A Theoretical and experimental study of neutron wave propagation in moderating media

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A Theoretical and experimental study of neutron wave propagation in moderating media
Alternate Title:
Neutron wave propagation in moderating media
Booth, Ray Sturgis, 1938-
Place of Publication:
Gainesville, Fla.
University of Florida
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Copyright Date:
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xv, 178 leaves : illus. ; 28 cm.


Subjects / Keywords:
Amplitude ( jstor )
Analyzers ( jstor )
Buckling ( jstor )
Data analysis ( jstor )
Eigenfunctions ( jstor )
Eigenvalues ( jstor )
Graphite ( jstor )
Neutrons ( jstor )
Thermalization ( jstor )
Time dependence ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF ( lcsh )
Neutrons -- Scattering ( lcsh )
Nuclear Engineering Sciences thesis Ph. D ( lcsh )
Nuclear reactors ( lcsh )
bibliography ( marcgt )
non-fiction ( marcgt )


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Thesis - University of Florida.
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April, 1965

Copyrinht by
Ray Sturgis Booth
Department of :Iuclear Engineering!
University ~of Floridla
19 65


The author wishes to express his sincere appreciation

to the members of his supervisory committee for their

assistance in the preparation of this dissertation. Special

recognition is due Dr. Rafael B. Perez for his continual

guidance and encouragement. His advice and profound in-

sight have been invaluable throughout all phases of this


The author is grateful to Dr. M. J. Ohanian for his

major contribution to the theoretical aspects of the

dissertation through stimulating discussions and the contri-

bution of useful computer codes. Important in the experi-

mental work has been the generous aid of Robert H. Hartley

and his careful and exacting experimental technique during

the acquisition of mutually valuable data.

Other persons whose contributions to this work are

gratefully recognized include Robert B. Razminas for the

sharing of his DIAGONAL-CORRECTOR code, the staff of the

University of Florida Computing Center for valuable

assistance in connection with many codes, George Fogle for

the contribution of his technical know-how to the experi-

ment, and Barbara Gyles, who took more than a typist's

interest in turning out the final manuscript.


The author is deeply indebted to his wife, Judy,

for her devoted encouragement, understanding, and patience.

Her valuable help in the preparation of the manuscript is

also appreciated.







. . . ii

. . . vi


. . . . .



. . . . xiii

* *





. . .1






. 11

. 58

. 99

EXPERIMENT ..............










. .

. .

. .

LIST OF REFERENCES ....... ........

APPENDIX . . . . .


. .










The Energy Points Used in the Discrete
Energy Representation and the Calculated
Values of the Diffusion Coefficient at
These Points . . . .

Parameters Used in the Numerical
Calculations . . . .

Ligenvalues of the Space and Energy
Dependent Boltzmann Operator Using the
SUMMIT Kernel and the MASSMK Kernel ....

The Experimental and Theoretical Values for
the p(n) Coefficients ...........

The Experimental and Theoretical Values
for the Q(n) Coefficients .........

The Static and Dynamic Parameters .....







Parameters as Determined from Various
Die-Away Experiments Compared with the
Results of This Work .........

. .134



1. Flow Diagram of Several of the Computer
Codes Used in the Theory ...... ..

2. Several Eigenfunctions of the Space and
Energy Dependent Boltzmann Operator
Using the SUMMIT Kernel. .........

3. Several Eigenfunctions of the Space and
Energy Dependent Boltzmann Operator
Using the MASSMK Kernel. .........

4. Real Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Four Energy Harmonics as Calculated
Using the MASSMK Kernel. .........

5. Imaginary Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Four Energy Harmonics as Calculated
Using the MASSMK Kernel..........

6. Real Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Three Energy Harmonics as Calculated
Using the SUMMIT Kernel..... ....

7. Imaginary Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Three Energy Hlarmonics as Calculated
Using the SUMMIT Kernel. .........

8. Real Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Two Space Hlarmonics as Calculated
Using the MASSMK Kernel.

9. Imaginary Component of the Complex Inverse
Relaxation Length for the Fundamental and
First Two Space Harmonics as Calculated
Using the MASSMK Kernel. .........



. Q






Figure Page

10. Arrangement of the Experiment and Details
of the Thermalizing Tank (from Refer-
ence 14). . * * 60

11. A Block Diagram of the Electronic Equip-
ment Used in the Wave Experiment. ...... 66

12. Time Sequence of Electrical Signals Associ-
ated with the Electronic Equipment Used in
the Wave Experiment . 68

13. A Block Diagram of Electronic Equipment
Used in the Pulse Propagation Experiment
(from Reference 14) .. . 71

14. Thermal Wave of the 200 ops Wave Experiment
for Several z Positions .. .... . 72

15. Thermal Pulse for the Pulse Propagation
Experiment for Several z Positions (from
Reference 14) . . * 74

16. Amplitude of the 400 cps Wave as Obtained
from the First Harmonic of the 200 ops Wave
Data and the Fundamental of the 400 ops
Wave Data .. ... .. .. * 82

17. Phase of the 400 ops Wave as Obtained from the
First Harmonic of the 200 ops Wave Data and
Fundamental of the 400 ops Wave Data. .. 83

18. Combined Amplitude Curves for Several
Frequencies . . . 93

19. Combined Phase Curves for Several of the
Frequencies .. .. .. * * 94

20. Comparison of the Theoretically Derived a
to the Experimental Results ......... 100

21. Comparison of the Theoretically Derived 5
to the Experimental Results ......... 101

22. Comparison of the Theoretically Derived 2aS
to the Experimental Results ........ 106



Figure Page

23. Comparison of the Theoretically Derived
22 52 to the Experimental Results for
Frequencies up to 1,400 cps. .. ... 107

24. Comparison of the Theoretically Derived
a2 52 to the Experimental Results for
Frequencies up to 400 cps. ... .. .. 108

25. The Ptn) as a Function of the Maximum Fre-
quency Included in the Least-Squares Fit 112

26. The Experimentally Determined Real Part of
1 versus Distance from the Source. .. .. 115

27. The Experimentally Determined Imaginary
Part of t versus Distance from the Source. 116

28. The Experimentally Determined a' and 5' as
a Function of Frequency. .. .. .. .. 117

29. The First Frequency Derivatives of 2aS and
a2 2 VerSUS . . . 118

30. A Pictorial Representation of the Die-Away
and Neutron Wave Experiments .. .. .. 126


8: Transverse buckling of the fundamental
space mode (cm-2)

B2 Square of the fundamental complex inverse
relaxation length for BJ=0. Also used as
a general eigenvalue in one part of the
theory (cm-2)

Bn Square of the complex inverse relaxation
length for B:=0 associated with the higher
energy modes (cm-2.)

B K) The Kh coefficient of the Maclaurin's
series expansion of 82 (Cm-2sec )

C, Diffusion cooling coefficient (cm sec'l)

D(E) Energy dependent diffusion coefficient

Do Energy a ve ra ged diffusion coefficient

E rleutron energy (ev)

7, Coefficient of the B* term in the power
series expansion of X in power of B2
(cm sec-1)

Lmn Matrix elements of the energy expansion
in eigenfunctions of the P operator
(cm-2 Sec)

M(E) Maxwellian distribution (number density)

P Energy and space dependent Boltzmann
P The coefficients of the power series fit
of the experimental data (cm' seen)

0(n) Coefficients of the power series expansion
of p2 (Cm2)n-1

LIST OF SYMOBLS (continued)

S, The nth ratio associated with the
fundamental eigenvalue

S(K) The Kt coefficient in the Maclaurin's
n series of Sn (secK)

t Time (sec)

u(I-J) Heaviside unit step function

vo Reference neutron speed for thermal
neutrons (2.198xl05 cm sec'l)

Wn(q,v) The nth Fourier width of the measured
pulse in the pulse propagation experiment

x,y,z Space variables of the rectangular
coordinate system (cm)

ao v Eac (sec-1)

6mn Kronecker delta

n Variable used in the discrete energy
representation (n = (E/KT)1/2/(1+(E/KT)1/2)

X Decay constant measured in the die-
away experiment or calculated from the
Q(nI with ao0 (sec-l)

1W Decay constant calculated from the Q(n)
with so0 (sec'l)

v,f Frequency (cps)

02 Squared fundamental complex inverse
relaxation length. Also used in one
section of the theory as the general
eigenvalue (cm'3)

noThe nth eigenvalue of the P operator

Zao Energy averaged macroscopic absorption
cross section (cm'l)

LIST OF SYMBOLS (continued)

Energy dependent macroscopic scattering
cross section (cm'l)

Energy dependent macroscopic absorption
cross section (cm-l)

Energy dependent second Legendre moment
of the angular dependent scattering cross

Energy dependent differential scattering
cross section (cm-l/ev)

Energy dependent macroscopic total
collision cross section (cm-1)

Neutron flux (n cm-2sec' eV-1)

Spatial eigenfunction used in the
space expansion

Summetrized neutron flux

The nth eigenfunction of the P Operator

The measured pulse of the pulse propa-
gation experiment

The nth Fourier moment of ~(q,t)

Frequency (rad/sec)

Laplacian operator

Laplacian operator for the x and y




e, ,(x,y)




lt (q,v)



Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy



Ray Sturg~is Booth

April, 1965

Chairman: Dr. Rafael B. Perez
Major Department: Nuclear Engineering

The space, energy, and time or frequency dependent

Boltzmann equation in the diffusion approximation has been

used within the framework of neutron thermalization theory

to predict thermal neutron wave propagation in a moderat-

ing medium. This theory was then compared with the

experiment conducted on a graphite pile.

In the theoretical section the energy dependence

was expanded in the space-energy eigenfunctions of the

Boltzmann operator, and the transverse-space dependence

was expanded in the orthogonal solutions to the Helmholtz

equation with homogeneous boundary conditions. It was

then shown that each of these space and energy modes thus

formed propagate with characteristic complex inverse


relaxation lengths. The perturbation method developed by

Perez was used to expand these complex eigenvalues in a

Maclaurin's series of the frequency variable. In cases

where this series did not rapidly converge a direct

solution of the complex secular determinant was obtained.

Two scattering kernels were selected for implementation

into the theory, the MASSMK kernel and the SUMMIT kernel

based primarily upon derivations by Parks. The numeri-

cal calculations of the theory were performed on an

IBM-709 digital computer and founded upon the previous

work of Ohanian.

In the experimental section an accurate technique

for conducting neutron wave experiments in multiplying or

diffusing media was presented along with the results of

the graphite experiment. The excitation of neutron waves,

both by modulated and pulsed sources, and the electronic

systems associated with their detection were discussed.

Special attention was given to the analysis of data for

which several computer codes were developed.

In the comparison of theory and experiment it was

shown that the neutron wave technique allows for the

separation of diffusion and thermalization effects, thereby

yielding a more exacting comparison of theoretical and

experimental results. Polynomial fits of functions formed


from the experimental results yielded coefficients to be

compared with the theoretically derived terms of the

perturbation expansion. The results of the thermal pulse

propagation experiment were also analyzed by a method

suggested by Moore, and confirmation of his predictions

was established.

The diffusion and thermalization parameters

determined from the data analysis were then compared with

results of several die-away experiments and with the

theory. The complementary relationships between the

neutron wave and die-away experiments were also presented.

The diffusion and thermalization parameters

determined in this work for graphite of density 1.60 gm/cm3

were vCa = (91.'1.)sec-1, vD = (2.16'.01)xl05 cm2sec-1,

C = (39._'2.)x105 cmksec-1, F = (12 .'2.)x107 cm6sec-1,



Placement of a modulated or pulsed source of thermal

neutrons at one boundary of a nuclear assembly creates a

disturbance which can be analyzed in terms of wave components

propagating from this source with frequency dependent phase

shift and attenuation factors. The importance of the

neutron wave experiment is based upon the fact that these

two independently measurable quantities are strongly dependent

upon the diffusion and thermalization properties of the

assembly. Thus accurate determinations of physical con-

stants and severe tests of theoretical models are possible

through an experimental determination of the frequency

dependent dispersive properties (dispersion law) of the


Several experimental methods for investigation of

moderating and multiplicative assemblies are now in exten-

sive use. They include, among others, exponential experiments,

"poisoning" experiments, and the die-away technique. It is

interesting to note that the neutron wave technique, which

is the subject of this dissertation, is related to each of

the three methods mentioned above. Basically, the neutron

wave experiment is an extension of the exponential experiment

in that the constant source of the latter is replaced by a

modulated or pulsed source. One then measures the complex

inverse relaxation length, p, of the neutron waves as they

travel through the assembly. An extrapolation to zero fre-

quency of the real part of p yields the same value, Bm a

obtained by the exponential experiment. Using a modulated

source, the material buckling is obtained through an extrap-

olation of a curve, which is nearly constant near the zero

frequency intercept, rather than by measuring a single

point on the curve. Also the imaginary part of p is still

available for determining another parameter or comparing

with the theory. Still another possibility with the neutron

wave experiment is the appearance of resonance phenomena

arising from the splitting of each spatial mode into various

energy modes which propagate with different complex inverse

relaxation lengths.

Increasing the frequency of modulation in a neutron

wave experiment can be considered equivalent to "poisoning"

the medium with a 1/v poison. Therefore, determining the

bucklings and relaxation lengths at several poison concen-

trations in the "poisoning" experiment and finding the complex

relaxation length as a function of frequency in the wave

experiment are complementary measurements. Moreover, the

frequency-dependent, energy spectrum of the neutron1 wave

experiment bears a close resemblance to the energy spectrum

associated with the "poisoning" experiment as a function of

the poison concentration. In the wave experiment, however,

solids as well as liquids can be homogeneously "poisoned."

The neutron wave technique and the die-away experi-

ment also bear a close resemblance. The modulus of the

complex inverse relaxation length can be thought of as a

negative buckling. Its relationship to w/v, the "poisoning"

term, is quite similar to the relationship between the geo-

metric buckling and the decay constant as measured by the

die-away technique. In the die-away experiment one determines

X, the lowest time-energy eigenvalue of the Boltzmann operator,

as a function of the geometric buckling. This method, though

quite effective for the determination of diffusion parameters,

has severe limitations when applied to the determination of

thermalization parameters unless the time-dependent, energy

spectrum is measured. At high bucklings, thermalization

effects cause the curve of X versus buckling to deviate

slightly from the linear relationship predicted by diffusion

theory. This second-order effect occurs precisely at the

range of buckling where the least confidence can be given

to the experimental results; for it is in this range of

buckling that the extrapolation distance is most important,

transport corrections become significant, and the energy

dependence takes the longest time to stabilize. Also an

asymptotic decay is never attained in some crystalline modera-

tors because of the so-called "trapped neutrons" which decay

in time much slower than those associated with the

fundamental eigenvalue. Thermalization effects are of

first-order significance in the neutron wave experiment.

Diffusion theory predicts that the real part of p2 should

be independent of frequency, whereas it is in fact nearly

linearly related to the square of the frequency. The

measurement of thermalization parameters via the neutron

wave technique can be accomplished in large systems where

angular dependence and extrapolation distances are of

comparatively minor importance.

The purpose of this dissertation is to present both

theoretical and experimental investigations of the propa-

gation of neutron waves in a moderating medium. The

Boltzmann equation in the diffusion approximation is used

as the theoretical model, and its energy dependence is

expanded in the space-energy eigenfunctions of the Boltzmann

operator. A perturbation method is used to expand the

square of the complex inverse relaxation length in a

Maclaurin's series of the jw variable and in the x variable

which is related to the frequency of excitation by the

expression w = xex. In the experimental section an accurate

technique for conducting neutron wave experiments in multi-

plying or diffusing media is presented along with the

experiment in graphite. Special attention is then given

to the analysis of the data thereby obtained and to its

comparison with the theory.

* Underlined numbers in parentheses refer to the List
of References.



The propagation of neutron waves caused by

oscillating a thermal-neutron absorber in a chain-reacting

pile was investigated by Weinberg and Schweinler (1)."

Weinberg and Wigner (2) derived equations for the atten-

uation distance, wave length, and propagation velocity as

a function of frequency for waves excited by a localized

source in an infinite diffusing or multiplying medium.

It was shown that the diffusion coefficient and the prompt

material buckling could be determined by measuring the

propagation characteristics of these neutron waves.

Experimental measurements of diffusion parameters in

graphite and heavy water were performed by Raievski and

Horowitz (3) by using a mechanical source and a frequency

range which varied from 50 ops to 500 ops. Their results

were in good agreement with theoretical predictions, but

no attempt was made to extend the theory to include therma-

lization effects.

Preliminary experiments in graphite and in water-

moderated subcritical assemblies were done by Uhrig (4,5).

He used one-group diffusion theory to derive equations

relating the prompt material buckling and diffusion co-

efficient to the neutron wave's phase shift and attenuation

and compared his results to experiments conducted on the

graphite, subcritical reactor at lowa State University (6).

A mechanical source was used in these experiments; and the

basic unit in the detection system was a dual, coincidence-

anticoincidence switching circuit which directed the detector

pulses to four slave scalers so that each scaler counted

during a different phase interval of the modulated source

(7). It was determined that the waves do follow the ten-

dency predicted by theory; however, the wave velocity was

in all cases higher than expected, with the deviation becoming

greater as the oscillation frequency increased.

The theoretical work of Uhrig was extended by Booth

(8) to a two-group diffusion theory model. It was shown

that when two energy groups are considered two thermal waves

propagate in the fundamental spatial mode, that the relative

magnitude of the Fermi age compared with the square of the

diffusion length is of predominant importance in determining

the propagating characteristics of the two waves, that K,

is a second-order effect with respect to these same charac-

teristics, and that the material buckling can be measured

from experiments conducted in the central region of the

assembly. Several methods for the detection of neutron

waves were also presented by Booth.

Perez and Ulhrig (9) investigated the propagation

of neutron waves in a moderating media with a one-group

diffusion model and a thermalization model. In the

thermalization model the energy dependence was expanded

in the Associated Laguerre Polynomials of order one. The

complex inverse relaxation length was expressed in a power

series of the ratio w/vD, and it was shown that the co-

efficients of this expansion are functions of the thermal-

ization and diffusion properties of the moderator. The

authors demonstrated that the neutron wave technique affords

a complementary method to the "poisoning" technique and that

the difficulties encountered in the die-away experiment

because of the trap effect should not be present in a

neutron wave experiment.

The diffusion and cooling coefficients of graphite

were determined experimentally by Perez, Booth, and Hartley

(10) by the neutron wave technique. The value of DO agreed

with previous measurements, but the value of CO was a factor

of two larger than that obtained by Starr and Price (11)

using the die-away technique. The propagation of neutron

waves in a subcritical assembly using multigroup diffusion

theory was investigated by Perez, Booth, Denning, and

Hartley (12). It was shown that for "n" groups each spatial

mode splits into "n"' damped waves of the same frequency but

with different phase shift and attenuation factors, and that

the interaction of these waves presents the possibility for

measurable resonance effects which are quite sensitive to

the lattice parameters of the system. An attempt was made

to demonstrate thermalization effects in moderators by the

reflection of neutron waves by Denning, Booth, and Perez

(13). In the experiment a sinusoidally modulated source

of neutrons was placed at one face of a 26.7 cm. long

graphite block with transverse buckling of 4.0 x 10-3 cm-2

which preceded a 62 cm. long heavy-water region of the same

transverse area. Theoretical investigations indicated that

if any reflection occurred at the graphite, heavy-water inter-

face it would have to be caused by energy modes which pro-

pagate only when the energy dependence of the neutrons differs

from a Maxwellian distribution. This was qualitatively con-

firmed by the experiment.

The propagation of thermal pulses was demonstrated by

H-artley (14) and Miley and Doshi (15). Hartley produced

thermal pulses of about one millisecond width by use of a

neutron generator and thermalizing tank, recorded the pulses

at various distances from the source, and compared the

results in the time domain to a one-group diffusion theory

model. An extension of the theory to that of Perez and

Uhrig met with severe difficulties when the inversion to

the time domain of the complicated Laplace-transformed flux

was encountered. The disagreement between the experiment

and the one-group diffusion theory showed the sensitivity

of the experiment to thermalization effects. Miley and

Doshi used the TRIGA Reactor as a source and thermal pulses

were produced of width ranging between 32 and 66 milli-

seconds. The measured asymptotic velocity of the pulse

yielded a value of Do; but because of the large pulse width

(i.e., lack of high frequency content), the more interesting

thermalization effects observed by Hartley were not detected.

Diffusion theory was compared with the experiment, and

corrections were made for the source distribution.

A method of comparing theory and experiment in the

neutron wave experiment with pulsed, thermal sources was

suggested by Moore (16) so that the inversion of the theory

to the time domain (which proved so difficult for Hartley)

would not be necessary. In essence, Moore suggested that

comparisons be made in the frequency domain by numerically

determining Fourier moments and widths of the data measured

in the time domain. Courier widths were defined and shown

to be proportional to the derivatives with respect to

frequency of the dispersion law of the medium. This theory

was shown to be especially applicable when the fundamental

solution of the dispersion law was expressed in the per-

turbation expansion introduced by Perez and Uhrig (9, 17).

Kunaish (18) used time, energy, and space dependent

neutron thermalization theory in the consistent Pl approx-

imation to develop a theory for comparison with neutron

wave and die-away experiments. The non-Maxwellian component

of the energy dependence was expanded in a Taylor series,

a heavy-gas scattering kernel was used, and the results pro-

vided an accurate description of the neutron wave experi-

ment for source frequencies up to 500 ops.



Presentation of the Theoretical Model
and the Method Used for its Solut-ion


Placement of a sinusoidally modulated or pulsed

source of thermal neutrons at one boundary of a moderating

medium generates thermal neutron waves which propagate from

the source with frequency and space dependent amplitude

and phase. The positional variation of this frequency

dependent thermal neutron flux is what one measures experi-

mentally, and it will receive the principal emphasis in

the theory which follows. Attention will be given to the

space, energy,and time or frequency dependent thermal

neutron flux in a moderating medium within the framework

of neutron thermalization theory. The main objective will

be to calculate the complex inverse relaxation length as a

function of frequency for the fundamental space and energy


The energy distribution of thermal neutrons in a

finite moderating medium deviates from a Maxwellian distri-

bution for several reasons. The preferential leakage of

higher energy neutrons from the assembly produces a

"cooling" effect upon the spectrum, and the preferential

absorption of slower neutrons causes a "hardening" effect.

The prediction of this spectrum is further complicated by

the thermal velocity of the scattering nuclei and the

chemical binding and crystalline effects in the moderator.

The probability of a scattering interaction between

neutrons and the moderator nuclei is expressed by the

differential scattering cross section Es(E',E) for energy

transfer from E' to E. It is the essential physical

parameter in the thermal region, and the prediction of

transient and asymptotic phenomena can be only as accurate

as this scattering kernel. For this reason, care has been

taken to select as accurate a scattering kernel as possible.

The more accurate scattering kernels are much too complicated

to be considered analytically; and when using them, one must

employ numerical methods.

The Boltzmann Equation in the Diffusion Approximation

To investigate the effect of the energy spectrum upon

thermal neutron wave propagation, one can use the Boltzmann

equation in the diffusion approximation. For a finite,

isotropic, homogeneous, and non-multiplying medium it is

expressed as

v at -D(E)V24(r E t)) [L (E) + Eg(E)]M(r,E,t) +

3[Et(E) Esi 3 ~





t = the time variable

r = the position vector

E = the energy variable

v = the speed variable

2 = the Laplacian operator

4-(r,E,t)drdE= the total flux of neutrons
at time t in the volume
element d? about point 3
which have energies between
E and E + dE (neutrons/cm2/

Ca(E)O(r,E,t)dE = the total number of absorptions
per unit volume about r and per
unit time at time t due to
neutrons with energy between
E and E + dE

Es(E)O(r,E,t)dE = the total number of scattering
collisions per unit volume about
r and per unit time at time t due
to neutrons with energy between E
and E + dE:

CB(E',E)4(rE',t)dE'dE the total number of scattering
collisions per unit volume about
i"and per unit time at time t
due to neutrons of initial energy
between E' and E' + dE' and of
final energy between E and E + dE

D(E) = the energy dependent diffusion
coefficient (cm.)

Within the diffusion approximation, D(E) is given by the





CtEt Cs( a ,E (3)


Csl(E)= the second Legendre moment of the
angular dependent scattering cross

TEs i II n ~~s(E0*G')d(n*G')

From the definitions of cs(E) and Es(E',E) it is evident


Cs(E) = I Zs(E,E')dE' (4)

In the experiment the source is placed near the

exposed, z=0 face of a long rectangular stack of cadmium-

covered moderator. Thus homogeneous boundary conditions can

be applied at extrapolated boundaries of the x and y variables,

the source condition is applied at z=0, and the flux can be

specified to vanish at an infinite value for the z variable

(i.e., end effects are considered negligible since measure-

ments will not be made at the far end of the pile).

Specifically, the boundary conditions that should be applied

to equation (1) for the solution of this problem are

4(Sa,y,z,E,t) = 0 (5)

$(x, Sb,z,E,t) = 0 (6)

Q(x,y,=,E,t) = 0 (7)

-D(E -- (x,y,z,E,t) =; 1/2 S(x,y,E,t) (8)

where S(x,y,E,t) is the space, energy, and time dependent

source of thermal neutrons.

Under these conditions, it will be advantageous to

expand the flux in an orthogonal set of functions in the

x and y variables. In view of the linear character of the

problem and since our main interest is focused upon the

frequency dependence of the flux, it can also be separated

into a steady part(due to the constant component of the

source) and an oscillatory part. Considering only the

oscillatory component of the solution to equation (1), we

can expand it in the form

O(r,E,t) =Re [ Z tem(x~y)q (zEI) exp(jwt)]


where 40,m(x,y) satisfies the Helmholtz equation with

homogeneous boundary conditions, which is expressed as

(V2 + B 2 ) 4 (xIy) =
x,y Itn tm (10)


B2 = the transverse buckling of the e,m
spatial mode and

Vxy the Laplacian operator for the x and y

The substitution of equation (9) into equation (1)

with the use of equation (10), multiplication of the result

by 4p,k(x~y), and integration over the x and y variables
results in

E-()Ip,k aE sc(E) + D(E) -2 3 mp,k(Z'E) +

+ I Es(E' ,E)6plk(Z'E')dE' = 0 (11)

In the experiment, one would like to measure only the

fundamental spatial mode so it is of major concern in the

theory. Since the equations for the various spatial modes

differ only by the magnitude of the transverse buckling,

one can, without serious loss in generality, remove the

subscripts. The symbols B~ and g(zE) are now understood

to represent the transverse buckling and the flux associated

with the fundamental spatial mode, respectively. It is also

convenient at this point to express equation (11) in a more

compact, operator formalism by use of the substitution*

* The author wishes to express his thanks to Dr. Frank J.
Munno for helpful discussions in connection with this

~VNE: 7/DE~


where M(E), the Maxwellian distribution, is expressed as

= C(E/kTj12




By substituting equation~S2i (1 at(2) into equation (11) and

dividing the result by vM(E)D(E), one obtains

C-2 8,E LS(E) + 92
D(E) D(E) az2


I (EE)~zE) vM(E)
w S(E,)(~

+ /

dE' 0


~-B2 + --

- ]~X(z,E) + PX(z,E) = 0



E (E',E) v'M(E')'
PX(z,E) = I dE'(dE(L=--~--

o () Z())

(E'-E)[--- + ]} x~z,E')
D(E) D(E)


Solution of the Eigenvalue Problem Associated with the
space and energy uependent doiramann Up~erator
It is instructive to consider equation (15) for the

case when the transverse leakage and frequency of excitation

are both zero (i.e., a semi-infinite plane with a constant

source). The incentive for this deviation comes from the

work of Ohanian (19) and Ohanian and Daitch (20) in their

theoretical investigation of the die-away experiment. We

shall assume that the solution to the equation thus formed

may be written as

XazE) = a Jl(E) exp[-p, z] (17)

with the understanding that the above expansion is convergent

only if the qn(E) form a complete set. Substitution of

the above expansion into equation (15) with both B~ and

w equal to zero results in

i apno n(E)exp(-pnOz Pnn(E)exp(-p oz)
n=1 n=1 (18)

If the initial assumption of completeness is correct, the

~n(E) will be linearly independent. Therefore, equation (18)
must hold separately for each value of "n" and reduces to

-no n,(E) =Pn (19)

Thu pn and 9 (E) are the eigenvalues and corresponding

eigenfunctions of the P operator. An inspection of this

operator with reference to the principle of detailed balance,

v'Cs(E',E)M(E')dE'dE = vLs(E,E')M(E)dEdE'

reveals that it is both real and symmetric. Thus its eigen-

functions must be real, satisfy the orthogonality condition,

I Q (E)9m(E)dE = 0 for n/m, (21)

and be linearly independent (21). This latter characteristic

assures that the expansion expressed by equation (17) and

the subsequent manipulations were valid.

The facility of the substitution expressed by

equation (12) and of the investigation of the semi-infinite,

zero-frequency problem now becomes evident. If one expands

the energy dependence of equation (15) in terms of the

eigenfunctions obtained above, equation (19) can be used

to exchange for the complicated P n,(E) expression its simple

equivalent -p2 9 (E). Thus the solution of equation (15) by
no n
this method is not directly complicated by the consideration

of realistic and therefore involved energy exchange kernels

since more difficult kernelS Simply result in different

eigenfunctions for use in the expansiOn. Or, expressed
differently, the complicated characteristics of the energy

exchange kernel are carried to the solution of equation (15)

by the eigenfunctions and eigenvalues obtained from the P

operator. Moreover, the homogeneous equations and secular

determinant resulting from the above mentioned expansion

will be much simpler than that obtained by using other

orthogonal functions which do not bear as close a relation-

ship to the actual solution.

The eigenfunctions and their corresponding eigen-

values also have physical significance. They are the

thermalization theory solution to the exponential experi-

ment in a semi-infinite plane. The fl(E) function is the

fundamental energy mode which is exponentially attenuated

with distance from the source, and its corresponding eigen-

value is the square of the asymptotic inverse relaxation

length. The one-group diffusion theory equivalent to this

eigenvalue is 1/L2. Thus n,(E) and pno represent, respec-

tively, the eigenfunctions and eigenvalues of the space and

energy dependent Boltzmann operator and are what one en-

counters when performing the exponential experiment in a

very large moderating medium. More important physical

significance can be assigned to these eigenvalues in the

context of this work when one recalls that the exponential

experiment can be considered a neutron wave experiment

?: This argument is based upon the assumption that the
difficulty involved in the solution to the eigenvalue
problem is not dependent upon the complexity of the
scattering kernel. This is, in general, the case since,
as explained in the second part of this chapter, a
discrete energy representation of the eigenvalue problem
is solved on a computer. Thus more involved scattering
kernels simply result in different numbers being read
as input to the code.

performed at zero frequency. From this viewpoint it becomes

apparent that the pno are the zero intercepts of the squared
complex inverse relaxation lengths plotted as a function of

frequency. A knowledge of these zero intercepts is useful
because their closeness to the fundamental eigenvalue is

quite indicative of the amount of higher-energy-mode con-
tamination predicted by the particular scattering kernel

under investigation.

Further insight into the nature of these eigen-
functions and eigenvalues can be obtained by a more detailed

investigation of the eigenvalue problem represented by

equation (19). With the P operator explicitly expressed
and some terms rearranged, equation (19) becomes

C 8 2 S#(E')d
D(E) ~"o nE ME1DE


For the special case in which Ea(E) = 0, the lowest eigen-

value is zero and its corresponding eigenfunction is

~VM(E)D(E); which is easily verified by direct substitution

and by use of the principle of detailed balance. This is
consistent with the fact that in an infinite medium with

no absorption the flux will attain an equilibrium Maxwellian

distribution in energy and a constant value in space. From

the arrangement of equation (22) it is evident that it will

be satisfied by two types of solutions:

1. P2o C Z (E)/D(E) for any value of E. This
results in discrete eigenvalues and the more common

continuous eigenfunctions as solutions.

2. pno = Zt(E)/D(E). In this case the expression
enclosed in brackets is zero. The right-hand side of

equation (22) is not necessarily zero, however, so that if

the eigenfunctions are to satisfy the equation in this

circumstance they must exhibit in part a singular behavior.

Moreover, condition (2) can be satisfied for any value of

the continuous variable E under the condition p2

[E (E)/D(E)]MIN. Thus above this limit point a continuous

eigenvalue spectrum is obtained.

Expansion of the FreQuency Dependent Eauation in
agen unctions of the pace and nery De endent
Boltzmann Operator

The energy dependence of the flux can now be developed

in terms of the eigenfunctions mentioned above. An exponential

dependence upon the z variable will be assumed so that the

solution to equation (15) can be expressed in the form

X~,E = AJl,(E)exp(-pz) (23)

*A more rigorous expression for this expansion would be

x(z,E) = IA(d) (d)(E +I A A(c)( 2) >(C)(E, 2 )dp2)exp[-pz]
n Jn o 0 0
n=1 CC (E)/D(E)]MIN

where the n(E) satisfy the eigenvalue equation, and they
have been normalized so that

S,( 90 m(E)dE = 6mn
o (24)

Substitution of equation (23) into equation (15) results


n 1 [-B+ p2 A (exp[-pz])q(EE) +

+ P(exp[-pz])P9 (E) = (25)

which, when multiplied by 9m(E) and integrated over all

energy, yields

C(p2 B2 -p2 )6 L jw]A =
I no mn mn n

m = 1,2,3,*** (26)

where the superecripts "d" and "c" represent, respectively,
the discrete and continuum sets. However, in the actual
solution a discrete energy formalism is used and calcula-
tions are performed only at a finite number of discrete
energy values. Eigenvalues in the continuum, which
actually are a function of energy, and their corresponding
eigenfunctions are obtained only at particular energy
points. Thus equation (23) correctly expresses the actual
expansion used, which is the discrete approximation to the
more rigorous and less convenient representation.

where use has been made of equations (19) and (24) and

Lmn m 9(E) [1/vD(E)] n(E)dE
o (27)

Equation (26) expresses in a compact form a set of linear,

homogeneous, algebraic equations for the modal amplitudes

An and the eigenvalues p2. One is able to solve the set

only for the relative magnitudes of the An, and the number

of eigenvalues which allow a solution will be equal to the

number of equations considered. Also there will be a

different set of ratios for each eigenvalue. The compatibility

condition for the above set of homogeneous equations yields

the secular determinant or dispersion law

S(B2 p2 )6 -jwb 6 (28)
no mn mn


B2 = p2 2 (29)

which can be used to determine the complex eigenvalues that

will allow solutions for equation (26). These eigenvalues

are the squared complex inverse relaxation lengths which

are to be compared with the experiment. It is satisfying

to note that for w=0 the secular determinant, equation (28),

reduces to a diagonal determinant yielding the p2 for the
B2 eigenValues.

Determination of the Fundamental Complex Inverse
Rel axati on Le~ngth from the Complex Homogenneous Equations

The perturbation method developed by Perez and

Uhrig (9) and systematized by Kunaish (18) will be used

to develop each of the eigenvalues and modal amplitudes

in a power series expansion of the frequency variable.

This power series representation is used because it allows

one to compare very easily various theoretical models

with one another and with the experimental results. Once

the coefficients are calculated, the eigenvalue can be

determined for any frequency within the radius of con-

vergence of the series. To initiate this expansion, the

first equation and the first term of each of the other

equations in the homogeneous set represented by equation

(26) are explicitly expressed. Considering the first "N"

equations of this set, one obtains

(B2 P10) L11hw LlnjwSn = 0

for m = 1 (30)

-Lmlj C (B2 80 mn mnjw ) Sn = 0
for m = 2,3,4,*** (31)

where each equation has been divided by Al so that the set

could be solved for the ratios

S, = A /Al n = 2,3,4,*** (32)

and for the eigenvalue B2. In this method, B2 and each

of its corresponding ratios, S are expanded in a Maclaurin's

series. Thus we let

I B(K) jwK
B2 C
K=0 K! (33)


Sn = (34)
K=0 K!


B B2
aK ju (35)


(K) K

naK je=0.. (36)

The Kth coefficient of B2 and those of each of the S, are

obtained from equations (30) and (31), respectively, after

differentiating them "K" times with respect to jw and

setting jw equal to zero.

It should be emphasized at this point that we are

presently expanding only the fundamental eigenvalue along

with its corresponding ratios. However, it will be shown

that the other N-1 eigenvalues and their ratios are obtained

in exactly the same way after rearranging the initial

equations and redefining some terms.

By letting jw=0 in equation (30) and equation (31),

it can easily be determined that




S, = 0

n = 2,3,4,***


Using the identity

A(x)B(x)] =

IK) AK(x)B(I-K)(x)




7 1!


one can differentiate equation (30) and equation (31) "I"

times to obtain at jw=0

- ull- ) 1 1
K=0 n=2


T ) (I-K)=0
K In K1 n

- 2 6
10 10

-L 6
11 II


L 6 -u(I-2) ( )(B(1)6 -L )S(I1
ml Il 1 an mn n

+ r

I j( (B(K) p2 6 )6 LS(I-K)= 0
K ~no KO an an K

-L 6

n=2 K=0

m = 2,3,L4,***


where u(I-2) is the unit-step function. By evaluating the

summation over "K" in equation (41) and rearranging terms,

one obtains


+ u(I-2) C : Ll Sn(I-1)


= 2 4
10 IO

+ L 4
11 Il


which can be used to determine each 8 if the S(I1

are known. Equation (42) can be simplified by separating

out from the rest of the series in the "K" summation the

terms for which K=0 and K=1. By performing this manipulation,
extracting the S coefficient from the term for which
K=0, and rearranging terms, an expression for the Sm

ratios is obtained


(P2 -p2 )
mo 10

N I-1
u(I-3) (K) (K)S~(I-K)
n=2 K=2
; m = 2,3,4,***

(P2 p2 )
mo 10

The summation over "K" in the above expression is extended
only to "I-1" since Sm is zero, and the step functions
are introduced because some terms were extracted from the


The procedure used for evaluating the B(I and S(I

is initiated by first determining B(1) and S(1) from

equations (43) and (44), respectively. Knowing these terms,

B()can be calculated from equation (43) and placed in

equation (lt4) in order to find the S(2). This process of

alternately using the two expressions to first calculate
BI then when all the B(j and S(J are known for

Then using the B(I in equation (33), the frequency dependent

complex inverse relaxation length for the fundamental mode
can be evaluated.

It may happen, however, that the series does not

converge very rapidly. In fact, this is the case for one

of the scattering kernels selected. For this situation, a

more convenient series can be obtained by a change of
variable of the form

jw = jxex


One can then derive a power series expansion for B2 in

terms of X. Recognizing that


andusing the procedure as explained with the w expansion,

one can obtain expressions for the B(I and the S(I for

the power series expansions in the x variable. As in the

w expansion

(0) 2
B =P
10 (47)


and all of the

are zero. For I = 1, one finds



( 9)

Sm =jLml/ pmo 10)

and, in general, for I i 2

m =2,3,4,**

N I-1

L E [K) jKLlnS~(I-K)
n=2 K=1

8 )

= jILll -


ix-x (jXX) = n

and I-1 N I-1
jI~ml KB(K)S'I-K) K)jKLmnS(I
K=1 n=2 K=1
S =
(P2 p2
mo 10

m = 2,3,4,*** (51)

In the determination of p2 as a function of frequency, both

of these power series expansions will be used.

An alternative to the series representation is to

solve the complex secular determinant directly for several

values of w. For reasons previously discussed, this method

is not as appealing as the ones derived above; and it will

be used only as a last resort, that is, for cases in which

both of the series converge slowly or not at all.

Determination of the Higher Order Complex
Inverse Relaxation Lengths

It was stated that the equations obtained for the
(I) (I)
B and the Sm were derived to evaluate only the funda-

mental p2. To calculate the other eigenvalues, one can

rearrange the equations expressed by equation (26) so that

the same expressions can be used to calculate coefficients

associated with all of the eigenvalues. To explain this

procedure, consider the case where m=3 in equation (26).

Allowing n to range from one to three, one obtains

C~2 B2 102 11 jA1 L12j 2 L13 jw3

- L21jWA1 + ~2 _: p20)0 L22jU1A2 L23jWA3 =0

-L jWA-L 2jwA2 C(p2 B 2 p2) Ljw]A = 0

Let us specify that the eigenvalue associated with the

second equation is desired (i.e., the eigenvalue whose

zero intercept is p20+ B ), If we exchange equations (53)

and (52) and rearrange terms in each of the equations, the
result is

C(p2 :2 2 :) L2jW]A2 LljWA1 L 3j.A3 = 0

-L 2jWA + Ep2 2 102 )-L jm)A L 3jwA =

-L 2j A2 LljWA1 [(pZ B2 2 ) 0, L3jWlA = 0

An inspection of these last three equations and the ones

from which they were derived indicates that both sets are

of exactly the same form since they differ only in sub-

scripts. Thus the same expressions used for the solution
of the fundamental eigenvalue are also valid for the second

if care is taken to correctly rearrange the L and p2
mn no
matrix elements and to assign subscripts to them correspond-

ing to their new positions before applying the equations.

By analogy, the arrangement of terms for calculating the

nth eigenvalue and its ratios is obtained by exchanging

the first and the nth equations and then exchanging the

first and nth terms in all of the equations.

Numerical Calculations Associated
with the 'Iheory


The derivations thus far presented are founded upon

the theoretical formalizations of Perez and 'Ihric (9) and

Ohanian and Daitch (20). The work of the former authors

indicated the method by which, beginning with the Boltzmann

equation in the diffusion approximation, one could formulate

a power series expansion for the complex inverse relaxation

length. They used Associated Laguerre Polynomials in ex-

panding the energy dependence, whereas here the eigenfunctions

of the space and energy dependent Boltzmann operator form

the orthogonal functions for this expansion. This more

elegant method has practical value since the secular deter-

minant used to determine the eigenvalues, B2, has a much

simpler form and one need not calculate the complicated

matrix elements of the scattering operator. As a by-

product of this method, one obtains all of the zero inter-

cepts of the Bn ie, n

For the determination of the B2 and their

corresponding eigenfunctions, use was made of the numerical

techniques developed by Ohanian for his calculation of the

eigenfunctions of the time and energy dependent Boltzmann

operator. In fact, after minor modification, several of

the computer codes developed by Ohanian were used in this


Evaluation of the Eigenvalues and Einenfunctions of
he Space and Energy UDeendent Boltzmann Operator
in_ the bittusiono Approximation

The Discrete Energy Formulation of the Eigenvalue

Problem -- In the discrete energy representation, equation

(15) becomes a set of "J" equations of the form

L, i) s,(E )
02 +(E.) = r K(E ,E.)" (E.)AE. 1 + (E )
noj= n (E ) D(E )

i = 1,2,3,***,J



K(EE i)
iVM(q)1D( )D(ED ) (59)

11=1 (60)

bEj depends upon the numerical integration scheme selected,
"J" is equal to the number of energy points, and the prime

on the summation indicates that the term involving

Es Ei'Ei) has been eliminated from the summation in the
interest of greater numerical accuracy (19). An inspection

of the above equations with all terms included reveals that

for i=j the term E (E.,E.) in the j summation in equation

(58) should be canceled by an identical term in the

expression .It was determined by Ohanian that
greater accuracy could be achieved by elimination of these

terms altogether before introducing equation (58) into the

computer. This is a very fortunate development since, as

explained later in this section, it was necessary to adjust

the diagonal elements of the scattering kernel calculated

by the SUMMIT code. Since the diagonal elements are not

needed in this numerical solution to the eigenvalue equation,

this modification should not have any direct influence upon

that solution.

The set of equations represented by equation (58)

can be more simply expressed as

no n -n


(J9 ) is a column vector composed of the
values of the @n eigenfunctions at
each of the ] energy points,

(bE) is a diagonal matrix containing the
weights for the numerical integration,

(P) is a square matrix representing the
P operator.

The elements of the (P) matrix are given by the expression


ijj D(E g) D(E )


We have now arrived at the equation which is to be

solved by the EIGENVALUE code. Note that the symmetry

of the P operator is preserved in the discrete representa-

tion and that since (AE) is simply a diagonal matrix to

provide weighting for the integration the numerical eigen-

value problem also contains a symmetric operator.

Evaluation of the Soatte~rin~g Kernel and F~ormulation

of the Symmetr~ized P Operator -- All of the theoretical

calculations were performed on an IBM 709 digital computer.

In essence, the implementation of the theory derived in the

preceding section was accomplished through the use of an

echelon of computer codes, beginning with one which calcu-

lated the scattering kernel and ending with the EIGENVALUE

code. Each code made use of the results of one or more of

its predecessors and, in turn, contributed to the codes
farther on in the chain. The format for the discussion in

this section can therefore be logically based upon the

flow diagram shown in Figure 1.~ In the discussion which

follows, pertinent information pertaining to the computa-

tional methods will be presented. For a complete discussion

of the workings and input-output information for the com-

puter codes used the reader is referred to the reports

submitted by the author to his committee supervisor.

One of the more important considerations in the

implementation of this numerical method was the determina-

tion of what variable to use in order to most advantageously

express the energy dependence. This problem was solved by

Ohanian (19) by the use of a variable defined as

n= (E/KT)1/2 / (1 + (E/KT)1/2) (63)

The 33 energy points, corresponding to 32 equally spaced

increments of the n variable, that he used were adopted for

this work and are listed in Table 1. Also shown in the

table are the D(E ) as obtained by the use of SUMMIT,

PINNACLE, and DIAGONAL CORRECTOR codes. For the numerical

integration encountered, except those in SUMMIT and

*This computation was done under the direct supervision of
Dr. M. J. Ohanian, with initial inspiration from Dr. R. B.
Perez. The author is grateful to Robert B. Razminas for
the use of his DIAGON~AL CORRFECTOR CODE and for his con-
tribution to the matching of several of the codes.






O o

,O O
v0 9$
popr 4

iJ C Cp a

E aH
H-(O b e
v0 ppy C


O r


to C



c (





0.67426600E 00
0.10345899E 01
0.17191599E 01



PINNACLE, Simpson's rule was used since most of the functions

to be integrated were generally smooth. Comparatively few

quantities were needed as input parameters, and the more

important ones are listed in Table 2.

Figure 1 shows the sequence of the numerical cal-

culations and the input and output of each code in the chain.

For the first task, that of calculating the scattering kernel,

two different models were used. The first, and more realistic,

kernel was formulated by Parks (22,23) and incorporated by

Bell (24) into a computer code called SUMMIT. The Pl kernel

was calculated from the output of SUMMIT by the PINNACLE

code. The code multiplied the angular dependent scattering

kernel from SUMMIT by the cosine of the scattering angle,

performed a Gaussian integration over the angle, and multi-

plied the result by an appropriate energy dependent factor.

The second scattering kernel considered was a free-gas energy-

exchange kernel obtained by the use of the MASSMK code (25).

Note that the calculations involving the MASSMK kernel

differed from those of the SUMMIT code only by the scattering

kernel Es(EiE~j). All other inputs to the OMEGA-ZERO SYMBOL
via DIAGONAL CORRECTOR (26) were the same as those obtained

from the SUMMIT kernel.

The scattering kernel calculated by SUMMIT does not

include the elastic contribution. For this reason, it was

necessary to correct the diagonal elements of this kernel

so that the scattering cross sections obtained from it were

consistent with values taken from the "barn book" (BNL-325).

Parameter Value Units



Mass of Graphite


0I (number density)













as FRE~E

This was accomplished in the DIAGONAL CORRECTOR code.

As shown in Figure 1, several other quantities were also

obtained from this code. The (P) matrix was calculated

using the OMEGA-ZERO SYMBOL code. This code was obtained

by modifying the SYMBOL code written by Ohanian to account

for the differences between the (P) matrix of equation (62)

and the one obtained by him for his eigenvalue problem.

The eigenvalues and eigenfunctions of the space and

energy dependent Boltzmann operator were computed by the

EIGENVALUE code using the Jacobi method of matrix diagonali-

zation (Appendix A of reference (19)), This was not the

same code as used by Ohanian but it was similar to it.~

The code used in this work was modified to accept the

diagonalization when the smallest diagonal element became

more than 1016 greater than the largest off-diagonal element.

Presentation of the Eigenfunctions and Eigenvalues -

Several of the eigenfunctions and all of the eigenvalues

computed by the method outlined above are presented,

respectively, in Figure 2 and Table 3 for the SUMMIT

kernel and Figure 3 and Table 3 for the MASSMK kernel.

Some problems were encountered in drawing the eigenfunctions

* The code was obtained by Dr. Ohanian and converted
for use on the University of Florida IBM 709 computer
by the author.








Log of Energy


( 11)

( 21)

ig. 2. Several Figenfunctions of the Space and Energy Dependent
Boltzmann Operator Using the SUMMIT Kernel.

g,. 3. Several Eigenfunctions of the Space and Energy Dependent
Boltzmann Operator Using the MASSMK Kernel.





0.50220024E 00
0.71235961E 00
0.10231198C OL
0.12931190E 01
0.48972284E 01


because they can have more than one singularity and in the

discrete representation it is difficult to distinguish

between multiple discontinuities and oscillations. The

possibility for multiple disconitinuities is attributed to

the many fluctuations in Eg(E)/D(E). This function can be

identical to a particular eigenvalue for several energy

values thereby causing the corresponding eigenfunction to

have several discontinuities. From the characteristics of

the eigenfunctions, it appears that only the fundamental

eigenvalue is discrete; and the remaining ones represent

particular eigenvalues in the continuum. As shown in Table

3, beyond p20 the larger eigenvalues associated with the
SUMMIT kernel are closer to the fundamental eigenvalues than

are the ones from the MASSMK kernel.

The Energy Expansion of the Freauency Dependent
boltzmann Lquation inth theDifusion Approximation

The expansion of the energy dependence of equation

(15) in terms of the eigenfunctions of the P Operator re-

quires the evaluation of only one matrix element, Lmns

expressed by equation (27). In the discrete energy repre-

sentation this integration becomes

L, = mi1 nE)g
i=1 v(Ei)D(Ei) (64)

where AEi is the weighting factor for the numerical integration
A L-MATRIX ELEMENTS code was written which evaluated these

matrix elements using both Simpson's rule and Weddle's rule

(27) and which also checked the orthonormality condition.

The numerical integration were evaluated using the n variable

since the eigenfunctions obtained from the EIGENVALUE code

were expressed in this variable and the discrete energy

representation was based upon the selection of equal n

increments for the evaluation of the energy points. It was

felt that since each of the continuum eigenfunctions exhibited

at least one discontinuity, and since D(Ei) is not a smooth

function, it would be necessary to apply very elaborate

integration schemes in order to obtain accurate results. If

any integration other than Simpson's rule was used, it was

found that for some of the eigenfunctions the orthonormality

check was not satisfied (i.e., the integration over energy of

all ~m i) times 9 (E ) was not necessarily much less than
the integration of all [$m(E )]2) ggypp fac isntsrrs
since the eigenfunctions obtained from the EIGENVALL'E code

were obtained with a Simpson's rule weighting function.

For this reason, it was felt that continuity of the numerical

methods, that is, the use of Simpson's rule throughout, would

result in the more consistent set of Lmn elements. Some

justification for this reasoning is found in the work of

Ohanian (19) since a similar integration was required in his

calculation of the cooling coefficient for water and accurate

results were obtained."

A pragmatic consideration of this calculation provides

some satisfaction since its use resulted in theoretical

curves which were in good agreement with the experiment.

Also the diagonal elements of the Lan matrix were, in

general, much larger than the off-diagonal elements since

in the evaluation of the former, the large discontinuity

of both eigenfunctions occurred at the same energy point.

Thus after a careful investigation of the integrands arising

in the evaluation of the Lmn matrix elements, some justifi-

cation can be obtained for their accuracy. But, in the

general case, the numerical integration of such "wild"

functions by Simpson's rule cannot be defended.

Evaluation and Presentation of the Complex
Inverse Relaxarion Lengrhs

With the Lmn matrix elements determined, one can

apply equations (43) and (4R) to evaluate the coefficients

of the expansion of p2 in powers of (jo) and equations (50)

and (51) to evaluate the same expansion in powers of x.

In cases where neither series converges rapidly, the

*Also in calculations: performed by Razmr~inas- mid the author
under the direction of Ohanian, the cooling coefficient
for graphite was calculated by Ohanian's method. It
agreed quite well with the one obtained by Honeck using
the same kernel.

eigenvalues were evaluated directly from the secular

determinant at particular values of frequency. Computer

codes were written to calculate the coefficients for both

the jw expansion and the X expansion of any of the 33 p2

along with their corresponding ratios, S If more than
a few terms were used in the jw expansion, however, it

was necessary to divide the equations by L11 and derive

expressions for the expansion in powers of Llljw. This

was because at high frequencies, wn became larger than

1038, the largest number acceptable to the computer. This

problem did not arise in the X expansion since for a

frequency of about 1000 ops X need be only about 7.
Problems were also encountered in the coefficients. For

the jw and Llljw expansions, the series could not be

carried past the fifteenth term since,in the former case,

underflow occurred in the computer and in the latter case,

overflow occurred. In the x expansion the coefficients did

not change as rapidly and no underflow or overflow problem

occurred; but since factorial 33 is nearly 1038, the series

was cut off at the 33rd term. This, however, was not the

limiting factor in the X expansion for it appeared that

after the 20th term round off errors built up rapidly.

Shown in Figure Ir and Figure 5 are the frequency

dependent real and imaginary components of the complex

inverse relaxation length for the fundamental energy mode

If m I I \ I -I I j

ce cu cy e- e- 0


O <






X *Hl




C *,

MO r


0 *j


o 'c- '

o ri
o OO


o COl

0t 0



on o

0~ UH

a .sa

E 0*

( O/pm) 3

and the first four energy harmonics as generated by using

the MASSMK kernel. Presented in Figure 6 and Figure 7

are these same eigenvalues up to the third harmonic ob-

tained by use of the SUMMIT code. Figure 8 and Figure 9

show the fundamental and first two space harmonics for

the MASSMK calculation. The transverse bucklings used in

this last set of curves are indicated on the figures, and
for all other curves a value of 5.96 x 104 om2 wsue

since this was the calculated value for the graphite pile

investigated. The omega expansion was used in the genera-

tion of all the MASSMK curves, and the method presented

previously for the generation of higher energy harmonics

was successfully applied with this scattering kernel. Six

terms were sufficient for convergence in this case. For

the SUMMIIT kernel, however, the omega expansion did not

converge when as many as 15 terms were used; and for this

scattering kernel the X expansion and the direct numerical

solution of the complex secular determinant yielded the

eigenvalues*" As shown in Figures 6 and 7, the 18-term

X expansion appears to have converged since agreement exists

between it and the direct solution for the fundamental

" The code used for this calculation was obtained by
Victor Cain from the SHARE Program Library. The code,
identification number NU EIG4.





0 *H
m CF0
o cXc
to HQ

o HO
o *H
Hr XG,

o ,
to a 0



E v
N *


r( rl

(~,U13) D

a Ukb
o OC


O cl

o *
o clm

o a Ur
O 4 C
m a

o mC

c 40

0 0 EC
rl v O 6



o v C


o do~-
*H O

Hi C/3



O *H

c u

H *H

O -

[2 *H



0 0

v, h

o k C
o OC-


0 J

o C

c CO


O Xa

o Eu
o Oa
rl Ua



AC *r

o OO

aC (
m rl
H *c
b^C a

O to
Hl 0

(ur~/peJ) t,

eigenvalue. The higher energy mode eigenvalues for the

SUMMIT kernel were calculated by the direct method.

Some concern can be caused by a careful inspection

of the higher energy modes. For higher frequencies, these

modes appear to be less attenuated than the fundamental

mode; and, therefore, modal contamination is predicted.

The less accurate MASSMK kernel does not predict as serious

energy mode contamination. One must recall, however, that

in order for these higher energy modes to propagate, they

must be excited by the source or the medium; and even in

the finite, absorbing medium to be investigated they will

account for only a small percentage of the energy spectrum

compared with the fundamental Maxwellian distribution.

Also Kunaish has shown that for the heavy mass scattering

kernel the total energy spectrum, calculated from as many

as nine of his energy harmonics, will propagate with a

complex inverse relaxation length which is within 1

per cent of that of the fundamental.

From the curves showing the higher spatial mode

eigenvalues, it can be concluded that for this experiment

they will be of secondary importance in comparison with the

higher energy modes. Calculations by Perez (9) and Kunaish

(18) predict that higher spatial contamination should be

reduced to a negligible amount at points farther than 20

cm. from the source. The calculations presented here

confirm these predictions.




The experimental techniques currently used at the

University of Florida for the production and detection of

neutron pulses and waves were developed over a period of

nearly four years through the efforts of many persons.

As a half-time graduate assistant on the neutron wave

project, the author has been intimately involved in all

phases of this developmental work. The methods for

analyzing the data have also evolved over a period of

years. The principal contribution of the author has been

in this area; for he has, under the guidance of his chair-

man and with suggestions from co-workers, written every

computer code required for the data analysis and analyzed

or co-analyzed all the data taken thus far in the neutron

wave project. Also this work represents by no means the

first, nor the last, experimental results to be derived

from the neutron wave project, as is evident from an in-

spection of the Review of the Literature.

The medium investigated in this experiment was a

rectangular pile of AGOT graphite 44.7 cm. by 71 cm. by

125 cm. covered, except on the source face, with 30 mil.

thick cadmium. The discussion of the experiment which

follows is divided into three general areas. Consideration

is given to the methods used to excite the thermal neutron

waves. Attention is then focused upon the systems employed

in their detection. The third section deals with the

analysis of data and will be given special emphasis.

The Source

The source of thermal neutron waves used in this

experiment was a Cockcroft-Walton type neutron generator

placed in a thermalizing tank as shown in Figure 10. The

neutron generator, manufactured by Texas Nuclear Corporation

of Austin, Texas, produced either a sinusoidally modulated

or a pulsed intensity of 14 Mev neutrons which were then

slowed down to thermal energy and directed to the graphite

pile by the thermalizing tank. The sinusoidally modulated

intensity of neutrons was obtained by varying the extraction

voltage of the ion source bottle, thereby controlling the

intensity of the deuterium beam which impinged upon the

tritium target. The pre-acceleration pulsing unit consists

of deflection plates, Einzel lens, and focus lens. This

unit deflects the deuterium beam to the base of the Einzel

lens during the wait time between pulses and allows it to

Fig,. 10.

Arrangement of the Experiment and Details
of the Thermalizing Tank ( from Re ference 14) .

pass on to the target for the duration of the pulse. For

the data obtained with the modulated source, the frequency

varied between 50 and 1226 cps, the ratio of maximum to

minimum neutron count rate was roughly 9 to 1, and harmonic

and 60 cycle contamination were quite small. For data taken

in the pulsing mode, the maximum to minimum thermal neutron

yield was about 1000 to 1; and the total pulse width was

about 1 millisecond at the a position in the graphite which

was closest to the source.~ More detailed discussions of

the sinusoidal and pulsed mode are given, respectively, in

References (8) and (14).

As previously mentioned, the thermalizing tank was

used to thermalize the 14 Mev neutron waves produced by the

neutron generator and then transport them to the external

port of the tank where the graphite medium was placed. This

was accomplished by placing light water, steel bricks, lead,

and graphite in the tank and facing the tank with a poured-

paraffin section. By systematically varying the dimensions

of the various materials and carving out a 8 cm. by 8 cm.

square hole in the center of the paraffin section, the

*' It should be mentioned that the attainment of such a
high-quality source was due primarily to the efforts
of Robert H. Hartley and George W. Fogle.

uncovered-to-covered count rate of a 1 atm. He3 detector

was optimized at 50 to 1 at the source interface of the

graphite. The initial thinking associated with this

thermalizing tank is presented in Reference (8). The

unwanted, but ever present, fast background emitted from

the tank was eliminated experimentally as discussed below

and in the section dealing with the data analysis.

The Accumulation of Data

It is fitting at this point to indicate precisely

what data are to be presented and how it was recorded. The

main objective of the neutron wave experiment is to measure

the frequency dependent complex inverse relaxation length,

p, of the fundamental space and energy mode. The most

obvious method to accomplish this objective is to modulate

Isinusoidally the neutron current which enters one boundary,

z=0, of the medium under investigation and to measure the

amplitude and phase of this neutron wave as it propagates

from this source. The slope of the semilog plot of amplitude

versus z is the real part of p, or the attenuation factor,

a; while a plot of the phase of the wave versus z is linear

with a slope equal to the imaginary part of p, or the phase

shift factor, 5. The source is modulated for as many fre-

quencies as desired by the experimenter. The process out-
lined above was used to obtain the wave data presented in

this work. Still another way of obtaining the same information

is to record the dispersion of a pulse of thermal neutrons as

it propagates from the source boundary. The amplitude and

phase of any frequency component of this pulse at a particu-

lar z position is obtained by numerically performing a Fourier

transform of the measured, time response at that position.

These methods are not new. The analogy in electrical theory

to the discussion above is that the frequency characteristics

of a "black box" can be determined by either directly measur-

ing its response at various frequencies or by performing a

Fourier transform of its unit impulse response. One method

is not necessarily superior to the other for all applications,

but in this measurement it was found convenient to obtain

most of the data by exciting the neutron waves with thermal


The pulse propagation data were collected by Hartley,

and first appeared in his master's thesis (14). The

purpose of his investigation was accomplished by considering

data in the time domain~ only. In this work the analysis of

Hlartley's data is extended beyond that which he completed

since Fourier transforms are performed under the Fguidance

of Reference (16) and the complex inverse relaxation length

is obtained. Actually, in the final comparison with the

theory, the wave data and pulse propagation data are combined

as discussed in the next section.

From the above discussion it is evident that it will

be necessary to discuss two data accumulation systems, that

associated with a sine wave input and the one used by

Hartley in his pulse propagation experiment. Before con-

sidering these systems individually, however, it is con-

venient to present those characteristics which were common

to both experimental set-ups.

In each of the experiments use was made of a movable

detector which could be located at several z positions, and

a stationary, standard detector placed in the thermalizing

tank. To facilitate accurate placement of the movable detector

and cause as little perturbation as possible, length-wise

holes were bored in several of the graphite stringers to

accommodate the detector and its preamplifier. For measure-

ments at a particular z position, a detector-filled stringer

was inserted into the graphite pile so that its center was

aligned with the transverse center of the pile and its length

was in the transverse plane. By using several of these

stringers with holes bored at different distances from their

front face, the detector could be moved from the source in

about 1 cm. increments from 1 cm. to 72 cm. Two measurements

were made at each position of the movable detector; one

where both epicadmium and thermal neutrons entered the

moderating medium, the uncovered run, and a second where

thermal neutron penetration was blocked with a cadmium shutter,

the covered run. The standard detector was used to correct

for source variations and to facilitate subtraction of the

epicadmium background from the movable detector data of

the uncovered run, thereby obtaining the thermal wave. The

same source (i.e., neutron generator and thermalizing tank)

was used in both experiments, and changing from the wave

experiment to the pulse experiment was accomplished by

changing accordingly the mode of operation of the neutron

generator. The same detector systems were also common to

both measurements. They consisted of He3 detectors and Nuclear

Chicago Corporation oreamnplifiers and scalers. The system

dead time was about 3 microseconds,which was found to be

adequate for our purposes. The arrangement of electronics

beyond the scalers was unique for each of the two experi-

ments and they will now be discussed separately.

Electronic Eguipment Associated with the Wave Experiment

A block diagram of the electronics used in the wave

experiment is shown in Figure 11. The main component of this

system is the multi-channel analyzer manufactured by Technical

Measurement Corporation. A modified 212 plug-in-unit is

used in the multi-channel analyzer so that it stores the

input pulses in a channel corresponding to their time of

arrival after an initial trigger. The analysis time of

each of the channels is stepwise variable from 10 to 2,560

microseconds so that the accumulation may cover a wide

range in frequency. The remaining circuitry is arranged so


Fig. 11. A Block Diagram of the Electronic Equipment Used
in the Wave Experiment.


Graphite Pile

Cadmium Sheet

that for two periods of the input sine wave the output

of the standard detector and that of the movable detector

are recorded, respectively, in the first half-memory and

second half-memory of the multi-channel analyzer. This is

accomplished by alternately allowing the pulses from each

detector to reach the 212 plug-in-unit, controlling their

storage in the multi-channel analyzer, and adjusting its

channel width so that the total sweep time is slightly less

than four periods of the input frequency.

A time-sequence diagram for the system is shown in

Figure 12. The HP202A function generator provides a pulse

at the positive peak of its sine wave output to the neutron

generator. This pulse train is fed to a Tektronix type

161 pulse generator which, when triggered by a sync. pulse,

produces another pulse whose width is almost as

long as two periods of the input sine wave. This output

pulse from the Tektronix type 161 pulse generator, which

is triggered by every other sync. pulse, serves two purposes.

It is used as a trigger for a Tektronix type 163 pulse gen-

erator whose output, connected to the trigger input of the

212 plug-in-unit, initiates the sweep of the multi-channel

analyzer. Thus the analyzer sweep always begins at the

same point in the cycle of the input wave. The pulse train

is also connected to the external trigger input of the

Tektronix type 53/54C plug-in-unit, and it controls the

switching of this gate circuit. Other inputs to the Tek-

tronix type 53/54C plug-in-unit are the pulses from the


3V53/54C M~emory Sig~nal

T 2T 3T 4T
Time (Sine Wave Periods)

Cycle Time of Elec-
tronic Equipment
Period of Input
Sine Wdave

Neutron Source

cync. Pulse from
HP202Ai Function

2T-Pulse from the
Tektronix 161 Pulse


12V Trigger Tnput from
Tektronix 163 Pulse

53/54C Output Sig~ndl

Fig. 12. Time Sequence of Electrical Signals Associated
with the Electronic Equipment Used in the Wave

-- II (20V

Standard Movable
Detector Detector

standard detector and those from the movable detector.

The Tektronix 53/54C plug-in-unit contains two identical

amplifier channels; and it is usually used with a dual

trace oscilloscope so that two signals, one connected to

input A and the other to input B, appear on alternate

sweeps of the oscilloscope. In this application the pulses

from the two detectors are allowed to pass alternately to

the input of the 212 plug-in-unit. An alternate output

from the 53/54C plug-in-unit is connected to the EXT input

of the multi-channel analyzer, and it i9 used to control

the half-memory in which the detector pulses are being

stored. Thus the 53/54C plug-in-unit controls which

detector pulses are being recorded by the multi-channel

analyzer and also controls the half-memory in which the

pulses are being stored. This eliminates the possibility

of the storage locations for the two detectors being inter-

changed during the accumulation of data.

Referring again to Figure 12, the functioning of the

detection system during one analysiS sweep of the multi-

channel analyzer is as follows:

1. A trigger from the Tektronix type 163 pulse

generator starts the multi-channel analyzer on its analysis

sweep. The pulses from the standard detector will be stored

in its first half-memory, and then it will stop and wait

for another trigger.

2. The next trigger will start the multi-channel

analyzer again, but now the Tektronix type 53/54C plug-in-

unit is passing only the movable detector signal and holding

the multi-channel analyzer in the second half of its memory.

3. The next trigger will again start the cycle,

which is repeated until enough statistics have been obtained.

Electronic Equipment Associated with the Thermal
_Neutron Pulse Propagation ELxperiment

A block diagram of the circuitry used by Hartley

is shown in Figure 13. For his experimentzonly the integral

counts of the standard detector were needed. The output of

the movable detector was recorded in the multi-channel analyzer

after shaping in a Tektronix type 163 pulse generator. The

sweep of the multi-channel analyzer was initiated by a signal

synchronized with the start of the neutron burst from the

neutron generator.

The Data Analysis


The focal point in the analysis of neutron wave

data is the counts per channel recorded for the movable

detector by the multi-channel analyzer. To indicate the

typical appearance of these data, the 200 ops thermal wave

is shown in Figure 14 for three of the positions used in

the wave experiment; and the thermal pulse at several

Fig.13.A Block Diagram of the Electronic Equipment Used
in the Pulse Propagation Experiment (from Refer-
ence 14).




0. 0

is IIF
Nv N


EC r:
*~ *

0( \I I N --

~O~ x u x ~auuP~3 ~ad s~uno3

positions is shown in Figure 15. In the wave experiment

these data are composed of a fundamental sine wave con-

taminated by harmonic distortion, and a Fourier series

expansion yields the amplitude and phase of the fundamental

thermal wave. The thermal pulse, on the other hand, was

considered to be composed of neutron waves in a continuous

spectrum from zero frequency up to a high frequency cutoff.

Thus the amplitude and phase of a particular frequency was

obtained by numerically Fourier transforming the pulse at

the frequency of interest. After the attainment of the

amplitude and phase points versus z position for several

frequencies, the data analysis and comparison with theory

can proceed in an identical manner for both the wave and

pulse propagation experiments. In fact, in this work the

amplitude and phase curves obtained by the pulse propagation

experiment and those from the wave experiment are normalized

together before determining their slope. In this way the

complex inverse relaxation length, p, arises from measure-

ments in both the time domain and the frequency domain. The

fact that these two sets of data can be normalized together

effectively before proceeding with the evaluation of p

is significant since it indicates that the basic assumptions

associated with their analysis were correct and also points

out the fundamental equivalence in these two methods of

excitation of neutron waves.

105 / z = 14.78 cm.

z = 29.67


a 1


z = 59.23 cm.

0 1200 2400 3600 4800 6000
Time (Microseconds)

Fig. 15. Thermal Pulse for the Pulse Propagation Experiment
for Several z Positions (from Reference 14).

In the discussion which follows in this section,

the data evaluation will be carried through the calculation

of the real and imaginary components of the complex inverse

relaxation length. The data acquired from wave experiments

will be considered separately from those of the pulse pro-

pagation experiment in obtaining the amplitude and phase as
a function of position and frequency. Then these data will

be normalized together for the evaluation of p.

The data presented here has been taken over a period

of two years in two separate wave experiments and in Hartley's

pulse propagation experiment. The curves of amplitude and

phase versus z position have been determined for 65 fre-

quencies ranging from zero to 1450 ops. For each of these
curves a minimum of 14 z positions have been taken; and in

many, where all three experiments have been normalized to-

gether, up to 40 z positions are considered. Moreover, to
obtain each of the 14 to 40 points on the 130 amplitude and

phase curves either a Fourier transform was performed over
the 256 channel output of the multi-channel analyzer or a

Fourier series expansion was made with between 50 and 120

channels per cycle.

The complex inverse relaxation lengths, obtained

from least-squares fits of the amplitude and phase curves,
were derived from a careful examination of from 2 to 8 fits

of each of these curves in which points with largest error

were systematically eliminated.- The presentation of all

of these data would be impractical. One can, however,

present some of the data which are indicative of the

results obtained and show sample calculations to indicate

the method of data analysis.

Determination of the Amplitude and Phase of the
Fundamental Thermal Wave of the Wave Data

For the initial analysis of the wave data use was

made of the University of Florida Non-Linear Least Squares

(UF-NLLS) code (28). Several new subroutines were written

for this code, and it was incorporated into another main

program to facilitate a Fourier series expansion up to

ten harmonics of the thermal neutron wave. The bulk of

the numerical calculations were performed in the WAVE-

ESTPAR subroutines which provided estimates of the para-

meters to the UF-NLLS code. The WAVE-ESTPAR subroutine will

now be discussed.

Information was first read into the code which

allowed it to calculate the period and number of channels

per cycle of the fundamental sine wave variation. The code

then read in the counts per channel recorded in the multi-

channel analyzer for the standard detector with no cadmium

placed between the source and the graphite pile (bare

standard data), for the standard detector with the cadmium

sheet in place so that only neutrons of epicadmium energy

entered the graphite (covered standard data), and for the

movable-detector counts per channel for the latter situation

(covered movable data). The data of interest, that arising

only from the thermal content of the source, are contained

in the movable detector counts per channel for the bare

run (bare movable data); but this information is contaminated

by counts arising from the epicadmium content of the source.

The fundamental amplitude, phase, and harmonic content up

to the tenth harmonic of the former three sets of data

were then calculated. From the two sets of standard data

the source variation from the covered to the uncovered

runs for each of the frequencies analyzed could be determined.

This information was then used to generate correction

factors to be applied to each of the frequency components

of the covered movable data. A set of data was generated

which represented the contribution of epicadmium neutrons

from the source to the bare movable data. This was

accomplished by individually multiplying the fundamental

and each of the harmonics of the covered movable data by

the ratio of the standards content of the same frequency for

the covered and uncovered runs and by using a subroutine

NEWCOV which synthesized the generated waves. This generated

epicadmium content of the bare movable data was then sub-

tracted from it to obtain the thermal neutron wave arising

only from the thermal neutron content of the source. Note

that this background subtraction has been made individually

for each frequency; therefore, changes in the harmonic

content of the source from one run to the next should not

affect the result. This elaborate method of background

subtraction was necessary because of the instability of

the neutron generator.

The fundamental component of the thermal wave and

as many of its first ten harmonics as desired were then

determinedsand the results were normalized to the bare

standard data. Corrections were also made for changes in

the constant component of the source as determined from

the total counts of the detectors for both runs. The data

were normalized so that the channel width of the multi-

channel analyzer, its total number of sweeps, counting

time, dead time between channels, and channels per cycle

could be varied without affecting the results. The print-

out of WAVE-ESTPAR included the normalized amplitude and

phase of the frequencies analyzed for the two detectors for

both runs so that any variation in the source was recorded.

The numerical methods used in WAVE-ESTPAR were rather

crude compared to those which could have been applied. The

correction for dead time between channels was made by cal-

culating the average of the counts accumulated in two con-

secutive channels, multiplying this by the dead time between

channels, and adding the result to the counts of the first

of these two channels.b The numerical integration associated

*The 212 plug-in-unit used for this experiment requires
10 microseconds to store the counts accumulated in a
channel before moving to the next channel. During this
storage time no input pulses are counted.

with the calculation of the Fourier coefficients were

approximated by a summation. The method can, however,

be justified for these data. The operation of the multi-
channel analyzer is such that it will store in each channel

all of the counts accumulated during the count-time of that

channel. In effect, the counts recorded in a channel are

the integral of the detector count rate over the time interval

allotted to the channel. If exact adjustment could be made

for the dead time between channels, the summation of the

counts accumulated in all the channels would be identical to

the integral of the count rate over the total sweep of the

multi-channel analyzer. The weighted summation performed by

the code approximates the exact integral over a period of

the fundamental sinusoidal variation by a weighted summation

of integrals over each channel. In each of these "channel

integrals" the sinusoidal function has been effectively moved
outside the integral, and the integrated count rate for the

time interval of the channel is multiplied by the value of

the sinusoidal function at the center of the time interval.

This approximation is mathematically expressed by

T N tj+r
I sin nwt y(t)dt sin nw(tj + t/2) I y(t)dt
oj=1 tj

where T is the period of the fundamental sinusoidal

variation of the count rate, y(t), r is the channel width,

ti is the time when the jth channel is opened; and it has
been assumed that the dead time was zero and that there were

exactly N channels per cycle. The accuracy of the above

expression is based primarily upon the value of N, which
varied between 50 and 120 for this work.

To prevent inaccuracies in the data, the harmonic

distortion, fast neutron content, and D.C. component of the

source were all held at as low a value as possible; and

every attempt was made to hold the neutron generator in

stable operation during all of the runs of a particular

frequency. These objectives were well realized, especially

for the second wave experiment; and for these data the

elaborate corrections seemed somewhat extreme. The original

purpose of WAVE-ESTPAR was to extract the thermal wave from

the bare movable data, keep track of variations in the source,

and provide estimates to the UF-NLLS code of the amplitude

and phase of the fundamental thermal wave. The UF-NLLS code

was then to be used to modify these initial estimates to

values which yielded the best fit to the experimental results.

Actually, WAVE-ESTPAR did such a good job of estimating the

parameters that changes in them by the UF-NLLS code were

made only in the third and fourth decimal places. In the

interest of saving computer time, the parameters calculated

by WAVE-ESTPAR were accepted for the majority of the data

To indicate the accuracy achieved by this technique,

Figures 16 and 17 show the amplitude and phase versus z

position of the 400 cycle data obtained from runs made

when the fundamental sinusoidal variation was 200 cps and

then 400 ops. For the 200 ops runs, the magnitude of the

400 ops distortion was about 3 per cent. As shown in the

Figures, the 400 ops results obtained from this small per-

centage of the thermal source agree quite well with that

for the 400 ops data. In fact, plotting the first harmonic

of a particular frequency became a very useful way to

determine if any mistakes had been made in the data input

since this small component of the measured signal was very

sensitive to inaccuracies or inconsistencies.

Analysis of the Pulse Data ~by Moore's Method"

Some enlightening derivations by Moore (16,17) provide

the guidelines for the analysis of data obtained from the

pulse propagation experiment. Moore expressed Fourier

moments, b,(q,v) ,of the measured data as

os -2wivt
In(+q,v) = I dt tn e rC(q,t)
..o (66)

* The author is indebted to Dr. M. N. Moore for several
stimulating discussions concerning the analysis of
pulse propagation data.

I I1I I 1I 1I
5 10 15 20 25 30 35 40

z ( cm. )

Fig. 16. Amplitude of the 400 ops Wave as Obtained from the
First Harmonic of the 200 cos Wave Data and the
Fundamental of the 400 ops Wave Data.

* Fundamental o
+ = First Harmonj

S= Normalization

,f 400 ops Exp
ic of 200 ops Exp




*=Fundamental of

+=First Harmonic
of 200 cps Exp

~jP $= Normalization Exp



1. 5



z (cm.)

Fig. 17. Phase of the 400 cps Wave as Obtained from
the First Harmonic of the 200 cps Wave Data
and the Fundamental of the 400 ops Wave Data.

where #($,t) is the experimentally measured pulse at space

point q as a function of time, t, and v is the frequency in

cycls pe secnd. e then defined Fourier widths, W (q,V),
by the expression


Wn0~,v) = E I dt(t-t)n e

4( ,t)]/q( ,v)



t = l9 oqv)


and proved that, neglecting higher space and energy mode
contamination, these widths are related to the frequency

derivatives of the system dispersion law by the expression

Wn (<,V 2)n I Gn **R2,93,B: ) +

+ d E-l~vBl 1(V,B )] J

n> 1



Gn 9n log A10 23 92'Q3,B )



~o(q,v) = Al0 ~)23 42,43,B )exp (E-al(V,B ) +igl(vB )]z 1