THE UNIVERSITY OF MICHIGAN
INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING
ON THE ANALYSIS OF GAMMA RAY PULSE HIEIGHT SPECTRA
Jacob Israel Trombka,
A dissertion submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy in The
University of Michigan
Department of Physics
1961.
September, 1961
IP-531

Doctoral Committee:
Professor Marcellius L. Wiedenbeck, Chairman
Professor Robert C Bartels
Professor Henry J. Gomberg
Professor William Kerr
Professor John S. King
Associate Professor Charles S. Simons

ACKNOWLEDGEMENT
It is a pleasure to express my gratitude to Dr. M. L. Wiedenbeck
for his kind and invaluable helpfulness and supervision throughout this
work. I am also indebted to my committee for their continued cooperation,
advice and assistance,
I wish to thank Dr. W. F. Miller and W. J, Snow of the Applied
Mathematics Division, Argonne National Laboratory, for their cooperation
and important work in performing the Monte Carlo calculation of the
pulse height distribution for spherical crystals. I am grateful to Mr,
R. W, Johnson of the Harshaw Chemical Company for his help in connection
with P*oblems in the preparation of spherical NaI(Tl) crystals.
The untiring patience and constant encouragement of my wife have
been a genuine contribution to the completion of this research. To my wife,
I wish to express my deepest appreciation,
The partial support of this work by the Michigan MemorialPhoenix Project and the United States Atomic Energy Commission is grater
fully acknowledged.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS o.o o o......o.o., o.......c o..... o o... ii
LIST OF TABLESo.... o.... eo.....*......O..o *...... V
LIST OF FIGURES........... o.. O.................... vi
I INTRODUCTION................ o o o.. o............... 1
IIo PROPERTIES OF SCINTILLATION DETECTOR SYSTEMSE............ 5
Ao The NaI Crystalo o O..0 o o 4........ o.......... 5O
B o The Multiplier Phototube O o o o o..... o....... o o.. 11
C o Construction of the NaI(T1) Detector................. 13
Do Factors Affecting Resolution,......................... 16
E. The Multichannel Pulse Height Analyser............ o 20
III. PROPERTIES OF PULSE HEIGHT SPECTRA........ oo........ 24
Ao Shapes of Gamma Ray Pulse Height Spectra. o.. o,..,.. 24
Bo Detection Efficiencies,o o oo..............o o......o, 32
IVo LEAST SQUARE ANALYSIS OF COMPLEX GAMMA RAY SPECTRA..... 47
A, Formulation of the Principal of the Least Squareo..,, 47
Bo Method of Obtaining Minimum for Least Squares Fit...oo 51
1o Incident gamma flux discrete in energy
(energy distribution known and intensities
required)o.. o.......... o o....o o o o o o o. o o o o o. 51
20 Discrete incident energy spectrum (both the
energy distribution and intensity of the
incident beam unknown)........................ 52
35 Continuous Incident energy spectra...o.......... 63
Co Error Calculation ooooooo.. o........,..o.o..oo...... 69
V. EXPERIMENTAL PROCEDURES AND RESULTS AND CONCLUSIONS. o... 72
Ao Detector - Analyser System........................... 72
Bo Measurement and Interpolation of Monoenergetic
Pulse Height Spectrun 0......0........................ 72
Co Experimental Measurements of Spherical Crystal, oo..o 83
Do The Experimental Determination of Discrete Spectrao.o 88
lo Monoenergetic Emitters,........00.0.0000.00 88
20 Polyenergetic Spectrao ooooooooooo.o.o..oo...ooo 95
iii

TABLE OF CONTENTS (CONT'D)
Page
Eo The Experimental Determination of the
Continuous Spectrum............................. 100
Fo Conclusion and Proposed Further Experimentationo 111
APPENDIX I................................................. 113
BIBLIOGRAPHY. o.. o..... o o. o o o o o o o o o o o o o o o o o o119
iv

LIST OF TABLES
Table Page
I The Values of a em for Various m's and (p/d)'s........ 39
II Intrinsic Total Efficiencies as a Function of
Distance and Energy for a 2" Spherical NaI(Tl)
Crystal. o. o o O. O o o o o o. a a o o o.. o o.o o o o o o 4 o 0 o o o 0 o o o.. o
III Peak to Total Value for 2" Spherical Crystal
Incident Upon the Parallel Beam of Gamma Rays...... 46
IV I3 Gamma Ray Spectra,.......... o o o.............o 97
V Results of Least Square Analysis of the W Spectrumo 99
VI Final Least Square Fitting Results for the Degraded
Pulse Height Spectra Normalized to an Incident Gamma
Intensity of 100 Gammals/secO o o,,OOOO C o o o.. o o..... 109
V

LIST OF FIGURES
Figure Page
1 Energy Level Scheme for NaI(Tl) Phosphor 0..00...00,,, 8
2 Emission Spectrum, Absolute Conversion Efficiency
Spectrum and Photocathode "Absorption" Spectrum
for NaI(Tl) Crystal and End Window (Sb Cs3) Photocathode Multiplier Phototubes..O.... c....... o...... oo 10
5 (a) Spherical Crystal Design..o..oo.,.......o.ooO oo 15
(b) Cylindrical Crystal Design.................... 15
4 Pulse Height Analyser Block Diagramo.oooo........... 21
5 Sc7 Gamma Ray on 3" x 3" NaI(Tl) Crystal. Source
at 3 cm....000..000.......00000000000000.00000..... 26
6 Theoretically Calculated and Experimentally Determined
Pulse Height Spectrum for Cs13 (0o661 Mev). Parallel
Beam Incident upon a 2" Spherical NaI(Tl) Crystal...... 27
Na24 Gamma Spectrum Source at 3 cm from 3" x 3"
NaI(Tl) Crystal........ o o o o o o o o o o o o o........ 29
8 Zn65 at 0.24 cm from NaI(T71) Crystal.o o o..... o o o o 31
9 (a) Pulse Height Spectrum of a Mixture of Cs137 and
Cr51..OO.00000000o0000000.00000......*. 33
(b) Pulse Height Spectra of Cs137 and Cr5l.......... 33
10 Source Detector Geometry for
(a) Cylindrical Crystal
and (b) Spherical Crystalo............................0 35
11 Total Intrinsic Efficiency as a Function of Distance
for a 2" NaI(Tl) Spherical Crystal. o 0000000,00,000 0000 42
12 Comparison of eTi between 2" Spherical Crystal and 1~"t
x 1" Right Cylindrical Crystal.........0..0.. 0..0.. 0 43
13 Intrinsic Peak Efficiency epi as a Function of Energy
for a Parallel Beam of Gamma Rays Incident Upon a 2"
Spherical NaI(Tl) Crystal.........o o... o. o, o o 45
14 Geometric Solution of Least Square Fitting Problemo.... 58
vi

Figure Page
15 Comparison of Gaussian Distribution with cos2kx
Distribution. Fitted at x = 0 and x =2a............ 67
16 Schematic Diagram of Detector-Analyser System.. 0..... 73
17 o2 as a Function of Energy for a 2" x 2" NaI(Tl)
Cylindrical Crystal 9.3 cm from the Top of the
Crystal......... o o o o o o o,.. o o 76
18 o- as a Function of Pulse Height for a 2" Spherical
NaI(Tl) Crystal..................... 77
19 Comparison of Theoretical and Experimentally Measured Photopeaks for a 2" x 2" NaI(Tl) Crystal........ 78
20 Experimentally Measured and Interpolated Compton
Continua for a 2" x 2" NaI(Tl) Crystal............ 80
21 Final Normalized Pulse Height Spectra as a Function
of Energy for a 2" x 2" Nal(T1) Crystal.......o.. o 81
22 Pulse Height Spectra as a Function of Energy for
a 2" Spherical NaI(Tl) Crystal...................... 82
23 The Ratio of the Areas Under the Photopeaks of
the Pulse Spectra for Point Sources at 5 cm and
10 cm from the Top of a 2" x 2" NaI(Tl) Right
Cylindrical Crystal........ o........... oo....... 85
24 The Ratio of the Intrinsic Peak Efficiency of a
2" Spherical Crystal to the Intrinsic Peak Efficiency
of 2" x 2" Right Cylindrical Crystal as a Function of
Energy.....Q..ao ooooeooooooo4OOOO.OO~~OO~~.~~.~ 87
87
198
25 Measured Pulse Height Spectrum Au o Bare Foil 10.0
cm from the Top of a 2" x 2" NaI(Tl) Right Cylindrical Crystal.o oo.o Oo...o o o..o.... 0aoo.oo.... 90.
26 Measured Pulse Height Spectrum Aul98, Cadmium
Covered Foil 10.0 cm from the Top of a 2" x 2"
Nal(T1) Right Cylindrical Crystal.... o.. ooo 91
27 I131 Pulse Height Spectrum; 2" x 2" NaI(Tl)
Crystal Source 10 cm from the Top of the Crystal.
The Amount of Each Monoenergetic Component Obtained
Using the Least Square Fitting Technique is Also
Indicated oo o...o...........o o o o o o o o o o 96
vii

Figure Page
187
28 Pulse Height Spectrum W 10 cm from 2" x 2"
in Nal(Tl) Crystal Detector. o o o o o............... 98
29 Illuminance at an Axial Point by a Circular Disk.,,o o. 102
30 Pulse Height Spectra of Degraded Radiation Due
to o661 Mev Gamma Rays Scattering Through 0"-11"
of Steel. A 2" Spherical NaI(Tl) Crystal Used
to Measure Pulse Height Spectra.........o o oo......o 104
31 Linear Absorption Coefficients and Energy Buildup
Factors for o662 Mev Gamma Rays Passing Through
Steel Slabs....... o...................... 0 o o o o 0 108
vaiii

CHAPTER I
INTRODUCTION
In recent years, the scintillation spectrometer has become an important instrument for the detection and energy measurement of gamma rays(1,2)
Other important experimental techniques used for gamma ray spectroscopy are
magnetic spectrometry and diffraction and reflection spectroscopyo These
latter techniques are used mainly for energy identification and are extremely
precise for this measurement. Both of these two methods require very high
source strengths with respect to those required for scintillation spectroscopyo This is the first important limitation. Furthermore, magnetic spectrometry and diffraction and reflection spectroscopy measurements can only
be performed for point source or parallel beam geometrieso Efficiency values
are very difficult and sometimes impossible to determine for these two systemso Thus the problem of determining gamma ray intensities is either extremely difficult or impossibleo Problems involved in studying the continuous gamma ray energy distribution would also be impossible using magnetic
spectrometry and diffraction and reflection spectroscopyo Also the instrumentation required for these two techniques is so difficult and expensive to
obtain, that their general use is greatly limited. Scintillation spectroscopy
therefore appears the best general method for the investigation of gamma ray
spectrao It is the purpose of this work to develop general analytic methods
for use in gamma ray spectroscopy and dosimetry using scintillation detectors.
NaI(Tl) crystal scintillators are particularly used for gamma ray
spectroscopy because of their comparatively high efficiency in converting the
energy of the gamma ray into scintillations~ Further, over a large energy
-a

-2range, there is a linear relationship between the amount of gamma ray energy
lost in the crystal, and the intensity of scintillations produced. Thus a
flux of gamma rays passing through such a crystal will produce a distribution
of scintillation intensities dependent upon the energy lost in the crystal.
A multiplier phototube is used to measure the intensity of these scintillations
and a voltage pulse whose height is proportional to the scintillation intensity
is produced at the phototube output. An analysis of the pulses as a function
of pulse height can be made using various types of pulse height analyserso A
so-called "pulse height" spectrum is obtained from such a measurement.
For the case of an incident monoenergetic flux and a well-defined
source detector geometry9 rather simple techniques have been developed to determine the absolute magnitude or intensity of the incident beam from the monoenergetic pulse height spectrumo(2,3) Since a gamma photon may lose its energy
to the NaI(Tl) crystal by photoelectric absorption, by Compton scattering,
and by pair production, and since there is a further smearing due to mechanics
in light collection and in the statistical variation in the gain of the phototube, a monoenergetic gamma flux incident upon a NaI(Tl) crystal produces a
distribution of pulse heights. The monoenergetic pulse height spectrum will
be characterized by a photo-peak, a Compton continuum, and annihilation escape
peakso The pulse height spectrum due toapolyenergetic gamma flux can be shown
as a summation of the pulse height spectra of the various monoenergetic components in the polyenergetic beam,
When the incident gamma flux is polyenergetic and when the source
detector geometry cannot be well defined, it becomes extremely difficult to
determine the incident gamma energy distribution from a measurement of the

-32
pulse height spectrum. It has been the purpose of this work to develop methods of analyzing such complex pulse height spectra for both those cases
wherein the source detector geometry can and cannot be specifiedo
A method for analyzing complex pulse height spectra for the case of
a well defined source detector geometry has been described by Stephenson and
Bell,(3) the so-called "peeling off" process. It was believed that a more
rigorous analytic technique than this was requiredo A least squares fitting
technique for the analysis of the data was therefore developed. In this the
pulse height spectrum due to a polyenergetic distribution is synthesized by
using a series of normalized pulse height distributions due to the monoenergetic components in the incident beam. Each of these monoenergetic pulse height
distributions is weighted so that their sum is a best fit to the experimentally
determined polyenergetic pulse height distribution.* The monoenergetic pulse
height spectra used in these calculations can be both theoretically calcu(4)
lated() and experimentally determinedo
A further experimental difficulty lies in the fact that both the
shapes of the monoenergetic pulse height distribution and the crystal efficiencies are dependent on source detector configurationo It was predicted
and found that by using spherical rather than cylindrical NaI(Tl) crystals
this dependence can be greatly reduced for many caseso
The above analysis has been used both for the measurement of discrete and continuous gamma ray energy distributionso In the case of a distribution continuous in energy, the monoenergetic component corresponds tco al ernegy
in region AE about Eo
* This technique has been programmed for the IBM 704 computero See
Appendix lo

-4These techniques have been applied to a number of problems; the determination of thermal neutron fluxes, the determination of complex gamma
decay schemes, and the determination of the linear absorption coefficients
and energy buildup factors from a measurement of the undegraded, and of the
degraded gamma pulse height spectra due to the scattering of gamma rays which
pass through a scattering media, Experimental results have been obtained,
and have been compared with theoryo

CHAPTER II
PROPERTIES OF SCINTILLATION DETECTOR SYSTEMS
Ao The NaI Crystal
Energy exchange between gamma rays and matter can take place in
many ways, but in terms of the scintillation process, there are only three
processes of major interest. They are photoelectric absorption9 Compton
scattering and pair production. Other interaction processes do not impart
a significant energy loss to the crystalo(2,5) Since it is the amount of
kinetic energy imparted to the secondary electrons by the interacting gamma
ray which is of interest in the scintillation process, it is important to
investigate the three above mentioned interaction processes in order to ascertain the amount of the total gamma ray energy given up as kinetic energy
to secondary electrons in each process.
Photoelectric absorption is considered at firsto This is the interaction of a gamma ray with the bound or orbital electronso All of the energy
of the gamma ray is lost in this interaction, but not all of the energy is
imparted to secondary electrons as kinetic energy. Some energy is required
to overcome the binding energy of the electron in the given shello'.6) Thus
the amount of energy available to produce scintillation is equal to the energy
of the gamma ray minus the binding energy of the electron. Of course, after
the photoelectric absorption, x-rays are produced with energies almost equal
to the binding energyo The absorption of these x-rays and their conversion
to the kinetic energy of secondary electrons will then reclaim, in a sense,
the lost energy.
-5

-6In the Compton scattering process, the electron may be treated as
unbound or free with both conservation of energy and momentum (7) The gamma
ray can be scattered through any angle with a diminution of energyo In terms
of the scintillation process, all the energy lost in scattering will be given
up to secondary electrons as kinetic energyo Furthermore, the gamma ray may
suffer one or a number of Compton scatteringso As the energy is degraded,
the probability of photoelectric absorption increaseso Thus the gamma ray
may lose only part of its energy (only Compton scattering) or its total energy
(Compton scattering followed by photoelectric absorption) to the crystal.
In the pair production process, a gamma ray loses its energy in the
field of the nucleus to produce an electron positron pairo The energy above
that which is required to produce the electron positron pair is imparted as
kinetic energy to the electron and positrono(8) This kinetic energy is then
available to produce scintillations. The positron annihilates with an electron and two 0.51 Mev gamma rays (most probably) are producedo These gamma
rays can either lose all or part of the energy by Compton scattering or photoelectron absorption in the crystal0 Thus, there is a possibility that the
gamma ray may lose any amount of its energy from the maximum energy down to
the maximum minus 1.02 Mevo
By any of the above mentioned processes or combination thereof, the
gamma ray loses all or part of its energy to the crystalo Having converted
this energy to high energy electrons, the crystal will then experience ionization due to an energy loss per unit path (dE/dx] of the electron in moving
through the crystal0 This is similar to the ionization of a gas by a charged
particleo In the case of the crystal one deals with an insulator in which
the band theory is applicable o(9,10)

-7The allowed values of energy for bound electrons in a perfect
NaI crystal belong to intervals of energy, the valence and conduction bands,
which are separated by unallowed intervals, the forbidden bandso Passage of
a high energy electron through the crystal excites electron-hole formation
in the valence bando
The NaI(Tl) crystal is not perfect and because of the presence of
thallous ions and lattice defects, there are localized permitted electron
levels that lie in the normally forbidden interval between the conduction
band and the valence bando Some of the levels are called trapping centers
and are due largely to lattice imperfections but are partly due to the presence of impurity ions. There are other allowed levels between the bands
called luminescence center levelso The mechanism of light emission (scintillations) is attributed to the existence of these centerso An energy level
diagram is shown in Figure lo(ll,12
It is believed that the luminescence centers consists of pairs of
thallous ions. These pairs have some of the properties of ordinary diatomic
moleculeso The energy E of those electrons within the radius of a pair of
thallous ions as a function of the inter-ion separation r is of the type
shown in Figure lo (The luminescence center levelo)
The lower curve A is for the electronic ground state and the upper
curve B is for an excited electronic stateo A and B have shapes that allow
the thallous ion to vibrate along r. Some of the vibrational levels are excited at room temperatureo By the Frank-Condon principle,(ll) changes in
electronic energy are represented by vertical lines at the extremes of the
ion vibration locus on the energy diagrams, whereas the ions are instantaneously stationary in the classical picture. In the transition of the electron

-8CONDUCTION BAND
TRAPPING LEVEL
Eb ~ A
NORMALLY
FORBIDDEN
->.(~ — ~\ —-~/ —— ~ ~REGION
w LUMINESCENCE
CENTER LEVELS
a.
h.
VALENCE BAND
Figure 1. Energy Level Scheme for NaI(Tl) Phosphor.

-9from Eb to Ea, a photon of energy Eb - Ea is emittedo Since transition
may occur at the extremes of these ion vibrations, one obtains photons whose
wavelengths lie in another broad band about energies corresponding to 3 ev
or 4100 Ao(13)
Electrons excited to the conduction band can fall to the valence
band through the trapping levels and the luminescence center levelso Transitions via the luminescence center levels are accompanied by the emission of
lighto This is the light of interest in the scintillation processo Transition by other levels are radiationless in terms of the scintillation process
NaI(Tl) has the property that the intensity of scintillation or
total light energy produced is proportional to the energy lost by gamma radiation, the fast particle, or particles that produced ito This is the property
of NaI(T1) that makes it useful in gamma-ray spectrometryo With respect to
gamma ray interaction, Figure 2 shows the emission spectrum and the absolute
conversion efficiency of the crystal ionization excitation into light emissiono(l4) The total light energy produced Ej, starting with the initial excitation energy of electron, Ee is
00
E = Ee f Cnp(X)dX (1)
where Cnp(X) is the conversion efficiency as a function of wavelength.
of the photons produced in the scintillation processo Cnp(X) does not depend
upon the initial excitation energy of electron Ee o 12,14) Thus there is a
linear relationship between the total light energy EQ produced and the excitation energy of electron Eeo

16- - 0
* 12 -- -1 -4900A 60 rCa
0Q \ < p 56 ma/wott O
-<G { \ ~0-.- 0s
- 8 —y —- -- -40C) 4u
W Caoo
3500 4300 5100 900
Figure 2. Emission Spectrum, Absolute Conversion Efiien Spectrum
Id Photocathode "Absorption" Spectrum for NaI(T1) Crystal
and End Window (Sb Cs3) Photocathode Multiplier Phototubes.

-11Finally, other properties of NaI(Tl)(1) of interest with respect
to the design of a detector are the following:
1o Transparency to its own luminescenceo
2o High relative light yield (compared to other scintillators)o
35 0.25 pi sec decay constant of scintillationo
4o Very hygroscopic.
B. The Multiplier Phototube
The intensity of the scintillation produced in the NaI(Tl) crystal
induced by the absorption and scattering of gamma rays have to be measured
accurately. Both DuMont 6292 and RCA 6342A multiplier phototubes were used
in this measurement. Light from the scintillating NaI(Tl) crystal passes
through the glass envelope and ejects photoelectrons from the photocathode.
The photosensitive material in the photocathode of the DuMont 6292 and RCA
6342A multiplier phototube is Cs-Sbo This material is used because of its
good photoelectric yield for the wave length of luminescent emission from
scintillation crystals. The photocathode absorption spectrum is also shown
in Figure 2. The photoelectrons are electrically accelerated to the first
dynode, where they eject secondary electrons. The process is cascaded in
ten stages so that the charge of electrons ejected from the tenth dynode and
collected on the anode is about 6 x 105 times the original charge from the
cathodeo This is true for the case where the potential difference per stage
is 105 volts (13) The fraction of photon energy which strikes the photocathode surface is Eg(Tp o Fp) where Tp is the transparency of the crystal and of the phototube optical seal, and Fp is the nonescape probabilityo

12For a well reflected crystal both Tp and Fp should be nearly unityo A
fraction of this energy will be absorbed in the photocathode, giving rise
to a number of low energy photoelectrons released inside the phototubeo The
efficiency of this conversion from photons to electrons depends on the absorption spectrum of SbCs3 and on the electronic stopping power of the photocathode and is measured by the sensitivity factor S(x), shown also in Figure
2o
Now dne/dE = S(X) or dne = S(x) dE but from (1) above,
dE = dE, = Ee Cnp(X) dX
So
dne = Ee Cnp (X ) S(X) d Tp Fp
or the number of photoelectrons ejected is ne and depends on the product
of the spectra of Figure 2.
00
ne = Ee J Cnp(X) S(x) dX (2)
If Fc is defined as the fraction of such electrons striking the first
dynode and if the subsequent phototube multiplication is M, the total charge
q produced at the anode, due to a single original gamma ray interaction is
q = n e e M Fc = Vo/C (3)
where e is electronic charge, the charge q is collected on the input
capacity C of an amplifier and measured as a voltage pulseo Thus it is
seen that the magnitude or height of the pulse is proportional to the atinitial excitation energy of electron Eeo However because of the statistical

-,3fluctuation in dE/dx, ne, M, and Fc, a distribution of voltage pulse
heights about some mean value Vo will be observed rather than a constant
pulse height as is indicated in Equation (3). This point will be further
considered later in this Chapter.
Co Construction of the NaI(Tl) Detector
Figure 3 shows the construction of the two scintillation detectors
used in this experiment. Most difficulties in the mounting and constructionof
scintillation detectors can be attributed to two properties of the NaI(T1)
crystals. First of all NaI is hygroscopic and in the presence of water vapor
and due to the release of free iodine, a yellow discoloration occurso This
discoloration interferes with the optical properties of the crystalo Therefore one must take great care to prevent any moisture from contacting the
crystal surface. Secondly, NaI(Tl) has a high index of refraction (rN 18)
in the wavelength region of its fluorescence. Because of its small critical
angle with respect to air, the light generated by the phosphor will be almost
totally reflected unless a good optical coupling is made with the phototube
glasso
The crystals are handled and polished in a dry box having a relative humidity less than 10o% The exit face should be highly polished to eliminate scratches and other flaws in this surfaceo The other sides are polished
mainly to clean the surface of water and free iodine, and to eliminate any
large scratches or flaws which might trap the light. These sides are left
so that one has diffusely reflecting surfaceso Specular reflections in the
crystal will cause the light to be trapped in the crystal, and thus remove
it from measurement by the photomultiplier tube0 Further in order to more

J14effectively return the light striking the top and lateral surface of the
crystal to the exit face, a diffusely reflecting material (a - aluminum
oxide) is packed around the crystalo The system is then housed in an
aluminum cano (See Figure 35) In case of the cylindrical crystal, the
aluminum housing is made of 0.010" aluminum, and the aluminum housing for
the spherical crystal is made of 0,020" aluminum, The only reason that a
0,020" aluminum housing was used for the spherical crystal was that thinner
housings were not available in the shape required. It is desirable to use.as thin a housing as possible so as to decrease the amount of scattering
from the materials surrounding the crystalso This scattering tends to perturb the measurement,
The exit face is optically coupled to the phototube envelope with
a material such as Dow-Corning DC-200 Silicone oil with a viscosity of 106
centistokes or white ophthomological petrolatum, Care must be taken in making this optical connection (ego,, one should attempt to exclude large bubble formation in the optical connection), The canned crystal is then hermetically sealed to the multiplier phototube using Apiezon Q vacuum waxo A
hyperdermic needle is inserted through the Apiezon seal and a vacuum appliedo
The needle is then withdrawn while pressing on the sealo This technique has
been described in more detail by Po Ro Bell, (2)
It was necessary to use a lucite light pipe for the spherical crystal in order to seal the exit face to the phototubeo An optical seal is
therefore necessary between the light pipe and the crystalo It is believed
that one of the reasons for the loss in resolution of the spherical crystal
can be attributed to difficulties in making proper optical connection between
crystal, light pipe, and phototube, It is rather difficult to polish the

-15RANGE OF ANGULAR
CONSTANCY OF PULSE AI205 REFLECTOR
HEIHT SPECTRA....No (T )I. LIGHT PIPE
— OPTICAL SEAL
ALUMINUM HOUSINGIEN Q
SEALING WAX
MAGNETIC
MULTIPLIER SHIELD
PHOTO TUBE
(a)
figure 3a. Spherical Crystal Design.
ALUMINUM
HOUSING
\! All0s /,
REFLECTOR
Nol (Tt)
-/APIEZON Q
SEALING WAX,*;; U~ l Mu METAL
OPTICALZ l / MAGNETIC SHIELD
SEAL
MULTIPLIER
PHOTO TUBE
(b)
Figure 3b. Cylindrical Crystal Design.

-16exit face of the sphere without deforming the sphere and it is also rather
difficult to eliminate air bubbles between the light pipe and the sphereo
Stray magnetic fields can have quite a large effect on the output
current of the phototubeo A mu metal magnetic shield is used to reduce
the effects of the earth's or other stray magnetic fieldso Further, any object at anode potential outside the glass envelope, which touches the glass
may produce spurious pulses. This effect can be prevented by either wrapping
the phototube with electrical tape or by wrapping the tube with a metal
shield which is held at cathode potential,
D, Factors Affecting Resolution
The gamma ray dissipates its energy in the phosphor producing high
energy photoelectrons. High energy photoelectrons produced in this way
produce pulses with varying magnitudes at the output of a scintillation countero A number of experimental and theoretical studies(15,16) of the widths
of the generated pulse height distributions have been performed. The following processes contribute most significantly to the broadening of the pulse
height spectrum.
1. Emission of Photons by the Phosphor
There is a statistical fluctuation in the number of photons per
scintillation, Other statistical variations may be attributed to a local
variation in luminescent efficiency of the phosphor due possibly to a none
uniform distribution of activity ions, to the fact that successive particles
lose different amounts of energy to the phosphor due to interaction, edge,
and scattering effects, and also to the luminescence process itself (ioeo,

-17conversion of absorbed energy to photon energy fluctuates for successive
particles)o
2o Collection of Emitted Photons by the Photocathode
Successive scintillations never occur at exactly the same position
in the crystal, and thus the photon collection efficiency of the photocathode
depends on the position where the scintillation is produced. Optical flaws
in the crystal and at the various optical seals further smears out the distributiono
30 Emission of Photoelectrons ne by the Photocathode
There is a statistical fluctuation of photoelectrons per scintillation released from the photocathodeo Further, there is a point-to-point
variation of photocathode response'also a random emission of thermal electrons
by the photocathode which adds to the variation in the pulse height distributiono
4o Collection of Photoelectrons by the First Dynode
Fc and Multiplication M by the Successive Stages
The variance due to the multiplication process can be shown to be
fundamentally statistical in nature(17) Further losses can be attributed to
the variations in the fraction of photoelectrons collected by the first dynode,
in the collection efficiencies of subsequent dynodes, and in dynode responseo
Other processes have also been considered such as the statistical
variation in dE/dx, and have been found to be negligible with respect to
those mentioned above. For further detail see References 13, 15, and 18o
Also it is shown in these references that the effects considered in (1) and
(2) above are small compared with those in (3) and (4)~ Therefore the

-18
theoretical calculation of energy resolution usually considers the statistical fluctuation in (3) and (4)(13,16,17,18) The statistical variation in
the number of photoelectrons ne produced at the photocathode can be considered to be Gaussian (l5l6,17,18) If Vo is the mean value of the pulse
height produced at the phototube anode due to the absorption of the gamma
ray energy Eo in the phosphor, then the probability P1 that a pulse of
height V1 is produced at the phototube anode due to the statistical fluctuation in ne is
(Vl-Vo)2
P1i = - = exp 2 (4)
ne(V1)
where a 2 is the dispersion in the photoelectron productiono Here we have
only considered the variance in neo
Now we consider the effect of the variance in the multiplication
Mo It is assumed that the distribution due to the statistical fluctuation
in M is Gaussiano(16 17) Therefore the probability that the phototube
multiplication of ne(Vl) produces a pulse corresponding to height of V2
is
2
(v2-Vl)
M(V2) e
P = - exp 2
2 M(V1)
2
where T is the dispersion due to the statistical variation in the multiplicationo
The probability P3 of obtaining a pulse of height V2 due to
the complete absorption of Eo energy is the product of Pi o P2 integrated
over all V1
- +00 -Vp-V0o)2 ] -(V2-Vl)2e d
P> = f exp 22 ] exp[ - - dV1
"~~~~~~ V

-19let e (v2-Vo) and x = (V- Vo)
P3 = exp[- x ] exp-(e)] dx
-00 2a2 2am2
exp+x 2an2 C 2 4n2 mC2 (2oan + 2o ) 2)
= exp-[(x - 2un nT N d
-cc 2ann2 + 2m2 (2an2 + 2am2)2 4 oan2m2
Let z = (x 2a2+ 2
2n2+ + 2 22
+00 2 2
P3 = exp[-E2/2an2 + 2am2] exp[- 4an2 m2 z21 dz
-00 han am
= Cexp[-C2/2an2 + 2am2] = Cexp[-e2/2a2] (6)
where a2 is the dispersion of the photoelectron rate and subsequent multiplication rate, respectivelyo C is a.constant obtained from the definite integral over zo
Analysis has shown (17,18) that
2 -_
an ne = C1Eo = C V
and
am2 ne f(M) = C2Eo C2 V
where f(M) is a function of the multiplication, and is constant for a
fixed Mo
a2 = C3Eo = C3 Vo (7)
Experimentally it has been found that relation (7) is true for
limited energy ranges (16)
The Gaussian form given in (6) describes the pulse height distribution generally, although one cannot use relation (7) to determine a2 in

-20generalo Over limited energy range, it has been found by this author that
the relation
a2 = CEn = C'Von (8)
can be used more generally~ n is determined experimentally. This equation
will therefore be used later for calculation (see Chapter V)
Eo The Multichannel Pulse Height Analyser
The voltage pulse distribution at the output of multiplier phototube due to the scintillation produced in the phosphor must now be measured
as a function of the magnitude of the voltage pulse. This can be done using
various types of pulse height analyserso A brief outline of the operations
of the multichannel analyser used in this experiment is given below. Detail
information concerning the exact design can be found in the operating manuall(19)
Figure 4 is a block diagram of the pulse height analysero The voltage pulses from the multiplier phototube is fed into a linear amplifiero The
amplified voltage pulse from the output of the linear amplifier is then fed
simultaneously to three different placeso The voltage is fed to the stretcher
which maintains this voltage as a bias on the discriminator The voltage signal also starts the address scaler which begins counting pulses from an oscillator. Finally, a ramp-generator is started by this amplified voltage
pulseo The ramp generator begins generating a linearly increasing voltage
which is fed to the discriminatoro When this voltage "ramp" exceeds the bias
voltage on the discriminator the discriminator sends a stop signal to the
address scaler which turns off the scaler counto The address scaler has
counted the number of oscillator pulses proportional to the magnitude of input voltage pulse, and thus to the pulse heighto The input pulse height

-21PULSE HEIGHT ANALYZER
BLOCK DIAGRAM
INPUT
( * PM TB)AMPLIFIER OSCILLATOR
(P.M. TUBE)
START STARTS
RAMP ADDRESS
STRETCHER
GENERATOR SCALER
STOP
DISCRIMINATOR
- MEMORY READ MEMORY
ADD ONE
WRITE MEMORY
RESET
Figure 4. Pulse Height Analyser Block Diagram.

-22information (data in analogue form) has been transformed into a number
(data in digital form)o The information is now stored in the address scalero
The number or "address", stored in the address scaler, corresponds
to a position in the memory. This "address" or memory position is then sent
to the read memory. Upon a signal from the discriminator, the number stored
in the memory at the "address" in the read memory unit is extractedo The
number extracted is equal to the sum of all the previous pulses of the same
pulse height as the one being presently analyzedo This number is increased
by unity in the add one unit, and is then returned to the memory at the appropriate address by the write memory unit. After this step the analyser is
reset to accept another pulse.
All the pulses are placed in the same pulse height channel, if
they differ in pulse height by a voltage increment less than the voltage
change of the ramp in one oscillator cycleo The pulse height interval AV
is given by
AV = Ramp slope
Oscillator frequency
Thus using this pulse height analyser one obtains the total number of pulses
in AV about V as a function of Vo The spectrum measured in this way is
called the pulse height spectrum.
The above discussion only describes part of the analyser operation,
The method of data readout, the problems of what happens when one pulse
arrives before another has been completely processed, (pulse pile up) and
other such problems have not been considered, These problems are handled
differently by various types of analysers, and it is believed that such discussion will not add significantly to the discussion of the problems

-23considered in this paper, although it must be noted that there is a possibility of distorting the shape of the pulse height spectra due to pulse pile
upo This problem will limit the source strength that can be used for this
measurement. The problem of pulse height distortion due to pile up for the
pulse height analyser used in this work is discussed in Reference 19o Care
was taken in performing this experiment, not to use sources strengths which
would distort the pulse height spectrum due to pulse pile upo

CHAPTER III
PROPERTIES OF PULSE HEIGHT SPECTRA
Ao Shapes of Gamma Ray Pulse Height Spectra
The pulse height spectrum obtained when monoenergetic gamma-rays
are detected using a scintillation system is never a line, but is of a
shape determined by the energy of the gamma ray and the source detector configuration. The shapes of these monoenergetic pulse height spectra are determined by the following factorso
1o The relative magnitude of the photoelectric, Compton, and
pair production cross sectionso
2o The losses and statistical fluctuation that characterizes
the crystal, light collection and photomultiplier system.
The second point was discussed in the previous section, and it
was indicated that one could describe this smearing for a given energy Eo
(or pulse height position Vo) by
y =i Am exp - E (9)
y = Am exp - - nE —
where y is the count rate at energy E or pulse height V9 and Am is
a constant.
The first case considered. is that wherein photoelectric absorption
predominates and Compton scattering and pair production can be considered
negligibleo In this process, as was discussed in the previous section, the
kinetic energy imparted to- a secondary electron is equal to the energy of
the gamma minus the electron binding energy. This binding energy can be reclaimed in terms of the scintillation process by the absorption of the x-rays
-A4

-25produced after photoelectric absorptiono There is also the possibility that
the x-rays may escape the crystal without being absorbedo The pulse distribution due to photoelectric absorption is characterized by two regions; the
region of total absorption (the photopeak) and the region of total absorption minus x-ray escape energy (the escape peak). This smearing plus the
Gaussian smearing discussed above yields a pulse height spectrum similar to
that shown in Figure 5(20) The distribution under the photopeak can be
described by Equation (9).
When Compton scattering becomes an important energy loss mechanism,
another region is observed in the pulse height spectrum, the so-called Compton
continuum. In terms of the scintillation process as was discussed in the previous section, all the energy lost in scattering will be given up to the electron as kinetic energy. The gamma ray may lose part of its energy to the
crystalo Furthermore, after suffering a Compton collision or a number of
Compton collisions it may then suffer a photoelectric absorption losing its
remaining energy. Thus there is both a possibility for the gamma ray to lose
all of its energy in the crystal, or part of its energy in the crystal and
part escaping the crystal at a diminished energy. Because of the multiple
scattering events, the shapes of the Compton continua can only be calculated
theoretically using Monte Carlo techniques.(4) The shapes of the Compton
continua have been theoretically calculated, and experimentally determinedo
There seems to be agreement between experiment and theory(4) In Figure 6,
the unsmeared Monte Carlo calculation obtained from Reference 4, the smeared
theoretical calculation and experimental determination of the pulse height
spectrum for a Csl157 (o66l Mev) are showno The experimental measurement was

-26100._-__'_II__
____~...1 -I
Sc GAMMA RAYS ON
3 x 3-IN. Nol CRYSTAL.
^~50~ _______ ______SOURCE AT 3 CM.
150KEV (PHOTOPEAK)
20
IO
- _
5 -
IODINE X-RAY
ESCAPE
1.0
0.1
0 200 400 600 800 1000 1200
PULSE HEIGHT
Figure 5. Sc47 Gamma Rays on 3" x 3" NaI(Tl) Crystal.
Source at 3 cm.

-276 —
4 - - I ___ ___ __ ___ __ PHOTOPEAK___
4-~~t -
3 -- ---- -- -- --- 180S BACKSCATTER PEAK - - - -
7
6.
W
IAJ
1 3
z
2-EXERIMENTAL DETERMI-ATION
0 -
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150) 160 170
PULSE HEIGHT
Figure 6. Theoretically Calculated and Ex~perimentally Determined
Pulse Height Spectrum for Cs10.^^l Mev). Parallel
Beam Incident upon a 2" Spherical Nal(Tl) Crystal.
73
* UNSMEARED MONTE CARLO CALCULATION
2 SMEARED THEORETICAL CALCULATION
10 20 30 40 50 60 70 80 90 100 110 120 130 140 ISO 160 170
PULSE HEIGHT
Figure 6. Theoretically Calculated and Experimentally Determined
Pulse Height Spectrum for Csl35((O.661 Mev). Parallel
Beam Incident upon a 2" Spherical NaI(T1) Crystal.

-28made using a 2" spherical crystalo The value a2 and n in Equation (9)
were determined experimentally by studying the shape of the photopeaks for
monoenergetic emitters as a function of energy (see Chapter V). The numbers obtained by Monte Carlo calculation(4) were smeared out using Equation
(9) and the experimentally determined values of a2 and n and are shown
in Figure 60 The difference between the theoretical and experimentally
determined pulse height spectrum can be attributed to the scattering of the
gamma rays from surrounding materials external to the crystalo When the
amount of material surrounding the crystal is decreased, there is much better
agreement between theory and experiment (4) There is an attempt being made
at present to include the effects of the scattering in the surrounding materials (eog., the materials used for canning, the reflector, and phototube)
in the Monte Carlo calculation by the Applied Mathematics Division Argonne
National Laboratoryo Hopefully with this scattering included in the calculation even better agreement will be obtained,
At energies higher than 2 Mev pair production becomes appreciableo
Two false "photopeaks" are then observedo Figure 7 is the pulse height spectrum of Na24o The gamma ray energies emitted by Na24 are 2.76 Mev and lo38
Mev. The three peaks of greatest pulse height are due, in order of increasing pulse height, to
1o Pair production with escape of both annihilation quanta
2. Pair production with the absorption of one annihilation
quantum
30 Pair production with absorption of both annihilation quanta,
and to total absorption by photoelectric effect or any combination of other effects leading to total absorptiono

-29Id
9
a - --
7 -1.38 MEV
6
r DOUBLE ESCAPE
AT 1.74 MEV
3 -- -- --- -- SINGLE ESCAPEAT 2.25 MEV
2r I I T r I- I 1 1 1T2.76 MEV
IC?
3- NoI CRSTAL
0 200 400 600 800 1000 1200 1400
6 6
PULSOURCE AT 3 CM. FROM,,
Figure 7. Na24 Gamma Spectrum Source at 3 cm from
733" x 3" NaI(T1) CrystalRYSTAL
0 200 400 600 800 1000 1200 1400
PULSE HEIGHT UNITS
Figure 7. Na Gamma Spectrum Source at 3 cm from
3" x )" NaI(Tl) Crystal.

-30In addition to the photopeak, iodine x-ray escape peak, Compton
continuum, and pair escape peaks, there are a number of other regions
characteristic of experimentally determined monoenergetic pulse height
spectra which are as follows
lo Multiple Compton scattering regiono Because of such scattering from materials surrounding the source and crystal which
degrades the primary energy, there is a continuous distribution of gamma rays incident upon the crystal with energies
less than the maximum energy, This tends to smear out the
true Compton continuum due to gamma rays of the undegraded
energy scattering in the crystalo
20 Annihilation radiation from the surroundingso Positrons
emitted from the source may annihilate in surrounding material.
Some of the 051 Mev gamma rays produced in such a manner
will reach the crystal and a pulse height spectrum characteristic of 0o51 Mev gamma rays will be superimposed on the monoenergetic pulse height spectrum (see Figure 8)~
30 Coincidence distribution. If two gamma rays interact with
the crystal during a time which is shorter than the decay
time of the light produced in the scintillation process, a
pulse will appear whose height is proportional to the sum
of the energy lost to the crystal by both interacting gamma
rays,
Since the interaction time for a single interaction or multiple interactions is shorter than the decay time of the light in the crystal, a
single gamma ray interacting with the crystal produces only one pulse, the

-31Dop- I - ZZE _ EF - - I1
8. 8_l__II_ _l -.6i Ml v l - I Iev M l l
6-.......y Aftnihllofon _ Photopeok _____
o 6
4
0.1 -- - - -
0 6 -- - -- - - -- —.............
Figure 8. Zn65 at 0.24 cm from NaI(Tl) Crystal.

-352
magnitude of the pulse height is effected by the type or number interactions
for a given gamma rayo Thus if the above mentioned coincidence effects are
negligible, the measured monoenergetic pulse height spectrum can be considered
as a distribution of the probability of energy loss as a function of energy
for the given gamma ray energy and geometrical configurationo In addition
the shape of the monoenergetic pulse height distribution depends on the
source detector geometry. By using spherical rather than cylindrical crystals, it will be shown later in this chapter that this geometric dependence
can be greatly reducedo This factor is of great importance in certain cases
where the source detector geometry cannot be specifiedo This point will be
discussed further later in this Chapter and also in Chapter V.
Again if coincidence losses are negligible, the pulse height distribution due to a polyenergetic gamma flux will be a summation of the pulse
spectra due to the various monoenergetic components in the polyenergetic
gamma fluxo A simple example is shown in Figure 9. The pulse height spectrum due to a source which is a mixture of Cs137 (0o661 Mev) and Cr51
(0~320 Mev) was measured, Figure 9ao The pulse height spectra of Cs137
and Cr5l were also measured separately obtaining the pulse height spectra
in Figure 9b, The sum of the monoenergetic spectra is also shown in Figure
9b, and is exactly the same as the spectrum obtained in Figure 9ao The
source strength of Cs137 and Cr51 were the same in both measurements,
Bo Detection Efficiencies
One pulse appears for every primary collision~ The number of
pulses produced independent of pulse height is equal to the number of primary collisions. Only the pulse height is affected by the multiple

.320 Mev.661 Mev
10.0
- -
1.0
z
0 Figure 9(a). Pulse Height Spectrum of a Mixture
O^ of Cs137 and Cr51.
The sum of the Pulse Height Spectro of
C" plus Cr I
Pls Het Pulee Height Spectrum Cs137
Pulee Height [/
Spectrum Cr7 / /
10.0
1.0
0 100 200 300 400 500 600 700
PULSE HEIGHT
Figure 9(b). The Pulse Height Spectra of Cs137 and Cr51.

4-p
collisions, The total area under the pulse height distribution AT (ioeo,
total number of interactions independent of pulse height) equals the number of primary interactions which have occurred
For a point source Io of gamma rays
AT = Io(l/4T)eTi (10)
where Q = 2jr(l-cos G) (see Figure 10), and -c pi the crystal detection efficiencyo
The nature of the detection efficiencies is now consideredo In
the following discussion pair production is not includedo The analysis
of course can be extended to include this process but in this paper, only
energies where this effect is either zero or negligible is considered. The
cross sections of interest therefore will be those for photoelectric absorption and Compton scattering.
Consider the case of a monoenergetic point sourceo It is assumed
that there is no scattering from surrounding source materialso
If Io gamma rays per unit time are emitted from the source
then Io(Q/4Ar) is the number of gamma rays incident upon the surface of the
detector, Further, if any interaction in which, a gamma ray produces scintillations in the crystal is considered an absorption interaction, then p,
can be defined as the linear absorption coefficient for first interactions
due to photoelectric and Compton interactionso With respect to Figure 10,
e -p is the noninteraction probability along the path p through the
crystal, and (l-el'P) is the probability of suffering a first interaction
along po Then
ETa = f (1-ei'P) dS/4j = the total absolute efficiency (ll)
Q

-35-,55-O \:e 8mo>\ Ih8 h
NoI(T) (
(o) (b)
Figure 10. Source Detector Geometry for (a) Cylindrical
Crystal and (b) Spherical Crystal.

-56This is the probability that a gamma ray emitted from the source will interact in the crystalo p is the path length in the crystalo The integration is carried out over the solid angle subtended by the source and surface of the crystal.
Then
ETA
~e T= = the total intrinsic efficiency (12)
Ti f dQ/4t
Both integrations are carried out over the solid angle described above.
eTi then is the efficiency factor discussed previously in Equation (l0)o
This efficiency factor has been studied as a function of source detector
geometry for a number of energies and for two differently shaped crystal
detectors, the right cylinder and the sphereo Values of eTa as a function of source crystal geometry for cylindrical crystals are available in
the literature (21l22,23)
Equation (12) has been. studied for spherical crystalso The resuits are outlined below. See Figure lObo
ETa = (1-ee P) dQ/4i
For spherical crystals
2a
eta = 1/2 (1-e -2a) f e _ p2 + 4(p2 _ a2) dp (13)
o 4P
where
h + a
and
p =2 Aa2 - 2 sin2Q

-57h is the height of the source above the crystal surface and a is the
radius of the sphere. The fraction of solid angle subtended is
Q/4i = l/2(l-cos G) = 1/2 (1 -( + a2) (14)
Let h = mao Then p = a(m + 1) and d = 2a. Using this substitution in
(13) and (14),
d -_i P 2
Ta = 1/2 (l-e -d ) - d2 (1 - 2) dp (1a)
I m-~ + (12 (m+1)2
Q/4 = 1/2 [1 - 1 - (14a)
(m+1)2
Now let
2 - [1 - 2
(m+l)2
then
Ta = 1/2 [(1 - e ) - /d e- 1 7 - + y22 (13b)
/4er = 1/2 (1 - y) (14b)
Now the intrinsic total efficiency is obtained from (13b) and (14b)
d 2
e ETa e ----- - e-le tf P + -72d2dp (15)
Ti 2 l-y d 0 (m+l)2
This can be rewritten as
d 2
t 1'! e [1 - 72 Y+ -2(l d)P (16)
Ti 1-7 o d(ml)2
The form of the integral, as m = 0 and as m —- o, (ieo, for the
source on top of the crystal and for a parallel beam source) is m = 0
Tio = f tep [1 - p/d] dp (17)
0

-38and m -
d _ 2
CTio = S ue"-P [1 - P ] dp (18)
o d2
0
Both of these expressions can be integrated and eTio and ETi- can be
obtained. For other cases the definite integral cannot be found; therefore
consider the following factor in the integral Equation (17)! - 2 _P
1 [ 1-
Fm ~ 1-y ) d2(m+l)2
to a good approximation, it has been found by the author that
m aom + alm (p/d) + a2m (p/d)2 + a3m (p/d)3
+ a4m (p/d)2 4 am (p/d) (19)
2=o
The value of a.m can be found for various values of m and (p/d)o The
results are presented in Table Io Using (19) in (16a) and using the following definition
Io = (1 - e ) = 5/d Id - eeid
Il = Io/nd - e- d 14 = 4/[d I3 - e-4d
12 211/kid - e-Pd
after integration, it is found that
2=4
ETi o- S ajmI (20)
(2=0
With these results ETi as a function of distance above a 2"
spherical NaI(Tl) crystal for a number of energies was calculatedo The results are presented in Table II and some of the results are plotted in

TABLE I
THE VALUES OF am FOR VARIOUS mYs AND (p/d)'s
~ p/d m = O m 005 m 0.1 m = O2 m = O05 m 1 O m = 20 m co
0 0 1 1 1 11 1 1 1
1 1/4 -1 0o50310 O,02772 0025523 OO008280 OO000508 OO000244 0
2 1/2 0 -2i488485 -2.024169 -L1658707 -l240o919 -L1088209 -l105268 -1
\J4
3 5/4 0 2o186250 lo595994 Oj748565 O0195592 0o58016 ooo00686 0
4 1 0 -0748075 -0597547 -O135381 OO037247 o0o49685 0025621 0

TABLE II
INTRINSIC TOTAL EFFICIENCIES AS A, FUNCTION OF
DISTANCE AND ENERGY FOR A 2 it SPHERICAL NaI(T1) CRYSTAL
Distance in
cm above Ti
2 ino Sphere E = 0,130 Mev 0o152 Mev 0o211 Mev 0o333 Mev Oo566 Mev 1o10 Mev 2o04 Mev
0o0 0o934 Oo902 0o804 o 637 0O485 0O372 03005
0o127 0O984 0o967 0O892 0O727 0O558 o0o46 Oo351
0O254 0o987 0o971 Oo900 0o738 o571 Oo 428 0O359
0o508 Oo989 Oo975 0o909 0o751 O 583 0.448 o5369
1o27 0o990 0o978 Oo918 0o764 0o598 0o464 Oo 79
2o54 Oo991 Oo979 Oo922 0o771 Oo6o4 o0469 Oo384
5~08 0o991 o0980 0924 0~774 06o08 0~470 0386
0o991 0.981 0o925 0.777 o.6lo 0.471 0o388

Figure 11o Theoretical calculations using Monte Carlo techniques have been
carried out(4) and the results are also plotted in Figure 11o In Figure 12
a comparison of ETi as a function of distance for two energies is made
between a 2" spherical and a 1-1/2" x 1" right cylindrical crystal.
It is seen in both Figures 11 and 12 that eTi is within 5% of the parallel
beam case for a distance of 127 cm from the top of a 2 " spherical crystal and therefore for all practical purposes is constant from this distance
to infinity, ioeo, the parallel beam caseo Such a constancy does not occur
until almost 20 cm from the top of a 1-1/2" x 1" NaI(Tl) right cylindrical crystal. The results for the right cylindrical crystal were obtained for
a point source placed on the axis of the cylinder at the various distance
above the crystal's faceo(21) These results for the right cylindrical crystal are angular dependent whereas of course this is not true for the sphereo
The results presented for other sized right cylinders yield similar resultso
In terms of the analysis, it is more accurate to study the area
under the photopeak onlyo (See Chapter V) This area can be determined much
more precisely than the total areao There are two major reasons for the
difficulty in obtaining the total areao First, it is rather difficult to
eliminate all scattering effects due to the surrounding materials. These
will appear as pulses in the Compton continuum. Secondly, pulse height
analyzers cannot detect all pulses down to zero pulse height, for below
certain pulse height levels, the equipment noise and the thermal noise of
the phototube completely interfere with the detection. Therefore, consider
Ap = the area under the photopeak
epi = the intrinsic peak efficiency
or, E. is the fraction of those gamma rays striking the crystal
face which are totally absorbedo

1.0
z.7
Z I 5 n i ] I I I I -HEIGHT ABOVE CRYSTAL s OD
U \ n- / r-HEIGHT ABOVE CRYSTAL = 1.27 CM.
E-.6.46' -' —- ---- -- -
W z I I I I l\ I \ / I j t I /T-HEIGHt AINOVE CRFYSTAL i. OO CM. lI
0..6.8 1.0 1.2 1.4 1,6 1.8
0 2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 22
ENERGY IN Mev
Figure 11. Total Intrinsic Efficiency as a Function of
Distance for 2" NaI(Tl) Spherical Crystal.

.9 -.8 PARALLEL BEAM 2 SPHERE -
3 | —— E.333Mev
M PARALLEL BEAM I- 1/2X 1" RIGHT CYLINDER
Z PARALLEL BEAM 2 SP HEREER.7 r.. - ~. _..........____ _ _.....
>. ~~-^~~~~~~~~~~~~~~E E.333 Mev
S L._______ _______ ____ ____ ____ _^ I._____
k I - 1-1 " RT- CYLINDER1.5.-'"..
z4 PARALLEL BEAM 2 SPHERE
O2 I2 3 4 S 6 7 8 9 10 1
PARALLEL BEAM I-I/2X I" RIGHT C LINDER
LL. ^ _____ ~-_ a E= 2.04
4 ~____ PARALLELEAM~2' SPHERE -_
~ 1 I I 1 1 I 1 - - I-I/2"X I RT. CYLINDER
0 I 2 3 4 5 6 7 8 9 10 I
DISTANCE FROM TOP OF CRYSTAL in cm.
Figure 12. Comparison of c Between 2" Spherical Crystal
and 1 1/2" x 1" right Cylindrical Crystal.

-44If both sides of (10) are divided by AT/Ap we obtain
Al =Io /4jr epi
where
epi = Ap/AT:Ti = PTETi (21)
PT is the peak to total ratio Ap/AT
For right cylindrical crystals the peak to total ratio does not
change significantly as a function of distance along the axis of the cyline
der (22)
der.
If the pulse height distribution should not change as a function
of distance and angle, then the peak to total value does not change. Experiments described in Chapter V using a 2 " spherical crystal indicate
that this holds true for the Si spherical crystalo Furthermore for a
given distance (in our case 6 cm) the pulse height spectrum did not change
as a function of angle as long as the source remained in the angular range
indicated in Figure 3ao
Theoretical calculations of the peak to total values have been
carried out by the Applied Mathematics Divisiong Argonne National Laboratory,
and the results are presented in Table IIIo
In Figure 135, the intrinsic peak efficiencies for spherical crystal
are plottedo These efficiencies are used in the calculation to be presented
in Chapter Vo

-451.0,.8
p,
pi' XXh=~^^r ^
INTRINSIC
PEAK
EFFICIENCY.3_.2 ",-.1.2.3.4.5.6.7.8.9 1.0
ENERGY in mev
Figure 13. Intrinsic Peak Efficiency ~pi as a Function of Energy
for a Parallel Beam Incident Upon a 2" Spherical NaI(T1)
Crystal.

-46TABLE III
PEAK TO TOTAL VALUE FOR 2 " SPHERICAL CRYSTAL INCIDENT
UPON THE CRYSTAL PARALLEL BEAM OF GAMMA RAYS
Energy in Mev Peak to Total (Photofraction)
0.142 00927
0.320 0.749
0.411 0.645
0.510 0.540
0.661 o.438
0.768 0.395
0.835 0.374
1.114 0.306
1.277 0.284

CHAPTER IV
LEAST SQUARE ANALYSIS OF COMPLEX GAMMA RAY SPECTRA
Ao Formulation of the Principal of the Least Square
In the previous chapter, it was pointed out that if a number of
gamma rays are incident upon a NaI(Tl) crystal, the pulse height spectrum
is made up of a summation of the photopeaks and Compton continua of the
various monoenergetic componentso The problem now is to find the contribution of each monoenergetic component in the polyenergetic pulse height
distribution. It was shown that for a monoenergetic source of energy En
of gamma rays that
A Xin
Ion - -i in (loa)
on 2
4 ETin CTin
where
ATn is the total area under the pulse height distribution
corresponding to energy En
Ion is the source strength in gamma rays per unit time for
energy En
Xin is the most probable interaction rate at pulse height i
due to the interaction of monoenergetic gamma rays of
energy En
ETin is the total intrinsic efficiency for a gamma ray of
energy En
The summation is used because the measured pulse height spectrum is a histogram; that is, one measures the total number of pulses in some pulse height
interval AVi about Vio
-47

-48Further
i=n
Nn = Xi = EI (10b)
Nn is the total number of gamma rays per sec emitted from the source which
interacts with the crystal. If we have a polyenergetic flux of gamma rays
with p monoenergetic components, the total number of gamma rays interacting with the crystal can be expressed as
n=p
N =, Nn = Xin = i/ti (22)
n=l n i i
in terms of its monoenergetic components and pi is the total count at
pulse height i in the polyenergetic pulse height spectrum for a time tio
Further
o = Bin (22a)
where Bin = Xinti or the most probable number of interactions occurring
in channel i due to a most probable interaction rate Xino
The problem now is to determine the Bin for all p monoenergetic
components. Then from (10) the source strength of each component can be determinedo
In order to obtain the Bin s, the shape of the pulse height spectra for each of the p energy components in the polyenergetic flux are determined.o These pulse height spectra are normalized so that the area under
the photopeak is unityo The area under the pulse height spectrum for energy
En is Nn where
Nn = X1Yin (25)

-49where
Yin is the most probable normalized interaction rate occurring
at pulse height i for the monoenergetic gamma rays of
energy En
and
Ain = Yinti (23a)
Now take the ratio of (22) to (23) and define
Nn Xin
6n = L in (24)
n LYin
further
Xin = PnYin
and now substituting in (22a)
Pi = Z PnAin (25)
n
Due to the variance in the determination of Pi and -nAin
PnAin cannot be simply determined from Equation (25)o An exact solution
cannot be found, therefore the following procedure was used to find the
most probable values of PnAino
The error xi assoicated with the measurement in channel i is
Xi Pi PnAin
n
If it is assumed that there are statistical fluctuations in the determination of Pi and PnAin and these fluctuations are Gaussian in nature, the
probability Pi that there will be an error xi which lies between xi
and xi + dxi associated with the measurement in channel i is
Pi = __ exp (26)
\/2jr ai 2a

-50where ai is the mean square deviation associated with the measurement
in channel io
The probability P that there will be m errors xl, x2, ooo xm
which lie between x1 and x1 + dx1, x2 and x2 + dx2, ooo x-imand dxim
associated with the measurements in the m channels is the product of m
terms like Equation (26), for the measurement in a given channel is independent of the measurement in the other channelso Thus
2
m X dxi
P = I Pi = (exp - )( (27)
i=1l i 2ai 2r
The differentials dxi are arbitrary and the ails are fixed, The validity
of this statement is considered in detail in Reference 24,
The criteria for obtaining the most probable values of PnAin is
that P be a maximum(24) (ioeo, that there is greatest certainty)o
p=2 2p.,2'..
XM = i (Pi - = (nAin) = ( Min. (28)
i 2ai i
(since ai is fixed and axi is arbitrary), ui is the statistical weight
and is equal to 1/2a2o In the simplest case wir l/Pij if the counting
time in each channel is constant and if it is assumed that there is no variance in Aino This then is the formulation of the Principle of the Least
Squareo If the variation in Pi and PnAin is systematic and non-random
in nature, then the application of this principle may lead to erroneous
results in terms of the inference of the gamma ray spectrum from the pulse
height spectrum. For certain cases, error calculations can also be carried
out (see Section C)o

-51Bo Method of Obtaining Minimum for Least Squares Fit
1. Incident Gamma Flux Discrete in Energy (Energy
Distribution Known and Intensities Required)
A number of algorithms can be used to obtain the minimum required
in Equation (28). The method of solution used depends on how much is known
about the incident fluxo The simplest case is considered firsto In this
case the gamma ray energy distribution is known, and it is desired to determine the intensity.
Pi is measured, and since the energy distribution is known the
monoenergetic components Ain are knowno The minimum is therefore obtained
by taking the partial derivative with respect 3k for each of the p monoenergetic components. Each derivative is then set equal to zeroo Thus
EM _-2 ci (Pi - iAin) Aik =0 (29)
6Pk i n
for k = 1, 2, ooo, po There are thus p linear equations to be solved
for the BISo Relation (29) can be expressed in matrix notations as
follows (25)
Awp - (mA)P -=0 (30)
where P is a vector of the Pk'So
A is a p by n matrix of the pulse height spectra
n is the maximum pulse height
a is the transpose of A, and
c is a diagonal matrix of the cuis o
Solving for P, it is found that
p= (XnA)1 A'P (31)

-52Once the Pts are known from (31) the Bin can be obtained from (24)o
The area under the photopeak for each monoenergetic component can then
be determined. This number divided by the intrinsic peak efficiency will
be equal to the number of gamma rays of energy En incident upon the
crystal. The calculation described in Equation (31) has been programmed
for the IBM 704 computer (see Appendix I for the FORTRAN listing of the
program), and can handle up to twenty monoenergetic pulse height spectra
and up to one hundred values for each pulse height spectrum.
The pulse height spectra required for this analysis is obtained
experimentally and theoretically (see Chapter V).
20 Discrete Incident Energy Spectrum (Both the Energy
Distribution and Intensity of the Incident Beam Unknown)
The difficulty in the application of the technique lies in the
method of obtaining the minimum. The minimization should be made with
respect to both Pn and Aino Ain is a function of both pulse height
and energy while Pn is only a function of energyo Since the pulse height
spectra (Ain) is not known analytically as a function of pulse height
and energy, it is extremely difficult to attempt to minimize Equation (28)
with respect to the Ain ( ioeo, numerical methods would introduce large
errors in the calculation)o Thereforelthe following method was used.
The energy spectrum under consideration is divided into discrete
increments. A monoenergetic pulse height distribution corresponding to
each increment is included, The energy components or increments will be
chosen depending on the photopeaks observed in the measured distribution,
and in those regions where the photopeaks are not obvlous)the energy region

-53is divided up depending upon the energy resolution of the system. The method
of choosing these increments is considered later in this chapter, and in
Chapter Vo Now ideally one can use relation (31) to obtain the values An
for the various energy components If a given energy component m is not
present, Am should be zero or the statistical variance in Pm should be
greater than Pm itself. The presence of these zeros in the inverse transformation leads to the possibility of obtaining negative solutions which in
turn leads to oscillating components in the solution of Equation (31)o This
problem is treated in detail in a work by Wo Ro Burruso(26) In this paper
it is pointed out that the source of error in unscrambling scintillation data
by the incremental technique [ioeo, simple inversion of Equation (25)] can
be attributed to an error amplification when the basic equations are solved
exactlyo As is stated in this paper, this amplification is caused by an attempt of the exact solution to restore rapidly fluctuating components in the
original gamma ray spectrum which have been attenuated below the statistical
error level by the instrumental responseo A first attempt to smooth out this
fluctuation was made by this author using the least squares technique described aboveo A further smoothing can be obtained by requiring not only
that Equation (28) lead to a minimum but that the solution for the Ps be
positive or zeroo
Before outlining the method of solution for this case, an example
is presented to help clarify the discussiono The method described above
[Equation (31)] and the method to be discussed below is applied to a simple
problem.

-54A measurement is made with a three channel pulse height analysero
The following data is obtained
Channel i Counts
1 3
2 2
3 3
It is known that the measured spectrum is some linear combination
of the following functions.
Normalized Count
Channel i Ail Ai2 Ai3
1 1 1 1
2 1 1 0
3 10 0
The following matrices are formed according to previous definitions
A= /1 1 1 \
1 1 0
\1 0 0 (32)
A = /1 1 1
1 1 0
1 0 0 (33)
-= / 1/3 0 0
0 1/2 0
\ O 1/3 (34)

-55P= 3
2
Then
(LA ) = 7/6 5/6 1/3
5/6 5/6 1/3
1/3 1/3 1/3 / (35)
(iAA)o-= / l -3 0o
- 5 -2
\ -2 / (36)
(%LA)- = / o O 1 \
0 1 -1
\ 1 -1 0 (7)
The P's or intensities of each component vector can be determir
from Equation (31)o
P (? /uA) 16p
0 0 1\ / 3/ 3
0 1 -1 2 = 1
1 -1 0 \ 3 1 / (38)
That is the sum of the vecotrs (19 1, 1), (1, 1, 0) and (1, 0, 0) which
yields the best fit to the permental data usng the least suae citeria
is 3(1, 1, l) - 1(1, 1, 0) + (1, 0, 0) = (3, 2, 3). The residual is zeroO

-56Now if it is known that the intensities must be positive, the least
square fit must be made so as to require Equation (28) to be a minimum with
the constraint p1>- 0, P2-> 0, and 3> Oo One could proceed by eliminating
the negative element and then reevaluating Equation (31)o That is, the following matrices are now formed
A =1 1
1 0
\ 1 0 / (39)
A = /1 1 1\
\ 1 0 0 1 (40)
and. p remains the sameo
Then
(AA) =/ 7/6 1/3
k 1/3 1/3 / (l1)
(AA) = 6/5 -6/5 >
K -6/5 21/5 (42)
(AoA)-A /o 3/5 2/5 \
K1 -3/5 -2/5) (43)
Using these matrices, p is found to be
= (AwA) Atup = /5 2/5 / 3
1 -3/5 -2/5 2
3 (44)
12 /5
\ 3/5 / (44)

-57These results indicate that the best fit requiring a positive definite solution is
2,4(1, 1 ) + 0(1, 1, 0) + o6(1, 0, 0) = (3. 2o4, 204)
The method just completed is rather simple but one must be careful
in the method of eliminating the negative components. A method for minimizing quadratics subject to various constraints is described by Bealeo (27)
The following is an application to the above problem, and is a simplification
of the work found in Reference 27o The simplification is possible since
when applying the technique to pulse height analysis, the only constraint required is that of a positive solution for the P's when minimizing the quadratico
In this method one forms the quadratic given in Equation (28)o For
the conditions in the problem discussed above, Equation (28) can be written
as
M = 8 - 6p1 - 42 - 2P3
+ 7 2 + 5 2 i2
71 62 3
+ 5- PlP2 + 2- l3 + - P2P3 (45)
3 3 3
M is to be minimized and also the constraint
and P1l 0O, P2> OY 93>- 0 is to be appliedo
A geometric solution of the problem can be considered with reference
to Figure 14o The method of solution proceeds in the following mannero
Start at point 0 (Figure 14), that is, assume the solution P1 = P2 = P3 = 0
Keeping p2= - 3 = 0 increase f1 in a positive direction. As P1 increases
along this direction M will decrease until we reach point Ao Point A is

-58B3
II
II
PLANE OF I
\ 2~'D I =0
Uz'o A\ \; />~~~~~~~~~~~~~~~~~~-'",'
" """"-/ \ \,' -'/'' \P L A \
I I
PLANE -^ ^ a.| /J'\
I I
%
%
*I~'
/%,,'~~~~~
d %I!,
%,~~~~~
I A'
I,,, I I I'
"' ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~% %%
/,
Figure 14. Geometric Solution of Least Square Fitting
Problem.
I Is
Ii I..,
I ~ s /.J"..... I.....,_"
\ ~ i ~
r I~~~~~~~/ -,
3~~~~~~~~~~~ C l / - "
~~~~~~~iur 1 4GomercSlto fLatSur itn
Problem

-59determined by taking the derivative of M with respect to 1 and setting
the derivative equal to zero with P2 = P3 = 0
M- = +- 6 + 7 1 2 + = 2 (46)
Then for
U1 = = 3 = 0, p = 18/7
The coordinates of A are (18/7, 0, 0). The plane ul = 0 in the space
described by P1, P2, and P3 contain all points P1 for which M is a
minimum given any values of p2 and 3o
Continuing the solution, it is found that increasing p1 any further will only increase Mo A change of basis is now made. Using (46), p1
is found in terms of ul, P2, and P3o
P]. = 6 (ul + 3 5- P2 - 3) (46a)
This substitution is now made in (44) and
o 2 2 6 2
M = 2 + B2 - D 3 + u1
7 7 7 7
+ 5 2 + 5 2 (45a)
21- 21 B3 21
Now keeping ul = P3 = 0, one changes p2 attempting to decrease
M; that is P2 is increased or decreased by moving along the line of intersection of the ul = 0 plane and 53 = 0 plane. This intersection is along
the line AKL indicated in Figure 14. The problem is how to determine which
direction to move along AKL so that M decreaseso This can be done by taking the derivative of M (45a) with respect to B2

-60aM = 2 + 10 2 + 4 p3 + 10 U (47)
&52 7 21 21 7
at
_ - 3 - u ~, 2 i -
That is (2 must go negative to decrease Mo This cannot be allowed because of the constraints therefore 52 must be made zero. In this case
= u2 (48)
and again substitution is made in (45a) and (46a)o The plane up = 0 is the
plane of all values of P2 given any Bl, (3 for which M is a minimum
within the constraints of the problem. Of course because of (48) this means
P2 0 for all values of p1, 3~
Starting again at point A an attempt is made to minimize M by
increasing P3 from zero P2 is increased along line ACJ, the intersection of the u] = 0 and up = 0 planes This insures that P1 and 2
will have values which will yield a minimum value of M within the constraint
B1l 0, p2?- 0 for any value of P3o Using (45a) and (48) and taking the
partial derivative of M with respect to P3, the direction of increase or
decrease of 3 and also the value of ~3 can be determined so as to minimize Mo
6M. 2. = - 2_ + + 2 + - ul = 2 u(49)
bP3 7 21 21 7
for u =u2 =u =0, 3/5
This value of 13 along with the value 32 = ul = 0 can be substituted in

-61(46a) to determine the value of
6 1 (3) 12
B1 7 5 5 5
This is point C on Figure 6 and corresponds to the intersections of the
u = 0, u =0, and u3 = 0 planes Thus the values of 1, P2, and 3
required to minimize(45) within the constraints that Plo 0, P2 0, and
53 0 are in order 12/5, 0, and 3/50 This is the same result as that
obtained in Equation (44)~ If the solution is continued ignoring the constraints, the absolute minimum (3, -1, 1) is obtained~ This is the point
H, the intersection of the u1 = O, u2 = 0, and u3 = 0 planes. The solution will be independent of the path taken to reach the solution.
The method applied to a case where there are n values of n
to be determined can be outlined as follows:
lo Using Equation (28) form the quadratic M = M(P1, P2, o0oo, n)
2~ Take the partial derivative of M with respect to P1, and
let 2ul =35 Let ul =2 = =3 = ooo = n 0 and solve for P1o If
1 > 0 then solve for p1 in terms of ul, P2 ~ooo Pno
Substitute this value of P1 into the equation for Mo Now
M = M(ul, P2, ooo Pn)o In this first time around the 1
chosen will always be positiveo
4o Now using the quadratic M found in step 3, the partial derivative of M is taken with respect to 20o Let 2u2 =
5o Let ul = u2 = 53 = o0o = n = 0 and solve for 52o If
P2> 0, then solve for p2 in terms of u1, u2, 3, o0, ~ n
and substitute this value in the equation for M step 3o If

-622 > 0, then one lets 2 = u2 and substitute this value of
P2 into M step 30
60 The above procedure is continued for all P'so At each step
the values for all the Bis considered up to that point are
determined. If any of these P's are negative one makes the
change of variable Pk = u'o Further if a previous change of
variable was made introducing a uj = u. must be eliminated from the function M using the relations uj = j and
uj = 6M before continuing the iterative process.
70 The n values of 3 are found after the last iteration by
letting all the u s and u+s equal to zero.
This solution is equivalent to the following matrix approacho Assume the measured distribution is made up of only two components (eog., the
Ail'S and Ai2's). One can use least square fitting to obtain the p1 and
2 from Equation (31)o It has been assumed the 3 = 4 = o = 5n = Oo
If Bl> 0 and 02 then one adds a third component (eogo, Ai3's) and
solves (31) for P1i, P2 and P3o If any of these P's are negative, one
sets that given (j equal to zero by eliminating the Aij components from
the matrix calculation involved in (31)o In this way, one of the n components are added at a time, the p's determined for each addition and if
any one of the Pis determined is negative, that P is set equal to zero
and its corresponding component eliminated from the matrix A before adding
another one of the n components to Ao The solution for the U's after all
n components have been added in the manner prescribed above, will give the
values for the 5's for which M is a minimum and 1-_ O~, 22 0, ooo,
ink 0o

-63In this way the energy distribution and intensity of the energies can be determined for the case where the incident flux is known to
have a discrete energy spectrum but the distribution is unknowno This
final method is similar to the "peeling off" method3 except that after
each addition of a monoenergetic component, the intensity distribution
for all energy components present is reevaluated, and also the fit is
made using least squares fitting. This technique could therefore be
described as a "dynamic least squares peeling off" process.
The number of monenergetic pulse height spectra used in performing the above will depend upon the energy resolution of the system,
If the energy distribution of the incident flux in some region is such
that the energy separation between the various energy components is
less than some fraction of the half widths of the photopeaks in this
region, it may be only possible to determine the total number of gamma
rays in this region without being able to uniquely determine the energy
distribution in this region. The half width of the photopeak is a measure of the energy resolution of systemo The problem of energy resol ution and the effects on the analysis will be discussed further in the
next section and Chapter V.
35 Continuous Incident Energy Spectrao
The measured pulse height spectrum p(V) due to a continuous
energy distribution of gamma rays p(E) can be written analytically in
the following manner.
E =E
n mrax
(V) Pd (E) A(V, En) de (0)
E -O
n

-64where A(V,En) is the smearing out function of the detection system. For
a given energy En, the smearing out function A(V,En) is characterized
by the monoenergetic pulse spectrum as a function of pulse height V,
A(V,En) is not only a function of energy, but is also a function of
source detector geometry. In Chapter 3, it was pointed out that the
monoenergetic pulse height spectrum could be described by the sum of a
Gaussian (the photopeak), a Compton continuum, and the various escape
peaks (ioeo, x-rays and annihilation). Other effects due to scattering
from the surrounding materials are also needed to describe this function
in the actual experimental situation, It is impossible to describe
A(V,En) analytically. Therefore this function is either measured experimentally using monoenergetic gamma ray emitters or calculated theoretically using Monte Carlo calculations(4) but in both cases one obtains
only an array of numbers describing the pulse height spectrum, and
still no analytic form is available.
The information desired is p(En) Equation (50), and since
A(VEn) is not known analytically, it is impossible to perform an inverse transformation to obtain p(En)o In fact even if A(V,En) were
known it may be extremely dilfficult, even impossible, to perform the
transform exactly. Therefore, it is impossible to perform an inverse
transformation to obtain p(En)o Therefore, it is necessary to see
whether or not some approximate method can be used to obtain P(En)
The following technique for obtaining P(En) is based on a
theorem in sampling theory (see References 26 and 28) and upon the

-65fact that there is a finite energy resolution of the detection system.
If a distribution has no oscillatory components with a frequency components greater than f, then the Shannon Sampling Theorem asserts
that samples at discrete points not further apart than 1 describes
max
the original function exactly, and in fact the original function may
28
be reconstructed from the samples8 If this theorem is applicable,
Equation (50) can be written as
pi = nin (51)
where pi is the measurement of the distribution p(V) in channel ijrn
is the total number of gamma rays in energy group AEn about En)and Ain
is the value of the nth sampling component (monoenergetic spectrum component) in channel i.
The sampling functions A(V,En) (monoenergetic pulse height
spectra) are written as arrays of numbers An, The shapes of these
distributions were discussed in Chapter III. Considering only the energies where pair production is negligible, these spectra can be
described by a Compton continuum and the photopeak. The Compton continuum is a more slowly varying function than is the photopeak. For
all practical purposes the photopeak can be described as a Gaussian,
Equation (9)
x2
y(x) = exp (9a)
2or2
m
where x = (V - Vm) in pulse height units and Vm is the pulse height
corresponding to En and Am in Equation (9) is unity. It is assumed

-66that y(x) is equal to zero, for those values of pulse height where
y(x) < 10-3 (Equation 9a). This assumption does not affect the analysis significantly, since these values y(x) will be lost in the noise
(background) of the system.
The frequency of oscillation of the photopeak is determined
in the following manner. Keeping in mind the above assumptions and further assuming that the Gaussian (Equation 52) can be closely approximated by a cos2kx distribution. (i.e., the amplitude of the higher frequency is negligible in terms of the analysis.) The comparison is
shown in Figure 15. The value of k is chosen so that for x = 0 and
for x =J2om (i.e., the e- point on the Gaussian) the amplitudes are
equal. k is found to be equal to t. The frequency fm can be
4.864a
found in the following manner
cos2kx = 2(1 + cos2kx) (52)
The frequency is therefore
f 2k _ 2 = 1 (53)
4.864Om 2.432am
The photopeak is assumed to contain the oscillatory component with the
maximum frequency. This frequency is given by Equation (53). From the
considerations briefly outlined above, and under the assumption also
made above, it is believed that Equation (51) can be used for the analysis of continuous spectra. A more complete discussion of the application of Shannon's sampling theorem to such problems can be found in
Reference 26 and 28.
n/2
Now from Equation (8) Chapter II, am = C Em where E is
mm m
th
the energy of the m energy component. For a given detector then,

-671.0 -.9 ~
~~~.8 ~?/
~~~~L.6o
I 1X l I I I | I I
/ \\
-.k
~~~~~~~.3 -— ~~~~~~~~Dstiuin -— =a =-'. I 1 I X X
- 2.432 o -- -- 1.946' —.486.486.973 1.459 1946 2.432-
X in units of 8'
Figure 15. Comparison of Gaussian Distribution with cos2kx
Distribution. Fitted at x = O and x = 2a.
Distribution. Fitted at x = 0 and x =4j2c.

-68am as a function of energy can be determined. Equation (53) can also
be writtenin terms of the full width at half maximum WI(l) This
width, also called the resolution width, is of importance in the definition of the resolution of the system l6) It can be shown that
W I
m 2a 5 (54)
m 2~354
or using this definition then
f = -- (53a)
1.03 W!
for all practical purposes the frequency is inversely proportional to
the full width of the photopeak corresponding to the energy E o The
criteria for the maximum separation (MoSo) between the sampling components used in Equation (51) then. can be found to be
M. S = - W i (51a)
2 f 2
max
That is., the sampling components should be chosen so that their separation in energy be no greater than the half width at half maximum, of
their photopeaks
The energy separation is energy dependent, thus the separation will increase for higher energies (see Equations 8 and 54).
Once the components are determined, the mp s in Equation (51.) can be
determined using the least squ:are fitting techniques described above
Equation (31.)
A similar calculation for determining the number of samples
pcri of the pulse height spectrun required to describe the pulse height
distribution has been carried out in detail by Burrus26 These calculations indicate that one must make at least two measurements per resolu

-69tion width to describe the pulse height spectrum exactlyo Thus the
gain, the high voltage and the pulse height increments AV must be
adjusted to fulfill this requirement when performing the experimento
Co Error Calculation
Once the ~ps in Equation (31) have been determined, it is
possible to determine mean square deviation in po If it is assumed
that the A.'s (ioeo, the pulse height spectra) are known without
13
error, this calculation is rather simple. Then due to the variation
in the measurement in Pi, there will be a corresponding mean square
deviation in the determination of the 83k1s25 Using Equation (29),
and (531)
PA IC~ — W P) X r i i(31a)
i v
where the following definitions are used, the matrix C (A c A) is
a symmetric matrix and the elements C of C are given by
vr
C =.i A, A, (55)
C 1 is the inverse of matrix of Co The elements of C are written
as C -o Thus
X' -
CC = I
where I is the identity matrix with elements IVX and
I C- C -1 (56)
remembering that both C are symmetric atries
remembering that both C and C-! are symmetric matrices.

-70Further
I =1 if v = X
VX
Ix = 0 if v X (57)
From Equation (31) it is seen that PX is a linear homogeneous function
of the countslunder the assumption that there is no error in the Aijo
Thus the mean square deviation oa(pi) corresponding to the variation
in p. can be written as
2(B)= IZ A.C -1 -1 A. A 2 )
vX' X y i a (Pi)
iv y
where
_ 1
i - ^l2^!'
2aor(pi)
Then
(r 2 ) - 2 - C % 1 Cy -1 A. D.
i v y7X iv i
o~r ~i
or
- Z C 1C tx c A A
2, v 7x LX i iv iy
Cf ( i j J2 Cx1Cx' Ccvx
2~Bi)'~' C C C
V t7
v Y
and from Equation (56) and (57)

-712 -i
a(Pi) - 1 -1= (58)
XX
2
that is a (pi) can be found from the diagonal elements of the C-1
matrixo The mean square deviation a (pi) in the simplest case where
one assumes that background is negligible is
i.2 pi (59)
Depending on the experimental situation, the corresponding ai. s can
be determined and used in the above situation. The probable error
can then be determined from the mean square deviation. The above considerations are only true if it is assumed that the Ai.'s are known
il
without error. The effects associated with the variation in Aij has
not been investigated in this worko

CHAPTER V
EXPERIMENTAL PROCEDURES AND RESULTS, AND CONCLUSIONS
Ao Detector Analyser System
A schematic diagram of The Detector Analyser system used to measure the pulse height spectra is shown in Figure 16.
The design and construction of the scintillation detector was considered in Chapter II (see Figure 3),
The voltage divider and associated regulated high voltage supply,
supplies the potential difference across the various dynodes in the multiplier phototubeo The design of the voltage divider can be found in Reference 20. A well regulated high voltage supply must be used in these measurements for small changes in the voltage will be reflected as larger changes
in the gain of the multiplier phototubeo
The output of the anode of the multiplier phototube is fed into
the linear amplifier and multichannel analysero The operation of this unit
was outlined in Chapter IIo A 256 channel pulse height analyser was usedo
An electric typewriter was used for the information readout.
Bo Measurement and Interpolation of
Monoenergetic Pulse Height Spectrum
In the last chapter, analytic methods for inferring the incident
gamma spectrum from a measurement of the pulse height spectrum were discussed,
From this discussion, it can be seen that a knowledge of the characteristic
pulse height spectra due to monoenergetic gamma ray emitters as a function
energy is of basic importance in the analysis. These functions cannot be
written in a closed analytic form as is indicated in Chapter III, therefore
-72

VOLTAGE DIVIDER
LINEAR
r4i | AMPLIFIER AND
SOURCE-.* INFORMATION
L —-PULSEF HWEOHW READOUT
ANALYSER
SCINTILLATION
DETECTOR
REGULATED
HIGH VOLTAGE
SUPPLY
0-1500 V.
Figure 16. Schematic Diagram of Detector-Analyser System.

-74it is necessary to either determine these functions theoretically using
Monte Carlo techniques,(4) or determine these functions experimentally.
Since the Monte Carlo calculations have become available only recently,
and. further since these calculations do not include the effects of scattering from the crystal containment materials, see Figure 6, it was found necessary to determine these functions experimentally.
The pulse height spectra of the following emitters were measured;
Co57 (014 Mev), Cr51 (032 Mev), Au198 (0411 Mev), Na22 (051 Mev),
Cs135 (0.662 Mev), Nb95 (Oo768 Mev), Mn54 (00842 Mev) and Zn65 (loll Mev)o
The spectra of Na22 and Zn65 are perturbed by the presence of 1l3 Mev gamma
ray in the case of Na22 and a 0o51 Mev gamma due to the emission of a positron in Zn65. The amounts of perturbation of the monoenergetic pulse height
spectra for 0o51 Mev and lll Mev due to the presence of these other energies is small, but an attempt was made to remove their effect by using the.available theoretical calculated shape functions for these energieso
The variation of the photopeak as a function of energy was first
considered, In Chapter II, it was argued that the photopeak should be
Gaussian in nature, and that for this Gaussian, a varies as a function of
energy in the following manner, 52 = a2Eno a was experimentally measured
from the photopeak of each of the monoenergetic emitters, and a2 was
plotted as a function of E on log-log paper. This measurement was carried
out for both a 2" x 2" cylindrical and a 2" spherical crystal. The measurements with the cylindrical crystal were made at 100 cm from the top of the
crystal, and the measurements with the spherical crystal were made at 3",
6", and 12" from the spherical crystal. For the 6" distance measurements

-75were taken at 0~, 45~, 90~, and 135~ (see Figure 3a). The pulse height
spectra did not change for these distances and did not change as a function of angle as long as the measurements stayed within the region shown
in Figure 3a.
From the log-log plots Figure 17 and Figure 18 both a2 and n
were obtained. A comparison between the calculated photopeaks using Equation (9) and the experimentally determined values of a2 and n, and the
experimentally measured photopeaks for 2" x 2" cylindrical is shown in
Figure 19. The calculated photopeak shapes are in good agreement with the
experimentally determined photopeak shapes. It is therefore believed that
Equation (9) can be used to describe the photopeak over the energy range
considered. A similar study was carried out for the 2" spherical crystal
and similar agreement was obtained.
The Compton continuum as a function of energy cannot be written
in an analytic form. It was therefore necessary to interpolate its shape
from the measured spectra.
The shape of the Compton continues for those energies which were
used in the later sections of this paper, and for which there were no monoenergetic emitters available, were obtained in the following manner. The
ratios of the height of the photopeak to the heights of a number of maxima
and minima in the Compton continuum (e.g., the ratio of the height of the
photopeak to the height of the maximum of the Compton continuum at the
pulse height corresponding to 180~ Compton scattering of the primary gamma
ray in the crystal. A consideration of the number-energy spectrum of
Compton electrons produced by primary photon, see References 2, 4, 5, and
17, indicates why such a maximum should be observed.) were recorded as a

-761.0
_ 6S</
8o, ---------- ----- — Mn
D C- - -SLOPE I
2 3 4 5 6 7 8 9
1.0 10.0
0~2
STANDARD DEV ATION
Figure 17. a2 as a Function of Energy ror a 2" x 2" NaI(Tl)
Crystal.
Crystal.

-77300 -— _____
200
150 - --
100
5 6 7 8 9-10 15 20 30 40 0 60 7080-90100 150 200
Spherical aT Crystal.
80 — NSLOPE *1.34
40
STANDARD DEVIATION, __
Figure 18. a5as a runction of Pulse Height for a 2"
Spherical Nal(Tl) Crystal.

Co57.4
5. o THEORETICAL
4 Mev Au 198.768.842
------ \- ----— __.768-.842No 22 c137 EXPERIMENTAL
Nb95 Mn 54.320 / /'-,Z
Cr 51
1.114 Mev
9' l1 f 1 r I I t I i U rI I I I
C;
2 - -^fi^ -t^ -^r'^^'-
8
A A
z.01
o.001 -
10 20 30 40 50 60 70 80 90 100 110 120 130 140
PULSE HEIGHT
Figure 19. Comparison of Theoretical and Experimentally Measured
Photopeaks for a 2" x 2" in NaI(T1) Crystal.

-79function of energyo The locations on the pulse height scale of these maxima
and minima as a function of energy were also determinedo The measured monoenergetic pulse height were then normalized so that the area under their
photopeaks were unityo The photopeaks were subtracted from the measured
pulse height spectra leaving the Compton continuao The Compton continua
obtained in this manner are shown in Figure 20 in dark lineso Next using
the information concerning the relative heights of the maxima and minima of
the Compton continua compared to their photopeaks, and the locations of
these maxima and minima as a function of energy as guides, the desired
Compton continua were interpolated from the measured Compton continuao
These interpolated Compton continua are shown in Figure 20 in light lineso
The calculated photopeak was then added to the interpolated Compton continuum to obtain the adjusted pulse height spectrum. These final adjusted
pulse height spectra as a function of energy for a point emitter at 100o cm
from the top of a 2" x 2" NaI(Tl) cylindrical crystal are shown in Figure
21o A similar analysis was carried out for the 2" spherical crystal, the
final results are shown in Figure 22~ Both sets of pulse height spectra
have been normalized so that the area under the photopeak is unityo Also
the pulse height spectra for the spherical was found to be relatively independent of source detector geometryo
In this manner, the monoenergetic pulse height needed for the
analysis in the following sections have been constructed. These spectra
include the effects of the scattering from the surrounding media.
Parenthetically it should be noted in Figure 17 and, Figure 18
that the a for the spherical crystal is greater than the a for the
cylindrical crystalo The ability to resolve two ganmma rays close in energy

9
8
7
6
5
~~~~id | | i i ~~~~~~~~~~~1 1~~~~~~~~~~~~~~~.i1114
3
0
8... _ I
8
0 7
Z 6
5
4
3
2
24 30 36 42 4& 54 60 66 72 78 84 90
36 42 4 PULSE HEIGHT2 84 90
PULSE HEIGHT
Figure 20. Experimentally Measured and Interpolated Compton
Continua for a 2" x 2" NaI(Tl) Crystal.

-81-.300.o0 -.284 M I-301 I1 _ t I 1 ll.364.440.637.200 I5 1 I I!7 I if i VT I!-l/ 1 — l = —.760.732.775.866.100
0 II 20 II 4!! lll 5I 60 70i 10i --- T T
IIII t! Ill l i1 A1111 Ill- - - f -.090.080.070.060'g A i21 ll. Fi i111 1 A Inel ll e Ps i v.020.010.008.007.006.005.001
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
PULSE HEIGHT
Figure 21. Final Normalized Pulse Height Spectra as a Function of
Ii I1Energy flor a 2" x 2" 1 NaI(T1) Crystal.l
Energy for a 2" x 2" NaI(Tl) Crystal.

0 z 6 ~ ~;20;EV
36 48 60 72 84 96 108 120 132 144 156 168 180.662 M-EV
PULSE HPULSE HEIGHT
SPECTRA AS A
FUNCTION OF
ENERGY FOR A
Figure 22. Pulse Height pectra as a on of ergy for a 2"
Spherical Na (Tl) Crystal.
iI
3f6 48 60 72 84 96 108 120 132 144 156 16e8 180
PULSE HEIGHT
Figure 22. Pulse Height Spectra as a Function of Energy for a 2"
Spherical NaI(T1) Crystal.

-85depends on these widths as was discussed in Chapter IVo The smaller is a
the better is the energy resolution, (16) Attempts were made to improve
the resolution of the spherical crystal detector, but because of the problems mentioned in Chapter II no better resolution was able to be obtainedo
Thus the spherical crystal was only used in cases where good resolution
was not necessary ( eogo, see Section E, Chapter V).
Co Experimental Measurements of Spherical Crystal Efficiencies
Both experimental and theoretical determinations of the efficiencies for cylindrical crystals have been carried out,(23,4,15 21,22,23) but
only theoretical calculations carried out by the author in Chapter III and
Monte Carlo calculations carried out by Miller and Snow(4) are available
for spherical crystalso In the calculations carried out in Chapter III,
only the total efficiencies are available, and the peak to total values or
the photofractions required for the determination of the peak efficiencies
[Equation (21)] are available in ANL 6318(4') for the case of a parallel
beam incident upon a spherical crystal. In ANL 5902( 23) photofractions
as a function of source detector geometry,(ioeo for point, disc, and line
sources,) are calculated and are found to be relatively independent of the
source detector geometry. (ioeo for a given energy this value does not
vary by more than +6% as a function of source detector geometry) This seems
to indicate that the ph.otofraction may be somewhat independent of the source
detector geometryo The constancy of the shape of the monoenergetic pulse
height spectra for 2" spherical crystals as a function of source detector
geometry was discussed in the preceding section. This seems to indicate
that the photofraction may be independent of source detector geometry for

-84spherical crystalso The following experiment was carried out to investigate this constancyo
The pulse height spectra of the monoenergetic emitter enumerated
in the preceding section were measured at 5 cm and 10 cm above a 2" spherical NaI(Tl) crystal. The sources were point sources. Then using relation
(21)
Ap5n = In pi5n A5/4r (21a)
AplOn = In EpilOn n10/4 (21b)
where Ap5n and AplOn are the areas under the photopeaks for the n-th
emitter at a distance of 5 cm and at a distance of 10 cm respectively above
the NaI(Tl) spherical crystal. 25/4[. and Q10/4t are the fraction of
solid angle subtended by detector with a point source at 5 cm and at 10 cm
from the top of the NaI(Tl) spherical crystal E pi5n and epilOn are the
intrinsic peak efficiencies for n-th emitter with the source at 5 cm and
10 cm from the top of the NaI(Tl) crystal, and In is the source strength
in gammas per second of the n-th emitter.
If it is assumed that the epis are independent of the source
detector configuration then
Epi5n = EpilOn
and
Api5n/ApilOn = Q5/n10 (61)
The results of this measurement are plotted in Figure 23~ The ratio of
Api5n/ApilOn are plotted as a function of energy and the ratio ~25/10

-850
u
0 Zn
5Cr 22 ~ n M -110
0-Cr No N
0 0.2 04 0.6 0.8 1.0 1.2 1.4
ENERGY IN MEV.
Figure 23. The Ratio of the Areas Under the Photopeaks of the Pulse
Height Spectra for Point Sources at 5 cm and 10 cm from
the Top of a 2" Spherical NaI(T1) Crystal as a Function
of Energy.

-86is indicated by the broken lineo The constancy asserted in the relation
holds within +7% over the energy range investigatedo A more direct measurement of the intrinsic peak efficiencies was obtained in the following
mannero The pulse height spectra of the various monoenergetic emitters
were measured, using a 2" x 2" right cylindrical NaI(Tl) crystalo The
solid angle was kept constant (iOeO, the sources were kept 9o3 cm from the
top of the crystal)o The intrinsic peak efficiencies epicn for 2" x 2"
are well known as a function of source detector geometryo(4l15121l22,23)
The area under the photopeak Apcn for each of the pulse height
spectra can be determined and then for the cylinder'Apcn = In(2c/4i) epicn (21c)
epicn was determined for the case of a point source 9~3 cm from the top
of the crystal. Taking the ratio of (21c) with (21a) and (21b)
Epi5n Ap5n 2c (62)
-picn Apcn Q5
EpinlO pnlO c (63)
epicn Apcn Q10
The results are plotted in Figure 24o The broken line shows the
results for the ratio of Epiojpic 9 3, the theoretical intrinsic peak
efficiency for a parallel beam incident upon a 2" spherical crystal,(4) to
epic 9 3 cmo In this case the deviation is no more than + 7% from the parallel beam caseo Thus these results and the theoretical calculations carried out in Chapter III, it is believed that the intrinsic peak efficiency
is relatively independent of the solid angle as long as there is no source
closer than one half the diameter of the crystalo

-87-.4 "
A E pi5/ pic
3 E — 0 Epic/ pic
C51 65
CrJ N22 137 95 5 Z65
NB C, Nb Mn Z
Cf) Episao
Ca:___- -- A _ _
w 198I I pI ic 9.3
-0.2 0.4 0.6 0.8 1.0 1.2 -.4
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
ENERGY
Figure 24. The Ratio of the Intrinsic Peak Efficiency of a 2" Spherical
Crystal to the Intrinsic Peak Efficiency of a 2" x 2" Right
Cylindrical Crystal as a Function of Energy.

-88D. The Experimental Determination of Discrete Spectra
1. Monoenergetic Emitters
The simplest case to consider is that of a monoenergetic emitter.
The energy of an unknown emitter is determined by measuring the pulse height
spectrum of a sample of known energy, and determining the point on the pulse
height scale corresponding to the position of the midpoint of the photopeak.
This pulse height will correspond to the energy of the emitter as discussed
in Chapter II. The series of monoenergetic emitters described in Section A
above can be used to calibrate the pulse height scale in terms of energy.
Using these emitters, the energy of an unknown monoenergetic emitter can be
determined from this calibration.
The source strength can also be determined from an analysis of
the monoenergetic pulse height spectrum. The "absolute" determination of
the thermal neutron flux using activation analysis is a problem wherein this
technique can be easily applied. This application is a digression from problems in gamma ray spectroscopy considered up to this point. However, it is
believed that the detailed description of the methods for absolute determinations of source intensity to be described below will indicate how to calculate the relative intensities needed in gamma ray spectroscopy. Furthermore the detailed descriptions of the application to neutron flux determination is included because such a description has not appeaed previously in
the literature. This method has been used for determining the thermal neutron flux in the core of the Ford Nuclear Reactor and has been found
to be extremely useful,
The following experiment was performed. Two gold foils, a bare
foil and a cadmium covered foil, were irradiated in the center of the core

-89of the Ford Nuclear Reactor. The pulse height spectra for these two foils
were measured using a 2" x 2" NaI(Tl) right cylindrical crystal. Figures
25 and 26 are the measured pulse height spectra. From relation (21), the
source strength in gamma per second Iy can be determined in the following manner.
I= Ap i -(21d)
Epi 4 EPA
where epA = ~pi ~ and is the total peak efficiency. The area under the
photopeak can be determined in two different ways. First analytically the
photopeak is assumed to be described by a Gaussian from Equation (9) Chapter
II
+CO IE - E012
Ap= Amax exp - - dE
-o 2a2
-Co~,, rrRm
max = L2 AE W1/2 (64)
where Amax is the count rate of the maximum of the Gaussian. In terms
of the measured pulse height spectrum Amax = CRm/ E where CRm is the
maximum count rate in the measured pulse height spectrum and AE is the
channel width.
2 W1/2
2Ln2
and W1/2 is the half width at half maximum.
Secondly, Ap can be determined numerically in the following
manner. The pulse height spectrum is plotted on three cylce semi-log paper.
The measured puLse height spectrum is really a histogram; that is, each
point corresponds to the total integrated number events in an increment
AE about E. Then Ap is just the sum of the count rates at each AE

-904
10o
8
6
2
II0C
8
6
z I
z [
I10
4..
2
I0 -- - -~
200 300 400 500 600 700 800
PULSE HEIGHT
1-j " -198
Figure 25. Measured Pulse Height Spectrum Au. Bare Foil 10.0
cm from 2" x 2" NaI(Tl) Right Cylindrical Crystal.

-914
10 -lo - -
B r
10
Figure 26. Measured Pulse Height Spectr-m Au-98, Cs —m — um covered
ApnumeCylindrical 16,1Crystal. C/2 min
It
70,o' 5/ — --
S — 1.90X le- IOApnumecol.5 16,111 C/2 mC
10200 300 400 500 600 700 900
7 - -- - F -- u r - S61.88 e P u/sec - ---
Cylindrical Crystal.

-92increment about the photopeako On the left side of the photopeak there is
interference by the Compton continuum, therefore it is necessary to extrapolate the photopeak under the Compton continuum (see Figures 25 and 26)o
The sum of the points over three decades will give Apo The contribution
of the area below three decades will add about 1% to the area under the
curve, and therefore this amount is ignored. Using Equation (21d), I
can be determinedo The value for epA was found in References 21 and 22.
For a 0o411 Mev gamma ray emitter (Aul98) epA = 0o714 x 10-2o The following is the irradiation data for the foils
Time of Removal of Foil
from Reactor to the time
-Foil Weight, M Irradiation, te of beginning the count, tw
Bare 0,098 gms 4 minutes 7805 hours
Cadmium
Covered Oo098 gms 5 minutes 78 5 hours
The activities of the bare and cadmium covered foils were determined using the techniques described aboveo The results are in gammarays/sec.
but this within less than 1% is the same as the disintegration note (see
the decay scheme of Au198)o
Foil A in dis/sec A/M in dis/sec/100 mg
Bare 3585 x 104 d/s 3093 x 104 d/s/100 mg
Cadmium
Covered lo88 x 104 d/s 192 x 104 d/s/100 mg
The following steps are used to calculate the sub-cadmium flux

-93a. Calculate Saturated Activity, As
A = -— A
(1-e-Xte)e-Xtw
where te is the exposure time
tw is the waiting time
and
X decay constant for gold = 0o313 x 10-5 sec1
1) Bare Foil Calculation
AB
AsB AB
AB = (l-e-Xte)e-Xtw
since te = 240 seCo
-et
(l-e - ) = te
(0313 x 10-5 secl1)(2o40 sec x 102)
= 0o751 x 10-3
tw = 78o5 hrs = 2~826 x 105 sec
e-xtw = 0o41
Thus
AsB = 1o27 x 108 d/s/100 mg
2) Cd Covered Foil Calculation
Since te = 300 sec
Xte = 0o919 x 10-3
Since the waiting time is the same
e-tw = 0o41
Thus 8
Ascd/M = 0o509 x 108 d/s/100 mg

-94bo Flux Calculation
The sub-cadmium neutron flux iscd is
AsB - Ascd = V ET esscd
where V is the volume of the foil and ZaT is the average thermal absorption macroscopic cross section for Au197 = No
A aAT
Since in the above calculation, the saturated activity has been
calculated per unit weight, the above relation can be written as
AsB - Ascd V NoP
M = M A ~AT scd
A aAT Oscd
where A is the atomic wto = 197 for goldo
Now
AT = aa = tl l
where ca is the most probable value of the microscopic absorption cross
section of gold = 96 barns
aAT = 96/lo12 barns = 8507 barns
It may be necessary to correct for the hardening of the MaxwellBoltzmann distributiono In the case of the FNR this correction is
aAT(o03) = 85o7 ~o025403 barns
= 7808 barns
Now the flux can be calculated
AsB - Ascd A
AsB - M Ncd
MB As = 0o761 x 109 d/s/gm

-95In order to have the correct units, one must have M in grams
0scd = 3514 x 109 neut/cm2 sec
The technique has been checked using standard foils calibrated
at the Argonne National Laboratory, and the results have agreed to within
+6%
2o Polyenergetic Spectra
The first case considered was that of an emitter whose energies
distribution was known but the problem was to determine the relative intensities of these energy components. The emitter chosen was I131 The
gamma ray energies in the spectrum of 113 were 0~722 Mev, 00637 Mev,
Oo364 Mev and 0284 Mevo Another gamma ray which is a possible contaminant
was also noticed at 0o5 Mevo The measured pulse height spectrum is shown
in Figure 27. A point source of I131 was placed 10 cm from the top of
a 2" x 2" NaI(Tl) crystal. The shapes of the monoenergetic pulse height
spectra were determined in section A, Figure 21. These spectra were
normalized so that the area under the photopeak is unityo (The curves
were normalized in this manner so that the PIs obtained would equal the
area under the photopeako) Thus it is only necessary to divide by the peak
efficiency to determine the intensity as is done in the case described in
the preceding sections. A least squares fit was made using Equation (31)o
The results are shown in tabular form in Table IV and graphically in
Figure 2'7 Results for two experimentally decay for I131 determined by
other investigators(29,30) are also presented to show the close agreement
with the results obtained using the least squares fitting techniqueo

NORMALIZED COUNT RATE
t.....,'......... I I I I iI
0
C ~~~~~~~~~~~~~~~~8^~
g0.
0
Figure 27. I131 Pulse Height Spectrum; 2" x 2" in NaI(Tl) Crystal
m
Source 10 cm from the Top of the Crystal. The Amount
of Each Monoenergetic Component Obtained Using the Least
Square Felting Technique is Also Indicated.
Square Felting Technique is Also Indicated.

-97TABLE IV
I131 GAMMA RAY SPECTRA
Energy Least Mean Square Other Experimental Results (29,30)
0~284 Mev 5o2 + 14% 6o0 4o2
0o364 Mev 100 + 9% 100 100
0,500 Mev* 0o5 + 20%
Oo637 Mev 10o2 + 12% 10 7o2
0~722 Mev 2~4 + 14% 3 2o4
* Possible contaminant
The errors were determined, by the method described in Chapter IVo
It was also assumed that the intrinsic efficiencies are known to within
+5%.o This assumed error was determined by comparing theoretical and experimental results obtained for the efficiencies of 2" x 2" NaI(Tl) crystals (2,21,22,23)
The next problem was to determine the energies and intensities
of the singlet spectrum of W187o The measured pulse height spectrum is
shown in Figure 28. Certain energies are easily identified from the resolved photopeaks. In Table V, the various energies that were assumed
present are tabulated, Also four iterations using the least square fitting
technique described in Chapter IV are presented in Table Vo
The negative P is obtained for the 0,440 Mev gamma. In terms
of the decay scheme of 187 it i found from other experimental and theoretical calculations that its intensity relative to the other gamma rays
present is almost zero,(Sl) and thus is lost in the background.

-98105
^480 Mev,-.686 Mev.552
Mev
104
Possibly.619 Mev these
-' Gommo Roys.735 Mev
mw I~~ \ ~f /.760 Mev
- "T /.775 Mev
101.866 Mev
Pu/se Height
Spectrum W'
/0 cm from 2"x2"
Nol (TI Crystal
Detector
10
200 400 600 800 1000
PULSE HEIGHT
Figure 28. Pulse Height Spectrum W187 10 cm from 2" x 2" NaI(T1)
Crystal Detector.

-99TABLE V
RESULTS OF LEAST SQUARE
ANALYSIS OF W187 SPECTRUM
Energy for Data 731 for Data 741 for Data 733 for Data 743
0o866 0o464 + o008 0o462 + o008 0o438 + o007 0o437 + o007
0.775 1l93 + o06 lo94 + o06 2o95 +.02 2o94 + o04
0o76o 172 + o09 169 + o09
0o735 Oo0245 + o06l5 0o0383 + o06l5 1oll + o03 1oll + o03
0o686 227 + o6 22o7 + o06 6218 o05 21o7 + o05
0o619 7o36 + o03 75533 + 03 7o42 + o03 7 40 + o03
0.552 6.02 i o03 6~02 + o03 6~17 + 003 6018 + o03
0o48o 25o9 + o06 25o6 + o06 26ol + o06 26o0 + o06
0o.440 -o6016o + o035 - 0223 + o036
0O301 0112 +.032 o5136 + o032 0oll5 + 032 0o123 +.032
0O256 0o867 + o04o 0o850 + ~04o 0o8o6 + o04o 0799 + O04c
An interesting result to be noted is that a variation in the
choice of energy components in one region does not seem to affect the
values of 3 in the other energy regions. Further, an important result
is obtained in the region of the 0O730 - 0O866 Mev region. In this region the energies cannot be resolved as separate (ioe., there is loss of
resolution due to the smearing out of the information by the detector),
but the results obtained show that the number of gamma rays in this region remain constant although the distribution in the region changes depending upon the components chosen to represent the region0 These

-100results seem to hold for other cases investigated. Thus it seems that when
one cannot resolve the energy spectrum in a given region, because of the
finite resolution of the detection system, only the total number of gamma
rays in the region can be determined, but the exact energy distribution in
that region cannot be determined, The ability to resolve two energies as
separate can be related to the width of the photopeak at half maximum.
This problem has been considered theoretically by Burrus (26) There seems
to be a minimum separation in the choice of monoenergetic components to be
used in a given region depending on the width at half maximum of the photopeako This choice will also depend greatly how well numerically (i.eo, to
how many significant figures) the monoenergetic pulse height spectra are
known, for the least squares analysis depends on differences in these numerical valueso
The results obtained in this calculation for W187 agree well with
other experimental and theoretical calculationso The results of this calculation are discussed further in a paper by RoGo Arns and MoLo Wiedenbeck (1)
In this paper the decay scheme of W187 is discussed in detail.
Eo The Experimental Determination of
The Continuous Spectrum
The analytic method was described in the previous Chapter therefore a particular problem will be described to illustrate the methodo
The problem under consideration will be that of determining the
degraded gamma energy spectrum due to the scattering of gamma rays while
passing through some medium. The measurements are made outside the scattering medium, In particular, the scattering of 0o662 Mev (Cs137) gamma rays
that pass through steel slabs was measuredo

- ~,0i d
-Jo-i. -
An attempt was made to simulate semi-infinite slab geometry with
a plane parallel monoenergetic beam of gamma rays incident upon the front
surface; so that the experimental results could be compared with theoretical resultso The parallel beam was obtained by placing a point source at
a large distance from a finite steel slabo The next problem is to determine
how large a slab is required to simulate a semi-infinite slabo The illuminance E (ioe,, the luminous or gamma flux incident per unit area) at an
axial point produced by a circular disk is determined assuming that the
source obeys Lambert's Lawo(32) The exit surface of the slab is assumed to
be the source of radiationo It is found that the illuminance E obeys the
following relationo(32)
E = B sin a (65)
where B is a constant and a is shown in the Figure 290
Now relation (65) can be rewritten in the following manner
E -= jB [ —--- (65a)
1+ (a)
In the limit of the semi-infinite slab a- oo and E = jtBo If it is
assumed that a point detector is used and if the detector is three inches
from the circular source disk of radius 1-1/2 feet, then the illuminance
at P due to such a disk will be
E = 0o97 AcB
Thus if the source disk were extended to inifinity the illuminance would
only be increased by 30, Thus the results measured with this disk should

-102\ige 2 l cADb
Figure 29. Illuminance at an Axial Point P by a Circular Disk.

-103not differ from the results obtained with a semi-infinite slabo In actual
practice a 2" spherical crystal was used and placed a distance such that
the center of the crystal was three inches away from the center of a 3 foot,
square slab of steelo For all practical purposes the sphere can be assumed
to be a pointo
Another problem now arises and that is that the source detector
geometry cannot be defined since the source of gamma rays in the degraded
spectrum is derived from the scattering of the primary beam by the media.
The scattering will not only degrade the primary energy, but also the direction of propagation of the scattered gamma ray will be different than
that of the primary beam. Therefore it is necessary to use a detector
whose response is independent of the source distribution. As is shown in
the previous Chapters, a spherical crystal can be used for this measurement as long as there are no sources closer than one half the diameter of
the crystal, and as long as there is no source behind the detectoro Thus
placing the spherical detector outside the scattering medium (the iron slabs)
and at a distance such that the center of the 2" spherical crystal is three
inches from the surface of the slab, fulfills the above requirementso In
this way the measurement in a semi-infinite slab geometry is simulatedo
Figure 30 shows the measured pulse height spectra for 0", 1/2"9
3/4", 1", 1-1/4", 1-1/2" thickness of steel slabs with a parallel beam
source of 0662 Mev gamma rayo The assumption that the continuous spectrum
can be written as a sum of monoenergetic spectra discussed in Chapter IV
is usedo Since the so-called "dynamic least square peeling off" process
has not as yet been programmed for computer computation, the following

-10410O
O" SLAB
1/2" IRON SLAB
3/4"IRON SLAB \
I" IRON SLAB \
I I/4"IRON SLAB-\\
10 Y S- I\ \
a~/W?^ 1t. 1
|_ DEGRADED RADIATION DUE TO',.661 MEV SCATTERING THROUGH \
- 0 -1-1/2 OF STEEL. A 2:"'
o_ SPHERICAL NaI (T) CRYSTAL'
- )SPHERICAL NoI (TJL) CRYSTAL
USED TO MEASURE PULSE \
HEIGHT SPECTRA,
10
I
40 50 60 70 80 90 100 110 120 130 140 150 160
PULSE HEIGHT
Figure 30. Pulse Height Spectra of Degraded Radiation Due to.661
Mev Gamma Rays Scattering Through 0"-1l" of Steel. A
2" Spherical NaI(Tl) Crystal Used to Measure Pulse
Height Spectra.

-1 o5iterative process was used. The energy region from 0.18 Mev to O.66 Mev
was divided into twenty equal increments o Pulse height spectra corresponding to the energy at the middle of each energy region were determined and
are shown in Figure 22. The choice was made such that the monoenergetic
components at the lower energy region increments were not separated in
energy by more than 0.7 times the widths of their photopeaks at half maximum. The half widths vary in the manner given in Equation (8)o Thus at
higher energies the energy separation will correspond to increments which
will be less than 0.7 times the width of the photopeaks at half maximum.
Experimentally it seems that as long as one keeps the energy separation in
the choice of monoenergetic less than the total width of the photopeaks at
half maximum, rather than the half width at half maximum, derived theoretically in Chapter IV, that one can describe the continuous distribution.
This result was determined by decreasing the number of monoenergetic components used to describe a given energy region, and for each choice of monoenergetic components calculating the number of photons in this region using
least squares analysis. This number remained relatively constant, and the
distribution described remained consistent until the energy separation was
greater than the width at half maximum of photopeaks.
Once the monoenergetic components were chosen for the analysis of
the pulse height spectra (Figure 30) a least squares analysis of the data
was carried out, and the f's or intensity of each monoenergetic component
was determined. Oscillatory solutions were obtained at the higher energy
region where there are four or five energy components per photopeak
half width. These oscillations can be attributed to the fact that the information in the energy increments is below the statistical noise or background level. That is, the more monoenergetic components are included the

-1io6smaller the AE increment, and thus the fewer events occurring in that
increment. Correspondingly there will be a greater statistical uncertainty in the number of events in that incremento Five iterations were
made eliminating the highest energy component containing a negative solution for each iteration. The elimination of a given energy component in
a region does not imply that this component is necessarily zero, it may
imply that the energy increment AE corresponding to this increment is
too small to infer statistically significant informationo The elimination of an energy component in a sense increases the width of the AE regions corresponding to the energy regions one step lower and higher in
energy from that energy component removedo
The fifth iteration yielded positive solutions for all components,
for all thicknesses of slabs considered and it was found that the separation in energy of the monoenergetic spectrawas no greater than 0o7 times the
half width of the corresponding photopeak throughout the energy region
analysed. The results of the least square analysis yields the area under
the photopeaks of each component because of the method of normalization of
the monoenergetic pulse height distributiono This number divided by the intrinsic peak efficiency corresponding to that energy gives the total number
of photons in that energy region. The final results are indicated in Table
VIo The results have been further normalized so that the total incident
gamma intensity for each slab thickness is 100 gamma rays/seco
The analysis of the pulse height spectra could not be extended
to energy region below that indicated in Table VI because of the scatter
from the surrounding media (ioeo, both the materials surrounding the crystal

-107in the detector reflection from the walls, floors and ceilings, and the
material supporting the iron slabs). This scattering greatly affects the
shape of the monoenergetic pulse height spectra in this low energy region
and is a function of source detector geometry. These scattering effects
must be greatly diminished in order to analyze this region.
Because of the presence of the scattering and because of the uncertainty in the knowledge of the monoenergetic spectrum, no attempt was
made to calculate the error involved in the results. The validity of the
results were investigated in the following manner. The linear absorption
coefficient for 0.662 Mev gamma rays and the energy buildup factor as a
function of thickness of steel were determined from the results in Table
VI and compared with theory.
The linear absorption coefficient was determined by plotting the
number of photons in the region of 0.662 Mev as a function of slab thickness, The photons in this region should correspond to the undegraded
gamma rays ( i.e., those gamma rays which have suffered no interaction when
passing through the scattering media). Any interaction should either totally
remove the gamma ray by complete absorption or by scattering out of the
given energy group. The results are plotted on a semi-log plot, and the
results should lie on a straight line on such a plot since the removal is
exponential in nature. The slope of this line is the removal cross section.
The results are plotted in Figure 31. The removal cross section p measured in this way is p = 0.525 cm-. Theoretical results for the mass absorption coefficients p/p for 0.66 Mev gamma rays are available in the
literature.(33) p/p = 0.073 cm2/gm for 0.66 Mev gamma rays. The density

-108LINEAR ABSORPTION COEFFICIENT AND
ENERGY BUILDUP FACTOR FOR.662 MEV
GAMMA RAYS PASSING THROUGH
STEEL PLATES
BE(X)
2.0 -
|ENERGY BUILDUP
0.6
0.59.6 0 ^S,~, ~u~ =.52 /CM.
04 -
LINEAR ABSORPTION
0.3- OF PRIMARY ENERGY.662 MEV
0.20.1 I I
1/4" 1/2" 3/4" 1" 1/4" 1 1/2"
THICKNESS OF STEEL.7 1.1 1.4 1.7 2.1
MEAN FREE PATH
Figure 31. Linear Absorption Coefficients and Energy Buildup Factors
for.662 Mev Gamma Rays Passing Through Steel Slabs.

-109TABLE VI
FINAL LEAST SQUARE FITTING RESULTS FOR THE DEGRADED PULSE
HEIGHT SPECTRA NORMALIZED TO AN INCIDENT
GAMMA INTENSITY OF 100 GAMMA S/SEC
E Mev PH Eo 1/2" 3/411 1" 1-1/4" 1-1/2"
Oo58 115 0 o 424 0o384 0o339 0o357 0305
Oo53 105 0 0o338 Oo425 Oo418 0o323 0o270
o 48 95 0 O 438 Oo473 Oo435 0o386 0o338
0o42 82~5 0 0.383 0o403 0 9 0393 47 0o300
0o38 75 0 Oo532 o 508 Oo 442 o 412 03o06
035.70 0 0o458 0o458 0o441 0o384 0o358
0o33 65 0 0o362 o038o 0o358 0o298 0o232
0o30 60 0 o 48o 0o 462 0396 00378 00316
0028 55 0 0o344 0O374 0O360 0o306 0o272
0O25 50 0 0O344 0o386 0o362 0.344 00306
0023 45 0 Ool48 00240 00326 00316 00314
0o20 40 0o0 oO o.492 0o548 0o562
of the steel used was 7~6 gm/cm3o Thus i = 00555 cm-lo The results obtained experimentally is 6% lower than the theoretical valueo This difference can be partially attributed to the inclusion of the coherent
scattering cross section in the theoretical results and to the forward
Compton scattered gamma rayso Since the detector does not subtend a zero
solid angle with respect to the scattered beam, coherently scattered gamma
rays and small angle forward Compton scattered gamma raysjwhose energy is

slightly changed from the primary energy,will be detected as photons in
the 0~66 Mev regionO This effect has been noted by other experimenterso(3)
Next the energy buildup factor BE(x) was calculated as a function of shield thickness in terms of lengths measured in mean free pathso
The energy buildup factor is defined as follows~(34)
Eo
j Ix (E,Eo) dE
BExo)X (66)
BE(x) = 0 (66)
Io (E 0
where Ix(E,Eo) dE is the differential energy flux measured due to the
scattered and unscattered flux of gamma rays with incident energy Eo, measured at the exit surface of the scattering media of thickness xo Ixo (Eo)
is the energy flux due to the unscattered flux of gamma rays measured at the
exit surface of the scattering media of thickness Xo
The integral can be easily evaluated from Table VI in the following mannero The total number of gamma rays in a region AE about the
average energy E are tabulated in this table Thus multiplying the average energy by the total number of photons in the AE about E and, summing over all E will yield the value of the integral in Equation (66)o
Monte Carlo calculations carried out by the Oak Ridge National Laboratory
indicate that the integration from 0 to 0O17 Mev adds only about 6% to the
integralo(35) Therefore this region can be excluded without affecting the
results significantlyo Ixo (Eo) is found simply by multiplying o066 by
the number in this energy groupo BE(x) can then be foundo The results
for BE(x) were compared with theoretical calculations carried out by
Peebles using a multiple scattering technique (36) In the energy region
from 0O5 Mev to lO Mev the shape of the energy buildup agree wello

-11 -
Further. BE(x) calculated by multiple scattering techniques at two mean
free paths is 2o0 which agrees well with the results obtained experimentally.
Fo Conclusions and Proposed Further Experimentation
The use of least square techniques in the analysis of the gamma
ray pulse height spectrum allows one to infer information concerning the
energy distribution and intensities of the gamma ray flux which produced
that pulse height spectrum. This least squares technique has been programmed for the IBM 704 computero Experimental results for both discrete and continuous energy distribution seem to agree well with theoretical calculations. A more general method of solution the so-called "dynamic
least squares analysis" is proposedo Because this method has not been programmed for computer use, the technique has been used only in a limited
wayo One of the major limitations in programming of this latter method for
the IBM 704 computer available at The University of Michigan is the lack
of sufficient fast memory capacity0 Further, the use of the least squares
fitting analysis also allows one to obtain some measure of the statistical
variance in the measurement0 It is believed that further work is required
to include the effect of the error inthe shape function of the monoenergetic
pulse height spectra on the analysiso
It has also been shown theoretically and experimentally that in
certain cases when the source detector geometry cannot be well defined,
that the use of spherical crystals allows one to make the measurement and
perform the analysiso The geometric limitation is that there be no sources
of radiation behind the detecto nor any sources closer to the spherical
detector than one-half the diameter of the sphereo

-112One of the great limitations in the use of this scintillation
method can be attributed to the lack of precise knowledge concerning the
shapes of the monoenergetic pulse height spectrao It is believed that it
would be rather fruitful to experimentally study this problem, compare
the results with the theoretical calculations(4), and develop a more precise method for obtaining these pulse height functionso
Another problem which must be considered carefully when performing these experiments is that of the perturbation of the pulse height spectra by the scattering from the surroundings. For fixed source geometry,
this may not be too great a problem. Although the scattering will greatly
distort the analysis at low energies, the effect is rather constant for
fixed geometry. The problem is of utmost importance in the analysis of
spectra in geometries which cannot be well defined since this effect can
be geometrically dependento Care must be taken to decrease the effects of
scattering from the surrounding walls, floors, ceilings, equipment containing
the experiment, and from the materials making up the detector itself (ioeo,
the canning material, light pipes, photomultiplier, voltage divider, and
preamplifiers)o

APPENDIX
FORTRAN STATEMENT OF LEAST SQUARE ANALYSIS37)
-115

-114C
C THI P I S R O A C L C J L T V ECTO ET JL T Q U A T iL T I X
C PRODUCT OF T" E INlVEJNRS OF T-!E Pi')l- (ATRAH'-':'LD —iA) /N
C THE PROD'"CT (ATP\t^O'A-R ). 0 LP' S L ARY AVIN T K, "AX
C _COLUNF'S \hn J'MAXX ROW S A) IN S RA) I I AS DATA. ATRRA.i IS TH_: L
C T RA'INSP OJSF A A L~ PHAI IRO IS A C'oLUMNJ ^!:t; ^4*:1ATRIX L i- AVIN, TG I^ V X A'I J'.S AN",L
C I S ALC R,5 ) I s 1 L'TA. ( IS A )IA'ON>\L F 1J T,I)xI L4/A TVIN"
-C 0DI)>IESINISj J ) U\AX BY J",AX, AN'D MIAY E %READ IN AS 3AT, CET "-)AL
C TO Tf UIl\!N I T MI`ATI:? RIXq O(R C/ALC UL'ATED_) FRO,- RHO. I N TiE LATTER CAiSE
_ _ _ _ _ _ _ _ _ _ _ DS_ _. A.___... __L___ -_._____I ___ _ L "'_- AT_ _ _I,_i E_ _ _ _ _,:_ —-.. —-----
C THE D A GN AL EL"' LN TS ECS T E I J P F A IS O i r OD;I XA I NG
C __ROW'LE' NTS OF RHO. INTrERl E I -IATE PRODL -T ARRAYS IN T-iE CALCUL',C TIO\' /'RE As FOLLJ\OWS..
C A'TRORA - A TRO. t.'_ALPHA
C LNL. N VRT = INVERSE OF ATROR,!A,
C * OTI-EIR'R'RAYS USED IN THE INVERSION CALCUL\TION ARE, AJX,C, AND UNIT.
C ANY OF TH-iEcE ARRAYS C\AN LE PRINTED OUT IF DESIRED LY SETTI.NG THC. ARPPR)OPRIATF CONTRO CCOINSTA`N T.TO 1, SEE BE-LO'L.
c
C TTHE SAviE AJLPHA ARRAY CA;\N bE.USED FOR CALCUJLATION WIITH SEVERAL
c SETS OF RH!- i AND t)OMEGA VALUES i3Y PROPLRLY SETTIN'G A CONTRiOL
C CONS rANT. SEE E3ELOW..
C
_______....__._........_.....^.______________ ----— _ - ----- -- - - -________ —- --- -~ ---- -__
INPUT TO THL PR'OGRA, CONSISTS OF T-HE FOLLOWI NG.
C
----.i- R'T';' -Tr:.; EN'T 2 -------------- ----- ---------------
C,__A)ONL CARD CONTAIN IN G.
C' 1 I.iALP.....ALN IDENT FYING NU.'ER FOR THE ALPHA AiRRAY
C CONSISTING OF NO MORE T lHAN 5 DCCI'lAL DIGITS.
C 2. * U >1 A X.
2"'. J.. " - -':- f\-. 77 -—,-U-',- --....TA
C ONE,HUD[)RED.
C T THAI TWEN-TY.
C B ) ANY NUM;'ErR:' — OF CA RDS CCNTA IN'I "[ N THE J AX*KMiAX ELENE,,TS OF ALPHA,
C SEVEN OF: CARlD DATA IS ENTERED AS A LINEAR STiRING, FIRST 3Y
C — RO~W,, T —'::t"HE.I 3Y" COLUFI',MN.
C READ ST TEJ-',EN T 5.
C A)ONE CARD CONTA-INING.
C 1. IDRHO. AN I DENTIFY IN NUMBLER FOR THE RHO ARRAY
C' CONSIST ING OF N'O MlORE THAN 5 DECIVAAL!DIGI TS..C. 22. NO.'E'G i..'S-T EOUAL TO 1 IF O0IEGA -IS TO RE READ'IN' AS.C DATA UJSIN INP'JT STATEMENT 418.'C "-. —-- SET ELQJ/AL TO ZEiRO IF OtMEGA IS TO 3E CALCULATL'D
C _______FROM.. TH-E- RHfO A RRiAi Y.
C *SET EQUAL TO 2' IF OMEGA IS TO 3LE THE UNIT;,ATIX.
-C —- 3.I? NSYM. S, T rQ, L TO 1'.,:'N ATROA IS KJ^ TO i_
C SYM Mf1 ETRIC SYSTE,'. TF SYJMMETRIC TLIF 4U.'I"' LR OF
C CALCULAT I ONS I?'VOLVD I N THE I iNVERS IiJ4 PROCESS
C IS CONSIL ERA\LY SMALLER (ABOUT HALF) TH-A
*.C__ R__i RE rUI R I. _SY E T I' F_______R I C..M',
C L N.SWTCH. SET EQUAL TO 1 IF A NEW ALPHA SE.T IS TO BE READ
C IN O3-FORE- THE NEXT (NOT THE CURRENT) RHO D)0ATA
C' """" ""'"- S ET IF NOT ElUAL TO 1 THE PROGR/A". ILL E-XP-CT

TO KEEP THE SAME ALPHA ARRAY FOR THE NEXT SET
OF RHOS.
5. NDUjP. _ SET EQUAL TO 1 IF A DUMP OF CORE _Ii_ DESIRED
AFTER THE. COMPUTATION FOR THE CURRENT RHO SET
HAS BEEN.COMPLETED........__...___.._.__HABEE_N__...LE_T_,._;____..............................
6. NOMEG2o SET EQUAL TO 1 IF IDOMEG AND A COMPLETE LISTING
OF ALL THE ELEMENTS OF THE OMEGA ARRAY ARE
TO BE PRINTED AS OUTPUT.
71. ^NALPHA. SET EQUAL TO 1 IF IDALPH AND AN COMPLET L LSTING
OF ALL THE ELEMENTS OF THE ALPHA ARRAY ARE
TO BE PRINTED AS OUTPUT.
8. NR RHO ~ SET EQUAL TO 1 IF IDRHO AND A COMPLETE LISTING
OF THE RHO ARRAY ARE TO BE PRINTED AS OUTPUT.
9. NATRAN. SET EQUAL TO 1 IF A COMPLETE LISTING OF THE
ATRAN ARRAY IS TO BE PRINTED AS OUTPUT.
10. NATROM. SET EQUAL TO 1 IF A COMPLETE LISTING OF THE
ATROM ARRAY IS TO BE PRINTED AS OUTPUT.
I. NATROR. S'T EQUAL T T IF 7 COM'PLET-' L-r - TING OF TRHEATROMR ARRAY I'S TO BE PRINTED AS OUTPUT.
12. NATROA. SET EQUAL TO1 IF A COMPLETE LISTING OF THE
AcTROMA ARRAY IS TO BE PRINTED AS OUTPUT.
~'l 37 -'N A.U'-X'_ --------:~S7f TA-rT^Tf'T-T-ofiT
AUX ARRAY IS TO BE PRINTED AS OUTPUT.
14. NC. * SET EQUAL TO 1 I F A ~COMPL'EE'- T I NG O F 1TH E..
C ARRAY IS TO BE PRINTED AS OUTPUT.
I~. NUNVRT. SET EQUAL TO 1 IF A. COMPLETE LISTING OF THE
UNVERT ARRAY IS TO BE PRINTED AS OUTPUTS
16. NPUNCH. SET EQUAL TO 1 IF IDALPH,'DOMEG, AND A COMPLET
LISTING OF THE UNVERT ARRAY ARE TO BE PUNCHED
_____-. —--------------— U__ __ ______..
B)ANf JUMBER OF CARDS CONTAINING THE JMAX ELEMENTS OF RHO,
SEVEN PER CARD. DATA IS ENTERED AS A LINEAR STRING, FIRST BY
ROW9 THEN BY COLUMN.
~_~_._T~_.L__ _IN__,__ ___C__LU _M _. _: ________- _ -..._.._._._._._... -__ _ __
READ STATEMENT 418.'- --' T-' —'...........-. — ---------...- -
A)ONE CARD CONTAINING.
1. IDOMEG. AN IDENTIFYING NUMBER FOR THE OMEGA ARRAY.
CONSISTING OF NO MORE THAN FIVE DECIMAL DIGITS.
B)ANY NUMBER OF CARDS CONTAINING THE JMAX NONZERO DIAGONAL
ELEMENTS OF OMEGA TO BE READ IN WHEN NOMEG1 IS SET EQUAL TO'1.
DATA IS ENTERED AS A LINEAR STRING, SEVEN PER CARD, FIRST BY
ROW THEN BY COLUMN..OUTPUT FROM THE PROGRAM CONSISTS OF THE FOLLOWING.
-_WIT!E _STATsMENT_ 1944._ _ _ __ _o_ ___
THIS IS AN UNCONDITIONAL OUTPUT STATEMENT WHICH PRINTS THE THREE
IDEETI`YTNG NUMBERS IDALPHIDRHO, AND IDOMEG, APPROPRIATE TO THE
CALCULATION, AND THE KMAX ELEMENTS OF THE FINAL VECTOR BETA.
WRITE STATEMENT 2137.
THIS_ IS AN, UNCONDITIONAL OUTPUT STATEMENT WHICH PRINTS THE THREE
IDENfTIFYING NUMBERS IDALPH,IDRHO, AND IDOMEG APPROPRIATE TO
THE CALCULATION, AND THE KMAX ELEMENTS OF A VECTOR MERIT.
THE ELEMENTS OF MERIT ARE COMPOSED OF THE PRODUCTS OF THE..CORRESPO4DI NG,ELEMENTS; OF BETA AND ATRC.MR.
WHEN THE CONTROL CONSTANTS HAVE THE APPROPRIATE VALUES. SEE.-ABPUNCH S _ ___N____ AB 2_5.__1_._
PUNCM STATEMENT 2251.

-116C TH.S IS A CONDITIONAL PUN'iCH S'lTLMirclT v'illCri 15 LXECU' lu. ILY
_______ S WHEN TH' rDONTROL CONSTAN-T NPUNCH HAS TH-. APPi OP I.T _Vl.A'______
C THIE FIRIST PUNCHED CARD WILL CONTAIN Tt - T',,wO IDENT TIFICIATION
--— __. ___. _ —I~.L.L__IU a.'l-A _I -LDO.. S L _ I -L.J.ALJ-T -F _-_C ELEMENTS OF UNVERT, FIVE PER CARD. RESJLTS'WILL rE PUNCHED
C AS A LIN.AR STRING, FIRST 3Y' ROW, T-.iEN LiY COLUL!-.'N.
C
C ADD I T1 ON T L C O^1E,'l1; EN^ TS,
C___ __-WHEN O'vEGA IS CALCULATED FROM RHO (SEE A0VE), ID G I S SET
C EQUAL TO IrRHO. WHEN OMEGA IS SET EQUAL TO THE UNIT MATRIX,
C IDOME( IS ARBITRARILY SET TO 11111.
C A SPECIAL RLOCK OF EM:,PTY LOCATIONS CALLED ADD HAS SEEN ADDED
~C__T.JA_'LOWL _A)DITIONAL INJTRJCTIONS TO BE ASSEMBBLED INIQ TH.E
C CAN BE AC.QMO DATED.
C________A1,AAO2,
C
DIMEN ION ALPHA( 100,20),~ATRAN(2,j10U), ATROM 1(20,100),RHO(1u0),
__ XONEGAlO 10) ATROMA(2 O,2 ), U NIT(2 ), C(2 C0,2 }) AUX(2,20),JUNVERT(2
X,20), ATRONR(20), BETA(20), ADD(100)
EUIVALENC (AT'R AN, ATRO C ), ( OMMEGA, AT RCHR ), ( OC'EGA ( 2 1),'ETA ),
X ( ATRAN ( 4 n i), AUX ), ( ATA7'T -6'-7), U'VET )
NONE = 1
2- READ I'NPUT TAPE 7,10,IDALPH JUMAX, KMAX, ((ALPHA(U,K), =i<=i 1,KAX),
XJ=1,JMAXI
--------- - - - - ---- - - - - - - -
10 FORMAT (15 I,13/ ( TIFo.Tu — r
5 READ INPUT TAPE 7,8, IDRHO, NOMEG 1NSYM, NSWTCH, NDUP,NOMEG2,NALPHA,
XN R H ON A TA N N A T )T' ATl ATT5~ A U XX N C 9uV TTJT;.x, j.,XJMAX )
8 FORMAT (i 5,q15 I 1/( 7F10.6 ) )
11 IF. (NALPHA-1) 1,16,1
1 6 I JRT I r, X 9, ( A, -HAr
X,J-= 1,JMAX)
-----— 1TBT'r rTT —- rr PT'^-5 ETETJTlWTr-FORTCPT — TEoR
X = I5/2!H NUMBER OF COLUMNS = I5///(7H ALPHA(I2,1H,I2,3H) =,
XE12.5, 11H ALPHA(I2,lH,I2,3H) =,E12.5, 11H ALPHA(I2,1H,I2,
X3H) =,E12.9, 11H ALPHA(I2,.1HI293H) =,E12.5))
1 IF (NRHO-1) 7,4,7
4 WRITE OUTPUT TAPE 6,19,.IDRHO,JMAX,NONE,(J,NONE,RHO(J),J=1,JiAX)
19 FORMAl (16H1INPUT DATA SET I59,1iH FOR RHO /2VH TNUMER — JF- OWS =
X I5/23H NUMBER OF COLUMNS = I5///(5f RHO(I2,1H,I2,3H) =E12.5))
7 IF ( CMEG1-1) 417,418,419
417 DO 45'.J=IJMAX --
467 ONEGA(JI = 1.O/RHO(J)
I_______O____ EG __= __I__HO__
GO TO 420
418 RFAD l1P'JT TAPE 7,425,IDOMIEG (OMEGA(J ),=1J=,MAX)
425 FORMAI (I5/(7F10.6))
____________________________________________ _ _ _ _____________ ____
GO TO 42\
419 DO 42: J=1J,UMAX
421 OMEGA(J) = 1.0
IDOMiEG = 11111
420 IF (NOMEG2-1) 17,229,17
229 WTRITE OUTPUT TAPE 6,231,IDOMEG,JMAX,J,AUAX,(J,J,lOMEGA(J), J=1,JAX).
231 FORQAT (27.H.1_ELEi;lENTS OF THE OMEGA SET 15 /2UHONUMBER OF RO'S
X = I5/23H NUMBER OF COLUMNS = I5/35HOALL NON-DIAGONAL ELEMENT
XS ARE ZEROQ//( 7 4O EGA( I2,JH,I2,31-) =,r12.5) )
17 DO 517 J=1',KMAX
DO 51? K=1,JUMAX__ _ _ _ _ _ _ _ _ _ _ _ _
517 ATRAN(J,K) = ALPHA(K,JU)
IF ( NATRAi-1i) 52,518,52Q
518 WRITE OUTP!IT TAPE 6,519, KifAX, JfUMAX, (.( J, K,ATRAN\ (:J, K), K= 1,J. AX ),

-117XJ=1, KAX)
519 FORMAT (33H1ELE;,EENTS OF THE ATRAN;f ARRAY /2OHONEU l-iER- OF RO/iS
X = I5/23H NJ:ER OF COLUMIS = I///(7 A1A(2r 3H)
X E 1 2,. 1 1 H A T R AN ( I 2 T T 11H 9,l I 2 9 3 H- ) H, E 1 2 R,? 1rl 65 T > Q i _ ( I 2 ~ 1 H I 2
XE12.3 H ATRAN( 2 12 H 23H) 1 =E1 A TRN ( I 2 1 1H I 2 i- ) = 2
X3H) =IE12.5, 1 H ATRAN(12,1H,12,3H) =,.E12.5))
520 DO 521 J=1,KMAX
DO 5.1 K=1,JM'AX
521 ATROt(J,K ) =ATRAN ( J, K )- -OMEGA-( K)
IF (NATRO",l - 1) 25,522,25
522 WRITE OUTPUT TAPE 6,524, KMAX, JMAX, ((J,.<,ATROM'(J,K),K=1,JMAX),
XJ=1, (MAX)
524 FORIMAT (33H1ELEMENTS OF THE ATROM ARRAY /2H N 2 H NUMBER OF ROWiS
X = I5/23H NUMB3ER OF COLUMNS = I5///(7H ATROM(12,1HI I2,3H) =
XE12. 1 iH ATROMi1( I2 1H I 2 3H) =, 12.5 11H A TR i ( I 2, 1H, I 2,
____3________________________5^__l,^^ _ ATRO 2 1H ^3J^=jJ__________2H________________
X3H) =;E12.,, 11H ATROM".(I2,1HqI2/,3H ) =,E12.5))
25 DO 30 J=1lKMAX
30 ATROMtiR(J) = 0.''0
DO 35 J=I, MAX
DO 35 I =1, j)AX
35 ATROMR(J) =ATRR + ATROROMR (J) ATRO ( J O I ) (I)
IF ( NATROP -1) 38,36,38
36 WRITE OUTPUT TAPE 6,37, K 7tAXNONE,"(J,,F.O7NEATKROiR;!R(J),J=i1,,,MVAX)
37 FOR'MAT (33H1ELEMENTS OF THE ATROMR ARRAY /20HiONlJNUMER OF ROWS
X = I 5/2 2 3H NUMEi-sER vl.-OF CUS = TS- /- / ( H AOT;gTR, rI 2 9 1H,F TT3HT - =,
XE14.7) )
38 DO 40 -J=,KM'AX
DO 40'K=1,KMAX
40 ATROM, (JK) = 0.0
DO.41 J=1I,KlAX
DO 4'. K=1, vAX
DO 41 I=1,JMAX
41 ATROM.A(J,:<) = ATROMA.(JK) + ATRO;( J I )-'ALPHA( I,K)
IF ( NATROA - 1 ) 45,43,45
43 WRITE OUTPUT TAPE 6,44, K~X, KTlAX ((J,K ATRi OM-A ( J, ) = 1 - i,KMAXT
XJ=1,KMAX)
_________.... ____________________________-________ _ _ —- - -_ — - - -__ ___ _ __ _ __ _
44 FORMAT (33H1-ELEMENTS OF THE ATROMiA ARRAY /2GHONUMtBER OF ROWS
X = I5/23H NUMBER OF COLUMNS = I5///(8H ATROMA(I2,1HI2,2H)=
XE12. 5 1Hi ATROM`A(I2,1HI2,3H) =,E 12.5 11H ATRO;iA(I 21HI 2,
X3H) =,E12. 9. 11H ATROMlA(I2,1HI'2,3H) =,E12.5))
45 DO 60 J=1,KMAAX
60 AUX(J;1) = ATROlA(J,1)
DO 65 K=2,KMAX
65 AUX( ItK) = ATROM".'A ( 1K)/ATROMA ( 1,1)
NDIAG = 2
69 K = _ NDIAG___________________
J = NDIAG
70 AUX(J.!K)= ATROMA(JUK)
IMAX = K —
Do 71 I=1,TMAX
71 AUX(JUK) = AUX(J,K) - AUX(J,I)*AUX(I,K)
IF (NSYt-! 1 _L_774, 771, 77_____
771 IF (JUI(). 773,774,773
77 3 AUX(t. J)- A J X (J <) / AUX( K<K)
___ IF (__-K i__ __AX) 70,7.J__74______
774 J = J 1 -
I F (J X 7 - 7 7 b 7 4'74 IF (NSYM- ) 75,78 75
__75 _ =__ NAG___
IF (NDIAG-< AX) 750,185,750

-1L8750 K = NDIAG + 1
76 AUX(J,K) = ATROMA(JK)
IMAX = J-1
DO 77 I=1,IMAX
77 AUX(J,K) = AUX(JK) - AUX(JgI)*AUX(IK)
AU" X ( Jt, K) AUX(,F'TiT'ATXT-JT —
K =K+1
IF (K-KMAX) 76,76,78
78 NDIAG = NDOAG +1
IF (NDIAG-KMAX) 69.699185
185 IF (NAUX-1) 849181,84
181 WRITECUT")UT TAPE 6,182T, rA-XT —-AX,((T,K,AXCJ, —),11,rl-AX7 *. 1
XKMAX)
182 FORMAT (33H1ELEMENTS OF THE AUX ARRAY /20HONUMBER OF ROWS
X = I5/2iH NUMBER OF COLUMNS =I5///(7H AUX(I2,1HI2,3H) =,
XE 12.59 11H AUX (I2 rHT2T93H T~7~-Z — 1H AUX(I2,..
X3H) =,E12.5, 11H AUX(I291HI293H) =,E12.5)')
84 DO 85 I=1,KMAX
85 UNIT(I) = 0.0
DO 98 K=1,KMAX
UNIT(K) = 1.O
C(19K) UNIT(1/AUX(1,1).DO 93 J=2,KMAX
C(J,K ) = UN I T (J)
IMAX t J-1
DO 9) I=i,IMAX
90 C(Jk) = C(J,K) - AUX(JI)*C(IK)
"""" JTT r —— J-'.R —J —J-T 1 ——.95 CONTINUE
---— T-' —' —---—' —---------------------------
98 CONTINUE
IF ( NC - I ) 1U1,99,1U1
99 WRITE OUTPuT TAPE 6,1009KMAX,KMAX, (J,K,C(J,K),K=1,KMAX),J=1KMAX)
-"'- TF^o'Mrlv p -T —( -T'3' -'E^'-''-K -E' —------ 2D —--— O --
X = I5/23H NUMBER OF COLUMNS = I5///(7H C(I291H,12,3H) =9
-----— XE1.- E 5-T-, —-------- rTr2-,T-H7 TT —-=T I2 7-.T -I —------— Cr 1
X3H) =,E12.5, 11H C(I2,1H,I2,3H) =,E12.5))
101 DO 19( K=1KMAX
J = KMAX
u —"- NV ET J —------------------------------------------
172 J=J-1
UNVERT J K) = TT ------------------------------------------------------
IMIIN = J+1
DO 175 I=IMIN,KMAX
175 UNVERT(J,K) = UNVERT(J,K) - AUX(JI)*UNVERT(IK)
-F (J.-1) 190,1'7' —-------------------
190 CONTINUE
IF ( NUNVRT-1 ) 193191l' —---— 93
191 WRITE OUTPUT TAPE 6,192, KMAXgKMAX,((JKgUNVERT(JgK),K=1,KMAX)JJ=
X1,KMAX)
- 192 FORMAT (_3H1ELEMENTS OF THE INVERT ARRAY /20HONUMBER OF ROWS
X = I5/23H NUMBER OF COLUMNS = I5///(8H INVERT(I2,1H,I2,2H)=
XE12.5, 11H INVERT(I2,1H,I2,3H) =,E12.5, 11H INVERT(I2,lH, I2
X3H) =,E12.5, 11H INVERT(I2,1H,I2,3H) =9E12.5))
193 IF ( NPUNCH-1) 119392251,1193
2251 PUNCH 2252,IDALPH,IDOMEG,((UNVERT(J,K),K=1,KMAX),J=1,KMAX)2252 FORMAT(1I(ITDALPH = I5,10HIDOMEG = I5/(5E4.6))
1.?93 J=1-J-MAX ----------------------------------------- - - - -
1193 DO 23 J=L_ C
293 BETA(J) = no
DO 194 J =1,KMAX
-- DQMAX____________________________________________A X
194 BETA(J) = PETA(J) + UNVERT(JI)*ATROMR(I)
1944 WRITE OUTPUT TAPE 6,195,IDALPH,IDRHOIDOMEG,(J,BETA(J),J.=1,KMAX)
195 FORMAT(33H1 ELEMENTS OF THE BETA VECTOR FOR /16H ALPHA DATA SET
XI5/16H RHO DATA SET I5/16H OMEGA DATA SET I5///(6H BETA(I2,
X3H) = E14.7))
DQ 2132 J =1,-KMAX —---------- ------
2132 ADD(J) = BETA(J)*ATROMR(J)
2137 WRITE OUTPUIT TAPE.6,2133, ]DALPH,IDRHO,IDOMEG, (JADD(J),J=1,KMAX)
2133 FORMAT (30H1FIGURE OF MERIT CALCULATION /16H ALPHA DATA SET I5
X/16H RHO DATA SET I5/16H OMEGA DATA SET I5///(7y MERIT(I2,
X 3H) = El4.7))
IF (.JDUMP - 1L 196,197,196
196 IF ( NSWTCH-1) 5,2,5
____197 CALL RR____________
*DATA

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