• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Theory
 Experimental method and analys...
 Experimental equipment
 Results
 Conclusion
 Appendices
 Bibliography
 Biographical sketch














Title: Spatially dependent transfer function of nuclear systems by cross correlation methods
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 Material Information
Title: Spatially dependent transfer function of nuclear systems by cross correlation methods
Physical Description: xi, 161 leaves : illus. ; 28 cm.
Language: English
Creator: Kylstra, Chester D., 1936-
Publication Date: 1963
Copyright Date: 1963
 Subjects
Subject: Nuclear engineering   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida, 1963.
Bibliography: Bibliography: leaves 157-160.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097957
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000568380
oclc - 13650214
notis - ACZ5111

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
    Theory
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
    Experimental method and analysis
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
    Experimental equipment
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
    Results
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 96a
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 101a
        Page 102
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        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 113a
    Conclusion
        Page 114
        Page 115
        Page 116
        Page 117
    Appendices
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
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        Page 128
        Page 129
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        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
        Page 156
    Bibliography
        Page 157
        Page 158
        Page 159
        Page 160
    Biographical sketch
        Page 161
        Page 162
Full Text











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 eight-stage, maximum length sequence,
pseudo-random 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 time-dependent Fermi age

and diffusion theories for a single-region, 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 non-multiplying 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 pseudo-random 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 non-multiplying 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 non-multiplying

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(1-P ) 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)
qr-coo 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 + TR-2) exp[-RRi ( + 2cm)] (56)
Res, .X+ n 2---1---2-. (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 non-multiolving 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 re-written


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 non-multiplying 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) T-imoo 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 pseudo-random 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 pseudo-random 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 4-J


-0






0
'41
to 4









41
4-1









00
1-1
G)t


'0 0
















4-
o d










0 941
41 2 d 0
u41 I to
I-r









"(D<
4J)C

-H C)
d0 0



m

C 14
CO ~.4. C)
14 C -61
IC2 4-I
-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 wide-band 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 pseudo-random 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


ti-e 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







l-4



0










Ci









C









c-ia
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 band-pass 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

pseudo-random 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 pseudo-random 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 10-4sec 71 = 0.5 P = 0
? = 5 x 103 counts/seo So = 5 x 107 neu/sec
T1 = 0.1 sec N = 255






Then


2 10-4 5.2 x 10"5 2.04 x 10-4
S= ---- +
T T T

3.56 x 10-4/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 pseudo-random source. As shown in
Reference (25), the variance of the correlator output is
smaller if a periodic pseudo-random 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]






N-i
+ exp(- Tn) n+l W2 n-1
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 pseudo-random 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 d-t

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 30-foot 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 d-t or

d-d reactions. The d-d 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 non-multiplying 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

L-shaped, 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 Minneapolis-Honeywell four-channel 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
Krohn-Hite ultra-low 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

twelve-bit 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.

















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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



4-J
Io











CC
0 0 w
CD C












o
) 0o
& o0












0

co












*-






0
R-4











-20 2



No. _
S-40 1 20 cm
2 50 cm
C 3 100 cm
L Lumped parameter

-60



-80






h I
0'




S-180 -
a- -






S-270 -
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



S-270 -




-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




S-180





S-270




-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 -



S-180 -




S-270 -




-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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