DETERMINATION
PARAMETERS
W AVE
OF HETEROGENEOUS
BY THE NEUTRON
TECHNIQUE
EMILE ANTHONY BERNARD
A DISSER'A iON I RSNL, ITED TO THIE GRADUATt COUNCIL 01"
1iLE UNITRSITrY or I LOrUDA
IN I'.AUTIALL I'LIILL.MENT OL T1HE hL:LQUIREMNENTS FOR TILE
D)EGICE OF" DOCTOR 01 IPILOSOP'IY
UNIVERSITY OF FLORIDA
19e6
To "B4i
TABLE OF CONTENTS
Page
ACKNtlOULEDGMHE TS . . . . . . . . .. iii
LIST OF TABLES . . . . . ... . . . . . vi
LIST OF FICURES . . . . . . . . .. . . vii
KEY TO S'Y OLS . . . . . . . . . . . .ix
ABSTRACT . . . . . . . . .. . . . xi
CHAPTER I INTRODUCTION . . . . . . . . .
Background . . . . . . . .. . ... .1
Scope and Objectives . . . . . . . . .
CHAPTER II ITHEORY . . . . . . . .... .. 6
Introduction . . . . . . . . . . 6
The Theorecical Model . . . . . . . .7
Solution of Eqatrion For Fast Neutrons .. .. .
Solucton of equation For Thermal Neucrons .. .. 1
Combinacior. of Fast and Thermal Neutron Soluticns 16
Nu.erical Scluti on . . . . . . . ... .17
7 Analysis . . . . . . . . . .. 20
Tie Critical Frequency . . . . . . . . 21
CHAPTER III EX'PEDIETL.L ?rTHOD. . . . . .... .26
Int ro jccion . . . . . . . . . ... .26
Prel imir ary procedures . . . . . . ... .51
Experimental Procedure . . . . . . ... .32
ACKNOWLEDGMENTS
The author wishes to express his thanks and appreciation to his
supervisory committee, especially to its chairman, Dr. M. J. Ohanian,
for their assistance and advice. Special thanks also go to
Dr. R. B. Perez who was chairman prior to his departure from the
University. Through his initial guidance and continued interest and
assistance he has remained closely associated with the work of this
dissertation. Appreciation is also extended to the Department of
Nuclear Engineering Sciences and to the U. S. Atomic Energy Commission
for their financial support of this dissertation.
Many other individuals rendered their time and services in support
of this dissertation and the author wishes to give recognition to
several of them. George Fogle gave invaluable assistance in the
electronic and technical problems encountered with the experimental
equipment. Jose Aldeanueva, Benny DuBois, Changmu Kang and Daniel
Meade were constantly available to assist in the data acquisition.
The competent assistance of Jim Burgess was very important in the
development of the computer programs used for data analysis.
Mrs. Edna Roberts was very helpful in the typing of the final manu
script.
The author wishes to thank especially his wife, Betty, who was
always willing co do everything she could and assisted in many ways
in the preparation of this dissertation.
TABLE OF CONTENTS (cont'd)
Page
CHAPTER IV DATA ANALYSIS AND RESULTS . . . . ... 6.
General . . . . . . . . ... .. .. 6.
Continuous Mode Analysis . . . . . . . .. 40
Heavy later Pulsed Experiment . . . . . . . 42
One Fuel Rod Pulsed Experim2nt . . . . . . 53
A Lattice of Fuel Rods . . . . . . . ... .75
CHAPTER V CONCLUSIONS AND RECOC'END..TIONS FOR FUTURE WORK 79
Conclusions . . . . . . . . ... . . 79
Reco mmenda t ns . . . . . . . . . ... . S.
APPENDICES
A MEASUPREENT OF RESOLUTION TILE: USING A PULSED
NEUTRON SOURCE . . . . . . . ... .. . 82
B DETERMINATION OF T.RGET CUP.REN'T PULSE WIDTH
AND MULTICHANNEL ANALYZE ? CU;!;;EL WIDT4. ...... 9. 9
C EXPERI.AL CHECKS . . . . . . . .92
D COCMPUTER P FROGP...M.S . . . . . . . . . 93
E EXPERIMENTai L A.IMPLITUDES ND FHK.'ES . . . ... 116
F NU'IERICAL CONSTAN;TS . . . . . . . . 13j
BIBLIOC RAPHY . . . . . . . . . . . . 155
BIOGRAPHIC.L SKETCH . . . . . . . . ... . 137
LIST OF TABLES
TABLE Page
1. COMPUTATION OF ALPHA AND XI AND CONTAMINATION
BY FIRST SPATIAL HARMONIC . . . . . ... 19
2. NUMERICAL ILLUSTRATION OF INTERSECTION OF
DISPERSION LAWS . . . . . . . . ... .24
3. CONTINUOUS MODE DECAY CONSTANTS . . . . .. 41
4. THEORETICAL VALUES OF ALPHA AND XI
FOR HEAVY WATER . . . . . . . . ... .46
5. EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATER . . . . . . . . . 47
6. EXPANSION COEFFICIENTS OF RHO SQUARED AND
THERMALIZATION AND DIFFUSION PARAMETERS
FOR HEAVY WATER . . . . . . . .... .55
7. EXPERIMENTAL VALUES OF REAL AND IMAGINARY
COMPONENTS OF RHO SQUARED FOR HEAVY WATER . . .. 56
8. THEORETICAL VALUES OF ALPHA AND XI FOR
HEAVY WATERONE FUEL ROD SYSTEM . . . . .. 62
9. EXPERIMENTAL VALUES OF ALPHA AND XI FOR
HEAVY WATERONE FUEL ROD SYSTEM . . . . . 63
10. EXPANSION COEFFICIENTS OF RHO SCUA.ED FOR
HEAVY WATERONE FUEL ROD SYSTEM . . . . .. 67
11. EXPERIMENTAL VALUES OF REAL AND IMAGINARY COMPONENTS
OF RHO SQUARED FOR HEAVY WATERONE FUEL ROD SYSTEM 71
12. HETEROGENEOUS PARAMETERS BASED ON INTERSECTION
OF DISPERSION LAWS . . . . . . . ... .77
15. DATA FOR RESOLUTION TIME DETERMINATION . . .. 87
LIST OF FIGURES
FIGURE
1. A SCHEMTIC VIEW OF THE HiETEROGENEOUS SYSTEM .
2. ILLUSTRATION OF DISPERSION LAW INTERSECTION .
5. EXPERIMENTAL ASSEMBLY . . . .
4. MOVABLE DETECTOR COUNTING SYSTEM . . . .
5. REFERENCE DETECTOR COUNTING SYSTEM . . .
6. METHOD OF ANA.LYSIS . . . . . . .
7. INTERRELATION OF COMPUTER PROGRAMS . . .
8. NORMALIZED NEUTROi; PULSES IN HEAVY WATER . .
9. EXPERLUEN FAL AM.PLITUDES FOR HEAVY WATER . .
10. EXPERILNTALI PH.'5ES FOR EEAVY AFTER . . .
11. EXPERIMENT.AL AND THEORETIC..L VALLES OF ALP?,
FOR HEAVY '.'ATI.. . . . . . . . .
12. EXPERIIE:I.NL AND THEORETICAL VALUES OF XI
FOR HEAVY :ATER . . . . . . . .
15. EXPERiLMENTAL .A:D THEORETICAL DISPERSIOI L.A.;S
IN RHO PLAkt: FO?. HEAY '.TER7E . . . ..
11. CC:P.RISO:I OF EX:?ERI::;T.L DIS3? AI' ON LA '15
IN FL'O PINE FOR HEA:.Y .ATE. . . . . .
15. Ik.GL .ARY CCMPO;NNT CF RiHO SQCUAl:AD FOR HEAVY ri
16. REAL COMPO:;:NET OF RO1 SCL'UAED FOR HEAVY LATER
17. EXPERIEN;TLL AND ThEOIrIC.AL DISPERSIC: LA'.:S
IN RiHO SC'UA.ED PL.iNE FOP. HE.VY WATER . . .
1.5. :;NO,'L.LI?ED THE?'t.L :NUTC.O ?ULSS FOP. HL:.'Y IATERCO E
FUEL ROD S ST . . . . . . . . .
Page
4
. . 25
. . 28
. 29
. . 30
. . 35
. . 49
*. 49
. . 50
S51
54
LIST OF FIGURES (cont'd)
FIGURE Page
19. EXPERIMENTAL AMPLITUDES FOR HEAVY WATERONE
FUEL ROD SYSTEM . . . . . . . . . . 60
20. EXPERIMENTAL PHASES FOR HEAVY WATERONE
FUEL ROD SYSTEM.. ....... ........... . 61
21. EXPERIMENTAL AND THEORETICAL VALUES OF ALPHA
FOR HEAVY WATERONE FUEL ROD SYSTEM . . . . .. 64
22. EXPERIMENTAL AND THEORETICAL VALUES OF XI FOR
HEAVY WATERONE FUEL ROD SYSTEM ........... 65
25. EXPERIMENTAL AND THEORETICAL DISPERSION LAWS IN
RHO PLANE FOR HEAVY WATERONE FUEL ROD SYSTEM ..... 66
24. IMAGINARY COMPONENT OF RHO SQUARED FOR HEAVY
WATERONE FUEL ROD SYSTEM . . . . . . ... 69
25. REAL COMPONENT OF RHO SQUARED FOR HFAVY
WATERONE FUEL ROD SYSTEM . . . . . . ... 70
26. EXPERIMENTAL AND THEORETICAL DISPERSIO:; LAWS IN
RHO SQUARED PLANE FOR HEAVY WATERONE FUEL ROD SYSTEM 75
27. EXPERIMENTAL AND THEORETICAL INTERSECTION OF
DISPERSION LAWS IN RHO PLANE . . . . . . 74
28. EXPERIMENTAL AND THEORETICAL INTERSECTION OF
DISPERSION LAWS IN RHO SQUARED PLANE . . . .. 76
29. ILLUSTRATION OF RESOLUTION TIME AND TIME
BETWEEN BURSTS ON COUNTING RATE ... . ... . . 84
50. COUNT RATE VERSUS TIME BETWEEN BURSTS . . . ... 88
31. REPRESENTATIVE DISPLAY OF COUNTING PATTERN . . . 94
viii
KEY TO SYMBOLS
A . . TlHEIULAL NEUTRON ABSORPTION TERN;
B . TRANSVERSE BUCKLING OF nth SPATL'L NODE
n
c . EXPANSION COEFFICIENT OF THEl'W'.L NEUTRON FLIU:
AT z 0
D .. DIFFUSION COEFFICIENT
J . ZERO OPDER BESSEL FUNC"LL)N
o
L~ .. DIFFUSION LENGTH SQUAI)
L . SLO:ING DO'.I TIM: FOP. FISSICN NEUTRONS
N . NOR .LIZATION CONSTANT
P . EXPANSION,, COEFFICIENT OF HiO SQU.A..RED IN PO'.TE.
n
SERIES OF (tw)
p . RESON;ANCE; ESCAPE PROBABILITY
q . SLOWING DO'IN DENSITiY
R . EXTRAPOLATED RADIUS OF EXPERI;.ENLT.L ASSEMBLY
r . RADLAL COORDINATE
r . RADIU C? FUEL ROD
o
S . THERM*.L ;EuTr?.N SOURCE YES.
S . E. EXPM;NSION COC.FFLCIE:,',07 SOLCE AT z 0
s . LAPLACE T:hANSFOPRM PA,?, TR 'lim .RESPECT TO ..GE
t .. TIME
u . L TH.3,.
v . V.LOC LT
KEY TO SYMBOLS (cont'd)
z . AXIAL COORDINATE
0 . REAL PART OF COMPLEX INVERSE RELAXATION LENGTH
6 . DIRAC DELTA FUNCTION
7 . FEINBERGGALANIN CONSTANT
T . NEUTRONS PER THERMAL NEUTRON ABSORPTION IN FUEL
e . FAST NEUTRON SLOWING DOWN DENSITY
. . IMAGINARY PART OF COMPLEX INVERSE RELAXATION LENGTH
p . COMPLEX INVERSE RELAXATION LENGTH AND LAPLACE
TRANSFORMnTTION PARAMETER WITH RESPECT TO z
E . MACROSCOPIC CROSS SECTION
T . FERMI AGE
S. NEUTRON FLUX
. . FOURIER TRANSFORM PARnMETER WITH RESPECT TO TIME
AND THE RADLAN FREQUENCY
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy
DETERIIINATION OF HETEROGENEOUS PARAMETERS
BY THE NEUTROI W.,VE TECHNIQUE
By
Enile Anthony Bernard
March 1963
Chairrmn: Dr. M. J. Ohanian
Major Departnnt: Nuclear Enginreerin Sciences
The neutron vvve technirqu has been used to study a heteroge.eous
sysLtem conisLinsl of a nacur! uraniun red surrounded by a r.oderator
of heavy "vater. By considering, only simle unit cell confizuracion,
the theoretical derivations are siTplified and :one can perform a
relntively si.ile and inexpensive experitent. The experimental
technique applied perit.s the. reasurenent of several heterozenecus
parr.~aters and ca be used to study ocl.er fu l,codrator s:s tes.
The FeinberGClnin Constrnct, 7, a measured. This r e.s.jre.en:
was acco:;pl ished b: first obct inning th e::perirM;n ta disoersLon la;.
for the unic cell. :Ie.tc, the theor'icical dispersion law is compuced
for various values of y. The theoretical dispersion law best fictir.,
the experimental one det.rn.nine t.e v.lue of 7.
The theory: i,:ca o..l us9ed ,s :h: .A eDi fus ion. odel. 3y usin;
a series of tranCsform ion an.d an e:.::nsion i r. transverse e i.gn
functions, the two coupled equations are solved for the thermal flux.
Inversion of the solution yields the theoretical dispersion law. This
dispersion law is interesting in that a critical frequency is pre 
dicted at which the dispersion law for the unit cell is equal to that
of the pure moderator. This intersection allows one to measure two
other heterogeneous parameters of the group Ls, Ith' q and p.
The thermal neutron pulse, as a function of axial position, was
numerically transformed from the time to the frequency domain. The
frequency dependent attenuation constants and phase shifts were then
obtained and the experimental dispersion laws were determined.
The real and imaginary parts of the square of the complex inverse
relaxation length were expanded in even and odd powers of the radian
frequency. From the expansion coefficients 7 was determined. However,
it was found that more accurate results were obtainable when 7 was de
termined from the dispersion law fit. The latter analysis yielded
the result:
7 = 0.25 + .02 cm.
The expansion coefficient analysis yielded the following results:
7(Po) = 0.29
and
7(P) = O.1 .
The dispersion law intersection was found to occur at
i
a = 0.0532 + .0005 cm
1
S= 0.0386 + .0034 cm
This corresponded to frequencies of 158 cps in the unit cell and
129 cps in the pure moderator dispersion laws. Evaluation of p and
Sth results in unrealistic values for their quantities indicating
that a more accurate model than AgeDiffusion Theory is required
for this analysis. Furthermore it is necessary to eliminate
variations in heavy afterr purity to improve the accuracy of the
experimental data.
The analyses of this work have been conducted primarily in
the p plane. It appears that more accurate and sensitive results
2
are obtainable when the analyses are conducted in the p plane.
Data to aid this type of future studies have been included.
CHAPTER I
INTRODUCTION
Background
The Department of Nuclear Engineering Sciences at the University
of Florida has been engaged in an extensive research program in the
field of neutron wave propagation since 1960. The original studies
dealt with the theoretical (1) and experimental (2, 16) determina
tion of the diffusion and thermalization parameters of homogeneous
moderating media. Successful results prompted studies of two region
moderating media (3), moderating media with localized absorbers (4),
a "homogenized" subcritical assembly (5, 6) and a reflected subcritical
assembly (7, 8, 11). The continued success of the neutron wave
technique has been demonstrated in recent conferences held at Karlsruhe
(9), Gainesville (10), San Diego (11) and Ann Arbor (12). Since past
endeavors have been successful it is of interest to investigate the
application of this technique to other systems.
In this work a heterogeneous modIeratorfuel system will be investi
gated using the neutron wave propagation technique. Cain (15) studied
the heterogeneous problem theoretically and computed the spatial and
frequency dependent neutron fluxes in various configurations. One of
the configurations consisted of two natural uranium fuel rods embedded
in a heavy w.'ater o.oderating medium. Calculation of the flux perturbation
showed no significant change due to the fuel rods. Experimental
measurements were predicted to be indeterminate unless enriched fuel
was used. Measurement of quantities other than the flux would have
to be made to obtain meaningful results with natural uranium fuel rods.
Corno (14) also studied heterogeneous systems theoretically and
proposed an experiment to study the heterogeneous characteristics of
a cylindrical fuel rod embedded in a moderating rediun. The ex.peri
ment involved the use of a highly enriched driving shell surrounding
che fuel rod. By measuring the exponential attenuation of the
fundamental mode for different driving shells, 7 and r can be determined.
7 is the Feir.bercGalanin Constant (15) and q is the number of neutrons
produced per thermal neutron absorption in the fuel. The technical
problems involved in designing and using such a driving shell mak a
simpler e;:riental configuration desirable.
Booth, Perez and Harcley (16) have dveloped a procedure for
neutron wave proag3ation experir.2r.cs. This procedure is a zgeeral one,
applicable co alrost any system. The experimental quencity measured
is the cc lIe:.: inverse rela.cation leng:h, p, as a function of crequen. y
*.he re
2 is the ia.licude a:te.zaiicn per unit i.enr.:th nd is the j'ase
shift per unit length. By masJring p it is then pos.ihle to obtain
the disperslc.n law', i.e., ; versus 2 with frequency as the parameter.
nalysis of t.he co.n lex inverse rela::acion length allo.s the e:x ri
cental d e;r.ir..tcioa of ie FeinF er:'C alanin Cfn.:ant for the feel rod
in chc s : e cc.m ;ris.on 'o.: :n d1. j s redic r.e b'
theory and the e..' er..na] dLs:ersi:,. lUa. can a!so be de This
method of analysis will be employedto study the heterogeneous system.
Scope and Objectives
The neutron wave technique has been used successfully to study
several systems. It has not been used experimentally to study
heterogeneous moderatorfuel systems and the question of its practi
cality in such cases remains unresolved. This study will investigate
the use of the technique on a system other than those already con
sidered. The scope of the neutron wave technique is to be extended
experimentally to include a method for investigating the heterogeneous
moderatorfuel system. The system is to be as simple as possible.
Heterogeneous parameters and their sensitivity to the neutron wave
parameters will be determined. In particular, the FeinbergGalanin
Constant, 7, will be measured. One natural uranium fuel rod embedded
in a moderating medium of heavy water, as shown in Figure 1., will be
used to illustrate the method.
Two other heterogeneous parameters of the group T, p, Tth and L
can also be measured. Their measurements are made possible by the
intersection of the dispersion laws of the moderator only and the
moderatorfuel systems.
A review of the literature failed to show the existence of an
established experimental method to measure 7 directly. 7 is usually
computed but complex designs often call for approximations which lead
to uncertainties in the results. The method presented here allows the
measurement of 7 to be made in a simple and inexpensive way using
one fuel rod.
130 cm
122 cm
c cm
122
FIGUirFL i.
A SC.ii .LATIC '.'IE.'; CO THE
HETEROGENEOuLS S'iYTE.
5
The effect on 7 when a lattice is considered needs to be examined.
Direct extrapolation from the one fuel rod case to that of a complex
lattice must be demonstrated experimentally. In the absence of such
data an extrapolation to the simple lattice configuration used by
Dunlap (5) will be investigated. If lattice effects are significant
the experimental method must be modified to make the measurement of
7 a valid one.
The potential of the method will be investigated and its exten
sion to determine other heterogeneous effects, such as interaction
and shadowing effects, will be considered. Versatility of the method
will also be discussed.
CHAPTER II
THEORY
Introduction
The theoretical model used in this analysis is the Age
Diffusion Model. Although this is a siinpliied approach it is
sufficient to exhibit the characteristics to be expected from a
more refined analysis. To be more accurate, thermalization
effects should be considered. However, Dunlap (6) found that ex
peirnental results for the sucritical assembly agreed, within
several percent, with AgeDiffusion Theory for frequencies up to
530 cps. .As will be seen in the developm.nt of this chapter,
the theoretical basis for the e.xerimental method is established
by cor.sidering trequecies less than 350 cLs. There:cre the
AgeDitrusion :odel will be used.
The purpos o: the theoretical model, in this case, is to
predict values for para.e cers 'hich are to be measured ex..:eri
me.cally. . ~Dif:tsion Theory, since it do:s irnore nhe rm.lij 
cion effects, cnr. not be expex:.ed to preiic: accjraze values.
.Ho.er :h2se :redictic..s are suf icient co .'eri'y tche .racricadity
of the experinent and do offer a comparison vih the experimental
rsults. Eecter comparisons till be obtained 'iith more accurate
theoretical models.
An important number associated with heterogeneous calculations is
the FeinbergGalanin Constant, 7 (15). This constant is defined as
the ratio of the net current of neutrons into a fuel rod divided by the
neutron flux on the surface of the rod. The value of 7 is determined
by two methods. The first method is to fit the experimental disper
sion law, using 7 as the fitting parameter. The second method is
2
based on expanding p in the dispersion law in a power series of iw
and equating coefficients of like powers. The first two expansion
coefficients will be related to 7. The results of the two methods will
be compared.
In the theoretical development of this chapter it will be shown
that the dispersion laws for the moderator only and the moderatorfuel
rod intersect. The frequency of the moderatorfuel rod dispersion law
at which the intersection occurs is designated the critical frequency.
c Experimentally this is important. By setting the real and
imaginary parts of the moderatorfuel rod dispersion law equal to those
of the moderator dispersion law and substituting the value of the
measured critical frequency and a and : measured at the critical fre
quency, two heterogeneous parameters can be determined. Further de
tail is presented later in this chapter.
The Theoretical "cdel
Two equations make up the AgeDiffusion Model. The time dependent
fast flux, 4 (x, u, t), in terms of lethargy, u, is described by the
equation
D(u) V O(x, u, t) + (u) o(x, u, c) =
[2.1]
1 OiL.(x u S(x, u, t) 8s(x. u, t)
v(u) 3z +u
8
t h
and the associated equation for the thermal flux, a (x,t) is
th 2 th th th
D c (::,t) + (x,t) =
a
th [2.2]
v a t + q(x,t ,) A(x,t)
th
where
D = the diffusion constant [cm]
a = the macroscopic absorption cross section (cn1]
I
v = velocity tcr. sec ]
q = slowing (own density [neutrons cm se ]
5 l
S = source of neutrons due to the fuel rod [neutrons cm sec
A = absorption of thermal neutrons in the fuel rod [neutrons cm sec 1
Sine cylindrical geomrry will be consiLdred,
x = x(r,z)
in both e atior.s.
SoluzLic of EauAntion For Fast N,eutrons The time dependence in
equLicon (2.1] is re.oved with a Fourier Trans or..aticn with respect:
to t. Since th6 El'\ results :ro. a he iJls.L ne'Ctron source which is
assumed co sLrc J c zero tLia
I(x,u,t) = 0; t < 0.
Under this condition the Fourier Transform of equation [2.1j is
D(u) 72 c(x,u,' w) + (u,) .(x,u,,) =2
a[2. ,]
ij ,, .3';:I u ,'.,
vI) (:x,u,.) + S.x, u,) 
v(') Cu
where n is che cransform parateter.
The source term, S(x,u ;1), represents the fission neutrons produced
by thermal neutron absorption in the fuel rod. It is assumed that
these source neutrons appear with lethargy zero, at the surface of
the fuel rod. The explicit form of the source term is then
S(x,u,co) = 6(u) s() 7 (r ro) th(x,h )
t (o) 2 T r
where
r = radius of the fuel rod [cm]
s (0) the probability of a neutron entering the slowing
Z (o) down process
S = the neutron yeild per thermal neutron absorption
7 = net therral neutron current into the rod [cm]
thermal neutron flux at the surface of the rod
5 = the Dirac Delta Function.
It is convenient to write equation [2.5] in terms of the slowing down
density, q(x,u,c). The infinite medium relation,
q(x,u,w) = ; t (u) (x,u,C),
is assumed since casuremrencs were made in heavy water which is a
weak absorber of neutrons. Substituting for f(x,u,c) and S(x,u,u)
equation [2.3] becomes
iu (x, u ,)__
( D(u) V + E (u) + ) (
a v(u) ( E(u)
[2.4]
_(.6( (o) 55(r r) h
u + 8(u) t (0) 1 7 2 r o h
A change of variables is now made from lethargy, u, to age, T, using
the definition
d "_ D(u) D(_)_
du = 5 Zt(u) t(
and the relations
t. (T)
6(u) 
Zt (.)
5(T)
equation [2.4] becomes
( V2 + 13(,w)) q(x,t,w) = 
+ 56() s( TI
Lt(o)
5(rr)
2 nr
[2.5]
th,
+ (x,> )
where
iw
) () + 
(B)= ul ve(T
D(B)
By multiplying equation [2.j] bv the integracing factor
f 1(r',W ) dt'
0
L 1
p(T,p) iL
p e s
where L is the slouing, doun tin and p is the resonance escape
prob3bility and recognizing2 hat
( ____ , :< (, ) .+(fL')
T p(,u) p P'0,') T + p (T,) '
aT
11
2 s(0) o 7 5(r r th
(v + ) e(xr,) = 6(T)) 8 r (x,W) [2.6]
e(x,T) = (0) p(Tr,c) 2 n r
where
eo(x,,c = T(,O )
p(r~w)
Equation [2.6] compares with equation [8.2.9a] of Beckurts and
Wirtz (17).
The next step is to remove the age dependence in equation [2.6]
using a Laplace Transformation with respect to T. The boundary
conditions
e(x,=o,co) = 0
and p(T=o,c) = 1
apply since no neutrons slow down past age zero and the resonance
escape probability at age zero is one. The former condition holds be
cause the source neutrons are taken into account by the source term
in equation [2.6]. The Laplace Transformation of equation [2.6] is
2 E (0) 5(r r ) th
V e(x,s,c) se(x,s,w) = s o 7 (r ) (x,w) [2.7]
t (o) 2 x r
where s is the transform parameter.
At this point the dependent variables, e(x,s,,) and t(x,u),
are expanded in terms of the radial eigenfunctions, J (B r ), given
by the solution of the Helmholtz equation,
(7 + B2) J (B r) = 0.
r n o n
The orthogonality condition is
R
J (B r) J (5 r) 2 n r dr = N 5
o o n o p p np
where N is the normalization constant and 5
p
R, the extrapolated boundary, is such that
J (B R) = 0.
o n
The expansions are
O(X,s,,L) = Z b (z,s,a') J (B r)
n n o n
is the Kronecker delta.
and
th ch
o (x,,) = (z,'i) Jo (Br).
n n on
By noting that
V2 J (B r) = 2 ( r)
r o n n o n
multiplying by
2 r r J (B r),
o p
integrating and applying the orthogonality condition, equation [2.7]
is reduced to the following equation for the expansion coefficients:
262 (z,s, ) + (z,s,) =
SP P
6Oz.
[2.8]
Z(o) q 7
(0) (o
" p
th
th (z,F) J (nro o(Bpro )
nL 0 n0 0 p0
The lasc s:ep in the solution to the fast flux equation. is co re
move the spatial dependency by a second Laplace Tran3sformation, this
time with respect to z. Th2 Laplace Transforniation of equation [2..]
is
2 2
p e (p,S,w) (B + s) (p,s,) =
(o) 7 th
() Ti 4 (p,w) J (Br ) J (B r)
t (o) N p
t p
where p is the transform parameter and the boundary conditions are
taken to be
d9 (z=o,s,w)
ep(z=o,s,w) = dz = 0
which neglects the gradient at z=o because the dispersion law. is
independent of its presence. The solution for the fast flux expansion
coefficients is
[2.9]
1 E,(o) i 7 th
p (P,s,
p s +(B2 2 E (o) N n no o p0
s + (B P ) t p
p
In order that the solution may be of use in solving for the thermal
flux, it is necessary to transform from the s domain back to the T
domain. The inverse transformation of equation [2.91 is
2 2
(B p2)
S(p, T,) = e s (o) 7 th(p,) J(B r ) (pro)
pn n o nro 0 0
C (o) N
n,p = 1,2, . k.
Solution of Equation for Thermal Neutrons It is also necessary to
solve the thermal flux equation, equation [2.2]. The procedure is
similar to that used to solve the fast flux equation. The order of
transformation is changed so that the boundary conditions may be more
readily applied. Before solving equation [2.2] it is noted that
A(x,t) 7 6(r ro oth(x,t)
2 r
which represents the absorption of neutrons in the fuel rod as losses
occurring at the surface of tle fuel rod. By representing the absorp
tions in this manner 'nd the source term in the manner described in
the previous section, the problem is reduced to a one region problem
with a source and absorber of neutrons.
Equation [2.2] is Laplace Transformed with respect to the z
variable:
Dth 2 ch 2 th th th th
(D p I V + E ) (r,p,t) + D p o (r,z=o,t)
r a
th h [2.10)
+ th o. (r,z=o,t) = 1 d (r,p,t)t)
dt Vth at th'
th
th 6(r r )
7 th(r,p,t) r ro
2 r
where p is the transforrim parrmeter. In this case neither the thermal
flux nor it derivative is zero at the origin. These terms are ex
pressed as
.th (r,zo,t) = S ( = S (t) J (B r)
dz 2
and
th
a (r,z=o,t) = c (c) J (B r).
n n an
Making these substitutions, usiLrg the eigenfunction expansions and
invoking the orchogonality condition as was done in the solution of
the equajion for the fast neutrons equation [2.10] beco m r;
th 2 thB2 zth}t h Dth th
(Dh + D B + h ) (p,t) + Dp (t) + D S (t)p
p a p p o p1
 __1 (p,t) + q (P,'tht)
th _t
E h n (p,t) Jo(Bnro) J (Br)
P
where
The time dependence is transformed out by a Fourier Transforma
tion, again using w as the transform parameter. The result is
2 2
(p + B +
P
th + iu
a
V th
th) h (p,O) + p c (C) + S (u) 5
Dth P p 0o
D
D th n n o no 0 p th p th
n n D h
P D
where
q (P,.thn,)) = P( th,") ep(p, ch, ).
q(r,p, th,t) = E q (p,T ,t) Jo(B r)
t n n th' Jo(Bnr
Combination of Fast and Therral ;Neutron Solutions The solution for
the thermal flux expansion coefficients is
2( p) 1 i teth(PW)
L p L2 Dth p p
th o
2 2
7 E S (o) ( P)rh
+ t L P(Tth') t e ) 1
P [2.11]
X Z Ot (p,,)J (B r )Jo ( r )= p c ('n) + S (mo)
n rL c, no J po p o pI
n,p = 1,2, . k
where the solution obtained for the fast flux expansion coefficients
has been substituted,
th
L
a = I
th 2
D L
th
and
th th
cVh o
Since the moderator to fuel ratio considered is large, the roba ility
of a source neutron being absorbed on its fir.t .ollision is very
small. Therefore, the approximation
Z (o)
S(o)
t
is made
Numerical Solution
Two cases of the solution will now be considered, n,p = 1 and
n,p = 1 and 2. The former case represents the solution for the
fundamental spatial mode while the latter illustrates the case which
includes the first spatial harmonic as well as the fundamental solu
tions. The latter was used in all the computations. The importance
of the first harmonic will be examined.
In both cases an inverse Laplace Transformation is required in
order to obtain the solution in z space. To simplify the inverse
transformation the poles of the transformed solution can be computed.
The solution in z space is then of the form
th(z,) = G.()e i()
p 1 i
where G.i()) is a constant, depending on the pole p (). The final
solution in terms of z will not be computed. Instead, the poles of
the transformed solution are conmuted, yielding the dispersion law.
2
For n,p = 1, p is computed from the equation
(B{ 2 02) 
1 2 D thj
th o
2 2) L
1(B t il s 2
+ pe 1 Jo (Bro) = 0 [2.12]
thN J
D N
For n,p = 1 and 5, o is comuted by setting the determinant of a
2 X 2 matrix ecual to zero. The elements of the matrix are
2 2
B = 2 + p 
(
__2_ ~qpe
t h,
D thN
21 = 7 pe
DtI L
A = 3, + p
2 L
1 i4O
2 th
L D
th o
2 2
B1 p2),
B
t h L
th
s 0 2
11 Jo (B1r ),
0 o o
2 2
(B p2) th
1 h
(B2 p2)th
2 2
< '^
1
th
s 1 Jo(Bro)J (B2r o ),
iwL
s
1 oJ (B r )J (B r ),
S2 o 1
 to)
th
D
o
(B2 p2)
(2 )ch iU
+ 7 .(ipe
Dt ,
"2
s N 2
1. J (B r )
S o 2 o
In each case soLutions *aire obtained us in :_ie 'c YZVE.V Code (See
Appendi:,: D) .
Results of :he cc.jtadicr. :cr. selected frequean.ie are given in
Table 1. The fundamental eigenv',luees changed in the fourth signifi
cant digit froi. t.he one ter. (n,p = 1) to the two ter (n,p = 1 and 2)
cas'c 3 Cc.: icr..' ions ere also ,d: for a ten cerm c:.:pan.icr (21) d n .
iunda..rte ei n.alu s differed by. less :han 3.5 rcr. :' ,0S of :he
one terry case. In the t 'o term cse the eien'alue of the first
harmonic code are of the same order of magnitude as the fundamental
19
r4
U o\ o\0 o\ o\ oP c\0 o.
ZZ o O. 0 rcO aD a. a
0O 0 * *
' 2 II 4 , ,4 . . 1
H< N
ZI
 O. o
Z II N N N N N N N
OU N
u
0 4 oo 4 r No
0O a rI k.0 0 Ie m
Z o C C C C 0 o
0 u. * * *
C 0 0 0 0 0 0 0
Z Z I I I I I I
0 0 U
C. ce 7 :1c c4 Mri cs4
< < < (n r cr
SI 4 N N o 7N N o
H1 "Z 00 0 0 0 0 m0 a.
O H  I I I I I
,' L O 4o ,4 O 0N L,.
zI < C
S  * * *O
< 0> C10. o o o o o o o
o E 0J0 C
OZ Z
< ' J < N 1 C
H < X 4 N
S Z X o o ) o o C o
a D0 < *. .
L u. C o o o o o o o
0 < I I I I I I I
z
0 Z
Z 0  0 N 4.
S U 0 4 N 0 ) N Lf
2 0 C4 4 " co N%
Z X o o o o o o o
...
0 I I I I I I I O
z 
LUJ cn r4 r4 N '4 LP, U % ^
C) C) C) O C) C) C) C._
~^ . . <*
N'\ OT O O n O0 Ocl O
0 I 1 I I I I I
LU C) 0 C) 0 C C; 0
0  CM C NN
U
Ct ^
eigenvalues. However, the experimental source was found to contain a
low percentage of first harmonic contamination.1 These results are
included in Table 1. in this case only the fundamental mode is
significant.
7 Analysis
Using equation [2.12] it is possible to analyze 7. 7 is varied
parametrically and the resulting dispersion law is computed and com
pared with the experimental one. The best comparison represents the
measured value of 7 within the limitations of the theoretical model.
If p in equation [2.12] is expanded in a power series of the
form
p2 n E) P
n n
and coefficients of like powers are equated, still another irezhod of
determinir ;' is available. Tha first two coefficients of the e:
pansion are *.elated to 7 as follows
1 + 7 X(l q p)
2 L2
PO = B [2.15]
0 1
1 + r p X th
"th
1. Thes results were obtained fro:.1, the e:.an..3 in ccefticients 0o
the cransverse ccnzinuous mode dacr at z = 10 ce. The ULFNLLS Code (20)
was used to Efi the data to the fundr.cntal and fi.rs: hrmnic r.:.des.
Con:2rLini tion a: z = 50 ca, rhe firs: data Doint ainalyzed, ,as
deter .ined . e*.* ua ir. t'.e e:.:cae..ial attc n'a.ior. of eac. male and
cO,.putin r ;, p 7r Cnc o.inatribactio.s. fhe p terZ .1La3 are based on
mai. n . 2 ol i tuda s and rpe rese.: r.a:i'ir.j.:. co.:3.min tcion.
and D + X(qpL (1 B2 + ))
and D t1 h th+ h
1 =2 [2.14]
1 + pyXTth( B th + P th)
where
J2 (B r)
X = 0 1 0
DthN
D 1
These expressions were obtained by separating the exponential term into
frequency dependent and independent terms, expressing the result as a
product of expcnentials and expanding each product in a MacLaurin Series.
The sensitivity of F0 and P to 7 was found to be
dP
0 4
Po 4
6P 3.35 X 10
and
3P
S3.1 X 10 ,
both P and P changing about the same a.ounL in the second significant
digit with respect to 7, yielding a 25% change in P and 7/ change in P1.
Comparison of the two 7 analyses with the experimental results
will be made in Chapter IV.
The Critical Freauenvc
To investigate the critical frequency equation [2.12] is again
considered, both for the moderator only and the moderatorfuel rod
systems. The right hand sides of both equations are zero so the left
2 tn
hand sides can be ecua:ed. L and D are constants of the moderator
th
th th
but D differs between the two systems because v is different.
o
2
However, this difference is small and they are assumed to be equal.
It is now assumed that the dispersion laws intersect
th
2. Subsequent analysis showed the difference in v to be less
than 2.
so that the two p's are equal. The relation is then obtained:
2 2
hc yJ (BC r ) P1 c th L
 c + o 1 o fpe I [2.15]
Dt thction a
o 1
iu
m
D th
where ishe frequency of the moderator only dispersion law asso
ciated with the intersection and
p = + i2
C C tC
ib LIt vlue uf p at the intersection. J is the critical frequency
dispersion law'.s intersect at the same values of an and but the
associated frequencies are different. By equating real and imaginary
;' 2 2
[BI (2 c th
r p cos (22 Tth L )e = 1
and
r) 3; v( e L( r a S0 1 0LI o S 'Ca31 3 1 ) t h q anci
both e_ onh aa O D
p s!n(2.c ,'.a L "e = c m 1
D 7J (Br )
atc obtair.ed. Compiuca ions performed for many conditions of inter
sec iton S3oJ t'3ht .. > .0 Therefore, the m.ne and cosine must be
Dcs3ii,.ve for a soluc t.ioi to e::L: because all che other quancities in
both eqatiors are positive. The angle
2a cT t Wc L
c th cs
must be in the first quadrant. If these two equations are divided,
one by the other, the result is
S DthN 2.16]
tan (2C T o L ) c m 1 (2.16]
s Dth J 2(Br )
o o 1 0o
By determining the experimental intersection and computing N1 and
J 2(Bro) the right hand side of equation [2.16] can be evaluated,
th
if v and 7 are known. The argument of the left hand side can then
be determined. This allows either T h or L to be evaluated in
th s
terms of the remaining parameters. Once the argument is found, either
of the two original equations can be used to solve for n or p in
terms of the other. The experimental results will be discussed in
Chapter IV.
The intersection of the dispersion laws is illustrated in
Table 2. and shown in Figure 2., where they are plotted for 7= 0.0
(pure moderator) and 7 = 0.6. These values were selected to give
a qualitative description of the intersection. Based on Cain's
results (13), the value of 7 was expected to be in this range.
The sensitivity of c to 7 was investigated by using the two
term expansion of equation [2.12]. It was found that a change of
0.1 in y produced a 2 cps change in w .
c
TABLE 2.
NUMERICAL ILLUSTPATIO;N OF
INTERSECTION OF DISPERSIO;J LA.S
GA;:.A = 0.0
FREQUENCY ALPHA XI
CCPS) (Cc1) (
294.0 0.07326 0.06260
295.0 0.07337 0.05272
296.0 3.07347 0.06284
297.0 0.07357 0.06235
298.0 0.07363 0.06309
::: INTc 'SZCTI
GA.'MMA = 0.6
FREQUENCY ALPHf XI,
(CPS) (C;.,, ) C c.1 )
305.0 0.07331 0.00626
306.0 0.07341 0.0627S
3: 06.5 0.07347 0.06264
307.0 0.07352 0.06239
305.0 0.07363 0.06301
0 100
oo
II
xi
0040
0,020
0
+ = 0.0
V 7 = 0.6
. 350 (
300 CPS
250 CPS
 300 CPS
0 CPS
200 CPS
150 CPS 
\o150 CPS
xa
g
100 cps
100 CPS
50 CPS <
50 CPS
V+
o'02
0d0A 4 Oo/
ALPHA I/CM
FIGURE 2.
ILLUSTRATION OF DISPERSION
LAW INTERSECTION
50 CPS
0G08
0.10
_1_ _L *
CHAPTER III
EXPERIMENTAL METHOD
Introduction
Two experiments were conducted. The first was in a pure
moderating medium of heavy water. The second was in the same system
with one fuel rod inserted as shown in Figure 1. The experimental
dispersion laws for both systems were determined using the pulse
propagation Lechnique (16) and converting the data into the frequency
domain in the u sal way:. A Texas Nuclear Corporation Neutron
Generator (Mode! 9505) was used as the source. Tne system in which
the measuremnarns were r'tade and the neutron ge.neracor which .:as used
have been descr ibed In deta il by Dunlap (5, 6). The 17 inch graphite
stack between the light water ther.alizing rank and the heavy a,, ter
tank 'as removed. This was done :o reduce the width of the therral
neutron :ulze and thus increase the hih frequency cencnr..
NIo data acouisit i on sys .e.T:z '.'re u'sed, one h'vinQ a r.o'.'able
drrtecor 'which was ucad for measure.emnt.s alon the axi:l lengci of
the assembly ani :he o'thn: beinj the reference detector whichc h .wa
used for nor.i.iizacion purposes. The reference deector was moved
to a position adjacen to the 2 rachite in Lhe chnernaizing assemrbi.y.
This was done to reduce the effects of source anisotropy. A brief
sketch of the set up is shown in Figure 3. Each detector system
contains basically the same components with the following exceptions:
(a) The movable detector system has a 12 inch He3
detector and the reference detector system has
a 6 inch He3 detector
(b) The movable detection system has additional
components for data acquisition and analysis
of the neutron pulse in time.
Thetwo detection systems are shown in Figures 4. and 5.
Considerable time was spent in becoming familiar with each
and every component. This step can not be overemphasized in im
portance for it proved invaluable in the immediate identification
of equipment failures. Several failures occurred and data acquisi
tion was terminated with a minimum collection of faulty data. The
use of oscilloscopes was likewise important in the verification of
proper pulses, monitoring signals and general trouble shooting.
Light water contamination in the heavy water moderator changed
between the two experiments. The first experiment was conducted at
99.5% purity while the second one was conducted at 99.0% purity.
The decrease in purity was caused by the several transfers of the
moderator that took place between the performance of the two
experiments.
1. The angular distribution of neutrons was found to be de
pendent on the am.pli:ad of the target current. Experin.ental data
showed tha: the variation of anisotropy for the range of target
current amplicudes used 'wa red,,ced to less than 1% when the reference
detector was positioned adjacent to the graphite.
Graphite
Light
"Wa cer
Experimental
System
Cadmium
"Shutter
Re ference
Detecto
Therma iz ing
Assemb l'.
FIGURE 3.
F :; F' E: RI;I L..L ~.L.
Atomic Instrument Co.
Regulated High Voltage
Power Supply Model 318
Texas Nuclear Corp.
12 in., 1 atm., He3
Texilium Detector
Hamner Electronics
Co., Inc.
PreAmplifier
Model 102
Hamner Electronics
Co., Inc.
Linear Amplifier
Model N318
Hamner Electronics
Co., Inc.
Scaler
Model NS11
Tektronix, Inc.
PreAmplifier
Power Supply
Type 127
IType CA 
Plugin Unit]
Technical Measurements
Corp.
Data Output Unit
Model 220C
Tally 420
Binary Tape
Perforator
HewlettPackard Corp.
Digital Recorder
Model J44561B
FIGURE 4.
MOVABLE DETECTOR COUNTING SYSTEM
Technical Measurements
Corp., Multichannel
Analyzer Digital Computer
Unit, Model CN1024/Model
212 Pulsed Neutron Logic
Unit
~C~
Atomic Instrument Co.
Super Stable High
Voltage Po'...'er Suppl''
Model 312
Texas Nuclear CorD.
6 in., 1 atm., He3
Texilium Detector
Hamner Electronics
Co., Inc.
PreAmpli ier
Model 1 102
Hamner Electronics
Co., Inc.
Linear .".molifier
Model N318
Hamner Electronic.
Co., Inc.
Scaler, ;i=,del TS11
FIGURE 5.
REFERENCE DE SECTOR COUNT'IliG SYSTL':.
Preliminary Procedures
Before any data were taken the following preoperational checks
and tests were performed on the counting systems:
(a) compatability of outputinput signals between
components
(b) detector plateau determination
(c) chisquared checks
(d) resolution time determination
(e) gamma ray discrimination checks
(f) minimizing of internal noise
(g) reverification of plateaus, chisquared check
and resolution time.
Step (g) is necessary to assure that changes made in the system in
the preceding steps have not affected the counting characteristics
of the systems. Adequate guidance in the completion of most steps
is easily obtained from the appropriate technical manuals and any
Nuclear Engineering laboratory manual. Resolution time determina
tion is described in Appendix A.
The last task to be accomplished before the actual experiment
is begun is the selection of the target current pulse width and re
petition rate for the neutron generator. Coincident with this is the
selection of a compatible channel width for the multichannel
analyzer. This procedure is described in Appendix B.
In the course of checking out the detector systems a limiting
count rate was observed. Increased source intensity produced no in
creased scaler count rate at approximately 200,000 and 250,000
counts per second for the reference and movable detector amplifiers
respectively. All data were checked to insure that the inximum
count rate during the neutron pulse peak did nor approach the
saturation rates.
Experimencai Procedure
In order to obtain the thermal neutron pulse propagation charac
ceristics through the assembly the cadmium difference Lechnique !as
employed (16), using the cadmium shutter located at the forward edge
of the assembly. The epicadmium contribution of the source vas
thereby eliminated.
Norrmally a completed neutron wave experiment consu;.es days ar.d
even veeks. During this tine components ray change enough to affe:
che data significantly. In order that one may have a standard of
comparison for reproducibility purposes a continuous mode run was
made at all the a:ial positions. These data can be acquired in
several hours, a period in which variations are n.?:ligible. If
necessary, normalii.ciion for minor variation: of che syster.,s can
then be made.
The conrinuous mode E::peirirent correspcnds to Ohna zro frecuenc'.
coponerL: of che r.eurron pulse in chat the amnplliude atcenucation per
unit Lenlrh are the same (16). It minor anronlies occur in the zelo
frequc'.ie : cc..onent o cthe pulsed da;c.a :he dj:: car be assily
normalized to the continuous m.ode dact, In chis manner, small changes
due co d;ytoday variations of the source, e.>., angular dL:cribu
cions, and counting sysrens, e.g., shifts in detector plateaus, can
be eli.:inaced. In this series of ex::perirer:ts, no normraliita io. jas
required.
33
The continuous mode data which were recorded at each position for
both the cadmium shutter up and down runs included:
(a) run time
(b) reference detector scaler counts
(c) movable detector scaler counts.
After the continuous mode experiments were completed, the pulsed
experiments were conducted. At this point the multichannel analyzer
which was set up in accordance with Appendix B was used. Once the
pulsed mode has been established, one proceeds to examine the propaga
tion of the thermal neutron pulses through the assembly, employing
the same cadmium difference technique used in the continuous mode case.
The pulsed mode data which were acquired for each run included:
(a) run time
(b) number of triggers
(c) reference detector scaler counts
(d) mcvable detector scaler counts
(e) multichannel analyzer printed output
(f) multichannel analyzer binary tape output.
Verification checks which were performed as soon as the data were re
corded are described in Appendix C.
During each run, two signals were continuously monitored with
oscilloscopes. One was the target current of the neutron gun. The
other was the input to the amplifiers of the movable detector systems.
Thus, two visual presentations were available to assure proper pulsed
operation. A secondary purpose of monitoring the amplifier input
signal was to measure the effective countinZ tite of the movable
detector. The effective counting time is discussed in Appendix C.
In both cadmium covered and bare pulsed runs, a minimum of
64,000 counts were accumulated in the peak channel of the multichannel
analyzer. This sets the time required for each run. Data were ac
quired every 5 cms from 50 through 90 ems. These limits are based on
the continuous mode analysis which showed definite deviations from a
simple exponential decay with distance for positions outside this
region. Deviation at the lower end was caused by the source while
end effects were significant at the higher end. The procedure em
ployed in determining the appropriate heterogeneous parameters is
outlined in Figure 6. The data analysis is described in Chapter IV.
0
0 C)
W co
Ea n
S.o o
c I 4 1S
Q
CHAPTER IV
DATA ANALYSIS AND RESULTS
General
Data analysis was accomplished with a modified version of the
Moore Moments Code (6) and a series of supplemental programs
written for the IBM1800. By using the 1800 the direct interface
capability bec:een the paper tape daca and the computer could be
utilized. A flow chart of the computer programs is shown in
Figure 7., while more detailed descriptions are given in Appendix D.
Purity of the heavy water was estimated to be 99.5 4 .2% for
the moderator only experiment and 99.0 + .2% for the one fuel rod
experiment. The 99.5. purity was based on the theoretical model
and the experimental data. This purity yielded the experi,.entally
observed values of Ci and s at near zero frequencies, where better
agreement exists betcw.en theory and experiment, when .: and were
computed from equation [2.12]. The 99.0 purity is based on the
difference of 0.5,: purity in samples from the two experiments w'.hich
were analyzed by the Departr.ent of Chemiscry. Purities were re
ported to be 99. +. .05, and 93.8 ~ .05, for the two experiments (1s).
In both experiments the heavy wacer temperacure was 2,1 + C.
Experimental errorsin measuring a and E verecompuced based on
the mean square deviation of the data points from the computed least
squares fit. Erors rani.ed from 0.;'. to 2.0' and the latter 'w:s
taken as the experimental error for all determinations of _ and
Resolution time measurements were conducted as described in
Appendix A. The subsequent analysis showed this method to be in
000
*HOj
0c Acn Ai
EP~ 0) C
00 0 X (n
H >L 0 )
0 4 x Q
04 a4P
H1 4J
w (n (n
rdcfn
**CQ) d) j
0 O
0 0 V0 0
< *H I
cr1 0
OPOEI L j~
CHmum
SI
o e
0 0 E E
'1 0 ,
S! o.. Q ) '4 )
30
14 0c 0 3 0 C4
r:Ir
4 .l 0 r. ^
H = ^ 0
2 0 3 ^ _j
C,
'JO
S 0
04 0 22 0n
F
^ w a i75
D 7
Q. 
C1)
,4
x
&1
Mq
Q)
(U7
44
H
c,
Q) 0)
H Q) OrO
HO) O4J
** 0 l
I !
,4
E4 0
SO
H J O
4J 0 H
0 0 7
00 0
H :: 0 0:
r1
CO J
cd30
error and a discussion of the error is also included in Appendix A.
DuBois (7) measured the resolution times of the two detector systems
just prior to this series of experiments. Resolutions times were
determined to be 1.42 and 1.77 microseconds for the movable and
reference detector systems respectively. Since the components of the
systems had not been changed, these resolutions times were used in
this analysis.
As was mentioned in Chapter Ill, the 17 incl. graphite stack
that had been located between the thermalizing and subcritical
assemblies was removed to obtain higher frequency content in the data.
There being less moderation, the asymptotic region of the assembly was
smaller. Dunlap (5) found the region to begin at 25 cms. Without
the graphite stack the asymptotic region began at 50 cns.
The asympLotic region was established by scanning the data.
The undesirable characteristics of higher spatial and energy modes at
axial positions near the source and end effects at positions far fro,,
the source tend to increase the computed values of a and . In region.
where neither effect is significant ninimun values of a and I are
comp.i ed. These cini.nu .'aLJes '..:er deernied by compluting and :
usin, a four pote: scan of the dai.. Then a five point scan w'as used
and thz rtiLia ;wi re ccnpared 'i'h the previous o;es. if they differed
by lesa than I, the additional data point was considered to be in the
asymptotic region. Progressively higher point scans were used until
the minimal differences e:.ceeded 1. The daca point produce irg this
1. See the speciall Controis Section of The .ALP?. proran des
cription in Appendi:: D.
deviation was excluded. The 1% criterion is based on the results of
Booth, Perez and Hartley (16) who found a 1% tolerance in the measure
ment of a and . A consistent asymptotic range, for all frequencies
in both experiments, was found to be 5075 cms. The end point was
the same as that determined by Dunlap (6).
In some cases deviating trends in the amplitudes and phases were
obviously present. Deviations at a given axial position increased or
decreased with increased frequency. In these cases the data points
were eliminated, beginning at certain frequencies. The criteria for
elimination were:
a) the obvious deviating trend at maximum frequency
b) the existence of a 0.25%2 difference in a and t between
the least squares computations which included and
excluded the data point in question.
Once a 0.25% deviation was observed the data for higher frequencies at
the given position were also eliminated.
In both experiments an expansion of the real and imaginary com
2
ponents of the experimental p in powers of o(rad/sec) was made. The
expansions are:
2 2 2 +
0 2 4
2. Daviations greater than. 0.25% produced significant dis
continuities in a and : 'hen subsecuenc data points were eliminated.
and
20( = P u) + Pgn5.
The P's were determined by using the UFNLLS Code (20). Data above
500 cps were excluded in the polynomial fitting because statistical
fluctuations were deemed too significant.
Perez, Ohanian and Dunlap (19) have found that greater sensitivity
of the dispersion law to ther;ralization and diffusion parameters
exists when it is considered in the p plane rather than in the p plane.
No analysis will be conducted in this manner because the experimental
errors appear to be more significant when the p dispersion law is
considered. The dispersion laws in the p plane will be shoun to
aid any future studies that may be conAlicted along these lines.
The remainder of this chapter will be divided into four parts.
The first part will deal with the continuous mode analysis. The pure
moderator and one fuel rod pulsed experic.nts will then be analyzed
and finally a lattice of fuel rods will be considered.
Continous Mode Ar.a!'.'sts
The continuous roe data serve, three purpoass:
1) it oFfers a strdard for conoirison with rhe zero
frequency component of the ther mal neutron pulse
2) it provides a standard for normlization of minor
ano rallies in the pulsed data due to source or count
ing s.ste; '.'ariat ion
5) it establishs a sta3nl. rd for co.oirison ~rith pref.io..s
and subsequent e:peririntt .
It becomes apparent that the continuous mode data are the key to the
reproducibility of the experiment.
At each data position the thermal ratio, TR, is computed from
the relation
N N
TR = mov 1 mov
Nref bare ref cd
where
N = the resolution corrected scaler counts
mov = the movable detector system
ref = the reference detector system
bare = the bare run data
cd = the cadmium run data.
TR, as a function of distance was fitted to an exponential function
using the least squares technique. The logarithmic decay constants
for each experiment are given in Table 3.
TABLE 5.
CONTINUOUS MODE DECAY CONSTANTS
Experiment Asymptotic Range Alpha (cm )
Heavy Water 5075 cms 0.0571
Heavy Water 5075 cms 0.0585
One Fuel Rod
As will be seen, there was no normalization required for anomalies
in the data. In this case the continuous mode data are used only for
comparative purposes.
Heavy 1 water Pulsed Experimnnt
The thermal neutron pulses obtained in the heavy water only
experiment are shown in Figure 8. The last 512 channels contain few
or no counts. These are insignificant compared to the peak counts
of appro::imately 45,000 counts and are not plotted. The pulses are
normalized to the 50 cn pulse. Each channel represents 50 micro
seconds. These pulses are the ones that were numerically Fourier
Transformed.
The Fourier Transformed flux:es and the least squares fit to the
anplitudes and phases are shown in Figures 9. and 10. The maximum
deviations of points included in the least squares fit Erom the least
squares fit are h", in 0 and 1. in ;. Theoretical values of 0. and 
computed fror. equation (2.12] are listed in Table b. while the result
ing experiTental values are listed in Table 5. Theoretical and ex
perir.ental vlu s of a are illustrated in Figure 11. A raaxinun devia
tion of 2.1"2 occurs at 300 cps. The smaller deviations at higher
frequencies is fortuicous. Statistical deviatio~is at these higher
freue.ncies are too significant to 3llow valid alaiLyses.
An illustration of , si.ilr :o che one of is given in
Fieure 12. In this case :he exo.eriantal data acree .i:h t heor:
within 1. 5'.
The heavy after disersion laws are shoin in Figures 1i. and 14.
The former shc:..s :r.e experiranta and theoretical discersion las.s of
this ''crk a.d the later suo's a co.crison uf chis ..'or'. with r.'.a
of 'niap ,. 3cLh Co:.arisoas i ire re;. .c:.ably co The better 'i;.r
frequency cort.cnt of chese data over cha: of Dcala.?'s (6. is decon
40,000
20;000
20 / 000Y
__J

LLI
LU
H
109000
0
50 CiMi
55 CM
.2 C.1;
5 CM
?3' C..
CHANNEL
iNME
FIGURE 9.
NORMALIZED NEUTRON PULSES IU HEAVY WATER
F
FI.
Li_
EL]
?Si
_, C i l
1000
100
0 CPS
50 CPS
100 CPS
150 CPS
200 CPS
250 CPS
300 CPS
2350 CPS
10+
0
X DAT', POiiT, OMITTED FRO;M LEAST SQUARES FIT
 LEAST SQL'UAES FIT TO FE:AAINIIG DATA POINTS
I I I i t
50
so 70 B08
AXIAL POSITION9 CMS
FIGUF'E 9.
E:*.PERItE[ITA..L APLI TUIDES
FOPR HEA',L WATER
SO 100
1' I I
I L.
0
B
DATA POINTS
X OMITTED FROM'LEAST
SQUARES FIT
100 CPS
150 CPS
200 CPS
250 CPS
300 CPS
350 CPS
 LEAST SQUARES FIT TO REMAINING DATA POINTS
I   II I L
60 70 SO
AXIAL POSITION CMS
50
100
EXPERIMENTAL PHASES FOR HEAVY WATER
4+ 50 CPS
4
LL
n
I
in
LJ
C
31
h~
t~ :
FIGURE 10.
i(40
;9T 0.
TABLE 4
TIEORET ICAL VALUES OF ALPHA A;:U XI
FOR HEAVY ,lATER
FREQUEIICY (CPS)
0.
10.
20.
'o.
40.
50.
GO.
70.
80.
90.
100.
110.
120.
130.
140.
150.
150.
170.
180.
190.
200.
210.
2, 0 0 .
220.
230.
2140.
250.
2G0).
270.
30.
290.
300.
310.
32 0.
33 J.
3 i 0.
350'.
ALPHA(I/Cil)
0.0375
0.0377
0.03 S
0.0393
0.0; 05
0.0410
0.0433
0. 0 4 47
0.01462
0.0477
0.0491
0.0506
0.0520
0.0534
0.0547
0.0561
0.0574
0.05"7
0.0500
0.0512
0.06214
0. 0636
3. 03 4 36
0. 0650
 0 6 7 1
0.063
0. 07 16
0.0726
0.0737
C. 0 75
0.0767
0.07 7 "
 0. T ,: .2
XI (1/Cl)
0.0000
0. 0041
0.00.81
0.011S
0.0153
0.01SG
0.0215
0.0243
0.0269
0.0294
0.0317
0.0339
0.0359
0.0379
0.039S
0.01415
0.0434
0.0451
0.0467
0.0433
3.0 099
0.05i0
0. 0 5
0.0543
0.0557
0.0570
0.05 c,'
0.0507
0.0 6 09
0.0622
0.0534
0O. r; 6
0 C 35
0.0570
0. 0 3 2 1
0. 0j n2
TABLE 5.
EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATER
FREQUENICY(CPS)
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
100.
110.
120.
130.
140.
150.
160.
170.
180.
190.
200.
210.
220.
230.
240.
250.
250.
270.
280.
290.
300.
310.
320.
330.
340.
350.
ALPHA(1/CIH)
0.0376
0.0378
0.0383
0.0391
0.0402
0.0414
0.0428
0.0443
0.04158
0.0473
0.0483
0.0503
0.0519
0.0534
0.0549
0.0564
0.0573
0.0593
0.0606
0.0619
0.0631
0.0643
0.06555
0.0669
0.0682
0.0696
0.0708
0.0719
0.0730
0.0741
0.0753
0.0762
0.0759
0.0776
0.0782
0.0726
XI (1/CM)
0.0000
0.0040
0.0079
0.0117
0.0153
0.0187
0.0218
0.0247
0.0275
0.0300
0.0324
0.0347
0.0368
0.0388
0.0407
0.0425
0.0442
0.0458
0.0473
0.048S
0.0503
0.051S
0.0533
0.0547
0.0561
0.0572
0.0582
0.0592
0.0603
0.0613
0.0623
0.0631
0.0541
0.0652
0.0664
0.0677
0 100
0' 080
Li
1<04G
I_
0_
<0o040
0 020
350 CPS
30C CPS
50 CPS
CPS
150 CPS
100 CPS
50 CPS
E:: PERIP.'E'iT.L ERROR
 A.EDIFFUSION THEORY
I E.:PERIUIENTAL DATA POIflTS
  I I
100 G00
FREOUEINCY s
300
CPS
FIGURE 11.
EXPERIrIENTAL .AD THEORETICAL
VALTrES CP AL.PHA. FOR FEA'Y WATERR
400
 EXPERIMENTAL DATA POINTS
 AGEDIFFUSION THEORY
I EXPERIMENTAL ERROR
350 CPS
300 CPS
50 CPS
CPS
150 CPS
100 CPS
CPS
.00 200 300
FREQUENCY CPS
FIGURE 12.
EXPERIMENTAL AND THEORETICAL
VALUES OF XI FOR HEAVY WATER
0 10(
0
0 010
LJ
N
0 042
0 04
0QCE~
0010
0o080
0o04
L00
OOCY
0 0P0
1 EXPERIMEiNTAL DATA POINTS
 AGEDIFFUSION THEORY
 EXPERIMENTAL ERROR
4t350 CPS
CPS
y 300 cPS
200 CPS
/150 CPS
100 CPS
50 CPS
ALPHA
0 .5
i/CM
0O
FIGURE 13.
EXPERIMENTAL AND THEORPETICAL DISPEFRSION;
LAWS I1E RHO PLACE FOR HEAVY WATER
0 oC00
0 o080
L0o080
Lj
+ EXPERIMENTAL DATA POINTS
X DUNLAP'S DATA (6)
 EXPERIMENTAL ERROR
EXPERIMENTAL ERROR
350 CPS
300 CPS
250 CPS\
200 CPS
50 CPS
300 CPS
250 CPS
200 CPS
150 CPS
0.040
0 020
K '150 CPS
100 CPS00 C
4 100 CPS
50 CPS C
\N50 CPS
AL'04 0 O/0
ALPHA I/CM
008
FIGURE 14.
COMPARISON OF EXPERIMENTAL DISPERSION LAWS
IN RHO PLANE FOR HEAVY WATER
0oi0
1 LII~ _
! 0 02
strated by the more consistent results in the 2!0510 cps range.
Results of the two experiments differ by 3.21 in 0, a phenomenon
attributable to a difference in heavy water purity.
2
The imaginary and real cciponents of p are shown in Figures 15.
and 16. respectively. The polynomial fits are included in each ploc.
Values obtained for the expansion coefficients are listed in Table 6.
Resultant values for the thermalizstion and diffusion pararmieters are
included. Values obtained by Dunlap (6) are also listeJ. PO and P1
agree within '(.1L and 1.27. respectively but the other coefficients
differ radically in magnitude and sign. An cxaninatiion of Figure
2
16. shows that the real component of p is erratic, e::hibiting
oscillations and increasing less rapidly than Dunlap's results (6).
The expcrinental values of the real and imaginary components of p2
are listed in Table 7. The e::perir.r.tal and Lheoretical dispersion
laws for heavy water in the p plane are sho.'ni in Figure 17. The
vertical line result of .iAeDiffusioa Theory is due Zo the neglect
of chermalizatior defects. The experimental dispersion 'aw obtained
by Dunlap (6) is also sho:n. The theoretical dispersion law' obzained
by Perez, Ohantin and Dunlap (1?) is also included. The later in
cludes thernali zai:J L effects. .art front :he shifC in 2 due to. a
difference in heav.'y ..: r purity, the to sets acree relatively ?!ll
at loit frecquenccie buc pronounced differerces occur at high fre
quencies. The anomalous behavior of tie heavy '..ater data in this
experiment is more apparent h.r.
In the hecercgeneous analysis the disperston la' in the p plane
is reqir. Thiss aree sc na' lv .'ell with :oth t.t.or: an
Dunlap's data (6).
0 0i
0o010
Li
[U
O0
c 1
00(
00(
+ EXPERIMENTAL DATA POINTS
 LEAST SQUARES FIT TO DATA
I EXPERIMENTAL ERROR
++\J 50
+ CPS
300 CPS
250 CPS
200 CPS
150 CPS
100 CPS
50 CPS
DO 1000 1500
OMEGAW RAO/SEC
FIGURE 15.
IMAGINARY COMPONENT OF RHO
SQUARED FOR HEAVY WATER
54
0 005
 EXPERIMENTAL DATA POINTS
LEAST SQUARES FIT TO DATA POINTS
0004
SEXPERIIIENTAL ERROR
II
x
FI
O003
CL
100 P 250 CPS 350 CPS
1ii03 CPS
0o001
&00200
00 2 ,00 '000 4 00 000
O!,'LEG\A QUnJAREO (RAOE)(RAD) / (SEC) (L
FIGU RE 16.
REAL CO[iPO,]EiT OF RIH
SQUARED FOR HEAVY W'.ER
Er)
/
TABLE 6.
EXPANSION COEFFICIENTS OF RHO SQUARED AND THERMALIZATION AND DIFFUSION
PARAMETERS FOR HEAVY WATER
QUANTITY
PO [cm2]
P [cm2 secj
P2 [cm sec ]
P3 [cm sec ]
2 4.
P4 [cm sec J
a0 Lsec I
Do [cm sec ]
C [cm4 sec
CO [cm sec _7
EXPER I MENTAL
VALUE
1.3746 x 103
6
5.0613 x 106
II
3.1374 x 10
15
8.443 x 10 15
4.3259 x 10 17
20
1.976 x 10
*
EXPERIMENTAL VALUE
OF DUNLAP (16, 19)
1.4713 x 103
5.0083 x 106
3.9684 x 10 1
1.8026 x 10 14
17
2.5045 x 10 1
20
1.996 x 105
3.73 x 105
*** Lack of high frequency content of the heavy water data prevented
a suiTable determination of C .
0
TABLE 7.
EXPERPIENTAL VALUES OF REAL AMID
IMAGIN'ARY COI1PO1JE;TS OF R;IO SQUARED
FOR HEAVY WATER
OMEGA(RPAD/SEC)
0.000
62.3 1
125.663
130.1195
25] .327
314.159
375.990
4 39 .322
502.65 4
515.43G
623.312
691.143
753.931
815.313
879.61 5
942.476
1005.30C
10 6:3. 14 0
1130.972
1193.304
1256. 636
1313 53
1332.29 9
11 5 13 1
1507.9u3
1570. 795
1633 62 L,
1696 4 5 3
1759 2 ) 0
1322.122
134 953
1947.785
2010 17
2073.U49
2136.231
219 113
OIE'lGA SQUARE)
0.0
3947 .3
15791.3
35530.5
63165. 3
93695.
14 2122.0
193443.3
252661.
319771. 6
394 733. 4
477637.9
5 G 438 .1
6671 3.9
773775.4
383262.5
1010645.14
11 04 92 4.2
1273 0908.4
1425163.1
1579133.7
1740994 7
1910751.7
203340L. 1
2273952.7
24G739G.0
2 6 7 35.
2877971. 1
3095191.9
3320123.14
3553050.7
37933 G .0
404251.6
4299190. 4
45G360..3
4 3 39 35 9
REAL
0.0014139
C.0011 159
0. 00110 3
0.0015970
0.00138S 1
0.0013714
0.00136)0
0. 001350()
0. 0013421
0.0013362
0.0013325
0.0013315
0.00134101
0. 0013';! '
0. 00135 3
0.0 61 746
0.0013?32
0.001416u
0. 0 9144 0 3
0.0')15514
0.0014597
0.0016553
0. 90174ESJ
O.OG14535
0.0014557
0.0014764
0. 00 I51
0. 001535J9
0.0016259
0. 0 0 1 G1 2 C 2
0. 00 160 2
0.0016927
0.0017350
0.0017 777
0.0013256
C.001.01 3I
C. 0, 1777 .
0. 0017 11'3
0 0016 C 7 3
I MAG I NARY
0.0000000
0.0003043
0.0006109
0.0009211
0.0012354
0.0015536
0.001274h6
0.0021977
0.0025221
0.002 '471
0.003172E;
0.0034932
0.0033271
0. 0041511)
0.0044734
0.0047949
0.0051153
0.005L360
0.0057505
0.0060573
0.0063002
0.0006659
0. 0 9 6 '^ 3
0.007327;';
0. 7606 2
0.00 79 712
0. 9i23253 :
0.090 5311
0.0038 207
0. 0 0 3 9 2 0 7
0.0091095
0.0093906
0.009622 :1
0.0090174 2'
0. 1001351
0.0103992
0. '1: c u
THERMALIZATION THEORY (19)
0
00
; a
CL
10,
ru
0o
0
150
CPS
+
+
100
CPS
+
4
300
CPS
0
250 CPS
7
200
CPS
150 CPS
+ EXPERIMENTAL DATA POINTS
100
CPS
V DUNLAP'S DATA (6)
l EXPERIMENTAL ERROR
50 CPS
00010 0'0015 0o0020 00025 0o003
(ALPHA) (ALPHA) (XI) (XI) 5 i/(CM) (CM)
FIGURE 17.
EXPERIMENTAL AND THEORETICAL DISPERSION
LAWS IN RHO SQUARED PLANE FOR HEAVY WATER
AGEDIF
One Fuel Rod Pulred Experiment
In Figure 1.. the thermal neutron pulses are shown Lor the various
axial positions. Again, the last 512 channels are not shown and each
channel represents 50 microseconds. All pulses are normalized to the
pulse at 50 cm.
Figures 19. and 20 show the Fourier transformed flux.es. At each
position considered in the final analysis, the amplitude and phase of
the frequency components are show.n. Included are the least squares
fits to the data. Deviations between the least squares fit and the
data points considered in the least square fic are small, the largest
being less than 5, in amplitude and less than I, in phase. The
theoretical results for 2 and are given in Table 8.and the e::peri
mrental results a 3e given in Table 9. The results of both are illus
trated in Figures 21.anl 22. The deviations at high frequencies
caused by the neglect of theroalizacion effec:s, are seen in the
illus :ra .ons. Taery under estimates and over estimates c at
high freq.encies. Msaxinj e'.'Lacicns, hw:.,. er, are less than 5.53 in
2 and 6.." in ;. These deviaions occur at 1,10 cps and 320 cps
respc:tive ly. The closer agreemnt bet':an theory and e:peri .nt 2a
higher reque.ncies is a:ain fortuitous. In the subsequent analyses
data above 500 cps, where mn::iirum deviations :.er 5.1", in Ct and
V.97 iL: r, were excluded.
Both the expeir:i:.1 aid theoretical dispersion lawJs :re so:'n
in Fi'ure .. TLe theoretical dispersicn .a.;: is ob:ained front
equtL.on [2.12] wich 7 = 0.25. This value of : ga.'ve agreemenc a:
zero frequency with the expert irental data aad the resulting
50s
40,
30 s00
20,00
10o
55 CM
60 CM
5 CM
70 CM
>
Z
LJ
a
22
En
EL
F
<
L,
F
5 CM
.90 CM
12 256 384
CHANNEL NUMBER
FIGURE 18.
NORMALIZED THERMAL NEUTRON PULSES FOR
HEAVY' WATERONE FUEL ROD SYSTEM
75 CM
 80 CM
0o cPs
50 CPS
100 CPS
150 CPS
200 CPs
250 CPS
300 CPs
350 CPs
+ F.PEP.IIE.T..L DATA POI[;TS
X DATA POINiTS OMITTED FROP.. LEAST SQUAFREL FIT
0i  LELAT SQUARES FIT TO PRE11.,IIIItIG POIDiT
1 o F 0 70 b sb 10o
AXIAL POSITION,9 /MS
F IGUFE 19.
EXPEP.IMElTAL AIIPLITULDES FOR
HEAVY WA.TEROC.E FUEL ROD SYSTErI
ioo10
10
LUQ
I 
l
I
L.
_j
10
\__0 CPS
I I I I t
t~i I ~t3
100CPS
150 CPS
+.I
200 CPS
250 CPS
CPS
X DATA POINTS OMITTED FROM LEAST SQUARES FIT
LEAST SQUARES FIT TO REMAINING DATA POINTS
I I I I I
3 S6 70 O
AXIAL POSITION CMS
FIGURE 20.
EXPERIMENTAL PHASES FOR HEAVY
WATERONE FUEL ROD SYSTEM
30 100
U3
m
E<
0i
[ 3

Or:
LL
Cfl
Cn
1
EL
4
E
%1
 ~
TABLE 3.
THEORETICAL VALUES OF ALPHA AND XI
FOR HEAVY WATERONE FUEL ROD SYSTEM
FREQU .IlCY(CPS)
0.
10.
20.
30.
40.
50.
60.
70.
0O.
90.
100.
110.
120.
130.
140.
150.
160.
170.
ISO.
190.
200.
210.
220.
230.
24 1 .
250.
260.
270.
280.
290.
300.
310.
320.
330.
340.
350.
ALPHA( 1/Cl l)
0.0379
0.0381
0.C387
0.03'J7
0.0400
0.01121
XI ( /CM)
0.0000
0.0040
0.0078
0.0115
0.0149
0.0121
0.0211
0.023S
0.0264
0.0232
0.0311
0.0332
0.0353
0.0373
0.0391
0.0400
0.0427
0.0444
0.04000
0.0475
0.0401
0.0506
0.0520
0. 0534
0. 0 '. :
0. 0562
0.0575
0.0523
0.0600
0.0513
0.0625
0.0537
0. 0643
0. 0600
0.01571
0.0 0 2
0.00434
0.00449
0.0463
0.0477
0. 0492
0.0500
0.0520
0.0533
0.0547
0.0560
0.0573
0.052G
0.0599
0.0611
0. 06 4
0.0536
0.0643
0o.n0653
0.0671
0.0332
0.0603
0.0704
0.0715
0.0725
0.0737
0.0747
0.0757
0.0762C
0.0772
0.0783
TABLE 9.
EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATERONE FUEL ROD SYSTEM
FREQUENiCY(CPS)
0.
10.
20.
30.
40.
50.
GO.
70.
80.
90.
100.
110.
120.
130.
140.
150.
160.
170.
180.
190.
200.
210.
220.
230.
240.
250.
260.
270.
2S0.
290.
300.
310.
320.
330.
340.
350.
ALPHA(1/CN)
0.0380
0.0382
0.0387
0.0395
0.0405
0.0418
0.0431
0.0446
0.0461
0.0476
0. 0491
0.0505
0.0520
0.0535
0.0549
0.0564
0.0579
0. 0593
0.0606
0.0619
0.0532
0.0646
0.0660
0.0674
0.06S9
0.0701
0.0714
0.0723
0.0741
0.0755
0.0771
0.0783
0.0791
0.0796
0.0799
0.0S00
XI (1/CM)
0.0000
0.0038
0.0076
0.0112
0.0147
0.0179
0.0209
0.0236
0.0262
0.0286
0.0309
0.0330
0.0351
0.0370
0.0388
0.0406
0.0422
0.0437
0.0452
0.0467
0.0482
0.0496
0.0510
0.0522
0.0534
0.0544
0.0555
0.0566
0.0576
0.0586
0.0532
0.0599
0.0604
0.0616
0.0G28
0.0551
0 10
010
0 06
CL
<0 04
0
50
SCPS
300 CPS
250 CPS
00 CPS
50 CPS
100 CPS
50 CPS
 EX:PERIMENT..L D..TA. POINTS
. .GEDIFFUSIO:; THEOR'Y
I
EXPERII.EiITAL ERROR
20'0
FREQUJENCYE CPS
FIGURE 21 .
EXPERItLE[ITAL AND THEORETICAL VALUES OF
ALPHA FOR HEAVY. WATERONE FUEL ROD SYSTEM
0 10
008,
1d
0G0
OaG2(
+ EXPERIMENTAL DA'A POINTS
AGEDIFFUSION THEORY
I
EXPERIMENTAL ERROR
\_3 50
t1
CPS
300 CPS
50 CPS
CPS
150 CPS
CPS
50 CPS
FREQUENCY,
CPS
FIGURE 22.
EXPERIMENTAL AND THEORETICAL VALUES OF
XI FOR HEAVY WATERONE FUEL ROD SYSTEM
01(
0 O.
Li
0 o
01
4 EXPERIMENTAL DATA POINTS
 AGEDIFFUSION THEORY
I
EXPERIMENTAL ERROR
350
CPS
200 CPS
o50 CP
100 CPS
50 CPS
O'04 0 '05
ALPHA 1/CM
FIGURE 23.
EXPERIMENTAL .MID THEOCF'.ETICA'L DISPERSIGCi LAJS IN
RHO FLAITE FOR E.AVY W.ATERONE FUEL ROD SYSTEM1
dispersion law is shown. Again, the deviation at high frequency is
seen. Since the determination of 7 is based on the zero frequency
data where theory and experiment better agree, the high frequency
deviation has no significant effect on the experimental value.
The sensitivity of the fit to 7 was determined by computing the
dispersion law for several values of 7. It was found that
6 7 = 22 6 a'
where a' is the zero frequency Q. Assuming that a is accurate to
+ 0.0008 (2.1% error) this sensitivity allows the determination of 7
to + 0.02. The feasibility of using the neutron wave technique to
measure 7 based on the dispersion law is demonstrated.
Attention is now turned to the polynomial expansion of p in
powers a). The expansion coefficients obtained from the UFNLLS Code (20)
are listed in Table 10.
TABLE 10.
EXPANSION COEFFICIENTS C'OF HO SCARED
FOR HEAVY WATER ONE FUEL ROD SYSTEM'
CO EFFI~ :. _:TS UFNLLS ?JSL1T
PO [cm2] 1.4285 X 10o
P1 [cm sec] L.354 X 10 '
P [cm2sec] 6.6175 X 10i
P, [cm sec ] 1.2051 X 101
P [cm 2sec ] 5.8555 X 1017
Figures 2h. and 25. showi the e::perirmental data and the leasr squares fit
for the imaginary and real parts of p2 respectively. The 23D fit has
a maximum deviation of ) .5, frotn the experimental data at j500 cps while
3 2
r' E, has a ma'imum deviation of 2.51 at 300 cps. Experimental
2 2
results for 2a; and a  are listed in Table 11.
In Chapter LI PO and P were related to 7. Using these relations,
equations (2.13] and [2.11], 7 vas computed to be 0.29 in terms of P0
and 0.l1 in terms of P The difference of these values from the 0.25
value obtained in the dispersion 12W fit are now discussed individually.
The P value of 7 is based principally on zero frequency data, P
0 0
being the zero frequency intercept of L' . It follows that fairly
good agreement should exist between the .two values of 7, 0.25 and 0.29,
both bein de:ermin?d from zero frequency data. Nonetheless, a 16''
difference e::ists in '. This d.s:repancyv s e.cplained by recalling
that P,, the zero frecuen'cy intercept was 2 cer.iF uced p rarareter in t'ae
least squares fit co the e:xperiental daJta. This value is less chan
the observed itercept, a phenor.encn w'hch is caused by the minimum
2 '.
occurring .n the 2 nta at apprcxirately 50 cps. The miniu..
has :,et to be e:: lined, :h.re be g. no es:ablisrhed tCeoretical basis
for tC e::i.Ste .ca. it m.Tiy oe che result of sc.Se s.sce.aj ic error. Zo.
seGq.n:l, che e..er:.i en:.l .val; of ?P is in question. It ito note
that if the value of the obser.'ed intercept is used, y is ccrmputed to
be 0.24 and this agrees 'ithin L', with the dispersion law result.
The greater :ev'.'itl o of e ? value of t has C':o c .auses. First,
the rela:ion deri.a:' for 7 n c ?r of ?, is dEj: e.' .1: on te fuil fra
quency raJn e, ? aein i e f:rst derL.ar.3 L" of the 2 ; fun :ion.
mere accurate model would predict a lo'.er v..'lue of P. l Lo:er 'alj:
l
+ EXPERIMENTAL DATA POINTS
LEAST SQUARES FIT TO DATA POINTS
EXPERIMENTAL ERROR
+ \__350
+ CPS
300 CPS
Li
0
=0
Li
O
FI
<
0
200 CPS
150 CPS
100 CPS
10 CPS
1000 15 00
OMEGA s RAD/SEC
FIGURE 24.
IMAGINARY COMPONENT OF RHO SQUARED
FOR HEAVY WATERONE FUEL ROD SYSTEM
Ic
0 02
50 CPS
0005
0004+
1j
)o .003
n_
50 002
<
0F
<
0 001
0'
OMEGA
+ EXPERIMENTAL DATA POINTS
 LEAST SQUARES FIT TO DATA POINTS
I
EXPERIMENTAL ERROR
300 CPS
350 CPS
250 CPS
200 C
N 150 CPS
100 CPS
000
SQUARE[],
A 4000 000
(RAO) /(SEC)(SEC)
FIGURE 25.
RE.L COMPONENT OF RHO SQUARED FOR
HE.'AVY WATEROHIE FUEL ROD SYSTEM
0030
(RAO)
 ~' ~L
TABLE 11.
EXPERIMENTAL VALUES OF REAL AND
IMAGINARY COMPONENTS OF RHO SQUARED
FOR HEAVY WATERONE FUEL ROD SYSTEM
OMEGA(RAD/SEC)
0.000
62.831
125.663
188.495
251.327
314.159
376.990
439.022
502.654
565.486
628.318
691.149
753.981
816.813
879.645
942.476
1005.308
1063. 140
1130.972
1193.804
1256.636
1319.468
1332.299
1445.131
1507.963
1570.795
1533.626
1696.450
1759.290
1822.122
1384.953
1947.735
2010.617
2073.449
2136.281
2193.113
OMEGA SQUARED
0.0
3947.8
15791.3
35530.5
63165.3
98695.8
142122.0
193443.8
252661.3
319774. 6
394783.4
477687.9
568488.1
667183.9
773775.4
338262.6
1010645.4
1140924.2
1279093.4
1425168.1
1579133.7
1740994.7
1910751.7
2033404.1
2273952.7
2467396.0
2663735.9
2877971.1
3095101.9
3320123.4
3553050.7
3793868.9
4042531.6
4299190.4
4563G96.8
4336095.9
REAL
0.001446G
0.0014455
0.0014415
0.0014363
0.0014315
0.0014232
0.0014231
0.0014304
0.0014354
0.0014433
0.0014526
0.0014643
0.0014774
0.0014921
0.0015112
0.0015343
0.0015676
0.0016016
0.0016230
0.0016544
0.001G773
0.0017161
0.0017604
0.0012203
0.0013927
0.0019571
0.0020113
0.0020363
0.0021691
0.0022590
0.0024522
0.0025514
0.0026108
0.0025451
0.00244G0
0.C021629
I MAG I NARY
0.0000000
0.0002949
0.0005913
0.0008903
0.0011934
0.0014990
0.0013065
0.0021147
0.0024241
0.0027320
0.0030399
0.0033485
0.0036562
0.0039662
0.0042771
0.0045890
0.0043936
0.0051967
0.0054947
0.0057953
0.0061051
0.0064234
0.0067373
0.0070574
0.0073555
0.0076470
0.0079417
0.0082553
0.0035456
0.0088723
0.0091399
0.0094011
0.0095803
0.0093191
0.01C0510
0.0104173
of P1 produces a lower value of 7. y computed from the PF relation
was found to be very sensitive to the value used for v th. 2,h80 m/sec
and 2,490 n/sec yielded results of 7 = 0.27 and 7 = 0,22 respectively.
The 2,470 m/sec is the average thermal neutron velocity in heavy water.
Insertion of the fuel rod would tend to increase the average velocity
because of preferential absorption of lower energy neutrons and pro
duction of fast neutrons in the fuel rod. Consequently, the correct
value of vth should be soiaewhat larger than that for the moderator,
resulting in a more consistent value of 7.
For the present, 7 = 0.25 .02 obtained from the dispersion law:,
is considered the r:ost accurate value. Assuming that Diffusion Theory
holds in the rod, 7 was computed to be 0.327 by Cain (13).
2
The experimenal and theoretical dispersion laws in the p2 plane
are sho'.'n in Figure 26. Tne high frequency difference is am:li.Lied.
The next analysis deals with the intersection of the dispersion
laws of the iieav,' 'ter outly and heavy waterone fuel rod systems.
This andly;,is is ha.pered sor.e,'hat by, the difference in heavy water
purity that existed between the two sets of data. This difference
Wjs :2:.er. into account: b'j reducing the 2 of the c .e fuel rod dat.,
0.00157 to cc,pns2:e for absorptions of the a.di:ional 0.5' li.it
water. This corre:cion was ob:.tin'ed by comoput ir. the zero frequency
a at the two concentrations of heavy water frori equation [2.12). The
intersection is then based on a 99.5' heavy water purity. The dis
persion law intersection is shio'n in Figure 27.'where both the ex.eri
r.in al and t;ec.:r ic ii intersections are sho:.r,. Tril or e ical '.L the
intersectio;. occurs t lO :ps, a value ch higher than the
e::perimencal inc.rsection at 1 .&, c's which was obtained by plotting
0 012(
0o010(
0r
O0a00(
u M
I
0000
0C 0010 00015
(ALPHA) (ALPHA)
350 CPS
/
\ 300 CPS
+. \250 CPS
+ 200 CPS
150 CPS
100 CPS
+ EXPERIMENTAL DATA POINTS
AGEDIFFUSION THEORY
10 CPS
EXPERIMENTAL ERROR
0 a0020 0 O25 0 0030
 (XI) (XI) 9 I/(CM) (CM)
FIGURE 26.
EXPERIMENTAL AND THEORETICAL DISPERSION LAWS IN RHO
SQUARED PLANE FOR HEAVY WATERONE FUEL ROD SYSTEM
.+
010
OoS
0 08
0
0 a04
0 02(
 MODERATOR ONLY DATA POINTS
V MODERATORFUEL ROD DATA POINTS
 AGEDIFFUSION THEORY
Li EXPERIMENTAL ERROR
I
300 CPS
250 CP
2003 C
'00 CPS
150 CPS
5' CPS
100 CPS
100 CPS
50 CPS
MODERATOR ONiLY
o04
005
ALPHA L
u uo
1/C1NI
FIGURE 27.
EXPERIMENTAL AIID THEORETICAL INTERSECTIONi
OF DISPERSION LA'.:S IN PJHO PLANT
2503 CPS
the observed dispersion laws for the heavy water only and heavy
waterone fuel rod systems. This discrepancy is due to the experi
mental error and the inadequacy of the AgeDiffusion Model. The
dominant experimental error is caused by the uncertainty of the heavy
water purity for it was found that
&U 7,000 5C
where C is the correction term applied for the purity difference be
tween the two experiments. C is approximate and introduces signifi
cant deviations. More accurate results would have been obtained if
the experiments had been conducted at the same heavy water purity.
On the other hand, an improved theoretical model would predict a
smaller value of we thereby producing better agreement between the
theoretical and experimental intersection. The intersection in the
p2 plane is shown in Figure 28. Uncertainty in the heavy water
results obscure this analysis but a more distinct intersection may
occur.
The combination of theoretical and experimental uncertainties
prevent the accurate dete:'mination of heterogenectus parameters based
on the intersection of the disper3ion laws. This is readily apparent
in Table ]2. where the experimental and predicted parameters are given.
A more accurate experiment and theoretical model are needed for
proper analysis of the dispersion law intersection.
A Lattice of Fuel Rods
7 has been measured for a one fuel rod system Will the value of
7 change when a lattice of fuel rods is considered? It is not possible
at this time to answer this question resolutely but che work of
0o0120
0 0100
0o 0080
'0 OOG
r
Oo0020
200
CPS
150
CPS
100
CPS
50.
CPS
(ALPHA)
300 +
CPS
+
+ v
+
+ v 250 CPS
+
+ V
_300 CPS
 200 CPS
150 CPS
100 CPS
+ 'IODEPRTOR OL iL' D.T.. POI'iTS
'50 C PS
(0'0015
(ALPHA)
V7 :MODERATORFULEL
POI ITS
ROD DATA
 AGEDIFFLITC.il THF.TOR
1E::PERIilEiTTAL ERROR
0o 20 00025 0 00t30
 (XI) (XI) I/ (CM) (EM)
F'IG'UJE 2,3.
EXPERILMEN[TA.L .liD THEORETICAL INTERSECTION
OF DISPERSION LAWS IN RHO SQUARED PL.iE
TABLE 12.
HETEROGENEOUS PARAMETERS BASED ON
INTERSECTION OF DISPERSION LAWS
EXPERIMENTAL VALUE
0.05515
0.03855
150.8
120.9
552.3
>. I
PREDICTED VALUE
1
0.07478
0.06375
31o.o1
310.01
510.141
118.82
.9992
These values were cb:ained from the AgeDiffusion Model.
These values .:ere obtained from N. J. Diaz by private comuni
cation.
VARIABLE
1
a (cm"I]
~c[cml]
c [cps]
r[ (cps]
P [cm2]
p
78
Dunlap (6) does offer a clue to the answer. His study included the
calculation of cell parameters of a lattice of fuel rods of the
same type as the one considered in this experiment. The calculated
parameters yielded a theoretical dispersion law which agreed reasonably
well With the experimental one. The agreement leads one to believe
that the calculated and true values of the parameters are nearly the
same. The value of L2 for the lattice will now be considered.
2 2
L of the lattice is related to 7 and L of the moderator (15)
and is very sensitive to 7 in that
L2 1.6 X 10 7.
Using Dunlap's data and = 0.25, L2 was computed to be 92.2 cm2
This compares within 0.6. of the 91.6.2 cm value co.iputed by Dunlap
(6). The close a[ree.ant in L and the agree.mer.t obtained by
Dunl=p between the theoretical and experir.ental dispecjion law in
dicate that :he 7 measured fcr one fuel rod jay be applicable to
a simple lattice of fuel rods.
CHAPTER V
CONCLUSIONS AND RECO:MiENDATIONS FOR FUTURE WORK
Conclusions
The neutron wave technique has been used to study a heterogeneous
system. 7 has been determined using the experimental and theoretical
dispersion laws and the expansion coefficients of 2 in a power series
of (iw). The intersection of the experimental dispersion laws of the
heavy water only and the heavy waterone fuel rod systems made possible
the determination of two other heterogeneous paramaters. Heterogeneous
parameters have been extracted from the neutron wave data. These para
meters are sufficiently sensitive to neutron wave parameters to allow
their determination by this technique. Accuracy of this technique has
been partially established in that Y can be determined through the dis
persion law fit to two significant digits. Further investigation of
accuracy was precluded because of the theoretical model and/or experimental
phencr.ena.
The exie:.sicn of the results obtained for v to a lattice has beer
investitel for the case cf a simple lattice. The diffusion length
squared for the lattice was co=puted based on the measured value of 7
and it was found to agree very well with previous results. The fact that
the correct value of the diffusion length squared can be computed from
the measured value of y is very sign ficant.
Tne exoerienta.l zechnicu has 'esn tested under the most adverse
conditions in that a .inimumL of fuel was used. Vith larger amounts
of natural uranium or with enriched uranium more pronounced character
istics will be exhibited. In these systems improved results should be
obtained.
Recccmendat ions
'Tne most imnorarnt recormendation is the development of a more
accurate theorelical model. Better accuracy at high frequencies would
improve the technique, predicting more accurate values for the appropri
ate parameters.
Future studies should be conducted to determine interaction and
shadoin; effect's on the heterogocneous rara..etcrs. These studies can
be accomplished by a study: equivalent to this one wh'here two, three or
more fuel rods are ir.serten at various locations in the assembly. AAn
intercstin an.d useful study. would be the jdternination of heter:enous
oarans.eter s as function of ne nu.'.ber of fuel rods in the asse.bi.'.
Still another a:.roz:h wsu'ld be the sta:." of t.te heterocenreous para
meters a: a functi on of the :enter lire se r'tio of the fuel rods.
.ne ar.aly:is of :'uJre e::eri..eants :.?. be nodifiedi in the follo'
in :':. Let t.e car::ric f: in ; cf the thore:'cal to 'the e::eri
centa_ UQi3_.i... ... '.":f i .t e: ., .2 : f...t.r.".i;." 'Ihen; in the
ev'alu'icn of ticE e;. .: s .. c.effi : ts,~; .(? .13) .n (Z2.1 ), 
is kr.'r. and :'c .t. r t:sr.xs ters can c cet r..ne..
The disperse or. law ir,tersettion aj not staiied in the lanse.
It ma be .rtr.'hle to do so in the f,:.ure. . artears that a imore
dirstinc t ii n :.::..O _.. r iclut:r.
time of the counting system is needed. The method described in Appen
dix A may serve as a starting point for the development of a suitable
method. It is important to determine the resolution time accurately
to 0.01 seconds because the experimental value of a and C are dependent
on the resolution time used. A more accurate resolution time yields
a more accurate experimental values of a and C and the other neutron
wave parameters.
Determination of heavy water purity proved to be a difficulty in
this experiment. Should heavy water or some other moderator with vari
able properties be used in future experiments, the physical properties
must be accurately determined and verified. Efforts should be made to
conduct the moderator only and moderatorfuel experiments under the
same conditions when such moderators are used. The error in determin
ing the physical properties must be minimized to improve the validity
of the data analysis.
APPE.1DIX A
1.EASU .ILJ:LiT 0?7 RSOLUJTIO; Ti ':3 USI;;G A PULSED
i;EUT3:I f SOURCE
The method presented here for determination of the resolution
time was found to be inadequate when performed in the manner described
herein. The method, however, is believed to be sound and with one
improvement good results should be obtained.
The difficulty concerns the rectangular shape of the neutron
pulse at the detector. The asymptotic decay constant of the thermalizing
apparatus was such that the individual pulses at the detector are not
sharp and the decay time consumes a great percentage of the time between
pulses. Consequently, an "effective" time between pulses exists rather
than the time between the end of one pulse and the beginning of the
subsequent one which was measured at the source. An improved procedure
to overcome this difficulty is discussed at the end of this Appendix.
Resolution time is defined as the minimum time between successive
neutron interaction events such that both events are registered by the
counting system. The definition is not changed if, instead of a single
neutron, a burst of neutrons, as shown in Figure 29. is used. Assuming
that the neutron bursts from the neutron generator are:
(a) the same
and
(b) near a square wave at the detector
a method for determiiing the resolution tir.e of the counting syste:rs
was developed which used the neutron generator. The major advantage
of this method is that the resolution time of the counting system in
its operating condition is determined.
The method is illustrated grachically in Figure 29. It is seen
(A) EFFECTIVE ;;U;SER OF CCONTS
REGISTE.PIED ?ER BURS'J
,I r
/IF
(B) LES3 :; FECTIV :I. '.. OF COL.;T
RiEGISTD.ED ?.' 3U.1"ST
ILL'S..'. IO.,:; l !S ,L'5 O:7 : 1 T.l : A 3 TI
BET.Z :i L..S S '.; CC.LC'::T ?.'TZ
I/
I
II I
11111
that when the time between the end of one burst and the beginning of
the next is less than the resolution time, , of the system the effec
tive number of neutrons counted per burst is less, because some of the
leading neutrons of the burst are not counted. Otherwise the effective
number of counts per burst is constant.
To verify assumption (a) the target current was monitored with
a Type 5h7 Tecktronix Oscilloscope with a 1Al Plugin PreAmplifier
unit. This PreAmplifier unit permitted double amplification of the
input signal by cascading the two channels of the PreAmplifier. This
monitoring method presented a trace of the target current which could
be monitored accurately enough for assurance of constant target current
and hence a constant burst of neutrons.
Since moderation will spread the neutron burst, the detector was
placed as near the source as practical. In this case, the detector
was placed in the reference detector position shown in Figure 3.
This arrangement was the optimum position for obtaining nearly square
pulses of thermal neutrons at the detector.
Tne resolution time of the system was believed to be of the order
of several microseconds. For this reason data were taken using a 10
microsecond target pulse initially separated by an 8 microsecond time
interval. Tne time intervals were measured by observing tne terminating
and initiating signals for the neutron pulses on an oscilloscope.
Tne time interval of separation was decreased until a nonlinear devia
tion was observed.
The integral number of counts were recorded by the detector system.
If the Li:e interval of pulse separation is greater than the resolution
time, the counts should increase linearly with decreasing time separa
Lion (increosins iepetition rate). '.Then the tire separation is less
than the resolution ti.me there should be a nonlinear deviation in the
scaler counts. Reprcsertative data are siven in Table 33. ani illustrated
in Figure 30.
In order to ovcrcome the difficulty of pulse spreading the genera
tor should be withdra''n from the ther,.alizing. asse bl, and a small
amount, for example several inches, of rodcrator placed between the
target and the detector. Although the sharpness of the detector pulse
has not been cc:.ipletely verified, a later experiment indicates that
this procedure 'ill yield suitable results for the resolution time.
TABLE 13.
DATA FOR RESOLUTION TIME DETERMIIN'ATION
TIME BETWEEN
BURSTS (ICROSECONDS)
AVERAGE COURTS
FOR ONE MINUTE
340,792
344,665
343,661
318,507
301,070
