN)dn
n
(3.75)
45
where Jn(r,E,t) is then the net current through a unit
area whose norm is N. Eq. (3.75) can be rewritten as
jn(r,E,t) n / *(r,n#E#t)n da
a
 N J(r,E,t)
(3.76)
which means that the net current, Jn(r,E,t), is the
4
projection on N of the vector net current
J(r,E,t) =
/ *(r,n,E,t)a da
(3.77)
The flux and the vector, a are given by
4>(i,a,E,t) = l l <^(r,E,t)Y^(a) (3.78)
m n
 *
a = Sine cos u + sine sin v + cose w (3.79)
where u, v and w are the unit vectors along the x, y
and zaxes, respectively, and y and e are the polar
4
and azimuthal angles of a in spherical geometry. From
Eqs. (2.7) and (2.12) it can be shown that
COS 0
(3.80)
46
sine cos f [Y^ (fi) Y^1 (&)1 /2Hi
(3.81)
sine sin ^ = [Y^(H)+Y^1(H)]/2jH^
(3.82)
The combination of Eqs. (3.77)(3.82) and the use of
the orthonormal property of Y^(n) expressed in Eq. (2.7)
we arrive to the expression
J(r,E,t) = u[*J(r,E,t)];(fE,t)]/2Hj
 v[(j,J(r#E,t)+4,~1(r#E,t) ]/2jH^
+ ^('E't)/H (3.83)
The moments, $n(r,E,t) satisfy the relations
ra
^(r,E,t)
 m(E)*Â¡J(,E
t)
(3.84)
*U,E,t)
1
S
*7
3
(j
3x
3
)
3y
R^E^i^U^t)
(3.85)
i1 (r,E,t)
i
= 7?
3
( +j
3x
3
)
3y
R^1(E,t)*(r,E,t)
(3.86)
(r,E,t)
i
=5 '
7
_i_ 1
Ri
3z 1
(E,t)*g(r,E,t)
(3.87)
Ci 4
47
and the scalar flux is obtained by
(r,E,t) = m(E)*(r,E,t)/H
= m(E) i> (r,E,t) (3.88)
Eqs. (3.83)(3.88) yield
(r,E,t) = m(E) [u ~ + v ~ +w] p" (E,t) \ (r,E,t)
 m(E) VR1 (E, t) i;(r,E,t)
(3.89)
In the case of N = w the net current through a unit area
perpendicular to the zaxis at (x,y;z = 0) is given by
Jn(r,E,t)
1 31
= ~ m(E) r, (E,t)
=0 3 z
,0 (3.90)
Having found the net current, the source
condition can be easily established. Let S(x,y,E,t) be
2
a plane source located at z = 0 evaluated in neutrons/cm
sec emitted in all inward directions. Equating the
neutron source strength to the net current we obtain the
source condition
48
m(E)
3
9 1
R, (E,t)*(r,E,t)
3z A
= S(x,y,E,t)
z=0
(3.91)
By factoring out the Maxwellian component of the source
and dividing Eq. (3.91) by ra(E) it becomes
1 3
3 3z
R^Ejt)
 X(x,y,E,t)
z=0
(3.92)
Finally, if we operate on Eq. (3.92) by R^(E,t) we
obtain the final form of the source condition
3 *
\p (r,E,t)
= 3R1(E r t)x(x,y,E,t)
z=0
(3.93)
where the operator R^(E,t) is given by Eq. (2.36). The
transformation of Eq. (3.93) into the domain of the
dimensionless energy variable, e, yields
3
r~ *(r,e,t) = 3R,(cft)x(x,y,e,t)
9z z=0 1 (3.94)
or, by using Eqs. (A.7) and (A.29),
3 >
i(r,c,t)
3 z
1 3
* 3[77 T~ + G(e) ]x(x,y,eft)
z=0 vn/e 91
(3.95)
This condition introduces two corrections on the source.
The first one is the transport correction given by the
49
derivative with respect to time and the second one is
the energy correction hidden in the energy dependence
of the operator G(e) and the energy derivatives
involved in it. When the transport correction is
neglected and a Maxwellian source is used the non
Maxwellian component of the source becomes independent
of energy. Then, the operator G(e) is reduced to
G(c)
[VÂ£) Vs(e)
1
3D (e)
(3.96)
and the condition of Eq. (3.95) becomes
D(e)
8 +
~~
8 z
= xUy#e#t)
z=0
(3.97)
or
D(e)
8 *
77 (r,e#t)
8 Z
= S(x,y,e,t)
z=0
(3.98)
which is essentially the condition used by Perez and
Uhrig (33) .
Before we use the source condition in Eq. (3.95)
to determine the so far unknown constants, C two
m, n
remarks should be made. The transport correction in
this condition has a strong effect when the source
frequency is high while its effect is small with low
50
frequencies. The energy correction is always persistent
except when a pure Maxwellian source is used, in which
case x(r,e,tj is constant and its derivatives are zero.
Determination of the Constants,
:n.m
The final step in the solution of the neutron
wave problem is to determine the constants, n, which
appear in the flux expression of Eq. (3.67). This
determination is carried out by using the source condi
tion given by Eq. (3.95). The substitution of ^(r,e,tr
from Eq. (3.67) into Eq. (3.95) gives
l l I Cm,nVx>Vy>0m,nzm,n!z>Ek<=>e:'t
m n k
= 3 [~ + G(e) ] X(x,y,e,t) <3.99)
v /e 3t
o
If we assume the separability of space, energy and time
dependence of x(x,y,Â£,t), i.e., if we express it by
X(x,y,e,t) = Xg(x) Yg(y)Eg(e)e^wt
(3.100)
Eq. (3.99) becomes
l l l Ck o* X (x)Y (y)Ek(e)
** L u m,n m,n m n 1
m n k
 G(e)]Xs(x)Ys(y)Es(c)
(3.101)
51
where the subscript s is used to indicate the source and
has no numerical values. To utilize this equation in
determining the unknown constants, Ck n, we first mul
tiply it by Xm<(x) Y {y]fdx dy and integrate it over
the whole extrapolated x and ydimensions. This
operation, along with the orthogonal properties
I3 X (x)
m
 a
X (x)dx
m
a
(3.102)
/. Yn Yn(y)dY = 6 Sn\n
 b
(3.103)
yields the equation
l Ck pk Ek(e) = a t_L + G(e)]E (e)
L m,n m,n m,n s
k
(3.104)
which is valid for all values of m and n, where we have
defined
co ao
am,n = s /a /b dy Xm(x)Xs(x)yn(x)Yg(x)
CO ao *
 a b
52
For given values of m and n Eq. (3.104) couples all the
unknowns, Ck corresponding to all the values of k.
in, n
This equation can be reduced to a coupled set of equa
tions by the multiplication by ra(e )L^(e^de and
k'
integration over e from 0 to .
k k
Cm,npm,ndk',k ara,n^A ^k' + 9k'^ (3.106)
k
where
d f = / m( e) L(1) (e) Ek(e)de
k k q k1
C = / m(e) (e) E (e)de
k* Q k* /7 s
6 = / ra(e) (e) G(e)E (E)de
k' k' s
o
(3.107)
(3.108)
(3.109)
Using the expression for Ek(e) given by Eq. (3.72) in
Eq. (3,107) and considering the orthonormality property
of the Laguerre polynomials, one obtains
/ m(e) (e) I ^ (e)de
o i '
= y R 6 = R
1 Z,k k' A k' ,k
z
(3.110)
53
Then, if we define
Dk'k=pk dv,
m,n v
m,n k' k P m, n Rk' k
Bm,n = m,n k* + V*
(3.111)
(3.112)
Eq. (3.106) takes the simple form
y Dk'tk ck = ak*
L m,n m,n m,n (3.113)
k
A more explicit form is

D'
m, n
D' 1
m, n
... D'L
m, n
C
m, n
B
m, n
D1,0
m, n
D1'1
m, n
... dx'l
m, n
C1
m, n
=
B1
m, n
*
dl'
m, n
m, n
... dl'l
m, n
c1
m, n
;
m, n
_ _
(3.114)
This linear inhomogeneous set applies for all values of
m and n and contains (L+1) equations with (L+1) unknowns,
C* which are completely determined by its solution,
m, n
At this stage we see that the solution for the
flux has been completely found, as given by Eq. (3.67).
All the constants have been determined. All the space
54
and energy eigenfunctions are known and given by Eqs.
(3.691 (3.775 The various inverse relaxation lengths,
pk n> are computed from the eigenvalues,, according
to Eq. (3.36). It is important to recognize that the
actual computational process must be carried out in
complex arithmetic because, in general, all the quan
tities involved in this computation are complex.
The actual scalar flux, as a function of space,
energy and time, is now given by
(r,e,t) =
Re il l l Ck X (x)Y (y)Zk (z)Ek(e)m(e)e^ut]
e L L L ra,n m n J m,n i
m n k
(1.115)
In order to compare Eq. (3.115) with the actual
experimental results, the neutron density has to be
computed. This arises from the fact that the experi
mental setup used by Booth (41) uses a 1/v detector
and, hence, it gives the energy integrated, or total
neutron density, as the output. The neutron density,
which is related to the flux by the relation
N (r, e,t)
4> (r,e ,t)
(3.116)
can be given by
55
N(r,e,t)
Re
[I I I
m n k
ra
_X (x) Y (y) Z
, n m n J i
(z)
v,
y?
(e)Ek(e)ejait]
(3.117)
The total neutron flux and total neutron density are
obtained simply by integrating Eqs. (3.115) and (3.117)
over energy.
It was mentioned at the beginning of this
chapter that the flux, or hence, the neutron density,
has the same time behavior as the source with a phase
shift which depends on the detector location and the
source frequency. Having found the flux and the
density we can now determine the amplitudes and phase
shifts for both of them using Eqs. (3.115) and (3.117)
which are rewritten as
(r,e,t)
Ret l Ef(c)(R^>n()<jB^n()}ej("t"bra>nZ) ]
m n k
(3.118)
N(r,e,t) =
k
Re[Â£ l l E^
m n k d m,n m,n
(3.119)
56
where
k _k i uk
p = a + jb
m,n m,n m,n
(3.120)
k
Ak (r) + jBk (r) = Ck x (x)Y (y) e amnZ
m,n mfn m,n m n
(3.121)
E^e) = m( e) Ek ( e)
f
(3.122)
1
Ek(e) m(e) Ek(e)
Q v /e
(3.123)
The indices f and d have been used to indicate the
neutron flux and neutron density, respectively. If we
write
f 4
(r,e,t) = Af(r,e) cos[(ut ef(r,e)] (3.124)
4 4 4
N(r,e,t) = Ad(rfe) cos [ut 0d(r,e)] (3.125)
then Af(r,e), ef(r,e), Ad(r,e) and 0d(r,e) signify,
respectively, the amplitude and phase shift of the neu
tron flux and density. If the righthand sides of Eqs.
(3.118] and (3.124) are equated and expanded, the
following expressions are obtained for the amplitude
and the phase shift:
57
A (r,E) =
I l (^) 2 } ( e) 2
m n k B'n m'n f
(3.126)
e.(r,e) =
I I I
m n k
Ak (r) sin bk zBk (r) cos z
m,n m,n m,n m,n
E*(e>
tan
1
l 11
m n k
*k k k ,k
Vr) cos bm,nz + Bm,Ar) sin bm,nz
Ek(e)
C
Similarly, we obtain the amplitude and phase shift for
the neutron density. These are found to be
Ad (r, e)
Af(r,e)
v^/e
(3.128)
(r,e) = 0f(r,e)
(3.129)
For the case of the total neutron flux and
density, Eqs. (3.118) and (3.119) are first integrated
over energy and then the same procedure discussed above
is used. This yields
A (r)
f
"l I
l
k
{Ak (rf+Bk (r)2}Ek
m,n m,n f
(3.130)
.127)
58
ef(r) =
I l l [{Ak (r) sin bk z
m n k L ra'n m'n
Bk (r) cos bk
ra,n m
tan1 r k
l l l (Ak
m n k L
)> ^
(r) cos b z + B (r) sin b
m,n ra,n
m
zECl
,n f J
(3.131)
A (r) =
d
III (r)2 + Bk (r)2 }e^2
m n k m'n m'n d
h
(3.132)
0d(r)
i 11K
(r) sin bk i Bk (r) cos bk z Ek
m,n m,n m,n I d
tan
1
l l l k,ni> * Â£.* Bm,n
rank1 J
(3.133)
where
Ef = r Ek(e)de
o
Ek = /" E^(c)dc
(3.134)
(3.135)
59
Thus we have developed the neutron flux and
density in all desired forms as functions of space,
energy and time. Furthermore, we have seen the
neutron traveling waves in the medium through the
expressions of the amplitudes and phase shifts of
various quantities of interest. In Chapter V we will
consider and discuss the results obtained from the
numerical computations using a heavy gas scattering
kernel. We will also compare these results with the
preliminary experimental results obtained by Booth
(41)
CHAPTER IV
PULSED NEUTRONS IN MODERATING SYSTEMS
The analytical treatment of the pulsed neutron
problem is similar to that of the neutron waves. A
major difference, however, arises in connection with
the extra multiplicity introduced by the second order
time derivative which introduces a quadratic term in
the time eigenvalue. For this reason and in order to
obtain a consistent set of conditions sufficient to
determine all the modal amplitudes, we use the invari
ance of the expectation value of neutron population,
* i
\$ <)>}, as our starting point. This involves the
v
knowledge of the adjoint flux which increases the
complexity of the problem.
The pulsed neutrons problem will be studied
here starting with Eq. (2.61). This linear inhomo
geneous equation is rewritten here for convenience.
[H( e) + J_ F(e)  + _1_ I A2] <^r,e,t)
vc at v02e at2 3
3 [G( e) + L ] xtf' e
v0 /F
60
61
where it should be recalled again that the functions
4>(r, e,t) and x(r,e,t) are the nonMaxweIlian com
ponents of the flux and the source, respectively.
The system configuration considered here consists of
a rectangular parallelopiped block of moderator whose
dimensions are 2a, 2b and 2c. The pulsed neutron
source is located at the center of this system, i.e.,
at x=0, y=0 and z=0. The boundary conditions associ
ated with this problem are:
(1) Symmetry of flux.
(2) Flux vanishes at the extrapolated
boundaries.
(3) Finite flux everywhere in the system.
Since these conditions must hold regardless of
the source energy distribution, it follows that they
must hold true for the nonMaxwellian component,
'l>(r,e,t), which contains the spacial and time depend
ence of the flux. Therefore, the above conditions can
be given by the following analytical expressions.
= ii(x,y ,z,e ,t)
(4.2)
s 4i(x,y ,z,e ,t)
(4.3)
i*(x,y ,z,e ,t)
= ifi(x,y ,z,e ,t)
(4.4)
62
4
= 0
(4.5)
= 0
(4.6)
i^(x,y ,c,e ,t)
a 0
(4.7 r
(x,y,z,e ,t)
= finite
(4.8)
where 2a, 2b and 2c are the extrapolated dimensions of
the system.
We will first solve the homogeneous part of
Eq. {4.1), i.e.,
3 2
[H(e) + 1_ F(e) JL + i iv2] tKr,e,t) = 0
Vo at vq2c 9t2 3 (4.9,
The solution of this equation is developed in cosine
functions for the spacial dependence, exponential
functions for the time dependence and Laguerre poly
nomials for the energy dependence, i.e.,
'Kr,e c X (x)Y (y)Z (z)T (t) E (e)
mnp ranP m n P m,n,p m.n,p
(4.10)
63
Considering the boundary conditions the functions
involved in this equation are given by
X (x)
m
= cos
2ml
2a
vn(y)
= cos
2nl
U
Zp( z)
= cos
2SzL
2c
(4.11)
(4.12)
(4.13)
T (t) =
m,n,p
E (e) =
mn ,p
exp(A t)
m ,n ,p
V A* l/ ^ ( e )
z m,n,p i
(4.14)
 (4.15)
where c and AÂ£ are constants to be determined
m#n/P m,n,p
and m, n and p are positive integers from 1 to some
desired maxima.
The combination of Eqs. (4.9)(4.15) gives the
equation
I c [H(e)
111,11 ,p in, n, p
A F(e) + A2 + I B2 ]
m,n,p m,n,p 3 m,n,p
X (x) Y (y) Z (z) T (t)
m n P m,n,p
E (O =
m,n ,p
0
(4.16)
64
where
A
in, n, p
X
m,n,p
(4.17)
B
ra,n,p
/ 2m1 \2
2ni
2 /2pl
it +
IT
l 2a /
2b
1 2c
("4.18)
Eq. (4.16) contains all the spacial modes corresponding
to all the combinations of m, n and p values. To make
this equation more useful we can separate it into a
set of uncoupled equations by operating on it with the
integral operator
0 =
dx
dy
dz Xm'^
v (y>
v(z)
(4.19)
and using the orthogonality property
/2q'1 ^
(2q1
cos TTU
cos
TTU
\ 2d 1
i 2d
d 6
q)q
(4.20)
which applies to each of X^Cx), Yn(y) and Zp(z). This
leads to the modal equation
Cm,n,p
2 _i
H(e)A F(e)+A e +
m,n,p v m,n,p
2
m,n,
P
(4.21)
65
Since the operators in brackets do not depend on time,
and hence do not operate on T(b) this equation can
m,n,p
be divided through by c T(t) The following
m, n, p m, n, p
equation follows
[H( e)
A F(e) + A2 e"1 + I B2 ] E (e) = 0
m,n,p m,n,p 3 m,n,p m,n,p
(4.22)
This equation applies for all the spacial modes, i.e.,
for all values of m, n, and p. We call this equation
"the modal equation" because its solution for given m,
n, and p determines E(e) A and hence T(t) ,
m,n,p m,n,p m,n,p
and therefore, completes the solution of the (m,n,p)
mode. To solve this modal equation, we first substi
tute for E(E) from Eq. (4.15) and then operate on
m, n,p
the resulting equation with the integral operator
P = / de m(e) L(e)
o
This leads to the equation
(4.23)
m *n ,p
y n A2
1> 1 m,n,p
n,,cB2 ]A^
1r1 m,n,p mn P
0
(4.24)
66
where ft = 0, 1, 2, .. ., and
a.,. = / m(e) L^;(e) H(e) L(1)(e) de
ft ft J Â£ Â£
O
(4.25)
0 t = 0/ m(e) L(l)(e) F(e) LllJ(e) de
(1)
Â£, ft Â£ i
(4.26)
T = / m( e ) L^Ce) i L(1)(e) de
ft, Â£ ft ft
O E
(4.27)
3_
nÂ£Â£ = m(e) L1)(e) de = 6
* >0 ft Â£ 3 *
(4.28)
Notice that the matrices a f B f y and n,,,^
Â£, Â£ ft j ft ftjft ,
are exactly the same as those for the neutron waves.
See Eqs. (3.28)(3.31) of Chapter III.
If the maximum value of the index, Â£ is L, then
Eq. (4.24) represents a set of (L+l) linear homogeneous
equations in (L+l) unknowns, n where Â£ Â£ =0, 1,
2, ..., L. The compatibility condition of Eq. (4.24)
is that its determinant be zero.
ft*,ft V,Â£ A
m,n,p
+ y.t A 2
ftft m,n,p
VftBm n n
1 mn p
= 0
V
= 0, 1, 2,
/
L
(4.29)
67
This equation will be called the eigenvalue equation
since its solution determines the time eigenvalues,
A which lead to the eigenfunctions, T(t) ,
m,n,p m,n,p
according to Eqs. (4.14) and (4.17). Each element in
the determinant of Eq. (4.29) is, in general, quad
ratic in A Therefore, the expansion of this
m,n,p
determinant is a polynomial in A with a maximum
m,n,p
power equal to 2(L+1). Hence, the solution of Eq.
(4.29) yields 2(L+1) eigenvalues, A which lead
m, n,p
to 2(L+1) time eigenfunctions
Tk(t) = exp(Ak t) (4.30)
m,n,p m,n,p
where
Ak = v Ak
m,n,p m,n,p
(4.31)
and where k = 1, 2, ..., 2(L+l). The multiplicity in
the time eigenvalues leads to the multiplicity not only
in the time eigenfunction but also in the energy modes,
since, for each eigenvalue, Ak there corresponds a
m, n,p
Â£ )C
set of constants, A as a solution of Eq. (4.24).
' m, n, p M
Each set of constants leads to the formation of an
energy mode given by
Ek(e) = l AÂ£k L(1)(e)
m,n,p k m,nfp *
(4/32)
68
For any eigenvalue, a / Eq. (4.24) can be solved for
m,n,p
Ai,k (*=1,2L) in terms of A'k or we can solve
m,n,p m,n,p
for the ratios
r'* = A*"k / A'k Â£ = 1,2 L (4.33)
m,n,p m,n,p m,n,p
from the set
L
1
ll
Ak + V
m, n p Â£ Â£
A*2
m,n ,p
Â£J
b2 ]RÂ£k
Â£ m,n,p m,n,p
A k
m,n,p
+
y
Â£ i o
Ak2 +
mn ,p
B2 ]
m,n ,p
Â£ 1 = 0,1,2,L1
(4.34)
where the equation corresponding to jt1 = L w^s neglected
in order to make the set, Eq. (4.34), with L inhomogeneous
ir 1C
equations and L unknowns, R (a 1, 2, .., L) .
m,n,p
A FORTRAN subroutine, DETEX, was developed to
expand a determinant, the general element of which is a
polynomial in an unknown, say x, of any size. The
result of this expansion is a polynomial in x with a
maximum power of I*J, where I is the size of the deter
minant and J is the size of its general element, i.e.,
the maximum power of x in this element. This sub
routine is listed in Appendix D. The use of DETEX for
Eq. (4.29) leads to an algebraic equation with a maxi
mum power of 2(L+l). Then a polynomial solver, POLY,
(48) is used to find all the roots of this equation,
69
i.e., Ak (k = 1, 2, ..., 2L+2). Having determined
m, n,p
these eigenvalues the set of Eqs. (4.34) is solved for
0 T,
the ratios, R using another subroutine, ELEM,
m,n,p
which was developed for this purpose utilizing the
Gauss elimination method discussed in (49). This
subroutine is also listed in Appendix D.
The whole computational process discussed
above is then applied for all the spacial modes, i.e.,
for all the values of m, n and p. Finally, the flux is
recombined using Eqs. (4.10)(4.15J, (4.17), (4.30)
(4.33), along with the relation
(?,e,t) = m(e) i)(r,e,t) (4.35)
This combination leads to the total flux expression
m,n,p mnP m n p mnP m,n,p
(4.36),
where we redefine
Ck
mn p
c
mn ,p
m,n,p
(4.37)
X (x)
m
3
COB
2a
(4.38)
70
12n 1
vnW =
COS 1
\ 2b
Tty
J (4.39)
Zp(z)
cos j
' 2p 1
( 2c
IT Z
1
1
1 (4.40)
Tk(t)
m,n,p
exp (
"Xm,n,p
t)
(4.41)
Ek(e)
m,n,p
l R4
L m
i
+ dl(1>
tn t P z
(e)
(4.42)
Notice that all these functions have been determined
except which will be found later from the prin
m,n,p
ciple of neutron conservation.
In the neutron waves case we saw that the source
condition, Eq. (3.95), leads to the set of Eqs. (3.114)
which are enough to determine all the constants, n
(k = 1, 2, ..., L+l), for each (m,n) spacial mode. In
the pulsed neutron case such a condition is not enough
to determine all not because of the additional
m, n, p
space index, p, but because of the greater multiplicity
in the energy modes, i.e., due to the fact that k takes
twice as many values as in the wave problem, k = 1, 2,
..., 2(L+l). A more general condition based on the neu
tron conservation principle will be used in this case.
The use of this principle requires the knowledge of the
71
adjoint equation and the adjoint flux which are
studied in the following section. 1
The Adjoint Flux Equation
The differential equation of the nonMaxwellian
component of the flux is given by Eq. (4.1). Substi
tuting for the operators H(e), F(e) and G(e) from Eqs.
(A.30)(A.32), this equation becomes
2
[h (e) + h> (e) ~ + h9(e)  + ff (e) + f.Ce)~
1 2 8e2 Vq1 1 3e
32 9 l 32 l
+ f (e) ""*7 ) + +A2] ij)(r,e,t) =
2 v 2e 3t2 3
o
+g2(e) e ,t) (4.43)
where f^Ce) and g^{e) are algebraic functions of
t. To find the adjoint flux equation the following
rule is used (50). This rule states that the adjoint
operator of the differential operator
m n d
/ v 3 3 3^
a(x,y,.., ,z) ... _
3xm 3yn 3zp
is given by
72
m+n+...p 3m 3n 3P
( 1) ... a( x ,y ,... z)
m n p
3x 3y 3 z
and its application to Eq. (4.43) gives the adjoint
equation
[ H + ( e) J_
'(e)  +
31
1 ii + I A2]'{'*(?,Â£ ,t)
31'
= 0
(4.44)
with the adjoint operators defined as
H + ( e) = hD(e)
2
JLh^Ce) + h2(e)
3e 3e 2
(4.45)
F+(e) = fo(e)
3 32
fx(e) + f2(e)
3 e 3 Â£ 2
(4.46)
In solving Eq. (4.44) for the adjoint flux the
same technique used in solving Eq. (4.9) for the
ordinary flux is followed and we obtain the solution
**(?,Â£,t) = l c+ Xm(x)Yn(y)Zp(z)T+(t) E+(e) m(e)
m,n,p m,n,p m,n,p m,n,p
(4.47)
73
where c+ are constant, Xfx), Y_(y) and Z (z)
m, n,p In 11 p
those given by Eqs. (4.11} (4.13) and
T+(t) = exp(A+ t)
m 9n }p m,n fp
E+ ( e)
m ,n ,p
l
+ Â£
!l.
m,n,p
t(1)r ^
Li
The use of Eqs. (4.11)(4.13), (4.47), (4.48) in
(4.44Y and then the application of the operator
oo
O = Â¡de L(1)(e)
o Â£'
lead to the adjoint modal equation
[H+(e)
A+ F+(e) + i A+ + \ B2 ]
m,n,p G m,n,p 3 m,n5p
E +
m,n,p
m( e )
0
where
A+ X+ /
m,n,p m,n,p
are
(4.48)
(4.49)
Eq.
(4.50)
(4.51)
(4.52)
74
The adjoint eigenvalue equation is then obtained by
using Eqs. (4.49) and (4.23) in Eq. (4.51).
u
Â£
A+ +^;tA+2 ]a+
m,n,p m,n,p m,n,p m,n,p
(4.53)
and the compatibility condition is
Â£(Â£
+ Y+i A+ + R2
Â£,Â£ m,n,p Â£;Â£ m,n,p Â£jÂ£ Dm,n,p
e + a+
where
l, = / H+(c) m(e) L^^(c) de
B+ = / L<1>(e) F+(e) m(e) ^^(t) de
If i It Â£
% w
T+, / L(^)(e) Im(e) L^i;(e) de = y ,
Â£',Â£ o' Â£' e Â£ Â£ Â£
(1)
n+ = / L^l)(e) m(e) L.(D(e) de = n
0 0 1 0 0 0
Â£Â£ 3 0' Â£
Â£; Â£
a 0
(4.54)
(4.55)
(4.56)
(4.57)
(4.58)
We can easily show that these adjoint matrices are the
rotational matrices of a g v and n .
Combining Eqs. (4.45) and (4.55) one can calculate the
matrix a+,
Â£ y Â£
75
a
I L(])U)ihAt)
o *'
m( e)
 JL h,(e) + (e) 3
3e 1 3 e 2 2
L(1)(e) de 4.59)
Â£
Two successive integrations by parts lead to the
expression l
* = / m(e) L^)(e)[h0(e)+h (e) l*h2(e) iL^UMe
n 1 9c2 4
3 E
/ rn ( e ) L(1)(e) H( e ) LC])(c) dE
0 0 I
(4.60)
Thus
Â£* Â£
'.l*
(4.61)
Similarly,
= 3
Â£ Â£ A Â£ 1
(4.62)
Due to the symmetry of y and n we can write
a ; z i; a
Â£ J Â£
Â£,Â£*
(4.63)
Â£ 1 Â£
= n
Â£,Â£'
(4.64)
76
Hence, Eqs. {4.53) and (4.54) become
= 0
(4.65)
and
A simple comparison between Eqs. (4.29) and (4.66),
keeping in mind that the rotation of a determinant
about its main diagonal does not change its roots, we
arrive at the conclusion that both the flux and its
adjoint have the same time eigenvalues, i.e.,
(4.67)
A
m,n,p
The essential difference is that the time eigenfunctions
of the flux are decaying exponentials, while those of
the adjoint are rising exponentials, as seen from Eqs.
(4.14} and (4.48). This result is expected because the
adjoint flux is the importance function and must rise
as the neutron flux decays where the remaining neutrons
become more important.
77
Thus, Eq. {4.66} does not have to be solved
since its roots are given by Eq. (4.67). Defining
the ratios
R* = / A*0*
rnjn jp mynyp iti ^ n ^ p
(4.68)
and solving Eq. (4.65) using all the eigenvalues one
by one, we can finally recombine the adjoint flux.
^*(r,e,t) =
l C+k X (x)
m,n,p,kmn*P m
Yn(y)
Zp(z)
. T+k(t) E+k(e) m(e)
mfn fp m jri jp
(4.69)
where we redefine
C+k c+ a+ o,k
m,n,p m,n,p m,n,p
(4.70)
T+k(t)
m,n,p
exp(Xk t)
m,n,p
(4.71)
E+k(e)
l R+l'k L1)(G)
m,n,p l
(4.72)
In this section we have derived the adjoint
flux equation, proved that it has the same time eigen
values, and finally, solved the adjoint equation except
78
+k
for the constants, C the determination of which
m,n,p
is not necessary as we will see later.
Neutron Conservation
In order to complete the solution for the
neutron flux the quantities, must be deter
m,n,p
mined. Before we do so it is helpful to discuss
qualitatively the time behavior of the neutron popula
tion in the system. As the source pulse has a finite
width there is some time interval during which neutrons
are continuously fed into the system. Consequently,
the neutron population at a certain position and a
certain energy rises to reach a maximum and then decays.
The time at which this maximum is reached depends on
the amplitude, the time dependence and the duration of
the pulse. Furthermore, the neutron density may reach
an intermediate steady state depending on the pulse
shape, as in the case of a broad rectangular pulse, or
a pulse with a broad flat portion. For these reasons,
and since the negative time exponentials alone can
describe the time behavior of the neutron population
only in the asymptotic region but not in the transient
region, the quantities, n must be functions of
time. Hence, the flux expression of Eq. (4.36) becomes
79
I
m,n,p,k
Xm(x) Yn(y) Z (z) Jk (t) E* (e)
m,n,p in,n,p
(4.73)
where
Jk(t) = Ck(t) x Tk(t) (4.74)
m, n, p m, n, p m, n, p
Consider the flux and the adjoint flux equations,
Eos. (4.1) and (4.44). Operating on the first by the
operator
Q+ = / dr / de /t dt tji*(r,e,t) (4.75)
re o
and on the second by the operator
Q =
_J dr / de /* dt i>(?>e ,t)
r e o
(4.76)
and then subtracting the resulting equations, we
obtain the integrodifferential equation
80
[Q+H* QH+**] v^[Q + F QF+ **]
O 91 H
1 + 1 32 1 92 ,
% 2 7 at2 Q2 3t2 **3j CQ A2 ^ Q A2
s nf. 1 3 j
 Q+ __ X + Q+ GX
o /e at
(4.77)
where all the arguments have been dropped for simplicity.
Introducing Eqs. (4.45) and t4.46) into Eq. (4.77) and
integrating by parts we can prove that
Q+Hi(i QH + ^* s 0
(4.78)
Q+F L i>QF+  ** r/ dr / [ ^*Fip] de
at 91 e t
,+ 3
(4.79)
Q+ k JL *Q I JL >>*
e at2 v 912
3 3
/ dr / [** 7 gt F 9t **lt de
(4.80)
Q+V2\p Qvzip* =
(4.81)
where in this proof we have used the condition that the
flux is zero at t=o and utilized the invariance of the
81
expectation of the flux, i.e.,
/ dr / \}>(r,e,t) = 0
at r e
(4.82)
at any time. Eq. (4.77) takes the form
i/"] de+ /dr/ de
v_
(4.83)
Next, we will assume for the source the expression
x(r,e,t) = E (e) T (t) 6(x) 6(y) 6(z)
s s
(4.84)
i.e., we assume a point source at the origin with
separable energy and time dependence. At this stage
we introduce Eqs. (4.47), (4.73), (4.36) and (4.84)
into Eq. (4.83). Performing the spacial and energy
integrals in the resulting equation and considering
the orthogonal property of the spacial modes we arrive
at the equation
82
I I C+k T+1Ct) JL. ak*k
m,n,p k',k mnP mnp[vo2 mnP
_L_ Ak
at mnP
\
t l.k;k
vo mn,P
Jk(t)
m,n ,p
aEc m^n,p ^m,n "mjn'jpj m,n,p3t mnPj
,+k1
ft +k
/ T (t)
+ d
1
T (t)dt
where
(4.85)
Jk(t) = Ck(t) Tk(t) (4.86)
m,n,p m,n,p m,n,p
k Jk
J
+ k V
E (e)
m( e)
I Ek(e) de
mn,p
o
m,n,p
e m,n,p
k\k
CO
= J
+k' \
E (e)
m( e)
F(e) Ek(e)
m,n,p
o
mn P
m,n,p
k
,n,p
00
 /
o
E+k{e)
m,n,p
m( e )
1 E (e) de
/T 3
k
CO
0
+k*
)
* /
E (e)
m(e)
G(e) E ( e )
m ,n ,p
o
m,n ,p
b
(4.87)
(4.88)
<4.89)
(4.90)
Equation <4.85) must be satisfied at any time regardless
1
of what the constants, C are. Therefore, the
m,n,p
83
coefficients of C1^ in both sides of Eq. (4.85) must
m, n, p
be equal. Hence, the following set is obtained.
I T+k(t) [JL akk {JL xk' }+ _1_ bk*k ]jk(t)
k ran,p v 2 m,ntp 9t m,n,p vq m,n,p m,n,p
= l /t T+k(t) { ck + dk h (t) dt
Bc k m*nP v0 m*n*P 3t m,n,p s
(4.91)
This equation holds true for all the values of m,n,p
and k'. Hence it represents (M.N.P.K) differential
equations in (M.N.P.K) unknowns, Jk n where M,N,P
and K are the total numbers assigned to m,n,p and k (or
k1), respectively. Rearranging Eq. (4.91) and integrat
ing the righthand side by parts it becomes
k' 1
l T
k vc2
k'k
a
Ak akk}] Jk(t)
= [C ck T+k( t) T (t) + { dk Ak ck } / Tk(t)Tc(t)dt]
a5c k vo oS
(4.92)
84
where the indices, m,n,p, have been dropped for
simplicity, If Tq(t) is a known function the integral
o
in Eq. (4.92) can be evaluated and this set of equa
tions can be solved for the unknown functions, Jk(t).
Formally, this completes the solution for the
flux in the pulsed neutrons problem. To go into more
detail in the solution a specific case has to be
treated. Assume that the pulse takes a rectangular
shape as a function of time. Analytically, this pulse
can be given by
Ts(t) = u(t) u (t T),
(4.93)
where x is the width of the pulse and u(t) is the
heaviside unit step function. With the aid of Eq. (4.93)
it can be seen that the righthand side of Eq. (4.92) is
a function of time in the range 0 < t < T while it
becomes a constant in the range t 3 t Therefore,
this set must be solved in both regions, where it takes
the two forms
l T+k(t) for t < i
abc k4 ^k
(4.94)
(4.95)
85
Multiplying these two equations by vQ2 and dividing
them by T+k (t) and then using the value of the latter
they can be reduced to the forms
l [Akk JL + Bkk]Jk(t)
* I Pk
k
exp [
( Xk + Xk)t] t < T
(4.96)
exp(
Xk't) t > T
(4.97)
where
Ak
' k
' =
ak',k ,
[
(4.98)
Bk
' 'k =
v0 (t>k'' kA k' ak''k)
(4.99)
Pk =
(dk/Ak)
abc
(4 0100)
II
(dk/Akck) exp^) .
abd
(4.101)
Taking the Laplace transform of Eqs. (4.96) and (4.97)
one obtains
86
I [Akk s + Bkk]Jk(s)
k
= l pk / (s Ak Ak<) t < t (4.102)
= y Qk / (s xk') t T
K (4.103)
If these algebraic equations are solved for Jk(e) then
the inverse Laplace transform
Jk(t) = L"1 / {ty (4.104)
gives the sought for functions which complete the
solution.
In Chapter V we will discuss some of the numerical
results obtained for graphite and compare these results
with those obtained experimentally by Starr and Price
(33) .
CHAPTER V
RESULTS AND CONCLUSIONS
The theoretical analysis of the theory of
neutron thermalization in moderating media in the
consistent P.^ approximation was developed in Chapter
II. This analysis is tested in two applications of
practical interest, the neutron waves and pulsed
neutron experiments as presented in Chapters III
and IV. In both applications the heavy gas scatter
ing kernel and the 1/v absorption cross sections are
used, and the first order associated Laguerre poly
nomials are utilized to express the energy dependence
of the flux. All the results presented in this
chapter are obtained for AGOT type graphite with
density equal to 1.67 gm/cm3.
Application to Neutron Waves Technique
The experimental arrangement under consideration
consists of a parallelopiped block of graphite with a
sinusoidally modulated plane neutron source at its xy
face. This source is assumed to have a Maxwellian
distribution in energy and a cosine shape in the x
and ydirections. A 1/v neutron detector is used to
87
88
measure the neutron density at various distances from
the source along the central zaxis of the graphite
assembly. The experimental technique and the data
analysis are fully discussed by Booth (41) who obtained
the amplitude and phase shift of the neutron density
at various positions in the graphite assembly for
several source frequencies. The errors in all Booth's
preliminary results used here are typically of the
order of 3 per cent.
A code, NWP, has been developed for the IBM709
to calculate all the quantities involved in this dis
cussion. A listing of this code and the subroutines
associated with it is found in Appendix D. In all the
computations the fundamental spacial mode alone was
considered. Therefore, the spacial indices are dropped
from all the quantities involved in this presentation.
The computational scheme follows the analytical steps
of Chapter III and the perturbation technique is used
to solve the eigenvalue equation, Eq. (3.35).
The use of Laguerre polynomials in the expansion
of the energy flux modes, E (e), leads to a number of
eigenvalues equal to the number of polynomials in this
expansion. Each eigenvalue, r^, is related to the
inverse relaxation length, pk through Eq. (3.36).
I
89 1
These eigenvalues were calculated using numbers of
polynomials ranging from 1 to 10. First one polynomial
is used and the fundamental eigenvalue, rQ# is obtained.
Then the number of polynomials is increased by 1 in
each step and the next higher order eigenvalue arises.
It was found that the use of L+l polynomials does not
change the L unperturbed eigenvalues obtained using L
polynomials, but merely gives rise to the L+lth eigen
value which is greater than all the previous ones.
Furthermore, the difference between two eigenvalues
of successive orders decreases with increasing orders
as seen in Fig. 1. Therefore, it is expected that the
highest ordered eigenvalues tend to be so closely
spaced that their distribution may approach a continuum.
The frequencydependent complex inverse
relaxation length, p of the fundamental energy mode is
plotted vs. source frequency in Fig. 2 and is compared
with the experimental values obtained by Booth. Based
on the analytical development of the neutron waves
problem in the diffusion equation by Perez and Uhrig (38)
calculation was made in the heavy gas and heavy crystal
models. The matrix elements for the later model were
obtained from Razminas (51) and the results are also
presented in Fig. 2. A very reasonable agreement
Inverse Relaxation Length
90
[
123456789 10
Number of Laguerre Polynomials, L
Fig. 1. The Steady State Inverse Relaxation Length
vs. Number of Laguerre Polynomials
O 100 200 300 400 500 600 700 800 900 1000
Source Frequency (cps)
Fig. 2. Components of the Complex Inverse Relaxation Length of the
Fundamental Mode vs. Source Frequency.
92
between the theoretical results of the approximation
and the experiment is obtained for frequencies up to
500 cps. For higher frequencies the theoretical curves
diverge from the experimental values giving a smaller
real component and a greater imaginary component of po,o
leading to less attenuation and greater phase shift,
respectively. I
A more appropriate comparison may be made between
the calculated and the experimental total neutron densi
ties. In Fig. 3 and Fig. 4 the amplitude of the neutron
density is plotted as a function of position for source
frequencies of 100 and 500 cycles per second. In both
cases the theoretical results agree very well with the
\
experimental values up to about 25 cm from the source.
The disagreement beyond this position becomes more
pronounced where the effect of the higher modes dis
appears, due to their fast attenuation, and the
fundamental mode dominates. This disagreement may be
reduced by including more modes in the combined solution
since the higher modes tend to raise the amplitude near
the source and hence increase the slope of the theoreti
cal curve toward that of the experimentally measured
one.
The theoretical phase shift of the neutron
density is in a much better agreement with the
Amplitude
Fig. 3. Amplitude of the Combined Neutron Density vs. Position along the
zaxis for Source Frequency of 100 cps.
Amplitude
r
Fig. 4. Amplitude of the Combined Neutron Density vs. Position along the
zaxis for Source Frequency of 500 cps.
95
experimental phase shift as seen from Fig. 5. It is
noted that in Fig. 5 all phase shifts are calculated
relative to the position at 4.89 cm which is the
closest experimental point to the source obtained by
Booth.
The energy spectrum of the total neutron
density at several positions was calculated and is
plotted in Fig. 6. Near the source the spectrum is
rather "cool." As the position of observation moves
away from the source into the system the spectrum
tends to approach the Maxwellian distribution. The
behavior of the spectrum in this fashion is justified
by the fact that, near the boundary of the system,
fast neutrons have a leakage probability through the
outer boundaries greater than that of the slow neu
trons .
Application to Pulsed Neutrons Technique
The asymptotic eigenvalues, of the neutron
flux in the P^ heavy gas approximation were calculated
for graphite using the DETEX routine to expand the
determinant of Eq. (4. 29) and the POLY routine which
solves the resulting equation. The fundamental time
decay constant, Xg = Ag vQ, is plotted against buckling
in Fig. 7. These results are compared with the experi
mentally measured values obtained by Starr and Price (33).
Phase Shift (rad.
r
Fig. 5. Phase Shift of Neutron Density vs. Distance from the Source.
Number of Neutrons per Unit
97
Dimensionless Energy Variable, e=E/T
Neutron Spectra at Various Positions along the
Central Axis of the Graphite Assembly.
Fig. 6.
Decay Constant, X, (sec
Fig. 7. Decay Constant, X of the Fundamental Mode of the Flux
in the Neutronpulsed Graphite System.
99
For large systems, small geometric buckling, the
theoretical results are very close to the experimental
ones. The range of agreement between the analytical
and experimental results may be considered up to
3 2
geometric buckling of the order of 3x10 cm
Calculations have been made in the diffusion
approximation (38) in the heavy gas model. The results
of these calculations almost coincide with those of
the P1 approximation throughout the range of buckling
included in Fig. 7. Therefore, the deviation of the
P1 results from the experimental values in this range
cannot be explained by an overcorrection which may
arise from the transport correction hidden in the
second order time derivative of the flux. However,
this overcorrection does exist, and becomes appreci
able in very small systems, i.e., for bucklings of
1 2
the order of 10 cm
Conclusions
The time, energy and spacedependent neutron
thermalization theory in moderating assemblies has
been formulated in the consistent P^ approximation.
This formalism was based on the assumption that the
nonMaxwellian component, >l>(r,E,t), of the flux is a
smooth function of the variable neutron energy, E, so that
100
\(r,E',t) can be well approximated by a Taylor series
expansion about E. This assumption seems reasonable
because the flux energy distribution in moderating
assemblies is very similar in shape to the Maxwellian
distribution and hence the component, iii(Â£,E,t)^ is not
expected to have sharp variation in E. This assump
tion was essential in order to evaluate the scattering
integral in the Boltzmann equation.
This theory was applied to describe the neutron
waves and the pulsed neutrons experiments. In both
applications the theoretical analysis was general from
the standpoint that it made no specific reference to
what types of scattering kernel and absorption cross
section should be used. The analysis in both cases
used the associated Laguerre polynomials to formulate
the energy dependence of the flux where the solutions
led essentially to eigenvalue problems.
The validity of the heavy gas model and its
compatibility with the approximation of the Boltz
mann equation were tested in the numerical solutions
obtained for these two applications. From the results
that have been presented here it can be concluded that
the heavy gas model in the P^ approximation can be used
in studying the pulsed neutrons experiment in rela
tively large moderating assemblies with bucklings up
101
3 2
to about 3x10 era In the neutron waves study the
heavy gas model in the P^ approximation can describe
the experiment with source frequencies up to about
500 cycles per second, or equivalently, with time
variations of periods down to about 2 milliseconds.
These results have interesting practical
repercussions in that if one is only interested in
the slow changing regimes of nuclear systems, the P^
heavy gas kernel approximation is proven to be fully
satisfactory. Then it does not pay off to go to more
sophisticated kernels unless the emphasis lies in
fast transients, in which case the flux distributions
become sensitive to the detailed behavior of the
scattering kernel according to Ohanian (32) .
This conclusion both confirms and extends the
results of de Sobrino (17) who concluded that in the
computation of the steady state regime of nuclear
reactors the heavy gas kernel approximation (Wilkins
Equation) was conveniently accurate.
Finally, it would be of much interest to
investigate the theory developed in this work using a
more sophisticated scattering kernel in order to
enlarge the applicability of this theory to fast
transients. Steps in this direction are being taken
by the author.
APPENDIX A
OPERATORS IN GENERAL FORMS
This appendix is devoted to the derivation of the
operators, R^E, t) which appear in Eq. (2.50) and H(E),
F(E) and G(E) which appear in Eq. (2.54). Then these
operators will be transformed into their equivalent forms
in g domain as they appear in Eq. '(2.61). The develop
ment of these operators in this appendix will not specify
the model of the energy dependent absorption and scatter
ing kernels. Therefore, the results can be used in any
model one would like to adopt in any specific application.
However, the development of these operators assume the
representation of the flux component by a three
term Taylor expansion about E. For consistency, there
fore, terms with derivatives up to the second order only
are kept in this development.
For convenience, the equations defining the
operators are rewritten here.
Rr^(E,t) = Qm(E) E .(E) (A. 1)
mt V(E) 3t
Qm(E) = Mm(E) + Mjj^(E) D + m2(E) D2 (A.2)
102
103
mJÂ¡(E) = Â¡4E' E)k E
kl E' m
The energy moments, (E), can be calculated only when
the energy dependence of the scattering kernels,
^m(E' * E} is known. Therefore, and since we desire
to develop the general forms of the operators, these
forms will explicitly contain the energy moments, M^E).
Since Rq(E,t) appears only in the product,
(E,t]Rq(E,t), it will not be calculated separately.
Consider the two equations resulting from using
m=0 and m=l in Eq. (A.l) .
R lE,t) = 0o(Er Z. (E) 2 (A.4)
t V(E) 3t
R,(E,t) Qi(E) E (E)  (A.5)
X v (E) 31
If Eq. (A.4) is premultiplied by Eq. (A.5} the resulting
equation and Eq. (A.5) can be expressed in the forms
1 3 1 3 2
Ri(E,t) R0(E,t) = H
1 v(E> at v2(E} at2
R. (E, t) = G(E)  L (A.7)
v(e) at
104
where
H(E) = ^(E) + QjlEJQqE) 0L(E)Zt(E) Zt(E) Q0(E)
(Pi. 8)
F(E) = 2Et(E) vEJQj^E) v1(E) Qq(E)
(A.9)
G^E) = Zt(E) QX(E)
(A. 10)
Notice that all the operators do not commute due to ,the
energy derivatives involved in them. See Eq. (A.2).
Using Eq. (A.2) in Eqs.(A.8), (A.9) and (A. 10), and
grouping the coefficients of the energy derivatives, we
can express the operators H(E), F(E) and G(E) in the
following equations:
H(Ei = h (E) + hx(E) 5 + h2{E) (A. 11)
3E 3 E2
3E 3 E2
(A.12)
(A.13)
where the functions h^(E), f^E) and g^fE) are found to be
105
ho(E) E 2(E) + CM(B) M(E)] S (E)
o a. oia
i d o d
 M(E) S (E) M 2(E) E (E)
1 dE a L 21
(A. 14)
h, (E) = C M^(E) + m1(E)] Z (E) 2M 2 (E) Z (E)
x u 1 a 1 dE a
+ M^(E) [M(E) M^E)] + M}(E) M (E)
1 1 dE
+ M12(E) 2 *Â£<*)
(A.15)
h2(E) = [ Mq2(E) + M12(E)3 T a (E) + mJ(E) mÂ£(E)
+ Mq2(E) r mJ(E) M(E) ] + Mj(E) M02(E)
O d 1 n d2 O
+ 2M 2(E) M"" (E) + M 2(E) M 2 (E) (A. 16)
1 dE o 1 dE2
f 0 (E) = 2la(E) + M (E) M(E) + ~ m];(E) M^CE)
X Je* x
(A. 17)
f1(E) = M(E> M*(E) + ^ M12(E)
(A.18)
106
f2(E) = Mq2(e) M12{EJ
(A.19)
g0(E) = ia(E) + Mg(E) M(E)
(A.20)
gL (E) = mJ(e)
(A.21)
g2(E) = Ml2(E)
(A. 22)
In all the algebra used in the above derivation we
neglected all derivatives of the third order and above,
in consistency with keeping up to second order deriva
tives in the Taylor series expansion of the flux. We
have also used the relation
I (EV = (E)
s
(A.23)
Thus, we have calculated the operators and put them in
general forms as functions of energy, cross sections and
energy moments.
In transforming Eq. (2.54) from the energy domain
to the domain of the dimensionless variable
e
E/T
(A.24)
107
the ith energy derivative gives rise to the factor (1/T)^
according to the relation
_3_ 2_ iL 1 i_ (A.25)
3E 3c 3 E T 3e
The transformed velocity gives rise to the term eV2
according to
v (E)
voe
(A.26)
The transformed flux equation was given by Eq. (2.61)
where the transformed operators are
1 3 i g
H(e) = H(E) = h0(E) + hx(E) + h2(E) (A.27)
3 e T 3 e
F(tÂ¡ =F(E) =lto(E) +1 +lj f2(E) lj]
/c /e 9e 1 3 c
(A.28)
2
G(e) = G(E) = g0(E) + g.(Er + \ 92(E) i (A.29)
T 1 3eT 2 2
_ 1/2
Notice that the e term, which arises from the
transformation of l/v(E) preceding the operator, F(E),
in Eq. (2.54), is now included in the operator F(e). If
we express the transformed operators by
108
H(e) = h0(c) + + h2(e)
3 G
F(e) = fQ(e) + fj^e) + f2(Â£)
3e
g(Â£) = g0(Â£) + gi(e) + g2ie)
Be
the functions (e ) f^(e) and g^ (e )
hi{c) = hi(E} /T1
f {t) = fi(E) /T
fz
gi ie) = g(E) /t1
3
(A.30)
3e2
32
(A.31)
3c2
2
_ 2
(A.32)
3e
are then given by
(A. 33)
(A.34}
(A.35)
where h^iE), f^fE) and g^iE) have been given in Eqs.
(A .14)(A.22) .
The results obtained so far are general,
regardless of what kernels we use. In Appendix B, we
will adopt the heavy gas model for the scattering kernel
and 1/v for the absorption cross section. We will
derive the energy moments, m(E), and then calculate all
the functions needed.
APPENDIX B
HEAVY GAS 1/v MODEL
In this appendix the heavy gas model is adopted
for the scattering kernel and the absorption cross
section is considered to be of the 1/v type On this
basis, then, the energy moments, m(E)^, and the
functions h^tE) f^CE) and g^E) will be calculated.
Heavy Gas Scattering Kernels
In,this rather idealized model the moderator is
treated as a monatomic gas whose atoms are heavy com
pared to neutrons. The advantages of using this model
are that it offers an analytical expression for the
scattering kernel and "it bypasses the complex con
siderations of neutron interactions with chemically
bound atoms, but retains much of the essential physics
of the thermalization process," according to Hurwitz,
Nelkin and Habetler (15). "It does not include,
however, the effect of 'jumping' across a narrow
resonance."
Based on the treatment of this model in (15)
the scattering kernels for the Pn approximations can
be given by
109
110
I (E * E 1 }
m
where
/"G(E,E' ,y ,t)
00
jt(E'E)
e
dt
(B.
G (E E V fe)
exp uj(jt t2T)
4tt
E' + E 2xXEE'}^ >]
(B.2>
The quantities involved in Eq. (B.l) and Eq. (B.2) are
defined as follows:
E = neutron energy before a scattering
collision
E' = neutron energy after a scattering
collision
x = cos (n, n')
3 = neutron direction before collision
= neutron direction after collision
0o = 4ira = bound atom cross section
a = scattering length
T = neutron energy corresponding to the
most probabie speed
v* = neutron to moderator mass ratio.
Assuming a twoterm Taylor expansion of the function
G(E,E', P ,t), i.e., neglecting terms in p and higher,
the kernels are then given by
Ill
o (E E 1 )
m
4ir
f1 xm dx
/ exp [j t (E '
 E}]
[1 + P { E'
+ E 2x (EE 1 ) :
} (jt t2T) ]
(B.3)
By integrating this equation for m=0 and m=l and using
the relations
1 n
 / (jt)n exp(jwt) dt = pjy 6 (w) = (w) (.4)
z T* a, dw
A = A E = E'E (B.5)
we obtain the P.^ scattering moments
1
E'
ao(E E1 ) = aQ
' \2
E
Cfi (A } + u (E+E) 6 ^ (A )+vT(E'+E)5 ^ (A)]
(B.6)
E' [6 (A) + T 6 (A )]
(B.7)
Moments of Scattering Cross Sections
The energydependent moments of the scattering
kernel are given by
mE) = I m(E + E') dE
E'
(B.8)
112
These can be obtained simply by integrating Eqs. (B.6)
and (B.7) with respect to E', using the relations
dE' = dA
(B.9)
/ f (w)
o
6^n^tw)dw = (l)n
w=o
(B.10)
The integration leads to
0O(E) = <1 2y + yT / 2B)oq
(B.ll)
(^(E) = (2y/3) aQ
(B.12)
These are
kernel.
obtained
by N, the
the microscopic moments of the scattering
The corresponding macroscopic moments are
simply by multiplying Eq. {B.ll) and Eq. (B.12)
number of atoms in one cubic centimeter.
Z0(E) = (1 2y + yT /2E)Eq
(B.13)
E1 (E) = = (2V3)Eq
(B.14)
l
113
Energy Moments
The energy moments defined by Eq. (2.35) are
/ (E' E)k
E'
Z (E * E ) dE '
m
or
mÂ£(E) = / Ak Im(E> E' ) dE'
kl E'
(B.15)
(B.16)
Introducing Eqs. (B.6) and (B.7) into Eq. (B.16) and
using the property in Equ (B.10), we obtain
M(E) = (1 2U + vT/2 E) ZQ (B.17)
M(E> = 2\i (2T E) l0 (B.18)
Mq(E) = 2 y T E z0 (B.19)
mÂ£(E) = 0 for k > 2 (B.20)
and
M(E) = (2u/3) ZQ (B.21)
M^{E) = (2 w/3) (B 2T) Zq
(B. 22 )
114
M^(E) = (2g/3) TE lQ
(B. 23 )
M*(E) = O
for k > 2
(B.24)
Notice that the moments, M^(E) are zero for all values
of k greater than 2. Therefore, the Taylor expansion of
the flux, Eq. (2.29), contains three terms only, i.e.,
up to second order derivatives, since the coefficients
of higher derivatives are identically zero in this case.
See Eq. (2.34).
The Functions, h^CE), f,. (E) and g^ (E)
Having found the energy moments we can calculate
these functions in the heavy gas model for a 1/v type
adsorption cross section.
T k
(B.25)
o
With rather lengthy but straightforward algebra, the
combination of Eqs (B 17) (B 25) and Eqs (A. 14) (A. 22)
of Appendix A leads to these equations:
(B.26)
115
(N O
IN
Ei
a
(N
co
F1
+
HCN
tTPw
Eh
O
id
qu
EhM
cn O
in
Eh
IN
a
IN
I
w
*^
A
IN
m
WEh
CM o
in
Eh
f"
a
(N
+
rHlN
wTeh
O
id
in
O
in
Eh
a
 O
+
>=*
. .
00
IT>
IN
IN
m
CQ
vv*
CN
W 1 Eh
CN 0
IN
CN
W
Eh
o
IN
CN 0
IN
IN
a
IN
Eh
'siln
M c
IN
l
1
a
rH
CT
WEh
r
1
r( IN
IN 0
+
IN
rHlN
WEH
id
IN
CN
Eh
a
l
I!
<'**%
W
IN
IN
Eh
a
CM
IN
a
^Iro
Eh I M
id
in
IN
Eh M
IN
a
11 IN
II
w'
''
c
X
. ,
n
O
H
co
Ht
CO
co
co
co
CO
CQ
CQ
5.
fi
CQ
V
0
IN
a
00 1 CO
, s
1
MEh
v /
i1
V*
0
IN
+
Eh
W
a
Eh W
0
^l ro
/
0
W
W  Eh
id
+
IN
0
0
+
CN  ro
IN
IN
1
W
Eh
CN
eTw
o
0
a
Eh
a
w
IN
O I _
IN
E*
Eh
iH 1 ^
^fco
a
a
1
1
r  CN
^co
IN  co
II
II
II
1!
II
.
^51
w
M
w
W
w
v^*
H
IN
o
H
CN
IH
Cn
CT
Cn
116
The Functions, h^e), f^CO and g^Ce)
These functions are obtained by direct substitution
of Eqs. (B .26) (B .34) in the transformation given in Eqs.
(A.33) (A.35) .
hJe5 = in E0 I a e_3/2 + 2 a e "1 + (1 Z y ^ 0 1 a e ~1/2
(B.35)
hi!0 = 2us2e1 .a, EoEaoel/2 + (4 u 2 4Â¡Â¡) E 2
+ 4,Io!;a Â£ 1/2 + 2U(1 ) r 2 c (B.36)
J o 3
h2 (e) = E0 E e1/2 9 y2 E2 + y 2 2y ) E 2 e
 j y2 E e2 (B 37)
f0U) = j y e3/2 + 2 1&q e"1 + (12d) Z0 e1/2 (b.38)
f1(e) = H y E Q e'1/2 + y E Q e 1/2 (B.39)
f2(e) = ju E0 eV2
(B.40)
117
(B.41)
gL(e)
(B.42)
g2(e)
(B.43)
APPENDIX C
CALCULATION OF THE MATRICES
^ i Â£ # ^ Â£ Â£ 11 Â£ Â£ Â£ i Â£ j Â£ Â£ Â£
In solving the modal equation of the flux for
neutron waves, Eq. (2.35}, and for the pulsed neutrons,
Eq. (4.22), the functions, E (e) and E (e) are
m, n iu/ n / p
expanded in terms of the normalized associated Laguerre
polynomials of the first order, L^(e). This gave
Â£
rise to the matrices, a . 3 . y and n / defined
in Eqs (3.28)(3.31). The application of the source
condition of Eq. (3.94) gave rise to two other matrices,
and 0l, defined in Eqs. (3.108) and (3.109).
Instead of treating each of these matrices
separately, we will find a general form which can be
applied to all of them and then treat this form in some
detail. This will cut down the amount of work in hand
ling the mathematics involved in all the computational
processes. The examination of these matrices and opera
tors involved in them [see Eqs. (A.30)(A.32)], enables
us to express these matrices in the form
V,,, f U> I^fe) AU> dE (C.l)
Â£ Â£ Â£
118
119
where v stands for a 0 ... and the operator
i \ l l Z i i
A(c) = ao(c) + a^e) + a2 (e) jf (C.2)
stands for H(e), F (e), ... The functions, a^(e),
which correspond to h^e), f^iejr, ..., [see Eq. {B.35)
(B.43)] can be given the form
a^e)
a. .
i#3
(j4)/2
G
(C. 3)
The reason for this notation is that j=l corresponds
to the smallest power of g, 3/2, that appears in the
functions, hu (c K f^Cc), ... The combination of Eqs.
(C.l), (C.2) and (C.3) leads to the general expression
10
l z
1 tao,ju2;
3=1
l + al,jV' + a2,jWtÂ¡l 1
(C.4)
where we define
IT
l \ l
r (j4)/2 (!) (1)
J e m(e) 1/,'(g) L^1'(e) dc
o
(C. 5)
V
Z)Z
/ e
o
(j4)/2 (l) d (!)
m(e) 1* (e) L (e)
Z' de j.
WD
Z\Z
r (j4)/2 (i)
/ e m(e) L (Â£)
l (1)
dÂ£2 h (a) de
(C. 7)
120
It remains now to calculate the matrices, U^. V"*
and WJ. This will be done after we discuss some of
l \ l
the essential properties of Laguerre polynomials.
The nonnormalized associated Laguerre
polynomials, ^(x), are related to the hypergeometric
functions, F(mnx), by the equation
(x) = F (m  n+1  x) [ (m + n) 1] ^ / ml nl (C.8)
where
F (m n x)
1 +
m
n 11
+ m(m+ 1) X2 +
n (n + 1) 21
m (m + 1) (m + 2 ) ... fm + p 1} xP
1 1 r t i i 11 1 f
n (n + 1) (n + 2) ... (n + p 1) p
n > 0 (C.9)
[See (52).] Using these relations we will develop a new
recursion relation connecting the Laguerre polynomials
of different orders. From Eq. (C.8)
n1 n
Ji (x) +// (x) = n F(m I n I x) + m F (m +1 I n+1 I x)
m ml 11 11
2 2
. [ (m + n) 1] / ml nl (m + n)
(C.10)
121
The use of Eq. (C.9) gives
nF{mnx) + m F (m+l n+11 x) =
(m+n) +
m + JEkHtii
n + 1 .
m(nHl) + m(m+l) (m+2)1
. n+1 (n+1) (n+2) J 21
m(m+l) (m+2) m(m+l) (m+2) (~m+3)
+
^n+1)(n+2)
(n+1) (n+2) (n+3)
31
(C.ll)
If the algebra is carried out, this relation takes the
form
nF(mnx) + m F (m+l n+11 x) = (m+n) F{mn+lÂ¡x) (C.12)
Introducing Eq. (C.12) in Eq. (C.10) one obtains
/n (x) +m = F(mn+lx) j(m+n) ij / ml nl (m+n)
(C.13)
Finally, a simple comparison of the righthand side of
Eqs. (C.8) and (C.13) yields the important recursion
relation
n
(x) = (m + n;
m
n1(x) + Ax)
L m m~*
(C. 14)
122
n
Due to the fact that X (x) does not exist for negative
in
indices and is unity for m=0 and n=0, we can reform kq.
(C.14), using the Kronecker delta function and Heavi
side unit step function, where it takes the final form
. n
X (x) = <5 6 + (m+n)
m / m,o n,o v
u (n1) X,n 
m
(C.15)
Three other important properties of the Laguerre
polynomials are given by the relations (53)
dp n +
jC (*) = (DP u(mp)/n P(x)
dxp m mp
n m+1 n 5 n .
(n+2m+lx) X (x) = L (x) + u(ml)(m+n) Â£ (x)
m m+n+1 m+1 ra1
(C.17)
CO 2
/ xn ex/n(x) dx = 6 [r (m+n+1)] / r (m+1)
o m' m, m
(C.18)
This last relation is the socalled orthogonal property.
If we define the orthonormal polynomials, L^(x) to
satisfy the relation
^ xn ex L (x) L (x) dx = 6 i
m m' m,m
o
(C.19)
123
then
_ n 1/2 3/2
L (x) = >/, (x) {m+iy] [r (m+n+1)] (C.20)
m m
The introduction of Eq. (C.20) in Eqs. (C.15),
(C.16) and {C.17) leads to the relations
1/2 1/2
Lm (xi = () u (nl)L^ (x) + (} u {m1) i/* (x)
m nri n m v m+n' ra l
+ 6 6
ra, o n, o
(C. 21)
Lm^x' = i_1'P u(mP>
dxp
T (m+1)
.r(m+lp)J
1/2
Ln+P(x!
mp ^
(C.22^
1/2
^m(m+n)j L^(x) ~ (n+2mlx) u(ml) L^_1(xj
T 1V2 n
 ^(m1) (m+n1) J u(m2) Lm_2(x) (C.23)
Suitable combination of Eqs. (C.21) and {C.22\
leads to the important relation
dx*
n / m \l/2 fdP a
L (x) = u(ml) u(nl) 
m 'm+n 1 LdxP hvP1
P1 i
Ln (C.24)
m1
124
Upon the substitution from Eq. (C.24) in Eqs.
(C.6) and (C.7) we find the recursion relations
VJ
Â£ J Â£
wJ
Â£ J Â£
() u (Â£1) IV UJ 1
^tl^ [ Â£ J Â£1 Â£ J Â£lJ
(Vs u (Â£1) iwJ VJ ]
li+lj L 4}tl Â£ J Â£1J
(C.25)
(C.26)
Thus the computation of VJ and VT1 becomes very
Â£ Â£ Â£ Â£
straightforward once we compute the matrix, To
Xr
calculate this matrix we will write the polynomials in
the form
L(1: (e)
Â£
r yi i
i=o *
(C.27)
Then,
U
Â£ J Â£
l l b^ /* Â£i' + t+(j4)/2m(E)d(
i'=o i=o
(C.28)
or
u
Â£ J Â£
i' i j+2i'+ 2i4
I I b > r( )
4C.29)
i'=o i=o
where r(x) is the wellknown gamma function
12 5
Having calculated VJ and
l\l
the
matrices, ljl# 6*,*, yl)lt V,i' and
follow immediately from the general form, Eq. (C.4), using
the proper functions, a Appendix D contains a list
i # J
ing of the FORTRAN Program, HGM, developed for all these
calculations.
i
APPENDIX D
COMPUTING CUUES
1HE SE CODES WERE UEVhLUPED FUR THE IBM709 COMPUTER.
Tut MAIN CODE IS THE NWP (NEUTRON WAVES PKOPOGATION) WHICH
WAS DEVELUPED BY THE AUTHUR FUR THE NUMERICAL COMPUTA IIUNS
OF I HE NEUTRON WAVES PROBLEM PRESENTED IN THIS WORK. IHIS
CODE HANDLES THE COMPUTATIONS REGARDLESS UF THE NATURE OF
THE SCATTERING KERNEL AND THE ABSORPTION CROSS SECTION OF
THE HOMOGENEOUS MODERATING ASSEMBLY. THE PRINCIPAL QUANTITIES
THAT ARE CALCULATED BY THIS PROGRAM ARE LISTED BELOW.
1STtADY SIATE EIGENVALUES AND INVERSE RELAXATION
LENGTH FJK EACH MODE OF THE FLUX
2FREQUENCY DEPENDENT EIGENVALUES AND INVERSE RELAXATION
LENGTH FUR EACH MODE
3SIEADY STAIL AND FREQUENCY DEPENDENT AMPLITUDE AND
PHASt SHIFT FUR EACH MODE OF THE FLUX OR THE DENSITY
AT THE DESIRED PUSITIONS
AmMPLITUDE AND PHASE SHIFT OF THE TUTAL NEUTRON FLUX
OR NEUTRON DENSITY VS. POSITION AND FREQUENCY
5MODAL AND TuTAL ENERGY SPECTRUM OF FLUX AND DENSITY
VS.POSITION AND FREQUENCY
THE INPUT DATA MATRICES OF THIS CODE ARE COMPUTED BY
THE SECOND CODE HGM (HEAVY GAS MATRICES). THIS CODE FuLLOWS
Tub ANALYTICAL STEPS OF APPENDIX C.
THE USAGE OF THESE CODES AND THE SUBROUTINES ASSOCIATED
WITH THEM IS cXPLAINED ONLY BRIEFLY. IN BuTH PROGRAMS THE
FOLLOW ING NOTATIONS ARE USED.
RP = REAL PART
IP IMAGINARY PART
PARI i
NWP AND ITS SUBROUTINES
1 INPUT PARAMETERS
FORMAI(5E14.6) IS USED FUR ALL FLOATING PulNT VARIABLES.
FORMA I (101 T) IS USED FUR ALL FIXED POINT VARIABLES
L = NUMBER uF LAGUERRc POLYNOMIALS
NUM = iORULK OF PERTURBATION
BUCK = TRANSVERSE BUCKLING
ni = number of energy points in spectrum umputatiun
NO = 0 FOR FLUX SPECTRUM
= i FOR DENSITY SPECTRUM
11 NT = INTLRVAL BETWEEN EACH TWO SUCCESSIVE ENERGY POINTS
ZDIC = A POSITIVE UR NEGATIVE QUANTITY TO DECIDE WHETHER OR
126
127
NOT SPACIAL COMPUTATIONS ARE DESIRED,RESPECTI VELY.
NZ = NUMBER OF PuSITHINS ON THE ZAXIS AT WHICH COMPUTATION
IS DESIRED
POINT(J) = THE ZVALUCS OF THESE POSITIONS
ALPHA,BETA,GAMMA,EI A,THETA AND ZETA ARE THE INPUT MATRICES
WHICH,IN THE CASE OF THE HEAVY GAS MODEL,ARE OBTAINED
FROM THE HGM CODE.
S ( M, N I = COEFFICIENT OF THE (N1ITH POWER OF E IN THE
(MI)TH LAGUERRE POLYNOMIAL OF THE FIRST ORDER
OMEGA = SOURCE FRCOUENCY IN CYCLES PER SECOND
THE DESIRED VALUES OF FREQUENCY ARE PUNCHED EACH ON
A StPERATE CARL) AND ARE RE AD ONE AT A TIME ACCORDING TO
STATEMENT NO. 210 SEE THE ( GO TO 210 ) STATEMENT AT THE
END OF THE NWP CODE.
2 UUI PUT VARIABLES
XS(K) = RP OF STEADY STATE EIGENVALUE UF THE (KITH MODE
YS(K) = IP UF STEADY STATE EIGENVALUE OF THE (KITH MODE
XlK,NU I = RP OF
Y(K,NU I = IP OF
RR(K,M,NU) = RP OF R
HI (K,M,NU I = IP OF R
X E V ( K I = RP OF
YCV(KI = IP OF
RHOR(K) = RP OF
RHOI(KI = IP OF
RRTKI(K) = RHOR(K)*RHOI(KI
R2MI21K) = RHOR(K)**2RH0l(K)**2
BR(K) = RP OF C
Ei 1 { K ) = IP OF C
XR ( K M ) = RP OF R
YR(K,M ) = IP UF R
AR(K,MI = RP OF C .R
A I (K MI = IP OF C .R
DAMP(KI AND OPHASC(K) =AMPLITUDE AND PHASE SHIFT,RELAI IVE
10 THE SOURCE ,CF THE (K) Th MODE OF NEUTRON DENSITY
TAMP TOTAL AMPLITUDE OF THE NEUTRON DENSITY
TUTW =TOTAL AMPLIODE OF THE NEUTRON DENSITY
NORMALIZED TO 1 AT ZERO FREQUENCY
TOTZ =TOTAL AMPLITUDE OF NEUTRON DENSITY
NORMALIZED 0 1 AT THE SOURCE
FAMP(K,NI AND FPHASÂ£(K,N) =AMPLI TUDE AND PHASE EHFT OF THE
(K) TH MODE AT THE (N) TH ENERGY POINT. THIS
APPLIES FOR BOTH FLUX AND DENSITY.(THE NUMBER ND
DICIDES WHICH ONL IS COMPUTED)
LAST OuLUMN In THE SPECTRUM OUTPUT CORRESPONDS TO THE TOTAL
FLUX OR DENSITY SPECTRUM,DEPEND ING ON THE VALUE UF ND.
i DEiCX
THIS SUBROUTINE EXPANDS A DETERMINANT OF SIZE LSD THE
GENERAL ELEMENT OF WHICH IS A POLYNOMIAL OF SIZE NSE IN
THE UNKNOWN EIGENVALUE, THE RESULT OF THIS EXPANSION
128
IS A POLYNOMIAL IN OF SIZE (NSE1 )*LSD+1. THE SIZE OF
A POLYNOMIAL MEANS HERE THE NUMBER OF TERMS IN THE
POLYNOMIAL,THAT IS,THE MAXIMUM POWER +1.
LSD =SIZE OF DETERMINANT
NSE = S l ZE UF ITS GENERAL ELEMENT ,MAXIMUM POWER +1.
DD(M,N,J) ^COEFFICIENT OF THE (Jl)TH POWER IN THE
lM,N)TH ELEMENT OF THE DETERMINANT.
AIK) ^COEFFICIENT OF THE (Ki)TH POWER IN THE RESULTING
EXPANSION. J=i,2...(NSEl)*LSD*I .
4EVADET
THIS SURUINE IS A PART OF THE DETEX CODE .
IT EVALUATES A DETERMINANT OF CONSTANT ELEMENTS.
LD = THE SIZE UF THE DETERMINANT
CD(M,N) = THE (M,N)TH ELEMENT OF THE DETERMINANT
DEI = THE VALUE OF THE DETERMINANT
sPoly
THIS SUBROUTINE FINDS THE ROOTS UF THE POLYNOMIAL
RESULTING FROM THE DETEX ROUTINE.
Nl =MAXIMUM POWER IN THE POLYNOMIAL
BUt(K) ^COEFFICIENT OF THE (KI)TH POWER, AIK) IN DETEX.
ROTRIJ) = RP OF THE (J)TH ROOT
ROOT I IJ) =IP UF THE IJ)TH ROOT
PART II
HGM AND ITS SUBROUTINES
THIS CODE COMPUTES THE INPUT MATRICES UF THE NWP CODE.
THE FIRST PUR I I ON UF THIS CODE IS MERELY A DIRECTORY PROGRAM
FOR THE SUBROUTINES WHICH FORM ITS ACTIVE PART.
THE UUIPUT OF THIS PROGRAM IS SELFEXPLANATORY. THEREFORE,
ONLY THE INPUT PARAMETERS ARE EXPLAINED.
L =NUMBER UF LAGUERRE POLYNOMIALS (SIZE OF MATRICES)
ss =
SA ^
UM =
NS =0 IF THE SMALLEST POWER OF E IN THE FUNCTIONS
IS 3/2
2 IF THE SMALLEST POWER OF E IN THE FUNCTIONS
IS 1/2
NCV =0 IF
= 1 IF
IS ENERGY INDEPENDENT
IS ENERGY DEPENDENT
n o o
129
MAIN PRUGRAM NWP (NEUTRUN WAVES PROROGATION)
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DIMENSION
DI ME NS I ON
DIMENSION
DIMENSION
DIMENSION
ALPHA! 10, 10),BETA! 10,10).GAMMA 110,10)
DELTA!10,10),XS(20),YS(20),CC(10,10,3)
C!20,21),CTR( 10,10),CTI( 10,10)
RSK!10,10),RS1(10,10),3UE(21)
A!20,21),G! 10,10), RR110,10,10)
RI(10,10,10),X(10,10) ,Y( 10,10)
XEV!10),YEV!10),XR(10,10),YR{10,10)
RHR!10),RHUI!10),THETA{ 10),ZETA(10)
ARI10,10),AIl10,10),BR(10),6 Il10)
DAMP! 10),DPHASE{ 10),FAMP( 10,51)
EPHASE!10,51),S(10,10),6(10,51)
SAMP(51)SPHASE!51),ETA(10,10)
RIURI 10) R2M 12 ( 10 ) ZPO IN T! 2 1) AMEG l 50)
UNIT NO. 1
INPUT DATA
1 FORMAT(1013)
2
FORMAT(5E14
.6)
READ
INPUT
TAPE
5,1 ,L
READ
INPUT
TAPE
5,1 NUM
READ
INPUT
TAPE
5,2,SUCK
READ
INPUT
TAPE
5,1,NE
READ
INPUT
TAPE
5,1,ND
READ
INPUT
TAPE
5,2,EINT
READ
INPUT
TAPE
5,2,ZDIC
READ
INPUT
TAPE
5,1,NZ
READ
INPUT
TAPE
5,2,(ZPOINT!J),J=1,NZ)
DO 3
M=l, 10
3
READ
INPUT
TAPE
5,2,(ALPHA!M,N),N=1,10)
DO 4
M=l, 10
4
READ
I NPUT
TAPE
5,2,!BETA!M,N),N=1,10)
DO 5
M= 1,10
5
READ
INPUT
TAPE
5,2,(GAMMA!M,N),N=1,10)
DO 2 3
M=l, 10
23
READ
INPUT
TAPE
5,2,(ETA(M,N),N=1,10)
DO 22
M= 1,10
22
READ
INPUT
TAPE
5,2,(S(M,N),N=1,10)
READ
INPUT
TAPE
5,2,(THETA(M),M=1,10)
READ
INPUT
TAPE
5,2,(ZETA(M),M=1,10)
WRITE OUTPUT TAPE 6,6
6 FORMAT!1H1////.52X, 10HALPHA!M,N)III)
DU 7 M=1, 1 0
7 RITE OUTPUT TAPE 6,8,(ALPHA(M,N),N=1,10)
8 FORMAT!// 5X.5E20.8/5X,5E20.8)
WRITE OUTPUT TAPE 6,9
9 FORMAT(1H1 ////,52X,10H BETA(M,N) ///)
DO 10 M=l,10
10 WRITE OUTPUT TAPE 6,8,!BETA!M,N),N=1 10)
WRITE OUTPUT TAPE 6,11
11 FORMAT( 1H1 ////,52X,10HGAMMA(M,N) ///)
o o o o o o
130
DO 12 M=l,iO
12WRITE OUTPUT TAPE 6,8,(GAMMA(M,N),N=1,1O>
WRITE OUTPUT TAPE 6,21,(THETA(M),ZETA(M},M=1,10)
21 FORMAT(1H1////,8X,8HTHETA(M),l5X,7HZETA(M)/iO(2E20.8/))
UNIT NO. 2
PRELIMINARY COMPUTATIONS
DO 14 M=11 10
DO 13 N= 1 10
13 DELTA!MN)=0.0
14 DELTA(M,M)=1.0
DO 60 J = 1 NE
EPS = EINT*FLOATF(Jl )
DN=1.0FLA TF(ND)/2.0
WELL=(EPS**DN)*EXPF(EPS)/{220000.0**ND)
DO 89 M = 1t L
E(M, J)=0.0
DO 98 N = 1 L
38 E(M J) = E(M J)+ S(M,N)* W E L L *(E P S *(N1)I
59 CONTINUE
60 CONTINUE
UNIT NO. 3
STEADY STATE EIGENVALUES
DO 16 M= 1 L
DO 15 N=1 L
CC(M,N,1)=ALPHA(MN)
15 CC(M,N,2)=ETA(MN)
16 CONTINUE
LP1 = L + 1
LM1 = L~1
L 2 = 2 L
L2P1 = 2*L + 1
L2Ml=2*Li
L2M2=2 *L2
CALL DETEX(L,2,CCB0E)
WRITE OUTPUT TAPE 6,17
17 FORMAT(1H1///,5X,21HSTEADY STATE SLUTI ON/5X,21( 1H*)//
1 5X,2H J,10X,6HB0E(J)/5X,2H,10X,6H )
WRITE OUTPUT TAPE 6,18,(J,BQElJ),J=1,LP1 )
18 FORMAT(17,E20.8 )
CALL POLY(L,BOE,XS,YS)
WRITE OUTPUT TAPE 6,19
19 FORMAT(// 5X,24HSTEADY STATE EIGENVALUES /5X,24(1H=)/
1 5X,2H K,11X,5HXS(K),13X,5HYS(K)/3X,2H,11X,
2 5(1H),13X,5( 1H) )
WRITE OUTPUT TAPE 6,20,(K,XS(K),YS(K),K = 1,L )
20 FORMAT(I7,2E20.8)
on non
131
UNIT NO. 4
PERTURBATION TECHNIQUE
DO 200 K= 1 L
DO 25 M= 1 L
DO 24 N=1,L
C TR ( M, N)=ALPHA(M,N)ETA(M,N)*XS(K)
24 CTI(M,N)=ETA(M,N)*YS(K)
25 CONTINUE
DU 27 M=1,LM1
ML=M+LM1
DU 26 N=1 LMl
NL=N+LM L
C(M,N)=CTR(M,N+1)
C ( ML f N ) =CT I ( M N + I )
C(M,NL)=C(ML,N)
26 C ( ML ,NL ) =C ( M, N )
C(M,L2M1)=CTR(M,1)
27 C(ML,L2 M1)= C TI(M,l)
CALL CLEM(L2M2,C,DET)
RSR(K, 1) = 1.0
RSI(K,1)=0.0
DO 26 J = 2 L
R$R(K,J)=C(Jl,L2MI )
1=J+L1
28 RSI(K,J)=C(I1,L2M1)
CALCULATION CF THE HOMOGENEOUS MATRIX
WHICH IS THE SAME FUR ALL DERIVATIVES
00 31 M =1 L
ML=M+L
U 29 N = i L
NL=N+L
A(MfN)=CTR(M,N)
A(ML,N)=CTI(M,N)
A(M f NL)=A(ML N)
2 9 A(ML NL)=A(M,N)
AIM,1)=0.0
A(M,LP1)=0.0
DO 30 N = 1 L
A(M,1)=A(M,L)ETA(M,N)*RSR(K,N)
30 A(ML 1)=AIML,1JETA(M,N)*RSI(K,N)
A(M,LP1)=A(ML,1)
31 A(ML,LP1)=A(M,1)
C CALCULATION OF THE DERIVATIVES OF
C THE EIGENVALUES AND THE RATIOS
DU 33 M1,10
DO 32 N=l,10
32 G{M,N)= 0.0
33 G(M, 1) = 1.0
G ( 1,2 ) = 1.0
DO 35 M=2, 10
DU 34 N=2,10
34 G(M,N}=C(Ml,N)+G(MlfNl)
35 CONTINUE
132
DO 100 NU=1,NUM
NU1=NU~1
RR!K,1,NU)=0.0
RI( K 1,NU)=0.0
DO 40 M= 1, L
ML=M+L
A(M,L2P1)=0.0
A{ML,L2P1)=0.0
DO 49 N= 11 L
AT=8ETA(M,N)DELTA(NU,1)+2.0*GAMMA(M,N)*DELTA{NU,2)
A(M,L2P1)=A(M,L2P1)RSR(K,N)*AT
A(MLL2P1)=A{ML,L2P1)RSI!K,N)*AT
IF(NUl) 38,38,36
36 CONTINUE
DO 47 MU=i,NU 1
NM=NU~MU
AT=(3ETA!M,N)DELTAlMU,1)+2.0*GAMMAlM,N)DELTA(MU,2)
1 ETA(M,N)X(K,MU))*G(NU,MU)
T = ETA t M,N)*YlK,MU)*G(NU,MU)
A(ML2P1)=A!M,L2P1)AT*RR(K,N,NM)+BTRI(K N NM)
A!ML,L2P1)=A{ML,L2P1)AT*RI(K,N,NM)BT*RRK,N,NM)
37 CONTINUE
38 CONTINUL
49 CUNIINUt
40 CUNINUL
CALL ELcM(L2,A,DET)
X(K,NU)=A(1.L2P1)
Y(K,NU)=A(LP1 L2P1)
DO 41 M= 2,L
RR(K,M,NU)=A(M,L2P1)
M L = M + L
41 RI(K,M,NU)=A(ML,L2P1)
100 CONTINUE
200 CONTINUE
WRITE OUTPUT TAPE 6,201,L,NUM
201 FORMAT!1H1///5X,27HTHE TIME DEPENDENT SOLUTION /
1 5X,27( 1H*)/>X,22HNU OF LAGUERRE POLYN =13/
2 5X,22UMAX DERIVATIVE ORDER =13 //)
WRITE OUTPUT TAPE 6,202,(M,M=1,7)
202 FORMAT(9X,7HX(K,NU),1X,7(4X,5HRR!K,I1,4H,NU)))
WRITE OUTPUT TAPE 6,203,(M,M=1,7)
203 FORMAT(9X,7HY(K,NU) IX,7(4X,5HRI!K,11,4H,NU) ))
DO 208 K=1,L
WRITE OUTPUT TAPE 6,204,K
204 FORMAT!// 5X,2HK= 12)
WRITE OUTPUT TAPE 6,205,XS(K),(RSR(K,M),M=1,L)
WRITE OUTPUT TAPE 6,205,YS(K),{RSI(K,M),M=1,L)
205 FUKMATIE17.6,7E14.6)
DU 206 NU=1,NUM
WRITE OUTPUT TAPE 6,207 X(K,NU),(RR(K,M,NU),M=1,L)
206 WRITE OUTPUT TAPE 6,205,Y!K,NU),(RI!K,M,NU),M=1,L)
207 FURMAT!/El 7.6.7E14.6)
208 CONTINUE
JQM=0
o a o
133
UNIT NO. 5
FREQUENCY DEPENDENT COMPUTATIONS
210 READ INPUT TAPE 5,2,OMEGA
JOM=JUM+I
UV=6.2832*UMEGA/( 2.2E+05)
DO 300 K=I L
XEV(K)=XS(K)
YEV(K)=YS(K)
DO 2 11 M= 1 L
XR(K,M)=RSR(K,M)
211 YR{K,M)=RSI(K,M)
P= 1.0
DO 250 NU=1f NUM
P = P V/F LOA T F(NU)
GO TO (213,215,217,219,213,215,217,219,213,215),NU
213 XEV(K ) = XEV(K) Y( K,NU)* P
YEV(K)=YEV(K)+X(K,NU)*P
DO 214 M= 1,L
XR(K,M)=XR(K,M)RI(K,M,NU)*P
214 YR(K,M)=YR(K,M)+RR(K,M,NU)*P
GO TO 221
215 XEV(K)=XE\/(K)X(K,NU)*P
YEV(K)=YEV(K)Y(K,NU)*P
DO 216 M=1,L
XR(K,M)=XR(K,M)RR(K,M,NU)*P
216 YR(K,M)=YR(K,M)RI(K,M,NU)*P
GU TO 221
217 XEV(K)=XÂ£V(K)+Y(K,NU)*P
YÂ£V(K)=YEVIK)X(K,NU)*P
DO 218 M=1,L
XR(K,M)=XR(K,M)+RI(K,M,NU)*P
218 YR(K,M)=YK(K,M)RR(K,M,NU)*P
GO TU 221
219 XEV(K)=XEV(K)+X(KNU)*P
YEV(K)=YEV(K)+Y(K,NU)*P
DU 2 20 M = 1,L
XRIK,M)=XR(K,M)+RR( K,M,NU)*P
220 YR(K,M)=YR(KM)+RI(K,M,NU)*P
221 CONTINUE
250 CONTINUE
300 CONTINUE
DO 305 K=1,L
AAA=XEV(K)+BUCK
B8Q=YEV(K)
CHI = SOR T F(AAA**2 + BB8*2 )
RHOR(K)= SQR T F(0.5* (CHI+AAA) )
RHOI(K)=SQRTF(0.5*(CHIAAA))
RRTRI(K)=RHOR(K)*RHOI(K)
R2MI 2(K)=RHORlK)** 2RHQI(K)**2
305 CONTINUE
o o o o
134
UNIT NO. 6
APPLICATION OF SOURCE CONDITION
TO DETERMINE FINAL CONSTANTS
DU 350 M= 1 i L
ML=M+L
C(M,L2P1)=3.0*THETA!M)
C(ML,L2P1)=3.0*OV*ZETA!M)
DO 349 K=I,L
KL=K+L
C(M,K)=RHUR(K)*XR(K,M)RHOI!K)*YR(K,M)
C!ML,K)=RHUR!K)*YR(K,M)+RHO I lK)*XR(K,M)
CIM,KL)=C(ML,K)
349 C(ML KL)=C(M L)
350 CONTINUE
CALL ELEM(L2 C DET)
DU 360 K=I L
KL=K+L
BR(K)=C(KfL2Pl)
BI (K ) =C(KL L2P1 )
DU 355 M=1 L
AR(KfM)=BR(K)*XRlK,M)BI(K)*YRlKM)
35 5 AI(K,M)=BR(K)*YR!K,M) + BI(K)*XR(K,M)
360 CONTINUE
WRITE OUTPUT TAPE 6,362,L,NUM,BUCK,OMEGA
362 FORMAT!1HI////5X27HTIME DEPENDENT SOLUTION FOR /
1 5X,28HTHE NEUTRON WAVE PRDPOGATI UN/
2 5X 2 8HN0. OF LAGUERRE POLYNOMIALS= 12/
3 5X,2BHMAXIMUM ORDER OF DERI VAT IVE = 12/
4 5X,20HTRANSVERSE BUCKLING= EL5.6 /
5 5X,20HFREQUEIMCY (IN CPS) = E15.8 /
6 5X,35(1H*) )
DO 365 K=1 L
WRITE OUTPUT TAPE 6,363,K,XEV{K),YEV(K),RHOR{K),RHOI(K)
l RRTR I (K)t R 2 MI 2(K),3R!K),BI!K)
363 FORMAT!/ 5X2HK=I2/5X,5HXEV = E16.8/5X,5HYEV =Â£16.3/
1 5X,5HRH0R=EI6.8/5X,5HRH0I=E16.8/
2 5X6HRRTRI = EI6.8/5X 6HR2MI2 = EI6.8/
3 5X,3HBR=EL6.8/5X,3HBI=E16.8 )
WRITE OUTPUT TAPE 6,364,tXR!K,M),YRlK,M),
I AR(K,M),AI(K,M),M=L,L)
364 FORMAT!/ 9X,7HXRlK,M),I3X,7HYR(K,M),13X,
1 7HAR!K,M),I3X,7HAI(K,M)/1014E20.8/))
365 CONTINUE
non
135
UNIT NU. 7
SPACE DEPENDENT COMPUTATIONS
************
WRITE OUTPUT TAPE 6,366,L,OMEGA
366 FORMAT(1H1///,5X,34HAMPLITUDES AND PHASE SHIFTS OF THE/
1 5X,34HTOTAL NEUTRON DENSITY AND THE FLUX/
2 5X,34HENERGY DISTRIBUTION VS. POSITION,1/
3 5X,18HF0R THE CASE UF L= 12/
4 5X,16HAND OMEGA(CPS)= E16.8/5X,34(1H*)//)
IFIZDIC) 401,401,367
367 CONTINUE
DO 400 J = i,NZ
Z=ZPUINT(J)
ASIAU=0.0
SUDAU=0.0
U 381 K=1,L
SC=CUSF(RHUI(K)*Z)
SS = SlNF(RH1IK)*Z)
EX=EXPF(RHOR(K)*Z)
QDR^O.O
CD I = 0.0
DU 369 M=1 L
0DR=QDR+ZETA{M)*AR(K,M)
369 GDI = QDI+ZE TAlM)*A11K,M)
DAMP(K)=EX*SORTF(QDR**2+QD1**2)/220000.0
DTI=QDR*SSUDI*SC
DTR=QDR*SC+ODI*SS
UPHASE(K)=ATANF(DTI/DTR)
IF(DTR) 370,371,371
370 UPHASE( K)=DPHASE(K)+3.14159
371 CONTINUE
ASIAD=ASIAU+DAMP(K)*COSF(DPHASE(K))
SUDAD= SUDAD+DAMP{K)*SINF(DPHASE(K))
IF(EINT) 385,385,375
375 CONTINUE
DU 380 N=1,NE
C'FR = 0.0
WFI=0.0
DO 372 M= 1 ,L
QFR=QFR+ARIK,M)* E(M,N)
372 QFI=QFI+AI(K,M)E(M,N)
FAMP(K,N)=EX*SQRTF(QFR**2+QF1**2)
FTI=QFRSSOFI*SC
FTR=OFR*SC+OFI*SS
FPHASE(K,N)=ATANF(FTI/FTR)
IFIFTR) 373,374,374
373 FPHASEU,N)=FPHASE(K,N)+3.14159
374 CUNTINUE
380 CONTINUE
385 CONTINUE
381 CONTINUE '
TAMP=SQRTF(ASIAD**2+SUDAD**2)
TPHASE = ATANF(SUDAD/AS I AD)
136
IF (AS I AD) 377, 378,378
377 TPHASE=TPHASE+3.14159
378 CONTINUE
IF(Jl) 388,387,388
387 AZÂ£RO=TAMP
388 TDTZ=TAMP/AZERO
IF(JQMl) 303,302,303
302 AOMEG(J) = T AM P
303 TOTW=TAMP/AOMEG(J)
DO 386 N=1,NE
HASSAN=0.0
SABtHA=0.0
DU 382 K=1,L
HASSAN = HASSAN+FAMP(K,N)*COSF(FPHASE(K,N))
382 SA3EHA=SABEHA+FAMP(K,N)*SINF(FPHASElK,N))
SAMP(N)=SQRTF(HASSAN**2+SABEHA**2)/TAMP
SPHASE(N)=ATANF(SABEHA/HAS SAN)
IF (HASSAN) 383,384,384
383 SPHASElN)=SPHASElN)+3.14159
384 CONTINUE
386 CONTINUE
WRITE OUTPUT TAPE 6,390,Z
390 FORMAT(//5X,5HAT Z= F6.2,19H CM FROM THE SOURCE /
1 5 X,30( 1H)// )
WRITE OUTPUT TAPE 6,391, (K,DAMP(K),DPHASt(K),K=1,L)
391 FORMATt5X,1HK,10X,4HDAMP,13X,6HDPHASE/10(I6,2E20.8/) )
WRITE OUTPUT TAPE 6,379,TAMP,TPHASE
1 ,TOTW,TPHASE,TOTZ,TPHASE
379 FORMAT( / 5X,4HTAMP,E17.3,E20.8/
1 5X,4HTOTW,El7.8,E20.8/
2 5X,4HTOTZ,E17.8,E20.8)
IF(EINT) 396,396,376
376 CONTINUE
WRITE OUTPUT TAPE 6,392,(K,K=1,L)
392 FURMAT(/5X,9HFAMP(K,E)/5X,11HFPHASE(K,E) /
i 6X,3HE K, 16,9110,10X,5HTCTAL )
DO 393 N = 1, N E
Â£1=EINT*FLUATF(N1)
WRITE OUTPUT TAPE 6,394,E1,(FAMP(K,N),K=1,U,SAMP(N)
393 WRITE OUTPUT TAPE 6,395,lFPHASE(K,N),K=1,LJ,SPHASE(N)
394 F0RMAT(/5X,F5.2,10E10.3)
395 FORMAT(10X,10E10.3)
396 CONTINUE
400 CONTINUE
401 CONTINUE
GO TO 210
END
137
SUBROUTINE DETEX (DETERMINANT EXPANSION)
SUBROUTINE DETEX(LSD,NSE,DD,A)
DIMENSION DO(10,10,3),CC(10,10,3),CD(10, 10),A(21) ,B(21 )
LD= L SO
L E = N S E
DO 30 M = I L 0
DO 20 N=1, L U
DO 10 K= L,LE
10 CC(M,N,K)=DU(MNK)
20 CONTINUE
30 CONTINUE
JMAX = LD*(LE1 ) + 1
DO 40 J = 1, JMAX
40 B(J)=0.0
DO 401 J 1 = 1 L E
ND= 1
N1 = J 11
JP=1+N1
IF(LDND) 301,101,301
101 CONTINUE
DO 201 M=1,L D
CD(M,i)=CC(M, 1,J1)
201 CONTINUE
CALL EVADET(LD,CD,DET)
B(JP)=B(JPJ+UET
301 CONTINUE
DO 402 J2=1,LE
ND=2
N2=J21
JP=1+N1+N2
IF(LDNO) 302,102,302
102 CONTINUE
DO 202 M =1,LD
CD(M,1)=CC(M,1,J1)
CD(M,2)=CC(M,2,J2)
202 CONTINUE
uALL EVADET(LD,CD,DET)
5(JP ) = B(JP)+DET
302 CONTINUE
DO 403 J3 = i,LE
ND= 3
N3 = J 3 1
JP=i+Nl+N2 + N3
IF(LDND) 303,103,303
103 CONTINUE
DU 203 M=I,LD
CD(M,l)=CC(M,l,Jl)
CD(M,2)=CC(M,2,J2)
CD{M,3)=CC(M,3,J3)
203 CONTINUE
CALL EVADETtLUfCDiOcT)
B(JP ) = B(JP)+DET
303 CONTINUE
DO 404 J4=1,LE
NU = 4
N4=J4I
JP=l*Nl+N2+N3+N4
IFILONO) 304,104,304
104 CUNT1 NUE
DU 204 M =I,L U
CU(M,I)=CC(M,I,J1)
CD(K,2)=CC(M,2,J2)
CO(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4)
'204 CONTINUE
CALL EVAUET( L D C 0, D E T )
8(JP)=B(JP)*UET
304 CONTINUE
DC 4 05 J5 = l, LE
ND = 5
N5 = J 5 I
JP=I+N1+N2+N3+N4+N5
IF(LDNU) 305,105,305
105 CUNT INUt
UU 205 H=1,L
CD(M,1)=CC(M, 1 ,J 1 )
CD(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4J
C0(Mt5)=CC(M,5,J 5)
205 CONTINUE
CALL EVADET(LD,CD,DET )
B(J P)=B(JP)+DET
305 CONTINUE
UU 406 J6=1,LE
NU = 6
\6J 61
JP=1+N1+N2+N3+N4+N5+N6
IF(LDNU) 306,106,306
106 CONTINUE
CD(M,1)=CC(M,1,J1)
CD(M,2)=CC(M,2,J2)
CD(M,3)CC(M,3,J3)
CD(M,4)CC{M,4,J4)
CD{M,5)=CC(M,5,J5)
CD(M,6)=CC(M,6, J6)
DU 2 06 M = 1,L U
206 CUNTINUL
CALL EVADET(LD,CO,DET)
B(JP)=B(JP+DET
139
306 CONTINUE
DU 407 J 7=1,LE
ND=7
N7 = J 7 1
JP=1+N1+N2+N3+N4+N5+N6+N7
IF(LOND) 307,107,307
107 CONTINUE
L)U 07 M= 1 LU
CD(M,1)=CC(M, 1,J 1 )
CD(M,2)CC(M,2,J2)
CD(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4)
CD(M,5)=CC(M,5J5)
CD(M,6)=CC(M,6,J6)
CD(M,7)=CC(M,7,J7)
207 CONTINUE
CALL EVAUET(LD,CD,DET)
B(JP)=B(JPJ+DET
307 CONTINUE
L'O 408 JS=i,LE
NQ = 8
N8=J81
JP=1+N1+N2+N3+N4+N5+N6+N7+N8
IF(LDNU) 308,108,308
108 CONTINUE
DO 208 M=1,LD
CD(M,1)=CC(M,1,J1 )
CD(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3,J3)
CD(M,4)=CC(M,4,J4)
CD(M,3)=CC(M,6,J3)
CD(M,6)=CC(M,6,J6)
CD(M,7)=CC(N,7,J7)
CD(M,8)=CC{M,8,J8)
208 CONTINUE
CALL EVADET(LD,CD,DET )
til JP)=B( JPJ + DET
308 CUNTINUE
BO 409 J9=1,LE
N D = 9
N 9 = J 9 1
JP=i+NHN2+N3+N4+N5+N6+N7+N8+N9
IF(LDND) 309,109,309
109 CONTINUc
DU 209 M=1,LD
CU(M,1)=CCIM,1,J1)
CD(M,2)=CC(M,2,J2)
C D { M 3 ) C C ( M 3 J 3 )
CD(M,4)=CC(M,4,J4)
CD(M,9)=CC(M,5,J5)
CD(M,6)=CC(M,6,J6)
140
CD(M,7)=CC(M,7,J7)
CD(M,8)=CC(M,8,J8)
CD(M,9)=CC(M,9J9)
209 CONTINUE
CALL EVADETILD,CD,DET)
B(JP)=B(JP)+DET
309 CONTINUE
U 410 J 10=l LE
ND= 1 0
N10 = N10 1
JP=l+N1+N2+N3+N4+N5+N6+N7+N8+N9+N10
IF(LDND) 310.110,310
110 CONTINUE
DO 210 M=1,LD
CO(M,1)=CC(M,1,J 1 )
C0(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3J3)
CD(M.4)=CC(M,4,J4)
CD{M,5)=CC(M,5,J5)
CD(M,6)=CC(M,6,J6)
CO(M,7)=CC(M,7,J 7)
CD(M,8)=CC(M,8,J8)
CD(M,9)=CC(M9,J9)
CD(M,10)=CC(M,10,J10)
210 CONTINUE
CALL EVADETILO,CD,OET)
{JP) = 8(J P)+DE T
310 CONIINUE
410 CONTINUE
409 CONTINUE
408 CONTINUE
407 CONTINUE
406 CONTINUE
405 CONTINUE
404 CONTINUE
403 CONTINUc
402 CONTINUE
401 CONTINUE
DO 50 J=1,JMAX
50 A(J)=B(J)
RE TURN
END
141
SUBRCJUT I NE EVADET
SUBROUTINE EVADET{LU,CD,DET)
DIMENSION GDI 10,10) ,HI 10,10)
DE T= 1.0
DU 100 K=I,LD
X=0.0
DO 15 I = K,LD
A=A6SF(X)ABSFICD(IK))
I F(A ) 5,10,10
5 X=CD(I,K)
I X= I
10 CONTINUE
15 CONTINUE
DU 20 J=K,LD
HIK,J)=CD(K,J)
CD(K,J)=CD(IX,J)
20 CD(IX,J)=H(K,J)
IF(IXK) 25,30,25
25 DET=DET*CD(K,K)
GO TO 35
30 DE T = DE T*CD(K,K)
35 B=CD(K,K)
DO 40 J=K,LD
40 CD IK,J)=CD(K,J)/B
DO 45 J = 1,LD
45 HIK,J)=CD(K,J)
DO 55 1=1,LD
R=CD(I,K)
DO 50 J=1,LD
50 GDI I,J)=CD( I,J)R*H(K,J)
55 CONTINUt
DO 60 J=1,LD
60 CD(K,J)=H(K,J)
100 CONTINUE
RETURN
END
142
SUBROUTINE ELEM {EL EM I NA TI UN METHOD)
SUBROUTINE E LE M(N,C,T )
DIMENSION HI 20,21),0(20,21)
KE=N+I
T= 1.0
DO 40 K=1,N
X = 0.0
UU 10 IK,N
IF(ABSE(X)ABSF(C(I,K))) 9,10,10
9 X=ClI,K}
I X= I
10 CONTINUE
DU 11 J = K,K E
H(K,J)=0{K,J )
0(K,J)= C( IX,J)
11 C(IX,J)=H(K,J)
1F(IXK) 17,18,17
17 T =T *ClK,K)
GO 10 l'>
18 T=T*C(K,K)
19 CON TINUE
AY=C(K,R)
DO 20 J = K,KE
20 C(K,J)=C(K,J)/AY
DO 2 1 J = 1,KE
21 H(K,J)=C(K,J)
DO 23 I=1,N
R = C( I,K )
UU 22 J=1,KE
22 C(I,J)=C(I,J)R*H(K,J)
23 CON INUE
DO 10 J = 1,K;
30 C(K,J)=H(K,J)
40 CONTINUE
RE TURN
END
143
SUBROUTINE PULY (POLYNOMIAL SOLVER)
SUBROUTINE PULY{NI,BE,ROTR,ROOT I)
DIMENSION BUE(81),ROOTR(80),RGOT I (80),CUE(81)
N 3 = NI
N2 = N1 + 1
00 I KUN=1 N2
NUK=N2+IKUN
1 COE(KUN)= BOc(NUK)
XCONI
= 0.
XC0N2
= 0.6
XC ON 3
= 0.66
XCN4
= 0.9
XCONb
= 1.0
XCUN6
= 2.0
XC0N7
= 4.0
XC0N8
= 0.5
XCUN9
= l.E20
XCONIO
=1.E7
XCONI1
= 1 E~5
N4 = 0
I = N2
19 IFCOEII)) 9 7,9
7 N4=N4+1
ROO r R(N4) = XCONI
R0UTKN4) = XCONI
1 = 11
IF(N4Ni)19,999,19
9 AXR=XC0N2
AX I = XCONI
L= 1
N3 = 1
K=1
ALP1R=AXR
ALP1 1 = 4X1
GO TO 99
11 BE T1R = TEMR
BE T1 I = T EM 1
AXR = XCCN3
ALP2R=AXR
ALP2I=AX I
M=2
GO TO 99
12 BE T 2 K= T E MR
BET2 I = T E MI
AXR = X C U N 4
AL P3K=AXR
AL P 3 I = AX I
M= i
GO TO 99
13 BE T 3R= T CMR
BET3I=TEMI
14 TE1=ALP1RALP3R
TE2=ALP1IAIP3I
rEb=ALP3RALP2R
TE6=ALP3IALP2I
Z 1 = T E 5 TEb
Z2= T E6 TE6
T EM = Z1 + Z2
Z1=TE1*TE5
Z2=TE2TE6
Z3 = Z1+ Z 2
TE3=Z3ZF EM
Z1= FE2* EEb
Z2=FE1*FE6
15 = 1 1Z2
TE4=Z3/FEM
TE7 = T E3 + XC0Nb
Z1=FE3*FE3
Z2 = FE4*FE4
FE9=Z1Z2
Z1 = XCQN6 TE3
FE10=Z1FE4
Zl=FE7*BET3R
Z2=FE4*BEF3l
DElb=ZlZ2
ZL=FE7*BEF3I
Z2=TE4*BEF3R
DE16=Z1+Z2
Z1 = FE3*8E F2R
Z2= FE4*BET2i
E11 = Z1Z2 + BET1RDE15
Zl=Tc3*BET2I
Z2= FE4*BET2R
FE12=Z1+Z2+BEF 1IDE16
TE7 = T E9XCGN5
ZL=FE9*8ET2R
Z2=rE10*BET2I
FE1=Z1Z2
ZL=rE9*BEr2I
Z2=FEL0*BEF2R
TE2=Z1+Z2
Z1=FE7*BET3R
Z2 = TE10*BET 31
lE13 = TElBEri RZ1+ Z2
Z1=FE7*BET3I
Z2=TE10*BET3R
TE14 = TE2ilET 11Z1Z2
Z 1 =l)E 1 5 F E 3
Z2 = DE 16 TE4
Tcl5 = Z 1Z2
Z1=DE15*TE4
Z2 = l)E16*FE3
F E L6=Z1+Z2
Z1=TE13*TE13
Z2= FE L4*F E 14
Z3=FE1 l*FElb
145
z4=fe2*fei6
Z5=Z3Z4
Z6=XCUN7*Z5
TE 1 = Z1Z2Z6
Z l = XC0N6 *TE13
Z2 = Z 1 *TE 14
Z3 = TÂ£ 12* FE 15
7.4=FE11*FE16
Z5 = Z 3 + Z4
Z6=XC0N 7*Z 5
TE2=Z2Z6
Z1=FE1*FE1
Z2= FE2*FE2
Z3=Z1+Z2
rEM=SQRTF(Z3)
IF(TE1)113,113,112
113 Zi=FEMFEl
72 = XCUN8 Z1
FE4 = SQR F F{Z2)
ZL=FE2/Ft4
TE 3 = XCONb Z 1
GQ FU 111
112 Zl=rCM+Tri
Z2=XCUN3*Z1
TE3 = SG,'RFF( Z2 )
I F r E2 ) 1 10,200,200
110 T E 3 = T E 3
200 Z1=TE2/Tc3
TE4=XC0N8*Z1
111 TE7=rE13+TE3
rE8=TE14+TE4
T E9= TE 13TE 3
FE 10F E14F E4
FE 1 = XC0W6*T E15
TE? = XC0i*6*TE16
Z1=FE7*F87
Z2= Ed*FE8
Z3=FÂ£9*FE9
Z4=F E10*TE10
IF ( Z 1 + Z2Z3Z4)204,204,205
204 E 7 = TF9
FE8=FE10
205 Zl=fE7*FE7
Z2=FE8*FE8
EMZ1+Z2
Zl=rEl*FÂ£7
Z2 = TE2* FE8
Z3 = Z 1 + Z2
IE3=Z3/FEM
Z 1 = FE2* E7
Z2 = T E1 F Â£8
Z3=Z1Z2
IÂ£4=Z3/TEM
146
Z1=rE3*Tfcb
L2T Â£4* E6
AXR=ALP3R+Z1Z2
Zl=rE3*TE6
Z2 = TE4;>TE5
AXIALP3I+Z1+Z2
ALP4R=AXR
AL P4I = AX I
M = 4
GO TO 9*
15 N6=l
30 IF{ABSF(HELL)+ABSF(BELDXC0N9)18,lb,16
16 TE7=ABSF(ALP3RAXR)+ABSFIALP3IAXI}
Z1 = A BSF(AXR)+ABSF(AXIJXC0N10
I F(rÂ£7/Zl)18,18,17
17 N 3 = i\l 3 + 1
ALP1R=ALP2K
ALP1 I = AL P2I
ALP2R=ALP3R
ALP2I = AL P31
AL P 3R=AL P4R
ALP3 1=ALP4I
BET1 1 = BET2I
BE T2R=BE T3R
BET2 I = BE T3 I
BE F 3R= TEMR
BET3I=TEMI
BEriR=BET2R
IF!N3100)14,18,18
18 N4=N4+L
ROOT R(N4)=ALP4R
ROUTI(N4)=ALP4I
N 3 = 0
41 IF(N4N1)30,999,999
30 IF(ABSFtROUTI(N4>)XCON11)9,9,31
31 GO rO(32,9),L
32 AXR = AL P1R
AXI=ALP1I
AL P 1 I =ALP 11
M= 5
GO TO 99
33 BETlR= TEMR
BET1I = TEMI
AXR= AL P2 R
AX I= AL P2 I
ALP2I=ALP2 I
K = 6
GO TO 99
34 BET2R=TEMR
bET2 I = TEMI
AX R= AL P3R
AX I=AL P3I
ALP 3 I=ALP3I
L = 2
147
M = 3
99TEMR=COE(1)
TEMI=XCUN1
100 I =
Z1=TEMRAXR
Z2 = TEM IoAX 1
TE1=Z1Z2
Z1=TEMI*AXR
Z2 = T EMR*AX I
TEMI=Z1+Z2
100 IEMR =TE1+CGE(1+1)
HE LL = TEMR
BE LL = TEM I
42 IFIN4}102,103,102
102 DO 101 1=1,N4
TEMI =AXRRUUTR( I )
TEM2 = AX IROOT I ( I )
Zl=TEMI*TEMI
Z2= T EM2 TEM2
IE1=Z1+Z2
Z1=TEMR*TEM1
Z2 = T EM I TEM2
Z 3 = Z 1 + Z 2
TE2 = Z3V TE1
Z1=TEMI*TEM1
Z2=TEMRTEM2
Z3=Z1Z2
fÂ£MI=Z3/TEl
101 TEMR=TE2
103 GO 10(11,12,13,13,33,34),M
999 CONTINUE
RETURN
END
148
INPUT DATA PROGRAM HGM (HEAVY GAS MATRICES)
DIMENSION ALPHA(10,10),BETA(10,10),GAMMA(10,ID),
1 ETA(10,10),THETA(10),ZETA(10),DELTA(10,10),
2 A(10,10),AA(3,10),BB(3,10),TT(10),U(10,10,10),
3 V(10,10, 10),W(10,10,10), C ( 5 0 ) G ( 5 0 )
COMMON ALPHA, BE I.'A, GAMMA, ETA, THETA, ZtTA, DELTA,
1A,AA,BQTr,U*V,W,C,G
1 FORMAT( 1013)
2 FORMAT(5E16.8)
3 FORMAT(1HP,5E14.6)
READ INPUT TAPE 5,1,L
CALL COLAP(L,A)
CALL MOLAPlL,A,U,V,W)
READ INPUI TAPE 5,2,SS,SA,UM
R=1.O/UM
100 READ INPUT TAPE 5,1,NS
NSING=NS3
WRITE OUTPUT TAPE 6,1 5,L,NS ING,SS,SA,UM
15 FORMAT( 1H1/////5X,32HMATRIX ELEMENTS FOR THE CASE OF /
1 5X32(1H=)//5X,2HL=I2/5X,19HSINGULARITY ORDER =13,2H/2/
2 5X.9HSIGMAS =F12.8/5X,9HSIGMAA =F12.8/5X,6HMASS =F5.1 )
READ INPUT TAPE 5,1,NCV
IF(NCV) 16,16,20
16 WRITE OUTPUT TAPE 6,17
17 FORMAT(5X,21HWITH CONSTANT SIGMAS )
CALL A8TCS(L,NS,SS,SA,R,AA,BB,TT)
GO TU 25
20 WRITE OUTPUT TAPE 6,21
21 FORMAT(5X.21HWITH VARIABLE SIGMAS )
CALL ABTVS(L,NS,SS,SA,R,AA,BB,TT)
25 CONTlNUt
CALL ABGETZlL,NS,AA,BB,TT,U,V,W,ALPHA,
1 BETA,GAMMA,ETA,THETA,ZETA)
DO 4 M= 1, L
4
WRITE
OUTPUT
TAPE
6,3,
(ALPHA(M,N),N=1,L)
DO 5
M= 1, L
5
WRI I E
OUTPUT
TAPE
6,3,
( BETA(M,N),N=1,L)
DO 6
M= 1, L
6
WRITE
OUTPUT
TAPE
6,3,
(GAMMA(M,N),N=1,L)
DU 7
M= 1 L
7
WRITE
OUTPUT
TAPE
6,3,
(ET A(M,N),N=1,L)
DO 8
M= 1, L
8
WRITE
OUTPUT
TAPE
6,3,
(A(M,N),N=1,L)
WRITE
OUTPUT
TAPE
6,3,
(THETA(M),M=1,L)
WRITE
OUTPUT
TAPE
6,3,
(ZETA(M),M=1,L)
oO TO
END
100
o c o o o on
149
SUBROUTINE COLAP
SUBROUTINE CULAP(L,A)
CALCULATION OF THE COEFFICIENTS OF I HE NORMALIZED
ASSOCIATED LAGUERRE POLYNOMIALS OF THE FIRST KIND
DIMENSION ALPHA! 10, 10),BETA( 10,10 I.GAMMA!10,10),
1 ETA!i 0,10),THETA!10),ZETA(10).DELTA!10,10),
2 A!LO,10),AA(3,L0),BB(3,L0),TT(L0),U(10,10,10),
3 V!10,10,10),W(10,10,10)C(50),G(50)
DI MENS I UN E( 10, 10)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1A,AA,BB,TI,U,V,W ,C,G
DO 2 N=i,L
DO 1 M=1,L
1 L(N,M)=0.0
2 CONTINUE
Ell, 1) = 1.0
L(2,1)=SQRTF{2.0)
b!2,2)=SQRTF(0.5)
DO 3 N=3,L
NM1=N1
NM2 = N2
X=FLOATFIN)
D E(N,1)=E(NM1, 1)*2.0*SQRTF((X1.0)/X)E(NM2,1)
1*SQR TF((X2.)/X)
3 CONTINUE
DO 5 N=3,L
NM1=N1
NM2=N2
X=F LOAT F{N)
DO 4 M = 2 L
MMl=M1
D 4 E N, M)=E(NM1,M)*2.0*SQRTF( (Xl.0)/X)E!NM2,M)
i*SQRTF((X2.0)/X)E!NMl,MMl)/SQRTF!X*(X1.0) )
5 CONTINUE
DC 12 M=1,L
DO 11 N = 1L
11 A!M,N)=fc(M,N)
12 CONTINUE
WRITE OUTPUT TAPE 6,6
6 FORMAT!1H1 /////,5X,31HC0EFFICI ENTS OF THE POLYNOMIALS)
7 FORMAT(I3.10F11.5 //)
DO 10 N =1,L
10 WRITE OUTPUT TAPE 6,7,N,(A{N,M),M=1,L)
RETURN
END
Q Q
150
SUBROUTINE MULAP
SUBROUTINE MOLAP(L,A,U,V,W)
DIMENSION ALPHA(10,10),BETA(10,10),GAMMA(10,10) ,
1 ETA10,10),THETA{10), ZE TA(10),DELTA!10,10),
2 A(10,10)AA(3,10)*8B(3,10)TT(10),U(10,10,10)
3 V(10,10,10),W( 10,10, 10) ,C(50)G{50)
D DIMENSION Bt100),D( 100),Y{ 10,10,10)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB, TT,U,V,W ,C,G
DO 5 M= 1 L
DU 4 N = 1 ,L
DU 3 1=1,10
U(M,N,I)=0.0
V(M,N, I ) = 0.0
3 W(M,N,I)=0.0
4 CONTINUE
5 CONTINUE
D(1) = 1. /724535
D ( 2 ) = 1.0
DO 10 1=3,60
IM 2 =I~2
X=FLOATE(I)
D D(I) = (0.5*X1.0)*D( I M2)
10 CONTINUE
L1=2*L1
DO 20 M=1,L
DO 20 N=1, L
DO 11 1=1,LI
D 11 b(I)=0.0
DO 12 1=1,L
DU 12 J = 1,L
T = A ( M, I ) A ( N J )
K=I + J1
D 6 l K ) = B ( K ) + T
12 CONTINUE
DO 14 1=1,10
D Y(M,N,I)=0.0
DO 14 J =1,L 1
K=I+2*J2
D 14 Y(M,N,I)=Y(M,N,I)+D(K)*B(J)
DU 17 M =1,L
DU 16 N =1,L
DO 15 1=1,10
15 U(M,N,I)=Y(M,N,I)
16 CONTINUE
17 CONTINUE
20 CONTINUE
DO 30 M=i,L
WRITE OUTPUT TAPE 6,22,M
22 FORMAT( 1H1 /////,52X,2HU(,I 2,10H,N,I = 1,10) ///)
DO 25 N= 1 L
WRITE OUTPUT TAPE 6,23,N
151
23 FORMAT(/ ,15X,2HN= *12)
25 WRITE OUTPUT TAPE 6,27,(U(M,N,I),I = 1.10)
27 FORMAT(5X,5F20.8)
30 CONTINUE
00 3 5 M= i L
DO 34 N = 2 L
NMI=N1
X=FLOATF(N)
XN=SQRTF( (X 1.0)/X)
DO 33 1 = 1, 10
V(M,N,I)=XN*(V(M,NMl,I)U(M,NMI,I))
33 W(M,N,I)=XN*(W(M,NMi,I)V(M,NMI,I))
34 CONIINUE
35 CONTINUE
DU 4 0 M =1,L
WRITE OUTPUT TAPE 6,36,M
36 FORMAT(1HI /////,52X,2HV(,I 2,1OH,N, I = 1, 10) ///)
DO 39 N=1 L
WRITE OUTPUT TAPE 6,23,N
39 WRITE OUTPUT TAPE 6,27,(V(M,N,I),1=110)
40 CONTINUE
DO 4 5 M =I.L
WRITE OUTPUT TAPE 6,42,M
42 F0RMATUH1 /////,52X,2HW(,12,10HtNf1=1,10) ///)
DO 44 N=I,L
WRITE OUTPUT TAPE 6,23,N
44 WRITE OUTPUT TAPE 6,27, (W(M,N.I)I = 1.10)
45 CONTINUE
RETURN
END
152
SUBRUTINt ABTVS
SUBROUTINE ABTVS(L,NS,SS,SA,R,AA,BB,TT)
DIMENSION ALPHA10,1),BETA(10,10),GAMMA10,10),
1 ETA( 10,10),T HE T A{10),ZETA(10),0ELTA(10,10),
2 A(1010),AA(3,10),BB(310),TT(1Q)U(10,10,10),
3 V(10,10i.0)W(10,10,10),C(50),G(30)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZÂ£TA,DELTA,
1 A,AA,BB,T,U,V,W,C,G
DO 4 N=l, 10
TT(N)=0.0
DO 3 M= 1,3
AA(M,N)=0.O
3 36(M,N)=0.O
4 CONTINUE
NSPl=NS+l
NSP2=NS+2
NSP3=NS+3
NS P4 = NS + 4
NSP3=NS+5
NSP6=NS+6
NS P 7 = NS + 7
NSP8=NS+8
AA(l,NSPL)=R*SS*SA/3.0
AA(1,NSP2)=SA**2
AA(1,NSP3)=SS*SA*(1.07.0*R/3.O)
AA(2,NSP2)=2.0MR*#2)*SS*SS
AA ( 2 NS P3 ) = 10 O*R* SS*S A/3. O
AA(2,NSP4)=SS*SS*(43.0*R/3.4.0)*R
AA(2,NSP5)=4.0*R*SS*SA/3.0
AA(2,NSP6)=2.0*R*SS*SS*(1.O10.0*R/3.0)
AA(3,NSP4)=9.0R*R*SS*SS
AA(3NSP5)=R*SS*SA*4.0/3.0
AA{3NSP6)=R*SS*SS*(44.0*R/3.02.0)
AA( 3 ,NSP8) = 4.0*RbR*SS*SS/3.0
BB(1,NSP1)=R*SS/3.0
BB (I,NSP2)=2.0*SA
BB(i,NSP3) = SS*l 1.O 7.O R/3.O )
8B(2,NSP3)=~10.0*R*SS/3.0
33(2,NSP3)= 4.0*R*SS/3.0
3B(3.NSP5)=4.0*R*SS/3.0
TT{NSP2)= R*SS/2.0
TT ( N 3 P 3 ) = S A
TT(NSP4)=SS*(1.08.O *R/3.O)
WRITE OUTPUT TAPE 6,5
5 FORMAT(IH1///,50X,7HAA(M N)//)
DO l M=1,3
7 WRITE OUTPUT TAPE 6,8,(AA(M,N),N=1,10)
8 FORMAT(10E12.4)
WRITE UUTPUT TAPE 6,9
9 FORMAT(///,50X,7HBB(M,N )//)
DO 10 M1,3
10wRIIE OUTPUT TAPE 6,8,(BB(M,N),N=1,10)
153
WRITE OUTPUT TAPE 6,11
11 FORMAT!///,50X.5HTTN)//)
WRITE UUTPUT TAPE 6,8,{TT{N),N=1,10)
RETURN
END
SUBROUTINE ABTCS
SUBROUTINE ABTCS{L,NS,SS,SA,R,AA,BB,TT)
DIMENSION ALPHA!10, 10),BETA!10,10),OAMMA(10,10),
1 ETA 110,10),THETA(10),ZETA(10),DELTA(10,10),
2 A(10,10),AA(3,10),BB(3,10),TT(10),U(10,10,10),
3 V! 10,LO, 10),W(10,10,10),C(50),G(50)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB,TT,U,V,W,C,G
DO A N=1,10
T T N ) = 0.0
DO 3 M= 1,3
AA(M,N)= 0.0
3 3B(M,N)=0.0
4 CONTINUE
NSPl = NS + 1
NSP2=NS+2
NSP 3 = NS + 3
NSP4=NS+4
NSP5=NS+5
NSP6=NS+6
NSP7=NS+7
NSP8=NS+8
AA( 1 ,NSPl)=R*SS*SA/6.0
AA( 1 ,NSP2)=SA**2
AA!1,NSP3)=SS*SA*(1. 01.0*R/3.0)
AA(2,NSP3)=10.0*R*5S*SA/3.0
AA!2,NSP4)=SS*SS*(16.0*R/3.0~4.0)*R
AA(2,NSP5)=4.0*RSS*SA/3.0
AA(2,NSP6)=2.0*R*SS*SS#1.04.0*R/3.0)
AA13,NSP4)=8.0*R*RSS*SS
AA!3,NSh5}=K*SS*SA*4.0/3.0
AAl 3,NSP6)=R*SS*SS*(32.0*R/3.02.0)
AA! 3.NSP8 )=4.0*R*R*SS*SS/3.0
BB(1,NSP1)=R*SS/6.0
BB(1 ,NSP2)=2.0*SA
BB(1,NSP3)=SS*( 1.01.0*R/3.0)
BB(2,NSP3)=10.0*R*SS/3.0
B 5!2,N S P 5)= 4.0*R*SS/3.0
BB(3,NSP5)=4.0*R*SS/3.0
TT(\'SP3)=SA
TT(NSP4)=SS*(1.02.0*R/3.0)
WRITE OUTPUT TAPE 6,5
5 FORMAT(1H1///,50X,7 HAA(M,N)//)
DO 1 M= 1 3
7 WRITE OUTPUT TAPE 6,8, ( AA(M,N),N = 1, 10 )
154
)
8 FORMAT(10E12.4)
WRITE OUTPUT TAPE 6,9
9 FORMAT!///,5UX,7HBB(M,N )//)
UO 10 M= i 3
10 WRITE OUTPUT TAPE 6,8,(BB(MN),N=1,10)
WRITE OUTPUT TAPE 6,11
11 FORMAT{///,50X,5HTT(N)//)
WRITE OUTPUT TAPE 6,8, (TT(N),N = 1 10}
RETURN
END
155
SUBROUTINE ABGE TZ
SUBROUTINE ABGETZ(L,NS,AA,BB,TT,U,V,W,
1 ALPHA,BET A,GAMMA,ETA,THETA,ZETA)
DIMENSION ALPHA!1G, 10),BETA(LG,10),GAMMA(10,10),
1 ETA!10,10),THETA!10),ZETA(10),DELTA( 10,10),
2 AllO,10),A A(3,10),B B(3,10),TT(10)*U(10,10,10) ,
3 V(10,10,10),W(10,10,10),C(50),G{50)
CUMMON ALPHA.BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB,TT,U,VW,C,G
DO 5 M= 1 L
DO 4 N= 1, L
ALPHA(M,N)=0.0
BE T A(M,N)=0.0
GAMMA(M,N)=0.0
4 ETA(M,N)=0.0
THE T A(M)=0.0
5 ZE T A(M)=0.0
DO 7 M= 1 L
DO N=i,L
6 DELTA!M,N)=0.0
7 DE LT A ( M M) = 1.0
NSP2=NS+2
NSP3=NS+3
NSP4=NS+4
DO 11 M=1,L
DO 9 N= 1 L
DU 8 J=l, 10
ALPHA!M,N)=ALPHA(M,N)+AA(1,J)*U(M,N,J)
1 + A A { 2 J ) V ( M, N J ) + AA { 3, J ) W { M M J )
BETA!M,N)=BETA(M,N)+BB(1,J)*U(M,N,J)
1 +BB!2,J)*V(M,N,J)+BB!3,J)*W(M,N,J)
8 CONTINUE
GAMMA{M,N) = U!M,N,NS P2)
9 CTA(M,N)=U(M,N,NSP4)/3.0
ZETA(M)=U(M,1.NSP3)
DO 10 J=l,10
10 THETA(M)=TH2TA(M)+TT(J)*U(M,1, J)
11 CONTINUE
WRITE OUTPUT TAPE 6,12
12 FORMAT(1H1 /////, 52X, 10HALPHA{M,N) )
DO 13 M=1,L
13 WRITE OUTPUT TAPE 6,14, (ALPHA(M,N),N=1,L)
14 FORMAT!// 5X,5E20.3/5X,5E20.8)
WRITE OUTPUT TAPE 6,15
15 FORMAT(1H1 /////,52X,10H BETAlM,N) )
DO 17 M=1,L
17 WRITE OUTPUT TAPE 6,14,( BETA(M,N),N=1,L)
WRITE OUTPUT TAPE 6,18
18 FORMAT!1H1 /////,52X,10HGAMMAlM,N))
DU 20 M=1,L
20 WRITE UUTPUT TAPE 6,14, (GAMMA(M,N),N = 1,L)
WRITE OUTPUT TAPE 6,21
156
21 FORMAT{1H1/////,52X,8HETAIM,N) )
U 22 M=1 L
22 WRITE UTPUT TAPE 6,14,(ETA(M,N),N=1,L)
WRITE OUTPUT TAPE 6,23
23 FORMAT(1H1/////,6X,8HTHETA(M),7X,7HZETA(M) //)
WRITE OUTPUT TAPE 6,24,(THETA
24 FORMAT{2F15.7)
RETURN
END
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24. Garelis, E. and Russell, J. L., Jr., "Theory of
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fusion Parameters of Graphite and Graphite Bismuth
by Pulsed Neutron Methods," BNL 719 (C32), Vol. Ill,
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Neutron Thermalization Problem," Nuclear Science and
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tion of Diffusion Cooling in Graphite by Measurement
of the Average Neutron Velocity," Nuclear Science and
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36. Raievski, V. and Horowitz, J., "Determination of the
Mean Free Path of Thermal Neutrons by Measurement of
the Complex Diffusion Length," Peaceful Use of Atomic
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critical Assembly," Proceedings of the University
Subcritical Assemblies Conference, TID7619, 161 (1961).
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Waves in Moderating Media," Nuclear Science and
Engineering 17, 90100 (1963j .
39. Perez, R. B., Booth, R. S., Denning, R. S. and
Hartley, R. H., "Propagation of Neutron Waves in Sub
critical Assemblies," Transactions. American Nuclear
Society 7, 49 (June 1964).
40. Perez, R. B., Booth, R. S. and Hartley, R. H., "Experi
mentation with ThermalNeutron Wave Propagation,"
Transactions, American Nuclear Society 6., 287
(November 1963) .
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42. Hetrick, D. L. and Seale, R. L., "Neutron Waves and
Vibrational Modes in Moderating Materials," Bull. Am.
Phys. Soc. (Abstract) 9, No. 2, 153 (1964).
43. Suramerfield, A., Partial Differential Equations in
Physics. Academic Press, Inc., New York, pp. 126129,
1949.
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McGrawHill Book Co., New York, p. 337, 1960.
45. Meghreblian, R. V. and Holmes, D. K., Reactor Analysis,
McGrawHill Book Co., New York, p. 351, 1960.
46. Weinberg, A. M. and Wigner, E. P., The Physical Theory
of Neutron Chain Reactors, The University of Chicago
Press, Chicago, p. 235, 1958.
47. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics. Part II, McGrawHill Book Co., New York,
p. 1010, 1953.
48. Milligan, J. L., "Polynomial Root Finder Program,"
Share Distribution Number SD# 1215.
161
49. Faddeeva, V. N., Computational Methods of Linear
Algebra, Dover Publications, Inc., New York, pp.
6574, 1959.
50. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics, Part II, McGrawHill Book Co., New York,
p. 875, 1953.
51. Razminas, R., Private Communication, 1964.
52. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics, Part II, McGrawHill Book Co., New York,
p. 552, 1953.
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p. 785, 1953.
BIOGRAPHICAL SKETCH
Hassan H. Kunaish was born in Zabadani, Syria,
in 1933. He completed his secondary education in
Damascus and received the bachelor's degree in Mathe
matical Sciences from the University of Damascus in
1958. In 1959 he received a scholarship from the
Soviet government for a study in nuclear physics at
the University of Moscow.
Sponsored by the American Friends of the Middle
East, Inc., he came to the United States and entered
the Graduate Scool of the University of Florida in
1960 where he received the degree of Master of Science
in Nuclear Engineering in June, 1962. During 1964 he
held a onethird time assistantship at the Department
of Nuclear Engineering while working toward the Ph.D.
degree at the University of Florida.
Hassan H. Kunaish is married to the former
Subhieh AbdulDayem and they have two children. He
is a student member of the American Nuclear Society,
a member of the American Friends of the Middle East,
Inc., a member of the Arab Students Organization in
the U.S.A., and a member of the Arab Club at the
University of Florida.
162
This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.
December 19, 1964
~* * r? y 1
Dean, College of Engineering
Dean, Graduate School
Supervisory Committee:
Date Due
Due Returned Due Returned
Aue rryf
AU6 '1 Tt
OCT ti 7J
SEP 2 9 711
LO
tNGINEERINQ
* PHYSICS
UBRAry
17
(2.52)
(2.53)
Eq. (2.50) is then the sought for thermalization model
of the neutron transport equation in the consistent
approximation with space, energy, and time dependence.
This equation is in a general operator form. To re
write it in a more explicit form we make use of Eqs.
(A.6), (A.7), (A.11), (A.12) and (A.13) which give
1 ala2
[H (E) + F (E)  + ~~
v(E) 3t v2(E) 3t2
a v
+ G(E) ] X(r,E,t)
(2.54)
along with the definitions
(2.55)
X (r ,E,t) = AT xg(r,E,t)
(2.56)
H
(2.57)
k
k
F(E) = l fk(E) 2
(2.58)
24
[H (e)
3 1 32 1
31 + VQ2e 3t2 3
*(r,e,t)=0
(3.2)
where, as noted before, ij>(r,e,t) is the nonMaxweIlian
component of the flux
4>(r,e,t) = m(e)
The corresponding component of the source is x(r,Â£,t)
where
> >
S(r,e,t) = ra(e) x(r,e,t) (3.4)
When the source is sinusoidal in time its
component x(r,e,t) is also sinusoidal and can be
expressed by
x(r,E,t) = SQ(r,e) + Re[S(r,e)e^wt] ^
where sQVr,Â£) and s(r,E)^ut are, respectively, the
timeindependent and dependent components of x(r,e,t) .
Similarly, the nonMaxweIlian component of the flux
has the same time behavior with the same angular fre
quency, f, or angular velocity, w = 2Ttf, but with some
angular phase shift. This phase shift (lag), which is
39
ak r(v* + l k r(v) = bk (v)
O I A L O I 0 0 \r O V
i ,ok
Â£=1
Â£' Â£ Â£ ,k Â£'
(3.59)
where
Â£' ,o
L
I R
Â£=0
(o)
Â£ ,k nl1 Â£
(3.60)
= n
(o) _
Â£ ,Â£ Â£ ,Â£ k Â£ ,Â£
Â£ > 0
(3.61)
bK,(v) = l [R*0}(8 ,6 +2y 6 } + u(v2)
Â£1 Â£,k Â£' fSL 1 ,v Â£',Â£ 2 v
Â£=0
v1
l ( Â¡Â¡ ) R(v"y) (8 6 +2y 5 n r(y)
p = ]_ Â£^k Z 11 1 y Â£fÂ£ 2, V Â£'fÂ£ k
where the Heaviside unit step function
U (v2) = 1 for v i 2
}]
(3.62)
for v < 2
(3.63)
indicates that the term following it exists only for
v > 2 In explicit matrix notations Eq. (3.59) takes
the form
16
by Rj^t), Eq. (2.41) by A(z}/V3, Eq. (2.42) by
B
the resulting set yields
[R1(E,t)Ro(E/t)  V2]
1 1 1
+ yj A(z) x?(r,E,t) + B(x,y)x., (r,E,t)
v 1 yr 1
/6
1
r
B*(x,y) x^1(r,E,t)
(2.46)
R]_(E,t) *(r,E,t) + X(r,E,t)
1 0 +
: A(z) (r,E,t)
y 3
(2.47)
Rx(E,t) (r,E,t) + x];(r,E,t) = B* (x ,y) *(,E,t)
R_^(E,t) tj;^l(r,E,t) + x^(r,E,t) g(x,y) tJ>(r,E,t)
(2.49)
(2.48)
l
If the source is assumed to be isotropic, the
o .>
only nonzero moment of the source will be S0(r,E,t)
or Xo(r>E,t). Then the set of Eqs. (2.46)(2.49) is
reduced to
(R1(E,t)R0(E,t) V2]
x x0(r,E,t) = 0
(2.50)
R,(E,t) *?(r,E,t) ~ A(z) ii(r ,E,t)
11 /T
(2.51)
o a o
133
UNIT NO. 5
FREQUENCY DEPENDENT COMPUTATIONS
210 READ INPUT TAPE 5,2,OMEGA
JOM=JUM+I
UV=6.2832*UMEGA/( 2.2E+05)
DO 300 K=I L
XEV(K)=XS(K)
YEV(K)=YS(K)
DO 2 11 M= 1 L
XR(K,M)=RSR(K,M)
211 YR{K,M)=RSI(K,M)
P= 1.0
DO 250 NU=1f NUM
P = P V/F LOA T F(NU)
GO TO (213,215,217,219,213,215,217,219,213,215),NU
213 XEV(K ) = XEV(K) Y( K,NU)* P
YEV(K)=YEV(K)+X(K,NU)*P
DO 214 M= 1,L
XR(K,M)=XR(K,M)RI(K,M,NU)*P
214 YR(K,M)=YR(K,M)+RR(K,M,NU)*P
GO TO 221
215 XEV(K)=XE\/(K)X(K,NU)*P
YEV(K)=YEV(K)Y(K,NU)*P
DO 216 M=1,L
XR(K,M)=XR(K,M)RR(K,M,NU)*P
216 YR(K,M)=YR(K,M)RI(K,M,NU)*P
GU TO 221
217 XEV(K)=XÂ£V(K)+Y(K,NU)*P
YÂ£V(K)=YEVIK)X(K,NU)*P
DO 218 M=1,L
XR(K,M)=XR(K,M)+RI(K,M,NU)*P
218 YR(K,M)=YK(K,M)RR(K,M,NU)*P
GO TU 221
219 XEV(K)=XEV(K)+X(KNU)*P
YEV(K)=YEV(K)+Y(K,NU)*P
DU 2 20 M = 1,L
XRIK,M)=XR(K,M)+RR( K,M,NU)*P
220 YR(K,M)=YR(KM)+RI(K,M,NU)*P
221 CONTINUE
250 CONTINUE
300 CONTINUE
DO 305 K=1,L
AAA=XEV(K)+BUCK
B8Q=YEV(K)
CHI = SOR T F(AAA**2 + BB8*2 )
RHOR(K)= SQR T F(0.5* (CHI+AAA) )
RHOI(K)=SQRTF(0.5*(CHIAAA))
RRTRI(K)=RHOR(K)*RHOI(K)
R2MI 2(K)=RHORlK)** 2RHQI(K)**2
305 CONTINUE
51
where the subscript s is used to indicate the source and
has no numerical values. To utilize this equation in
determining the unknown constants, Ck n, we first mul
tiply it by Xm<(x) Y {y]fdx dy and integrate it over
the whole extrapolated x and ydimensions. This
operation, along with the orthogonal properties
I3 X (x)
m
 a
X (x)dx
m
a
(3.102)
/. Yn Yn(y)dY = 6 Sn\n
 b
(3.103)
yields the equation
l Ck pk Ek(e) = a t_L + G(e)]E (e)
L m,n m,n m,n s
k
(3.104)
which is valid for all values of m and n, where we have
defined
co ao
am,n = s /a /b dy Xm(x)Xs(x)yn(x)Yg(x)
CO ao *
 a b
35
Furthermore, the whole solution must be repeated for
every value of w since we are interested in the solution
for a wide range of source frequencies.
Solution by the Perturbation Technique
This technique is based on Feenberg's
perturbation method discussed in (47) A modified
version of this technique was developed and used by
Perez and Uhrig (38) to solve a similar problem.
We will use this technique here to solve Eq. (3.35).
First, Eq. (3.35) is rewritten as
L
l
1=0
+ 8
i a
A + y
0
where
(3.44)
R* = At/Ao. (3.45)
Then, it is solved for the case of A = 0 by the exact
method, i.e.,
L
1
1=0
[a
,' Z
 n
i' i
r(o)]A(o)
l
0
(3.46)
with the compatibility condition
V ,i ~ V ,i
\
r(o)  o
(3.47)
I
31
with i' = 0, 1, 2, ..., L. This represents a set of
Â£
L+l homogeneous equations with L+l unknowns, n,
Â£ = 0, 1, L.
In order for this set to have a solution, its compati
bility conditions
A + y
(3.34)
l', Â£=0, 1, 2, ..., L must be satisfied. This
equation will be referred to as the eigenvalue equation
and r will be referred to as the eigenvalue. From this
equation we recognize that r is completely "space
independent" because it is a function of A and the
matrices a,, B , v c, and n ., all of which
are "spaceindependent." Consequently, and for similar
reason, the coefficients, are "spaceindependent,M
m, n
as is seen from Eq. (3.33). This means that all the
spacial modes have the same eigenvalues and the same
coefficients which lead to the formulation of the
eigenfunctions as we will see later. On this basis the
indices m and n, which indicate the (m,n) spacial mode,
can be omitted from Eq. (3.34) which is rewritten as
L
l
Â£=0
[V,* +
B A
Â£ ,Â£
2 _
V,*n\
(3.35)
Â£' = 0,1,2, ,L
21
/ P]>0) "s^o)dlJo
= / Es(vJo)d;jo = Es*o
(2.70)
1
Then the operators, RQ(E,t) and Ri(E,t), take the
energyindependent forms
1 3
R (t) = l E =
o s t v at
1 8 v

R (t) qEs Et v 3t
 (x + k *)
'3D v 31'
where
(2.71)
(2.72)
D = 1/3(Zfc p0Eg)
(2.73)
is the diffusion coefficient. The energyindependent
flux equation is obtained by the substitution of Eqs.
(2.71) and (2.72) in Eq. (2.50).
3D a2 1 3
v2 3t2 + v <1+3EaD)at +
*(rrt)
1 +
3D 3 ~
v at
x(rt)
(2.74)
CHAPTER V
RESULTS AND CONCLUSIONS
The theoretical analysis of the theory of
neutron thermalization in moderating media in the
consistent P.^ approximation was developed in Chapter
II. This analysis is tested in two applications of
practical interest, the neutron waves and pulsed
neutron experiments as presented in Chapters III
and IV. In both applications the heavy gas scatter
ing kernel and the 1/v absorption cross sections are
used, and the first order associated Laguerre poly
nomials are utilized to express the energy dependence
of the flux. All the results presented in this
chapter are obtained for AGOT type graphite with
density equal to 1.67 gm/cm3.
Application to Neutron Waves Technique
The experimental arrangement under consideration
consists of a parallelopiped block of graphite with a
sinusoidally modulated plane neutron source at its xy
face. This source is assumed to have a Maxwellian
distribution in energy and a cosine shape in the x
and ydirections. A 1/v neutron detector is used to
87
Number of Neutrons per Unit
97
Dimensionless Energy Variable, e=E/T
Neutron Spectra at Various Positions along the
Central Axis of the Graphite Assembly.
Fig. 6.
Date Due
Due Returned Due Returned
Aue rryf
AU6 '1 Tt
OCT ti 7J
SEP 2 9 711
136
IF (AS I AD) 377, 378,378
377 TPHASE=TPHASE+3.14159
378 CONTINUE
IF(Jl) 388,387,388
387 AZÂ£RO=TAMP
388 TDTZ=TAMP/AZERO
IF(JQMl) 303,302,303
302 AOMEG(J) = T AM P
303 TOTW=TAMP/AOMEG(J)
DO 386 N=1,NE
HASSAN=0.0
SABtHA=0.0
DU 382 K=1,L
HASSAN = HASSAN+FAMP(K,N)*COSF(FPHASE(K,N))
382 SA3EHA=SABEHA+FAMP(K,N)*SINF(FPHASElK,N))
SAMP(N)=SQRTF(HASSAN**2+SABEHA**2)/TAMP
SPHASE(N)=ATANF(SABEHA/HAS SAN)
IF (HASSAN) 383,384,384
383 SPHASElN)=SPHASElN)+3.14159
384 CONTINUE
386 CONTINUE
WRITE OUTPUT TAPE 6,390,Z
390 FORMAT(//5X,5HAT Z= F6.2,19H CM FROM THE SOURCE /
1 5 X,30( 1H)// )
WRITE OUTPUT TAPE 6,391, (K,DAMP(K),DPHASt(K),K=1,L)
391 FORMATt5X,1HK,10X,4HDAMP,13X,6HDPHASE/10(I6,2E20.8/) )
WRITE OUTPUT TAPE 6,379,TAMP,TPHASE
1 ,TOTW,TPHASE,TOTZ,TPHASE
379 FORMAT( / 5X,4HTAMP,E17.3,E20.8/
1 5X,4HTOTW,El7.8,E20.8/
2 5X,4HTOTZ,E17.8,E20.8)
IF(EINT) 396,396,376
376 CONTINUE
WRITE OUTPUT TAPE 6,392,(K,K=1,L)
392 FURMAT(/5X,9HFAMP(K,E)/5X,11HFPHASE(K,E) /
i 6X,3HE K, 16,9110,10X,5HTCTAL )
DO 393 N = 1, N E
Â£1=EINT*FLUATF(N1)
WRITE OUTPUT TAPE 6,394,E1,(FAMP(K,N),K=1,U,SAMP(N)
393 WRITE OUTPUT TAPE 6,395,lFPHASE(K,N),K=1,LJ,SPHASE(N)
394 F0RMAT(/5X,F5.2,10E10.3)
395 FORMAT(10X,10E10.3)
396 CONTINUE
400 CONTINUE
401 CONTINUE
GO TO 210
END
LO
tNGINEERINQ
* PHYSICS
UBRAry
27
2m~l
Xj^x) = cos TIX
2 I
(3.15)
Yn(y) cos
2nl
*y
2b
(3.16>
Zm,nU) exP<pm,nz> Uexp<2pm(n(Sz) )]
exp(~pm#nz)
(3.17)
E
m,n
(e)
I
l
a* t*1*
A Lo
m,n 1
U)
(3.18^
The terra in brackets is the end effect correction due
to the finite size of the system in the zdirection and
p is the inverse relaxation length of the (m,n)
ran
spacial mode. The energy dependence of the (m,n) mode',
Em n^e^ exPanc^ed in the normalized associated
Laguerre polynomials of the first order, (e) .
These polynomials and some of their properties are
discussed in Appendix C.
The substitution of the solution given by Eq.
(3.14) in Eq. (3.7) gives the equation
64
where
A
in, n, p
X
m,n,p
(4.17)
B
ra,n,p
/ 2m1 \2
2ni
2 /2pl
it +
IT
l 2a /
2b
1 2c
("4.18)
Eq. (4.16) contains all the spacial modes corresponding
to all the combinations of m, n and p values. To make
this equation more useful we can separate it into a
set of uncoupled equations by operating on it with the
integral operator
0 =
dx
dy
dz Xm'^
v (y>
v(z)
(4.19)
and using the orthogonality property
/2q'1 ^
(2q1
cos TTU
cos
TTU
\ 2d 1
i 2d
d 6
q)q
(4.20)
which applies to each of X^Cx), Yn(y) and Zp(z). This
leads to the modal equation
Cm,n,p
2 _i
H(e)A F(e)+A e +
m,n,p v m,n,p
2
m,n,
P
(4.21)
159
24. Garelis, E. and Russell, J. L., Jr., "Theory of
Pulsed Neutron Source Measurements," Nuclear Science
Engineering 16, 263270 (1963) .
25. Garelis, E., "Theory of Pulsing Techniques," Nuclear
Science and Engineering 16., 263270 (1963) .
26. Vertes, P., "Some Problems Concerning the Theory of
Pulsed Neutron Experiments," Nuclear Science and
Engineering 16, 363368 (1963).
27. Daitch, P. B. and Ebeoglu, D. B., "Asymptotic and
Transient Analysis of Pulsed Moderators," Nuclear
Science and Engineering 17, 212219. (1963).
28. Maiorov, L. V., "Asymptotic Distribution of Thermal
Neutrons Far from a Planar Source," BNL 719 (C32),
Vol. IV, 13751390 (1962).
29. de Saussure, G., "The Neutron Asymptotic Decay
Constants in a Small Crystaline Moderator Assembly,"
BNL 719 (C32), Vol. IV, 11581179 (1962).
30. Corngold, N., "The Phase Integral Method in Neutron
Thermalization," BNL 719 (C32), Vol. IV, 10751102
(1962).
31. Ohanian, M. J. and Daitch, P. B., "Eigenfunction
Analysis of ThermalNeutron Spectra," Nuclear Science
and Engineering 19, 343352 (1964).
32. Ohanian, M. J., "Eigenfunctional Analysis of Thermal
Neutron Spectra," Ph.D. Dissertation, Rensselaer
Polytechnic Institute, Troy, New York (1963).
33. Starr, E. and Price, G. A., "Measurement of the Dif
fusion Parameters of Graphite and Graphite Bismuth
by Pulsed Neutron Methods," BNL 719 (C32), Vol. Ill,
10341073 (1962).
34. Koppel, J. U., "A Method of Solving the Time Dependent
Neutron Thermalization Problem," Nuclear Science and
Engineering 16, 101110 (1963).
35. Starr, E., Honeck, H. and deVilliers, J., "Determina
tion of Diffusion Cooling in Graphite by Measurement
of the Average Neutron Velocity," Nuclear Science and
Engineering 18, 230235 (1964).
o o o o
134
UNIT NO. 6
APPLICATION OF SOURCE CONDITION
TO DETERMINE FINAL CONSTANTS
DU 350 M= 1 i L
ML=M+L
C(M,L2P1)=3.0*THETA!M)
C(ML,L2P1)=3.0*OV*ZETA!M)
DO 349 K=I,L
KL=K+L
C(M,K)=RHUR(K)*XR(K,M)RHOI!K)*YR(K,M)
C!ML,K)=RHUR!K)*YR(K,M)+RHO I lK)*XR(K,M)
CIM,KL)=C(ML,K)
349 C(ML KL)=C(M L)
350 CONTINUE
CALL ELEM(L2 C DET)
DU 360 K=I L
KL=K+L
BR(K)=C(KfL2Pl)
BI (K ) =C(KL L2P1 )
DU 355 M=1 L
AR(KfM)=BR(K)*XRlK,M)BI(K)*YRlKM)
35 5 AI(K,M)=BR(K)*YR!K,M) + BI(K)*XR(K,M)
360 CONTINUE
WRITE OUTPUT TAPE 6,362,L,NUM,BUCK,OMEGA
362 FORMAT!1HI////5X27HTIME DEPENDENT SOLUTION FOR /
1 5X,28HTHE NEUTRON WAVE PRDPOGATI UN/
2 5X 2 8HN0. OF LAGUERRE POLYNOMIALS= 12/
3 5X,2BHMAXIMUM ORDER OF DERI VAT IVE = 12/
4 5X,20HTRANSVERSE BUCKLING= EL5.6 /
5 5X,20HFREQUEIMCY (IN CPS) = E15.8 /
6 5X,35(1H*) )
DO 365 K=1 L
WRITE OUTPUT TAPE 6,363,K,XEV{K),YEV(K),RHOR{K),RHOI(K)
l RRTR I (K)t R 2 MI 2(K),3R!K),BI!K)
363 FORMAT!/ 5X2HK=I2/5X,5HXEV = E16.8/5X,5HYEV =Â£16.3/
1 5X,5HRH0R=EI6.8/5X,5HRH0I=E16.8/
2 5X6HRRTRI = EI6.8/5X 6HR2MI2 = EI6.8/
3 5X,3HBR=EL6.8/5X,3HBI=E16.8 )
WRITE OUTPUT TAPE 6,364,tXR!K,M),YRlK,M),
I AR(K,M),AI(K,M),M=L,L)
364 FORMAT!/ 9X,7HXRlK,M),I3X,7HYR(K,M),13X,
1 7HAR!K,M),I3X,7HAI(K,M)/1014E20.8/))
365 CONTINUE
CHAPTER I
INTRODUCTION
The theory of neutron thermalization has been
given much emphasis in recent years. The interest of
investigators has been focused on two objectives.
The first objective is the theoretical and experi
mental understanding of the scattering law of thermal
neutrons in various materials, while the second
objective is to use the best available information
about the scattering law in order to study the details
of the neutron thermalization process and the neutron
spectra which are important in determining the
behavior and properties of nuclear systems. The
mathematical framework of the latter is the develop
ment of an adequate solution for the Boltzmann
equation describing the neutron transport phenomenon.
The work presented in this dissertation can be
classified in the second category.
In solving this equation investigators followed
several approaches and based their solutions on certain
assumptions in order to reduce the complexity of the
mathematics involved in the problem. Effort was made,
1
127
NOT SPACIAL COMPUTATIONS ARE DESIRED,RESPECTI VELY.
NZ = NUMBER OF PuSITHINS ON THE ZAXIS AT WHICH COMPUTATION
IS DESIRED
POINT(J) = THE ZVALUCS OF THESE POSITIONS
ALPHA,BETA,GAMMA,EI A,THETA AND ZETA ARE THE INPUT MATRICES
WHICH,IN THE CASE OF THE HEAVY GAS MODEL,ARE OBTAINED
FROM THE HGM CODE.
S ( M, N I = COEFFICIENT OF THE (N1ITH POWER OF E IN THE
(MI)TH LAGUERRE POLYNOMIAL OF THE FIRST ORDER
OMEGA = SOURCE FRCOUENCY IN CYCLES PER SECOND
THE DESIRED VALUES OF FREQUENCY ARE PUNCHED EACH ON
A StPERATE CARL) AND ARE RE AD ONE AT A TIME ACCORDING TO
STATEMENT NO. 210 SEE THE ( GO TO 210 ) STATEMENT AT THE
END OF THE NWP CODE.
2 UUI PUT VARIABLES
XS(K) = RP OF STEADY STATE EIGENVALUE UF THE (KITH MODE
YS(K) = IP UF STEADY STATE EIGENVALUE OF THE (KITH MODE
XlK,NU I = RP OF
Y(K,NU I = IP OF
RR(K,M,NU) = RP OF R
HI (K,M,NU I = IP OF R
X E V ( K I = RP OF
YCV(KI = IP OF
RHOR(K) = RP OF
RHOI(KI = IP OF
RRTKI(K) = RHOR(K)*RHOI(KI
R2MI21K) = RHOR(K)**2RH0l(K)**2
BR(K) = RP OF C
Ei 1 { K ) = IP OF C
XR ( K M ) = RP OF R
YR(K,M ) = IP UF R
AR(K,MI = RP OF C .R
A I (K MI = IP OF C .R
DAMP(KI AND OPHASC(K) =AMPLITUDE AND PHASE SHIFT,RELAI IVE
10 THE SOURCE ,CF THE (K) Th MODE OF NEUTRON DENSITY
TAMP TOTAL AMPLITUDE OF THE NEUTRON DENSITY
TUTW =TOTAL AMPLIODE OF THE NEUTRON DENSITY
NORMALIZED TO 1 AT ZERO FREQUENCY
TOTZ =TOTAL AMPLITUDE OF NEUTRON DENSITY
NORMALIZED 0 1 AT THE SOURCE
FAMP(K,NI AND FPHASÂ£(K,N) =AMPLI TUDE AND PHASE EHFT OF THE
(K) TH MODE AT THE (N) TH ENERGY POINT. THIS
APPLIES FOR BOTH FLUX AND DENSITY.(THE NUMBER ND
DICIDES WHICH ONL IS COMPUTED)
LAST OuLUMN In THE SPECTRUM OUTPUT CORRESPONDS TO THE TOTAL
FLUX OR DENSITY SPECTRUM,DEPEND ING ON THE VALUE UF ND.
i DEiCX
THIS SUBROUTINE EXPANDS A DETERMINANT OF SIZE LSD THE
GENERAL ELEMENT OF WHICH IS A POLYNOMIAL OF SIZE NSE IN
THE UNKNOWN EIGENVALUE, THE RESULT OF THIS EXPANSION
TIME, ENERGY AND SPACEDEPENDENT
NEUTRON THERMALIZATION THEORY
By
HASSAN H. KUNAISH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1964
o o o o o o
130
DO 12 M=l,iO
12WRITE OUTPUT TAPE 6,8,(GAMMA(M,N),N=1,1O>
WRITE OUTPUT TAPE 6,21,(THETA(M),ZETA(M},M=1,10)
21 FORMAT(1H1////,8X,8HTHETA(M),l5X,7HZETA(M)/iO(2E20.8/))
UNIT NO. 2
PRELIMINARY COMPUTATIONS
DO 14 M=11 10
DO 13 N= 1 10
13 DELTA!MN)=0.0
14 DELTA(M,M)=1.0
DO 60 J = 1 NE
EPS = EINT*FLOATF(Jl )
DN=1.0FLA TF(ND)/2.0
WELL=(EPS**DN)*EXPF(EPS)/{220000.0**ND)
DO 89 M = 1t L
E(M, J)=0.0
DO 98 N = 1 L
38 E(M J) = E(M J)+ S(M,N)* W E L L *(E P S *(N1)I
59 CONTINUE
60 CONTINUE
UNIT NO. 3
STEADY STATE EIGENVALUES
DO 16 M= 1 L
DO 15 N=1 L
CC(M,N,1)=ALPHA(MN)
15 CC(M,N,2)=ETA(MN)
16 CONTINUE
LP1 = L + 1
LM1 = L~1
L 2 = 2 L
L2P1 = 2*L + 1
L2Ml=2*Li
L2M2=2 *L2
CALL DETEX(L,2,CCB0E)
WRITE OUTPUT TAPE 6,17
17 FORMAT(1H1///,5X,21HSTEADY STATE SLUTI ON/5X,21( 1H*)//
1 5X,2H J,10X,6HB0E(J)/5X,2H,10X,6H )
WRITE OUTPUT TAPE 6,18,(J,BQElJ),J=1,LP1 )
18 FORMAT(17,E20.8 )
CALL POLY(L,BOE,XS,YS)
WRITE OUTPUT TAPE 6,19
19 FORMAT(// 5X,24HSTEADY STATE EIGENVALUES /5X,24(1H=)/
1 5X,2H K,11X,5HXS(K),13X,5HYS(K)/3X,2H,11X,
2 5(1H),13X,5( 1H) )
WRITE OUTPUT TAPE 6,20,(K,XS(K),YS(K),K = 1,L )
20 FORMAT(I7,2E20.8)
28
I [H(e) + AF (e) + 7 V2 (P2 _b2 )]
e 3 m,n Jm n'
m,n
, .cm,Xm(x)Yn(y)Zm(n(^)Em(n(c) O (3.19J
where
A
and
B?
im,n
(3.20)
(3.21)
is the transverse buckling of the (m,n) spacial mode.
Eq. (3.19), which contains all the spacial modes, can
be separated into a decoupled set of equations each of
which involves one mode only. This decoupling is
achieved by operating on Eq. (3.19) by the special
integral operator
P
as 
Ja dx / dy (x) Yn, (y)
a 6
(3.22)
This operation and the recognition of the orthogonality
properties,
component, */(r,E,t), is expected to be a smooth function
of energy, E, and hence its value at a different energy,
E1, can be well approximated by a finite Taylor series
expansion about E. This analysis and the detailed
balance principle are used in order to evaluate the
scattering integral. The resulting form of the trans
port equation is a fourth order differential equation
with respect to time, energy and space.
This theory is used in two applications of
practical interest, the neutron waves and the pulsed
neutrons experiments. The use of the cosine functions
in expressing the spacial dependence of the neutron flux
leads, in both cases, to the modal equation of the flux
as a function of time and energy. The energy dependence
of the flux modes is then developed in terms of the
associated Laguerre polynomials. This procedure leads
to two similar complicated eigenvalue problems.
The source condition in the neutron waves problem
is established from the P^ approximation and is used to
determine the modal constants. In order to account for
transient behavior of the pulsed systems the principle
of neutron conservation is used and the final solution
of the time dependent flux is obtained by the technique
of Laplace transformation.
i
vi
13
where the energy moments are
Mk (E)
m
;rr!:m(E)ivE>= in 1 (e'_e) zm(E*E,)dE'
E'
(2.35)
Furthermore, the integrals of Eq. (2.34) can be written
in operator form as
l(,E,t) = mtEjQ^Ej^r^t)
n
(2.36)
where the operators Q^E) are defined as
Qm = I 'VDk
k=o
(2.37)
The use of Eqs. (2.22), (2.27), (2.28) and
(2.36) in Eq. (2.21) yields:
RmiEft) ij;^(r,E,t) + *
f (m+lrn) (m+l+n) ^ n + (mn) (m+n) **
A(z)j^{ _} t;m(r,E,t) + { }
(2m+3)(2m+l)
(2m+l)(2n+l)
+ B(x,y)
2
x 'P^_1 (r ,E,t)
(m+n+2)(m+n+1) ^ n+1
(2m+3)(2m+l)
1 V+l (r,E,t) {
(mn)(mn1) %
) >
(2m+l) (2ml)
n+l,> ^ ..X
* *ml(r'E't)
88
measure the neutron density at various distances from
the source along the central zaxis of the graphite
assembly. The experimental technique and the data
analysis are fully discussed by Booth (41) who obtained
the amplitude and phase shift of the neutron density
at various positions in the graphite assembly for
several source frequencies. The errors in all Booth's
preliminary results used here are typically of the
order of 3 per cent.
A code, NWP, has been developed for the IBM709
to calculate all the quantities involved in this dis
cussion. A listing of this code and the subroutines
associated with it is found in Appendix D. In all the
computations the fundamental spacial mode alone was
considered. Therefore, the spacial indices are dropped
from all the quantities involved in this presentation.
The computational scheme follows the analytical steps
of Chapter III and the perturbation technique is used
to solve the eigenvalue equation, Eq. (3.35).
The use of Laguerre polynomials in the expansion
of the energy flux modes, E (e), leads to a number of
eigenvalues equal to the number of polynomials in this
expansion. Each eigenvalue, r^, is related to the
inverse relaxation length, pk through Eq. (3.36).
I
66
where ft = 0, 1, 2, .. ., and
a.,. = / m(e) L^;(e) H(e) L(1)(e) de
ft ft J Â£ Â£
O
(4.25)
0 t = 0/ m(e) L(l)(e) F(e) LllJ(e) de
(1)
Â£, ft Â£ i
(4.26)
T = / m( e ) L^Ce) i L(1)(e) de
ft, Â£ ft ft
O E
(4.27)
3_
nÂ£Â£ = m(e) L1)(e) de = 6
* >0 ft Â£ 3 *
(4.28)
Notice that the matrices a f B f y and n,,,^
Â£, Â£ ft j ft ftjft ,
are exactly the same as those for the neutron waves.
See Eqs. (3.28)(3.31) of Chapter III.
If the maximum value of the index, Â£ is L, then
Eq. (4.24) represents a set of (L+l) linear homogeneous
equations in (L+l) unknowns, n where Â£ Â£ =0, 1,
2, ..., L. The compatibility condition of Eq. (4.24)
is that its determinant be zero.
ft*,ft V,Â£ A
m,n,p
+ y.t A 2
ftft m,n,p
VftBm n n
1 mn p
= 0
V
= 0, 1, 2,
/
L
(4.29)
116
The Functions, h^e), f^CO and g^Ce)
These functions are obtained by direct substitution
of Eqs. (B .26) (B .34) in the transformation given in Eqs.
(A.33) (A.35) .
hJe5 = in E0 I a e_3/2 + 2 a e "1 + (1 Z y ^ 0 1 a e ~1/2
(B.35)
hi!0 = 2us2e1 .a, EoEaoel/2 + (4 u 2 4Â¡Â¡) E 2
+ 4,Io!;a Â£ 1/2 + 2U(1 ) r 2 c (B.36)
J o 3
h2 (e) = E0 E e1/2 9 y2 E2 + y 2 2y ) E 2 e
 j y2 E e2 (B 37)
f0U) = j y e3/2 + 2 1&q e"1 + (12d) Z0 e1/2 (b.38)
f1(e) = H y E Q e'1/2 + y E Q e 1/2 (B.39)
f2(e) = ju E0 eV2
(B.40)
37
This expansion is restricted by the condition
U)
v0 (rad/cm) < 1
(3.53)
( V ) ( V )
Once we find r and R the solution is obtained
l ,k
from Eqs. (3.49) and (3.50) by the substitution of the
numerical value of a.
It remains now to determine the derivatives
r(v) and R^v This determination is done step by step
from Eq. (3.48). First, the first derivative of the
set (3.48) with respect to A at A = 0 is equated to
zero and solved for r,^ and R^ in terms of and
k A,k k
, (o)
'a ,k
which have been determined from Eqs. (3.46) and
(3.47). Then the same process is used for the second
derivative of Eq. (3.48)^ to solve for I*/2) and R^2^ in
k i ,k
terms of r^^, and R^, and so on. Suppose
k k i,k Â£,k
now that we have found r ^p) and R ^ ^ for y = 0, 1, 2,
k , ,k
..., v1 and want to find and R^v,^. Applying the
k l ,k
rule
9x'
[A(x) B (x) ]
l {i)AU)(x, B(VB,(X,
y=o
(3.54)
CHAPTER IV
PULSED NEUTRONS IN MODERATING SYSTEMS
The analytical treatment of the pulsed neutron
problem is similar to that of the neutron waves. A
major difference, however, arises in connection with
the extra multiplicity introduced by the second order
time derivative which introduces a quadratic term in
the time eigenvalue. For this reason and in order to
obtain a consistent set of conditions sufficient to
determine all the modal amplitudes, we use the invari
ance of the expectation value of neutron population,
* i
\$ <)>}, as our starting point. This involves the
v
knowledge of the adjoint flux which increases the
complexity of the problem.
The pulsed neutrons problem will be studied
here starting with Eq. (2.61). This linear inhomo
geneous equation is rewritten here for convenience.
[H( e) + J_ F(e)  + _1_ I A2] <^r,e,t)
vc at v02e at2 3
3 [G( e) + L ] xtf' e
v0 /F
60
67
This equation will be called the eigenvalue equation
since its solution determines the time eigenvalues,
A which lead to the eigenfunctions, T(t) ,
m,n,p m,n,p
according to Eqs. (4.14) and (4.17). Each element in
the determinant of Eq. (4.29) is, in general, quad
ratic in A Therefore, the expansion of this
m,n,p
determinant is a polynomial in A with a maximum
m,n,p
power equal to 2(L+1). Hence, the solution of Eq.
(4.29) yields 2(L+1) eigenvalues, A which lead
m, n,p
to 2(L+1) time eigenfunctions
Tk(t) = exp(Ak t) (4.30)
m,n,p m,n,p
where
Ak = v Ak
m,n,p m,n,p
(4.31)
and where k = 1, 2, ..., 2(L+l). The multiplicity in
the time eigenvalues leads to the multiplicity not only
in the time eigenfunction but also in the energy modes,
since, for each eigenvalue, Ak there corresponds a
m, n,p
Â£ )C
set of constants, A as a solution of Eq. (4.24).
' m, n, p M
Each set of constants leads to the formation of an
energy mode given by
Ek(e) = l AÂ£k L(1)(e)
m,n,p k m,nfp *
(4/32)
12
To put the integrals In into a more workable form, the
m
detailed balance principle will be used. This prin
ciple states that an equilibrium is established for
the product of the Maxwellian distribution and the
scattering kernel between any two energy intervals,
i.e. ,
mE'jZgtE'fE) = m (E) Eg (E*E 1 ) (2.31)
Notice that the relation (2.31) is completely inde
pendent of the amount of absorption in the medium and
the size of the medium. Eq. (2.31) can be general
ized for the moments of the kernel and is then stated
as
ra(E')E (E+E) = m(E)Â£ (E+E') (2.32)
mm v
Eq. (2.30) can be combined with Eq. (2.32) to yield
I^(r,E,t) = m(E) / Â£ (EE1) l L (EE)k
m E' m k=o k!
x DktÂ£(r,E,t) (2.33)
m
or
I^(,E,t) = ra(E) l
k=o
Mn
k,n,> .
D *m(r,E,t)
(2.34)
107
the ith energy derivative gives rise to the factor (1/T)^
according to the relation
_3_ 2_ iL 1 i_ (A.25)
3E 3c 3 E T 3e
The transformed velocity gives rise to the term eV2
according to
v (E)
voe
(A.26)
The transformed flux equation was given by Eq. (2.61)
where the transformed operators are
1 3 i g
H(e) = H(E) = h0(E) + hx(E) + h2(E) (A.27)
3 e T 3 e
F(tÂ¡ =F(E) =lto(E) +1 +lj f2(E) lj]
/c /e 9e 1 3 c
(A.28)
2
G(e) = G(E) = g0(E) + g.(Er + \ 92(E) i (A.29)
T 1 3eT 2 2
_ 1/2
Notice that the e term, which arises from the
transformation of l/v(E) preceding the operator, F(E),
in Eq. (2.54), is now included in the operator F(e). If
we express the transformed operators by
121
The use of Eq. (C.9) gives
nF{mnx) + m F (m+l n+11 x) =
(m+n) +
m + JEkHtii
n + 1 .
m(nHl) + m(m+l) (m+2)1
. n+1 (n+1) (n+2) J 21
m(m+l) (m+2) m(m+l) (m+2) (~m+3)
+
^n+1)(n+2)
(n+1) (n+2) (n+3)
31
(C.ll)
If the algebra is carried out, this relation takes the
form
nF(mnx) + m F (m+l n+11 x) = (m+n) F{mn+lÂ¡x) (C.12)
Introducing Eq. (C.12) in Eq. (C.10) one obtains
/n (x) +m = F(mn+lx) j(m+n) ij / ml nl (m+n)
(C.13)
Finally, a simple comparison of the righthand side of
Eqs. (C.8) and (C.13) yields the important recursion
relation
n
(x) = (m + n;
m
n1(x) + Ax)
L m m~*
(C. 14)
8
H~n 3 [(m+n)l/(mn)!] (2.11)
dn
Pra(w) 5 sinne Pm(M) (2.12)
m dyn m
P^n(u) = (l)n[(mn)i/(m+n)l]PÂ¡J(y) t (2.13)
Pm(^(
(mn) 1 n infipvM
.> I 7=TT7 V*>*Â£<>o3
nm
(2.14)
where
(2.15)
(2.16)
y = COS 9 = Si z
y' = COS 01 = SI1 Z
and where e,
azimuthal and longitudinal angles of the directions
si and si'.
The moments of the flux, the source and the
scattering kernel are given by
n(i,E,t) = / ^(n) (r,n,E,t)dn (2.17)
m m
SI
sÂ£(i,E,t) = / Ym(8)S(r,n,E,t)dn
SI
(2.18)
92
between the theoretical results of the approximation
and the experiment is obtained for frequencies up to
500 cps. For higher frequencies the theoretical curves
diverge from the experimental values giving a smaller
real component and a greater imaginary component of po,o
leading to less attenuation and greater phase shift,
respectively. I
A more appropriate comparison may be made between
the calculated and the experimental total neutron densi
ties. In Fig. 3 and Fig. 4 the amplitude of the neutron
density is plotted as a function of position for source
frequencies of 100 and 500 cycles per second. In both
cases the theoretical results agree very well with the
\
experimental values up to about 25 cm from the source.
The disagreement beyond this position becomes more
pronounced where the effect of the higher modes dis
appears, due to their fast attenuation, and the
fundamental mode dominates. This disagreement may be
reduced by including more modes in the combined solution
since the higher modes tend to raise the amplitude near
the source and hence increase the slope of the theoreti
cal curve toward that of the experimentally measured
one.
The theoretical phase shift of the neutron
density is in a much better agreement with the
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF FIGURES iv
ABSTRACT V
Chapter
I. INTRODUCTION 1
II.THE THERMALIZATION MODEL OF THE CONSISTENT
PL APPROXIMATION OF THE NEUTRON TRANSPORT
EQUATION 5
III. NEUTRON WAVES IN MODERATING SYSTEMS 23
IV. PULSED NEUTRONS IN MODERATING SYSTEMS .... 60
V. RESULTS AND CONCLUSIONS 87
APPENDIXES
A OPERATORS IN GENERAL FORMS 102
B HEAVY GAS 1/v MODEL 109
C CALCULATION OF THE MATRICES
Bn', l' yi\l . i 118
D COMPUTING CODES 126
LIST OF REFERENCES 157
BIOGRAPHICAL SKETCH 162
iii
83
coefficients of C1^ in both sides of Eq. (4.85) must
m, n, p
be equal. Hence, the following set is obtained.
I T+k(t) [JL akk {JL xk' }+ _1_ bk*k ]jk(t)
k ran,p v 2 m,ntp 9t m,n,p vq m,n,p m,n,p
= l /t T+k(t) { ck + dk h (t) dt
Bc k m*nP v0 m*n*P 3t m,n,p s
(4.91)
This equation holds true for all the values of m,n,p
and k'. Hence it represents (M.N.P.K) differential
equations in (M.N.P.K) unknowns, Jk n where M,N,P
and K are the total numbers assigned to m,n,p and k (or
k1), respectively. Rearranging Eq. (4.91) and integrat
ing the righthand side by parts it becomes
k' 1
l T
k vc2
k'k
a
Ak akk}] Jk(t)
= [C ck T+k( t) T (t) + { dk Ak ck } / Tk(t)Tc(t)dt]
a5c k vo oS
(4.92)
Phase Shift (rad.
r
Fig. 5. Phase Shift of Neutron Density vs. Distance from the Source.
120
It remains now to calculate the matrices, U^. V"*
and WJ. This will be done after we discuss some of
l \ l
the essential properties of Laguerre polynomials.
The nonnormalized associated Laguerre
polynomials, ^(x), are related to the hypergeometric
functions, F(mnx), by the equation
(x) = F (m  n+1  x) [ (m + n) 1] ^ / ml nl (C.8)
where
F (m n x)
1 +
m
n 11
+ m(m+ 1) X2 +
n (n + 1) 21
m (m + 1) (m + 2 ) ... fm + p 1} xP
1 1 r t i i 11 1 f
n (n + 1) (n + 2) ... (n + p 1) p
n > 0 (C.9)
[See (52).] Using these relations we will develop a new
recursion relation connecting the Laguerre polynomials
of different orders. From Eq. (C.8)
n1 n
Ji (x) +// (x) = n F(m I n I x) + m F (m +1 I n+1 I x)
m ml 11 11
2 2
. [ (m + n) 1] / ml nl (m + n)
(C.10)
6
E (E) $ (r ,E, t) dr dadE =
S(r,S,E,t) drdadE
/ dE'J da' es(e'*e,')
e' a'
x ,E' ,t) drdadE
the loss rate due to
absorption, scattering
out and slowing down,
rate of contribution
from all sources in the
space dr da dE,
contribution into dr da
dE due to scattering
from all directions a'
and all energies E'.
The total space, energy and time dependent scalar flux
is obtained by integrating the directional flux over
all directions
(Hr,E,t) = / (r,a,E,t) da
a
(2.2)
In order to eliminate the angular (direction)
dependence from Eq. (2.1) the flux, the source, and
the scattering kernel are expanded in terms of the
spherical harmonics
4> (r ,a,E,t)
m
 I l Or,E,t> Y"(fl)
(2.3)
ra=o n=m
s(r,a,E,t) = l l s"(r,E,t) y"()
(2.4)
m=o n=m
89 1
These eigenvalues were calculated using numbers of
polynomials ranging from 1 to 10. First one polynomial
is used and the fundamental eigenvalue, rQ# is obtained.
Then the number of polynomials is increased by 1 in
each step and the next higher order eigenvalue arises.
It was found that the use of L+l polynomials does not
change the L unperturbed eigenvalues obtained using L
polynomials, but merely gives rise to the L+lth eigen
value which is greater than all the previous ones.
Furthermore, the difference between two eigenvalues
of successive orders decreases with increasing orders
as seen in Fig. 1. Therefore, it is expected that the
highest ordered eigenvalues tend to be so closely
spaced that their distribution may approach a continuum.
The frequencydependent complex inverse
relaxation length, p of the fundamental energy mode is
plotted vs. source frequency in Fig. 2 and is compared
with the experimental values obtained by Booth. Based
on the analytical development of the neutron waves
problem in the diffusion equation by Perez and Uhrig (38)
calculation was made in the heavy gas and heavy crystal
models. The matrix elements for the later model were
obtained from Razminas (51) and the results are also
presented in Fig. 2. A very reasonable agreement
73
where c+ are constant, Xfx), Y_(y) and Z (z)
m, n,p In 11 p
those given by Eqs. (4.11} (4.13) and
T+(t) = exp(A+ t)
m 9n }p m,n fp
E+ ( e)
m ,n ,p
l
+ Â£
!l.
m,n,p
t(1)r ^
Li
The use of Eqs. (4.11)(4.13), (4.47), (4.48) in
(4.44Y and then the application of the operator
oo
O = Â¡de L(1)(e)
o Â£'
lead to the adjoint modal equation
[H+(e)
A+ F+(e) + i A+ + \ B2 ]
m,n,p G m,n,p 3 m,n5p
E +
m,n,p
m( e )
0
where
A+ X+ /
m,n,p m,n,p
are
(4.48)
(4.49)
Eq.
(4.50)
(4.51)
(4.52)
52
For given values of m and n Eq. (3.104) couples all the
unknowns, Ck corresponding to all the values of k.
in, n
This equation can be reduced to a coupled set of equa
tions by the multiplication by ra(e )L^(e^de and
k'
integration over e from 0 to .
k k
Cm,npm,ndk',k ara,n^A ^k' + 9k'^ (3.106)
k
where
d f = / m( e) L(1) (e) Ek(e)de
k k q k1
C = / m(e) (e) E (e)de
k* Q k* /7 s
6 = / ra(e) (e) G(e)E (E)de
k' k' s
o
(3.107)
(3.108)
(3.109)
Using the expression for Ek(e) given by Eq. (3.72) in
Eq. (3,107) and considering the orthonormality property
of the Laguerre polynomials, one obtains
/ m(e) (e) I ^ (e)de
o i '
= y R 6 = R
1 Z,k k' A k' ,k
z
(3.110)
Amplitude
r
Fig. 4. Amplitude of the Combined Neutron Density vs. Position along the
zaxis for Source Frequency of 500 cps.
84
where the indices, m,n,p, have been dropped for
simplicity, If Tq(t) is a known function the integral
o
in Eq. (4.92) can be evaluated and this set of equa
tions can be solved for the unknown functions, Jk(t).
Formally, this completes the solution for the
flux in the pulsed neutrons problem. To go into more
detail in the solution a specific case has to be
treated. Assume that the pulse takes a rectangular
shape as a function of time. Analytically, this pulse
can be given by
Ts(t) = u(t) u (t T),
(4.93)
where x is the width of the pulse and u(t) is the
heaviside unit step function. With the aid of Eq. (4.93)
it can be seen that the righthand side of Eq. (4.92) is
a function of time in the range 0 < t < T while it
becomes a constant in the range t 3 t Therefore,
this set must be solved in both regions, where it takes
the two forms
l T+k(t) for t < i
abc k4 ^k
(4.94)
(4.95)
2
however, in order for the problem to retain certain
aspects of its physics depending on the goal of the
investigator.
'ft
Wigner and Wilkins (A/ 2.) established the
monoatomic gas model of the scattering kernel on
which investigators (3. 17} and many others, based
their works later on. Using this model, Barnard et
aJL. (16) studied the time dependent neutron spectra
in graphite varying the moderator to neutron mass
ratio. The analytically calculated spectra for a
fictitious mass ratio of 33 were in good agreement
with the experimental results while for the mass ratio
of 12 this agreement disappeared at times less than
300 y sec after the pulse and for energies below 0.1
ev.
Corngold et cil, {!_) and Shapiro (14) showed
that, in the analysis of the time dependent thermaliza
tion problem in the neutron pulse technique, the
consideration of a discrete set of eigenvalues is not
adequate and continuous eigenvalues should also be
considered for complete analysis. However, the ques
tion of how important these continuous eigenvalues and
* Underlined numbers will be used to indicate
references cited in the List of References of this
dissertation.
O 100 200 300 400 500 600 700 800 900 1000
Source Frequency (cps)
Fig. 2. Components of the Complex Inverse Relaxation Length of the
Fundamental Mode vs. Source Frequency.
APPENDIX A
OPERATORS IN GENERAL FORMS
This appendix is devoted to the derivation of the
operators, R^E, t) which appear in Eq. (2.50) and H(E),
F(E) and G(E) which appear in Eq. (2.54). Then these
operators will be transformed into their equivalent forms
in g domain as they appear in Eq. '(2.61). The develop
ment of these operators in this appendix will not specify
the model of the energy dependent absorption and scatter
ing kernels. Therefore, the results can be used in any
model one would like to adopt in any specific application.
However, the development of these operators assume the
representation of the flux component by a three
term Taylor expansion about E. For consistency, there
fore, terms with derivatives up to the second order only
are kept in this development.
For convenience, the equations defining the
operators are rewritten here.
Rr^(E,t) = Qm(E) E .(E) (A. 1)
mt V(E) 3t
Qm(E) = Mm(E) + Mjj^(E) D + m2(E) D2 (A.2)
102
113
Energy Moments
The energy moments defined by Eq. (2.35) are
/ (E' E)k
E'
Z (E * E ) dE '
m
or
mÂ£(E) = / Ak Im(E> E' ) dE'
kl E'
(B.15)
(B.16)
Introducing Eqs. (B.6) and (B.7) into Eq. (B.16) and
using the property in Equ (B.10), we obtain
M(E) = (1 2U + vT/2 E) ZQ (B.17)
M(E> = 2\i (2T E) l0 (B.18)
Mq(E) = 2 y T E z0 (B.19)
mÂ£(E) = 0 for k > 2 (B.20)
and
M(E) = (2u/3) ZQ (B.21)
M^{E) = (2 w/3) (B 2T) Zq
(B. 22 )
10
ln(,E,t) = / I (E'+E) n(,E',t)dE' (2.22)
m m m
E*
A(z) =
3
(2.23)
3 z
{
B (x ,y)
3 3
3x +3 3y
(2.24)
B*(x,y)
3 _3_
3x 3y
(2.25)
In an infinite moderating medium with no
absorption the energy spectrum of the neutron flux is
given by the Maxwellian function
m(E) = (E/T2)e"E//T (2.26)
where
T = the energy corresponding to the
most probable speed,
= Boltzmann constant x Kelvin
temperature (in energy units).
On the other hand, the energy spectrum in a finite
medium with a small absorption cross section, com
pared with the scattering cross section, is not
Maxwellian but similar to it in the general shape.
APPENDIX D
COMPUTING CUUES
1HE SE CODES WERE UEVhLUPED FUR THE IBM709 COMPUTER.
Tut MAIN CODE IS THE NWP (NEUTRON WAVES PKOPOGATION) WHICH
WAS DEVELUPED BY THE AUTHUR FUR THE NUMERICAL COMPUTA IIUNS
OF I HE NEUTRON WAVES PROBLEM PRESENTED IN THIS WORK. IHIS
CODE HANDLES THE COMPUTATIONS REGARDLESS UF THE NATURE OF
THE SCATTERING KERNEL AND THE ABSORPTION CROSS SECTION OF
THE HOMOGENEOUS MODERATING ASSEMBLY. THE PRINCIPAL QUANTITIES
THAT ARE CALCULATED BY THIS PROGRAM ARE LISTED BELOW.
1STtADY SIATE EIGENVALUES AND INVERSE RELAXATION
LENGTH FJK EACH MODE OF THE FLUX
2FREQUENCY DEPENDENT EIGENVALUES AND INVERSE RELAXATION
LENGTH FUR EACH MODE
3SIEADY STAIL AND FREQUENCY DEPENDENT AMPLITUDE AND
PHASt SHIFT FUR EACH MODE OF THE FLUX OR THE DENSITY
AT THE DESIRED PUSITIONS
AmMPLITUDE AND PHASE SHIFT OF THE TUTAL NEUTRON FLUX
OR NEUTRON DENSITY VS. POSITION AND FREQUENCY
5MODAL AND TuTAL ENERGY SPECTRUM OF FLUX AND DENSITY
VS.POSITION AND FREQUENCY
THE INPUT DATA MATRICES OF THIS CODE ARE COMPUTED BY
THE SECOND CODE HGM (HEAVY GAS MATRICES). THIS CODE FuLLOWS
Tub ANALYTICAL STEPS OF APPENDIX C.
THE USAGE OF THESE CODES AND THE SUBROUTINES ASSOCIATED
WITH THEM IS cXPLAINED ONLY BRIEFLY. IN BuTH PROGRAMS THE
FOLLOW ING NOTATIONS ARE USED.
RP = REAL PART
IP IMAGINARY PART
PARI i
NWP AND ITS SUBROUTINES
1 INPUT PARAMETERS
FORMAI(5E14.6) IS USED FUR ALL FLOATING PulNT VARIABLES.
FORMA I (101 T) IS USED FUR ALL FIXED POINT VARIABLES
L = NUMBER uF LAGUERRc POLYNOMIALS
NUM = iORULK OF PERTURBATION
BUCK = TRANSVERSE BUCKLING
ni = number of energy points in spectrum umputatiun
NO = 0 FOR FLUX SPECTRUM
= i FOR DENSITY SPECTRUM
11 NT = INTLRVAL BETWEEN EACH TWO SUCCESSIVE ENERGY POINTS
ZDIC = A POSITIVE UR NEGATIVE QUANTITY TO DECIDE WHETHER OR
126
33
leads to the formation of the energy eigenfunction or
energy mode, E (e), through Eq. (3.18) which becomes
m, n
L
Ek (e) = Ek(e) = l A L(1) (e) (3.37)
m,n 4=o Â£'k l
Each set, A is the solution of Eq. (3.35) using
l ,k
the value for F, i.e.,
L
1 [aA + rk^A4,k=0 (3*38)
l=o
or
L
Â£ V ,4 (A'Wk = 0 {3.39)
1=0
This homogeneous set of L+l equations with L+l unknowns
can be solved for L unknowns in terms of the other. If
Eq. (3.39) is divided by A it can be rearranged in
O / JC
the form
L
E 4* ,4(A,rk)R4,k = D4,,o(Ark) (3.40)
4=1
with i
= 0, 1, 2,
L1.
12.
158
Takahashi, H., Space and Time Dependent Eigenvalue
Problem in Neutron Thermalization," BNL 719 (C32),
Vol. IV, 12991330 (1962).
13. Purohit, S. N., "Time Dependent ThermalNeutron
Energy Spectra in a Monoatomic Heavy Gas," ORNL
60744 (1960).
14. Shapiro, C. S., "Time Eigenvalues and Degenerate
Kernels in Neutron Thermalization, BNL 8433 (1964) .
15. Hurwitz, H., Jr., Nelkin, M. S. and Habetler, G. J.,
"Neutron Thermalization, I. Heavy Gaseous Moderator,"
Nuclear Science and Engineering JL, 280312 (1956) .
16. Barnard, E., Khan, N. A., McLatchie, R. C. F., Poole,
M. J. and Tait, J. H., "Time Dependent Neutron
Spectra in Graphite," Nuclear Science and Engineering
.17, 513522 (1963).
17. de Sobrino, L. and Clark, M., Jr., "Comparison of the
Wilkins Equation and Higher Order Approximations for
Solid Moderators," Nuclear Science and Engineering 10,
384387 (1961).
18. Purohit, S. N., "Neutron Thermalization and Diffusion
in Pulsed Media," Nuclear Science and Engineering 9.,
157167 (1961).
19. Koppel, J. U., "Time Dependent Space Independent
Neutron Thermalization," Nuclear Science and Engineer
ing 12. 532541 (1962).
20. Williams, M. M. R., "Space Energy Separability in
Pulsed Neutron Systems," Journal of Nuclear Energy 17,
55 (1963) .
21. Beckurts, K. H., Nuclear Instruments and Methods,
Vol. II, 144168 (1961).
22. Meadows, J. W. and Whalen, J. F., "Thermalizatiqn and
Diffusion Parameters of Neutrons in Zirconium Hydride,"
Nuclear Science and Engineering 13, 230236 (1962).
Profio, A. E. and Eckard, J. D., "Investigation of
Neutron Moderation with a Pulsed Source," Nuclear
Science and Engineering 19, 321328 (1964).
23.
54
and energy eigenfunctions are known and given by Eqs.
(3.691 (3.775 The various inverse relaxation lengths,
pk n> are computed from the eigenvalues,, according
to Eq. (3.36). It is important to recognize that the
actual computational process must be carried out in
complex arithmetic because, in general, all the quan
tities involved in this computation are complex.
The actual scalar flux, as a function of space,
energy and time, is now given by
(r,e,t) =
Re il l l Ck X (x)Y (y)Zk (z)Ek(e)m(e)e^ut]
e L L L ra,n m n J m,n i
m n k
(1.115)
In order to compare Eq. (3.115) with the actual
experimental results, the neutron density has to be
computed. This arises from the fact that the experi
mental setup used by Booth (41) uses a 1/v detector
and, hence, it gives the energy integrated, or total
neutron density, as the output. The neutron density,
which is related to the flux by the relation
N (r, e,t)
4> (r,e ,t)
(3.116)
can be given by
TE2=ALP1IAIP3I
rEb=ALP3RALP2R
TE6=ALP3IALP2I
Z 1 = T E 5 TEb
Z2= T E6 TE6
T EM = Z1 + Z2
Z1=TE1*TE5
Z2=TE2TE6
Z3 = Z1+ Z 2
TE3=Z3ZF EM
Z1= FE2* EEb
Z2=FE1*FE6
15 = 1 1Z2
TE4=Z3/FEM
TE7 = T E3 + XC0Nb
Z1=FE3*FE3
Z2 = FE4*FE4
FE9=Z1Z2
Z1 = XCQN6 TE3
FE10=Z1FE4
Zl=FE7*BET3R
Z2=FE4*BEF3l
DElb=ZlZ2
ZL=FE7*BEF3I
Z2=TE4*BEF3R
DE16=Z1+Z2
Z1 = FE3*8E F2R
Z2= FE4*BET2i
E11 = Z1Z2 + BET1RDE15
Zl=Tc3*BET2I
Z2= FE4*BET2R
FE12=Z1+Z2+BEF 1IDE16
TE7 = T E9XCGN5
ZL=FE9*8ET2R
Z2=rE10*BET2I
FE1=Z1Z2
ZL=rE9*BEr2I
Z2=FEL0*BEF2R
TE2=Z1+Z2
Z1=FE7*BET3R
Z2 = TE10*BET 31
lE13 = TElBEri RZ1+ Z2
Z1=FE7*BET3I
Z2=TE10*BET3R
TE14 = TE2ilET 11Z1Z2
Z 1 =l)E 1 5 F E 3
Z2 = DE 16 TE4
Tcl5 = Z 1Z2
Z1=DE15*TE4
Z2 = l)E16*FE3
F E L6=Z1+Z2
Z1=TE13*TE13
Z2= FE L4*F E 14
Z3=FE1 l*FElb
86
I [Akk s + Bkk]Jk(s)
k
= l pk / (s Ak Ak<) t < t (4.102)
= y Qk / (s xk') t T
K (4.103)
If these algebraic equations are solved for Jk(e) then
the inverse Laplace transform
Jk(t) = L"1 / {ty (4.104)
gives the sought for functions which complete the
solution.
In Chapter V we will discuss some of the numerical
results obtained for graphite and compare these results
with those obtained experimentally by Starr and Price
(33) .
103
mJÂ¡(E) = Â¡4E' E)k E
kl E' m
The energy moments, (E), can be calculated only when
the energy dependence of the scattering kernels,
^m(E' * E} is known. Therefore, and since we desire
to develop the general forms of the operators, these
forms will explicitly contain the energy moments, M^E).
Since Rq(E,t) appears only in the product,
(E,t]Rq(E,t), it will not be calculated separately.
Consider the two equations resulting from using
m=0 and m=l in Eq. (A.l) .
R lE,t) = 0o(Er Z. (E) 2 (A.4)
t V(E) 3t
R,(E,t) Qi(E) E (E)  (A.5)
X v (E) 31
If Eq. (A.4) is premultiplied by Eq. (A.5} the resulting
equation and Eq. (A.5) can be expressed in the forms
1 3 1 3 2
Ri(E,t) R0(E,t) = H
1 v(E> at v2(E} at2
R. (E, t) = G(E)  L (A.7)
v(e) at
95
experimental phase shift as seen from Fig. 5. It is
noted that in Fig. 5 all phase shifts are calculated
relative to the position at 4.89 cm which is the
closest experimental point to the source obtained by
Booth.
The energy spectrum of the total neutron
density at several positions was calculated and is
plotted in Fig. 6. Near the source the spectrum is
rather "cool." As the position of observation moves
away from the source into the system the spectrum
tends to approach the Maxwellian distribution. The
behavior of the spectrum in this fashion is justified
by the fact that, near the boundary of the system,
fast neutrons have a leakage probability through the
outer boundaries greater than that of the slow neu
trons .
Application to Pulsed Neutrons Technique
The asymptotic eigenvalues, of the neutron
flux in the P^ heavy gas approximation were calculated
for graphite using the DETEX routine to expand the
determinant of Eq. (4. 29) and the POLY routine which
solves the resulting equation. The fundamental time
decay constant, Xg = Ag vQ, is plotted against buckling
in Fig. 7. These results are compared with the experi
mentally measured values obtained by Starr and Price (33).
o c o o o on
149
SUBROUTINE COLAP
SUBROUTINE CULAP(L,A)
CALCULATION OF THE COEFFICIENTS OF I HE NORMALIZED
ASSOCIATED LAGUERRE POLYNOMIALS OF THE FIRST KIND
DIMENSION ALPHA! 10, 10),BETA( 10,10 I.GAMMA!10,10),
1 ETA!i 0,10),THETA!10),ZETA(10).DELTA!10,10),
2 A!LO,10),AA(3,L0),BB(3,L0),TT(L0),U(10,10,10),
3 V!10,10,10),W(10,10,10)C(50),G(50)
DI MENS I UN E( 10, 10)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1A,AA,BB,TI,U,V,W ,C,G
DO 2 N=i,L
DO 1 M=1,L
1 L(N,M)=0.0
2 CONTINUE
Ell, 1) = 1.0
L(2,1)=SQRTF{2.0)
b!2,2)=SQRTF(0.5)
DO 3 N=3,L
NM1=N1
NM2 = N2
X=FLOATFIN)
D E(N,1)=E(NM1, 1)*2.0*SQRTF((X1.0)/X)E(NM2,1)
1*SQR TF((X2.)/X)
3 CONTINUE
DO 5 N=3,L
NM1=N1
NM2=N2
X=F LOAT F{N)
DO 4 M = 2 L
MMl=M1
D 4 E N, M)=E(NM1,M)*2.0*SQRTF( (Xl.0)/X)E!NM2,M)
i*SQRTF((X2.0)/X)E!NMl,MMl)/SQRTF!X*(X1.0) )
5 CONTINUE
DC 12 M=1,L
DO 11 N = 1L
11 A!M,N)=fc(M,N)
12 CONTINUE
WRITE OUTPUT TAPE 6,6
6 FORMAT!1H1 /////,5X,31HC0EFFICI ENTS OF THE POLYNOMIALS)
7 FORMAT(I3.10F11.5 //)
DO 10 N =1,L
10 WRITE OUTPUT TAPE 6,7,N,(A{N,M),M=1,L)
RETURN
END
14
(ran+2)(mn+1)M
n (m+n) (m+n+1) > n_!  *
+ B*(x,v) I } ij t (r,E,t) ( j
2 [ (2m+l)(2ml) m1 (2m+3)(2ra+l)
x C<'E't)
(2.38)
where
1 3
R (E,t) = Q (E) Z. (E) ~
m m t v
31
(2.39)
Hence, we have obtained the general form of the thermali
zation model of the Boltzmann equation in its spherical
harmonics formulation, where its validity is only limited
by the approximation involved in the Taylor expansion of
^(r,E',t) about E. The operators (^(E) and Rm{E,t)
are discussed and calculated in Appendixes A and B.
PjL Approximation
When the index, m, in Eq. (2.38) is given the
values 0 and 1, the index n takes the values il, 0, and 1.
Then, the infinite set of Eq. (2.38) is reduced to the
following four equations which couple the moments of
the flux in the socalled P^ approximation.
Ro(E't)'t,o(^E,t)+xo(^,E,t)4)^ A(Z)4'l(^E't>
+ (i)*5 B(x,y)^(,E,t)()^ B*(x,y)^(irE,t) (2<40)
(2.41)
82
I I C+k T+1Ct) JL. ak*k
m,n,p k',k mnP mnp[vo2 mnP
_L_ Ak
at mnP
\
t l.k;k
vo mn,P
Jk(t)
m,n ,p
aEc m^n,p ^m,n "mjn'jpj m,n,p3t mnPj
,+k1
ft +k
/ T (t)
+ d
1
T (t)dt
where
(4.85)
Jk(t) = Ck(t) Tk(t) (4.86)
m,n,p m,n,p m,n,p
k Jk
J
+ k V
E (e)
m( e)
I Ek(e) de
mn,p
o
m,n,p
e m,n,p
k\k
CO
= J
+k' \
E (e)
m( e)
F(e) Ek(e)
m,n,p
o
mn P
m,n,p
k
,n,p
00
 /
o
E+k{e)
m,n,p
m( e )
1 E (e) de
/T 3
k
CO
0
+k*
)
* /
E (e)
m(e)
G(e) E ( e )
m ,n ,p
o
m,n ,p
b
(4.87)
(4.88)
<4.89)
(4.90)
Equation <4.85) must be satisfied at any time regardless
1
of what the constants, C are. Therefore, the
m,n,p
71
adjoint equation and the adjoint flux which are
studied in the following section. 1
The Adjoint Flux Equation
The differential equation of the nonMaxwellian
component of the flux is given by Eq. (4.1). Substi
tuting for the operators H(e), F(e) and G(e) from Eqs.
(A.30)(A.32), this equation becomes
2
[h (e) + h> (e) ~ + h9(e)  + ff (e) + f.Ce)~
1 2 8e2 Vq1 1 3e
32 9 l 32 l
+ f (e) ""*7 ) + +A2] ij)(r,e,t) =
2 v 2e 3t2 3
o
+g2(e) e ,t) (4.43)
where f^Ce) and g^{e) are algebraic functions of
t. To find the adjoint flux equation the following
rule is used (50). This rule states that the adjoint
operator of the differential operator
m n d
/ v 3 3 3^
a(x,y,.., ,z) ... _
3xm 3yn 3zp
is given by
75
a
I L(])U)ihAt)
o *'
m( e)
 JL h,(e) + (e) 3
3e 1 3 e 2 2
L(1)(e) de 4.59)
Â£
Two successive integrations by parts lead to the
expression l
* = / m(e) L^)(e)[h0(e)+h (e) l*h2(e) iL^UMe
n 1 9c2 4
3 E
/ rn ( e ) L(1)(e) H( e ) LC])(c) dE
0 0 I
(4.60)
Thus
Â£* Â£
'.l*
(4.61)
Similarly,
= 3
Â£ Â£ A Â£ 1
(4.62)
Due to the symmetry of y and n we can write
a ; z i; a
Â£ J Â£
Â£,Â£*
(4.63)
Â£ 1 Â£
= n
Â£,Â£'
(4.64)
160
36. Raievski, V. and Horowitz, J., "Determination of the
Mean Free Path of Thermal Neutrons by Measurement of
the Complex Diffusion Length," Peaceful Use of Atomic
Energy Conference, Vol. 5, p. 42 (1955).
37. Uhrig, R. E., "Oscillatory Techniques in a Sub
critical Assembly," Proceedings of the University
Subcritical Assemblies Conference, TID7619, 161 (1961).
38. Perez, R. B. and Uhrig, R. E., "Propagation of Neutron
Waves in Moderating Media," Nuclear Science and
Engineering 17, 90100 (1963j .
39. Perez, R. B., Booth, R. S., Denning, R. S. and
Hartley, R. H., "Propagation of Neutron Waves in Sub
critical Assemblies," Transactions. American Nuclear
Society 7, 49 (June 1964).
40. Perez, R. B., Booth, R. S. and Hartley, R. H., "Experi
mentation with ThermalNeutron Wave Propagation,"
Transactions, American Nuclear Society 6., 287
(November 1963) .
41. Booth, R. S., Private Communication, 1964.
42. Hetrick, D. L. and Seale, R. L., "Neutron Waves and
Vibrational Modes in Moderating Materials," Bull. Am.
Phys. Soc. (Abstract) 9, No. 2, 153 (1964).
43. Suramerfield, A., Partial Differential Equations in
Physics. Academic Press, Inc., New York, pp. 126129,
1949.
44. Meghreblian, R. V. and Holmes, D. K., Reactor Analysis.
McGrawHill Book Co., New York, p. 337, 1960.
45. Meghreblian, R. V. and Holmes, D. K., Reactor Analysis,
McGrawHill Book Co., New York, p. 351, 1960.
46. Weinberg, A. M. and Wigner, E. P., The Physical Theory
of Neutron Chain Reactors, The University of Chicago
Press, Chicago, p. 235, 1958.
47. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics. Part II, McGrawHill Book Co., New York,
p. 1010, 1953.
48. Milligan, J. L., "Polynomial Root Finder Program,"
Share Distribution Number SD# 1215.
Q Q
150
SUBROUTINE MULAP
SUBROUTINE MOLAP(L,A,U,V,W)
DIMENSION ALPHA(10,10),BETA(10,10),GAMMA(10,10) ,
1 ETA10,10),THETA{10), ZE TA(10),DELTA!10,10),
2 A(10,10)AA(3,10)*8B(3,10)TT(10),U(10,10,10)
3 V(10,10,10),W( 10,10, 10) ,C(50)G{50)
D DIMENSION Bt100),D( 100),Y{ 10,10,10)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB, TT,U,V,W ,C,G
DO 5 M= 1 L
DU 4 N = 1 ,L
DU 3 1=1,10
U(M,N,I)=0.0
V(M,N, I ) = 0.0
3 W(M,N,I)=0.0
4 CONTINUE
5 CONTINUE
D(1) = 1. /724535
D ( 2 ) = 1.0
DO 10 1=3,60
IM 2 =I~2
X=FLOATE(I)
D D(I) = (0.5*X1.0)*D( I M2)
10 CONTINUE
L1=2*L1
DO 20 M=1,L
DO 20 N=1, L
DO 11 1=1,LI
D 11 b(I)=0.0
DO 12 1=1,L
DU 12 J = 1,L
T = A ( M, I ) A ( N J )
K=I + J1
D 6 l K ) = B ( K ) + T
12 CONTINUE
DO 14 1=1,10
D Y(M,N,I)=0.0
DO 14 J =1,L 1
K=I+2*J2
D 14 Y(M,N,I)=Y(M,N,I)+D(K)*B(J)
DU 17 M =1,L
DU 16 N =1,L
DO 15 1=1,10
15 U(M,N,I)=Y(M,N,I)
16 CONTINUE
17 CONTINUE
20 CONTINUE
DO 30 M=i,L
WRITE OUTPUT TAPE 6,22,M
22 FORMAT( 1H1 /////,52X,2HU(,I 2,10H,N,I = 1,10) ///)
DO 25 N= 1 L
WRITE OUTPUT TAPE 6,23,N
30
where the matrices involved in this equation are defined
by the following:
/ m(e) L(*} (e)H(e) LU) (e)de
o Â£ Â£
(3.23)
Â£' ,Â£
r m(e) (e)F(e) (E)de
(3.29)
Â£' ,Â£
f m(e) L(1) (e) L(1) (e)de
Â£1 e Â£
(3.30)
Â£ Â£ 3
= 1 / m(e) L(1) (e) L(1)(e)de = 6 (3.31)
Â£ i jj, 3 Â£ Â£
and where
r
p2 B2
m ,n im, n
(3.32)
If the Laguerre expansion of Eq. (3.18h contains
L+l polynomials, i.e., if Â£ takes the values 0, 1, 2,
..., L, then, the set of Eqs. (3.27) can be rewritten as
l i
Â£=0
Â£' ,Â£
+ 3
Â£' ,Â£
A + y
2 _
V Â£
r]Am,n= (333)
Inverse Relaxation Length
90
[
123456789 10
Number of Laguerre Polynomials, L
Fig. 1. The Steady State Inverse Relaxation Length
vs. Number of Laguerre Polynomials
Decay Constant, X, (sec
Fig. 7. Decay Constant, X of the Fundamental Mode of the Flux
in the Neutronpulsed Graphite System.
18
3k
G(E) = l gk(E) (2.59)
k SEk
where fk(E), hk'{E) and gk(E), which are algebraic
functions of the energy and the macroscopic properties
of the medium, are given by Eqs. (A.14} (A.22) and
where the derivative operators, ak/3Ek / arise from the
Taylor expansion of the flux.
In Eqs. (2.57) (2.59) all the energy
derivatives are kept and the truncation of the expansion
depends on the model of the scattering kernel used and
on the energy moments, M^(E), which appear as coeffici
ents of the derivatives, 3k/3Ek, in the Taylor expan
sion. See Eqs. (2.34) and (2.37). In the heavy gas
model scattering kernel, which is applied in this work
for numerical computation, the moments, Mk(E}, are
zero for k greater than 2. Therefore, only the first
and second derivatives are used.
It is very convenient to transform Eq. (2.54)
from the energy domain, E, into the domain of the
dimensionless variable, e given by
e = E/T (2.60)
This transformation leads to
85
Multiplying these two equations by vQ2 and dividing
them by T+k (t) and then using the value of the latter
they can be reduced to the forms
l [Akk JL + Bkk]Jk(t)
* I Pk
k
exp [
( Xk + Xk)t] t < T
(4.96)
exp(
Xk't) t > T
(4.97)
where
Ak
' k
' =
ak',k ,
[
(4.98)
Bk
' 'k =
v0 (t>k'' kA k' ak''k)
(4.99)
Pk =
(dk/Ak)
abc
(4 0100)
II
(dk/Akck) exp^) .
abd
(4.101)
Taking the Laplace transform of Eqs. (4.96) and (4.97)
one obtains
203 CONTINUE
CALL EVADETtLUfCDiOcT)
B(JP ) = B(JP)+DET
303 CONTINUE
DO 404 J4=1,LE
NU = 4
N4=J4I
JP=l*Nl+N2+N3+N4
IFILONO) 304,104,304
104 CUNT1 NUE
DU 204 M =I,L U
CU(M,I)=CC(M,I,J1)
CD(K,2)=CC(M,2,J2)
CO(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4)
'204 CONTINUE
CALL EVAUET( L D C 0, D E T )
8(JP)=B(JP)*UET
304 CONTINUE
DC 4 05 J5 = l, LE
ND = 5
N5 = J 5 I
JP=I+N1+N2+N3+N4+N5
IF(LDNU) 305,105,305
105 CUNT INUt
UU 205 H=1,L
CD(M,1)=CC(M, 1 ,J 1 )
CD(M,2)=CC(M,2,J2)
CD(M,3)=CC(M,3,J 3)
CD(M,4)=CC(M,4,J4J
C0(Mt5)=CC(M,5,J 5)
205 CONTINUE
CALL EVADET(LD,CD,DET )
B(J P)=B(JP)+DET
305 CONTINUE
UU 406 J6=1,LE
NU = 6
\6J 61
JP=1+N1+N2+N3+N4+N5+N6
IF(LDNU) 306,106,306
106 CONTINUE
CD(M,1)=CC(M,1,J1)
CD(M,2)=CC(M,2,J2)
CD(M,3)CC(M,3,J3)
CD(M,4)CC{M,4,J4)
CD{M,5)=CC(M,5,J5)
CD(M,6)=CC(M,6, J6)
DU 2 06 M = 1,L U
206 CUNTINUL
CALL EVADET(LD,CO,DET)
B(JP)=B(JP+DET
58
ef(r) =
I l l [{Ak (r) sin bk z
m n k L ra'n m'n
Bk (r) cos bk
ra,n m
tan1 r k
l l l (Ak
m n k L
)> ^
(r) cos b z + B (r) sin b
m,n ra,n
m
zECl
,n f J
(3.131)
A (r) =
d
III (r)2 + Bk (r)2 }e^2
m n k m'n m'n d
h
(3.132)
0d(r)
i 11K
(r) sin bk i Bk (r) cos bk z Ek
m,n m,n m,n I d
tan
1
l l l k,ni> * Â£.* Bm,n
rank1 J
(3.133)
where
Ef = r Ek(e)de
o
Ek = /" E^(c)dc
(3.134)
(3.135)
76
Hence, Eqs. {4.53) and (4.54) become
= 0
(4.65)
and
A simple comparison between Eqs. (4.29) and (4.66),
keeping in mind that the rotation of a determinant
about its main diagonal does not change its roots, we
arrive at the conclusion that both the flux and its
adjoint have the same time eigenvalues, i.e.,
(4.67)
A
m,n,p
The essential difference is that the time eigenfunctions
of the flux are decaying exponentials, while those of
the adjoint are rising exponentials, as seen from Eqs.
(4.14} and (4.48). This result is expected because the
adjoint flux is the importance function and must rise
as the neutron flux decays where the remaining neutrons
become more important.
62
4
= 0
(4.5)
= 0
(4.6)
i^(x,y ,c,e ,t)
a 0
(4.7 r
(x,y,z,e ,t)
= finite
(4.8)
where 2a, 2b and 2c are the extrapolated dimensions of
the system.
We will first solve the homogeneous part of
Eq. {4.1), i.e.,
3 2
[H(e) + 1_ F(e) JL + i iv2] tKr,e,t) = 0
Vo at vq2c 9t2 3 (4.9,
The solution of this equation is developed in cosine
functions for the spacial dependence, exponential
functions for the time dependence and Laguerre poly
nomials for the energy dependence, i.e.,
'Kr,e c X (x)Y (y)Z (z)T (t) E (e)
mnp ranP m n P m,n,p m.n,p
(4.10)
161
49. Faddeeva, V. N., Computational Methods of Linear
Algebra, Dover Publications, Inc., New York, pp.
6574, 1959.
50. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics, Part II, McGrawHill Book Co., New York,
p. 875, 1953.
51. Razminas, R., Private Communication, 1964.
52. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics, Part II, McGrawHill Book Co., New York,
p. 552, 1953.
53. Morse, P. M. and Feshback, H., Methods of Theoretical
Physics, Part II, McGrawHill Book Co., New York,
p. 785, 1953.
110
I (E * E 1 }
m
where
/"G(E,E' ,y ,t)
00
jt(E'E)
e
dt
(B.
G (E E V fe)
exp uj(jt t2T)
4tt
E' + E 2xXEE'}^ >]
(B.2>
The quantities involved in Eq. (B.l) and Eq. (B.2) are
defined as follows:
E = neutron energy before a scattering
collision
E' = neutron energy after a scattering
collision
x = cos (n, n')
3 = neutron direction before collision
= neutron direction after collision
0o = 4ira = bound atom cross section
a = scattering length
T = neutron energy corresponding to the
most probabie speed
v* = neutron to moderator mass ratio.
Assuming a twoterm Taylor expansion of the function
G(E,E', P ,t), i.e., neglecting terms in p and higher,
the kernels are then given by
45
where Jn(r,E,t) is then the net current through a unit
area whose norm is N. Eq. (3.75) can be rewritten as
jn(r,E,t) n / *(r,n#E#t)n da
a
 N J(r,E,t)
(3.76)
which means that the net current, Jn(r,E,t), is the
4
projection on N of the vector net current
J(r,E,t) =
/ *(r,n,E,t)a da
(3.77)
The flux and the vector, a are given by
4>(i,a,E,t) = l l <^(r,E,t)Y^(a) (3.78)
m n
 *
a = Sine cos u + sine sin v + cose w (3.79)
where u, v and w are the unit vectors along the x, y
and zaxes, respectively, and y and e are the polar
4
and azimuthal angles of a in spherical geometry. From
Eqs. (2.7) and (2.12) it can be shown that
COS 0
(3.80)
41
1. The unperturbed equation, Eq. (3.46), and
its compatibility condition, Eq. (3.47),
are solved and the unperturbed quantities
rand r() are determined,
k H,k
2. For each value of k, the perturbation
equation, (3.59) or (3.64), is solved for
all values of v in increasing order up to
the desired maximum value. Each time
v) and are calculated in terms of
k L ,k
r(vi) and y = 0, 1, 2 v1.
k i. ,k
3. r and R are calculated from Eqs.(3.50) and
k l ,k
(3.50) using the desired value for A or w.
4. Finally, the energydependent flux modes,
or eigenfunctions, Ek(e), are formed
Ek(Â£) Ao,k R1(kLi1>U) <365>
l0
It should be emphasized that this perturbation technique
has been used here only in the energydependent equation
(3.44), leading to the determination of the energy modes,
Ek(Â£) .
Before we move on to further developments we
ought to compare the two methods. The exact method has
69
i.e., Ak (k = 1, 2, ..., 2L+2). Having determined
m, n,p
these eigenvalues the set of Eqs. (4.34) is solved for
0 T,
the ratios, R using another subroutine, ELEM,
m,n,p
which was developed for this purpose utilizing the
Gauss elimination method discussed in (49). This
subroutine is also listed in Appendix D.
The whole computational process discussed
above is then applied for all the spacial modes, i.e.,
for all the values of m, n and p. Finally, the flux is
recombined using Eqs. (4.10)(4.15J, (4.17), (4.30)
(4.33), along with the relation
(?,e,t) = m(e) i)(r,e,t) (4.35)
This combination leads to the total flux expression
m,n,p mnP m n p mnP m,n,p
(4.36),
where we redefine
Ck
mn p
c
mn ,p
m,n,p
(4.37)
X (x)
m
3
COB
2a
(4.38)
15
Ri (E;t) \j)i" (r,E,t) +x^(r/Ert) = (i)^ B* (x,y) < (r,E,t)
6 (2.42)
R, (E,t)iA1(r,E,t)+x71(r,E,t)=(i)2 B(x,y) i>(r,E,t)
1 60 (2.43)
The scalar flux is obtained from Eq. (2.2)
$ (r ,E, t) = /
Q
m
 / I I Oi'E't>YS<)d(!
m=o n=m
= 0 (r ,E,t)  Hq Air 4>o (r fE,t)
= AT m(E) *g(r,E,t) (2.44)
where Eq.s (2.2), (2.3), (2.7), (2.10), (2.12) and
(2.27) were used.
Similarly,
S(r,E,t) = ATT xn(E) xÂ§(r,E,t) (2.45)
Since the scalar flux is easily obtained from ^g(r,EA)
according to Eq. (2.44) the set of Eqs. (2.402.43)
will be solved for ^g(i,E#t). Multiplying Eq. (2.40)
80
[Q+H* QH+**] v^[Q + F QF+ **]
O 91 H
1 + 1 32 1 92 ,
% 2 7 at2 Q2 3t2 **3j CQ A2 ^ Q A2
s nf. 1 3 j
 Q+ __ X + Q+ GX
o /e at
(4.77)
where all the arguments have been dropped for simplicity.
Introducing Eqs. (4.45) and t4.46) into Eq. (4.77) and
integrating by parts we can prove that
Q+Hi(i QH + ^* s 0
(4.78)
Q+F L i>QF+  ** r/ dr / [ ^*Fip] de
at 91 e t
,+ 3
(4.79)
Q+ k JL *Q I JL >>*
e at2 v 912
3 3
/ dr / [** 7 gt F 9t **lt de
(4.80)
Q+V2\p Qvzip* =
(4.81)
where in this proof we have used the condition that the
flux is zero at t=o and utilized the invariance of the
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
TIME, ENERGY AND SPACEDEPENDENT NEUTRON
THERMALIZATION THEORY
By
Hassan H. Kunaish
December, 1964
Chairman: Dr. Rafael B. Perez
Major Department: Nuclear Engineering
The time, energy and spacedependent neutron
thermalization theory in moderating assemblies is
developed in the consistent P^ approximation of the
Boltzmann equation of neutron transport. In order to
evaluate the scattering integral involved in this
equation the neutron flux is analyzed into two com
ponents according to the expression
$ (r, E, t) = m(E) {Â¡i (r, E, t) .
The Maxwellian function, m(E), represents the energy
spectrum of neutrons in a nonabsorbing infinite medium.
In a finite medium with a small absorption cross section
the energy spectrum of the neutron flux resembles a
Maxwellian distribution. Therefore, the nonMaxwellian
v
153
WRITE OUTPUT TAPE 6,11
11 FORMAT!///,50X.5HTTN)//)
WRITE UUTPUT TAPE 6,8,{TT{N),N=1,10)
RETURN
END
SUBROUTINE ABTCS
SUBROUTINE ABTCS{L,NS,SS,SA,R,AA,BB,TT)
DIMENSION ALPHA!10, 10),BETA!10,10),OAMMA(10,10),
1 ETA 110,10),THETA(10),ZETA(10),DELTA(10,10),
2 A(10,10),AA(3,10),BB(3,10),TT(10),U(10,10,10),
3 V! 10,LO, 10),W(10,10,10),C(50),G(50)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZETA,DELTA,
1 A,AA,BB,TT,U,V,W,C,G
DO A N=1,10
T T N ) = 0.0
DO 3 M= 1,3
AA(M,N)= 0.0
3 3B(M,N)=0.0
4 CONTINUE
NSPl = NS + 1
NSP2=NS+2
NSP 3 = NS + 3
NSP4=NS+4
NSP5=NS+5
NSP6=NS+6
NSP7=NS+7
NSP8=NS+8
AA( 1 ,NSPl)=R*SS*SA/6.0
AA( 1 ,NSP2)=SA**2
AA!1,NSP3)=SS*SA*(1. 01.0*R/3.0)
AA(2,NSP3)=10.0*R*5S*SA/3.0
AA!2,NSP4)=SS*SS*(16.0*R/3.0~4.0)*R
AA(2,NSP5)=4.0*RSS*SA/3.0
AA(2,NSP6)=2.0*R*SS*SS#1.04.0*R/3.0)
AA13,NSP4)=8.0*R*RSS*SS
AA!3,NSh5}=K*SS*SA*4.0/3.0
AAl 3,NSP6)=R*SS*SS*(32.0*R/3.02.0)
AA! 3.NSP8 )=4.0*R*R*SS*SS/3.0
BB(1,NSP1)=R*SS/6.0
BB(1 ,NSP2)=2.0*SA
BB(1,NSP3)=SS*( 1.01.0*R/3.0)
BB(2,NSP3)=10.0*R*SS/3.0
B 5!2,N S P 5)= 4.0*R*SS/3.0
BB(3,NSP5)=4.0*R*SS/3.0
TT(\'SP3)=SA
TT(NSP4)=SS*(1.02.0*R/3.0)
WRITE OUTPUT TAPE 6,5
5 FORMAT(1H1///,50X,7 HAA(M,N)//)
DO 1 M= 1 3
7 WRITE OUTPUT TAPE 6,8, ( AA(M,N),N = 1, 10 )
APPENDIX C
CALCULATION OF THE MATRICES
^ i Â£ # ^ Â£ Â£ 11 Â£ Â£ Â£ i Â£ j Â£ Â£ Â£
In solving the modal equation of the flux for
neutron waves, Eq. (2.35}, and for the pulsed neutrons,
Eq. (4.22), the functions, E (e) and E (e) are
m, n iu/ n / p
expanded in terms of the normalized associated Laguerre
polynomials of the first order, L^(e). This gave
Â£
rise to the matrices, a . 3 . y and n / defined
in Eqs (3.28)(3.31). The application of the source
condition of Eq. (3.94) gave rise to two other matrices,
and 0l, defined in Eqs. (3.108) and (3.109).
Instead of treating each of these matrices
separately, we will find a general form which can be
applied to all of them and then treat this form in some
detail. This will cut down the amount of work in hand
ling the mathematics involved in all the computational
processes. The examination of these matrices and opera
tors involved in them [see Eqs. (A.30)(A.32)], enables
us to express these matrices in the form
V,,, f U> I^fe) AU> dE (C.l)
Â£ Â£ Â£
118
141
SUBRCJUT I NE EVADET
SUBROUTINE EVADET{LU,CD,DET)
DIMENSION GDI 10,10) ,HI 10,10)
DE T= 1.0
DU 100 K=I,LD
X=0.0
DO 15 I = K,LD
A=A6SF(X)ABSFICD(IK))
I F(A ) 5,10,10
5 X=CD(I,K)
I X= I
10 CONTINUE
15 CONTINUE
DU 20 J=K,LD
HIK,J)=CD(K,J)
CD(K,J)=CD(IX,J)
20 CD(IX,J)=H(K,J)
IF(IXK) 25,30,25
25 DET=DET*CD(K,K)
GO TO 35
30 DE T = DE T*CD(K,K)
35 B=CD(K,K)
DO 40 J=K,LD
40 CD IK,J)=CD(K,J)/B
DO 45 J = 1,LD
45 HIK,J)=CD(K,J)
DO 55 1=1,LD
R=CD(I,K)
DO 50 J=1,LD
50 GDI I,J)=CD( I,J)R*H(K,J)
55 CONTINUt
DO 60 J=1,LD
60 CD(K,J)=H(K,J)
100 CONTINUE
RETURN
END
I
7
v
Es (E'+Ejft+ft)
Z3(E'E,w0)
" 2m+l
= l ~ EmE,,E>VJ)
m=o
ra + +
 I I VE"E>Ym
m=o n=m
(2.5)
where
w0 = cos 0O = ft*ft* (2.6)
is the cosine of the scattering angle in the labora
tory system.
The spherical harmonics, ^(3), the Legendre
m
polynomials, pm(y0)< and their associated functions,
P^(u)/ satisfy the following relations, see refer
ences (43) and (44) .
YmW 2 s HmP>>ejlW
(2.7)
(V* (2*8)
f n + n Â¥ Â¥
f (ii)yJ(n)dQ
m p
ft
6 6
mp nq
H
n
m
1/2
(2.9)
[ (2m+l) (mn) /4ir (m+n) 1 ]
(2.10)
34
The new unknowns appearing in this equation are the
ratios
Ri,k Aa,k//Ao,k
where
Ro,k = Ao,k/Ao,k = 1
(3.41)
(3.42)
Notice that the equation corresponding to i = L was
omitted from the set represented by Eq. (3.40) in
order to equalize the number of variables to the number
of inhomogeneous equations, i.e., to remove the degen
eracy of the solution. The set of Eqs. (3.40) can be
solved for all the ratios, R and the energy modes,
X, f ix
Ek(e), can be formed by recombining Eqs. (3.37) and
(3.41).
Ek (e)
L
l
1=0
(3.43)
The exact method that has been discussed yields the
correct solution as a function of A or w. The whole
computational process must be carried out in complex
algebra due to the fact that A and r are complex.
108
H(e) = h0(c) + + h2(e)
3 G
F(e) = fQ(e) + fj^e) + f2(Â£)
3e
g(Â£) = g0(Â£) + gi(e) + g2ie)
Be
the functions (e ) f^(e) and g^ (e )
hi{c) = hi(E} /T1
f {t) = fi(E) /T
fz
gi ie) = g(E) /t1
3
(A.30)
3e2
32
(A.31)
3c2
2
_ 2
(A.32)
3e
are then given by
(A. 33)
(A.34}
(A.35)
where h^iE), f^fE) and g^iE) have been given in Eqs.
(A .14)(A.22) .
The results obtained so far are general,
regardless of what kernels we use. In Appendix B, we
will adopt the heavy gas model for the scattering kernel
and 1/v for the absorption cross section. We will
derive the energy moments, m(E), and then calculate all
the functions needed.
68
For any eigenvalue, a / Eq. (4.24) can be solved for
m,n,p
Ai,k (*=1,2L) in terms of A'k or we can solve
m,n,p m,n,p
for the ratios
r'* = A*"k / A'k Â£ = 1,2 L (4.33)
m,n,p m,n,p m,n,p
from the set
L
1
ll
Ak + V
m, n p Â£ Â£
A*2
m,n ,p
Â£J
b2 ]RÂ£k
Â£ m,n,p m,n,p
A k
m,n,p
+
y
Â£ i o
Ak2 +
mn ,p
B2 ]
m,n ,p
Â£ 1 = 0,1,2,L1
(4.34)
where the equation corresponding to jt1 = L w^s neglected
in order to make the set, Eq. (4.34), with L inhomogeneous
ir 1C
equations and L unknowns, R (a 1, 2, .., L) .
m,n,p
A FORTRAN subroutine, DETEX, was developed to
expand a determinant, the general element of which is a
polynomial in an unknown, say x, of any size. The
result of this expansion is a polynomial in x with a
maximum power of I*J, where I is the size of the deter
minant and J is the size of its general element, i.e.,
the maximum power of x in this element. This sub
routine is listed in Appendix D. The use of DETEX for
Eq. (4.29) leads to an algebraic equation with a maxi
mum power of 2(L+l). Then a polynomial solver, POLY,
(48) is used to find all the roots of this equation,
115
(N O
IN
Ei
a
(N
co
F1
+
HCN
tTPw
Eh
O
id
qu
EhM
cn O
in
Eh
IN
a
IN
I
w
*^
A
IN
m
WEh
CM o
in
Eh
f"
a
(N
+
rHlN
wTeh
O
id
in
O
in
Eh
a
 O
+
>=*
. .
00
IT>
IN
IN
m
CQ
vv*
CN
W 1 Eh
CN 0
IN
CN
W
Eh
o
IN
CN 0
IN
IN
a
IN
Eh
'siln
M c
IN
l
1
a
rH
CT
WEh
r
1
r( IN
IN 0
+
IN
rHlN
WEH
id
IN
CN
Eh
a
l
I!
<'**%
W
IN
IN
Eh
a
CM
IN
a
^Iro
Eh I M
id
in
IN
Eh M
IN
a
11 IN
II
w'
''
c
X
. ,
n
O
H
co
Ht
CO
co
co
co
CO
CQ
CQ
5.
fi
CQ
V
0
IN
a
00 1 CO
, s
1
MEh
v /
i1
V*
0
IN
+
Eh
W
a
Eh W
0
^l ro
/
0
W
W  Eh
id
+
IN
0
0
+
CN  ro
IN
IN
1
W
Eh
CN
eTw
o
0
a
Eh
a
w
IN
O I _
IN
E*
Eh
iH 1 ^
^fco
a
a
1
1
r  CN
^co
IN  co
II
II
II
1!
II
.
^51
w
M
w
W
w
v^*
H
IN
o
H
CN
IH
Cn
CT
Cn
43
In both methods the energy eigenfunctions take
the same form, as shown by Eqs. (3.43) and (3.65).
Having determined the energy dependence of >J>(x,y,z,e) by
either one of these methods we can easily rewrite Eq.
(3.14) in the form
4> (x,y ,z,
o l
l Ck X (x) Y (y)Zk (z)Ek(e)
L m,n m n m*n
(3.67)
m,
n k
where we
redefine
ck
m,n
c A
m,n o
(3.68)
n
'x
X
cos (
2m1 \
TTX
2a
(3.69)
Yn(y) =
2nl ,
in 7 n \
tUo t
try J
2b
\ J / U j
Zk (z)
m,n
= exp (
k
"PmrnZJ
(3.71)
Ek(e) =
l R
; *
k Â£ '
(3.72)
It remains now
V
to determine the constants, CTO _.
ill / XI
This
will be done later from the source condition developed
in the following section.
72
m+n+...p 3m 3n 3P
( 1) ... a( x ,y ,... z)
m n p
3x 3y 3 z
and its application to Eq. (4.43) gives the adjoint
equation
[ H + ( e) J_
'(e)  +
31
1 ii + I A2]'{'*(?,Â£ ,t)
31'
= 0
(4.44)
with the adjoint operators defined as
H + ( e) = hD(e)
2
JLh^Ce) + h2(e)
3e 3e 2
(4.45)
F+(e) = fo(e)
3 32
fx(e) + f2(e)
3 e 3 Â£ 2
(4.46)
In solving Eq. (4.44) for the adjoint flux the
same technique used in solving Eq. (4.9) for the
ordinary flux is followed and we obtain the solution
**(?,Â£,t) = l c+ Xm(x)Yn(y)Zp(z)T+(t) E+(e) m(e)
m,n,p m,n,p m,n,p m,n,p
(4.47)
42
the advantage of yielding the functions E (e without
any approximation at any desired frequency and without
any restriction on the maximum value of ui that can be
used. The disadvantage of this method is that all the
computational processes must be performed in complex
algebra and all the computational steps must be
repeated for each value of m. On the other hand, the
perturbation technique introduces some error in the
solution due to the approximation hidden in Eqs. (3.49)
and (3.50). The value of this error diminishes rapidly
as the number of terms kept in these equations in
creases. The other disadvantage of this method is the
restriction given by Eq. (3.53), namely
< VQ (3.66)
It is obvious, of course, that the solution is improved
with decreasing values of u. The advantages of the
perturbation technique are that the algebra involved in
it is easier to handle in actual computations and that
the computation does not have to be repeated for each
frequency. The computational scheme used in this work
utilizes the perturbation technique.
154
)
8 FORMAT(10E12.4)
WRITE OUTPUT TAPE 6,9
9 FORMAT!///,5UX,7HBB(M,N )//)
UO 10 M= i 3
10 WRITE OUTPUT TAPE 6,8,(BB(MN),N=1,10)
WRITE OUTPUT TAPE 6,11
11 FORMAT{///,50X,5HTT(N)//)
WRITE OUTPUT TAPE 6,8, (TT(N),N = 1 10}
RETURN
END
49
derivative with respect to time and the second one is
the energy correction hidden in the energy dependence
of the operator G(e) and the energy derivatives
involved in it. When the transport correction is
neglected and a Maxwellian source is used the non
Maxwellian component of the source becomes independent
of energy. Then, the operator G(e) is reduced to
G(c)
[VÂ£) Vs(e)
1
3D (e)
(3.96)
and the condition of Eq. (3.95) becomes
D(e)
8 +
~~
8 z
= xUy#e#t)
z=0
(3.97)
or
D(e)
8 *
77 (r,e#t)
8 Z
= S(x,y,e,t)
z=0
(3.98)
which is essentially the condition used by Perez and
Uhrig (33) .
Before we use the source condition in Eq. (3.95)
to determine the so far unknown constants, C two
m, n
remarks should be made. The transport correction in
this condition has a strong effect when the source
frequency is high while its effect is small with low
148
INPUT DATA PROGRAM HGM (HEAVY GAS MATRICES)
DIMENSION ALPHA(10,10),BETA(10,10),GAMMA(10,ID),
1 ETA(10,10),THETA(10),ZETA(10),DELTA(10,10),
2 A(10,10),AA(3,10),BB(3,10),TT(10),U(10,10,10),
3 V(10,10, 10),W(10,10,10), C ( 5 0 ) G ( 5 0 )
COMMON ALPHA, BE I.'A, GAMMA, ETA, THETA, ZtTA, DELTA,
1A,AA,BQTr,U*V,W,C,G
1 FORMAT( 1013)
2 FORMAT(5E16.8)
3 FORMAT(1HP,5E14.6)
READ INPUT TAPE 5,1,L
CALL COLAP(L,A)
CALL MOLAPlL,A,U,V,W)
READ INPUI TAPE 5,2,SS,SA,UM
R=1.O/UM
100 READ INPUT TAPE 5,1,NS
NSING=NS3
WRITE OUTPUT TAPE 6,1 5,L,NS ING,SS,SA,UM
15 FORMAT( 1H1/////5X,32HMATRIX ELEMENTS FOR THE CASE OF /
1 5X32(1H=)//5X,2HL=I2/5X,19HSINGULARITY ORDER =13,2H/2/
2 5X.9HSIGMAS =F12.8/5X,9HSIGMAA =F12.8/5X,6HMASS =F5.1 )
READ INPUT TAPE 5,1,NCV
IF(NCV) 16,16,20
16 WRITE OUTPUT TAPE 6,17
17 FORMAT(5X,21HWITH CONSTANT SIGMAS )
CALL A8TCS(L,NS,SS,SA,R,AA,BB,TT)
GO TU 25
20 WRITE OUTPUT TAPE 6,21
21 FORMAT(5X.21HWITH VARIABLE SIGMAS )
CALL ABTVS(L,NS,SS,SA,R,AA,BB,TT)
25 CONTlNUt
CALL ABGETZlL,NS,AA,BB,TT,U,V,W,ALPHA,
1 BETA,GAMMA,ETA,THETA,ZETA)
DO 4 M= 1, L
4
WRITE
OUTPUT
TAPE
6,3,
(ALPHA(M,N),N=1,L)
DO 5
M= 1, L
5
WRI I E
OUTPUT
TAPE
6,3,
( BETA(M,N),N=1,L)
DO 6
M= 1, L
6
WRITE
OUTPUT
TAPE
6,3,
(GAMMA(M,N),N=1,L)
DU 7
M= 1 L
7
WRITE
OUTPUT
TAPE
6,3,
(ET A(M,N),N=1,L)
DO 8
M= 1, L
8
WRITE
OUTPUT
TAPE
6,3,
(A(M,N),N=1,L)
WRITE
OUTPUT
TAPE
6,3,
(THETA(M),M=1,L)
WRITE
OUTPUT
TAPE
6,3,
(ZETA(M),M=1,L)
oO TO
END
100
46
sine cos f [Y^ (fi) Y^1 (&)1 /2Hi
(3.81)
sine sin ^ = [Y^(H)+Y^1(H)]/2jH^
(3.82)
The combination of Eqs. (3.77)(3.82) and the use of
the orthonormal property of Y^(n) expressed in Eq. (2.7)
we arrive to the expression
J(r,E,t) = u[*J(r,E,t)];(fE,t)]/2Hj
 v[(j,J(r#E,t)+4,~1(r#E,t) ]/2jH^
+ ^('E't)/H (3.83)
The moments, $n(r,E,t) satisfy the relations
ra
^(r,E,t)
 m(E)*Â¡J(,E
t)
(3.84)
*U,E,t)
1
S
*7
3
(j
3x
3
)
3y
R^E^i^U^t)
(3.85)
i1 (r,E,t)
i
= 7?
3
( +j
3x
3
)
3y
R^1(E,t)*(r,E,t)
(3.86)
(r,E,t)
i
=5 '
7
_i_ 1
Ri
3z 1
(E,t)*g(r,E,t)
(3.87)
81
expectation of the flux, i.e.,
/ dr / \}>(r,e,t) = 0
at r e
(4.82)
at any time. Eq. (4.77) takes the form
i/"] de+ /dr/ de
v_
(4.83)
Next, we will assume for the source the expression
x(r,e,t) = E (e) T (t) 6(x) 6(y) 6(z)
s s
(4.84)
i.e., we assume a point source at the origin with
separable energy and time dependence. At this stage
we introduce Eqs. (4.47), (4.73), (4.36) and (4.84)
into Eq. (4.83). Performing the spacial and energy
integrals in the resulting equation and considering
the orthogonal property of the spacial modes we arrive
at the equation
143
SUBROUTINE PULY (POLYNOMIAL SOLVER)
SUBROUTINE PULY{NI,BE,ROTR,ROOT I)
DIMENSION BUE(81),ROOTR(80),RGOT I (80),CUE(81)
N 3 = NI
N2 = N1 + 1
00 I KUN=1 N2
NUK=N2+IKUN
1 COE(KUN)= BOc(NUK)
XCONI
= 0.
XC0N2
= 0.6
XC ON 3
= 0.66
XCN4
= 0.9
XCONb
= 1.0
XCUN6
= 2.0
XC0N7
= 4.0
XC0N8
= 0.5
XCUN9
= l.E20
XCONIO
=1.E7
XCONI1
= 1 E~5
N4 = 0
I = N2
19 IFCOEII)) 9 7,9
7 N4=N4+1
ROO r R(N4) = XCONI
R0UTKN4) = XCONI
1 = 11
IF(N4Ni)19,999,19
9 AXR=XC0N2
AX I = XCONI
L= 1
N3 = 1
K=1
ALP1R=AXR
ALP1 1 = 4X1
GO TO 99
11 BE T1R = TEMR
BE T1 I = T EM 1
AXR = XCCN3
ALP2R=AXR
ALP2I=AX I
M=2
GO TO 99
12 BE T 2 K= T E MR
BET2 I = T E MI
AXR = X C U N 4
AL P3K=AXR
AL P 3 I = AX I
M= i
GO TO 99
13 BE T 3R= T CMR
BET3I=TEMI
14 TE1=ALP1RALP3R
79
I
m,n,p,k
Xm(x) Yn(y) Z (z) Jk (t) E* (e)
m,n,p in,n,p
(4.73)
where
Jk(t) = Ck(t) x Tk(t) (4.74)
m, n, p m, n, p m, n, p
Consider the flux and the adjoint flux equations,
Eos. (4.1) and (4.44). Operating on the first by the
operator
Q+ = / dr / de /t dt tji*(r,e,t) (4.75)
re o
and on the second by the operator
Q =
_J dr / de /* dt i>(?>e ,t)
r e o
(4.76)
and then subtracting the resulting equations, we
obtain the integrodifferential equation
Ci 4
47
and the scalar flux is obtained by
(r,E,t) = m(E)*(r,E,t)/H
= m(E) i> (r,E,t) (3.88)
Eqs. (3.83)(3.88) yield
(r,E,t) = m(E) [u ~ + v ~ +w] p" (E,t) \ (r,E,t)
 m(E) VR1 (E, t) i;(r,E,t)
(3.89)
In the case of N = w the net current through a unit area
perpendicular to the zaxis at (x,y;z = 0) is given by
Jn(r,E,t)
1 31
= ~ m(E) r, (E,t)
=0 3 z
,0 (3.90)
Having found the net current, the source
condition can be easily established. Let S(x,y,E,t) be
2
a plane source located at z = 0 evaluated in neutrons/cm
sec emitted in all inward directions. Equating the
neutron source strength to the net current we obtain the
source condition
This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.
December 19, 1964
~* * r? y 1
Dean, College of Engineering
Dean, Graduate School
Supervisory Committee:
57
A (r,E) =
I l (^) 2 } ( e) 2
m n k B'n m'n f
(3.126)
e.(r,e) =
I I I
m n k
Ak (r) sin bk zBk (r) cos z
m,n m,n m,n m,n
E*(e>
tan
1
l 11
m n k
*k k k ,k
Vr) cos bm,nz + Bm,Ar) sin bm,nz
Ek(e)
C
Similarly, we obtain the amplitude and phase shift for
the neutron density. These are found to be
Ad (r, e)
Af(r,e)
v^/e
(3.128)
(r,e) = 0f(r,e)
(3.129)
For the case of the total neutron flux and
density, Eqs. (3.118) and (3.119) are first integrated
over energy and then the same procedure discussed above
is used. This yields
A (r)
f
"l I
l
k
{Ak (rf+Bk (r)2}Ek
m,n m,n f
(3.130)
.127)
128
IS A POLYNOMIAL IN OF SIZE (NSE1 )*LSD+1. THE SIZE OF
A POLYNOMIAL MEANS HERE THE NUMBER OF TERMS IN THE
POLYNOMIAL,THAT IS,THE MAXIMUM POWER +1.
LSD =SIZE OF DETERMINANT
NSE = S l ZE UF ITS GENERAL ELEMENT ,MAXIMUM POWER +1.
DD(M,N,J) ^COEFFICIENT OF THE (Jl)TH POWER IN THE
lM,N)TH ELEMENT OF THE DETERMINANT.
AIK) ^COEFFICIENT OF THE (Ki)TH POWER IN THE RESULTING
EXPANSION. J=i,2...(NSEl)*LSD*I .
4EVADET
THIS SURUINE IS A PART OF THE DETEX CODE .
IT EVALUATES A DETERMINANT OF CONSTANT ELEMENTS.
LD = THE SIZE UF THE DETERMINANT
CD(M,N) = THE (M,N)TH ELEMENT OF THE DETERMINANT
DEI = THE VALUE OF THE DETERMINANT
sPoly
THIS SUBROUTINE FINDS THE ROOTS UF THE POLYNOMIAL
RESULTING FROM THE DETEX ROUTINE.
Nl =MAXIMUM POWER IN THE POLYNOMIAL
BUt(K) ^COEFFICIENT OF THE (KI)TH POWER, AIK) IN DETEX.
ROTRIJ) = RP OF THE (J)TH ROOT
ROOT I IJ) =IP UF THE IJ)TH ROOT
PART II
HGM AND ITS SUBROUTINES
THIS CODE COMPUTES THE INPUT MATRICES UF THE NWP CODE.
THE FIRST PUR I I ON UF THIS CODE IS MERELY A DIRECTORY PROGRAM
FOR THE SUBROUTINES WHICH FORM ITS ACTIVE PART.
THE UUIPUT OF THIS PROGRAM IS SELFEXPLANATORY. THEREFORE,
ONLY THE INPUT PARAMETERS ARE EXPLAINED.
L =NUMBER UF LAGUERRE POLYNOMIALS (SIZE OF MATRICES)
ss =
SA ^
UM =
NS =0 IF THE SMALLEST POWER OF E IN THE FUNCTIONS
IS 3/2
2 IF THE SMALLEST POWER OF E IN THE FUNCTIONS
IS 1/2
NCV =0 IF
= 1 IF
IS ENERGY INDEPENDENT
IS ENERGY DEPENDENT
48
m(E)
3
9 1
R, (E,t)*(r,E,t)
3z A
= S(x,y,E,t)
z=0
(3.91)
By factoring out the Maxwellian component of the source
and dividing Eq. (3.91) by ra(E) it becomes
1 3
3 3z
R^Ejt)
 X(x,y,E,t)
z=0
(3.92)
Finally, if we operate on Eq. (3.92) by R^(E,t) we
obtain the final form of the source condition
3 *
\p (r,E,t)
= 3R1(E r t)x(x,y,E,t)
z=0
(3.93)
where the operator R^(E,t) is given by Eq. (2.36). The
transformation of Eq. (3.93) into the domain of the
dimensionless energy variable, e, yields
3
r~ *(r,e,t) = 3R,(cft)x(x,y,e,t)
9z z=0 1 (3.94)
or, by using Eqs. (A.7) and (A.29),
3 >
i(r,c,t)
3 z
1 3
* 3[77 T~ + G(e) ]x(x,y,eft)
z=0 vn/e 91
(3.95)
This condition introduces two corrections on the source.
The first one is the transport correction given by the
29
/a X (x) X (x) dx = a
i m' m m',m
a
(3.23)
b
/ Yn'(y)Yn(^)dy = b6nl,n (3.24)
b
leads to the modal equation
[H(e)+4F(e)+ i 42 (P/5tfln>n)] Em,n(e)=0 (3.25)
ra,n = 1,2,3,
where we divided through by the constant C .
Ill /XI
In order to determine the coefficients, A ,
m, n
we will operate on Eq. (3.25^ by the energy integral
operator
0 = / de m(e) (e)
Xr
o
(3.26)
where m(e) is the Maxwellian function previously
defined by Eq. (2.26K* This operation leads to the
following set of algebraic linear homogeneous equations.
I [a + 0 A + y A2
Z l' ,1 l\i f ,1
n r]
Â£' ,i
m,n
= 0
(3.27)
147
M = 3
99TEMR=COE(1)
TEMI=XCUN1
100 I =
Z1=TEMRAXR
Z2 = TEM IoAX 1
TE1=Z1Z2
Z1=TEMI*AXR
Z2 = T EMR*AX I
TEMI=Z1+Z2
100 IEMR =TE1+CGE(1+1)
HE LL = TEMR
BE LL = TEM I
42 IFIN4}102,103,102
102 DO 101 1=1,N4
TEMI =AXRRUUTR( I )
TEM2 = AX IROOT I ( I )
Zl=TEMI*TEMI
Z2= T EM2 TEM2
IE1=Z1+Z2
Z1=TEMR*TEM1
Z2 = T EM I TEM2
Z 3 = Z 1 + Z 2
TE2 = Z3V TE1
Z1=TEMI*TEM1
Z2=TEMRTEM2
Z3=Z1Z2
fÂ£MI=Z3/TEl
101 TEMR=TE2
103 GO 10(11,12,13,13,33,34),M
999 CONTINUE
RETURN
END
122
n
Due to the fact that X (x) does not exist for negative
in
indices and is unity for m=0 and n=0, we can reform kq.
(C.14), using the Kronecker delta function and Heavi
side unit step function, where it takes the final form
. n
X (x) = <5 6 + (m+n)
m / m,o n,o v
u (n1) X,n 
m
(C.15)
Three other important properties of the Laguerre
polynomials are given by the relations (53)
dp n +
jC (*) = (DP u(mp)/n P(x)
dxp m mp
n m+1 n 5 n .
(n+2m+lx) X (x) = L (x) + u(ml)(m+n) Â£ (x)
m m+n+1 m+1 ra1
(C.17)
CO 2
/ xn ex/n(x) dx = 6 [r (m+n+1)] / r (m+1)
o m' m, m
(C.18)
This last relation is the socalled orthogonal property.
If we define the orthonormal polynomials, L^(x) to
satisfy the relation
^ xn ex L (x) L (x) dx = 6 i
m m' m,m
o
(C.19)
142
SUBROUTINE ELEM {EL EM I NA TI UN METHOD)
SUBROUTINE E LE M(N,C,T )
DIMENSION HI 20,21),0(20,21)
KE=N+I
T= 1.0
DO 40 K=1,N
X = 0.0
UU 10 IK,N
IF(ABSE(X)ABSF(C(I,K))) 9,10,10
9 X=ClI,K}
I X= I
10 CONTINUE
DU 11 J = K,K E
H(K,J)=0{K,J )
0(K,J)= C( IX,J)
11 C(IX,J)=H(K,J)
1F(IXK) 17,18,17
17 T =T *ClK,K)
GO 10 l'>
18 T=T*C(K,K)
19 CON TINUE
AY=C(K,R)
DO 20 J = K,KE
20 C(K,J)=C(K,J)/AY
DO 2 1 J = 1,KE
21 H(K,J)=C(K,J)
DO 23 I=1,N
R = C( I,K )
UU 22 J=1,KE
22 C(I,J)=C(I,J)R*H(K,J)
23 CON INUE
DO 10 J = 1,K;
30 C(K,J)=H(K,J)
40 CONTINUE
RE TURN
END
106
f2(E) = Mq2(e) M12{EJ
(A.19)
g0(E) = ia(E) + Mg(E) M(E)
(A.20)
gL (E) = mJ(e)
(A.21)
g2(E) = Ml2(E)
(A. 22)
In all the algebra used in the above derivation we
neglected all derivatives of the third order and above,
in consistency with keeping up to second order deriva
tives in the Taylor series expansion of the flux. We
have also used the relation
I (EV = (E)
s
(A.23)
Thus, we have calculated the operators and put them in
general forms as functions of energy, cross sections and
energy moments.
In transforming Eq. (2.54) from the energy domain
to the domain of the dimensionless variable
e
E/T
(A.24)
11
If the Maxwellian function is factored out from the
flux moments, i. e.,
$n(r,E,t) = m(E) ij>n (r,E,t) (2.27)
m ra
the component, ij>n(r,E,T) is expected to be relatively
m #
smooth.
Similarly, the source moments are analyzed:
SÂ¡Â¡(r,E,t) = m(E) x^(r,E,t)
(2.28)
To evaluate the integrals,, the flux
components,^(r,E',t) are expanded in Taylor series
about E.
00 1 k k
^{r,E,,t) = l \E'E) d nÂ£(,E,t)
kiC i
=o
(2.29)
where
k n,* n *
D *m(r,E,t) r *m(r,E',t)
9E'K
E'=E
Using Eqs. (2.27) and (2.29) in Eq. (2.22) the
integrals become
" I E (E'E)m
E k=SQ k!
Dkipn (r,E,t) dE'
(2.30)
156
21 FORMAT{1H1/////,52X,8HETAIM,N) )
U 22 M=1 L
22 WRITE UTPUT TAPE 6,14,(ETA(M,N),N=1,L)
WRITE OUTPUT TAPE 6,23
23 FORMAT(1H1/////,6X,8HTHETA(M),7X,7HZETA(M) //)
WRITE OUTPUT TAPE 6,24,(THETA
24 FORMAT{2F15.7)
RETURN
END
63
Considering the boundary conditions the functions
involved in this equation are given by
X (x)
m
= cos
2ml
2a
vn(y)
= cos
2nl
U
Zp( z)
= cos
2SzL
2c
(4.11)
(4.12)
(4.13)
T (t) =
m,n,p
E (e) =
mn ,p
exp(A t)
m ,n ,p
V A* l/ ^ ( e )
z m,n,p i
(4.14)
 (4.15)
where c and AÂ£ are constants to be determined
m#n/P m,n,p
and m, n and p are positive integers from 1 to some
desired maxima.
The combination of Eqs. (4.9)(4.15) gives the
equation
I c [H(e)
111,11 ,p in, n, p
A F(e) + A2 + I B2 ]
m,n,p m,n,p 3 m,n,p
X (x) Y (y) Z (z) T (t)
m n P m,n,p
E (O =
m,n ,p
0
(4.16)
146
Z1=rE3*Tfcb
L2T Â£4* E6
AXR=ALP3R+Z1Z2
Zl=rE3*TE6
Z2 = TE4;>TE5
AXIALP3I+Z1+Z2
ALP4R=AXR
AL P4I = AX I
M = 4
GO TO 9*
15 N6=l
30 IF{ABSF(HELL)+ABSF(BELDXC0N9)18,lb,16
16 TE7=ABSF(ALP3RAXR)+ABSFIALP3IAXI}
Z1 = A BSF(AXR)+ABSF(AXIJXC0N10
I F(rÂ£7/Zl)18,18,17
17 N 3 = i\l 3 + 1
ALP1R=ALP2K
ALP1 I = AL P2I
ALP2R=ALP3R
ALP2I = AL P31
AL P 3R=AL P4R
ALP3 1=ALP4I
BET1 1 = BET2I
BE T2R=BE T3R
BET2 I = BE T3 I
BE F 3R= TEMR
BET3I=TEMI
BEriR=BET2R
IF!N3100)14,18,18
18 N4=N4+L
ROOT R(N4)=ALP4R
ROUTI(N4)=ALP4I
N 3 = 0
41 IF(N4N1)30,999,999
30 IF(ABSFtROUTI(N4>)XCON11)9,9,31
31 GO rO(32,9),L
32 AXR = AL P1R
AXI=ALP1I
AL P 1 I =ALP 11
M= 5
GO TO 99
33 BETlR= TEMR
BET1I = TEMI
AXR= AL P2 R
AX I= AL P2 I
ALP2I=ALP2 I
K = 6
GO TO 99
34 BET2R=TEMR
bET2 I = TEMI
AX R= AL P3R
AX I=AL P3I
ALP 3 I=ALP3I
L = 2
BIOGRAPHICAL SKETCH
Hassan H. Kunaish was born in Zabadani, Syria,
in 1933. He completed his secondary education in
Damascus and received the bachelor's degree in Mathe
matical Sciences from the University of Damascus in
1958. In 1959 he received a scholarship from the
Soviet government for a study in nuclear physics at
the University of Moscow.
Sponsored by the American Friends of the Middle
East, Inc., he came to the United States and entered
the Graduate Scool of the University of Florida in
1960 where he received the degree of Master of Science
in Nuclear Engineering in June, 1962. During 1964 he
held a onethird time assistantship at the Department
of Nuclear Engineering while working toward the Ph.D.
degree at the University of Florida.
Hassan H. Kunaish is married to the former
Subhieh AbdulDayem and they have two children. He
is a student member of the American Nuclear Society,
a member of the American Friends of the Middle East,
Inc., a member of the Arab Students Organization in
the U.S.A., and a member of the Arab Club at the
University of Florida.
162
55
N(r,e,t)
Re
[I I I
m n k
ra
_X (x) Y (y) Z
, n m n J i
(z)
v,
y?
(e)Ek(e)ejait]
(3.117)
The total neutron flux and total neutron density are
obtained simply by integrating Eqs. (3.115) and (3.117)
over energy.
It was mentioned at the beginning of this
chapter that the flux, or hence, the neutron density,
has the same time behavior as the source with a phase
shift which depends on the detector location and the
source frequency. Having found the flux and the
density we can now determine the amplitudes and phase
shifts for both of them using Eqs. (3.115) and (3.117)
which are rewritten as
(r,e,t)
Ret l Ef(c)(R^>n()<jB^n()}ej("t"bra>nZ) ]
m n k
(3.118)
N(r,e,t) =
k
Re[Â£ l l E^
m n k d m,n m,n
(3.119)
+ a
I
9
Zm(E'E) = /1Pm(y0) Es(EUE,n.n)du (2.19)
I
The use of Eqs. (2.3)(2.5) and the properties of
the spherical harmonics expressed by Eqs. (2.7)(2.14)
along with
3 3 8
7 = cose + sine cos V + sine sin y
3z 3x 3y (2.20)
leads to the formulation of a set of coupled integro
differential equations relating the moments of the flux
up to any order of approximation, i.e.,
I ~nt(E) ]*Â£(?,E,t)+s5J(r,E,t)+lÂ£(r,E,t) =
V 31
A(Z)
(m+ln) (m+l+n) ** (mn)(m+n) *5 + 1
) .+ E#t)+ } (r,E,t)
(2m+3)(2m+l) m+1 (2m+l)(2ml) m"1 J
+7 B(x,y)
(2m+3)(2m+l) w+1 (2m+l)(2ml) m1 J
+ie*(x,y) j
(m+n)(m+n1) h n1 ^ ^ r(ran+2)(mn+1) h>n]_ +
(2ra+l)(2ml)
HT* Ciir'E.t)!
(2m+3)(2m+l)
m+ltr'E't)]
(2.21)
where
56
where
k _k i uk
p = a + jb
m,n m,n m,n
(3.120)
k
Ak (r) + jBk (r) = Ck x (x)Y (y) e amnZ
m,n mfn m,n m n
(3.121)
E^e) = m( e) Ek ( e)
f
(3.122)
1
Ek(e) m(e) Ek(e)
Q v /e
(3.123)
The indices f and d have been used to indicate the
neutron flux and neutron density, respectively. If we
write
f 4
(r,e,t) = Af(r,e) cos[(ut ef(r,e)] (3.124)
4 4 4
N(r,e,t) = Ad(rfe) cos [ut 0d(r,e)] (3.125)
then Af(r,e), ef(r,e), Ad(r,e) and 0d(r,e) signify,
respectively, the amplitude and phase shift of the neu
tron flux and density. If the righthand sides of Eqs.
(3.118] and (3.124) are equated and expanded, the
following expressions are obtained for the amplitude
and the phase shift:
114
M^(E) = (2g/3) TE lQ
(B. 23 )
M*(E) = O
for k > 2
(B.24)
Notice that the moments, M^(E) are zero for all values
of k greater than 2. Therefore, the Taylor expansion of
the flux, Eq. (2.29), contains three terms only, i.e.,
up to second order derivatives, since the coefficients
of higher derivatives are identically zero in this case.
See Eq. (2.34).
The Functions, h^CE), f,. (E) and g^ (E)
Having found the energy moments we can calculate
these functions in the heavy gas model for a 1/v type
adsorption cross section.
T k
(B.25)
o
With rather lengthy but straightforward algebra, the
combination of Eqs (B 17) (B 25) and Eqs (A. 14) (A. 22)
of Appendix A leads to these equations:
(B.26)
65
Since the operators in brackets do not depend on time,
and hence do not operate on T(b) this equation can
m,n,p
be divided through by c T(t) The following
m, n, p m, n, p
equation follows
[H( e)
A F(e) + A2 e"1 + I B2 ] E (e) = 0
m,n,p m,n,p 3 m,n,p m,n,p
(4.22)
This equation applies for all the spacial modes, i.e.,
for all values of m, n, and p. We call this equation
"the modal equation" because its solution for given m,
n, and p determines E(e) A and hence T(t) ,
m,n,p m,n,p m,n,p
and therefore, completes the solution of the (m,n,p)
mode. To solve this modal equation, we first substi
tute for E(E) from Eq. (4.15) and then operate on
m, n,p
the resulting equation with the integral operator
P = / de m(e) L(e)
o
This leads to the equation
(4.23)
m *n ,p
y n A2
1> 1 m,n,p
n,,cB2 ]A^
1r1 m,n,p mn P
0
(4.24)
44
Neutron Net Current and Source Condition
Before we establish the source condition the
neutron net current has to be developed in terms of the
neutron flux. To do so we first define the neutron net
current. Let dA be an infinitesimal area whose norm is
the vector N located at r and consider the directional
>  *
flux, $(r,n,E,t) where the vector Q makes with N the
anglevQ given by
cos 6 = Q*N (3.73)
The net current can then be defined by the following
relation
Jn(r,E,t) dA = the net number of neutrons
with energy E which pass
through the area dA in all
possible directions.
This definition is equivalent to the analytical
statement
J (r,E,t)dA = dA / $ (rf,E,t) (lN)d2 (3.74)
.
where dA(n N) is the apparent area seen by the
neutrons traveling in the direction, ii, or
Jn(r,E,t) = / 4 (r,n,E,t) (nN)dn
n
(3.75)
123
then
_ n 1/2 3/2
L (x) = >/, (x) {m+iy] [r (m+n+1)] (C.20)
m m
The introduction of Eq. (C.20) in Eqs. (C.15),
(C.16) and {C.17) leads to the relations
1/2 1/2
Lm (xi = () u (nl)L^ (x) + (} u {m1) i/* (x)
m nri n m v m+n' ra l
+ 6 6
ra, o n, o
(C. 21)
Lm^x' = i_1'P u(mP>
dxp
T (m+1)
.r(m+lp)J
1/2
Ln+P(x!
mp ^
(C.22^
1/2
^m(m+n)j L^(x) ~ (n+2mlx) u(ml) L^_1(xj
T 1V2 n
 ^(m1) (m+n1) J u(m2) Lm_2(x) (C.23)
Suitable combination of Eqs. (C.21) and {C.22\
leads to the important relation
dx*
n / m \l/2 fdP a
L (x) = u(ml) u(nl) 
m 'm+n 1 LdxP hvP1
P1 i
Ln (C.24)
m1
99
For large systems, small geometric buckling, the
theoretical results are very close to the experimental
ones. The range of agreement between the analytical
and experimental results may be considered up to
3 2
geometric buckling of the order of 3x10 cm
Calculations have been made in the diffusion
approximation (38) in the heavy gas model. The results
of these calculations almost coincide with those of
the P1 approximation throughout the range of buckling
included in Fig. 7. Therefore, the deviation of the
P1 results from the experimental values in this range
cannot be explained by an overcorrection which may
arise from the transport correction hidden in the
second order time derivative of the flux. However,
this overcorrection does exist, and becomes appreci
able in very small systems, i.e., for bucklings of
1 2
the order of 10 cm
Conclusions
The time, energy and spacedependent neutron
thermalization theory in moderating assemblies has
been formulated in the consistent P^ approximation.
This formalism was based on the assumption that the
nonMaxwellian component, >l>(r,E,t), of the flux is a
smooth function of the variable neutron energy, E, so that
70
12n 1
vnW =
COS 1
\ 2b
Tty
J (4.39)
Zp(z)
cos j
' 2p 1
( 2c
IT Z
1
1
1 (4.40)
Tk(t)
m,n,p
exp (
"Xm,n,p
t)
(4.41)
Ek(e)
m,n,p
l R4
L m
i
+ dl(1>
tn t P z
(e)
(4.42)
Notice that all these functions have been determined
except which will be found later from the prin
m,n,p
ciple of neutron conservation.
In the neutron waves case we saw that the source
condition, Eq. (3.95), leads to the set of Eqs. (3.114)
which are enough to determine all the constants, n
(k = 1, 2, ..., L+l), for each (m,n) spacial mode. In
the pulsed neutron case such a condition is not enough
to determine all not because of the additional
m, n, p
space index, p, but because of the greater multiplicity
in the energy modes, i.e., due to the fact that k takes
twice as many values as in the wave problem, k = 1, 2,
..., 2(L+l). A more general condition based on the neu
tron conservation principle will be used in this case.
The use of this principle requires the knowledge of the
117
(B.41)
gL(e)
(B.42)
g2(e)
(B.43)
137
SUBROUTINE DETEX (DETERMINANT EXPANSION)
SUBROUTINE DETEX(LSD,NSE,DD,A)
DIMENSION DO(10,10,3),CC(10,10,3),CD(10, 10),A(21) ,B(21 )
LD= L SO
L E = N S E
DO 30 M = I L 0
DO 20 N=1, L U
DO 10 K= L,LE
10 CC(M,N,K)=DU(MNK)
20 CONTINUE
30 CONTINUE
JMAX = LD*(LE1 ) + 1
DO 40 J = 1, JMAX
40 B(J)=0.0
DO 401 J 1 = 1 L E
ND= 1
N1 = J 11
JP=1+N1
IF(LDND) 301,101,301
101 CONTINUE
DO 201 M=1,L D
CD(M,i)=CC(M, 1,J1)
201 CONTINUE
CALL EVADET(LD,CD,DET)
B(JP)=B(JPJ+UET
301 CONTINUE
DO 402 J2=1,LE
ND=2
N2=J21
JP=1+N1+N2
IF(LDNO) 302,102,302
102 CONTINUE
DO 202 M =1,LD
CD(M,1)=CC(M,1,J1)
CD(M,2)=CC(M,2,J2)
202 CONTINUE
uALL EVADET(LD,CD,DET)
5(JP ) = B(JP)+DET
302 CONTINUE
DO 403 J3 = i,LE
ND= 3
N3 = J 3 1
JP=i+Nl+N2 + N3
IF(LDND) 303,103,303
103 CONTINUE
DU 203 M=I,LD
CD(M,l)=CC(M,l,Jl)
CD(M,2)=CC(M,2,J2)
CD{M,3)=CC(M,3,J3)
Ill
o (E E 1 )
m
4ir
f1 xm dx
/ exp [j t (E '
 E}]
[1 + P { E'
+ E 2x (EE 1 ) :
} (jt t2T) ]
(B.3)
By integrating this equation for m=0 and m=l and using
the relations
1 n
 / (jt)n exp(jwt) dt = pjy 6 (w) = (w) (.4)
z T* a, dw
A = A E = E'E (B.5)
we obtain the P.^ scattering moments
1
E'
ao(E E1 ) = aQ
' \2
E
Cfi (A } + u (E+E) 6 ^ (A )+vT(E'+E)5 ^ (A)]
(B.6)
E' [6 (A) + T 6 (A )]
(B.7)
Moments of Scattering Cross Sections
The energydependent moments of the scattering
kernel are given by
mE) = I m(E + E') dE
E'
(B.8)
119
where v stands for a 0 ... and the operator
i \ l l Z i i
A(c) = ao(c) + a^e) + a2 (e) jf (C.2)
stands for H(e), F (e), ... The functions, a^(e),
which correspond to h^e), f^iejr, ..., [see Eq. {B.35)
(B.43)] can be given the form
a^e)
a. .
i#3
(j4)/2
G
(C. 3)
The reason for this notation is that j=l corresponds
to the smallest power of g, 3/2, that appears in the
functions, hu (c K f^Cc), ... The combination of Eqs.
(C.l), (C.2) and (C.3) leads to the general expression
10
l z
1 tao,ju2;
3=1
l + al,jV' + a2,jWtÂ¡l 1
(C.4)
where we define
IT
l \ l
r (j4)/2 (!) (1)
J e m(e) 1/,'(g) L^1'(e) dc
o
(C. 5)
V
Z)Z
/ e
o
(j4)/2 (l) d (!)
m(e) 1* (e) L (e)
Z' de j.
WD
Z\Z
r (j4)/2 (i)
/ e m(e) L (Â£)
l (1)
dÂ£2 h (a) de
(C. 7)
152
SUBRUTINt ABTVS
SUBROUTINE ABTVS(L,NS,SS,SA,R,AA,BB,TT)
DIMENSION ALPHA10,1),BETA(10,10),GAMMA10,10),
1 ETA( 10,10),T HE T A{10),ZETA(10),0ELTA(10,10),
2 A(1010),AA(3,10),BB(310),TT(1Q)U(10,10,10),
3 V(10,10i.0)W(10,10,10),C(50),G(30)
COMMON ALPHA,BETA,GAMMA,ETA,THETA,ZÂ£TA,DELTA,
1 A,AA,BB,T,U,V,W,C,G
DO 4 N=l, 10
TT(N)=0.0
DO 3 M= 1,3
AA(M,N)=0.O
3 36(M,N)=0.O
4 CONTINUE
NSPl=NS+l
NSP2=NS+2
NSP3=NS+3
NS P4 = NS + 4
NSP3=NS+5
NSP6=NS+6
NS P 7 = NS + 7
NSP8=NS+8
AA(l,NSPL)=R*SS*SA/3.0
AA(1,NSP2)=SA**2
AA(1,NSP3)=SS*SA*(1.07.0*R/3.O)
AA(2,NSP2)=2.0MR*#2)*SS*SS
AA ( 2 NS P3 ) = 10 O*R* SS*S A/3. O
AA(2,NSP4)=SS*SS*(43.0*R/3.4.0)*R
AA(2,NSP5)=4.0*R*SS*SA/3.0
AA(2,NSP6)=2.0*R*SS*SS*(1.O10.0*R/3.0)
AA(3,NSP4)=9.0R*R*SS*SS
AA(3NSP5)=R*SS*SA*4.0/3.0
AA{3NSP6)=R*SS*SS*(44.0*R/3.02.0)
AA( 3 ,NSP8) = 4.0*RbR*SS*SS/3.0
BB(1,NSP1)=R*SS/3.0
BB (I,NSP2)=2.0*SA
BB(i,NSP3) = SS*l 1.O 7.O R/3.O )
8B(2,NSP3)=~10.0*R*SS/3.0
33(2,NSP3)= 4.0*R*SS/3.0
3B(3.NSP5)=4.0*R*SS/3.0
TT{NSP2)= R*SS/2.0
TT ( N 3 P 3 ) = S A
TT(NSP4)=SS*(1.08.O *R/3.O)
WRITE OUTPUT TAPE 6,5
5 FORMAT(IH1///,50X,7HAA(M N)//)
DO l M=1,3
7 WRITE OUTPUT TAPE 6,8,(AA(M,N),N=1,10)
8 FORMAT(10E12.4)
WRITE UUTPUT TAPE 6,9
9 FORMAT(///,50X,7HBB(M,N )//)
DO 10 M1,3
10wRIIE OUTPUT TAPE 6,8,(BB(M,N),N=1,10)

