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1964 The ompC ound Elastic Scattering of 3.15 Mev Neutrons by Calcium-40. Zorawar Khangura Singh Louisiana State University and Agricultural & Mechanical College
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Recommended Citation Singh, Zorawar Khangura, "The ompoundC Elastic Scattering of 3.15 Mev Neutrons by Calcium-40." (1964). LSU Historical Dissertations and Theses. 999. https://digitalcommons.lsu.edu/gradschool_disstheses/999
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SINGH, Zorawar Khangura, 1928- THE COMPOUND ELASTIC SCATTERING OF 3.15 MEV NEUTRONS BY CALCIUM-40.
Louisiana State University, Ph.D., 1964 Physics, nuclear Please Note: Name in vita is Zorawar Singh Khangura. University Microfilms, Inc., Ann Arbor, Michigan THE COMPOUND ELASTIC SCATTERING OF 3.15 MEV NEUTRONS BT CALCIUM-40
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physios
by Zorawar Khangura Singh B.A., The Punjab University, 1950 M.So., The M.U. Aligarh, 1954 B.T., The Punjab University, 1957 August, 1964 ACKNCJWLBDfflfSNT
Thtt author wishes to express hie gratitude to Dr« Dorr C* Ralph for his valuable suggestions and critical discussions on this work* He also wishes to thank Carol J* Spahn and James Benham for their assistance and support in the Van de Graaff laboratory*
ii TABLE OF CONTENTS
Page ACKNOWLEDGMENT, • • ...... 11 TABLE OF CONTENTS...... I l l LIST OF TABLES...... v LIST OF FIGURES,...... vi ABSTRACT...... r i i
CHAPTER I . INTRODUCTION 1.1 The Bohr Rypothesis of Compound N ucleus...... 1 1.2 The Optical Model ...... 2 1.3 Shape-elastic and Compound-elastic Scattering...... 4 1.4 Measurement of Compound-elastic S cattering ...... 5 I I . EXPERIMENTAL APPARATUS 2.1 Introduction ...... • • • • ...... 8 2.2 The Pulsed Neutron Source.•••••••.••• ...... 8 2.3 Neutron Produolng Target.. •••• 9 2.4 Scattering Samples •••••• ...... 12 2.5 Neutron Detector ...... 12 2.6 The Experimental E lectronics...... ••••• 14 2.7 Neutron Monitor ...... • • • • • • • ...... 18 2.8 Performance of the Tiae-of-Flight Spectrometer••••••• 19 I I I . THE EXPERIMENTAL PROCEDURE
3*1 Introduction...... 22 3.2 Effective Energy of Neutrons Produced from a Gas Target.....e.•••••••• ...... ••••• ...... 22 3.3 Measurement of Angular D istributions ...... 25 3*4 Separation of Elastically Scattered Neutrons . in the Time Spectra...... 29 3*5 The Calculation of the Angular Distribution.••••••••• 29 IV. CONCLUSIONS 4*1 The Differential Elastic Cross Section...... 40 • i l l CHAPTER Page 4*2 Compound-elastic Cross Sectio n ...... 42
SELECTED BIBLIOGRAPHY...... 44 APPENDIX 1. Flux Calculations ...... 46 2. Attenuation Factor...... *..*...... ••••••••• 48 3. Fortran for K_...... ••••...•• ...... 48 4* Fortran for ?Ea?fr(Qg«r.En) ... ••••• 49 5. Table for D(d,n)He3 Cross Se c t i o n ...... 54 VITA...... 55
C
iv LIST OF TABLES
Table Page
I. Neutron Energy Distribution in the Gas Cell ...... 27 II. Differential Cross Sections...... 38 III . D(d,n)He^ Cross Sections ...... 54
V LIST OF FIGURES
Figure Page
1 Level Sehoaes of Caloiue^O and Potaseium-39 ...... 7 2 Pulsed Neutron Source ••••••• 10 3 Gas Target Assembly ...... •••••••••••••• 11 4 S oatterers ...... 13 5 Experimental Geometry...... 15 6 Block Diagram of Time-of-Flight C irc u itry .... •••••• 16 7 Tunnel Diode Discrim inator.•••••• ...... •••••••«. 17 S Monitor Ciouitry ...... • • • • • ...... 20 9 Monitor Shield ...... 21 10 D(dfn)He3 Reaction Cross Section vs ...... 26 11 Unconnected Time-of-Flight Spectrum of Calcium-40.•••••••• 30 12 Time-of-Flight Spectra of Calcium-40 and Potassium-39 with Background Subtracted ...... •••••••• 31 13 Direct Beam Time Spectrum in the Forward D irection. •••••• • 32 14 Polyethylene and Caloium-40 Speotra a t Low Bias Setting. •• 35 15 The Differential Elastic Cross Sections of Ca-40 and K-39* 36 16 The Differential Compound-Elastic Scattering Cross Section of Caloium-40 ...... •••••••••• •••• 37 17 Flux Calculations ...... ••••• 50
vi ABSTRACT
An attempt has been made to measure the differential compound elastic scattering of neutrons of energy 3*15 Mev by calcium -40 using a method based on the dissimilarity between the energy level schemes of potassium-39 and calcium-40, and based further on the assumption that shape elastic scattering of neutrons from these two nuclei is essentially the same. Since potassium-39 w ill exhibit very small compound elastic scattering compared to calcium-40 the difference in differential elastic scattering cross sections of these two nuclei will represent, to a close approximation, differential compound elastic scattering by calcium-40. The measured compound e la stic cross section of calcium-40 appears larger a t angles below 50° and sm aller above 1 1 0 ° than pre dicted by Hauser-Feshbach theory. The experimental data curve agrees fa irly well with theory between 50° and 1 1 0 ° • The measured total compound elastic cross section is 1. 0 &0.05 barns and th at found by theory is 1.0 barn. The detailed fit betweon the measured and computed differential compound elastic cross section is not good. This may due in part to the choice of single level parameters used in the theoretical calculations. That the general magnitude of the compound elastic cross sections is correct, is supported by the good agreement of the total cross sections. The assymmetry of the d iffe re n tia l compound ela stic cross section presumably resu lts from a breakdown of the assumption of identical shape elastic scattering in the calcium-40 and potassium-39.
vii CHAPTER I
INTRODUCTION 1*1 The Bohr Hypothesis of Compound Nucleus In 1936 N. Bohr suggested that nuclear reactions take place in two stages:^ first, the formation of a compound nucleus; and second, the subsequent decay of this compound nucleus. The cross section for a particular nuclear reaction may then be expressed as the product of two factors; first the cross section for the formation of the compound nucleus—*a quantity, for fast neutrons, always close to the geometrical area—and second, a quantity which measures the probability that the compound nucleus decays in the mode in question. The creation of the compound nucleus implies that the compound nucleus has an independent existence and that lasts for time long compared with the duration of the co llisio n . As pointed out by Bohr, in a ll probability, i t s lifetim e w ill be long enough fo r i t to lose all memory of its mode of formation when finally it is relieved of its excess energy by the emission of another particle. Immediately following the instant of collision this energy is rapidly shared out among tho constituents of the nucleus, thus the average energy of any one of them is insufficient to enable that particle to leave the nucleus and the compound nucleus finally emits a particle only when
^Neils Bohr, "Neutron Capture and Nuclear Constitution," Nature. CXXXVII (1936), 344.
1 2
6h« improbable event ooours that a large fraction of the energy of axoitation ia again conoantratad in ona oonatituant. The mode of decay o dapanda only on the angular momentum and tha energy and ita binding energy to tha compound nucleus• This is called Bohr Hypothesis. To calculate the probability of a particular reaction it is necessary to make an additional assumption. Tha siapiest ia to assume th at -the amission of tha different constituents of the nucleus, neutrons, - protons and more complex bodies likeoC -partioles, may be treated on an equal footing and that the probability of evasion of a specified par ticle may be calculated on. a statistical basis.
1 .2 The Optical Model Instead of considering the individual collisions which the incident particle must make within the nucleus it is possible to consider, alternatively, the interaction of that particle with the nucleus as a whole. This interaction is usually represented by a potential well. Thus calculations of the scattering and absorption of a nucleon resembles the calculations of the scattering and absorption of a beam of light by a refracting and absorbing sphere. This is the optical model* it cam give an account only of the gross features of nuclear reactions; it cannot account for the sharp resonances or for the competition between various nuclear reactions. The inadequacy of the statistical theory led to the interest in the 2 optical model by Feshbaoh, Porter and Weisskopf. They suggested the
2H. Feshbaoh, C. E. Porter and V. F. Weisskopf, "Model for Nuclear Reactions with Neutrons," Physical Review. XCII .(1954), 443. 3
^ model, called the cloudy crystal ball model, which describee only the features of nuclear reactions after averaging over the resonances of the compound nucleus* This model predicts only the shape-elastic scattering and the compound nucleus formation cross oection. It 4 ' O o° considers only the°conditions in the entrance channel* Hence the compound nucleus formation is considered as an absorption of the ■ ••• ,, •'**'* 0 incident beam, although part of it leads to an elastic scattering process* These conditions are‘described by means of a one-particle problem* The nucleus i s represented by a potential w ell having both re a l and imaginary components given by
V : V0tiW 0 fo r r< R and V s 0 fo r r> R
where R o S r AV3
In this expression VQ and WQ are respectively the.depths of the real and imaginary part of the nuclear potential in Mev, rQ is a constant and A is the atomic number* This complex po ten tial ac ts upon the incoming neutron* The scattering which the neutron suffers is the shape elastic scattering and the absorption which is caused by the imaginary part is the compound nucleus formation* The imaginary part of the potential function Indicates the strength and location of the processes that lead to an energy exchange between the incident neutron and the target nucleus* The real part represents the average potential energy of the neutron within the nucleus. Feshbach et* al* applied th e ir cloudy cry stal b a ll model to calculate the total cross section, shape elastic cross section and the cross section for the formation of the compound nucleus for neutrons, each as a function of energy and mass number, and also the angular 4 dependence of the elastic scattering* The experimental cross section exhibit resonances at lower energy but ire smooth functions in the higher energy regions where the experimental resolution is inadequate to resolve the closely spaced levels* The one-particle problem does not apply to the resonance region because of the rapid fluctuations* However, the averages of the cross sections taken over an interval, which included many resonances, were shown to be the cross sections belonging to a new scattering problem called the ngross-structure,n . problem* Thus the average properties of neutron resonances, in particular the strength function /^^/D, where is the average neutron width and D is the average level spacing, are connected with the gross structure problem and can be predicted by this model* The experimental neutron total cross sections were well reproduced in the energy region between 0*1 and 3 Mev. The angular dependence of the scattering cross section at 1 Mev was fairly well reproduced by this model* . The values of the strength function and the reaction cross section at 1 Mev showed only the qualitative agreement between theory and experiment*
1*3 Shape-Elastic and Compound-Elastic Scattering Elastic scattering is defined as scattering in which the kinetic energy in the center of mass system does not change* The elastic scattering cross section (Qie) is defined by the equation!
^•t ^e ' ^ne where ^.m total cross section,
C and ne * nonelastic cross section*
The e la stic cross section is composed of two parts, the compound 5
elastic cross section and another part, known as the shape-
elastic cross section (Q*)BO
* • o
The two parts of the elastic cross section have very different origin, for while the amplitude of the shape-elastic part is coherent with the incident wave, the other, the compound-elastic amplitude, is incoherent. Since the neutrons scattered by these,twoprocesses have the sane energy, a separate measurement of the compound-elastic scatterin g poses a d iffic u lt experimental problem. Eisberg, Yennie and Wilkinson have proposed one possible method for measuring compound-elastic scattering of protons by measuring the time delay between;the bremsstrahlung 3 associated with the formation and decay of the compound nucleus, but no one has yet done this experiment.
1.4 Measurement of Compound-Elastic Scattering The method used in this experiment, which is easier experimen tally but applicable only for a relatively small number of nuclei, is based upon the fact that shape-elastic scattering of neutrons is predicted fairly well for all nuclei by a potential well whose radius is R«r0A^/3. xhus the calculated shape-elastic scattering based on th e o p tical model is a slowly varying function of the mass number. Nuclides of approximately the same mass number w ill thus have the same shape-elastic neutron cross section, but not necessarily the same compound-elastic cross section. On the basis of an optical model
. M. Eisberg, D. R. Yennie, and D* H. Wilkinson, ”A Brem sstrahlung Experiment to Measure the Time Delay in Nuclear Reaction,” Nuclear Physics. XVIII (i960), 338. 6 of th« nucleus; the elastic scattering of neutrons by compound nucleus formation can be followed d irectly by comparing the angular d is tri butions of elastically scattered neutrons by elements which .are ■ close to each other In the magnitude of the shape elastic scattering cross section. At energies of several Mev there will be negligible compound-elastic scattering for most nuclei since there are many levels to which the compound nucleus will decay. Thus It follows that Inelastic scattering will predominate. If, however, there are no levels, or only a few such levels, compound-elastic differential cross section may contribute significantly to the elastic differential cross section. Since calcium-40 and potassium-39 have nearly the same atomic weight and have different energy level schemes, and since calcium-40 has no inelastic neutrons in the energy range intended, they are particularly suitable for making a measurement of compound-elastic scatterin g . The level schemes of these nuclei are shown in Figure 1. 3 90 3 5-62 5-60 5-24 3 -27 -27 3 4*' 3-73 3-35 40 40 Co OF Fig. 1 SCHEMES LEVEL 39
8 9
3-95 3-61 3 3*03 2-82
Mev CHAPTER II
EXPERIMENTAL APPRATU3
2*1 Introduction In this experiment the differential cross sections of neutrons elastically scattered by calcium and potassium at 3«15 £0.15 Mev neutron energy were measured. The equipment used In this experiment was the Louisiana State University time-of-flight spectrometer. This spectrometer was similar to the one developed by Crahberg et. al. at Los Alamos Scientific Laboratory.^ It utilised the radio frequency pulsed beam technique and variable path magnetic ion bunoher that produces the neutrons during short pulses of the accelerated beam on 2 3 a neutron producing target. 9 Since the neutrons are produced for a ‘ very short duration the time difference between the time of particle production and the time of neutron detection can be measured. The measured time difference allows the energies of the neutrons to be measured and neutrons of different energies to be separated. The schematic diagram of the experimental arrangement is shown in Figure 5*
2.2 The Pulsed Neutron Source
The Van de Graaff accelerator of Louisiana State University has
^L. Cranberg and J. S. Levin, "Neutron Scattering at 2.45 Mev by a Time-of-Flight Method," Physical Heview. CIII (1956;, 343* ^R. C. Mobley, "Proposed Method fo r Producing Short Intense Monoenergetio Ion Pulses," Physical Revieir. LXXXVIII (1952), 360.
gri E c ia r S been used to produce a beam of deuterons of energy 0.5 £0.005 Mev. The deuteron beam after maee analysis was swept by a 6.8 megacycle radio frequency potential applied to a set of deflecting plates. The swept beam was chopped by passing it through a set of plates. Then the chopped beam was bunched into short intense ion pulses by passing i t through the ion buncher. Further it was focussed by means of a set of electrostatic quadrupole lenses and then was collimated before it was allowed to fall on the target. The arrangement is shown schematically in Figure 2. The detailed description of the arrangement is given in reference 3. In pulsing the beam two things had been considered, the pulse duration and the frequency of pulsing. The burst duration was between two to three millimicroseconds with the average target current of ten microamperes collimated at 3/d inch. Since for a given beam duration the beam current on the target is proportional to the frequency, high frequencies (1 to 10 megacycles) are generally used. The limiting value of the frequency is determined by the difference in the time of arrival of the neutrons (or gamma rays) of interest to the observer. In this experiment a deflecting frequency of 6.8 megacycles were used.
•3 Neutron Producing Target In order to obtain 3.15 Mev neutrons the D(d,n)Ho^ reaction was used. The deuterons were accelerated and allowed to enter a deuterium gas cell through a nickel foil 25 microinch thick. The target assembly is shown in Figure 3. The gas cell was 2/5"*5/8" diameter and i t was cooled by an air blast. The window was supported by a grid with a close packed hexagonal array of holes each of 1/32 inch diameter. The cell was <§vacuated and was filled with deuterium Q 10
Beam
ANALYSING
MAGNET
R . F 0E ELECTING
PLATES
CHOPPING SLIT
Mobley Buncher
Collimator
GAS TARGET ELECTROSTATIC LENS (QUADRUPOLE) Fig. 2 COOLlNO .AIR OUTLET
HI FOIL
0534 COLLIMATOR
2/5 x 5/8" HI6H VACUUM ■CAM
6 AS INLET
COOLlNO AIR INLET GAS TARGET ASSEMBLY
Fig. 3 gas. A pressure of one atmosphere was maintained in the deuterium cell 8 The neutron yield was of the order of 10 neutrons per seoon*
2*4 Scattering Samples
V Each, sample was a oylinder two inches long and 5/8 inch in diameter* The oaloium sample weighing 15*3 grams contained 96*92 per cent of oalcium-40. It was sealed in a brass shell of thickness 0*006 inch with a 0*030 inch thick aluminum cap* The weight of the brass sh ell was 5*0 grams and th a t of the aluminum cap was 1*0 gram* Sim ilarly the potassium sample weighing 8*3 grams contained 93*1 per cent of potassium-39* It was sealed in a brass shell of the same thickness as that for calcium* Potassium sample was sealed in the presence of an inert atmosphere* The brass shell which contained potassium weighed 3*65 grams and i t s aluminum cap made out of 0*010 inch thick aluminum foil weighed 0*05 gram* Identical blank shells sealed with aluminum caps were used for "OUT" spectra* The soatterers are shown in Figure 4*
2*5 Neutron Detector The neutron detector consisted of a 2 inches long and 1*5 inches diameter plastic scintillator optically coupled to a 56AVP photomultiplier tube and was shielded from the room scattered neutrons gamma rays and other background by means of a large shield which consisted of a lead cylindrical shell 5 inches thick surrounded by a mixture of paraffin and lithium carbonate (Figure 5)* The purpose of the lithium carbonate was to capture the neutrons by means of the . Li^(n,tt)H3 reaction. An adjustable wedge constructed of iron and tungsten was used to screen radiation coming directly from the neutron I SEAL (VINYL PLASTIC)
BRASS SHELL
BLANK SCATTERER o Fig. 4 u source* The photomultiplier tube m s also shielded magnetically by a „ mu-metal shield and was cooled by ice water in order to increase the signal-to-noise ratio. The noise level was about 5 percent of the maximum proton recoil pulse height* The gain of the detector was checked from tim e-to-time by means of cobalt-60 and cesium -137 sources* The high voltage of the photomultiplier tube was also checked frequent ly and maintained constant* The slow pulse for the pulse height information was obtained from the twelfth dynode and the fast pulse for the tlmt 'lnformation was obtained from the anode of the photo multiplier tube* The entire detector and shielding arrangement was alligned at the height of the accelerator beam to point directly at the scattering sample*
2*6 The Experimental Electronics A block diagram of the time-of-flight electronics is Shown in Figure 6* The fast output at the anode of the photomultiplier was fed to a tunnel diode discriminator in order to reject signals less than the chosen height. The detailed schematic diagram of the tunnel diode discriminator is shown in Figure 7* Then the pulses were fed to two Hewlett-Packard 46OB and 46OA wideband amplifiers in cascade* The amplified pulses were fed to a time-interval to pulse-height converter* The time to pulse height converter measures the difference between the time of arrival of a signal from the neutron detector at the start input and the time of arrival of a signal from ths radio frequency pulsing system at the stop input* This timing information was converted to a pulse whose height was proportional to the measured time differ ence. The electronic circuit on,the input of the time to pulse height 1
SO% PARAFFIN ATTENUATOR
50% U m iM CARBONATE
LEAD TARGET Y7 7 7 A 120 CM.
LEAD SCATTERER
•3 C
0 10 20 30 CM. \ ■ ■ » 1
EXPERIMENTAL GEOMATRY Ul
Fig. 5 16
"NEUTRON OETtCTOR
FU STIC CRYSTAL rH P .
H . V
SLOW n w ' »
C . F TUNNEL MODE 0 -2 0 0 a SEC DISC. DELAY
• < p *
H P 4SO * H P 4 0 0 0 NON-OVERLOAD AMP AMP
AMP I I H.P 4 0 0 A H P 4S0A DISC AMP AMP H4I 2000, 3.0^*8 EC I INPUT •ATE TMC TMC INROT 4 00-CHANNEL PRINTER X-Y PLOTTER ANALY I LOCK DIAGRAM OF TIME-OF-FLIGHT CIRCUITRY Fig. 6 + 9 V 900 300 300 NEGATIVE 1:2 720 3 3 0 POSITIVE W f TO TO TUNNEL DIODE DISCRIMINATOR FlGl 7 18 converter carved to chape the pulsed for the converter, which ic <.7 designed to handle short, fact rising pulses of sufficient amplitude. o ' * * The fast output of the photomultiplier tube met the requirements of the start input of the time to pulee height converter in rise time and in duration and needed only to be sufficiently amplified by faet rising linear amplifiers, suoh as the Hewlett-Packard 4*60. The sinusoidal wave form from the pulsing oscillator had to be altered into a short pulse suitable for driving the stop input of the converter* This was done by driving two Hewlett-Packard 460B and 46OA amplifiers in cascade to saturation, giving an approximately square wave* The signal was then clipped in time' with a shorted coaxial cable about 125 oms* long* The resulting wave form which drove the stop input of the converter was consisting of alternate spikes of about 6*2 volts high and about 20 nanosecond duration. The gate signal to the pulse height analyser, of linear output neutron detector, was amplified through a nonoverload amplifier, discriminated to reject the small signals* The output of the discrimina tor was further shaped by using a pulse shaping circuit* It was then fed into the gate input of the pulse height analyser* The pulse height analyser was gated to discriminate against small pulses from the neutron deteotor* The spectrum thus obtained was analysed by the TMC 400 channel analyser and was printed by the printer. 2*7 Neutron Monitor i In order that all scattering measurements could be normalised to a constant number of neutrons impinging upon the scattering samples, it was necessary to monitor the flux of the neutron source. The monitor r was a 1” x 2" diameter plastic sointillator optically coupled to a 56AVP photomultiplier. The pulse from the anode of the photomultiplier a * fV-' ' * r“ was fed through a oathode follower to a nonoverload amplifier with a discriminator output which drove the scaler. The discriminator was adjusted to aooept most of the pulses caused by neutrons from the * • targ e t but high enough so' th a t the counting ra te would be reduced to a few percent if the particle beam was off the target. The monitor circuitry, is shown in Figure 8. The monitor shield consisted of a cone filled with paraffin and borax in equal ratio and was lined with 3 inches thick lead. I t was hung from the ceiling and was situated to view only the neutron source as shown in Figure 9. The monitor photomultiplier tube was also shielded magnetically and was oooled with ice water. The monitor was kept s till throughout the experiment. The high voltage and the gain of the monitor were checked frequently and were kept constant throughout the experiment. 2.8 Performance of the Time-of-FUght Spectrometer The performance of the Time-of-Flight speotrometer was quite satisfactory throughout the experiment for the purpose of the present work. The time resolution was six nanoseconds. The monitoring uncertainty was fraction of a percent. The shielding arrangement and electronic equipment provided the best signal to noise ratio. The noise level was about five percent of the maximum proton recoil pulse height and was sufficiently low for measuring differential cross sections to within five percent. PLASTIC •CINTILATOR RCA 6 9 4 2 BASE 1 «•' . m * i i H.V. POWER NON-OVI AMPL NEUTRON MONITOR FIG. 8 50 % BORAX / PLASTIC SCINTILATOR 5 0 % PARAFFIN OEUTRON BEAM SCATTERER TARGET O 10 20 30 MONITOR SHIELD CM E! & £______CHAPTER III THE EXPERIMENTAL PROCEDURE 3.1 Introduction The purpose of the experiment was to measure the average differential elastic cross sections of calcium -/»0 and potassium -39 in the vicinity of 3*15 Mev since the theory is applicable to the average cross sections and does not reproduce the resonance structure* The type of information obtained is dependent on the energy spread of the neutrons employed in the experiment. In this experiment the neutron energy spread obtained by the suitable choice of target thickness was 300 Kev* This spread was large enough for the measure ment of suitably averaged cross sections of both elements* 3*2 Effective Energy of Neutrons Produced from a Gas Target In order to find out the effective energy of the neutrons produced from gas targets, it is necessary at the first step to find out the energy loss AE of a beam particle as it traverses the window which may be computed by the following equation: -A E sC N AX Negative sign on the le f t hand side of the equation represents the energy loss, g is the atomic stopping cross section of the window 22 material^, N is the number of atoms per cm3., AX is the thickness of the f o il. Since the loss in energy of the beam particle in traversing through the whole length of the gas cell is not linear, also the corresponding neutron energy and the scattering cross section are not linearly related to the incident energy. Therefore, the following computations were made to find out the effective neutron energy. Let the gas cell be divided along its length into a number of sub cells and consider the i£& cell whose thickness is AX^. If E^ is the incident energy and E j^ the emergent energy of the particle from th is sub c e ll, then = E^ - A E ^ The energy loss AE^ of the particle in traversing the sub cell under consideration can be computed from the equation: AtE^ s (dE/dx)i A x ± where (dE/dx)^ is the specific energy loss of the particle in the -gas whose incident energy is E^. The emergent energy E ^i is the incident energy for (i+l) thy sub cell. Thus the incident energies for a l l the sub c e lls were computed and the corresponding neutron energies En^*s were found from the tables fo r D(d,n )He3 reaction2. ^S. Flugge, HMndbuch der Phrsik. Corpuscles and Radiation in Matter II, (Berlin: Spinger-Verlag,-1957 ) X$£lv, pp. 193-213. 2 J. B. Marion, I960 Nuclear Data Tables. Part 3. (Washington: U.S. Government Printing Office, 19&)). o (■) Since the neutron energy En is a function of D(d,n)He^ cross section - fX9 .?• „ 0 • » Qi (% ), incident deuteron energy E^, specific energy lose (dE/dx)^ of the deuteron and the thickness of the sub cell, therefore, the average effective energy ER of neutrons can be found from the following equation: (dE/dX)i AXi iil^(% ) (dE/dX)i **i where k is the number of sub c e lls. The loss of energy AE in the nickel foil of thickness 25 microinch was .140 Mev and thus the deuteron beam of energy 500 Kev was left with the incident energy of 0.360 Mev to enter the gas cell. To find the effective energy of the neutrons produced in the gas cell, it was divided into eleven equal parts along its length, and the sample calculations can be done by considering one of the sub cellsy Its thickness A was 0.1cm. The energy E-^ at the entrance of this cell was 0.360 Mev, the specific energy loss .(dE/dX)^ corresponding to th is incident energy was 0.210 Mev/cm.,^ and the D(d,n)He^ cross section was 12.75 millibarns/ster.^ The particle lost 0.021 Mev in traversing the sub cell. The emergent deuteron energy from the first sub cell or the incident energy to the second sub cell was 0.339 Mev. The neutrons of energy 3*30 Mev were produced from th is c e ll.^ Sim ilarly the neutron.. energies E *s from a l l the sub ce lls were computed and are shown in 3 Flugge, op. c it., pp. 193-213. Slarion, fip. c it. P Nuclear Data Tables. 25 Table I. The effective neutron energy E^ calculated through the above equation w t 3*15 Mev. The energy spread caused by the length of the gas cell was computed as follows. From the graph, A e s e -E " i [ n The arithmetic mean AE of these P values given by the equation: is the energy spread caused by the gas cell. The effective neutron energy thus obtained was 3 «15 £0.15 Mev. The overall energy resolution o f the spectrometer was 330 Kev. This included the spread due to the flight path, detector responee and the efficiency of the electronics etc. 3*3 Measurement of Angular D istributions The Van de Graaff accelerator was adjusted for the stability of beam current. The bunoher and the associated equipment was adjusted to obtain a burst of short duration of about two to three millimicrosecond and the beam was collimated and focused to get a maximum target current of about ten to twelve mioroamperes on the target. The energy of the beam was regulated at 0.5 £0.005 Mev. The slow and the fast signal channels of the detecting system were adjusted for high counting rate. The timing system was adjusted for better time resolution which was six nanoseconds. The experimental set up was precisely alligned. The sample o r |l*0)» mk/8t. 25 2.7 E T O EEG V COS SECTION CROSS VS ENERGYNEUTRON (DATA TAKEN FROM 2) REF. 3.3 3.9 o m TABLE I NEUTRON ENERGY DISTRIBUTION IN TIE GAS CELL No. 4X± c r \ ) Ei (H ) 1 S . (i) (cm) (Mev) (mb/st) (Mev/cm;) (Mev) . 1 0.1 .360 12.75 .210 3.30 2 0.1 .339 11.95 .215 3.27 3 0.1 .317 11.15 .225 3.23 4 0.1 .294 10.30 .233 3.20 5 0.1 .270 9.35 .245 3.16 6 0.1 .245 8.35 .260 3.12 7 0.1 .219 7.20 .277 3.08 a 0.1. .181 5.30 .297 3.01 9 0.1 .151 3.95 .320 2.95 10 0.1 .119 2.70 .347 2.89 n 0.1 .084 1.60 .380 2.79 holder was positioned to hold the scattering sample centered 2 * 25"Inches In the forward direction with respect to the beam direction and.with cylindrical scatterer axis perpendicular to the beam arid coaxial with the axis of rotation of the detector. The shield of the neutron detec tor was pivoted directly below the scatterer so that the scattered neutrons could be detected at any angle between 0° and 140° with respect to the direction of the incident flux. A flight path of 120cms. was used. The monitor and its shield were hung from the ceiling and were positioned so that the monitor viewed only the neutron source. The discriminator of the monitor was adjusted to discard the radiation other than that from the neutron source. The time scale was 1.2 nano seconds per channel on the pulse height analyser. With these adjustments made in the equipment, the time-of-flight spectra were recorded at different angles between 20° and 140°• Each spectrum taken for calcium and potassium was normalized to the number of direct beam counts in the forward direction. Six runs were taken and direct beam counts were recorded after every two angles during each run. At each angle the spectra were taken with the scatterer "IN” and scatterer "OUT*1. The "OUT1* spectrum was subtracted from the "IN" spectrum in order to find out the number of neutrons elastically scat tered for a constant number of monitor counts. The differential cross sections obtained were corrected for the difference in the number of nuclei per cm? for the two elements. Small changes in the beam energy, while not greatly effecting the energy of the neutrons produced at the target caused the elastic peak on the time spectra to drift due to different arrival time of the deuterons at the target. During the course of the data runs the machine energy had to be adjusted quite frequently. This error was kept to a minimum by a close watch of the o,29 time spectra while they were accumulated, and discarding those in t i which the large shifts were observed. The error due tothe variation in gain, due to the change in temperature of the cooling system and due to the change in room temperature, was minimised by normalising the data runs to the direct beam counts taken at approximately the same time. 3.4 Separation of Elastically Scattered Neutrons in the Time Spectra For the incident neutron energy of 3.15 Mev calcium-40 had no inelastic scattering because the first excited state 3*35 Mev is higher than the incident energy and therefore, can not be excited. Potassium-39 has first and second excited states at 2.53 Mev and 2.82 Mev respectively which were weakly excited and so it had some inelastic scattering. The time spectra of calcium -40 and potassium-39 were fitted together with reference to their corresponding gamma peaks. The counts due to inelastic scattering in the potassium -39 spectrum were easily distinguishable from the elastic peak. The total number of elastic counts were added by considering equal number of-channels on both sides of the elastic peaks. The uncorrected time spectrum of calcium-40 is shown in Figure 11. The time spectra of caloium-40 and potassium-39 with background subtracted are shown in Figure 12.. In Figure 13 is shown the direct beam time-of-flight spectrum to which the elastic counts of calcium -40 and potassium -39 were normalised. 3.5 The Calculation of the Angular Distributions In order to convert the relative angular distributions to absolute elastic differential cross sections, the relative number of neutrons scattered at 40° from calcium was compared to the relative UN CORRECTED TIME SPECTRUM j OF CALCIUM-4 0 •IN 1 OUT * • 4 0 * ► » 0 U ~ VntliiWiiHrfiMwWF11** ' m u —-u10------20^ ^ 30------$------&------&------fer ~~ "' t o ------to ------ifco- COUNTS / CHANNEL 400 300 •00 900 too 100 . 10 CHI SO 5* W V ■ " 40 SO s • s FIS. FIS. •0 Co40 AND K5* , BACKGROUND , K5* AND Co40 SUBT. CORRECTED OF SPECTRA TIME 12 En-3.I5±0.I5MEV f * 0 4 « O 70 TIME/ CHANNEL* l.2n« TIME/ •0 « - K” - « ■Co40 • •0 -TIME 100 110 0 O 10 COUNTS/CHANNEL x 2xlC" 14 22 10 6 3 *3 10 CHANNELS 20 \ SPECTRUM TIME BEAM DIRECT IN THE FORWARD DIRECTION DIRECTION FORWARD IN THE 3. 5 .1 3 = t 15 5 .1 0 Mev 100 33 number of neutrons detected from the direct beam in the forward «3> direction for the same nunber of monitor counts. The equation used to calculate the absolute elastic differential cross section of calcium-40 at 40° was: c, y^ d»* cDB n, K„ A *a53p(e.T ,Ed) A t I The derivation and explanations of the symbols used in the equation a re given in appendix. The computer programs for some parts of the equation are also given in the appendix. The absolute angular dis tributions by direct beam method are shown in Figure 15. The conversion was also done by polyethylene method. The relative number of neutrons . scattered at 40° from calcium-40 at low bias settings was compared to the re la tiv e number of neutrons scat tered at 40° from the hydrogen part of (CH 2 )n sample of the same s iie and at the same bias settings for the same number of monitor counts. The two spectra are shown in Figure 14* The equation used for this method was: g L _C. % C H KCH y (?) * CH «.«.*. " and ® h(0) 5 oosO- TT* where Cs and CH are the number of counts, N8 and NH are the number of nuclei per cm?, for the scatterer and hydrogen respectively. Ks and Kch are the attenuation factors for the scatterer and (CHg^ sample, «V»> is the differential cross section for hydrogen at the angle 0 o 34 5 and CTn. p the total neutron-proton cross section. The relative, efficiency £ . 0/ was calculated by using Cranberg’s efficiency ourve.^ The reason for using Cranberg*s efficiency curve is that our plastio scintillator was approximately of the sane else as that of Craribergfs and the shape of the part of the curve used in this work was V • nearly the same for different bias settings. Evidently the relative efficiency from any one of the curves was approximately the same. The o absolute elastic cross section of oaloium-40 at 40 found by this method was fifteen percent higher than that found by the direct beam method. These calculations were repeated using carbon. The d iffe re n tia l cross section of carbon at 40° for 3*1 Mev neutrons was obtained from 7 B the published angular distributions. * This absolute value of the cross section was in agreement w ith the d irect beam method value w ithin 2.4 percent. The satisfactory agreement of the carbon calculation suggests that the disagreement in the hydrogen comparison results from the assumptions made in attempting to correct the deteotor efficiency for the lower energy neutrons. The angular distributions shown in Figure 15 were calculated by direct beam method. The difference of the two angular distributions is shown in Figure 16, which represents the differential compound ^Marion, op. c i t . . Nuclear Data Tables. ^Crariberg, op. c i t .. Physical Review. C III, p. 343. 7 R. W. Meier, P. Soheroer and G. Trumpy, "Elastishe Streuung und Polarlsationseffekte Von D-D- Neutronen an Kohlenatoff." Helvetica Physloa Acta. KVII (1954), 577. 8 J. E. Wills, Jr., J. K. Bair. H. 0. Cohn and H. B. Willard, "Scattering of Fast Neutrons from and Fl9.n Physical Review. CUC (1958), 8 9 1 . ------NEUTRON SPECTRUM FROM ELASTIC PEAK OF Co40 POLYETHYELENE BACKGROUND SUBTRACTED 40* » * 4 0 * (LOW BIAS) (LOW BIAS) MKUTROMS SCATTKRCO PROM HYOROOCR^ MKUTROMS SCATTERED PROM CARRON, 10 SO40 SO •0 TO C p i o. 14 THE DIFFERENTIAL ELASTIC CROSS SECTIONS OF Ca*°AND K FOR S .IS i 0.15 MEV NEUTRONS . Co40 * K » x DIFFERENCE 4 M .0 io 140 ito ..100 , 4 H DFEETA COMPOUND DIFFERENTIAL ELASTIC THE CTEI6 RS SCIN F Ca*o OF SECTION CROSS SCATTERIN6 416 1 .4 r FOR FOR 15 ± 0. 5 .1 0 ± 5 .1 3 NEUTRONS i • EXPERIMENT • THEORY MEV i ' m 38 TABLE II DIFFERENTIAL CROSS SECTIONS IN BARNS/STERADIAN E lastic Compound E lastic Cos(tfcm) Ca-40 __ K-39 Ca-40 20 ° .936 1.026*0.009 0.610* 0.008 0. 416* 0.0012 30° .859 0.763*0.007 0.438*0.006 0.325*0.009 40° .755 0.49210.005 0.304*0.005 0.188*0.007 50° .627 0.296*0.004 0.193*0.004 0.103*0.006 60° .481 0.172*0.003 0.122*0.003 0.050*0.004 70° .319 0.103*0.003 0.076*0.003 0.027*0.004 80° .149 0.071*0.003 0.056*0.003 0.015*0.004 90° -.025 0.069*0.003 0.053*0.003 0.016*0.004 100° -.197 0.068*0.002 0. 047* 0.002 0.021*0.003 110° -.364 0. 074* 0.002 0.Q47t0.002 0.027*0.003 120 ° -.518 0. 087* 0.002 0 .060* 0.002 0.027*0.003 130° -.657 0.115*0.003 0.073*0.003 0.042*0.004 140° -.776 0.149*0.004 0.099*0.004 0. 050* 0.006 elastic scattering cross seotion of oaloium-40. Also in Figure 16, is shown the theoretical an^ilar distribution of compound elastic scatter- ing of caloium-40. The experimental curve is in general agreement with the theoretical curve. dHAPTER IV CONCLUSIONS 4*1 Differential Elastic Cross Sections The differential elastic cross sections of calcium-40 and potas- sium-39 are shown in Figure 15* The error bars are statistical errors which range from one percent to five percent* The estimate of the rela tiv e accuracy of the measurements can be made from the smoothness of the curves and the consistency of the data. The consistency of the data throughout the experiment for all the runs was very encouraging. The inaccuracies contributing errors were as followst a,) The statistical uncertainty in monitoring was 0.15 percent and that in the direct beam was 0.28 percent. Thus the uncertainty in the direct beam counts introduced by monitoring was 0.32 percent. b) Machine background was less than one tenth of a percent. c) The machine in s ta b ility contributed the error in several ways: changes in the beam position at the target could have caused a change in the scattering angle by t 3 percent but the consistency of the data shows that this source could not lead to a greater error. d) Scattered neutrons lose about 8 percent of their energy at 140° in comparison to the direct beam energy. This loss of energy reduces the experimental values of the cross sections by upto 3 percent at back angles. e) The secondary scattered neutrons from the shielding wedge or the collimator were more probable at the smaller angles than at greater o. - 1 41 * - o angles. The magnitude ^Meir, et. al., op. cit.. Helvetica Physica Acta, p. 577. S fills , ot# a 1#| 2E* cl^bt| GIX| p# 89Xe 42 180 ^ 2jf f "5 Sin 0 do " J d * 0 The total elastic cross sections thus obtained are 2.8 2*0.04 barns for calcium-40 and 1.81.0.04 barns for potassium-39. Since there is no inelastic scattering in calcium -40 at this energy the total elastic cross section is also its total cross section. The total elastic cross section for calcium-40 found by Popov is 28 percent higher than this value. His differential elastic cross sections for both the elements are higher at all angles than the one's found in this experiment. 4*2 Compound E lastic Cross Section If both the optical model and the Hauser-Feshbach theory were valid for the neutron scattering at 3.15 Mev, the difference in the elastic scattering of the two elements should be compound elastic scattering (according to the optical model) and symmetric about 90° (according to Hauser-Feshbach theory). For an approximate check on the symmetry and the magnitude of the differential compound elastic dis tribution, computations were made on the basis of Hauser-Feshbach theory using single level param etersThe measured compound elastic oross section of calcium-40 at 3*15 Mev (Figure 16) appears larger at angles below 50° and smaller above 110° than predicted by Hauser-Feshbach theory. The experimental data curve agrees fairly well with theory 3v. I . Popov, "Angle D istribution of 3.1 Mev Neutrons E lastically Scattered on Al, Si, K, Ca arid Th," Neitronnaya Fiaika. (Moscow: Gosatomadat, ed. Krupchithkogo, p. A., 1961). Sr. Hauser and H. Feshbach, "The Inelastic Scattering of Neutrons" Physical Review. LXXXVII (1952). 366. 0 © (•"' -■ <••• 43 O , between 50° and 110°. Within the aeeumptione of the theory, the fluctuations in the total cross sections are dus^t 6>resonance formation of the compound nucleus and fluctuations in the elastic differential cross sections are again due to resonance in the formation of the compound nucleus and in this case appear as a change in the compound elastic scattering. The published angular distributions of elastic scattering of 3.1 Mev neutrons on calcium and potassium by V. I. Popov has cross section of potassium larger than that of calcium between 60° and 90°.^ This means, according to the assumptions of this experiment, the compound-elastic cross section for calcium is negative in that region which is meaningless. This makes one tend to disbelieve thatr'dkta. The total compound elastic scattering cross section for calcium-40 for 3*15 Mev neutrons found by Hauser-Feshbach theory using single level parameters is 1 .0 barn and the difference in the total elastic cross sections of calcium -40 and potassium -39 found by this experiment is 1.0 £ 0.05 barns. It is therefore concluded that, although the detailed fit between the computed and measured differential compound elastic cross section is not good, this may be due in part to the choice of single level parameters in the Hauser-Feshbach calculations. That the general magnitude of the compound elastic cross sections is correct, is supported by the good agreement of the total cross sections. The assymmetry of the differential compound elastic cross section presumably re su lts from a breakdown of the assumption of id en tical shape elastic scattering in the calcium -40 and potassium- 39* ^Popov, op. c i t .. Neitronnaya F isika. o SELECTED BIBLIOGRAPHY Batchelor, R., Gilboy, W. B., Purnell, A. D. and Towle, J. H. "Improve ments in the Fast Neutron Time-of-Flight Technique using Pulse Shape Discrimination in an Organic Phosphor," Nuclear Instruments and Methods. VIII (I960), 1*6. Bjorklund, F. and Fernbach, S. "Optical-Model Analysis of Scattering of 4*1-,7-, and 14-Mev Neutrons," Physical Review CIX (1958)* 1295* Blatt, J. and Biedenharn, L. C. "Nautron-Proton Scattering with Spin Orbit Coupling. I. General Expressions," Physical Review. LXXXVI (1952), 399. Bohr, N. "Neutron Capture and Nuclear Constitution," Nature. CXXXVII (1936), 344. Bfyster, J. R., Walt, M. and Salme, E. W. "Interaction of lJD-,1,.77-. 3.25-, and 7.0-Mev Neutrons with Nuclei," Physical Review. CIV (1956), 1319. Brandenberger, J. D. "Compound-Elastic Scattering of Fast Neutrons by Lead-206," (Unpublished Ph.D. Dissertation, The University of Texas, Austin, 1962). Brooks, F. D. "A Scintillation Counter with Neutrons and Ganaa-Ray Discriminator," Nnoliar Instruments and Methods. IV (1959), 151. Cranberg, L. "Time-of-Flight Technique Applied to Fast Neutron Measure ments." International Confrence on the Peaceful uses of Atomic Energy. 1955. Cranberg, L. and Lelfin, J. S. "Neutron Scattering at 2.45 Mev by a Time-of-Flight Method," Physical Review. CIII (1956), 343. Day. R. B. "Ganma-Rays from Neutron Inelastic Scattering." Physical Review. CII (1956), 767. Eisberg, R. M., Yennie, D.-R. and Wilkinson, D. H. "A Bremsstrahlung Experiment to Measure the time Delay in Nuclear Reactions." Nuclear Physics, XVIII(1960),338. Feshbach, H., Porter, C. E. and Weisskppf, V. F. "Model for Nuclear Reactions with Neutrons," Physical Review. XCVI (1954), 448. Flugge, S. "Nuclear Reactions," Handbuch der Physlk XL, Berlins Spinger-Verlagj 1957. pp. 356-357. 0 44 Friedman, F. L. and WiesskOpf, V. F. "The Compound Nuoleua." Mail Bohr and Develoonent of Physics, ad. F aull.W . New York: KoGrew-Hill Book cS™“in7.71*55. Hauser, W*juid Feehbaoh, H. "Inelastic Scattering of Neutrons,w Physical Review. LXZXVII (1952), 366. Kant, D. W., Puri, S. P., Snowdon, S. C. and Buohar, W, P. "Interaction of 3*7 Her Neutrons with Medium Weight Nuclei." Physical Ravi aw. CXXV (1962). 331. Marlon, J. B. and Fowler J. L., Fast Neutron Physios Part I and Part II New York) Intersoienoe Publishers, Inc., I960. Marion, J. B. I960 Nuclear Data Tables. Part 3. Washington! U. S. Gov ernment Printing Office, I960. . Mobley, R. C. "Proposed Method for Froduoing Short Intense Monoenerget- ic Ion Pulses," Physical Review. LXXXVIII (1952), 360. - r Mobley, R. C. "Varlable-Path Magnetic Ion Bflmcher," Review S cien tific Instruments, XXXIV (1963), 256/ Nailer, J. H. and Good, W. M. "Time-of-Flight Technique," Fast Neutron Physios. Part I ed. Marion, J. B. and Fowler, New Yorks Intersoienoe Publishers, m e,, I960. pkhuysen, P. L ., Brandenberger, J . D. and Smith, W. R. "Compound-Eketic Scattering of Fast Neutrons by Lead," Bulletiii Aaerioan Physioal Sooi- ety,YI (1961). 375. Okhuysen, P. L. and Prud'hoomo, J . T. "Compound-Elastic Scattering of 4.2 Mev Neutrons in Lead." Physioal Review. CXVI (1959), 986. Popov, V. I. "Angular Sistrlbution of 3*1 Mev Neutrons Elastically Scattered by Al, Si, K, Ca and Th," Neitronnaya Fiska. ed. Krupohitskogo, P. A., Moscow: Gostamisdat, 1961. - Preskit, C. A. and Alford, W. P. "Elastic Scattering of Protons by V, Cr, Fe and Co," Physioal Review. CXV (1959), 389. Rybakov, B. V. and Sidorov, V. A. Fast Neutron Speotrosoopy. ed. Vlasov, N. A., New Yorks Consultants Bureau, Ino., I960. Vincent, L. D. "The Interaction of 4*1 Mev Neutrons with Sulfur, Calci um, Molybdnum, Antimony, Barium and tferoury," (Unpublished Ph.D. Dissertation, The University of Texas, Austin, I960). Weber, W.,_ Johnstone, C. W. and Cranberg, L., "Time-to-Pulse Height Converter for measurement of Millimicrosecond Time Interval," Review Scientific Instruments. XXVII (1956), 166. _ Wolfenstein, L. "Conservation of Angular Momentum in the Statistical Theory of Nuclear Reactions," Physioal Review. LXXXII (1951), 690. o o o o APPENDIX l.Flux Calculations Consider a small element of volume in the scatterer and in the target as shown in the Figure i. Then the neutron flux 0 ^ intercepted by the scattering element from the target element is given by 0TSs n DAAT nTAxTQ #(0ST#ED) where 2 n^a number of beam deuterons/sec/cm. A a t « area of cross section of volume element in the gas target. nTr number of nuclei/cm? in the target. Ax,pS thickness of the volume element in the ta rg e t. Q" *D(d,n)He^ cross section. 0 Q s angle subtended by the radius vector from £>T scatterer-to-target with the direction of the incident deuteron beam coming from the accelerator. E^« deuteron energy at the point under consideration in the target. Then the number of scattered counts C detected by the detector is given by do* ^ CiA > nD ^*T "T 4 *t 46 „ - w *' a x » thickness of volume element in the scatterer* s ^sT » eblid angle subtended by the scattering element on targ e t element* ^■Ds % solid angle subtended by the detector on the scattering element* do* * differential cross section for the scattered partioles. d a ^ (E) * Efficiency of the detector* K « attenuation factor of the scatterer. s Similarly, the number of direct beam counts Cgg detected by the detector in the forward direction is given by AA,p Oj, AXp 0 ^*(Ogiji,Eg) ^^dT £ (E ) where 0DTs angle subtended by the radius vector from detector-to- ta rg e t. S^DT s solid angle subtended by the detector on the target element* Thus, if £ lI)8a* £ l DT xnd AAT, ^x^,, A x# are constant, then c . i f t /.iftn. ax.k. Y y o t V Ep) n . T °DB ^ ®’ ^®DT,ED^ Since 0 ^ is very small, then < r ( 8DT.BD) X ff(E D) Therefore, _ dcr /dXl n 8 K8 4 x 8 > CT(0.t>Ed) & sT cdb ^ 1 YLz*=±------S i s l where 0 was the angle between the path of the incoming neutron and the direction of the incident deuteron beam as measured from the center of the target. The quantity F^ is given by the expression 0 where is the distance that a neutron, which left the target at an angle 0^, traversed in passing through the scatterer. The distances a^ were determined graphically for ten values of 0^ (i.e. m * 10). In performing the graphical summation, the a ^ 's were taken in equal increments of 0^, ( i.e . 0-^- 02 ~ ^3------e tc .) Hence m 3. Fortran for K s c c ATTENUATION FACTOR DIMENSION A(10) 7 READ 1,X 1 FORMAT (F5.4) 6 DO 2 1 :1 ,1 0 49 2 READ 3, A(l) 3 FORMAT (F5.4) DO 8 IS 1,10 8 A (I) =A(I) * 2,54 SUM cO. DO 4 I 81,10 4 SUM«SUM+(1./A(I)) * (l.-EXPF(-x*A(l))) Y«SUH/ (10.*x) . PUNCH 5, Y 5 FORMAT (2HYs , F4.2) GO TO 7 END In the above program there were three sets of a^fs. All were measured in Inches. I s n# and m "10* The machine w ill punch three attenuation factors as Y, the average value of Y can be considered as the overall value of K • 8 4. Fortran Y orJ^O ^sT*^ CVsT Before the actual computer program was written the expression d O ‘(° aT#ED) J"18t was derived into a simplified form as follows. The scatterer was divided paralled to its length into n slices of equal thickness AX, see Figure ii(a) and each slice was divided into p x q rectangular elements, p divisions along Z-axis and q along Y-axis. The origin being at the center of the n£& slice. Only one quarter of the nt& slice is shown sub-divided in Figure ii(b). Consider any point p in the target at a distance of X from the n origin 0. Let D be jtys.distance of P from the back of the scatterer as 50 TARGET SCATTERER FIG. i END VIEW FIG. II FLUX CALCULATIONS Fig. 17 shown In Figure 11(c)* R is the radius and Y is half the length of the - ' 1L scatterer* Then for the n— slice: A x X s D- (2r>-l)------2 Y s Y n [r2 {r- (2n-l) t L J 2 | ^ : The radius vector from the center of the (p,q)~ element in the n— slice to the point P is given by ) l / 2 •n .p ., * [ < + -en }2 * { (2” « ] The angle subtended by this radius vector with x~axis is sT c n,p,q „ e Coil ( ------n Xn ) n,p,q and the solid angle subtended by the (p,q)^& element at the point P is 3 YnZn n»P»q found from the Table III* Thus SCEjj)* n»P#q By changing the position of the point P in the target we have a s > ; » (e„). The computer program is as follows: c c FLUX CALCULATIONS DIMENSION XD(5,200), ITH(200) 52 10 READ11,ND, DX, R,Y,ITHMX,(D(I),I»1,ND) 11 F0RMAT(I2,8X,3E5.0,15,525.0) 1*1 20 READ22, ITH(l),(XD( J,’l),J*l,ND) IF(ITH(I)-ITHMX)21,30,21 21 1*1*1 G0T020 30 S«0.° DO 72 1*1,ND SEDaO. D073N*1,10 T*0 T1*2*N-1 XN«D(I)-T1*.5*DX ZN»SQRT(R*&-T1*. 5*DX)**2) D070IP«1,4 T1«2*IP-1 T1«XN*XN*{ Tl*ZN/8.0)**2 D070IQtl,4 T2*2#IQ-1 RSML»(T1 (2*Y/8.)**2)**.5 THETA«ARCOS(XN/RSML)*57.2957795 OMEGA*XN*Y*ZN/ (16 .*RSML**3 ) C FIND SIGMA FROM TABLE j*2 , 38 XT«ITH(j) IF(XT-THETA)50,40,40 40 XT2sITH(j-l) 53 M*j-1 SIGMA s XD( I»M)+(THETA-XT2)*(XD( I, j-l)-XD( I, J ) )/(XT2-XT) PRINT 71,THETA,SIGMA, OMEGA GOTO 70 50 J«J*1 IF(ith(j )- ithmx )38,6o ,6o 60 PAUSE *■ <» GOTO 10 u 70 T«SIGMA#OMEGA*T SEDrSED+T 73 PUNCH 71,D(D,XN,T 71 F0RMAT(3E18.8) 72 S^S+SED PUNCH 71,3 STOP 22 F0RMAT(I2,2X,3F5.2,F6.2) END TABLE III D(d,n)He3 CROSS SECTION IN mb/ST 0.156 0.206 0.270 0.362 Mev 8 ' XD 6.71 6.91 7.11 7.51 cm. 00 4.30 6.33 9.19 11.75 01 4.30 6.33 9.19 11.74 02 4.29 6.32 9.18 11)72 03 4.29 6.32 9.17 11.68 04 4.28 6.31 9.15 11.65 05 4.28 6.29 9.12 11.63 06 4.27 6.27 9.09 11.59 07 4.26 6.25 9.05 11.54 08 4.25 6.22 9.02 11.46 09 4.23 6.19 8.97 11.40 10 4.20 6.16 8.92 11.33 11 4.18 6.13 8.86 11.24 12 4.16 6.09 8.79 11.15 13 4.13 6.05 8.72 11.04 14 4.11 6.00 8.64 10.92 15 4.09 5.95 8.56 10.78 16 4.05 5.89 8.49 10.67 17 4.03 5.84 8.40 10.55 18 4.00 5.78 8.30 10.42 19 3.97 5.71 8.20 10.28 20 3.95 5.65 8.10 10.15 21 3.91 5.57 8.01 10.00 22 3.88 5.52 7.89 09.84 23 3.85 5.45 7.78 09.68 24 3.82 5.39 7.67 09.52 25 3.78 5.31 7.56 09.36 26 3.73 5.24 7.44 09.22 27 3.69 5.16 7.32 09.06 28 3.65 . 5.09 7.19 08.90 29 3.60 5.02 7.06 08.72 30 3.57 4.93 6.96 08.54 31 3.51 4*86 6.82 08.35 32 3.47 4.78 6.70 08.16 33 3.43 4.70 6.57 07.98 34 3.38 4*62 6.46 07.80 35 3.33 4.54 6.34 07.64 VITA Zorawar Singh Khangura son of S. Shiam Singh Khangura and Sardarni Harnam Kaur was born on April 3, 1928, at the village Latala of Ludhiana district in Punjab, India. He graduated from Government High School, Gujjarwal (Punjab University, Lahore) in 1945 and attended D. M. College, Mbga (Punjab University, Lahore) from 1945 to 1947. After the partition of the Punjab in 1947* he enrolled in Government College Ludhiana (Punjab University, Solan) which he attended from 1948 to 1950. He received his Bachelor of Arts degree from the Punjab University, Solan in 1950. He was employed as a teaching assistant from 1950 to 1952 at Lyallpur Khalsa College, Jullunder. In 1952 he enrolled in the Muslim University Aligarh and received his Master of Science degree in jjhysics in 1954. From 1954 to 1959 he was employed as a college teacher in the Punjab and New Delhi. During this period, by working part time, he earned a bachelor's degree in teaching from The Punjab University, Chandigarh in 1957. He married former Miss SwarnJeet Kaub Padda, daughter of Brigadier Sant Singh Padda, of village Khiranwali of Kapurthala district in Punjab on February 8, 1959. He came to the United States of America in September 1959 ahd enrolled in the Graduate School of Louisiana State University, Baton Rouge in fall of 1959. Since then he has been working towards a degree of Doctor of Philosophy in physics, for which degree he is a candidate now in August 1964. 55 EXAMINATION AND THESIS BEPORT Candidate: Zorawar Khangura SinghJ Major Field: Physics Title of Thesis: THE COMPOUND ELASTIC SCATTERING OF 3.15 MEV NEUTRONS BY CALCIUM - **0 Approved: Major Professor and Chairman Dean of the Graduate School EXAMINING COMMITTEE: \ Date of Examination: July 29. 1964