SPATIALLY DEPENDENT TRANSFER
FUNCTION OF NUCLEAR SYSTEMS
BY CROSS CORRELATION METHODS
By
CHESTER D. KYLSTRA
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, 1963
ACKNOdLEDGMENTS
The author wishes to express his appreciation to the
members of his graduate committee for their advice and
assistance. In particular, Dr. R. Uhrig and Dr. R. Selfridge
have provided continued aid and guidance during this
study.
The author wishes to thank the technicians and grad
uate assistants of the Department of Nuclear Engineering
for their help. Aid in checking out equipment, building
special circuits, and running the experiments was provided
by: F. Primo, H. Diaz, K. Fawcett, R. Lyttle, T. McCall,
A. Tuthill, R. Hartley, R. Kavipurapu, G. Fogle, P. Hunter,
d. Nelson, J. Mlaller, D. Butterfield, J. Moore.
Special thanks are due his wife, Pat, for the many
hours she spent drawing and inking figures, inserting
Greek letters in the text, proofreading, and typing the
manuscript.
The author also wishes to extend his thanks to Dr.
M. Moore, Dr. R. Perez, and Dr. R. Dalton for their advice
and many helpful comments.
TABLE OF CONTENTS
ACKNOLEDGENTS . . . . . . . . .
LIST OF TABLES . . . . . . . . .
LIST OF FIGURES ..................
ABSTRACT .....................
Chapter
I. INTRODUCTION . . .
II. THEORY ........
Transfer Function
Statistical Theory
III. EXPERIMENTAL METHOD AND
Experimental Method
Page
ii
v
vi
ix
4
4
. . . . . 4
. . . . 30
ANALYSIS . . 45
. . . . 45
Data Analysis . . . . . .
IV. EXPERIMENTAL EQUIPMENT . . . . .
Input System . . . . . . .
Nuclear Systems . . . . . .
Output System . . . . . .
V. RESULTS . . . . . . . . .
Theoretical Results . . . . .
Experimental Results . . . . .
VI. CONCLUSION ...............
Appei
ndix
A. FOURIER TRANSFORM OF A PERIODIC DELTA
FUNCTION ................
B. COMBINATION OF TRIGONOMETRY FUNCTIONS,
EQUATION (50) . . . . . . .
iii
53
62
62
66
70
75
75
87
114
119
121
TABLE OF CONTENTS (Cont'd)
Appendix Page
C. ROOTS OF THE CHARACTERISTIC EQUATION . 123
D. COMPUTER CODES . . . . . 129
E. COUNT RATE CIRCUIT DESIGN . . ... 151
BIBLIOGRAPHY ................... 157
BIOGRAPHY ................... .. 161
LIST OF TABLES
Table Page
1. Count Rate Meter . . . . . ... 71
2. Nuclear Parameters Used for the Theoretical
Results . . . . . . . ... 76
3. Number of Terms Required for Four Per Cent
Error in Equation (57) . . . ... 86
4. Design Criteria . . . . ... 153
5. Count Rate Circuit Components . . .. 155
LIST OF FIGURES
Figure Page
1. Definition of functions for statistical
theory . . . . . . . ... 35
2. Block diagram of the experimental equip
ment . . . . . .. . ..... . 46
3. Typical maximum length sequence and its
auto correlation function . . . 48
4. Auto power spectrum of the maximum length
input sequence . . . . . ... 51
5. An eightstage, maximum length sequence,
pseudorandom binary signal generator 64
6. Light water subcritical assembly . . 67
7. Heavy water subcritical assembly . . 69
8. Block diagram of the data acquisition
system . . . . . . . . 74
9. Theoretical transfer function for light
water . . . . . . . ... 77
10. Theoretical transfer function for light
water, z = 100 cm . . . . .. 79
11. Theoretical transfer function for heavy
water . . . . . . . ... 80
12. Theoretical transfer function for a
light water subcritical assembly . .. 82
13. Theoretical transfer function for a
heavy water subcritical assembly . .. 83
14. Auto power spectrum of the shift
register . . . . . . . . 89
15. Cross correlation of two detectors,
light water . . . . . ... 91
16. Cross correlation of two detectors,
heavy water . . . . . ... 93
vi
LIST OF FIGURES (Cont'd)
Figure Page
17. Cross correlation of two detectors, light
water subcritical assembly ..... .94
18. Cross correlation of two detectors, light
water subcritical assembly ..... .97
19. Auto power spectrum of Detector 1, light
water subcritical assembly ..... .98
20. Amplitude of the cross power spectra,
light water subcritical assembly . . 99
21. Phase shift of the cross power spectra,
light water subcritical assembly . .. 100
22. Amplitude of the transfer function
between two detectors, light water
subcritical assembly . . . . .. 102
23. Phase shift of the transfer function
between two detectors, light water
subcritical assembly . . . . .. 103
24. Cross correlation between two detectors,
heavy water subcritical assembly . . 105
25. Cross correlation between two detectors,
heavy water subcritical assembly . . 108
26. Auto correlation of Detector 1, heavy
water subcritical assembly ..... . 109
27. Amplitude of the cross power spectra,
heavy water subcritical assembly . . 110
28. Phase shift of the cross power spectra,
heavy water subcritical assembly . .. 111
29. Amplitude of the transfer function
between two detectors, heavy water
subcritical assembly . . . . .. 112
30. Phase shift of the transfer function
between two detectors, heavy water
subcritical assembly . . . . .. 113
31. Typical roots of the characteristic
equation . . . . . . .... 125
32. Count rate circuit diagram ..... . 152
LIST OF FIGURES (Cont'd)
Figure Page
33. Amplitude of the transfer function for
the count rate circuit . . . .. 156
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
SPATIALLY DEPENDENT TRANSFER FUNCTION
OF NUCLEAR SYSTEMS USING CROSS CORRELATION IETHDDS
By
Chester D. Kylstra
December 21, 1963
Chairman: Dr. Robert E. Uhrig
Major Department: Nuclear Engineering
The concept of a transfer function for a nuclear sys
tem is extended to include spatial effects. The general
equation is derived using the timedependent Fermi age
and diffusion theories for a singleregion, isotropic,
homogeneous medium. The fluctuation of the thermal neu
tron density at any point in the assembly is related to
the variation of the fast neutron source intensity.
The general transfer function equation is specialized
for several cases, including the case of a point source in
a cylindrical medium. Theoretical curves are calculated
for multiplying and nonmultiplying media and compared with
the commonly used lumped parameter transfer function.
The results indicate, in general, that the lumped
parameter model predicts the correct behavior of the
nuclear system only if the output detector is located a
specific distance from the source. If the detector is
located elsewhere, the lumped parameter model is not
capable of accurate results.
Experiments were performed on light and heavy water
subcritical assemblies to measure the spatially dependent
transfer function between two detectors. The input to the
assemblies was provided by a neutron generator which was
turned on and off in a pseudorandom manner by the maxi
mum length output sequence generated by a shift register.
Two neutron detection systems provided analog output
voltages proportional to the neutron density fluctuations
at different locations in the assembly. The voltages were
sampled at a high rate by a data acquisition system, which
stored the data on magnetic tape. The data was analyzed
by cross correlating the two outputs and calculating the
cross power spectra.
Statistical equations are developed which show that
the transfer function may be obtained by computing the
ratio of the cross power spectra for two different runs,
with one detector moved to a new location for the second
run. This method of statistical analysis has several
advantages. Among them is the ability to eliminate the
effects of the source spectra and the effects of the mea
suring system from the results.
A comparison of the experimental and theoretical
transfer functions provided inconclusive evidence as to
the validity of the theoretical model. Other effects, such
as reflection of the neutrons from the surroundings, pre
vented complete agreement. The equations used to predict
the behavior of a light water system clearly must take
into account the effects of a distributed source, rather
than a point source. The heavy water data was affected
by the small size of the system and the nearness of other
structures, but the theoretical curves still agreed very
well with the experimental data.
CHAPTER I
INTRODUCTION
The principal purpose of this study is to develop a
spatially dependent transfer function for nuclear systems,
using the time dependent Fermi age and diffusion theories.
The effects of neutron energy, time, space, and delayed
neutrons are considered. The experimental spatially de
pendent transfer function of a light water and a heavy
water subcritical assembly was measured and compared with
the theoretical transfer function. The multiplication
factor of both systems was varied from zero to near one.
The data collected for the measurement of the ex
perimental transfer function was processed by cross cor
relation methods. A secondary purpose of this thesis is
to demonstrate the advantages and accuracy of the parti
cular method used.
The transfer function equations commonly used for nu
clear reactors are derived from the time dependent dif
fusion equation, after the spatial dependence has been re
moved by assuming that the flux shape is the fundamental
spatial mode. The reactor is treated as a "black box",
or as a lumped parameter model (1, 2). This transfer
function works surprisingly well (.), especially for low
frequencies where the neutron flux appears to respond
1
simultaneously throughout the entire reactor and for cri
tical systems, where the band width is narrow and the
higher frequencies are greatly attenuated.
A reactor, however, is not actually a lumped para
meter system. The distance between the input and output
devices, as well as their relative locations in the sys
tem, can have a large effect on the measured attenuation
and phase shift of a disturbance as it propagates through
the system. As higher frequencies are present and of in
terest, as in neutron wave and random noise experiments,
the time and spatially dependent solution of the neutron
fux behavior is needed to adequately describe experimen
tal results. Weinberg and Schweinler (4), in 1948, were
the first to discuss the solution for a reactor driven
by an oscillating absorber. Other papers (j5 6) have
presented results for neutron wave propagation in both
multiplying and nonmultiplying media, showing the dis
persion nature of the process, and the attenuation and
phase shift of the neutron flux disturbance.
Further evidence of the spatial effect on the pro
pagation of a neutron flux disturbance was provided by
Badgley (Z) and Boynton (8). In their work with random
noise measurements, both used a transfer function for a
two point reactor model to represent the twin slab Uni
versity of Florida training reactor. Boynton (8) was
able to measure the transit time of a neutron flux dis
turbance as it travelled between the two slabs.
Statistical correlation techniques were used as
early as 1946, by deHoffman (9), to measure the dynamic
parameters of a chain reactor. This work was extended
by Albrecht (10) and Veles (11) to include the reactivity,
neutron lifetime, and delayed neutron effects in the the
oretical auto correlation function of the neutron density
fluctuations.
Moore (12 13) showed ho the power spectrum of the
reactor noise is related to the reactor transfer function,
where the power spectrum and the auto correlation function
are Fourier transform pairs. Power spectra measurements of
the reactor noise were performed by Cohn (14) and Griffin
(15) to obtain the reactor dynamic parameters as expressed
by a theoretical transfer function. Griffin was able to
compare the noise results with pile oscillator data.
Certain disadvantages inherent in the reactor noise
power spectrum and auto correlation function measurements,
particularly the uncertainty in the system input, the
effects of external instrument noise, and the lack of
phase shift information, are not present in the cross
power spectrum and the cross correlation function. Sev
eral investigators (8, 16 17.), using random reactivity
inputs and cross correlating the input and output signals
of the nuclear reactor, have measured the reactor dynam
ic ,parameters quite easily, even with input signals that
were small compared to the power level of the reactor.
CHAPTER II
THEORY
The theory presented in this chapter consists of
two parts. First, the general spatially dependent trans
fer function for nuclear systems is developed. It is
then adapted for several special cases and applications.
It is written for a point source, compared with the common
lumped parameter transfer function, and converted to an
alternate form that converges faster for large systems.
This conversion is easily made for a nonmultiplying
medium, but for a multiplying medium, the solution of a
complex transcendental equation is required. A completely
analytical solution in terms of the elementary functions
is not possible, but numerical methods may be used to ob
tain the theoretical transfer function for specific ap
plications.
In the second part, the statistical theory that re
lates the cross correlation and cross power spectrum of
three neutron detectors to the spatially dependent trans
fer function is developed.
Transfer Function
The theoretical spatially dependent transfer func
tion relates the thermal neutron density fluctuations at
4
5
the point r in a nuclear system to the fluctuations of an
artificial source of fast neutrons having a known spatial
distribution.
In the model used, the nuclear system, which is iso
tropic and homogeneous, consists of a single region. The
driving function or input to the system is a time varying
source of fast neutrons. The fast neutrons experience a
slowing down phase, represented by the time dependent Fermi
age theory. Upon reaching thermal energy, the neutrons en
ter a diffusion phase at constant energy, represented by
the time dependent diffusion theory. If the medium is a
multiplying one, additional neutrons are created by fission,
and join the source neutrons in the slowing down process.
It is assumed that the neutrons from the artificial
source, the prompt neutrons from fission, and the delayed
neutrons have the same energy at birth. This energy is
defined as the zero point on the lethargy scale.
The transfer function is limited by the same restric
tions and assumptions inherent in the Fermi age and dif
fusion theories, which may be found in Reference (19),
and by the mathematical approximation used in representing
the neutron source condition.
Basic equations
The time dependent Fermi age equation for contin
uous slowing down, absorption, and leakage (, 18) is
1 b)0(r,u,t) 'Oq(r,u,t)
= 7 t u Za(u) 0(rut)
+ D(u) V2(r,u,t) + S,(r,t) 6(u)
6
+ U f (1 f) t(r,t) 6(u) + ii(rt) 6(u)
i=1 (i)
u 2! 0
Neutrons starting the slowing down process at zero leth
argy are contributed by an artificial source Sg(r,t),
by prompt fission, V .f E (l)0) (t(r,t), and by the
decay of the neutron precursors XiCi(r,t). The slowing
down neutron flux is 0(r,u,t), at position r, at lethargy
u, and time t, in units of neutron /cm2 sec (unit lethargy).
The neutron velocity is v(u), corresponding to lethargy u,
and q(r,u,t) is the slowing down density at r and t, in
units of (neutrons passing lethargy u) /cm3 sec. The
macroscopic absorption cross section, a(u), and the dif
fusion coefficient, D(u), are both functions of the neutron
lethargy. The average number of neutrons produced per fis
sion is symbolized by U, the thermal fission cross section
is tf, the fast fission factor is C, and f3 is the frac
tion of neutrons produced by fission that appear as delayed
neutrons. The thermal flux, 0t(r,t), is not a function of
lethargy, but is a function of position r and time t, with
units of neutrons/cm2 sec. The six neutron precursors are
represented by Ci(r,t), i = 1 to 6, with units of precur
sors/cm3, and the decay constants are qi, with units 1/sec.
The Dirac delta function, 6(u), is used to insure that all
the neutron sources are considered mathematically as an
initial condition in lethargy.
The time dependent diffusion equation is (2. 18)
b t(rt) ,att(r,t) + DtV2t(r,t)
tt t
+ q(r,ut,t) (2)
with the symbols previously defined. The subscript t sig
nifies the value of the variable at thermal energy. .If
desired, the model used here could be changed to include
an artificial thermal neutron source. The source term
would be added to the diffusion equation.
The equations for the six delayed neutron precursors
are (2 18)
OC l (r,t) V3 fC0. iir,
Ci(rt ) i BIu O t(r,t) Aici(r,t)
i = 1 to 6 (3)
where
6
ZAi = (3a)
i=l
The three basic equations describing the neutron be
havior are coupled. The diffusion equation is coupled to
the slowing down equation by q(r,ut,t), the precursor equa
tion is coupled to the diffusion equation by
Pi UV fC( t(r,t), and the slowing down equation is coupled
to both the diffusion equation and precursor equation by
[LfE(1P ) t(rt) + i6 iCi(rt)] 6(u).
The slowing down equation may be written in terms
of q(r,u,t) by using an approximation relating 0(r,u,t)
and q(r,u,t) (18)
q(r,u,t) = Et(u) .(r,u,t) (4)
This is a reasonable assumption if Fs(u) 0(u) is slowly
varying with u, and .s(u) 'a(u) (18).
The average loss in the logarithm of the neutron
energy per collision, E, has units of (average gain in
lethargy) per collision, and Zt(u) is the total cross
section at lethargy u.
The relation between the thermal neutron density and
the thermal flux is (18)
t (rt) = vtnt(r,t) (5)
Separation of time varyina and steady state compon
ents. The time varying quantities in the basic equations
may be separated into steady state components and time
dependent components. The latter vary positively and neg
atively about the steady state value. Using the A to
represent this variation, the quantities of interest,
after substitution of Equations (4 and 5) in the first
three equations, are
nt(r,t) = nto(r) + An (rt)
q(r,u,t) = qo(r,u) Aq(r,u,t)
Ci(r,t) = Cio(r) + ACi(r,t)
a(r,t) = Sao(r) + Sa(r,t) (6)
Since the basic equations are linear with respect
to the time variable, they may be separated into the
steady state and the time varying equations.
For the steady state components, the slowing down
equation is
b)q (r,u) (u) q (r,u) D(u)
o+ a 0 qo(r,u)
o = u Et(.) g Et(u)
+ sao(r) 6(u) + Ef El )) vtnto(r) 6(u)
6
+ AiCo(r) 6(u) (7)
i=1
the diffusion equation is
0 = atVtnto(r) + DtvtV2nto(r) + q(r,ut) (8)
and the precursor equations are
0 = liUTfEvt"to(r) XiCio(r)
i = 1 to 6 (9)
The time varying components satisfy the following
equations. The slowing down equation is
1 bAq(r,u,t) A)q(r,u,t)
EZt(u) v(u) 10t Ou
E (u) D(u) .
S q(r,u,t) + g u) 2 Aq(r,u,t)
+AS,(r,t) 6(u) + V f E(1 )0) vtAnt(r,t) 6(u)
6
+ iAc, i(r,t) 6(u) (10)
i=l
the diffusion equation is
" OA (r,t)
t = att Ant(r,t) + DtvtV2 Ant(r.t)
+ Aq(r,ut,t) (11)
and the precursor equations are
.(r,t) iU fVt Antcrt) i(iA (rt)
i = 1 to 6 (12)
Solutions of time dependent eqations
Equations (10, 11, and 12) are now solved for the
transfer function, which is defined as the variation or
perturbation in the thermal neutron density, Ant(r,L.),
at position r, caused by a variation in the artificial
source, ASa(r,LJ) 6(u).
Laplace transform in time. A Laplace transforma
tion of Equations (10, 11, and 12) is now performed, which
transforms the equations from the time domain to the fre
quency domain and removes the time derivatives from the
equations. The time varying components are restricted to
bounded functions, which may be composed of sinusoidal
components. Thus the Laplace transform converges with the
real part of the Laplace variable, s = + jW, equal to
sero, or s = J W (19).
11
Using the initial condition that all the time de
pendent variables are zero at time equal zero, which
eliminates the transient solutions, Equations (10, 11,
and 12) become respectively
j uAq(r,u,) bAq(r,u, W ) .a(u) Aq(r,u,W)
EZt(u) v(u) 1 u Z t (u)
+ u 2 Aq(r.u.,U) + AS (r.W) 6(u)
D(u)
+ U ZfC(1 3G) vt Ant(r,W) 6(u)
6
+ .ki Aci(rW 6(u) (10a)
i=1
j W Ant(r,L) = atvt Ant (r,LW) + DtvtV2 Ant(r,W)
+ Aq(r,ut,LJ) (lla)
j( Aci(r,J) = pi) ,fC vt Ant('r,() Ai ACi(r,(J)
i = 1 to 6 (12a)
The independent time variable, t, is replaced by an
independent frequency variable, LJ, in all the terms.
Spatial expansion. The slowing down density, the
thermal neutron density, the precursor density, and the
artificial source are now expanded in terms of complete,
orthogonal functions that satisfy the Helmholtz equation
(19)
v2 H(rB) = 1 H(rBa)
over the volume of the system. The function H(r,B ) may
be sines and cosines for Cartesian coordinates, and for
cylindrical coordinates the function may be Bessel func
tions and sines and cosines. The requirement that the
slowing down density, thermal neutron density, and pre
cursor concentration be equal to zero at the extrapolated
boundaries of the system is satisfied by H(r,B ). The
pp
eigenvalues, B2, which correspond to the familiar buckling
in reactor analysis terminology, may be composed of one
component for each space dimension; for example
B2 = 82 2 2 (14)
P nl + 8n2 + 3 (14)
in general, or
B = [n T ][T]2"' l"
p 1 2a J b 2 [ 2c J
nl, n2, n3 all odd (14a)
for a rectangular parallelpiped.
Performing the expansions, the four frequency depen
dent variables become
Aq(r,u,J) = Fp(u,J) H(rBp)
p
Ant(rJ) = h p(W) H(r.p)
p
Ac (r,W) = Z ip(W) H(r,B )
p
ASa(r,LJ) = Z ,(W) H(r.B_) (15)
p
The bar symbolizes the expansion coefficients.
To solve Equations (10a, 11a, and 12a) for the trans
fer function, it is necessary that the spatial distribu
tion of the artificial neutron source be known. Therefore,
9p may be found by using the orthogonality property of
H(rBp) (19).
fASa(r,W) H(r.B,) d(Vol) = f p(U) H(r, B )
Vol Vol p
(*) H(r,Bp) d(Vol)
p= (W) p (16)
since (19)
fH(r,Bp) H(r,B ) d(Vol)= A
Vol
B = B
p p
= 0
B B (17)
A is a constant dependent on p only.
Substituting Equation (15) into Equations (10a, lla,
and 12a), and using Equation (13) where applicable, the
slowing down equation becomes
.H(r,Bp) [q(uU) i + Eat(u)
tLg (u) v(u) + ^tZ(u)
D(u) B2 q (u,W ) 6 6
+ vUZf(1 vtp( ) + p(W)) 6(u)] = 0 (18)
the diffusion equation becomes
H(r.B) [ip( jW) iW + Zatt + DtvtBP1
and the precursor equations become
ZH(r,BpU)[ p {i + + ,i U)f Ev'tp())] = 0
p
i = 1 to 6 (20)
By use of Equation (13), the spatial divergence term
has been replaced by BP H(r,Bp) from the Helmholtz equa
tion. Equation (18) is the only one still containing a
derivative.
Since the sums in Equations (18, 19, and 20) must
equal zero independently of r, each coefficient of
H(r,Bp) must be equal to zero (19). Thus, the terms en
closed by the square brackets can be set equal to zero
individually.
Slowing down solution. The slowing down equation has
now been reduced (from Equation (18)) to
b (u, =  (uu + t
L uEt (u) v (u) t (u)
+ ~t + (LW) + Ef C( ,0) vtRp( )
6
+ Z6ip(wJ &(u) (21)
=1 u a 0
where the bracket multiplied by 6(u) is the initial con
dition on p (u,LJ).
The solution to this equation, subject to the boundary
conditions, is easily found and is
u du
p(u,LJ) = qp(oLJ) exp 3J L) f
I o g t(u) v(u)
U (u) du uD(u) du 1
Sa u) 2 j (22)
The first integral in the exponential function can be
identified as the average time required to slow down from
zero lethargy to lethargy u. This time is defined as
IT(u), with the slowing down lifetime to thermal lethargy
defined as
NI(ut) = (23)
The exponential function raised to the power repre
sented by the second integral is known in reactor analysis
as the resonance escape probability P(u) (18).
The third integral is defined in reactor analysis as
the Fermi age, T(u), (18). The Fermi age to thermal
lethargy is defined as
T(ut) = T (23a)
Thus, Equation (22) becomes
p(u.JW) = qp(O,J) P(u) exp[jUWI (u) r(u) '] (24)
The slowing down density at zero lethargy, qp(0,OJ), may
be found from the initial condition
Wp(0,L ) = pL(W) + UC f C(l f) vt p(W)
6
+ Ziip (W) (25)
i=1
Precursor terms. A relation between the precursors
and the thermal neutron density is provided by Equation
(20)
i(l ) = (26)
jG) + + i
The number of unknown variables in the basic set of equa
tions may be reduced by substituting this relation into
Equation (25), obtaining
vt j6 
qp(O.U) = 3p(LJ) + Evl/+1 np(t27J)
(27)
17
For convenience, Equation (27) may be simplified by
defining
c 1 + (28)
i=liw +
Slowing down density terms. Substitution of Equation
(27) into Equation (24), then Equation (24) into Equation
(19) further reduces the basic equations by eliminating the
plowing down density, qp(u,(), resulting in
p(LW)[jLJ + zatVt + DtvtB] + [(
+ EZf vtnp(W) (] P(u exp[JJJ TB ] = 0 (29)
Solving for n (LJ), and using the following defini
tions (18)
L2 D
Eatv
UEfP(ut) (30)
k =E (30)
kzat
for the square of the thermal diffusion length, thermal
lifetime in an infinite medium, and multiplication factor
in an infinite medium, respectively, gives
1p(LJ) P(ut)
p ) = (1 L + jP) exp(j T + TB) /
(31)
General transfer function
An expression for the thermal neutron density may be
constructed using Equations (15 and 31).
Ant(rJ) = P(ut) s I c (j) H(r,B,)
(a + L2 2 + jWs) exp(jiWt
P
+ TBp) k k a
(32)
P (j) is found using Equation (16) and Ac is defined by
Equation (28).
This is the general transfer function, showing the
perturbation in the neutron density at location r caused
by a source A Sa(r,L)).
General point source transfer function
The transfer function can be obtained in a more ex
plicit form if the artificial neutron source is a point
source located at ro. Then, from Equation (16)
p(W) Ap = fASa(rW) 6(r ro) H(r.B_) d(Vol)
Vol
= ASa(ro,'U) H(ro, Bp) (33)
The correct form of the Dirac delta function corresponding
to the geometry of interest must be used.
Therefore, Equation (32) becomes
Ant (r,U)
ant (rJ) = G(r, ro'U)
Asa (r 0,)
A1a =zN H(ro,B ) H(r,B,)
Nut)'es Z P [(1 + L2B + jU ) exp(jUr
pp
+ 72) km ] (34)
Thus, the perturbation of the thermal neutron density at r,
caused by a fast neutron source at ro, is equal to an in
finite sum. Each term of the sum is composed of the pro
duct of the orthogonal eigenfunctions, divided by the
characteristic equation for the Fermi age, diffusion
theory model used in this study. The characteristic equa
tion corresponds to the dispersion function of classical
physics. When set equal to zero, this function is called
the dispersion law (42). Equation (34) is also the Green's
function for this model, and could be used as a kernel in
integrating over a source occupying a finite volume (27).
Equation (34) can be written in another form, using
the more easily measured quantities
I = Ls (35)
s 1 + L2B2
k exp(7 TB (36
keff 1 + L2B2 (36)
1
which are the thermal neutron lifetime and multiplication
factor for a finite medium.
Then
G(r,rT,W) = [P(ut) i exp(TB )]
1 H(ro, B) H(r,B
L 1 +L + L2 B
+ T(B% B)] k c (37)
There are several interesting features about the
transfer function as written in Equations (34 and 37).
The denominator, which is the characteristic equation of
the system, is readily available for stability and control
studies. The transcendental equation can be solved numeri
cally for its roots, or an approximate form can be used
(see Appendix C).
The approach used above to find the expansion coef
ficients for the source may still be used even if the
source is located on one face of the assembly. The eigen
function, H(r,B ), would then be defined over the actual
medium and its mirror image, putting the source at the
center of the mathematical problem.
The quantity defined by Equation (28), Pc, is simpli
fied at the two extremes of the frequency range. As the
frequency goes to zero, Pc approaches one. As the fre
quency goes to infinity, P. approaches 1 Only in
the frequency range 0.001 cps to 0.5 cps is P3 changing
significantly. Thus, Pc may often be replaced by its lim
iting value, depending on which frequencies are of interest.
Both forms of the transfer function, Equations (34
and 37), have the disadvantage that the convergence of the
series is slow, particularly if the size of the system is
greater than the diffusion length and/or the Fermi age.
Since one of the assumptions of the Fermi age and diffusion
theories (18) is that the system is large compared to the
characteristic nuclear dimensions, many terms must be in
cluded for a numerical analysis of Equation (34) or Equa
tion (37). An alternate form of the transfer function that
converges faster as the size of the system increases is de
veloped after the next section.
Comparison with the lumped parameter transfer function
It is interesting to note that Equation (34) can be
readily converted to the lumped parameter transfer func
tion commonly used for nuclear reactors.
First, let all the quantities associated with the
Fermi age model go to zero, that is, T to 0, ut to 0, and
Tr to 0. Second, all terms in the series beyond p=l may
be ignored, or better yet, it may be assumed that the in
tensity of the artificial source is given by
Sa(r.J) = AS() H(rB1) (38)
which is the fundamental mode of the system. Then p is
zero for all p greater than one, and
Ant(r,W ) H(r,Bl)
AS ((Lu 1 + 8L2+ jW k2
H(rB) (9)
J + 1eff
If the thermal neutron density is measured at the
center of the system, where H(0,B1)=1, then Equation (39)
is identical with the lumped parameter transfer function
(5) for a subcritical system; otherwise, they differ by
a constant. As keff goes to one, the correct form for a
critical reactor is obtained
An(r, J) H(r,3 (40)
*' (40)
LS (W)) JW 1 6 iA
As i=() + Ai
Alternate form for the point source transfer function
The general transfer function may be converted to
another form that converges much faster for large systems.
The procedure followed is known as the inversion theory of
series (20), and as Poisson's summation formula (4). In
general, the method may be applied in three dimensions in
any coordinate system. However, for the specific approach
described below, it is necessary that the eigenvalue of
Equation (13) be periodic.
Specialized problem. In order to show the inversion
in some detail, the geometry of the medium is specified as
cylindrical, the same geometry as that used in the experi
ments (see Chapter IV). Since the axial eigenvalue is the
only one that is periodic in cylindrical coordinates, the
inversion is performed in one dimension only.
The artificial neutron source is located on the
centerline of the experimental assembly and approximated
by a point source. The origin of the coordinate system
is located at the source.
For the cylindrical geometry described in Chapter IV,
the eigenfunction is
m71z
H(rB) = Jo(Un) cos (41)
with
2 = B2 + B2
p n m
= R [ [2.]
m = 1 to o odd only
n = 1 to ao (42)
The variable P is the radial coordinate divided by the
maximum radius, R. The nth zero of the Jo Bessel function
is n, and c is the distance from the source to the end of
the assembly in the positive z direction, even if the source
is on one face of the cylinder. Under the assumptions
listed above, the solution is valid only for 0 cz c if
the source is not in the center of the medium.
Using Equation (16) to find A and !p, Reference (21)
gives for cylindrical geometry
24
mf mz m7T z
Ap = R2 Jo(n) Jo(Up) cos os 
LVol 2c 2c
(*) 27Kpdpdz (
= R27 J (U (n)43)
For the point source at r = 0,
p(U) = R2 [fAcS() 6() 6(s) Jo'(np,
mTmz
(*) cos 2Tpdpdz
2c
= R2A S((J) (44)
therefore,
2 AS((J)
Sj2((J) = (45)
and Equation (32) becomes
(ro, Ant (r,J) 2P(ut) Is
AS((w) c
S(.) (VnP) cos a sin2
=l l J (UJn) [(1 + L2B2 Ji s) exp(jLJI
+ "Bp) k.c ] (46)
where sin2(T ) has been included in the numerator so that
m may take all values, from one to infinity.
25
Inversion procedure. The first step is to replace Bm
in Equation (46) by a continuous variable, g2. To com
pensate for this replacement, Equation (46) must now be
multiplied by the periodic delta function
CID
S6( ) (47)
2c
q=oo 2
The sum over the index m is replaced by one half of the
integral over 9 from minus infinity to plus infinity.
Equation (46) becomes
P(u) 2 Jo(UnW )
G(r,o,L) = s ZJ(n)
n
GD
oo d5 cost sin2 c Z 6( 2)
[(1 + L 2B + L22 jJWs)exp(j1fT
+ B2 + T2) c Pc] (48)
The periodic delta function may now be replaced by its
Fourier expansion. It is shown in Appendix A that
Z 5(6 J) = fexp(j4mc) (49)
qrcoo m=oo
Assuming that the summation over m and the integration
commute, their order may be interchanged. The integral is
seen to have the form of a Fourier transform in the
variable.
Since Equation (48) is an even function of the vari
able, the cosine functions may be replaced by exp(j 0), or
vice versa. Doing this, the summation
CD
f() Z cost a sin2 c cos 4mcg (50)
m=co
may be reduced to a simpler form as shown in Appendix B.
The result is
co
f() = (I)m cos( + 2cm) (51)
m=oo
CO
S(1)1 exp[j (z + 2cm)] (52)
m=CD
Equation (48) becomes
co
P(ut) js Jo(VnD)
G(r,o,(J) =
n=1
.I m [1 Lexp[j2 (+ 2cm)] d
(*) ) (1) Im   T"  ^ 
[( + L Bn + L2 2 +J ) exp(jj
m=co co
+ TB2 + T2) k (53)
Thus, the original cosine eigenfunctions have been replaced
by positive and negative "sources" located periodically
along the z axis.
The integral in Equation (53) may be evaluated by the
residue theorem of complex variables (19), as applied to
real integrals. The main steps in performing the integration
are:
1. The real variable 4 is replaced by the complex
variable C.
2. The poles of the integrand are found, corresponding
in this case to the roots of the denominator of Equation (53)
DF = (1+ L2 L2 2 JU ) exp(jLW + cT
+ 72) 4 (54)
3. The residues are evaluated at the poles in the
upper half plane when z + 2cm is positive and at those in
the lower half plane when z + 2cm is negative. The variable
z is restricted to values between o and c.
4. The value of the integral is equal to the sum of
the residues multiplied by 27Tj.
The roots of the denominator cannot be found in terms
of elementary functions, since the denominator is a trans
cendental equation having an infinite number of complex
roots. Representing these roots by
i = JRi = i(Ai 4 jBi)
i = oo to aC (55)
allows a symbolic solution of the integral, with the small
est value of Ri corresponding to i = 0. Appendix C pre
sents a discussion of the characteristic equation, its
roots, and how they may be found.
The residue contributed by the roots of DF are
Re exp(J/r TB + TR2) exp[RRi ( + 2cm)] (56)
Res, .X+ n 212. (56)
2JRi [L + 7(1 + L uB e L2 R + jl )]
where L'Hospital's rule has been used. It is possible for
a root of DF to lie on the real axis, if keffg 1. For this
case, the residue is equal to one half the value obtained
using Equation (56).
Summing the residues, the transfer function becomes
G(r,o,U) = P(ut)fs exp(JUir) T 2(nP)
n=l
exp(TB2 + TR2) f
S R[L2+ Tr+ (1 + L2 B OL2 +j [
i=co
+ (1) {exp [R (z + 2cm)] t exp [(z 2cm)] (57)
m=1
Thus, the transfer function for a point source in a
cylindrical medium, with a detector located at r, is equal
to the product of the sum over n, the radial coordinate
index; the sum over i, the roots of the characteristic
equation (Equation (54)) in the upper half plane; and the
sum over m, the position index on the z axis.
The transfer function (Equation (57)) for systems that
are large compared to the characteristic nuclear dimensions
(T and L2) will ordinarily be adequately represented by
just a few of the possible number of terms.
Alternate form for the nonmultiolving medium transfer
function
For the case where k, is equal to zero, Equation (46)
may be easily inverted, producing a result written in terms
of the elementary functions. The problem will be restricted
29
to a cylindrical geometry containing a point source, as
discussed in the previous section.
Inversion Procedure. The procedure followed in the
previous section is applicable to the case for which koo
is equal to zero. Thus Equation (53) may be rewritten
P(ut s Q no( nP)
(r,o,U) T exp(j W eB8)
71 Z j2
n=1
O) YO ( exp( T7rE2) exp [j (z + 2cm)] d(
Sf 1 + L2B + L2 2 + U f
m=on oo S
The roots of the denominator are easily found, and are
= + (1 + L 2B + j ) (59)
L
= ((n + j Tn) (60)
where
[[(1 + L2B2 2 + 2 2 + ] (1
n 2L2 J
and
[[(1 + L2B)2 + 22 d (1 + L2 (62)
=n sn (62)
n L 2L2
Integrating around the upper half plane when z + 2cm
is positive, and the lower half plane when z + 2cm is
s3
negative, the transfer function becomes
G(r,o, ) = L(ut)2 exp[ + jUf .)
S 1) n [e( +z( n + [n)p
n=l
+ (1)m 111 exp[(2 + 2cm) ((X + j n]
m=l
+ exp[(+z 2cm) C(( + J )J (63)
Equation (63) applies for a nonmultiplying medium contain
ing a fast source located on the centerline of the assembly
and for 0 z < c.
Appendix D contains a description of the computer code
used to evaluate Equations (57 and 63), and Chapter V con
tains the theoretical results.
Statistical Theory
The statistical correlation and power spectrum equa
tions used in obtaining and processing the experimental data
are developed in this section.
If a nuclear system, represented by its spatially de
pendent transfer function, is driven by a random input (the
fast neutron source), the output is also a random function.
The statistical relations between the output and the input,
or between several outputs, may be expressed in terms of
the system's transfer function.
In this study, the cross power spectrum is calculated
from the cross correlation of two outputs, which are the
neutron density variations at two different locations in the
medium. Repeating the experiment with one output relocated
and using the equations developed in this section, the ex
perimental transfer function of the medium can be obtained.
Definitions
There are several definitions that are basic to the
statistical analysis of random functions. One that is very
important is the cross correlation function (22)
312(T) Timoo ai" fl(t) f2(t + 7) dt (64)
T
lim= T 1 c( ) f (t + T) dt (64)
T
It is seen that this is just the average over all time t of
the product of two functions, with one function delayed
relative to the other by the time T. The cross correlation
function is frequently written
21( ) = f (t) f2(t + f (65)
to signify the averaging process. For a stationary pro
cess, the time average may be replaced by an ensemble
average if desired (22).
The Fourier transform of the cross correlation func
tion is (22)
(L ) = f 1() exp(j(JT) dr (66)
cO
and is defined as the cross power spectrum density.
An important relationship in system analysis is the
convolution integral, which can be used to relate the sys
tem characteristics and the input to the output of the sys
tem (22)
00
f(t) = h(U) f (t U) dV (67)
Co
where f (t) and fi(t) are the input and output functions,
respectively, and h(U) is the impulse response function of
the system (22).
The impulse response function and the system frequency
function, H(W), of the system are related by the Fourier
transform, and they form a Fourier transform pair (22)
h(t) = fr H(( ) exp(j(t) dW
co
CO
H(W) = f h(t) exp(jCJt) dt (68)
00
For the class of functions considered here, the system
frequency function and the transfer function, defined by
the Laplace transform, are essentially the same thing,
since h(t) is equal to zero for times less than zero, and
the real part of the Laplace variable is zero.
Equation (67) may be used to show the relationship
between the cross correlation of the input and output of
a system and its impulse response function. If fl(t) and
f2(t) in Equation (64) represent the input and output sig
nals, fi(t) and fo(t), respectively, then substitution of
Equation (67) for fo(t) gives
T co
io = lsm i dt f(t) / h(U) fi(t
+ T V) d? (69)
Assuming that the two integration commute, their order may
be interchanged. Realizing that Equation (64) also defines
an auto correlation function if fl(t) and f2(t) are the
same, Equation (69) becomes
)(io ) = Jh(u) )ii(T h) d) (70)
co
This equation has the same form as Equation (67), but re
lates the statistical correlation functions rather than the
actual input and output.
The Fourier transformation of Equation (70) may be
performed by multiplying both sides of the equation by
exp(j Lt) dT/27( and integrating over all time T. Using
Equation (68) for H(W) gives
(l (J) = H(LJ) )ii(L1) (71)
Equation (71) relates the system frequency function
and the auto power spectrum of the input to the cross
power spectrum of the input and output of the system (22).
34
Cross power spectrum between two outputs
Referring to Figure 1, the experimental setup is re
presented by an input function, 3(t); three spatially de
pendent transfer functions between the source and detectors
1, 2, and 3; three measuring system transfer functions; and
a black box that performs the cross correlation of the in
coming functions. See Chapter IV for a description of the
experimental equipment.
Functions fl, f2, and f3 represent the neutron density
variation at each of the detectors when the variation is
caused by the source, while f4, f5, and f6 are uncorrelated
variations in the neutron density at the detectors, or noise
as far as the experiment is concerned. Functions f7, f8'
and f9 are instrument noise added to the signal as it passes
through the measuring system. Functions flO, f11 and f12
are the resultant signals that are cross correlated. From
this system, the experimental transfer function is obtained
SAD3 ) nt (r3,ro, J) (72)
Ant (r2, ro, U)
Using Equation (67), the neutron density variation at
each detector is
fDl(t) = fl(t) + f4(t)
OD
= fhl(u) S(t u) du + f4(t) (73)
00
Figure 1. Definition of functions for statistical theory
fD2 (t) = f2(t) + fg(t)
oD
= h2( v ) S(t v ) dv + f5(t) (74)
CD
fD3 (t) = f3(t) + f6(t)
co
= h3(w) S(t w) dw I f6(t) (75)
co
The signals which are cross correlated, fl0, fll. and
f12, are affected by the impulse response function of the
measuring system and are
D
f(t = f7(t) + fhml() dx [f4(t x)
OD
+ h2(vu) du (t y v u) (76)
f11(t) = f8(t) + f hm2(y) dy[fs(t y)
00
+ f h2(v) 3(t y v) dv] (77)
en
CD
+ fh3(w) S(t w) dw] (78)
co
Thus, the functions that are cross correlated are related
to the system input through two impulse response functions,
one for the nuclear system and one for the measuring system,
and to the noise present in the two systems.
Equation (64) may be used to obtain the two cross
correlation functions
lim 1 f
D1D2 TT) = T IT Ot) fll(t + T) dt (79)
T
and
T
DlD() = T o f10 (t) fl2(t + T) dt (80)
T
Substituting Equations (76 and 77) in Equation (79)
results in
T
(T) = lim f dt [f (t)
DID2 T a 2T
T
00 OD
+ hml(x) dx ff4(t x) + f h(u) du S(t x u)j
oo00 CD
(') [f(t + T) + f hm2(y) dy f5(t + y)
co
co
+ fh2(v) dv S(t + T y v)
00
Performing the multiplication, separating terms, and using
the definition of the cross correlation function (Equation
(65)), gives
DlD2( T) = f7(t) f8(t +
00
+ hml(x)
00
(D
+ hm(Y)
03
00
+ hml(x)
00
fhm2()
OD
co
+ f l()
9
c l
dx f4(t') f8(t'+ + x)
dy f7(t) f5(t t + Y)
ix f hl(u) du S(t') f8(tI+ T+ x + u)
00
dy f h2(v) dv f7(t) 3(t + y V)
OD
ODD
_00
00 C
x f hm2 (Y) dy fh2(v) dv
0o CI
(.) f 4(t') SWt + r +. x y v)
co aD
dx Jh.2(Y) dy f hl(u) d.
00 IM
(.1 )ItI) f 5(W + T +l x y ul
OD CD 00 00
+ fhml(x) dx f hm2(y) dy f hl(u) du f h2(v) dv
CO o D0 co
(*) S(t') S(t'+ + x y + u v) (82)
Now, if f4, f5, f6, f7, fg, and fg, are uncorrelated noise
functions, and satisfy (22)
CD
f(t) dt = 0 (83)
o0
then the cross correlations in the first eight terms of
Equation (82) are zero. This illustrates one of the advan
tages of the cross correlation function, the ability to
eliminate the uncorrelated noise from the final answer.
Equation (82) is now
(1D2(Ti = hml(x) dx hm2(y) dy hl(u) du
OD OD
oo
(*) h2(v) dv 0gss(7 + x y + u v) (84)
co
The cross power spectrum density of D1DD2(T) may be
obtained by using Equation (66). Multiplying both sides of
Equation (84) by exp(j UT) dT/27T and integrating over
T gives
D1lD2( =7) D2 exp(jTLJ) dT (85)
00
Using the variable
/ = T+ x y +u v (86)
in Equation (85) gives
DlD2 (L) = hml(x) exp(j(Jx) dx
oo
CD
( _)fi hm2() ep(JLJy) dy f hl(u) exp(JCJu) du
o co
(C (D
(*) h,(,) exp(jJv) dv f P (FL) exp(jLWJ) dA/
oo 2 M (87)
Using Equations (66 and 68), Equation (87) becomes
mD1D2(W) = Hml() H2 ) 1() H2() ss 88)
The cross power spectrum of two detectors depends on the
transfer functions of the two measuring systems, the transfer
function of the nuclear system, and the auto power spectrum
of the source.
Moore (38) has independently developed a general equa
tion for the auto power spectra of a noise field that is
similar to Equation (88). Moore, however, ignores the ef
fects of the measuring system.
Repeating the same development as that above for
DID3(WM ) gives
DlD3(L)) = H l(C) H ((1) H*(W) H S() ()([W) (89)
If Equation (89) is now divided by Equation (88), the result
is
41
DlD3(W) H_ (W)) H3 (GJ) H (W) H (W) 4)ts(W)
DID2(W) 4 1(LW ) Hm2(L) H*((J) H2() ss((J)
(90)
It is clear, if the third measuring system has the same
transfer function as the second, and if ss(WJ) is the
same for both cross power spectrum measurements, that
Equation (90) reduces to
DD ) H 32(W)) (91)
TD1D2(U) H2(LJ)
where
G(r3,r, U)
H32 () ,
G(r2, ro,W)
SAnt(r3,ro, (92)
Ant (r2, roJ)
the same as Equation (72).
Equation (91) indicates that the transfer function of
an unknown system may be accurately determined by the ratio
of the cross power spectra between detectors.
If DI1D2(7T) and (D1D3(T ) can be measured at the
same time, there is no restriction on the source behavior
or on the measuring system. In actual practice, it is
highly desirable to have the band width of the measuring
system wider than the band width of the unknown system, but
this is not critical when using the above approach.
If the two cross correlations cannot be performed at
the same time, it is necessary to use a pseudorandom source
to insure that it repeats the same input for both measure
ments. The frequency content of the source should be wide
enough to insure that the entire band width of the unknown
system is excited.
It is interesting to note that if the band width of
the measuring system is wider than the band width of the un
known system, and if the source is a white noise over the
band width of the unknown system, both 5Is(WJ) and H (jW)
are effectively constant. Then Equation (88) becomes
D1D(W ) = = l. H1(W) H2(U) Ks
= K' JH()2 H 21(J() (93)
Any phase shift in (D1D2(W() is caused only by H21(j ),
while the attenuation of tD1D2(J) depends on Hl(W)H(L) .
Even if the transfer function of the measuring system is not
flat, nor the source white, the phase shift in (lDD2(L/) is
still caused only by H21(W).
Since the above assumptions about the measuring system
and source may not be valid, it is usually better to use
Equation (91), even with the requirement of an additional
measurement.
Auto power spectrum of one output
The auto power spectrum of Detector 1 is of interest.
If ND1D2(W) and DD3LDS(W) are measured at two different
times, DlD1((W ) may be checked to insure that the fre
quency content of the source was the same for both measure
ments. If ss(WJ) is different, (DID1(LJ) may be used as
a reference in comparing DIPD2(GJ) and (DID3((W).
The relation between PDDl11(() and (ss(W) may be
obtained using the same procedure as used in the previous
section. The definition (from Equation (64)) for the auto
correlation is
T
D1DI(T) T i f (t) f 10(t + T) dt (94)
T
Substituting Equation (76) for fl0(t), or better yet, using
Equation (82) and replacing all the 5, 8, and 2 subscripts
by 4, 7, and 1, respectively, gives 0 DIDl(T). If all the
cross correlation functions between the various noise func
tions and the source are equal to zero (22), then
DIDl(T) = 077(T + fhml(x) dx hml(y) dy
oo aD
(*) 044(T + x y) + hml(x) dx f hm(y) dy
(*) h (u) du f hl(v) dv 5ss(T + x y + u v
(95)
To obtain the auto power spectrum, both sides of Equation
(95) are multiplied by exp(jUlT) dT/27T and integrated
over T (Equation (66)). Then, using the same reasoning
presented in the previous section (Equations (85, 86, 87,
and 88)),
44
D1Dl ) = 77 J IH ml (W 2 44(W)
+ (Hl(L)2 IH1(CJ)2 (W 9 6)
If the noise inputs are white, their power spectra are con
stant (22). Thus
ND1D1(W) = K7 + H(J)( 2 K44
+ (h(W )2 1 l(W )12 css(W (97)
So, even if ,ss(W) cannot be checked directly, DDIDl(W)
will be the same for each run if 5ss(W ) is also the same.
CHAPTER III
EXPERIMENTAL METHOD AND ANALYSIS
The analysis, procedures, and considerations nec
essary for the experimental verification of the spatially
dependent transfer function equations developed in Chap
ter II are presented and discussed in this chapter. 'The
method used to obtain the experimental data is discussed
first, followed by a description of the data analysis.
The experimental equipment is described in Chapter IV.
Experimental Method
The experimental data was obtained using the method
outlined in the statistical theory section, in Figure 1
and in the development of Equations (88 and 90).
The flow of information through the experimental
system is shown in Figure 2. The shift register gen
erates a pseudorandom binary signal, which is used to
turn the neutron generator on and off, producing the in
put signal. The input signal travels through the nuclear
system to the neutron detectors. The neutron detection
system is a pulse system which produces an output pulse
of constant height for each neutron detected. The pulse
rate is measured by a fast response count rate circuit
o 4J
0
0
'41
to 4
41
41
00
11
G)t
'0 0
4
o d
0 941
41 2 d 0
u41 I to
Ir
"(D<
4J)C
H C)
d0 0
m
C 14
CO ~.4. C)
14 C 61
IC2 4I
4 0 ~t,
Ud
0"~"
47
which produces an analog voltage proportional to the neu
tron density at the detector. This signal is conditioned
by a wideband band pass filter to remove the do compon
ent and the components higher in frequency than one half
the sampling rate of the data acquisition system. The
conditioned signal is sampled by the data acquisition sys
tem, and the samples are stored on magnetic tape in digi
tal form. The data is processed by an IBM 709 computer
at the University of Florida Computing Center, using the
procedure discussed in the second section of this chapter.
Input signal
Equations (89 and 90) show that almost any input
signal may be used to measure the experimental transfer
function, as long as it has the desired band width. 'ihe
input actually used is discussed below.
The input to the nuclear system is controlled by a
predetermined pseudorandom binary signal which turns the
neutron source on and off. The signal is generated by an
eight stage shift register, with the correct feedback cir
cuitry for producing a maximum length sequence of 255 A t
intervals.
'The maximum length sequence generated by a shift
register has a special auto correlation function, which
is shown in Figure 3. The integration required to cal
culate the auto correlation function (Equation (64)) is
performed over an integral number of periods, T1, of the
shift register. The function 05s (T) has a constant
0 5 10 15
TIME, n
20
= t / At
25 30
0 5 10 15 20 25 30
DELAY TIME, n = T / A
Figure 3. Typical maximum length sequence and its auto
correlation function
15
10
5
H
o 0
5
tie Ti 
negative component of 1/N magnitude, N being the number
of intervals in the sequence. With every TI, the period
of the sequence, the shape of ss(T) approaches a delta
function. This characteristic of 0ss(7) has been used
(16) to measure the impulse response of a reactor by cross
correlation of the input and the output. If 5ss(T U)
in Equation (70) is replaced by a delta function, the cross
correlation becomes
co
0o(7) = h(t) 6r( t) dt
OD
h(T) At (98)
if the settling time of h(') is less than T1. The im
pulse response is obtained as a function of the delay
time.
This particular property of the input signal is not
used directly in this study for several reasons. First,
the determination of the spatially dependent transfer
function requires accurate measurements to show the phase
shift and attenuation of the neutron disturbance as it
passes the detectors. Thus, the approximation of ss( T)
by a delta function is not an accurate one for this work.
Second, the system being measured, represented by
h(T), must have a band width narrower than 0ss(T) if
Equation (98) is to be a good approximation of Equation
(70). Another way of stating this is that the time con
stant of the system must be less than the period of the
input sequence, usually at least five times less.
50
Third, h(T) should not change significantly over one
At interval, the average width of the triangle on the
Oss(7) curve. This means that the time constant of the
system must be several times larger than At.
The auto power spectrum of the source, ss (W )
(Figure 4), has several interesting features. If Oss(T
is treated as an periodic function, which is valid if the
settling time of the system is less than the period of the
input sequence TI, the spectrum is continuous and is (using
Equation (66))
2 I1 t 2
277 Li/\t
(99)
However, if the settling time of the system is greater
than T1, ss(T) is periodic, and the auto power spectrum
is discrete (using the periodic Fourier expansion, Equa
tibn (A2))
1 sin lAt 2
ss(nl) = 2.
N ( 21At (100)
The fundamental frequency, W1, corresponds to T1. The
locus of the discrete spectrum components has the same
shape as Equation (99), the continuous case. The half
power frequency is approximately 0.44/At for both cases.
It is clear that the period of the input sequence T1 should
be larger than the settling time of the system if at all
51
C
n
C
Ln
0
l4
0
Ci
C
cia
HE
I I I
Cr
C C, D C~ C
P8TULO c f~la.TnadO~'~ i .O
possible, so that the input spectrum will appear continuous
to the system being investigated.
Sampling theory
The perturbations in the neutron density caused by
the input signal and uncorrelated noise are presented in
analog form by the measuring system and sampled at a high
rate by the data acquisition system.
Since the analog signal is sampled a finite number of
times, it cannot be completely reconstructed. In terms of
frequency, the reconstructed signal cannot have a frequency
content higher than one half the sampling rate. On the
other hand, if the analog signal has frequencies greater
than one half the sampling rate, they will contribute false
information on the amplitudes of lower frequencies through
the process known as aliasing (28). Thus, the sampling rate
should be at least twice as high as the bandpass of the un
known system to obtain the maximum amount of information.
If this condition is not met, because of limitations in the
sampling rate or lack of interest concerning the high fre
quency response, the input to the sampling equipment must be
filtered to eliminate the frequency components above one
half the sampling rate.
If it is desired to process data already recorded, but
only the low frequency components are of interest, it is
possible to skip samples, using only every nth sample. Now,
however, it might be necessary to numerically filter the
data to remove any high frequency components present above
one half the new, or effective, sampling rate.
The variance of the sample mean can be used to esti
mate the minimum number of samples needed to have the cor
rect function average. Lee (22) shows
S= 1 o loll
Olt , Or (101)
where ( is the variance of the sample mean, (f is the
variance of the function being sampled, and n is the number
of samples. Thus, as a large number of samples is taken,
? approaches zero. The error in may be taken as" TVn.
Data Analysis
The data stored on magnetic tape by the data acquisi
tion system corresponds to fl(t) and f2(t), the two output
functions, in Equation (64). The equations used to numeri
cally process this data and compare the experimental and
theoretical spatially dependent transfer functions are dis
cussed in this section.
Experimental correlation functions
The equations developed in the statistical theory sec
tion cannot be used directly in a practical experiment, since
the data is processed digitally, and the integration times
must be finite.
Integration time errors. Equation (79) may be approx
imated by
T
Dl('r T f fD (tl fll(t + T) dt (102)
o
for a finite integration time T.
The integration time necessary to obtain a reasonably
accurate answer for DIDo2(T) has been investigated by oth
ers. Balcomb (16) treated the cross correlator as a band
pass filter, and showed that the "improvement factor" is at
least
IF = (SIN) ut (103)
(S/N)i VI
in
IF is defined as the ratio of the signal to noise ratio at
the input to the same ratio at the output of the cross cor
relator, T is the integration time, and L is the settling
time of the system. The above derivation is based on a
cross correlator designed to utilize Equation (98). By
making T > L, the uncorrelated noise is averaged over a
longer period than the correlated signal. Since the un
correlated noise is assumed to satisfy Equation (83), its
contribution to the answer is decreased with greater T.
While Equation (103) defines an improvement factor, it
gives no information on the amount of noise present before
the signal enters the cross correlator.
Since the input signal discussed earlier is a periodic
pseudorandom function, no additional information is gained
about the system by integrating for times longer than one
period of the input. However, the S/N ratio may be sub
stantially improved by averaging the uncorrelated noise over
several periods.
Rajagopal (17) approached the error problem by con
sidering the ratio of the cross correlation function for an
infinite T to the square root of its variance for a finite
T. Considering a simple lag filter as the unknown system
with a white noise input, the signal to noise ratio from the
cross correlator, as defined in Reference (17), is
s 0o exp(a T)
= (104)
0[ 0 (7,T T +Vexp(aT)]
for T =7. Thus, as in Equation (103), the SI/ is pro
portional to the square root of the integration time T.
As the delay time 7 approaches the time constant of the
lag system, 1/a, the integration time T becomes very large
for what seems to be a reasonable error limit. For example,
for S/N equal to 100, T equals 105 seconds for aT = 0.625.
For S/N equal to 2, T becomes a reasonable 43 seconds. Thus,
even though the cross correlation function is still quite
large (exp(0.625) = 0.535), the error, as defined in Equa
tion (104), can also be quite large for reasonable integra
tion times. Equation (104) is specialized, however. For
inputs that have a finite band width, the error will be
much less than that calculated by Equation (104).
Stern (23) presents a detailed analysis of three
sources of error for the cross correlation function as re
presented by Equation (70). This analysis is applicable to
the cross correlation calculations performed herein.
Defining the relative error as (23)
IR o0,, 7o'] (105)
Pio( To)
Stern (f.) shows that, for TI/N < : ToC C TR,
2 TB r TRN 2
R2 = + R 2 + + (106)
2T 2TP1?T nT 2TT77 11 TR]
TR is the time constant of the unknown system, and T1 is
the period of the pseudorandom input signal. The number
of At intervals in the input signal is N, 7 is the duty
cycle of the input (0.5), and i is the average counting
rate. The average source strength is So, and
L zp \O
A =[797 41] (107)
where 1 is the number of neutrons emitted per fission and
p is the reactivity of the nuclear system.
The terms on the right in Equation (106) are due to
the finite integration time T, the finite counting rate,
and the natural neutron population fluctuation, or the un
correlated noise (23). Equation (106) may be used to esti
mate the required integration time T for any value of R.
As an example a medium of light water with the fol
lowing typical values may be considered.
T = 2 x 104sec 71 = 0.5 P = 0
? = 5 x 103 counts/seo So = 5 x 107 neu/sec
T1 = 0.1 sec N = 255
Then
2 104 5.2 x 10"5 2.04 x 104
S=  +
T T T
3.56 x 104/T (108)
Therefore, for an error of one per cent, the integration
time T is 3.56 seconds.
This is in sharp contrast to the 105 seconds obtained
from Equation (104) for T/TR equal to 0.625. However,
there are several differences between the two approaches,
preventing them from being compared directly. Rajagopal
(17) used an input having an infinitely wide spectrum,
while Stern (23) used an input having a finite band width.
Rajagopal considered only the finite integration time
error, for variable 7, with a simple lag system. Stern
considered three error sources, but for a fixed To.
Probably the greatest difference is that Stern con
sidered a periodic pseudorandom source. As shown in
Reference (25), the variance of the correlator output is
smaller if a periodic pseudorandom binary input signal is
used, rather than a purely random input. The variance is
reduced even more if the integration time is equal to an
integral number of periods of the input. Thus it would
appear that the purely random input is not the best input
signal for experimental work. The infinitely wide band
width adds unnecessary noise, requiring greater integration
times to reduce the error.
It is clear that the error analysis of cross cor
relation functions is not an exact procedure. However,
consideration of the discussion presented above, parti
cularly that of Stern (23), enables a reasonable estimate
of the error to be made.
Digital analysis. The actual calculation of the cross
correlation function was done numerically, using the Uni
versity of Florida IBM 709 computer. The sampled output
functions from the unknown system were read into the com
puter and the cross correlation function calculated at
discrete delay times by (from Equation (102)) (31)
I
fDlD2(nAT) E A fl0(iAt) f11(iAt + nAr)
n = 0 to N
I
A AZ) 11(A f10(iAt + nA 7)
i=1
n = 0 to N (109)
The integration time T is IAt, the delay time is nAT ,
and A is a normalizing factor. The delay time interval,
A, is equal to At and is determined by the sampling
rate of the data acquisition system. N is usually much
less than I. The functions a D1D2(T) and pDID3(T) are
calculated the same way, using the appropriate output func
tions in Equation (109).
Experimental Power Spectra
The power spectra are calculated by performing a
numerical Fourier transformation of the correlation
59
functions (from Equation (66)). Balcomb (16) investigated
the effects of several interpolation schemes on the Laplace
transform of a function having the form f(nAx) = exp(anAx).
The result was compared with the known transform of exp(ax).
His conclusion was that a linear interpolation was as ac
curate as higher order polynomial interpolations, and was
much superior to no interpolation of f(nAx) at all.
Since 0D1DD2(T ) was expected to have a functional
form
CD
f(7) = A exp(aJ71) (110)
i=l
it was decided to use a linear interpolation of b(1 ).
Therefore, the numerical form of the Fourier transform
becomes
D1D2(W ) fn+ {( n p +
7n)] exp(jLWT) dT (111)
where n = DlD2( Tn) and = 7n+1 n = constant.
Performing the integration and simplifying the sum results
in
DlD2(U) exp(j U '.NT [ A N 2]
Ni
+ exp( Tn) n+l W2 n1
n=N+l
+ exp(j TN N N (112)
10DID3(L) and 0 DL1(J) are calculated by substituting
the appropriate 0 (7) in Equation (112). A Fortran sub
routine TRNS was programmed to calculate the power spectrum
using Equation (112). It was included as part of the gen
eral program (31) for processing the experimental data.
See Appendix D for a description of TRNS.
Experimental transfer function
The transfer function is calculated using a modified
form of Equation (91). Although Equation (91) is derived
using three outputs from the system (Figure 1), the actual
experimental setup (Figure 2) contains only two detection
systems. By making two separate recordings, with a detector
first in the D2 position and then in the D3 position,
H23(LJ) may be calculated using D0101(W) as a normalizing
factor.
DIDS3()
H,23 () = [DDI(] Run 2 (91a)
^DID 1())JRun I
The numerical errors that accumulate as the data is
processed cause H23(W) to become less accurate as the
power spectra are attenuated at the higher frequencies.
However, as noted with Equation (93), the phase shift of
the system's transfer function can be found directly from
DDID2() or 4D1DS(W).
Calculation of dynamic parameters
The determination of the dynamic reactor parameters
was aided by the use of a Fortran code, "General Least
Squares Program", originally written at Oak Ridge National
Laboratory by Bussing and extensively modified since by
Cockrell (24) for use by the Nuclear Engineering Depart
ment, University of Florida.
Input to the program is the experimental data and the
theoretical function that is to be fitted. The program is
informed which parameters in the theoretical function are
to be varied to minimize the variance between the data and
the function.
As the theoretical function, Equation (57) was incor
porated into a main subroutine CALC, and several minor
subroutines, CRT, GUEST, GUESS, TERM, ERROR, and FINAL.
The computer codes are discussed in detail in Appendix D.
It is not possible to determine the five parameters
s,, ST2, L2' T, and koo simultaneously. The dependence
of the transfer function on them is such that no single
combination of Is, Pr, L2, T, and koo exists for mini
mizing the variance. However, if two or three of the
parameters are known from other sources, the remaining
ones may be estimated using the General Least Squares
program.
CHAPTER IV
EXPERIMENTAL EQUIPMENT
The equipment used to generate the input signal and
to measure the output of the subcritical assembly is
described in this chapter. A schematic of the measuring
system is shown in Figure 2.
The experiments using the light water subcritical
assembly were conducted at a different location than the
experiments using the heavy water subcritical assembly.
Thus, some items of equipment were different for the two
experiments.
Input System
The input system contains two major components: the
shift register and the neutron generator.
Shift Register
The pseudorandom binary input signal is generated
by a shift register with "modulo 2" feedback. By using
the proper feedback logic, the maximum length sequence
can be obtained. This sequence has the special auto
correlation function described in Chapter III. Peterson
(26) discusses the correct feedback circuitry for obtain
ing the maximum length sequence for shift registers
62
containing up to 34 stages. The number of intervals, N,
contained in a maximum length sequence generated by a
shift register having m stages is
N = 2m 1 (113)
References (23. 25) describe in some detail the charac
teristics of a shift register used to generate a pseudo
random signal.
The shift register used in the present study has eight
stages, producing a sequence having 255 intervals before
repeating. A schematic of the shift register is shown in
Figure 5. The At time interval is controlled by a free
running multivibrator, whose speed may be varied from 1 cps
to 10 KC.
Neutron Generator
A Norelco neutron generator was used for the experi
ments with light water, and a Texas Nuclear Corporation
neutron generator was used for the heavy water experi
ments.
Norelco neutron generator. The Norelco neutron gen
erator (32) produces neutrons of 14 MEV energy by the dt
reaction, with an average source strength of 108 neutrons
per second.
The neutron source tube contains a replenisher sys
tem which emits hydrogen isotopes when it is heated by an
electric current. The isotopes are ionized in a Penning
ion source and moved to the one stage accelerating system
64
4.
:j
00
4J
'a
41
dc
al
.0) ;;
If
0)
by the ion source voltage. The accelerating system imparts
enough energy to the ions (125 KV) to overcome the electro
static potential of the deuterium and tritium nuclei al
ready embedded in the titanium target, thus causing the
nuclear reaction. The size of the source tube is 3" x 20".
The neutron generator may be pulsed by controlling
the ion source voltage, with a maximum pulsing rate of
about 3 KC at a 50 per cent duty cycle. The signal gen
erated by the shift register is used to control the ion
source voltage through an intermediate voltage matching
amplifier.
A high voltage transformer that supplies the accel
erating voltage is located close to the source tube. A
control rack, containing the electronic equipment and ion
source voltage transformers, is connected to the source
tube by a 30foot cable. This length allows the rack
and operator to be located a safe distance from the neutron
generator.
Texas Nuclear Corporation neutron generator. The TNC
neutron generator produces neutrons by either the dt or
dd reactions. The dd reaction was used for the heavy
water experiments.
While the TNC neutron generator is physically differ
ent from the Norelco neutron generator, the sequence of
events required to produce neutrons is the same.
The replenisher system is a pressurized tank con
taining deuterium gas. The deuterium is ionizing in a
Penning ion source and is moved to the acceleration system
by the ion source voltage. The high energy ions leave the
acceleration system and strike the deuterium, titanium
target, producing neutrons.
The shift register controls the pulsing of the neu
tron generator through an intermediate circuit. The random
width pulse from the shift register is differentiated, and
on and off pulses are supplied to the neutron generator
to control the beam deflection plates. The plates are
located between the ion source and the acceleration system.
A higher pulsing rate is possible with this generator than
with the Norelco generator.
Nuclear Systems
A light water subcritical assembly and a heavy raterr
subcritical assembly, each containing various amounts of
natural uranium, were used as the test media.
Light Water Subcritical Assembly
The light water subcritical assembly consists of a
right cylinder tank containing water, an aluminum support
grid, and natural uranium fuel rods. The main features
of the tank are shown in Figure 6.
The neutron generator was installed inside a 3 1/4
inch horizontal pipe in the bottom of the tank, such that
the neutron source was on the longitudinal axis of the
assembly. With no fuel elements, the effective radius of
the tank is 52 cm. When the uranium is added, the core is
Water i Lattice
Reflector
Nleut ron    
Source
Tube
 38.9
Figure 6. Light water subcritical assembly
surrounded by a water reflector. For a full loading of
fuel, the effective radius of the core is 37 cm. The core
is immediately above the neutron generator and extends to
the surface of the water, 142.9 cm from the source.
The support grid holds the uranium in a triangular
lattice 4 cm on a side. Two hundred four fuel elements
make up a full load, with one diagonal row of elements
missing to leave room for the neutron detectors. The
natural uranium slugs are housed inside 1 inch aluminum
tubing, which is held upright by the grid.
The volume fractions for the full load are: FH2O
0.42133, FAL= 0.16058, and FUR= 0.41809.
Heavy Water Subcritical Assembly
The heavy water subcritical assembly (Figure 7) is
essentially the same as the light water subcritical as
sembly. The main differences are: a. The target of the
neutron generator is placed inside a 2 1/2 inch pipe
through the center of the tank. b. The lattice spacing
is 12 cm, resulting in volume fractions of: FD20= 0.9357,
FAL= 0.01784, and FUR= 0.04646, for a full loading of 55
fuel elements, c. The lattice occupies the entire volume
of the D20 tank. Thus, the effective radius is 59 cm for
both the nonmultiplying and multiplying cases. d. The
water level is maintained at 60.8 cm above the source.
Neutron
Source
Tube
Figure 7. Heavy water subcritical assembly
Output System
The variation of the neutron density is sensed by the
output system. The major components in this system are the
detection and amplification equipment, the count rate cir
cuits, the band pass filters, and the data acquisition
system.
Neutron Detectors
The neutron detectors were placed inside aluminum
conduit to protect them from the water. The conduit was
Lshaped, supporting the detectors in a horizontal posi
tion above the neutron generator. The height of the in
dividual detectors could be easily changed by vertical
adjustment of the conduit.
The detectors have a 12 inch active length, filled
with He3 to 4 atmospheres of pressure.
Pulse Amplifiers
For the light water experiment, preamplifiers and
linear amplifiers manufactured by Hamner Electronics
Inc. were used. Preamplifiers and linear amplifiers
made by Nuclear Chicago were used for the heavy water
experiment. Both systems supplied constant height, one
microsecond long voltage pulses to the count rate circuits
at a rate proportional to the neutron density at the de
tectors.
Count Rate Circuit
The count rate circuit was specially designed for
this study. The specifications were: a. a fast response
circuit, whose output would duplicate the neutron density
variation as closely as possible; b. little ripple, or
decay of the output voltage between input pulses; and
c. adequate output voltage (2 to 10 volts).
The count rate circuit has five counting ranges,
which are shown with their measured break frequencies in
Table 1. Appendix E contains detailed information on
the count rate circuit, including the design criteria,
a circuit diagram, and a plot of the transfer function of
the circuit.
TABLE 1
Count Rate Meter
Range Counts/sec. 3 db point
1 100K to 250K 5400 cps
2 40K to 100K 2000 ops
3 20K to 50K 1000 cps
4 10K to 25K 600 cps
5 4K to 10K 350 cps
FM Tape Recorder
A MinneapolisHoneywell fourchannel tape recorder
was used for storing the output from the count rate cir
cuits during the heavy water experiments and for trans
porting the information to the data acquisition system.
The tape recorder has a band width of 0 to 10 KC
at a recording speed of 30 inches per second; a speed of
1 7/8 inches per second is also available. An internal
calibration circuit and compensation channel allows the
original signal to be reproduced with less than one per
cent error.
Band Pass Filters
KrohnHite ultralow frequency band pass filters
were used to condition the analog signal from the count
rate circuit. The high pass filter was set at .02 cps
to remove the low frequency components from the signal.
The low pass filter was set to remove the frequencies
higher than one half the sampling rate of the data
acquisition system.
Data Acquisition System
The main components of the data acquisition system
are shown in Figure 8. The multiplexer samples two, four,
six, or eight input analog signals at a rate of 15,000
samples per second in the low density tape mode. The
twelvebit binary analog to digital converter accepts an
input from each channel in turn of 4.096 to 4.096 volts.
The format unit calculates and includes in the final bi
nary word a lateral odd parity bit. The unit also cal
culates the longitudinal parity check sum for inclusion
at the end of each record, and performs other control
functions.
73
The resultant computer words are written on tape
by the recorder unit, in a form compatible with the IBM
709 computer. The system can also produce a high density
tape for the IB'I 7090 computer, at a sampling rate of
28,571 samples per second.
0
c t7
r, co
P.
4J
C)
4J
.4 0
.
t: )
4
I C)
C) A
C .4.
0 1
U 0
CHAPTER V
RESULTS
The results from the theoretical and experimental
study of the spatially dependent transfer function of a
nuclear system are presented in this chapter.
Theoretical Results
The computer code SPAT, described in Appendix D,
was used to calculate the transfer function between a
thermal neutron detector and a point source of fast neu
trons (Equation (57)) for four cases, presented in Figures
9 through 13. The same geometry described in Chapter II
was used, with values of c = 150 cm, and R = 50 cm. The
values of the nuclear parameters used in the calculations
are listed in Table 2.
The series in Equation (57) were terminated when the
magnitude of additional terms was less than one tenth per
cent of the sum of the absolute values of the proceeding
terms. The curves labeled L in the figures represent the
lumped parameter transfer function for the nuclear sys
tems.
TABLE 2
Nuclear Parameters Used for the Theoretical Results
Figure L2, cm2 T, cm2 fs,/sec frt/.sec ko
9, H20a 8.12 30 200 10 0
10, H20a 8.12 30 200 10 0
11, D20a 13,700 110 40,000 46 0
12, H20 + Urb 1.61 40 200 10 0.923
13, D20 + Urc 87.2 133 300 40 1.15
a. References (2C 18)
b. References (18. 29)
c. References (301 33. 3)
The variation of the phase shift with frequency and
with distance from the source is of interest. At 100 cm
from the source, the phase shift in light water (Figure 9,
Part D, and Figure 10) is 720 degrees at 800 cps and is
1440 degrees at 2400 cps. In other words, four complete
cycles of the 2400 cps neutron wave are located between
the source and the detector. As the detector is moved
closer to the source, the transfer function experiences
a maximum negative phase shift (Figure 9, Part C). A
point is reached where the phase shift is always positive
(Figure 9, Part B). The data for heavy water, Figure 11,
also shows a positive phase shift. The multiplying media
data, Figures 12 and 13, shows a great reduction with fre
quency in the amount of the phase shift.
1 20 cm
2 50 cm 3
40 3 100 cm
L Lumped parameter
60 
80
0 .L
90 
180
270 1
360
S 100 200 500 1000 2000 5000 1001
FREQUENCY, cpa
Figure 9, a and b. Theoretical transfer function for light
water
90 I
a>0) \2
180 No. z
1 20 cm
2 50 em
3 100 cm
W L Lumped parameter
270
360
0
I L
3
So 1 I I
180 
270 
360
50 100 200 500 1000 2000 5000 10(
FREQUENCY, cps
Figure 9, c and d. Theoretical transfer function for light
water
79
40
,
o3
4J
Io
CC
0 0 w
CD C
o
) 0o
& o0
0
co
*
0
R4
20 2
No. _
S40 1 20 cm
2 50 cm
C 3 100 cm
L Lumped parameter
60
80
h I
0'
S180 
a 
S270 
I
360
20 50 100 200 500 1000 2000
FREQUENCY, cps
Figure 11, a and b. Theoretical transfer function for heavy
fater
81
90
CL
90 2 . .
S1 20 cm
2 50 cm
180 3 100 cm
L Lumped parameter
S270 
360 
90
S\3
180 
270 
360
II II I
20 50 100 200 500 1000 2000
FREQUENCY, cps
Figure 11, c and d. Theoretical transfer function for heavy
water
0
1
20 
40 3
No. z
60 1 20 cm
2 50 cm
3 100 cm
L Lumped parameter
80 I 1
0
9G 2
S180
S270
360
50 100 200 500 1000 2000 5000 10000
FREQUENCY, cps
Figure 12, a and b. Theoretical transfer function for a
lijht watp subcritical assembly
83
0
20
2
1 20 cm 3
40 2 50 cm
S 100 cm
SL Lumped parameter
60 
80 I I
0
90 
S180 
S270 
360
I I I
20 50 100 200 500 1000 2000
FREQUENCY, cps
Figure 13, a and b. Theoretical transfer function for a
heavy water subcritical assembly
The behavior of the phase shift clearly illustrates
the effect of frequency on the wave length and wave velo
city. A discussion of the dependence of the wave velo
city and wave length of a thermal neutron disturbance on
frequency may be found in References (5, 6).
A comparison of the spatially dependent transfer
function, SDTF, and the lumped parameter transfer function,
LPTF, shows, in general, that the lumped parameter model
is not an adequate representation of the nuclear system
(see Figures 9 through 13). However, it appears that the
two models can give similar results if the output detector
is located at a specific distance from the source.
As the multiplication factor approaches one, the two
models give essentially the same attenuation and phase
shift for z 20 cm. As ka, approaches zero, the heavy
water case can still be represented by the LPTF for
z = 20 to 30 cm and for frequencies less than 1000 cps,
but there is a significant difference between the SDTF
and the LPTF for the light water case.
McInerney (40) has shown that diffusion theory gives
the correct spatial dependence for fast neutrons when the
reciprocal diffusion length, 1/L, is approximately equal
to the total cross section, Et. McInerney's development
helps explain why the LPTF, which is derived from diffusion
theory, cannot be used with light water but can be used
in a light water moderated reactor at specific distances
from the source. As uranium is added to the water, 1/L
approaches Et, fulfilling the requirement stated above.
In using Equation (57) to determine the parameters
of a nuclear system from experimental data, it is of in
terest to know the number of terms that must be used to
adequately fit the data. An indication of the effects
of changes in z, media, and frequency on the number of
terms was obtained by considering the four cases listed
in Table 2. The results presented in Table 3 were ob
tained by assuming the average experimental error to be
four per cent and using this percentage as the accuracy
criterion in SPAT.
As expected, only the first term of Equation (57)
is needed when the detector is at a sufficient distance
from the source. The higher spatial modes are attenuated
more than the fundamental mode and can be neglected at
low frequencies. At higher frequencies, though, the
higher spatial harmonics are not attenuated much faster
than the fundamental mode. They must be included in
the equation even for large z. Figure 31 illustrates
part of the reason for this effect.
The fundamental root, ,o, which is much smaller than
the other roots at a frequency of 0 cps, is only slightly
smaller at 2000 cps. At higher frequencies, at least
three terms in the i series must be included in the equa
tion, and possibly several terms from the n series (see
Equation (57)). The spatial modes contributed by the m
series are usually quickly attenuated, so that only the
fundamental m term need be included.
TABLE 3
Number of Terms Required for Four Per Cent Error
in Equation (57)
Case Freauency. cos
0 10 100 1000
Figure 9, H20
z = 20 cm 6 7 7 7
a = 50 cm 4 4 4 5
z = 100 cm 2 3 3 3
Figure 11, D20
z = 20 cm 3 3 3 5
z = 50 cm 2 2 2 3
z = 100 cm 1 1 1 2
Figure 12, H20 + Ur
z = 20 cm 3 3 3 7
z = 50 cm 2 2 2 3
z = 100 cm 1 1 1 2
Figure 13, D20 + Ur
z = 20 cm 3 3 6 9
z = 50 cm 1 1 2 2
z = 100 cm 1 1 1 2
As the multiplication factor approaches one, the
fundamental mode becomes dominant, making fewer terms
necessary for most systems, even close to the source.
Unfortunately, it is more difficult to make measure
ments in that portion of the system which can be represented
by only one term of the equation, since there are fewer
neutrons present and the counting rate is lower.
Experimental Results
The methods used to check the accuracy of the equip
ment and the computer calculations are discussed first.
The experimental data follows.
Calibration
Equation (90) shows that the effects of the measuring
system on the input signals are cancelled when the ratio
of the cross power spectra between two detectors is cal
culated. Thus a calibration of the measuring system is
not necessary in the usual sense of the word. However,
it is important to know the amount of noise that is added
to the input signal by the equipment and to determine the
extent of the numerical errors accumulated in the data
during the computer processing.
The computer code was checked by correlating and
transforming known mathematical functions. Once the pro
gram was debugged, the error between the calculated and
the theoretical results (correlations and power spectra)
was less than one per cent.
The next step was to sample the output from the
shift register with the data acquisition system, and pro
cess the data in the computer. The auto correlation of
the recorded function exhibited the characteristic trian
gular spike at 7 = 0 (see Figure 3), but it did not
remain at a constant 1/N (0.00392) level for the other
delay times. Instead, the auto correlation function
fluctuated between values of 0.005 and 0.02. This fluc
tuation is caused partly by numerical error in the calcu
lation and partly by the discrete sampling interval of the
input voltage. The majority of the variation, though,
is caused by noise added to the signal by the electronic
equipment. The equipment involved in this test was the
shift register, an intermediate Schmidt trigger, and the
data acquisition system.
Figure 14 shows the auto power spectrum of the shift
register and the theoretical curve for the idealized sig
nal. The approximately 1 db gain in the 20 to 200 cps
frequency range is caused by the low frequency fluctuations
noted in the auto correlation function. If the results
presented in Figure 14 are used as a calibration test, it
appears that 13 to 15 db is the approximate limit for
accurate calculation of the power spectrum.
There are several other items to consider when judging
the accuracy of the experimental results. The number of
computational steps a particular sampled function has gone
through affects the accuracy of the function. As the
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