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Graduate School
1964
The Compound Elastic Scattering of 3.15 Mev
Neutrons by Calcium-40.
Zorawar Khangura Singh
Louisiana State University and Agricultural & Mechanical College
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Singh, Zorawar Khangura, "The Compound Elastic Scattering of 3.15 Mev Neutrons by Calcium-40." (1964). LSU Historical
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65—3398
SINGH, Zorawar Khangura, 1928THE COMPOUND ELASTIC SCATTERING
OF 3.15 MEV NEUTRONS BY CALCIUM-40.
Louisiana State U niversity, Ph.D., 1964
P h y sics, nuclear
P lea se 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 D issertation
Submitted to the Graduate Faculty of the
Louisiana S tate U niversity and
A g ricu ltu ral and Mechanical College
in p a r tia l fu lfillm e n t o f the
requirements fo r the degree of
Doctor of Philosophy
in
The Department of Physios
by
Zorawar Khangura Singh
B.A., The Punjab U niversity, 1950
M.So., The M.U. A ligarh, 1954
B .T ., The Punjab U niversity, 1957
August, 1964
ACKNCJWLBDfflfSNT
Thtt author wishes to express h ie g ratitu d e to Dr« Dorr C* Ralph
f o r h is valuable suggestions and c r i t i c a l discussions on th is work* He
also wishes to thank Carol J* Spahn and James Benham fo r th e ir assistan ce
and support in th e Van de Graaff laboratory*
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENT, • • ...........
11
TABLE OF CONTENTS....................................... .............................. . ...................
Ill
LIST OF TABLES.................................................................................................
v
LIST OF FIGURES,..
.............
vi
ABSTRACT...........................................................................................................
rii
CHAPTER
I.
INTRODUCTION
1.1
1.2
1.3
1.4
II.
1
The Bohr Rypothesis of Compound N u c l e u s ...................
The O ptical Model..............................................
2
S hape-elastic and Compound-elastic S c a t t e r i n g . . . . . . . .
4
5
Measurement of Compound-elastic S catterin g .......................
EXPERIMENTAL APPARATUS
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
In tro d u ctio n........................
• • • • ..............................
The Pulsed Neutron Source.• • • • • • • . • • • .............................
Neutron Produolng T a rg e t..
••••
S catterin g Samples
• • • • • • .......................
Neutron D etector...................
The Experimental E l e c t r o n i c s . . . . . . . . . . . . ................ • • • • •
Neutron Monitor
.........
• • • • • • • .........
Performance of th e T iae-o f-F lig h t Spectrometer• • • • • • •
8
8
9
12
12
14
18
19
I I I . THE EXPERIMENTAL PROCEDURE
3*1 In tro d u c tio n .. . . .........
3 .2 E ffectiv e Energy o f Neutrons Produced from
a Gas T a r g e t . . . . . e . • • • • • • • • .............. • • • • • .........................
3.3 Measurement of Angular D istrib u tio n s ............................
3*4 Separation of E la s tic a lly S cattered Neutrons
.
i n the Time S p e c t r a . . . . . . ...............
3*5 The C alculation of th e Angular D is trib u tio n .• • • • • • • • •
IV.
22
22
25
29
29
CONCLUSIONS
4*1 The D iffe re n tia l E la stic Cross S e c tio n ........................
•
ill
40
CHAPTER
Page
4*2 Compound-elastic CrossS e c t i o n . . . . . .................
SELECTED BIBLIOGRAPHY...................................................................................
42
44
APPENDIX
1.
2.
3.
4*
5.
Flux C alculations....... ..................................................
A ttenuation F a c t o r . . . . . . . . . . . * . . * . . .................. • • • • • • • • •
Fortran fo r K_. ........ • • • • . . . • • ......................
Fortran fo r ?Ea ? fr(Qg«r.En)
...
• • • • • 49
Table fo r D(d,n)He3CrossS e c t i o n . . . . . . . . . ......................
VITA.....................................................................................................................
C
iv
46
48
48
54
55
LIST OF TABLES
Table
Page
I.
Neutron Energy D istribution in the Gas C ell..................
27
II.
D iffe re n tia l Cross S ectio n s...............
38
III.
D(d,n)He^ Cross Sections......................
54
V
LIST OF FIGURES
Figure
Page
7
1
Level Sehoaes of Caloiue^O and Potaseium-39..................
2
Pulsed Neutron Source
3
Gas Target Assembly................................
4
S o a tte re rs................
5
Experimental Geometry.
6
Block Diagram of Time-of-Flight C i r c u itr y ....
••••••
16
7
Tunnel Diode D iscrim inator.• • • • • • .................................. •••••••«.
17
•••••••
10
••••••••••••••
11
13
...................
15
S
Monitor C io u itry.................................
• • • • • ..........
20
9
Monitor S hield.................................................... ................................
21
.............
26
10
D(dfn)He3 Reaction Cross Section vs
11
Unconnected Time-of-Flight Spectrum o f Calcium-40.• • • • • • • •
12
Tim e-of-Flight Spectra of Calcium-40 and Potassium-39
with Background Subtracted..................
••••••••
31
13
D irect Beam Time Spectrum in the Forward D irectio n . • • • • • • •
32
14
Polyethylene and Caloium-40 Speotra a t Low Bias S e ttin g . ••
35
15
The D iffe re n tia l E la stic Cross Sections of Ca-40 and K-39*
36
16
The D iffe re n tia l Compound-Elastic Scattering Cross
Section of Caloium-40................ • • • • • • • • • •
••••
37
•••••
50
17
Flux C alculations................
vi
30
ABSTRACT
An attem pt has been made to measure th e d if f e r e n tia l compound
e la s tic sc atte rin g of neutrons of energy 3*15 Mev by calcium-40 using
a method based on the d iss im ila rity
between the energy le v e l schemes
o f potassium-39 and calcium-40, and based fu rth e r on the assumption
th a t shape e la s tic scatterin g o f neutrons from these two nuclei i s
e s se n tia lly the same. Since potassium-39 w ill ex h ib it very small
compound e la s tic scatterin g compared to calcium-40 the d ifferen ce in
d if f e r e n tia l e la s tic sc a tte rin g cross sectio n s of these two nuclei w ill
rep resen t, to a close approximation, d if f e r e n tia l compound e la s tic
sc atte rin g by calcium-40.
The measured compound e la s tic cross se ctio n o f calcium-40
appears la rg e r a t angles below 50° and sm aller above 110° than pre­
dicted by Hauser-Feshbach theory. The experimental data curve agrees
f a ir ly w ell w ith theory between 50° and 110° • The measured t o t a l
compound e la s tic cross section i s 1. 0 &0.05 barns and th a t found by
theory is 1.0 barn. The d etailed f i t betweon th e measured and computed
d if f e r e n tia l compound e la s tic cross section i s not good. This may due
i n p a rt to the choice of single le v e l param eters used in th e th e o re tic a l
c a lcu latio n s. That the general magnitude of th e compound e la s tic cross
sectio n s i s co rrect, i s supported by th e good agreement o f the to ta l
cross sectio n s. The assymmetry of th e d if f e r e n tia l compound e la s tic
cross section presumably re s u lts from a breakdown o f the assumption o f
id e n tic a l shape e la s tic sc a tte rin g in the calcium-40 and potassium-39.
vii
CHAPTER I
INTRODUCTION
1*1 The Bohr Hypothesis of Compound Nucleus
In 1936 N. Bohr suggested th a t nuclear reac tio n s take place in
two stag es:^ f i r s t , the formation o f a compound nucleus; and second,
the subsequent decay of th is compound nucleus. The cross section fo r
a p a rtic u la r nuclear reac tio n may then be expressed as the product
o f two fa c to rs; f i r s t the cross section for the formation of the
compound nucleus—*a q u an tity , fo r f a s t neutrons, always close to the
geom etrical area—and second, a quantity which measures the p ro b ab ility
th a t th e compound nucleus decays in the mode in question.
The crea tio n of the compound nucleus im plies th a t the compound
nucleus has an independent existence and th a t l a s t s fo r time long
compared w ith the duration o f the c o llis io n . As pointed out by Bohr,
in a l l p ro b a b ility , i t s life tim e w ill be long enough fo r i t to lose
a l l memory o f i t s mode o f formation when f in a lly i t i s reliev ed o f i t s
excess energy by th e emission o f another p a r tic le . Immediately
following th e in s ta n t o f c o llisio n th is energy i s rap id ly shared out
among tho co n stitu e n ts o f the nucleus, thus the average energy o f
any one of them i s in s u f f ic ie n t to enable th a t p a r tic le to leave the
nucleus and th e compound nucleus fin a lly emits a p a r tic le only when
^Neils Bohr, "Neutron Capture and Nuclear C o n stitu tio n ,"
Nature. CXXXVII (1936), 344.
1
2
6h« improbable event ooours th a t a larg e fra c tio n o f the energy of
a x o ita tio n ia again conoantratad in ona oonatituant. The mode of decay
o
dapanda only on the angular momentum and tha energy and i t a binding
energy to tha compound nucleus• This i s called Bohr Hypothesis. To
c a lc u la te the p ro b ab ility of a p a rtic u la r reactio n i t i s necessary to
make an ad d itio n al assumption. Tha s ia p ie s t i a to assume th a t -the
amission of th a
d iffe re n t co n stitu en ts of th e nucleus, neutrons,
- protons and more complex bodies lik eo C -p artio les, may be tre a te d on an
equal footing and th a t the p ro b ab ility o f e v a s io n of a specified p a r­
t i c l e may be calcu lated on. a s t a t i s t i c a l b a s is.
1 .2 The O ptical Model
Instead of considering th e individual c o llisio n s which the
in c id en t p a r tic le must make w ith in the nucleus i t i s possible to
consider, a lte rn a tiv e ly , th e in te ra c tio n of th a t p a r tic le w ith the
nucleus a s a whole. This in te ra c tio n i s usually represented by a
p o te n tia l w ell. Thus calcu latio n s o f th e sc atte rin g and absorption
o f a nucleon resembles the ca lcu latio n s o f th e sc atte rin g and
absorption of a beam of lig h t by a re fra c tin g and absorbing sphere.
This is th e o p tic a l model* i t cam give an account only of the gross
fe a tu re s o f nuclear reac tio n s; i t cannot account f o r the sharp
resonances or f o r th e com petition between various nuclear reac tio n s.
The inadequacy o f th e s t a t i s t i c a l theory led to the in te re s t in the
o p tic a l model by Feshbaoh, P o rte r and Weisskopf.
2
They suggested the
2H. Feshbaoh, C. E. P o rte r and V. F. Weisskopf, "Model fo r
Nuclear Reactions w ith Neutrons," Physical Review. XCII .(1954), 443.
3
^ model, called the cloudy c ry sta l b a ll model, which describee only the
featu re s of nuclear reactio n s a f te r averaging over th e resonances of
the compound nucleus* This model p re d ic ts only the sh a p e-ela stic
s c a tte rin g and the compound nucleus formation cross oection. I t
4
'
O
o°
considers only the°conditions in the entrance channel* Hence the
compound nucleus formation i s considered as an absorption o f the
■ •••
,,
•
'
*
*
'
*
0
inciden t beam, although part of i t leads to an e la s tic sc a tte rin g
process* These conditions a r e ‘described by means of a o n e-p article
problem* The nucleus i s represented by a p o te n tia l w ell having both
r e a l and imaginary components given by
V : V0 tiW 0
and
V
s
where
R
oS
fo r r< R
0
fo r r> R
r AV3
In th is expression VQ and WQ are resp ectiv ely the.depths o f the r e a l
and imaginary p art of th e nuclear p o te n tia l in Mev, r Q i s a constant
and A i s the atomic number* This complex p o te n tia l a c ts upon the
incoming neutron* The sc a tte rin g which th e neutron s u ffe rs i s the
shape e la s tic sc a tte rin g and the absorption which i s caused by the
imaginary p art i s the compound nucleus formation* The imaginary p art of
the p o te n tia l function Indicates the stren g th and lo catio n o f the
processes th a t lead to an energy exchange between the in cid en t neutron
and the targ et nucleus* The r e a l p a rt rep resen ts the average p o te n tia l
energy o f the neutron w ithin the nucleus.
Feshbach et* al* applied th e ir cloudy c ry s ta l b a ll model to
calcu late the to ta l cross section, shape e la s tic cross sectio n and the
cro ss section for the formation of th e compound nucleus fo r neutrons,
each as a function of energy and mass number, and also the angular
4
dependence o f the e la s tic scattering* The experimental cross section
ex h ib it resonances a t lower energy but i r e smooth functions in the
higher energy regions where the experimental reso lu tio n i s inadequate
to resolve th e closely spaced levels* The o n e-p article problem does
n o t apply to th e resonance region because o f the rapid fluctuations*
However, the averages of the cross sectio n s taken over an in te rv a l,
which included many resonances, were shown to be the cross sections
belonging to a new sc a tte rin g problem called the ng ro ss-stru ctu re,n .
problem* Thus the average p ro p erties o f neutron resonances, in
p a r tic u la r th e stren g th function /^^/D , where
is the average
neutron width and D i s th e average le v e l spacing, are connected with
th e gross stru c tu re problem and can be predicted by th is model* The
experimental neutron to t a l cross sectio n s were w ell reproduced in the
energy region between 0*1 and 3 Mev. The angular dependence o f the
sc a tte rin g cross sectio n a t 1 Mev was f a ir ly w ell reproduced by th is
model* . The values o f th e stren g th function and th e reactio n cross
sectio n a t 1 Mev showed only the q u a lita tiv e agreement between theory
and experiment*
1*3 Shape-Elastic and Compound-Elastic S catterin g
E la s tic sc a tte rin g i s defined as sc atte rin g in which the
k in e tic energy in the center of mass system does not change* The
e la s tic sc a tte rin g cross section (Qie) i s defined by the equation!
^•t
where
^ e ' ^n e
^ . m t o t a l cross sectio n ,
C
and
ne* n o n elastic cross section*
The e la s tic cross sectio n is composed of two p a rts, the compound
5
e la s tic cross sectio n
and another p a rt, known a s the shape-
e la s tic cross section ( Q B
*O)
*
•
o
The two p a rts of the e la s tic cross section have very d iffe re n t o rig in ,
fo r while the amplitude o f the shape-elastic p art i s coherent w ith the
in c id e n t wave, th e o th e r, the compound-elastic am plitude, i s incoherent.
Since the neutrons scattered by these,tw oprocesses have the sane energy,
a separate measurement o f the compound-elastic s c a tte rin g poses a
d if f i c u lt experimental problem. Eisberg, Yennie and Wilkinson have
proposed one possible method fo r measuring compound-elastic s c a tte rin g
o f protons by measuring the time delay between;the bremsstrahlung
associated with th e formation and decay of the compound nucleus,
3
but
no one has y et done th is experiment.
1.4 Measurement of Compound-Elastic S cattering
The method used in th is experiment, which i s e a s ie r experimen­
ta l l y but applicable only for a re la tiv e ly small number o f n u clei, i s
based upon the f a c t th a t shape-elastic scatterin g o f neutrons i s
predicted f a ir ly w ell fo r a l l nuclei by a p o te n tia l w ell whose radius
i s R « r 0A^/3. xhus the calculated shape-elastic s c a tte rin g based on
th e o p tic a l model i s a slowly varying function o f th e mass number.
Nuclides of approximately the same mass number w ill thus have the
same sh ap e-elastic neutron cross section, but not n ec essarily the
same compound-elastic cross section. On the basis o f an o p tic a l model
.
M. Eisberg, D. R. Yennie, and D* H. W ilkinson, ”A Brem­
sstrahlung Experiment to Measure the Time Delay in Nuclear R eaction,”
Nuclear Physics. XVIII (i960), 338.
6
o f th« nucleus; th e e la s tic sc a tte rin g o f neutrons by compound nucleus
formation can be followed d ire c tly by comparing th e angular d i s t r i ­
butions o f e la s tic a lly scattered neutrons by elements which .are
■
close to each o th er In the magnitude of the shape e la s tic scatterin g
cross sectio n . At energies o f several Mev there w ill be neg lig ib le
compound-elastic sc a tte rin g fo r most nuclei since th ere are many levels
to which the compound nucleus w ill decay. Thus I t follows th a t In e la stic
sc a tte rin g w ill predominate. I f , however, there are no le v e ls , or only
a few such le v e ls , compound-elastic d iffe re n tia l cross sectio n may
contribute sig n ific a n tly to the e la s tic d if fe re n tia l cross sectio n .
Since calcium-40 and potassium-39 have nearly the same atomic weight
and have d iffe re n t energy le v e l schemes, and since calcium-40 has no
in e la s tic neutrons in the energy range intended, they are p a rtic u la rly
su ita b le fo r making a measurement o f compound-elastic sc a tte rin g . The
le v e l schemes of th ese nuclei are shown in Figure 1.
LEVEL
SCHEMES
OF
40
3 90
5-62
5-60
3 -27
5-24
4*'
3 -9 5
3 89
3-61
3 -7 3
3-3 5
Mev
3*03
2 -8 2
39
40
Co
F ig .
1
CHAPTER I I
EXPERIMENTAL APPRATU3
2*1 Introduction
In th is experiment the d if fe re n tia l cross sections of neutrons
e la s tic a lly scattered by calcium and potassium a t 3 «15 £ 0 .1 5 Mev
neutron energy were measured. The equipment used In th is experiment
was th e Louisiana S tate U niversity tim e -o f-flig h t spectrom eter. This
spectrom eter was sim ilar to th e one developed by Crahberg e t . a l . a t
Los Alamos S c ie n tific Laboratory.^ I t u tilis e d the radio frequency
pulsed beam technique and v aria b le path magnetic io n bunoher th a t
produces th e neutrons during short pulses o f th e accelerated beam on
2 3
a neutron producing ta r g e t. 9 Since the neutrons are produced fo r a
‘ very short duration the time d ifferen ce between the time o f p a r tic le
production and th e time of neutron detectio n can be measured. The
measured time difference allows th e energies o f the neutrons to be
measured and neutrons o f d iffe re n t energies to be separated. The
schematic diagram of the experimental arrangement i s shown in Figure 5*
2.2 The Pulsed Neutron Source
The Van de Graaff a c c e le ra to r of Louisiana S tate U niversity has
^L. Cranberg and J . S. Levin, "Neutron S catterin g a t 2.45 Mev
by a Tim e-of-Flight Method," Physical Heview. CIII (1956;, 343*
^R. C. Mobley, "Proposed Method fo r Producing Short Intense
Monoenergetio Ion P u lses," P hysical Revieir. LXXXVIII (1952), 360.
gri
E c ia r
S
been used to produce a beam o f deuterons o f energy 0.5 £0.005 Mev. The
deuteron beam a f te r maee an aly sis was swept by a 6.8 megacycle radio
frequency p o te n tia l applied to a s e t o f d eflectin g p la te s . The swept
beam was chopped by passing i t through a s e t o f p la te s . Then the
chopped beam was bunched in to sh o rt in ten se ion pulses by passing i t
through the ion buncher. Further i t was focussed by means o f a s e t o f
e le c tro s ta tic quadrupole lenses and then was collim ated before i t was
allowed to f a l l on the ta rg e t. The arrangement i s shown schem atically
i n Figure 2 . The d eta ile d d escrip tio n of the arrangement i s given in
referen ce 3.
In pulsing the beam two th in g s had been considered, the pulse
duration and the frequency o f p u lsin g . The b u rst duration was between
two to th ree millimicroseconds w ith the average ta rg e t current o f ten
microamperes collim ated a t 3/d inch. Since fo r a given beam duration
th e beam current on the ta rg e t i s p ro portional to the frequency, high
frequencies (1 to 10 megacycles) are generally used. The lim itin g
value o f the frequency i s determined by the d ifferen ce in the time o f
a r r iv a l o f the neutrons (or gamma ray s) o f in te r e s t to th e observer.
In th is experiment a d eflectin g frequency o f 6.8 megacycles were used.
•3 Neutron Producing Target
In order to obtain 3.15 Mev neutrons the D(d,n)Ho^ reactio n
was used. The deuterons were accelerated and allowed to en ter a
deuterium gas c e ll through a n ick el f o i l 25 microinch th ic k . The
ta r g e t assembly i s shown in Figure 3 . The gas c e ll was 2 /5 " * 5 /8 "
diameter and i t was cooled by an a i r b la s t. The window was supported
by a grid with a close packed hexagonal array of holes each o f 1/32
inch diam eter. The c e ll was <§vacuated and was f i l l e d 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
O U TLET
HI FOIL
COLLIMATOR
0534
2/5 x5/8"
HI6H
VACUUM
6 AS INLET
COOLlNO
INLET
AIR
GAS TARGET ASSEMBLY
F ig . 3
■CAM
gas. A pressure o f one atmosphere was maintained in the deuterium c e ll
8
The neutron y ie ld was of th e order o f 10 neutrons per seoon*
2*4 S cattering 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. I t was sealed in a brass s h e ll o f thickness
0*006 inch w ith a 0*030 inch th ic k aluminum cap* The weight of th e
brass s h e ll was 5*0 grams and th a t of th e aluminum cap was 1*0 gram*
Sim ilarly the potassium sample weighing 8*3 grams contained 93*1 per­
cent o f potassium-39* I t was sealed in a brass s h e ll o f th e same
thickness as th a t fo r calcium* Potassium sample was sealed in th e
presence o f an in e r t atmosphere* The brass s h e ll which contained
potassium weighed 3*65 grams and i t s aluminum cap made out o f 0*010
inch th ic k aluminum f o i l weighed 0*05 gram* Id e n tic a l blank s h e lls
sealed w ith aluminum caps were used fo r "OUT" spectra* The so a tte re rs
are shown in Figure 4*
2*5 Neutron D etector
The neutron d etecto r consisted of a 2 inches long and 1*5
inches diameter p la s tic s c in til la to r o p tic a lly coupled to a 56AVP
photom ultiplier tube and was shielded from th e room sc a tte re d neutrons
gamma rays and other background by means o f a la rg e sh ield which
consisted of a lead c y lin d ric a l s h e ll 5 inches th ic k surrounded by a
mixture o f p a ra ffin and lith iu m carbonate (Figure 5)* The purpose of
the lith iu m carbonate was to capture th e neutrons by means o f th e
. L i^(n,tt)H 3 reac tio n . An ad ju stab le wedge constructed o f iro n and
tungsten was used to screen ra d ia tio n coming d ir e c tly from th e neutron
I
SEAL (VINYL PLASTIC)
B R A SS SHELL
BLANK
SCATTERER
o
F ig. 4
u
source* The photom ultiplier tube m s also shielded m agnetically by a
„ mu-metal sh ield and was cooled by ic e water in order to increase the
sig n a l-to -n o ise r a tio . The noise le v e l was about 5 percent o f the
maximum proton re c o il pulse height* The gain o f the detector was
checked from tim e-to-tim e by means o f cobalt-60 and cesium-137 sources*
The high voltage o f the photom ultiplier tube was also checked frequent­
ly and maintained constant* The slow pulse fo r the pulse height
inform ation was obtained from the tw elfth dynode and the f a s t pulse
f o r the tlm t 'lnform ation was obtained from the anode o f the photo­
m u ltip lie r tube* The e n tire d etecto r and shielding arrangement was
a llig n e d a t the height o f the a c ce lerato r beam to point d ire c tly a t
th e sc a tte rin g sample*
2*6 The Experimental E lectronics
A block diagram o f the tim e -o f-flig h t ele c tro n ic s i s Shown in
Figure 6 * The f a s t output a t the anode o f the photom ultiplier was fed
to a tunnel diode discrim inator in order to r e je c t sig n als le s s than
the chosen h eig h t. The d etailed schematic diagram o f the tunnel diode
discrim inator i s shown in Figure 7* Then the pulses were fed to two
Hewlett-Packard 46OB and 46OA wideband am p lifiers in cascade* The
am plified pulses were fed to a tim e-in te rv a l to pulse-height converter*
The time to pulse height converter measures th e d ifferen ce between the
tim e of a r r iv a l o f a sig n al from the neutron d etecto r a t the s t a r t
in p u t and th e time of a r r iv a l of a sig n al from ths radio frequency
pulsing system a t the stop input* This timing inform ation was converted
to a pulse whose height was proportional to th e measured time d if f e r ­
ence.
The electro n ic c ir c u it on,the input o f the time to pulse height
1
SO% PARAFFIN
ATTENUATOR
5 0 % U m iM
CARBONATE
LEAD
Y
7 7 7
A
TARGET
120 CM.
SCATTERER
LEAD
•3
0\
EXPERIMENTAL
GEOMATRY
Fig. 5
10 ■ 20
»
■
C
30
CM.
1
Ul
16
"N EU TR O N
O E T tC T O R
H
.
r
P
F U S T IC
CRYSTAL
H . V
SLOW
n
w'
C .
F
»
TUNNEL MODE
0 - 2 0 0 a SEC
DELAY
DISC.
•
<p
*
NON-OVERLOAD
H P 4SO *
AMP
AMP
H.P
DISC
<f
HP 4000
AMP
I
I
400A
HP
AMP
4S0A
AMP
i
COINC DENCE
11
ANTIC;o in
ANIiLY
MO OHM CLIPPHM
3
T /H
STOP.
INPUT
H,P
4S0A
AMP
ANALYSER
.OUTPUT
H4I 2 0 0 0 ,
3 . 0 ^ * 8 EC
I INPUT
•A TE
INROT
TMC
TM C
4 0 0 - CHANNEL
X-Y
ILOCK
PLOTTER
PR IN TER
ANALY
DIAGRAM OF TIME-OF-FLIGHT
F ig . 6
CIRCUITRY
CA SLE'
120 CM
+ 9
V
900
300
NEGATIVE
1:2
720
330
W f
TO
TUNNEL
300
DIODE
FlGl 7
PO SITIV E
TO
DISCRIMINATOR
18
converter carved to chape th e pulsed f o r th e converter, which ic
<.7
designed to handle sh o rt, fa c t ris in g pulses of s u f fic ie n t am plitude.
o
'
* The f a s t output of th e photom ultiplier tube met th e requirements o f th e
s t a r t input o f th e time to pulee height converter i n r i s e time and in
duration and needed only to be s u ffic ie n tly am plified by f a e t ris in g
lin e a r am p lifiers, suoh as the Hewlett-Packard 4*60. The sinusoidal
wave form from th e pulsing o s c illa to r had to be a lte re d in to a short
pulse su ita b le fo r driving the stop input o f the converter* This was
done by driving two Hewlett-Packard 460B and 46OA am p lifiers in cascade
to sa tu ra tio n , giving an approximately square wave* The sig n al was
th e n clipped in time' w ith a shorted coaxial cable about 125 oms* long*
The re su ltin g wave form which drove the stop input o f th e converter
was con sistin g of a lte rn a te spikes of about 6*2 v o lts high and about
20 nanosecond d u ratio n .
The gate sig n a l to the pulse height an aly ser, o f lin e a r output
neutron d etecto r, was am plified through a nonoverload am p lifier,
discrim inated to r e je c t th e small signals* The output o f th e discrim ina­
t o r was fu rth e r shaped by using a pulse shaping c irc u it* I t was then
fed in to th e gate in p u t of the pulse height analyser* The pulse height
an aly ser was gated to discrim inate ag ain st small pulses from the
neutron deteotor* The spectrum thus obtained was analysed by th e
TMC 400 channel analyser and was p rin ted by th e p r in te r .
2*7 Neutron Monitor
i
In order th a t a l l sc a tte rin g measurements could be normalised
to a constant number o f neutrons impinging upon th e sc a tte rin g samples,
i t was necessary to monitor the flu x o f th e neutron source. The monitor
r
was a 1” x 2" diameter p la s tic s o in tilla to r o p tic a lly coupled to a
56AVP photom ultiplier. The pulse from th e anode o f th e photom ultiplier
a
*fV
-'
'
*
r“
was fed through a oathode follow er to a nonoverload am p lifier w ith
a discrim inator output which drove th e s c a le r. The discrim inator was
adjusted to aooept most o f th e pulses caused by neutrons from th e * •
ta rg e t b u t high enough so' th a t the counting r a te would be reduced to
a few percent i f the p a r tic le beam was o ff th e ta rg e t. The monitor
c irc u itry , i s shown in Figure 8 . The monitor sh ield consisted o f a
cone f i l l e d w ith p a ra ffin and borax in equal ra tio and was lin e d w ith
3 inches th ic k le ad . I t was hung from the c e ilin g and was situ a te d
to view only th e neutron source as shown in Figure 9 . The monitor
photom ultiplier tube was also shielded m agnetically and was oooled
w ith ic e w ater. The monitor was kept s t i l l throughout the experiment.
The high voltage and th e gain of th e monitor were checked frequently
and were kept constant throughout the experiment.
2.8 Performance o f the Time-of-FUght Spectrometer
The performance o f th e Time-of-Flight speotrometer was q u ite
s a tis fa c to ry throughout the experiment fo r th e purpose o f th e present
work. The time reso lu tio n was six nanoseconds. The monitoring
u n certain ty was fra c tio n o f a p ercen t. The sh ield in g arrangement
and electro n ic equipment provided th e b est sig n a l to noise r a t i o . The
noise le v e l was about fiv e percent o f th e maximum proton r e c o il pulse
height and was s u ffic ie n tly low fo r measuring d if f e r e n tia l cross
sections to w ithin fiv e p ercen t.
PLASTIC
•C IN T IL A T O R
RCA 6 9 4 2
BASE
1
« •'
.
m *
i
NON-OVI
H.V. POWER
i
AMPL
NEUTRON
FIG. 8
MONITOR
50 %
BORAX /
50 %
PARAFFIN
P LA STIC
SCINTILATOR
OEUTRON
SCATTERER
MONITOR
BEAM
TARGET
SHIELD
E!& £ _____________
O
10
20
30
CM
CHAPTER III
THE EXPERIMENTAL PROCEDURE
3.1 Introduction
The purpose of the experiment was to measure the average
d if fe re n tia l e la s tic cross sections o f calcium-/»0 and potassium-39
in the v ic in ity o f 3*15 Mev since th e theory i s applicable to the
average cross sectio n s and does not reproduce the resonance structure*
The type of inform ation obtained i s dependent on the energy spread of
the neutrons employed in the experiment.
In th is experiment the
neutron energy spread obtained by the su ita b le choice of ta rg e t
thickness was 300 Kev* This spread was larg e enough fo r th e measure­
ment o f su itab ly averaged cross sectio n s o f both elements*
3*2 E ffectiv e Energy of Neutrons Produced from a Gas Target
In order to find out the e ffe c tiv e energy o f the neutrons
produced from gas ta rg e ts , i t i s necessary a t the f i r s t step to fin d
out the energy lo ss AE of a beam p a rtic le as i t trav erses th e window
which may be computed by the following equation:
-AEsCN
AX
Negative sign on the l e f t hand side o f the equation rep resen ts the
energy lo s s , g i s the atomic stopping cross section of the window
22
m aterial^, N i s the number o f atoms per cm3., AX i s th e thickness of
the f o i l .
Since the lo ss in energy o f the beam p a r tic le in trav ersin g
through the whole length o f the gas c e ll i s not lin e a r , also the
corresponding neutron energy and the sc atte rin g cross sectio n are
not lin e a rly re la te d to th e in cid en t energy.
Therefore, the following
computations were made to fin d out the e ffe c tiv e neutron energy.
Let the gas c e ll be divided along i t s length in to a number
of sub c e lls and consider the i£& c e ll whose thickness i s AX^.
If
E^ i s the incident energy and E j ^ the emergent energy o f the p a r tic le
from th is sub c e ll, then
= E^ - A E ^
The energy lo ss AE^ of th e p a rtic le in trav e rsin g the sub c e l l under
consideration can be computed from the equation:
AtE^ s (dE/dx)i
A x±
where (dE/dx)^ i s th e sp e c ific energy lo ss o f the p a r tic le in the
-gas whose in cid en t energy i s E^.
The emergent energy E ^ i i s the
incident energy fo r (i+ l) thy sub c e ll.
Thus the in cid en t energies
for a l l the sub c e lls were computed and the corresponding neutron
energies En^*s were found from the ta b le s fo r D(d,n )He3 reac tio n 2.
^S. Flugge, HMndbuch der P hrsik. Corpuscles and Radiation in
M atter I I , (B erlin: Spinger-Verlag,-1957 ) X$£lv, pp. 193-213.
2J . B. Marion, I960 Nuclear Data Tables. P art 3.
(Washington: U.S. Government P rin tin g O ffice, 19&)).
o
(■)
Since the neutron energy En i s a function o f D(d,n)He^ cross section
»
- f9
X
.?• „
0 •
Qi (% ), incident deuteron energy E^, sp e cific energy lo se (dE/dx)^
o f the deuteron and the thickness
o f the sub c e ll, th e re fo re,
th e average e ffe c tiv e energy ER o f neutrons can be found from the
following equation:
(dE/dX)i AXi
i i l ^ ( % ) (dE/dX)i * * i
where k i s the number o f sub c e lls .
The lo ss o f energy AE in the n ickel f o il o f thickness 25
microinch was .140 Mev and thus the deuteron beam o f energy 500 Kev
was l e f t with the in cid en t energy of 0.360 Mev to enter th e gas c e ll.
To find the e ffe c tiv e energy o f
the neutrons produced in the
gas c e ll, i t was divided in to eleven equal p a rts along i t s length, and
th e sample ca lcu latio n s can be done by considering one o f the sub cellsy
I t s thickness
A
was 0.1cm. The energy E-^ a t th e entrance o f th is c e ll
was 0.360 Mev, the sp e c ific energy lo ss .(dE/dX)^ corresponding to th is
in c id e n t energy was 0.210 Mev/cm.,^ and th e D(d,n)He^ cross section was
12.75 m illib a rn s /s te r.^ The p a rtic le lo s t 0.021 Mev in trav ersin g the
sub c e l l . The emergent deuteron energy from the f i r s t sub c e ll or the
in c id e n t energy to th e second sub c e ll was 0.339 Mev. The neutrons o f
energy 3*30 Mev were produced from th is c e ll.^ S im ilarly th e neutron..
energies E *s from a l l the sub c e lls were computed and are shown in
3
Flugge, op. c i t . , pp. 193-213.
S lario n , fip. c i t . P Nuclear Data Tables.
25
Table I . The effe c tiv e neutron energy E^ calcu lated through th e above
equation w t 3*15 Mev.
The energy spread caused by the length o f the gas c e ll was
computed as follow s. From th e graph, <T(% ) verses
(Figure 10), P
d iffe re n t values of ^ ( E ^ ) were chosen and th e ir corresponding values
o f &E
were found, where
"i
Ae s
e -E
"i [ n
The arith m etic mean AE of th ese P values given by th e equation:
i s th e energy spread caused by the gas c e l l . The e ffe c tiv e neutron
energy thus obtained was 3 «15 £0.15 Mev. The o v e ra ll energy reso lu tio n
o f th e spectrometer was 330 Kev. This included th e spread due to the
f l i g h t path, d etecto r responee and the efficien cy of th e electro n ic s e tc .
3*3 Measurement o f Angular D istrib u tio n s
The Van de Graaff ac ce lerato r was ad ju sted fo r th e s ta b ili ty of
beam c u rre n t. The bunoher and the asso ciated equipment was adjusted to
o b ta in a b u rst o f short duration o f about two to th ree millimicrosecond
and th e beam was collim ated and focused to g et a maximum ta rg e t current
o f about te n to twelve mioroamperes on th e ta r g e t. The energy o f the
beam was regulated a t 0.5 £0.005 Mev. The slow and the f a s t sig n al
channels of the d etectin g system were adjusted fo r high counting r a te .
The tim ing system was adjusted fo r b e tte r time reso lu tio n which was six
nanoseconds. The experimental se t up was p recisely a llig n e d . The sample
o
NEUTRON
ENERGY
VS
CROSS
SECTION
r |l*0)» mk/8t.
(DATA TAKEN FROM REF. 2)
25
2.7
3.3
3.9
o
m
TABLE I
NEUTRON ENERGY DISTRIBUTION IN TIE GAS CELL
No.
.
4X±
c r \)
(mb/st)
(Mev/cm;)
( i)
(cm)
Ei
(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
( H )1
S
(Mev)
.
holder was positioned to hold the sc a tte rin g sample centered 2*25"Inches
In th e forward d irec tio n with respect to th e beam d ire c tio n and.with
c y lin d ric a l sc a tte re r ax is perpendicular to the beam arid coaxial with
th e axis o f ro ta tio n o f th e d etecto r. The sh ield o f the neutron detec­
to r was pivoted d ire c tly below the s c a tte re r so th a t the sc atte re d
neutrons could be detected a t any angle between 0° and 140° with
respect to the d ire c tio n o f the in cid en t flu x . A f lig h t path o f 120cms.
was used. The monitor and i t s sh ield were hung from the c e ilin g and
were positioned so th a t th e monitor viewed only th e neutron source. The
discrim inator of the monitor was adjusted to discard the ra d ia tio n
other than th a t from th e neutron source. The time scale was 1.2 nano­
seconds per channel on the pulse height analyser.
With these adjustments made in th e equipment, the tim e -o f-flig h t
spectra were recorded a t d iffe re n t angles between 20° and 140°• Each
spectrum taken fo r calcium and potassium was normalized to the number
o f d ire c t beam counts in th e forward d ire c tio n . Six runs were taken
and d ire c t beam counts were recorded a f te r every two angles during
each run. At each angle th e spectra were taken with the s c a tte re r "IN”
and s c a tte re r "OUT*1. The "OUT1* spectrum was subtracted from the "IN"
spectrum in order to find out the number o f neutrons e la s tic a lly scat­
te re d fo r a constant number of monitor counts. The d if f e r e n tia l cross
sections obtained were corrected fo r the difference in the number of
n uclei per cm? fo r the two elements. Small changes in th e beam energy,
while not g re a tly effe c tin g the energy o f the neutrons produced a t the
ta rg e t caused the e la s tic peak on the time spectra to d r i f t due to
d iffe re n t a r r iv a l time o f the deuterons a t the ta rg e t. During the
course of the data runs th e machine energy had to be adjusted q u ite
freq u en tly . This e rro r was kept to a minimum by a close watch o f the
,29
o
time spectra while
they were accumulated, and discarding those in
t
i
which the large s h if ts were observed. The e rro r due to th e v a ria tio n
in gain, due to the change in temperature o f th e cooling system and
due to the change in room tem perature, was minimised by normalising
the data runs to the d ire c t beam counts taken a t approximately the
same tim e.
3.4 Separation o f E la s tic a lly S cattered Neutrons in the Time Spectra
For the in cid en t neutron energy o f 3.15 Mev calcium-40 had
no in e la s tic sc a tte rin g because the f i r s t excited s ta te 3*35 Mev i s
higher than the in cid en t energy and th e re fo re , can not be ex cited .
Potassium-39 has f i r s t and second excited s ta te s a t 2.53 Mev and 2.82
Mev resp ectiv ely which were weakly excited and so i t had some in e la s tic
s c a tte rin g . The time sp ectra of calcium-40 and potassium-39 were f itte d
together with reference to th e ir corresponding gamma peaks. The counts
due to in e la s tic s c a tte rin g in the potassium-39 spectrum were e a sily
distin g u ish ab le from th e e la s tic peak. The t o t a l number o f e la s tic
counts were added by considering equal number of-channels on both sides
o f the e la s tic peaks. The uncorrected time spectrum o f calcium-40 i s
shown in Figure 11. The time spectra o f caloium-40 and potassium-39
with background subtracted are shown in Figure 12.. In Figure 13 i s
shown the d ire c t beam tim e -o f-flig h t spectrum to which th e e la s tic
counts of calcium-40 and potassium-39 were normalised.
3.5 The C alculation o f th e Angular D istrib u tio n s
In order to convert th e re la tiv e angular d is trib u tio n s to
absolute e la s tic d if f e r e n tia l cross se ctio n s, the re la tiv e number of
neutrons scattered a t 40° from calcium was compared to th e r e la tiv e
UNCORRECTED
j
OF
TIME
SPECTRUM
CALCIUM-4 0
•IN
1 OUT
* •4 0 *
►
»
0
U
~
VntliiWiiHrfiMwWF11**
'
—-u10-------20
^
^ 30------- $ ------- &------ -&------- fer ~~ " ' t o ------ to------- ifco-
mu
0
CORRECTED TIME SPECTRA OF
C o40 AND K5* , BACKGROUND SUBT.
O « 4 0 * f En-3.I5±0.I5M E V
TIM E/ CHANNEL* l.2n«
• ■ Co40
« - K”
•00
COUNTS
/ CHANNEL
900
400
300
.
to o
100
s•
"■ V W 5*
10
SO
40
SO
•0
CHI
70
•0
•0
100
110
-TIME
FIS. 12
O
DIRECT BEAM TIME SPECTRUM
IN THE FORWARD DIRECTION
\
= 3.15
t 0.15
Mev
COUNTS/CHANNEL
22
14
3
10x10
6
*3
2xlC"
10
20
CHANNELS
100
33
number of neutrons detected from the d ire c t beam in the forward
«3>
d ire c tio n fo r the same nunber o f monitor counts. The equation used to
calcu late the absolute e la s tic d if f e r e n tia l cross sectio n of
calcium-40 a t 40° was:
c , y ^ < F ( E d)
d»* cDB
___________________
n , K„ A * a 5 3 p ( e . T , Ed) A t I
The derivation and explanations of th e symbols used in th e equation
a r e given in appendix. The computer programs fo r some p a rts o f the
equation are also given in the appendix. The absolute angular d is­
trib u tio n s by d ire c t beam method are shown in Figure 15.
The conversion was also done by polyethylene method. The
r e la tiv e number o f neutrons . sc atte re d a t 40° from calcium-40 a t low
b ias se ttin g s was compared to the r e la tiv e number o f neutrons scat­
te re d a t 40° from th e hydrogen p art of (CH2 )n sample of th e same s iie
and a t the same b ias se ttin g s fo r the same number o f monitor counts.
The two spectra are shown in Figure 14* The equation used fo r th is
method was:
g L _C. % C H KCH y (?)
*
and
CH « . « . * .
® h(0) 5
"
oosOTT*
where Cs and CH are the number of counts, N8 and NH are th e number of
n u clei per cm?, fo r the s c a tte re r and hydrogen re sp e c tiv e ly . Ks and
Kch are the atten u atio n facto rs for th e s c a tte re r and (CHg^ sample,
« V » > i s th e d if f e r e n tia l cross sectio n fo r hydrogen a t th e angle 0
o
34
and CTn. p
efficie n cy
th e t o t a l neutron-proton cross se ctio n .
£ .0 /
5
The rela tiv e,
was calcu lated by using Cranberg’ s efficien cy
ourve.^ The reason fo r using Cranberg*s efficien cy curve i s th a t our
p la s tio s c in til la to r was approximately of the sane e lse as th a t o f
Craribergf s and th e shape of the p a rt o f the curve used in th is work was
V •
nearly the same fo r d iffe re n t b ias s e ttin g s . Evidently th e re la tiv e
efficie n cy from any one o f the curves was approximately th e same. The
o
absolute e la s tic cross section o f oaloium-40 a t 40 found by th is
method was f if te e n percent higher than th a t found by th e d ire c t beam
method. These ca lcu latio n s were repeated using carbon. The d if f e r e n tia l
cro ss section o f carbon a t 40° fo r 3*1 Mev neutrons was obtained from
7 B
th e published angular d is trib u tio n s . * This absolute value o f the
cro ss sectio n was in agreement w ith th e d ire c t beam method value w ithin
2 .4 percent. The s a tis fa c to ry agreement o f th e carbon c a lcu latio n
suggests th a t th e disagreement in the hydrogen comparison re s u lts from
th e assumptions made in attem pting to co rrect the d eteo to r efficien cy
f o r th e lower energy neutrons.
The angular d istrib u tio n s shown in Figure 15 were calcu lated
by d ire c t beam method. The d ifferen ce o f th e two angular d istrib u tio n s
i s shown in Figure 16, which rep resen ts th e d if f e r e n tia l compound
^Marion, op. c i t . . Nuclear Data Tables.
^Crariberg, op. c i t . . P hysical Review. C III, p . 343.
7
R. W. Meier, P. Soheroer and G. Trumpy, "E la stish e Streuung
und P o larlsa tio n se ffek te Von D-D- Neutronen an K ohlenatoff." H elvetica
Physloa Acta. K V II (1954), 577.
8
J . E. W ills, J r . , J . K. B a ir. H. 0 . Cohn and H. B. W illard,
"S catterin g o f F ast Neutrons from
and Fl9.n P hysical Review. CUC
---------------(1958), 891.
NEUTRON SPEC TR U M FROM
POLYETHYELENE
40*
ELASTIC PEAK OF Co40
BACKGROUND SUBTRACTED
»*40*
(LOW BIAS)
(LOW BIAS)
MKUTROMS
SCATTKRCO PROM
HYOROOCR^
MKUTROM
S
SCATTEREDPROM
CARRON,
10
SO
40
SO
•0
TO
C
p i o. 14
THE DIFFERENTIAL ELASTIC
CROSS SECTIONS OF Ca*°AND K
FOR S . I S i 0.15 MEV
NEUTRONS
.
*
Co40
K»
x DIFFERENCE
THE DIFFERENTIAL COMPOUND ELASTIC
SCATTERIN6 CROSS SECTION OF Ca*o
FOR 3.15 ± 0.15 MEV
NEUTRONS
•
EXPERIMENT
4M
• i THEORY
4 , ..100
ito
r.4 1 6
140
i'm
38
TABLE I I
DIFFERENTIAL CROSS SECTIONS IN BARNS/STERADIAN
E la stic
Compound E la stic
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
e la s ti c sc atte rin g cross seotion o f oaloium-40. Also i n Figure 16, i s
shown the th e o re tic a l a n ^ ila r d is trib u tio n o f compound e la s tic s c a tte rin g o f caloium-40. The experimental curve i s in general agreement w ith
th e th e o re tic a l curve.
dHAPTER IV
CONCLUSIONS
4*1 D iffe re n tia l E la stic Cross Sections
The d if fe re n tia l e la s tic cross sectio n s o f calcium-40 and potassium-39 are shown in Figure 15* The e rro r bars are s t a t i s t i c a l erro rs
which range from one percent to fiv e percent* The estim ate o f the
re la tiv e accuracy of the measurements can be made from th e smoothness
of the curves and the consistency of the d ata. The consistency o f the
data throughout the experiment fo r a l l the runs was very encouraging.
The inaccuracies contributing erro rs were as followst
a,) The s t a t i s t i c a l uncertainty in monitoring was 0.15 percent
and th a t in th e d ire c t beam was 0.28 p ercen t. Thus th e u n certain ty in
th e d ire c t beam counts introduced by monitoring was 0.32 percent.
b) Machine background was le ss than one te n th of a percent.
c) The machine in s t a b il ity contributed the e rro r in several ways:
changes in the beam p o sitio n a t th e ta rg e t could have caused a change in
th e sc a tte rin g angle by t 3 percent but th e consistency of th e data
shows th a t th is source could not lead to a g reater e rro r.
d) Scattered neutrons lo se about 8 percent of th e ir energy a t
140° in comparison to th e d ire c t beam energy. This lo ss o f energy reduces
the experimental values o f the cross sectio n s by upto 3 percent a t back
angles.
e) The secondary sc atte re d neutrons from th e shielding wedge or
the collim ator were more probable a t the sm aller angles than a t g reater
o. -
1
*
-
41
o
a n g les. The magnitude <Sf systematic erro rs from t h i s source are
d if f ic u lt to evaluate.
When the oross sectio n s of the two• elements
were extracted
-*
•
from the data the erro rs which appear were magnified. The uncertainty
introduced by the method of subtracting background from th e time
sp ectra was between one and two percent a t th e forward angles, between
two and fiv e percent a t th e interm ediate, angles and between th re e to
four percent a t the back an g les. This introduced an uncertainty in the
d ifferen ce curve up to 25 percent. The absolute values of the cross
sectio n s of calcium-40 a t 40° determined by d ire c t beam method and by
comparison w ith carbon method agree to each o th er w ithin 2.4 percant.
The d ire c t beam method involves a step by step measurement o f d ire c t
beam neutrons in the forward d ire c tio n , th e measurement of th e so lid
angle subtended by th e s c a tte re r on the ta r g e t, th e published D(d,n)Hp
oross sectio n s as a function o f energy and th e atten u atio n f a c to r. The
comparison w ith carbon method involves th e r e la tiv e efficien cy o f the
d etecto r the atten u atio n fa c to r and the published value of th e d if f e r e n tia l cross section o f carbon. 1 *2 Since both methods involve many
fa c to rs , i t i s hard to evaluate the accuracy of th e absolute determi­
n atio n .
The d if f e r e n tia l e la s tic cross sectio n curves shown in Figure 15
were extrapolated upto 0° and 160°• The ju s tif ic a tio n in th e extrapola­
tio n i s the smoothness o f the curves. The t o t a l e la s tic cross sections
o f both the elements were calcu lated by in te g ra tin g the curves through
the following equation using Simpson*s ru le :
^Meir, e t . a l . , op. c i t . . Helvetica Physica Acta, p. 577.
S f i l l s , ot# a1#| 2E* cl^bt|
GIX|p# 89Xe
42
180 ^
2jf f " 5 Sin 0 do
" J d*
0
The to ta l e la s tic cross sections thus obtained are 2.8 2*0.04 barns for
calcium-40 and 1 .8 1.0.04 barns fo r potassium-39. Since th ere i s no
in e la s tic sc a tte rin g in calcium-40 a t th is energy th e to ta l e la s tic
cross section i s also i t s to ta l cross sectio n . The t o t a l e la s tic cross
section fo r calcium-40 found by Popov i s 28 percent higher than th is
value.
His d if f e r e n tia l e la s tic cross sections fo r both the elements
a re higher a t a l l angles than the o n e's found in th is experiment.
4*2 Compound E la stic Cross Section
I f both the o p tic a l model and the Hauser-Feshbach theory were
v a lid fo r the neutron sc atte rin g a t 3.15 Mev, the d ifferen ce in the
e la s tic s c a tte rin g o f the two elements should be compound e la s tic
sc a tte rin g (according to the o p tic a l model) and symmetric about 90°
(according to Hauser-Feshbach th eo ry ). For an approximate check on the
symmetry and the magnitude o f the d if f e r e n tia l compound e la s tic dis­
trib u tio n , computations were made on the b asis of Hauser-Feshbach theory
using single le v e l p a r a m e te r s T h e measured compound e la s tic oross
section of calcium-40 a t 3*15 Mev (Figure 16) appears la rg e r a t angles
below 50° and sm aller above 110° than predicted by Hauser-Feshbach
theory. The experimental data curve agrees f a ir ly w ell with theory
3v. I . Popov, "Angle D istrib u tio n of 3.1 Mev Neutrons E la s tic a lly
S cattered on Al, S i, K, Ca arid Th," Neitronnaya F iaik a. (Moscow:
Gosatomadat, ed. Krupchithkogo, p. A., 1961) .
S r. Hauser and H. Feshbach, "The I n e la s tic S catterin g o f Neutrons"
Physical Review. LXXXVII (1952). 366.
0
©
(•"'
<••
-■
43
O
,
between 50° and 110° .
Within the aeeumptione o f the theory, the flu c tu a tio n s in the
t o t a l cross sectio n s are dus^t6 >resonance formation of the compound
nucleus and flu ctu a tio n s in the e la s tic d if fe re n tia l cross sections
are again due to resonance in the formation of the compound nucleus and
in th is case appear as a change in the compound e la s tic sc a tte rin g .
The published angular d istrib u tio n s of e la s tic sc a tte rin g of
3 .1 Mev neutrons on calcium and potassium by V. I . Popov has cross
section of potassium la rg e r than th a t o f calcium between 60° and 90°.^
This means, according to the assumptions of th is experiment, the
compound-elastic cross section fo r calcium i s negative in th a t region
which i s meaningless. This makes one tend to d isbelieve thatr'dkta.
The to t a l compound e la s tic sc a tte rin g cross section fo r
calcium-40 fo r 3*15 Mev neutrons found by Hauser-Feshbach theory using
sin g le le v el parameters i s 1.0 barn and the d ifferen ce in the to ta l
e la s tic cross sectio n s o f calcium-40 and potassium-39 found by th is
experiment i s 1.0 £ 0.05 barns. I t i s th erefo re concluded th a t, although
th e d etailed f i t between the computed and measured d if f e r e n tia l compound
e la s tic cross sectio n i s not good, th is may be due in p a rt to the choice
o f single le v e l parameters in th e Hauser-Feshbach c a lc u la tio n s. That
the general magnitude of the compound e la s tic cross sections i s c o rre c t,
i s supported by the good agreement of the to ta l cross se ctio n s. The
assymmetry o f the d if f e r e n tia l compound e la s tic cross sectio n presumably
r e s u lts from a breakdown o f the assumption o f id e n tic a l shape e la s tic
s c a tte rin g in th e calcium-40 and potassium-39*
^Popov, op. c i t . . Neitronnaya F isik a .
o
SELECTED BIBLIOGRAPHY
Batchelor, R ., Gilboy, W. B ., P u rn ell, A. D. and Towle, J . H. "Improve­
ments in th e Fast Neutron Time-of-Flight Technique using Pulse Shape
Discrim ination in an Organic Phosphor," Nuclear Instruments and Methods.
VIII (I960), 1*6.
Bjorklund, F. and Fernbach, S. "Optical-Model Analysis o f S cattering o f
4*1-,7-, and 14-Mev Neutrons," Physical Review CIX (1958)* 1295*
B la tt, J . and Biedenharn, L. C. "Nautron-Proton S catterin g w ith Spin
O rbit Coupling. I . General Expressions," Physical Review. LXXXVI (1952),
399.
Bohr, N. "Neutron Capture and Nuclear C o n stitu tio n ," Nature. CXXXVII
(1936), 344.
B fyster, J . R ., Walt, M. and Salme, E. W. " In te ra c tio n o f lJD -,1,.77-.
3 .2 5 -, and 7.0-Mev Neutrons w ith N uclei," Physical Review. CIV (1956),
1319.
Brandenberger, J . D. "Compound-Elastic S catterin g o f Fast Neutrons by
Lead-206," (Unpublished Ph.D. D isse rtatio n , The U niversity o f Texas,
Austin, 1962).
Brooks, F. D. "A S c in tilla tio n Counter with Neutrons and Ganaa-Ray
D iscrim inator," Nnoliar Instrum ents and Methods. IV (1959), 151.
Cranberg, L. "Tim e-of-Flight Technique Applied to F ast Neutron Measure­
ments." In te rn a tio n a l Confrence on th e Peaceful uses o f Atomic Energy.
1955.
Cranberg, L. and Lelfin, J . S. "Neutron S catterin g a t 2.45 Mev by a
Time-of-Flight Method," Physical Review. C III (1956), 343.
Day. R. B. "Ganma-Rays from Neutron In e la s tic S c a tte rin g ." Physical
Review. CII (1956), 767.
E isberg, R. M., Yennie, D.-R. and Wilkinson, D. H. "A Bremsstrahlung
Experiment to Measure the time Delay in Nuclear R eactions." Nuclear
Physics, XVIII(1960),338.
Feshbach, H ., P o rte r, C. E. and Weisskppf, V. F. "Model fo r Nuclear
Reactions w ith Neutrons," Physical Review. XCVI (1954), 448.
Flugge, S. "Nuclear R eactions," 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 a u ll.W . New York: KoGrew-Hill Book
cS™“in7.71*55.
Hauser, W*juid Feehbaoh, H. " In e la s tic S catterin g o f Neutrons, w Physical
Review. LXZXVII (1952), 366.
Kant, D. W., P u ri, S. P ., Snowdon, S. C. and Buohar, W, P. "In terac tio n
o f 3*7 Her Neutrons w ith Medium Weight N uclei." Physical Ravi aw. CXXV
(1962). 331.
Marlon, J . B. and Fowler J . L ., Fast Neutron Physios P art I and P art I I
New York) Intersoienoe P ublishers, In c ., I960.
Marion, J . B. I960 Nuclear Data Tables. P a rt 3 . Washington! U. S. Gov­
ernment P rin tin g O ffice, I960. .
Mobley, R. C. "Proposed Method fo r Froduoing Short Intense Monoenergeti c Ion P u lses," Physical Review. LXXXVIII (1952), 360.
- r
Mobley, R. C. "V arlable-Path Magnetic Ion Bflmcher," Review S c ie n tific
Instrum ents, XXXIV (1963), 256/
N a ile r, J . H. and Good, W. M. "Tim e-of-Flight Technique," F ast Neutron
P hysios. P a rt I ed. Marion, J . B. and Fowler, New Yorks Intersoienoe
P u b lish ers, m e ,, I960.
pkhuysen, P. L ., Brandenberger, J . D. and Smith, W. R. "Compound-Eketic
S catterin g o f F ast Neutrons by Lead," B u lletiii Aaerioan Physioal Sooiety,Y I (1961). 375.
Okhuysen, P. L. and Prud'hoomo, J . T. "Compound-Elastic S catterin g o f
4 .2 Mev Neutrons in Lead." Physioal Review. CXVI (1959), 986.
Popov, V. I . "Angular S istrlb u tio n of 3*1 Mev Neutrons E la s tic a lly
Scattered by A l, S i, K, Ca and Th," Neitronnaya F iska. ed. Krupohitskogo,
P . A ., Moscow: Gostamisdat, 1961. P re s k it, C. A. and Alford, W. P. " E la stic S catterin g of Protons by V,
C r, Fe and Co," Physioal Review. CXV (1959), 389.
Rybakov, B. V. and Sidorov, V. A. F ast Neutron Speotrosoopy. ed. Vlasov,
N. A., New Yorks Consultants Bureau, In o ., I960.
Vincent, L. D. "The In te ra c tio n of 4*1 Mev Neutrons w ith S ulfur, C alci­
um, Molybdnum, Antimony, Barium and tferoury," (Unpublished Ph.D.
D isse rtatio n , The U niversity o f Texas, A ustin, I960).
Weber, W.,_ Johnstone, C. W. and Cranberg, L ., "Time-to-Pulse Height
Converter fo r measurement of Millimicrosecond Time I n te r v a l," Review
S c ie n tific Instrum ents. XXVII (1956), 166. _
W olfenstein, L. "Conservation o f Angular Momentum in the S ta tis tic a l
Theory of Nuclear R eactions," Physioal Review. LXXXII (1951), 690.
o
o
o
o
APPENDIX
l.F lu x C alculations
Consider a small element o f volume in th e s c a tte re r and in the
ta rg e t as shown in the Figure i . Then th e neutron flu x 0 ^ in tercep ted
by the sc a tte rin g element from the ta rg e t element i s given by
0TSs n DAAT nT AxT Q #(0ST#ED)
where
2
n^a number o f beam deuterons/sec/cm .
A at « area o f cross section o f volume element
in the gas ta rg e t.
nTr number of nuclei/cm? in the ta r g e t.
Ax,pS thickness o f th e volume element in the
ta rg e t.
Q "
*D(d,n)He^ cross sectio n .
0 Q
s
£>T
angle subtended by th e rad iu s vector from
s c a tte re r- to - ta r g e t with the d ire c tio n o f the
incident deuteron beam coming from th e a c c e le ra to r.
E^« deuteron energy a t the point under consideration
in the ta r g e t.
Then the number of sc atte re d counts C detected by the d etecto r i s given
by
do*
CiA
>
nD ^*T "T 4 *t <r<«sT.ED)n»
^
•fi(E)Ks O D,
dO
where
3
ngm number of nuclei/cm . of s c a tte re r.
46
"
„
*'
-
w
» thickness of volume element in the scatterer*
s
^ s T » eb lid angle subtended by th e sc a tte rin g element on
ax
ta rg e t element*
^■Ds %so lid angle subtended by th e d etecto r on the sc a tte rin g
element*
do*
d
* d if f e r e n tia l cross sectio n fo r the sc a tte re d p a rtio le s .
a
^ (E) * E fficiency of th e detector*
K « atten u atio n fa c to r of th e s c a tte re r.
s
S im ilarly , the number of d ire c t beam counts Cgg detected by th e d etecto r
i n the forward d ire c tio n i s given by
AA,p Oj, AXp 0 ^*(Ogiji,Eg) ^^dT £ ( E )
where
0DTs angle subtended by the radius vector from d e te c to r-to ta r g e t.
S^DT s so lid angle subtended by th e d etecto r on th e ta rg e t
element*
Thus,
i f £ l I)8a * £ l DT xnd AAT,
^ x ^ ,, A x# are constant,
then
c.
°DB
i f t
/.if tn . a x .k .Y
^
y
o
tV
Ep) n . T
®’ ^®DT,ED^
Since 0 ^ i s very sm all, then
< r ( 8DT.BD) X ff(E D)
Therefore,
_ dcr /d X l n8
cdb
In order to c a lc u la te
^
K8 4 x 8 >
CT(0.t>Ed) & sT
1 <F (Ed)
K8 and ^ !Q*(Q«T»Ep) r i 8p the computer program
was w ritte n as shown in th e following sections*
2 , Attenuation Factor
The atten u atio n fa c to r Kfl for a s c a tte re r 2** x 5/8" a t 2.25
inches from the ta rg e t was obtained from the expression
YLz*=±------------S is l
where 0 was the angle between the path o f th e incoming neutron and the
d ire c tio n of the incident deuteron beam as measured from the center o f
th e ta r g e t. The quantity F^ i s given by the expression
0
where
i s the distance th a t a neutron, which l e f t the ta rg e t a t an
angle 0^, traversed in passing through the s c a tte r e r . The d istan ces a^
were determined g raphically fo r ten values o f 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-----------m
3. F ortran fo r K
s
c
c
ATTENUATION FACTOR
DIMENSION A(10)
7
READ 1,X
1
FORMAT (F5.4)
6
DO 2 1 :1 ,1 0
e tc .) Hence
49
2
READ 3 , A(l)
3
FORMAT (F5.4)
DO 8 I S 1,10
8
A ( I ) =A(I) * 2,54
SUM cO.
DO 4 I 8 1,10
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 th e above program th e re were th ree se ts o f a^f s . A ll
were measured in Inches. I s n#
and m "10* The machine w ill punch
th ree a tte n u a tio n fa c to rs as Y, the average value of Y can be considered
a s the o v e ra ll value of K •
8
4. F ortran Y o r J ^ O ^ s T * ^ CVsT
Before the a c tu a l computer program was w ritte n the expression
d O
‘( ° aT#ED) J " 1 8t was derived in to a sim p lified form as follow s. The
s c a tte re r was divided p aralled to i t s length in to n s lic e s of equal
th ick n ess AX, see Figure i i ( a ) and each s lic e was divided in to p x q
rectang u lar elements, p d iv isio n s along Z -axis and q along Y -axis. The
o rig in being a t the center of the n£& s lic e . Only one quarter o f the
nt& s lic e i s shown sub-divided in Figure i i ( b ) .
Consider any point p in the ta rg e t a t a distance o f X from the
n
o rig in 0 . Let D be jtys.distance of P from the back o f the s c a tte re r as
50
TARGET
SC A TT ER ER
FIG. i
END VIEW
FIG. II
FLUX CALCULATIONS
Fig. 17
shown In Figure 11(c)* R i s the radius and Y i s h a lf the length o f the
-'
1L
sc a tte re r* Then fo r the n— s lic e :
Ax
X s D- (2r>-l)-------2
Y s Y
n
[ r 2 { r - (2 n -l)
tL
J2 |^ :
The radius vector from the center of the ( p ,q ) ~ element in th e n —
s lic e to the point P i s given by
• n .p .,
-en }2 * { (2” «
*[< +
)l/2
]
The angle subtended by th is rad iu s vector w ith x~axis i s
sT c
n ,p ,q„ e Coil
( ------ nXn )
n,p,q
and the so lid angle subtended by the (p,q)^& element a t the point P i s
3
Yn Zn
n»P»q
found from the Table III*
Thus SCEjj)*
n»P#q
By changing the p o sitio n of the p oint P in th e ta rg e t we have
as > ; »
(e „ ) .
The computer program i s as follow s:
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,N D )
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)38,6 o,6 o
60
PAUSE
GOTO 10
70
*
■
u
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
8
00
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
'
XD
0.156
0.206
0.270
0.362
Mev
6.71
6.91
7.11
7.51
cm.
4.30
4.30
4.29
4.29
4.28
4.28
4.27
4.26
4.25
4.23
4.20
4.18
4.16
4.13
4.11
4.09
4.05
4.03
4.00
3.97
3.95
3.91
3.88
3.85
3.82
3.78
3.73
3.69
3.65
3.60
3.57
3.51
3.47
3.43
3.38
3.33
6.33
6.33
6.32
6.32
6.31
6.29
6.27
6.25
6.22
6.19
6.16
6.13
6.09
6.05
6.00
5.95
5.89
5.84
5.78
5.71
5.65
5.57
5.52
5.45
5.39
5.31
5.24
5.16
. 5.09
5.02
4.93
4*86
4.78
4.70
4*62
4.54
9.19
9.19
9.18
9.17
9.15
9.12
9.09
9.05
9.02
8.97
8.92
8.86
8.79
8.72
8.64
8.56
8.49
8.40
8.30
8.20
8.10
8.01
7.89
7.78
7.67
7.56
7.44
7.32
7.19
7.06
6.96
6.82
6.70
6.57
6.46
6.34
11.75
11.74
11)72
11.68
11.65
11.63
11.59
11.54
11.46
11.40
11.33
11.24
11.15
11.04
10.92
10.78
10.67
10.55
10.42
10.28
10.15
10.00
09.84
09.68
09.52
09.36
09.22
09.06
08.90
08.72
08.54
08.35
08.16
07.98
07.80
07.64
VITA
Zorawar Singh Khangura son o f S. Shiam Singh Khangura and
Sardarni Harnam Kaur was born on A pril 3 , 1928, a t the v illa g e Latala
o f Ludhiana d is t r ic t in Punjab, In d ia. He graduated from Government
High School, Gujjarwal (Punjab U niversity, Lahore) in 1945 and attended
D. M. College, Mbga (Punjab U niversity, Lahore) from 1945 to 1947. After
th e p a rtitio n of the Punjab in 1947* he enrolled in Government College
Ludhiana (Punjab U niversity, Solan) which he attended from 1948 to
1950. He received h is Bachelor of Arts degree from the Punjab U niversity,
Solan in 1950. He was employed as a teaching a s s is ta n t from 1950 to 1952
a t Lyallpur Khalsa College, Jullunder. In 1952 he enrolled in the Muslim
U niversity Aligarh and received h is 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 th is period, by working p a rt tim e, he
earned a b ach elo r's degree in teaching from The Punjab U niversity,
Chandigarh in 1957. He married former Miss SwarnJeet Kaub Padda,
daughter o f B rigadier Sant Singh Padda, of v illa g e Khiranwali o f
Kapurthala d i s t r i c t in Punjab on February 8, 1959. He came to the United
S tates o f America in September 1959 ahd enrolled in the Graduate School
o f Louisiana S tate U niversity, Baton Rouge in f a l l of 1959. Since then
he has been working towards a degree of Doctor of Philosophy in physics,
fo r which degree he i s 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