AN ABSTRACT OF THE THESIS OF
Larnar'\,Villiam Coleman for the Ph. D. in
(Name
Date thesis
Physic
l5%"".7
)
is presented
s
(Major)
JuIy 31, 1963
GAMMA- GAMMA DIRECTIONAL CORRELATION
TitIe
STUDY OF BARIUM-134
Abstract approved
Redacted for Privacy
,/(Ivfajor professor)
Gamma-gamma directional correlations were rneasured
in Bal34
^tten
angles between !0 degrees and 180 degrees in ten
degree increments using a scintillation spectrometer. The spec-
trometer which used NaI crystals, differential energy selection,
and standard coincidence techniques was tested by measuring the
directional correlation of the two cascade gamrna rays i., Ni60 to
be
in satisfactory agreelrrent with published results and with theo-
retical predictions. C"I34 decays by beta ernission to forrn several excited states below 2 Mev in Bal34 *hi"h subsequently decay
to
the ground state by the ernission of galrllrla rays. Two correlation
measurements were performea i., g"134. The first, the 'toverall,
correlation, consists of the 797 Kev-605 Kev,
559 Kev-797
802 Kev-563 Kev,
Kev, and the 802 Kev-[S6l Xev] -605 Kev correlations.
The second, the rrseparatedrr correlation, contains tt.e 7 97 Kev-505
Kev and the 802 Kev-[ 563 Kev] -505 Kev correlations. / The experimental correlation coefficients are consistent with angular rnornentum assignrnents of Z, Z, 4,4 to the 605 Kev, 1168 Kev, L401 Kev,
and 1970 Kev excited states respectively. These states are assumed to have positive
parity. An analysis of the data in terrns of
the electric quadrupole to rnagnetic dipole rnixing ratios of the 569
Kev and the 553 Kev radiations reveals that, within the limits of
experirnental error, each radiation contains at least a 50 percent
magnetic dipole admixture. The other garrrrrra rays involved are
assumed to be essentially pure electric quadrupole. This result
contradicts the asyrnmetric rotor rnodel of the nucleus, which, aIthough
it accurately predicts the measured energies of the states
with these angular rrlorrenturn and parity assignments in B^L34,
does not allow rnagnetic dipole transitions between the rotational
state s.
GAMMA - GAMMA DIRECTIONAL CORRELATION
STUDY OF BARIUM-I34
by
LAMAR WILLIAM COLEMAN
A THESIS
subrnitted to
OREGON STATE UNIVERSITY
in partial fulfillment of
the r'equirements for the
degree of
DOCTOR OF' PHILOSOPHY
August 1963
APPROVED:
Redacted for Privacy
Associate Professor of Physic
s
In Charge of Major
Redacted for Privacy
Chairrnan of the
partrnent of Physrc
s
Redacted for Privacy
Dean of the Graduate School
Date thesis is presented
JuIy 31, 1963
Typed by Jolene Hunter Wuest
ACKNOWLEDGMENTS
The author wishes to express his gratitude to his research
director, Dr. Larry Schecter, for proposing the problern, for
many enlightening discussions concerning all aspects of this work,
and
for continued and conscientious help and encouragement while
on leave in
Europe. He would like to thank Dr. E. A. Yunker for
his continued interest and support, Drs. Harry Easterday and
David Nicodemus and Mr. Raymond Sornrnerfeldt for rnany heipful
discussions about the work, Dr. George Trigg and Mr. Timothy
Keliey for interesting conversations regarding the theory, Mr.
Richard Siemens for his help in reducing the data and checking
calculations, and Mr. Jack McKenzie of the staff of the Oregon
State University cyclotron for his help in the production of calibra-
tion and testing sources. This work was perforrned under the auspices of the United States Atornic Energy Cornrnission.
TABLE OF CONTENTS
Page
I
INTRODUC TION
SINGLE PARTICLE AND COLLECTIVE NUCLEAR
MODELS
l5
THEORY OF DIRECTIONAL CORRELATION
34
THE ANGULAR MOMENTUM OF ELECTROMAGNETIC
RADIA TION
45
SOURCE PREPARATION
53
EXPERIMENTAL APPARAT
EXPERIME NTA
L
57
US
MEASURE MENT
S
63
CONCLUSIONS
7Z
BIBLIOGRAPHY
83
APPENDIX I
Geornetrical Correction Factors to W(+)
88
APPENDIX II
Treatment of the Data
Least Squares Analysis of the Data
93
97
LIST Or. FIGURES
Page
Figure
I
2
LEVEL AND DECAY SCHEME Or 8a134
7
THE ASPECTS OF A DIRECTIONAL
36
CORRE LA TION E XPERI MENT
PULSE HEIGHT SPECTRUM
Or
C"134
56
BLOCK DIAGRAM OT' THE SCINTILLATION
SPEC TROMETER
59
PULSE HEIGHT SPECTRUM OF Cs134 TOR
CHANNELS I AND 2 IN THE ''OVERALLII
CORRELATION
67
PULSE HEIGHT SPECTRUM OT' C"134 FOR
CHANNELS 1 AND 2 IN THE I'SEPARATEDII
CORRELATION
67
7
g"l 34 "ovERALL" CoRRELATIoN
69
8
Brl 34 !'SEPARATED" CoRRELATIoN
70
EXPERIMENTALLY ALLOWED 6 -, . VERSUS
EXPERIMENTALLY ALLOWED';;;
79
GEOMETRY OF THE CRYSTALS FOR THE
FINITE DETECTOR SIZE CORRECTION
89
9
l0
LIST OF. TABLES
Page
TabIe
OBSERVED GAMMA AND BETA RAYS IN THE
DECAY or Csl34
4
MEASURED AND THEORETICAL CONVERSION
COET'FICIENTS T'OR BAI34 GAMMA RAYS
8
THEORETICAL AND MEASURED GAMMA RAY
INTENSITY RATIOS FROM ROTATIONAL STATES
IN na134
33
DIREC TIONAL CORRELATION COEFFICIENTS
AND ANISOTROPY OF
7
A-I
A-3
7I
CALCULATED COEFF'ICIENTS FOR THE
''SEPARATED" CORRELATION IN BaI34
74
CALCULATED COEFFICIENTS FOR THE
IIOVERALL'I CORRELATION IN 8a134
75
THEORETICAL AND MEASURED GAMMA RAY
INTENSITIES IN 8a134
8l
SAMPLE CALCULATION OF *iRi(140), di,
d? , and *. d?
96
1
A-Z
BaI34
11
CALCULATION SHEET FOR THE LEAST
SQUARES ANALYSM OT' THE EXPERIMENTAL
FOR THE ''OVERALL'' CORRECOEFFICIENTS134
LATION OF Ba'
r00
CALCULATION SHEET T'OR THE LEAST
SOUARES ANALYSIS OF. THE EXPERIMENTAL
COEI.FICIENTS F'OR THE I'SEPARATEDI'
CORRELATION Or na134
104
GAMMA - GAMMA DIREC TIONAL CORRE LATION
STUDY OT' BARIUM- I34
INTRODUC TION
The atornic nucleus is a collection of nucleons which can
exist in various discrete states. Each of these states can be de-
scribedbya set of pararneters which includes the energy, angular
momentum, parity, mean life, and electric and rnagnetic moments.
A primary concern in low energy nuclear physics is the deterrnina-
tion of these pararneters. The experirnentalist atternpts to measure these quantities to aid the theorist in constructing a useful
rnathernatical description of nuclear levels and their properties,
that is, a nuclear rnodel, which,agrees in its predictions with
measured quantities and which can hopefully be used to predict in-
formation about other pararrreters and other nuclei. The electric
and magnetic rnornents of the ground states of many nuclei have
been measured with high precision whereas the deterrnination of
the mornents of excited states is rnore difficult. Knowledge of the
angular morrrenta and parities of these states is necessary for the
development of the theory of the structure of the states and their
electric and rnagnetic properties or, for the testing of models.
It is generally accepted that the ground states of even-even
nuclei have zero angular lTrolrtenturn and even parity. The even
parity is a definite consequence of the shell rnodel of nuclei in that
Z
an even number of nucleons in any state results in a configuration
of even parity. The absence of measurable rnagnetic rnoments in
the ground states of even-even nuclei can be considered as evidence
of zero nuclear angular rnomenturrr. Ground state static electric
quadrupole mornents have not been observed in even-even nuclei,
a further consequence of zero angular momentum in the ground
state.
Four methods that have been used to determine the angular
rnornenta and parities of excited nuclear states are indicated below:
1.
A nucleus rrray de-excite by internal conversion or if
sufficient energy is available by internal pair production. The rnagnitudes of the pair production and internal conversion coefficients depend upon the changes
in angular mornentum (A J) and parity (A n') in the transition. The ground state of an even-even nucleus has
spin zero and even parity so a careful measurernent of
one or both of these coefficients allows a definite assignrnent of angular momentum and parity to each of the excited states.
z.
The angular rnornenta and parities of nuclear states
can also be inferred frorn beta decay studies. The
shape of the beta spectrum and its log ft value depend
upon the angular rnornenturn and parity change in the
decay. If the spin and parity of the decaying state of
the parent nucleus is known, selection rules rnay be
used to assign a possible angular rnomentum and parity to the resulting state of the daughter nucleus Produced by the transition.
3.
If two or rnore gamrna rays are ernitted in cascade a
rrreasurement of the directional correlation between
thern deterrnines the rnultipolarities of the radiations
and places restrictions on the possible values of the
changes of angular mornenta between the nuclear states
involved.
4.
Polarization-correlation measurerrrents with the
garrrrrra rays emitted during a nuclear de-excitation
deterrnine the rnagnetic or electric nature of the
radiation.. Together wj.th a directional correlation
experiment this fixes the parity of the garnrna ray
which in turn fixes the change in parity of the nuclear leveIs.
2.3 year C.134 decays by beta-ray ernission to forrn excited levels of the even-even a^I34 nucleus. Several Ievels exist
below 2.0 Mev in g.i34 resulting in cornplicated beta"- and garn-
rr,a-ray spectra which make the task of deterrnining the level
scheme
e:rgy
difficult.
Sorne of the gamrrra
rays are very close in en-
to one another creating an experirnental probl.ern in resolu-
tion in atternpting to deterrnine what energies are present. A
nurnber of workers have investigated the beta- and garnrna-tay
spectrurn of C"134. Garnrn a-ray yield and energy deterrninations
have been rnade frorn spectrographl.c studies of the internally and
externally converted electrons and by scintillation techniques.
The beta-ray spectrurn has been studied with rnagnetic spectro-
rneters in atternpts to deterrnine the nurnber of branches, their
end point energies and log
ft values, and relative intensities.
The
results of a nurnber of these studies are presented in Table l.
The references to the various investigators are listed across the
top using the number by which they appear in the bibliography.
The energies of the garnrna- and beta-ray branches that are
TABLE
1.
OBSERVED GAMMA AND BETA RAYS IN THE DECAY oF Cr134
Gamma
Energy
(Kev)
1r7)
(
s1)
(
40)
473
::; )
60s
7e7
x1zo1
x(100)
X(100)
)x
x
x
x
x
802
)x
( 13)
(
30)
(
31)
(s)
1168
1365
1570
20)
(22)
(3)
x
x
x
x
x
x
x( 3. s)
x{ 1. 8)
x(4.0)
x
xtzt)
x(e, 4)
)Nznr
ilf]
x(100) xloo)
x
x
x
x
xo.e)
x(4.0)
x{3.0)
xs.3)
x4.6)
x( 1oo)
x( 1o0)
x( 12.8)
xloo)
xe1)
x
x
x( 18)
960
1038
(
x
x
x
x(1,3)
x(72) \x,szl
r
x( tr)
x(0.6)
x( 1. s)
x(s. o)
x(3.0) x12.2)
x(s.0) x(3.3)
x(0.
12)
Beta
Energy
15"9----86
X(2s)
x2s)
x2s)
x( 10)
2LO
x(3)
4to
x(6)
6s8
X(7s)
x(7s)
x(7s)
x(81)
8.8
683
x32)
6.4
x\24',)
6.5
x(28)
6. 3
x(27)
6.2
x{3)
8.6
x(s)
9.0
x(so)
9.2
x( 13)
9.9
x\72)
8.9
?( 1. O)
10.9
x(s)
9.3
x(s6)
9.0
?(
6. 0)
9.9
x(3)
9.6
x(66)
8. 8
10)
18)
x(100)
x(103)
x(8)
x(
x(
x(1)
x(2)
x3)
(
48)
x( 1. 4)
x(10)
x( 14)
x(100)
x(80)
x(11)
x( 1. s)
x(1.6)
x(l.s)
x(3.4)
x(0. oe)
5
believed to be present are listed in the left hand colurnn. An entry
X indicates that a beta ray or garrrma ray of that energy was observed by the workers under whose reference nurnber the entry
appears. A bracket connecting two or rnore energies indicates
that the separate corrrponents were not resolved. The relative in-
tensities when measured appear in parentheses after the entries.
For the beta ray corrlponents the log ft values when rneasured appear just below the entries. The results indicate two principal
beta ray branches and at least nine weII identified garrlrra ray ener gie s.
On the basis of these rrreasurements together with coincidence studies and correlation data various leveI schernes have
been proposed
for Bul34. one of the earliest was suggested
Elliott and Bell (I7, p.
1396
980) who placed excited levels
by
at794 Kev,
Kev, and L964 Kev which de-excited by the ernission of four
gamrna
rays. Keister, Lee, and Schrnidt (31, p. 453) were able to
set an upper lirnit of one percent for the decay of Crl34 by K-
capture frorn the absence of rrreasurable Auger electrons.
Bertolini (5, p. 280) arrived at the same conclusion frorn the absence of detectable Xenon
X-rays. Further work on the de-excita-
tion spectrurn of C"134 led to the proposal of other decay schernes
(13, p. 4451, (30, p. l0ZZ), (5, p. 280l., (3I, p. 455l,, (20, p.855)
6
none of which was colrrpletely compatible with
all of the experirnen-
tal inforrnation. Most workers agree, however, on the rnain features of the decay by.placing levels at 605 Kev, I158 Kev, l40I
Kev, and 1970 Kev. Bashilov et al. (3, p. 60) proposed a level
sequence and decay scherne which was supported by the work of
Girgis and van Lieshout (zz, p. 67zl with one additional level added
at
1
570 Kev and also by the recent work of Segaert et
al.
(48,
p. 761. It is this level scherne that is currently accepted as being
the rnost satisfactory. Figure I displays the order of the levels
and the proposed garruna ray transitions of interest in this work
according to this scheme.
The angular rnornenturn and parity assignrnents shown on
the level diagrarn (Figure 1) have been made on the basis of garnrrla ray correlation rneasurernents, beta decay studies, and rneas-
ured internal conversion coefficients. The ground state angular
rnornenturn and
even
parity (.rr ) is taken to be 0* as g^134 is an even-
nucleus" The internal conversion coefficient for a given
gamrra ray energy' depends on the electric or rrragneti.c nature of
the radiation and its rnultipolarity. Tab1e Z surnrnaxizes sorne of
the rneasured internal conversion coefficients and the K/L ratios
for Flul34 garruna rays. The garnrna ray energies are listed
across the top of the table and the numbers listed in the left colurnn
ICs
134
€1
= 86 Kev
(27%)
FZ= +tO Kev (9%l
1970 Kev
(6t%l
Q = osa Kev
t77O Kev
1641 Kev
1570 Kev
4+
1401 Kev
e+)
1168 Kev
1<5x10-10 secl
605 Kev
Stable Ba134
Figure I. LEVEL AND DECAY SCHEME Of g"134.
8
TASLE
2.
Gamma
Energy
( Kev)
MEASURED AND THEORETICAL CO}TVERSION COEFFICIENTS FOR g"134 CAMMA RAYS
473
569
563
605
797
802
1038
1168
1365
o*x103
ss)
5.0
(40)
(0. 1)
5.67
(
(
31)
8.6
(3.6)
5.19
6.6
1.5
4.7
(0.3)
(s)
(
1.0)
o.62
(0.03)
.s*#
2.5t
1.1
2.5
(0.7)
0.85
0. 55
(0. 18)
(0. 10)
o.77
(0.1)
(0. 2)
orx1o3
(3e)
(3)
(20)
s.6
46
( 1)
2.6
9.2
(t.2)
5.3
(0.s)
2.6
(0.s)
( 1)
10
o.49
2.4
(0.3)
7.2
2.6
(
-s
-10
6.4
(0.8)
7.7
(0. 8)
31)
(0. 6)
(3)
( 1)
-10
7.3
s
(0.6)
1.
0.4s
s.O
KlL
(13)
2
8.0
0.8
7.O
7.6
(0.s)
(0.6)
-10
(0.0s)
t.4
(o.s)
o.9o
-10
8.4 10
(0.6) ( 0.4)
K/L+M
(
6.6
3s)
7.8
Theorv
oKx1o3
3.20
E1
9.85
E2
13. O
M1
42.O
M2
Assign- El,z)
ment M( 1)
2.20
6.25
2.15
6. 10
1.85
5. 10
8. 50
8. 45
7.OO
26.O
2s. 5
Et2)
M(1)
M( 1)
E(2)
20. 5
q2)
1. 04
1.01
o.62
2.60
3.60
2. 55
t.45
3.57
2.OO
9. 50
9.45
4.90
Et
2)
F{,2)
M1)
E( 1)
E( 2)
M( 1)
0.48
1. 13
1.60
3.75
Etz)
E( 1)
M1)
0.39
0.85
t.20
2.60
El,z)
E( 1)
9
refer to bibliography references. The nurnber in parenthesis below an entry is the experimental uncertainty in the entry. The
theoretical a-, values given in the table are due to Rose (45, P.
K
and include the effects of finite nuclear size and
641
screening. Pos-
sible rnultipole assignrnents on the basis of the conversion coefficients are given.
The angular rnornenturn of the ground. state
been rneasured to
of C"134 h""
be J = 4 by an atornic bearn rnagnetic resonance
experirnent by Bellarny and Srnith (4, p. 33) and by Jaccarino et al.
(29, p. 6761. The angular rnornenturn and the rnagnitude of the
dr/, proton-neu/Ztron configuration which predicts positive parity for the ground
rnagnetic rnornent are consistent with a gl
state of crl34
.
The beta ray branches indicated
in Figure I are the ones
that are required frorn garnrra ray studies so that other apparently
observed beta groups are not included. There is no rnechanisrn
for including in the scherne the 683 Kev beta branch reported
sorne
workers. At the high energy
end the
by
internal conversion
lines rnask the character of the continuous spectrum. As indicated
in Table I the 86 Kev branch has a 1og ft value of about 5.4 and in
addition appears to have a linear Ferrni plot. This branch has an
intensity
of. Z7
percent and corresponds to an allowed transition
10
(AJ=0,+ 1, no) allowing an assignrnent of 3, 4, or 5 with positive
parity to the 1970 Kev level in gul34. The other intense corrrponent, the 658 Kev branch, has a log ft value of 9 and an allowed
shape permitting possible assignrnents of
3, 4, or
5
with either
parity to the 1401 Kev level. The identification of other rnuch less
intense beta branches depends on being able to rerrrove successive
straight line cornponents frorn the Fermi plot. This is difficult in
cases where rrlore than one weak corrrponent rnay be present and
could be rnisleading if one or Irtore of the components did not have
an allowed shape. On the basis of the reported log
ft values alone,
these components could be as much as second forbidden (A J=lZ,
*3, no). Waggoner (55, p. 4Z5l has reported a slight deviation
from allowed shape for the 658 Kev beta group. If the shape of
the beta spectrurn is allowed theory predicts a syrrrmetric betagarrrrna
correlation. Stevenson and Deutsch (53, P.
no beta-garrrrra anisotropy
1203) found
in Crl34 with the 658 Kev beta group.
Further i.nforrnation about the angular rrlornentum of the
levels and the rnultipolarity of the radiations is available frorn
directional and polarization correlation experiments. Brady and
Deutsch (10, p. 870), in an early experirnent, were unable to de-
tect a non-isotropic directional correlation with the garnrna rays
frorn CrL34. They were able to rneasure a correlation with
Co60
l1
which has a sirnpler decay scherne. In a following experiment with
scintillatiur counters Brady and Deutsch (11, P. l54I)detected
a
directional correlation at three angles. Using irnproved scintillation counters these workers (12, P. 558) rerneasured the correlation at seven angles and frorn the sirnilarity in the shape of the
curve to that for Co60 suggested that the angular rtornenturn
change in the rnain sequence (the 797 Kev-605 Kev cascade) may
be
4, Z, 0, with both radiations being quadrupole. They were un-
able to detect any change in the correlation using a solid or a liq-
uid source with or without a 104 gauss rnagnetic field
and. con-
cluded that the life time of the interrnediate state of the cascade
must be less than I0-8 sec. In a sirnilar experiment Beyster and
Wiedenbeck (6,
p. 4I1), again in cornparison with the Co60 curve,
interpreted their data on the basis of a sirnple decay scheme containing just three garrrrla rays and suggested a quadrupole-quadrupole correlation for the 605 Kev-797 Kev cascade. The 569 Kev
radiation was also assurned to be quadrupole allowing an angular
mornenturn of
4, 5, or 6 for the 1970 Kev state. None of these
experirnents ernployed energy discrirnination so that all garnrna
rays were detected in each counter. Using NaI(Tl) scintillation
counters and integral energy selection Robinson and Madansky
(43, p. 604) rneasured the directional correlation by requiring
a
L2
coincidence between one of the garrrrra rays and a 660 Kev beta
and found agreerrrent with a basic quadrupole-quadrupole
correla-
tion with an angular mornenturn sequence 4, Z, 0 in the 505 Kev797 Kev cascade. They assumed a decay scherne in which the 797
Kev radiation Ieads to the ground state. Frorn a measurernent of
the rroverallrr correlation, that is, accepting aII gamrna ray ener-
gies, they assign an angular mornenturn of five to the
1970 Kev
level with the 569 Kev radiation being dipoie. Kloepper and Lennox (33, p. 696) also indicate a probable 4(Z\Z(Z)0 correlation for
the rnain decay sequence of C"134, but it is not clear what garrrma
ray energies were accepted in the experiment frorn which this conclusion was drawn. Klerna (32, p. 66) used differential energy
discrimination to rneasure the J355 Kev-605 Kev directional cor-
relation. Interpreting his data on the basis of a different
scherne
(3I, p.
4551, K1erna assigns
3 to the
decay
1970 Kev level by
allowing an El + MZ rnixture in the 1365 Kev gamma tay to fit his
data. The rneasured correlation function however, agrees best
with an assignrnent of J = 4 to that level with the I365 Kev garnma being quadrupole. Everett and Glaubman (18, p. 955) report
rneasuring the
1
36
5 Kev-605 Kev
,
802 Kev- I 168 Kev
, ard 7 97
Kev,605 Kev correlations and find that they are all consistent with
a quadrupole-quadrupole decay and allow assignrnents of 2t, Z*,
l3
++
4', 4 ' to the 605 Kev, I168 Kev, 140I Kev, and 1970 Kev states
respectively. It is pointed out that by allowing rnixed rnultipoles
for the first radiation the data can also be interpreted as 3'?-0,
3-I-0, or 2-Z-0. The effects of
t.]ne 569
Kev and the 563 Kev ra-
diations which contribute to the 505 Kev photopeak are not rnentioned nor is the contribu'lion of the 797 Kev garrrrna ray to the
802 Kev range of the
spectrurn. On the basis of sorne proposed
decay schernes these energy overlaps would not contribute to the
correlations. Stewart, Scharenberg, and Wiedenbeck (54' p. 69I)
perforrned a detailed directional correlation experirnent with
C"I34. They report no observable difference in the rneasured
function for solid or Iiquid sources. They rneasured an rroverallrl
correlation, the 797 Kev-605 Kev, the
570 Kev-605 Kev
1365 Kev-605
Kev, and the
correlations. On the basis of their results they
assign J = Z to the 505 Kev level, J
=
4 to the 1402 Kev level
and J = 4 to the I970 Kev leve1 and report that the 559 Kev radia-
tion is 94 percent quadrupole and 5 percent dipole. These work-
ers interpret their data on the basis of a sirnplified decay scheme
which contains only the 569 Kev, 797 Kev, 605 Kev, and 1365 Kev
garnrrra rays and ornits the level
at 1168 Kev so that the effects of
other cascades which contribute to each of their rneasurements
and which rnay affect the interpretation have been neglected.
l4
Wintersteiger (57, p. 79) has measured the life time of the first
excited state in Bal'n ,obe < 5x10-I0
"u..
by the de}ayed coinci-
dence rnethod.
Metzger and Deutsch (38, p. 557) rneasured the polariza-
tion-direction correlation in g^134 at the average garnrrra ray en-
ergy. The results are of the forrn for an EZ-EZ cascade which
rnay be characteristic of the intense 797 Kev-505 Kev cascade
which contributes rnost heavily to the counting rate. Williams
and Wiedenbeck
(55, p.
8ZZ'1,
frorn a similar experirnent, con-
cluded that if both the 797 Kev and the 605 Kev radiations are
quadrupole, one is EZ and the other is M2. Kloepper (33, p.
697),
reported a polarization correlation result that supports the assignrnent
of.
EZ to both the 797 Kev and 605 Kev gamrna rays.
In sumrnary, it is quite well established from previous
work that the 605 Kev and 797 Kev radiations are electric quadrupole and that the angular rnornenta and parities of the 605 Kev and
I401 Kev leve1s
^r"
Z* and 4+
""sp"ctively.
The angular rnornenta
and parities of the 1168 Kev and 1970 Kev levels and the rnultipo-
larities of the 802 Kev,
569
Kev, and 563 Kev radiations are less
certain although the 1970 Kev le.vel probably has angular mornentum 4. The conclusions drawn frorn ITrany previous correlation
rneasurernents can be questioned because of the ornission of sorne
15
contributing cascades in the interpretation and because, in rnost
cases, of large statistical errors. A nuclear model has been Proposed which explains sorne of the features of the g^L34 level
scherne. It is felt that a reinvestigation of the gamrrra-gamma
directional correlation and its interpretation on the basis of
a
rnore cornplete decay scherne will be useful in clarifying the un-
certainties in angular momenturn and rnultipole assignrnents to
the levels and radiations, and in strengthening some assignrnents.
The cornparison of the experirnental results with those calculated
on the basis of the decay scherne can be applied to test the Pre-
dictions of the model.
l6
SINGLE PARTICLE AND COLLECTIVE NUCLEAR MODELS
A complete understanding of nuclear phenornena involves
a description of the nucleus as a system of nucleons interacting
with a corrrrrlon rneson field
The nucleus and sofi)e of
its ProPer-
ties can, however, be rather well described in terrns of a system
of well defined protons and neutrons with certain forces between
thern. The meson origin of these forces does not play an essential part in the behavior of nuclei at low energy so that the theory
of nuclear structure is not concerned with the theory of the force
itself but accepts its existence and its properties as exPerirnental
fact. Once an expression is given for the nuclear forces aII problerns of nuclear structure are those of solving a Schrddinger equa-
tion for A particles. This equation cannot be solved in general
so
a resort to approxirnations, that is, to nuclear rnodels is necessary.
Because of this lirnitation an attempt
is rnade to use as
little detailed inforrnation of the forces as Possible and to concentrate on qualitative features. This procedure introduces concepts
such as
radius, shape, and potential well which can be defined
and rneasured and which are useful
for a description of the observ-
ed facts but whose connection with the fundarnental forces
is
vague.
L7
This ernphasis brings about the introduction of rnany kinds of nu-
clear models. A rnodel is sirnply a stressing of certain features
which can account for the phenornenon under consideration. Frorn
a model it rnust be possible without prohibitively lengthy calcula-
tion to predict various observable properties of nuclides in
a
systernatic way.
The sirnplest nuclear rnodels have their basis in trying
to understand the experirnentally inferred shelI structure and mag-
ic nurnbers in analogy with the structure of atorns.
'Whenever
the nurnber of protons or neutrons reaches any of certain values
called rnagic nurnbers the nucleons forrn a closed and exception-
ally stable system. The extreme single particle model of nuclei
describes the nucleus as a spherical potential well in which the
nucleons rnove as noninteracting Particles and thus ignores any
nucleon correlations. This potential is the average effect of all
the other nucleons on a single one, has a certain size and depth,
and includes a spin-orbit
interaction.
Such a potential with a suit-
ably chosen forrn produces a shell structure and predicts the cor-
rect location of the rnagic nurnbers. This rnodel describes the
ground states of nuclei by specifying the nurnber of nucleons in
each quanturn state assurning that the proton and neutron states
fill independently. In the ground state the nucleons are thought of
t8
as being paired off to zero ar,g:ular rnornenturn by a strong pairing
force so that the values of the nuclear pararneters are deterrnined
solely by a single unpaired nucleon in odd A nuclei. The nucleus
can be irnagined then to be rnade up of a core which contains all
the particles in closed she1ls and a cloud which consists of those
nucleons in unfilled shells. The simplest case would be a nucleus
in which the cloud consisted of a single particle. Then the rnodel
allows a deterrnination of the nuclear angular rnomenturn and parity
for nuclei with one odd particle as being those of the odd particle.
The angular rnornentum of odd-odd nuclei cannot be predicted because
it cannot be determined what coupling of the particlest
angu-
lar rnornenta gives the lowest energy state. This single particle
approxirnation gives the Schmidt iirnits for the ground state nuclear
rnagnetic rnornents. In general the experirnental values lie between these
lirnits but differ greatly frorn them. The quadrupole
rnornents predicted by this rnodel are corrrpletely erroneous
(4 1,
p.I45l.
In atternpting to interpret the large arnount of data available
on energies, angular rnornenta,
parities, and transition probabiii-
ties, it becornes clear that the extrerne single particle rnodel is
inadequate. The next higher approxirnation, which will be terrned
the shell model, is to consider a nucleus with a larger nurnber of
19
nucleons in the cloud" Low energy nuclear properties are attributed to the particles outside of closed she11s. The assumption of
a strong pairing force is relaxed to allow recoupling of the angular
rnornenta of the ntlcleons beyond closed
shells. This is necessary
in order to obtain a larger nurnber of states as required by experi*
rnent than are available in the single particle rnodel. It is also assurned that those nucleons outside of closed sheIIs do not perturb
each other very rnuch frorn single
particle states. The interaction
is strong enough to rernove degeneracy but not so strong (compared to the spin-orbit interaction) that j ceases to be a good
quanturn
nurnber. This assurnption can also be relaxed and a su-
perposition of the wave functions of two or three pure single partic1e states taken to describe a
state. This process is terrned con-
figuration rnixing. In the few places where detailed calculations
have been made the rnodel has been found to give an excellent de-
scriptionof the spins and parities of low-lying excited states. In
trying to understand the energies and angular rnornenta of these
states it is necessary to stay in the vicinity of closed shells or
else the nurnber of competing configurations is overwhekning.
The interpretation of excited states by rneans of this type of coup-
ling scherne has been tested only for cases involving two or three
identical nucleons (or holes) outside of (or within) closed shells.
20
Adding rnore par:ticles presents a huge arnount of computational
labor compounded by the fact that configuration mixing, including
configuration i.nteractions between protons and neutrons, and with
the core, has a strong effect on the quantitative spacing of the
states and is not a feasible approach" The Bul34 nucleus with
Z -- 56 and N = 78 has six protons outside of a closed shell
and
four holes in a closed neutron shell producing a fairly cornplicated
systern.
Proper configuration rnixing is able to bring the theoretical
and experirnental rnagnetic mornents into agreernent. In general
the rneasured quadrupole rnornents agree very poorly with the shell
rnodel predictions. Near closed shells sorrle nuclear quadrupole
rnornents can be accounted
for with the aid of suitable configura-
tion rnixing.
In the rare earth region the quadrupole rnornents becorne
as rnuch as 30 ti.rnes the single particle values. This is a difficul-
ty which leads to the c:onclusion that collective effects are irnportant and that it is not a valid approxirnation to consider nucleons
as moving independently in a spherically syrnrnetric potential.
Nuclear rnornents are sensitive to configuration rnixing and collec-
tive effects so that an independent particle rnodel does not give
very good values for: garrrrra ray decay probabilities"
The
2l
formuLas for these single particle transition rates are simple,
however, and are often used as reference values for cornparing
experirnental data. The results apply quite well for low A. Elec-
tric quadrupole transitions are usually strongly
enhanced over
the si.ngle particle values in both light and heavy nuclei which again
leads to the collective notion and the large collective quadrupole
rnornents. ltt is not possible then to attribute all nuclear properties to the nucleons in unfilled shells.
It is necessary to consider correlations in nucleon rnotion
as exernpl"j.fied by configuration rnixings, and an interaction with
the cLosed-sheII configurations in the core rnay also be irnportant.
The shell rnodel has its basis in an average static field generated
collectively by all the particles but a rnore cornplete description
rnust include the variations of the field associated with collective
oscillations. Relatively srnall residual interactions introduce irnportant correl"ations in the rnotion of the particles outside of closed
shells. The energy spectra of nuclei with rnany-particle configurations show features which -vary in a systematic way frorn nucleus to nucleus. These regularities are associated with the fact that
a rnajor part of the correlations between particles rnay be des-
cribed in terrns of ordered collective rnotion of the nucleons corresponding to variations in the shape of the nucleus, for exarnple.
zz
This leads to a generaltzation of the sheIl rnodel in which the bi.nding field is no longer considered as a static isotropic potential,
bu.t
rather as a variable field which rnay take shapes di"ffering
from spherical syrnrnetry.
The
first problern is to detenrnine the nuclear equilibriurn
shape. The equilibriurn
shape and the character of the col.lective
rnodes of rnotion of the nucleons firay be understood as the result
of the cornpetition between the deformirrg tendency of the individual
nucleons and the effect of the pairing forces. The pairi.ng forces
tend to couple two eqr:.ivalent nucleons to a state cf zeto total an-
gular rnornenturn, that is,
a. spheri.caLly
syrnrnetric state. In the
region of closed shell,s these forces dorn'Lnate and the nu.clear
equilibriurn shape is spherical. The addition of nue leons in un-
filled shells j.ncreases the irnpcrtana;e of the tendency to deforrnation, a coherent effect of all these nucleons. The
rir1cleus can ac-
qui.re a nc,nspherical equilibrir:rrn shape and possesses a large
quadrupole rnornent" The quadrupole rnornents are srnall in the
region of closed shelis and becorne larger with the addition of
more particles. These trends can be accounted for then in terrns
of the tendency of the partic)"e stru,cture to produce coilective deformation of the nucl"eus. For
suc:h
nuclei the colk,ctive rnotion
separates i.nto rot.ational and vib,rational parts. The fonrnr:r
23
corresponds to a rotation of th.e nuclear orientation at constant
shape, while the la.tter corresponds to oscillations about the aniso-
tropic equilibriurn shape. The rotational concept arises frorn the
fact that the correlated particle motion rnay tre such that the pat-
tern of particles changes slowly rnaintaining a fixed shape but with
the orientation in space altering. Looked at fr"orn the outside, the
nucleus of constant shape is rotating. It is natural to attribute
kinetic energy and angular rnornenturn to this rotatirrg rnass and to
consider the rest of the nuclear kinetic energy and angular rnornentum to be internal, The equilibrium shape of nuclei can be obtained by calculating the energy of the nucleons as a function of the
shape of the
field and finding the
shape that gives rninirnurn
energy.
For the description of nuclear structure then, the dynamics
of the nucleus is considered in terms of collective and intrinsic
rnodes of
excitation. The collecti.ve rnodes are a.ssociated with de-
forrnations and the lowest rnodes clf this type correspond, in the
case of nuclei with sufficiently many pa.rticles outside of closed
shells, to rotations of th.e spheroidal shape. The states produced
by coller:tive vibrati.ons are at a.n ener:gy' of 1 or 2 Mev in heavy
strongly deforrned nuclei (A : 250) and increase in energy as A
decreases. For nuclei
ne-'.ar
closed shells the e ollective rnotion
24
corresponds to vibrations about the spherical equilibriurn shape.
Experirnental evidence indicates the absence of rotational states
in spherical nu.clei at least for low excitations. The intrinsic
rnodes represent the rnotion of the nr;.cleons in a fixed
field corres-
ponding to the nuclear equilibriurn shape and have an average spac-
ing of -100 Kev. This rnotion is subjected to the influence of the
pairing forces and in the case of odd-A nuclei the low lying intrin-
sic states rnay often be described in terrns of excitations of the
last unpaired nucleon. For configurations sufficiently far rernoved
frorn closed shells the nucleus acquires a Iarge equilibriurn deforrnation resulting frorn the deforrning effects of rnany nucleons and
an approxirnate solution
is obtained by considering first the rela-
tively fast rnotion of the particles with respect to the deforrned
nuclear field considered as fixed in space, and then the relatively
slow vibration and rotation of the entire systern. If the coupling
of the intrinsic and collective rnotion is srnall, the problern is usually treated by perturbation theory. A considerable body of data
reveals the existence of a rotational structure in the nuclear exci-
tation spectrurn. The sirnple rotational motion is characteristic
of the strongly deforrned nuclei and is thus especially well defined
in regions far
r:ernoved
frorn closed shells. The rotational struc-
?5
distances between shell closings are great and where the deforrna-
tions ar:e especially large. In this region this approach has rnet
with good success in describj.ng the low lying states of even-even
nuclei. Mention will be rnade of three such rnodels which atternpt
to describe the low lying states in even-even nuclei with the Parti-
cular aim of fi.nding a suitable description of the excited states of
gul34.
One of the
first collective rnodels of thip type useful inde-
scribing nuclear structure was the Bohr-Mottelson rnodel (9, P.
1).
This rnodel assurnes that rnost of the nuclear deforrnation resides
in the cloud of nucleons outside of closed sheIls although its influence tends to deforrn the core
slightly. In addition only nuclear
shapes which ha.ve axial syrnrnetry are considered. This ellip-
soidal nucleus can then, in addi.tion to vibrating, also perforrn
rotations producing rotational spectra. These collective rnotions
produce excited states at energies lower than the particle excita-
tion energies. The rnornents of inertia of the nucleus which apPear
in the rotational Hamiltonian are calculated in terrns of one pararTteter assurning that the nucleus behaves as an incompressible
rigid body. For two equi.valent particles both the particle forces
and the coupling to the deforrnation forrn a ground state particle
configuration of zero arrg:ular rnornenturn so that in even-even
26
nuclei the ground state particle configuration has angular rrlomenturn zero. If this configuration is not excited the low iying rota-
tional leveIs of a syrnrnetric nucleus rotating about an axis perPen-
dicular to the nuclear syrnrnetry axis are pure and are characterized by a rotational quanturn nurnber J. The possible rotational
quanturn states of the nucleus are restricted by the reflection syrn-
rnetry of the deforrnation which restricts the collective states to
even parity and excludes odd values of
J. Since the ground state
particle configuration in even-even nuclei possesses even parity
the collective states for this particle configuration will have even
parity. For even-even nuclei the lowest rotational
and
I
'
-1T
parities
J"
= 0',
I
L
Z' , 4'
r
, 6' ,
band has spins
at energies of
l'rZ
Erot=7;--J(J+l)
where I = (Z/5)MA(AR)Z is the rnornent of inertia for a spheroidal
deforrnation of the incornpressible rnodel, M is the nucleon rrtass,
A is the nuclear rnass nurnber, and AR is the difference between
the rnajor and rninor serni-axes of the spheroid. Such a rotational
band can be
built on an excited particle state or a vibrational state.
Very high excitations are thought of as vibrational excitations of
the core.
For highly deforrned nuclei which possess rotational states
the garnrna ray transition probabilities to states of the sarne
27
family obey simple relations. For electromagnetic transitions of
multipole order L the ratio of the reduced transition probabilities
is the square of the ratio of two Clebsch-Gordan coefficients.
These probabilities depend on the static mornents of the nuclear
states. For low energy transitions on the particle rnodel MI radiation will strongly predorninate over the E2 when both are possible.
The E2 transition probabitity depends upon the square of the quad-
rupole rnoment so the collective deforrnation leads to an enhancernent of the E2 probability which rnay lead to an appreciable ad-
mixture of E2 in laf l - I transitions. The E2 transition proba-
bilities from the excited, ?* state to the ground state in some eveneven nuclei have been
determined. The quadrupole rnoments cal-
culated frorn these show the expected trend, decreasing regularly
with the approach to closed shell configurations.
This type of rotational description works weIl in the re-
gion 160<A<185 and A>Z?5. Many even-even nuclei show evidence of a rotational structure but with a rrrore cornplicated col-
lection of states than the simple 0*, Z*, 4*, 6+,
quence predicted by Bohr and
rotational se-
Mottelson. For exarnple, there are
r++
often two l ' states, two 4' states, excited 0' sta.tes, and states
with odd J. These states can sornetirnes be fitted by building
rotational band on a suitable vibrational or particle excitation.
a
28
This can be done assurning a rotational band of an axially syrnme-
tric nucleus built on a vibrational state if the ratio of the energy
of the first 4t state to that of the first 2l state is greater than
3.27. In BaI34 tti" ratio is Z.3I and such a fit cannot be rnade.
It would be convenient to develop these states naturally as rrrernbers of the g::ound state rotational band and also to be able to fit
those states which cannot be explained with this model.
Davydov and Filippov (14,
p.
237) describe the energy lev-
els of even-even nuclei in terms of rotations of a deforrned nucleus which do not involve changes of its internal state. The re-
strictionof axial syrnrnetry is rernoved and it is found that this affects the rotational spectrum of the axial nucleus slightly although
additional rotational states of 2+, 3*, 4+,
appear. If the de-
viations frorn axial syrnrnetry are srnall these leveIs lie very high
but becorne lower as the deviations frorn syrnrnetry are increased.
tr'or exarnple, two 2* states appear and the rati.o of their energies
varies frorn infinity to two depending on this asymrnetry.
The
three rnornents of inertia are not independent but are deterrnined
frorn two pararneters, one of which rneasures the deviation frorn
axial syrnrnetry, and the other characterizes the deforrnation.
For this reason the approach is sornetirnes referred to as the restrictedasyrnrnetric rotor rnodel. The positions of the energy
29
levels are also deterrnined in terrns of these parameters. The
value of the pararneter describing the deviation frorn axial sym-
rnetry is restricted to a certain range. In other language this re-
striction all.ows fi.tting of energy levels with this rnodel only when
the ratio of the energy of the first 4* state to that of the first 2*
state is greater than 2.67 136, p. 553). When this condition is
rnet the Davydov-Filippov rnodel explains the data on energy levels
quite well for even-even nuclei far from closed shells with a few
exceptions. In Ba"n ,n. above condition is not rnet, since the
ratio is
Z"
3L, so that a consistent set of pararneters cannot
be
found with whj.ch to describe the known nuclear energy leve1s.
Ratios of garnrna ray reduced transition probabilities for E2 and
MI transitions between
sorrre of these rotational states have been
calculated in terrns of the pararneters of the rnodel. The agreernent with experirnental ratios
of.
EZ reduced transition probabili-
ties is very good in a few eases. The general trends of the ratios
of garnrna ray transition probabilities as a function of the ratio of
the energies of the states are predicted correctly. The results
predicted by thi-s model are, in general, in better agreelrlent with
experirnental results for even-even nuclei than are those given by
the Bohr - Mottelson rnodel.
For not too strongly deforrned nuclei the rapid rotational
30
motion will tend to distort the shape of the nucleus, that is produce
a centrifugal distortion. This effect is termed a rotation-vibration
interaction.
Such a
rotation-vibration interaction has been intro-
duced in the Davydov-Filippov model and results in improved agree-
ment with experirnental data for nuclei far away frorn closed shells
(36,
p. fi61.
An attempt to extend the validity of this rotational descrip-
tion has been presented by Mallrnann (36, p. 535). In this rnodel
aIso, rotational modes of excitation of the nucleus are considered.
The nucleus
is again considered to be asyrnmetric but in a more
general way in that the three rnoments of inertia are treated as independent parameters to be determined by cornparison with experi-
ment and no atternpt is rnade to determine them frorn first princi-
ples. A rotation-vibration interaction is included in the Mallrnann
rnodel as a perturbation. It is also assurned that the rotational
rnotion of the even-even nucleus can be treated quasi-adiabatically.
That is, it is assurned that the excited states due to other degrees
of freedom have energies appreciably higher than the energies due
to the rotational motion even though the particle structure will
be
perturbed because of the inability of the particles to follow the ro-
tations completely adiabatically. The departure of actual nuclei
from this behavior is assumed to be srnall and can be accounted
3l
for in terrns of a srnall perturbation. This rnodel is then referred
to as the quasi-adiabatic general asyrnrnetric rotor rnodel. A
general asyrnrnetric rotor rnodel without the rotation-vibration
correction has been considered but did not result in irnproved
agreement with experirnent over the restricted asyrnrnetric rotor
rnodel (36, p. 5361. The Mallrrann rnodel has as pararneters the
three nuclear rnornents of inertia and the strength of the rotation-
vibration perturbation which is added as a first order correction
to the pure asyrnrnetric rotor energy leve1s. With this approach
the cornparison of the predicted levels with experirnentally known
energy leveIs in even-even nuclei in the range 40 < A < 250
shows very good agreernent, usually within experirnental
error.
It rnust be pointed out however that in alrnost every case other
experirnental levels exist which do not fit well in this or other
rotational schernes. This suggests that the general asyrnrnetric
rotor Harniltonian is an essential part of a more cornplete Hamiltonian which includes other degrees of freedorn. The energies
of the two excited,2* levels and the two 4* levels in BaI34 "ho*r,
in Figure I are fitted very well as rotational levels with this
rnodel. The 15?0 Kev, I641 Kev, and I770 Kev levels i.,8"134
cannot be unarnbiguously accounted
for in the ground state rota-
tional band. The garnrna ray transition probabilities are
iz
calculated by Mallmann with the rotation-vibration interaction
omitted. The rotational states all have the sarne parity so that
the norrna.l selection rules forbid El, E3,
, jldz, M4,
transitions. Additionally now the rnodel forbids MI transitions
states. A survey of the experimental E2
I2
tt".r"ition in some even-even
to MI rnixing ra.tio in the 'r*
between the rotational
nuclei (36, p. 559) Iends support to this selection rule except
near closed shells. This leaves EZ, M3, 84, M5,
transitions
as possibilities of which only the EZ transitions are considered as
the oth.ers are too slow to cornpete favorably. The calculated garnrrla ray transition probabilities and ratios of garnrna ray intensities
between the rotational levels of even-even nuclei assuming pure E2
radiation are compared to experirnentally deterrnined values for
110
< A <
250
(36, p. 5661. The agreernent of the theory with
experirnent is within experirnental error with a few exceptions.
Although the energv Ieve1s of rr knownlt angular rnornenturn and
parity are well described by the rnodel the calculated intensity
134
ratios in Ba--- do not, in general, agtee well with the experimen-
ta1
values. Table 3 displays the cornparison. The notation 'r-,
for exarnple, rneans the second state of angular rnornenturn
Z
reached rn order of ascending energy. It is observed that the two
ratios departi.ng the rnost frorn the rneasured values are both low
and are those which contain the 559 Kev and 553 Kev radiations
33
Table
3.
THEORETICAL AND MEASURED GAMMA RAY INTENSITY RATIOS FROM ROTATIONAL STATES tN gal34.
Transition
ratio
z,
m
m
*'z
24
ModeI
prediction
Measured
1.5
7.0
2.7
2.I
0.35
2.3
563
m8
24 *zZ
24
Energy
ratio (Kev)
802
Bs
s69
-r4
*lz
TI_s
each of which
is a transition between states of the same angular
momenturn.
Of the various rnodels discussed it appears that the rnodel
proposed by Mallrnann is the one best able to describe the states
of known angu.Iar rnornentum and parity in the 8^134 nucleus and
its predictions will be cornpared with the experirnental results
of
the directional correlation experirnents as a further test of its va-
lidity for this nucleus, particularly with respect to the additional
selection rule irnposed on garrrrna ray transitions.
34
THEORY OF DIRECTIONAL CORRELATION
The probability of ernission of a particle
or quanturn frorn
a nucleus depends, in general, on the angle between the nuclear
spin (angular rnornenturn) axis and the direction of ernission.
Under ordinary circurnstances, however, the total radiation frorn
a collection of radioactive nuclei is isotropic because the nuclei
are randornly oriented in space. An anisotropic radiation pattern
can be observed only frorn a collection of nuclei which are not ran-
dornly oriented. The nuclei can be oriented in space by placing
the sarnple at low ternperature in a strong rnagnetic field. As
a
result of the interaction between the rnagnetic field and the nuclear
rnagnetic rnornent alignrnent will take place. The angular distribu-
tion of the ernitted radi.ation is then rneasured with respect to the
direction of the applied field. The interaction of an electric field
gradient with the nuclear quadrupole rnornent can also be ernployed
to produce an orientation of the nuclei in space.
Another rnethod consists in picking out only those nuclei
whose spins happen to lie in a preferred direction. This case can
be realized experirnentally if the nuclei decay by successive ernission of two radiations , L arrd Z, through a sufficiently short lived
interrnediate state. The observation of I in a specified direction
35
then selects an ensernble of nuclei which h.as a nonisotropic distri-
bution of spin orientations. The foll-owing radiation, Z, ther.
shows, in the usu.al case, a directional correlation, or nonisotrop-
ic radiation pattr:rn, with respect to radiation 1. This is the technique ernpi.oyed in the present experirnent.
Figure 2 shows the essentj.al aspects of a directional correlation experj.rnent. The
angu1ar:
correlation function W(+) is
the proba.bility that if r:adiation I is ernitted in a direction R,I thrt
radiation 2 wj11 be ernitted in a direction *-Z at an angle + with
respect
to R,.
,l
Experirnentally the nurnber of coincidences be-
tween radiations I and 2, N(e), is determined as a function of e,
the angle included by the axes of the two counters. Because of
the finite solid angles of the detectors these nurnbers N(0) are
averages of the tr:ue correlation W(+) over a range of angles A0
distributed around 0. As a result N(0) rnust be properly corrected and norrnaLized to
yreld W.*r(S). The cornparison of
W"*O(Q)
with the theory finally gives the desired inforrnation about the
properties of the nuclear leveIs and the radiations.
It is necessary to understand why there is a correlation at
all, that is, why the coincidence
counti.ng
rate depends on
+.
Irnagine a transition in which radiation is ernitted frorn a state
characteri.zed by quanturn nu.rnbers j, rn leading to a state jt
,
36
THE EXPERIMENT
THE CASCADE
Coincidence
Circuit
I
N(e)
THE THEORY
THE RESULT
L
.2
i
i
r
L
+
(D
z
I
L
=
I
L
I
180
13s
e
Figure
2.
1
.o
I
90
135
0
THE ASPECTS OF A DIRECTIONAL
CORRELA TION EXPERIMENT
37
rnt
" The radiation will have angular rnomenturn L, projection
quanturn nurnber M = rn
- rnl ,
and
will or will not result in
a
parity change between the levels depending upon the type of radia-
tion. The parity
change
is deterrnined by a factor (-1)L for elec-
tric radiation carrying angular rnornenturn L, and (-1)L*I for rnagnetic radiation carrying angular rnornenturn L. Several L values
ranging frorn l:-:' J to j+jt rnay be possible but the rnultiptricity
of L values is of no irnportance for this argulnent so the discussion will be in terrns of a single value of L. In practice the rnag-
netic substates of the level j are equally populated so that transitions are observed frorn all substates m to all substates rnl .
The angular distribution of the ernitted radiation, for given M,
depends
on +, but the surnrnation over all substates (or surnrna-
tion over M) is independent of +. Therefore, the radiation observed in a single transition frorn a non-oriented source is isotr opic.
For the observation of the two radiations in coincidence
the situation is changed. For a fixed M the radiation is not iso-
tropic because a particular direction i.n space is being singled out'
This is the quantization axis with reference to which M, as well
as rn and
rnr
, is rneasured. If of all the radiation ernitted in
aII directions only that proceeding in a specified direction is
38
observed, then this is equivalent to selecting a particular set of
M values. Further, if the axis of quantizatirrn is chosen to lie
along the direction of ernission of the first radiation, M will be
+1 with M= 0 ornitted. This is a result of the fact that a transverse wave corresponds to a photon which can have intrinsic spin
only parallel or antiparallel to the propagation direction and that
only the zeto component of the n orbitalll angular rnornenturn is
radiated in the z*direction. That the sarne conclusion can be
reached frorn classical field theory is dernonstrated later on although the separation of the angular rnornenturn into an
rr
intrinsic
spinrr and an n orbitaln part is not rnade. The result is then that
not all values of M are surnrrred over and in the final state not all
rnr values are equally populated. If this final state for the first
transition is also the initial state for the next transition, then the
ensuing radiation is anisotropic by virtue of the fact that it orig-
inates frorn unequally populated substates.
The general theory of angular correlation was established
by Harniiton (26, p. IZZ). Further developrnent and refinernent of
the theory has resulted rnainly frorn the work of Harnilton (?7,
p. 7821, Lloyd (34, p. 716), Alder (2, p. 235), Racah (4?., p.
Falkoff (1!, p" 98) and Goertzel lZ3, p. 897). Biedenharn and
Rose have presented a detailed review
article of the subject
910),
39
(7, p.
7 29l."
The theory of directional correlation can be sumrnarized in
the following way. Let the initial nuclear state be described by a
state function (j, -11, the interrnediate state by (i *1, and the
final state Uy (iZ^rl.
The
jrs are the angular rnornenta of the var-
ious nuclear levels, and the rnrs are their projections on the quantization axis. It is assurned that the perturbation of the interrnediate
state by extra nuclear fields is negligible. This will be true if the
lifetirne of the interrnediate state is srnall cornpared to the nuclear
precession period produced by an external coupling. An application
of second-order perturbation theory (44, p. 168) gives the prob-
abiiity that a nucleus decaying through the states I * i * ? will
ernit radiation I in direction RI followed by radiation
Z
in the di-
rection i.Z. This probability is the correlation function Wt*rEr)
given by
w(\K.) = s
\-
rn)(i rnlHr lir,r,r) l'
lrr.,
L
rlHzli
*1*z-
(r)
is the interaction
"t
Harniltonian responsible for the ernission or absorption of radiation
where constant factors have been ornitted.
I and H^Z-is that for the ernission or absorption of radiation 2. The
factor S indicates that a surnrnation is also rnade over unobserved
quantities, for exarnple, the state of polarization of the radiation.
40
Equation (I) rnay be brought into a rnore usable forrn by carrying
out sorne simplifying rnanipulations. First the rnatrix elernents
are split into multi.pole cofilponents according to the L, M,
and
parity of the ::adiati.on. A transforrnation is then perforrned rotating the coordinate systern of quantization over into the coordinate systern of the
radiation. fhe rnatrix elernents of distinct rnul-
tipole orders are then split into geornetrical factors and reduced
rnatrix elernents using the .Wigner-Ec.:kart theorern (44, P.
85).
This gives the dependerrce on the rnagneti.c quanturn nurnbers and
hence on geornetrical factcrs separately by rneans of Clebsch-
Gordan coefficients. Finally, with the help of relations among
the Clebsch-Gordan coefficients (44, p. 35) the total exPression
can be reduced by rneans of Racah algebra (44,
P. L73). The re-
sult for the correlation functj.on for pure rnultipole transitions is
t'ma*
w(S)
=
tL
A P (cos
lzl
Q)
v=0
where v takes on only even values and t,,."* is given by
s
rnalle s t of. Zj, ZLL,
ZLr. Pr.(cos
Q)
is the Legendre polynornial
of order v. The coefficients A' are given by
A
v - Fv(LlilJ)Fv(Lzizi)
the
4T
where
i,- j - I
F (Ljlj) = (-) '
(zj
+rf /zrrl+ r)c(LLv; l, -1)w(jjLL;vjr).
C(LLv;1, -tr) is a Clebsch-Gordan coefficient with 7= i +iand
jLL;
coefficient
r) is a Racah coefficient. The Racah
W(abcd;ef) is zero unless each of the three triangle rules A(abe),
\M(j
vj
A (cde), A(acf), and A(bdf)
are satisfied. It is this fact that deter-
rnines the restrictions on trrrr* in Equation (2). Tabulations of
the functions
",.
are available (7, p.
7461 f.or
various
j values. Thus, the constants Ar. in the expansion
",
jI,
and
depend on
five pararneters: Ll, LZ' iy j, ir, so that a comparison between
the theoretical and experirnental correlation functions can give in-
forrnation about these pararneter s.
If one of the radiations in the cascade consists of a rnixture
of different multipole orders, I if (Lf Li) i &Z) iZ] , the correlation
function has the forrn (7, p. 747\
w(Q) = wr(+) + 62 wrr(Q) + zo
wm(+).
(3)
62 is the ratio of the total angle-integrated intensity of the Llpole to that of the Lr-pole radiation. In general, the real rnixing
ratio 6 is defined as the rati.o of the reduced rnatrix elernents for
the Lr pole to the L pole radiation,
42
(jllL'll:,1
O-
ullL ll:,t
wI(+) and w,,(+) are given by Equation
wT(+) =
,;
F,.(L.j
rj)
) "rrt"ri ri)
wrr(Q) =
l'[
),u
(2).
p.r(cos 0)
j) r-rtr,rj rj) Prr(cos s)
"rt"[j,
Wm(+) is the contribution due to the interference between L,
and
Li and is given by
wrrr(+)
"-
r
LFvlLrLijri)
Fv(Lzjrj) Pr.(cos s)
v
with
1,(\Hiri) = (*)i-ir-'[(.i + l)(2L+I)(zlr +r11r/z
x G,.(L,Li
itj)
and
G..(lLijij) = c(LtLlv;
The
1,
-I)w(jj
"r"i;
vjr).
function Gv(LLt jlj) is tabulated by Biedenharn and Rose
(7, p. 749) f.ox the case Lr - L + I.
The expiicit forrn of W(+) for the case in which both radia-
tions of the cascade are rnixtures of two rnultipoles is given by
Rose 146, p. 477|"
43
It is possible, because of the finite energy resolution
of
the detecting sy'stern, that two or rnore correlations frorn different cascades consisting of radiations of nearly the same energy
rnay contribute to the experirnentally measured function. The
theoretical expression for such a comPosite correlation rnay
be
found by incoherently cornbining the separate contributing functions
with proper weights. These weights will be deterrnined by the rel-
ative garrurra ray intensities of the radiations involved.
In the event of a cascade of the form jO(\)jt(LZ)j Z(.L3li3
a directional correlation between the first and third radiations,
the I'cross overrt correlation, can be fileasured with the interrnediate radiation, LZ, unobserved. The correlation function is
given by Equation (2) with
Ar. = Frr(Ltioit)Fv(L3i
,rrrW,
so that the coefficients depend upon the angular nlomenta of the
states and upon the rnultipolarities of the radiations, including
that of the unobserved radiation. If either of the observed radia-
tions is rnixed, the correlation function is calculated as indicated
above
for the ordinary correlation, Equation (3). If the inter-
rnediate unobserved radiation is rnixed the correlation function is
44
given by
w(s) = w(Lz) +6zw(L:)
where 62 is the ratio of the intensity of the Lt-pole to that
of
the L-pole in the interrnediate radiation.
Another quantity often used in angular correlation work is
the anisotropy A defined as
A=
''r
-
w(I8o) -w(go)
-
wiE--
If the highest coefficient in the expansion for
becorne
s
lzA
z
8 - 4A
+ 5A4
z+
3A4'
1M(+)
is An this
45
THE ANGULAR MOMENTUM OF
ELEC TROMAGNETIC RADIA TION
In order to calculate the angular rnornenturn carried by
the electrornagnetic field it is convenient to use the spherical wave
solutions for the field (28, p. I13). The solution of the scalar
wave equation (the Helrnholtz equation) in spherical coordinates
can be wri.tten
*(*) =
I, trtr "j'J"f ')e"l + elra n(rz)(o,)l yr m(0, +)
(r
)
!-
are spherical Hankel functions and k - ,/c.
I
The coefficients Orr., will be determined by the boundary condi-
where the njt'z)
tions. The Yl,,,(e, +) are the spherical harmonics which are solutions of the angular part of the wave equation in spherical coordi-
nates. In operator notation this can be written
?
L-y"
lrrl
=
!(t
+ 1)y"ltTr
where Lz = Lzxyz
+ Lz + LZ
L=
*
a,,d
L=-i(rxV).
Defining
xy+ iL , the following equations hold:
L__
LY
+ lrn
Yl rrr*,
,
(?)
LY=rnY
"z- lrn - "'tlrn'
The orthogonality of the spherical harrnonics
(-
J "fu Yl'..,' & =
6l
!.' 6*rrr"
is expressed
by
(3)
46
In a source free region, assurning harrnonic fields,
Maxwellrs equations are
V x E=ikB
+
V 'E=0
vx
++
V
+
B=_ikE
(4)
B=o
Frorn these equations it can be shown that E and B separately
satisfy the vector Helrnholtz equation
?
)-
(v-+k")F=O.
The spherical wave rnultipole solutions for 6 and E
(5)
"rn
be ob-
tained frorn Equation (5) which indicates that each rectangular
cornponent
++
of E or B satisfies the scalar Helrnholtz equation,
so each cornponent can be represented by an exPression of the
forrn Equation (l). These can be cornbined to yield the vectorial
result, for exarnple, for C.
E = )L t rrtllnl')ru'l *Ila r',12)tt'll yrm(e, Q)
!.
(5)
rrn
In evaluating the coefficients frr.. the divergence condition on B
rnust be satisfied. This leads to a condition that the E field is
transverse to the radius vector, resulting in a special set of electrornagnetic fields,
Er.r, = f, (kr) r tro, (e' +)
*
+
i
F
x
B
---V
"lrn
"lrn - k '
(7)
47
where
r, (kr) = Af ')nlt)to"l + af z)r,f ')(o").
(8)
These represent solutions in which the rnagnetic field is perpendic-
ular to the radius vector and are called electrrc multipole fields.
The calculation for E, instead of B, gives an alternative
set of rnultipole fields in which E is perpendicular to the radius
ve
ctor.
f lrn
f, (kr)
.i*
-'V
k
B
lm
" "rrr,
x
E
(0, +)
(e)
!.rn
These are called rnagnetic rnultipole
fields. These two sets of
fields forrn a cornplete set of vector solutions to Maxwellrs equations (28, p. IZ0).
The vector spherical harrnonics are defined by
x, (e, +)
L
IITI
Y"
-YIYT
(r 0)
(e, +)
with *O = 0 and the orthogonality property
('*
+
J*i"r,,,' Xrr, dQ = 6rr, 6rrrr.r,
(t r)
The general solution of Maxwellls equations will be a cornbination
of the two types of fields.
E=
r+
LIa"(1,
l, rn
-f -- \-. i
*)fl (k')*m - f a*(1, rn)v x g, (kr)irr,,]
Lt f ""(1,
!"
,rrt
.. .=?
rn)v x f, (kr)Xrr,,
+
a*(l
(I2)
, rr-lst
lt"l*lr-l
48
The coefficients ar(l
.
rn) and a*(1, rn) specify the arnounts of
the electric and rnagnetic fields, respectively, and will be deterrnined by the sources and boundary conditions. The radial func-
tions f, (kr) and
S,
(kr) are of the forrn Equation (8).
The rnultipole fields of a radiating source can be used to
calculate the energy and angular mornenturn carried off by the
radiation. For exarnple, consider an electric rnultipole field.
From Equation (12), the fields which satisfy the boundary conditions of finiteness at the origin and outgoing waves at infinity are
Errr, = arlt,r.)hlI)1t .)*rrr, .-i<''rt
(ts1
i x *B
Elrn - jv
k
lrn
The tirne-averaged energy is
Considering the radiation zone where the two terrns in the inte-
grandare equal on the average
u
=
and. using Equation (13)
* Jl"",r,'n)l'lnl')(o")lt*r,ni;.
av.
for B,
(L4al
With the orthogonality relation for the vector spherical harrnonics,
Equation ( I I ), thi s bec orne
u = #Jl"rrr,")
s
l'lnl' )(r.") I z ,zd.r.
(l4b)
49
The tirne-average of the Poynting vector, giving the energy flux
in the field is
-+
S
= 8n
^: Re(E x
B'k).
==)
The average Erorrrenturn density is then S/.", so the time-averaged
angular rnornenturn carried by the field is
+lC_++
tul =
,ft
J
*"tr x (E x B'r')l dV.
Expanding the triple vector product and recalling that ?'E =
O
for this electric rnultipole fieId, the integral becornes
Substituting for E frorn Equation (13) this becornes
+tf_++
M = :8no \
J
n"[g,k(r x _iV).B
Using the operator identity i
I
c
14 = i
8n rr.: J\
++
dV.
= -ii-V
nuIn,:.1f,.e)]
the integral can be written
dV.
Using the expression for E frorn Equation (13),
fr
=
*-
J 1"",r, *) I 'lnl')(o,) lzn.r*;;f ir,-lu,r.
i'*"
Frorn the definition of *,,TITT
, (*:1.
,(M
TM
iY, IITT ),
) = (Y'i:
ITr}
so
ft = # J lr",r, *) l21r,f l)tr.') l2n"(yi-;iyr.,y"zd"d
Equations (2) and (3) show that only tine z-component of il
n.
exists.
50
Mz
--
# J Iar(1, ") I 'lnl')(or)lzrnRe(Yf' YrJaorzar
M,
=
# t lar(r,*)l'lnl')(r")lz,zd,
For a rnagneti.c
(1, rrr)
(15)
multipole, a"(1, rn) is replaced by
ar(/, rn). Comparing Equation (I4b) for the energy with Equation
(15),
M
z
U
rn
m?r
r., - tu,-r '
^'rr-= -1, -l +1,
,!
This is interpreted quanturn mechanically to rnean that
radiation frorn a rnultipole of order (1, m) carries off
corrrponent of angular rnornenturn
t}:,e
z-
mtr per photon of energy
?ro.
Furtherrnore, since according to this serniclassical calculation,
only the z-cornponent of the angular momenturn exists, the ratio
of the square of the total angular rnornenturn to the square of the
energy is
MZ
ITT
-=
U
-o)
Z
This result arises frorn the above calculation using classical fields.
Quanturn rnechanically
it rnight be expected that iZ = !.(!.+IlhZ,
as
is indeed the result given by a quanturn electrodynarnical calculation. A quantized field can be introduced by treating the amplitudes ar(l, rn) (or a*(1, rn)) as quanturn rnechanical operators
5I
which create and destroy photons of type (1, rn) (41, p. 2971. The
calculation is then perforrned in the same general way as in the
classical case. DeWitt and Jensen (16, p. 268) have perforrned
the calculation for a quantized rnultipole field containing N photons
and find,
L-z - Nmh, lz = {*'..,'+ N[l(t+l) - t,-\]hz
and
2?
Ui = (hN<,r)"
rn - -!, -!.+L, ......,
!.
where I is the rnultipole order of the radiation.
For large N (the classical case) this result gives
lrrn -ZZ
L
m
N*oo?= J
agreeing with the classical calculation. For a single photon,
however, the rigorous treatrnent shows that
*)?
L" = l(l + 1)h"
L z =rnh
for the square ar.d z-corrrponent of the angular rrlorrlenturn carried
off by a photon of the rnultipole field (1, rn).
The question to be answered in connection with a directional
correlation experirnent concerns t}l.e z-cornponent of the angular
rnornenturn carried off by the
first
garnrrra
ray. From the forrn of
the fields, Equation (I2), the angular dependence of the Poynting
5?.
vector will be given by l*rrrte, +) I 2. The direction of propagation of the first radiation is definedas the z-axis, that is, the di-
rection e - o. irrrtt,S) has the property thatir*t0,0)
for all I.
=6rn,* I
Therefore, energy will be transmitted in the z-direction
only when rrr = * I and will thus carry z-component of angular
mornentum of *h and no other value. Therefore, even this sernicl.aseical argument shows clearly that the choice of the direction
of propagation of the first radiation to be the z-axis fulfills the
conditions which are reguired to see an anisotropy, namely, that
the populations of the substates of the interrnediate state will not
be equal.
53
SOURCE PREPARATION
In order to obtain good statistics in the true coincidence
rate, the chance coincidence rate should be kept as small as possible compared to the true coincidence rate. The counters rnust
be
positioned so as to subtend a large solid angle and a source of ap-
propriate activity ernployed in order to rneet this condj.tion and stil1
get a sufficient nurnber of
true coincidence counts in a reasonable
tirne. Consider a source which ernits two garnrna rays in
and whose strength
cascade
is No disintegrations per second. The nurn-
ber of single counts recorded in each channel per second rnay
be
written as N.1 = N011
w.e. where w,1 is the solid angle subtended by
counter i at the source and .i is the detection efficiency of chan-
neI i for quanturn i. The nurnber of chance coincidence
counts
per second is given by
?
N.h = ZTNrN2 = ZTNo-wr*Z"l"z
where
Z'I
apparatus.
is the experirnentally determined resolving tirne of the
The nurnber of true coincidence counts per second is
Nrr,r" = No*l*rw(S)"re,
where W(0) is ttre correlation function.
dences to chance coincidences
is then
The ratio of true coinci-
54
N
true
w(+)
N_zTN
cho
Frorn this equation then, for a given ZT, No rnust be as srnall as
possible (consistent with a sufficiently high coincidence countino
rate) in order to rnake the true-to-chance ratio as large as possible.
The size and surroundings of the source are very irnportant
in angul"ar correlation work. If a quanturn is scattered in material
near the source it rnay lose its original direction and srnear out
the correlation function, an effect which rnay be reduced by decreasing the thickness of the source and of the walls of the source holder.
If the source is not cylindrically syrnrnetric about the rotation axis of
the rnovable counter the absorption can be angularly dependent. On
the basis of expressions presented by Aeppli, et al. (1, p. 339) for
a thin cylindrical source surrounded by scattering rnaterial,
Frauenfelder (21, p. 149) gives the critical source and absorber
thicknesses for various rnaterials. For a lucite container for 500
Kev garnrrra rays, the critical waIl thickness is about 2.00 rnrn.
The walls of the lucite containers used
in this experirnent were
0.79 rnrn thick. The critical source thickness is on the order of
1.00 rnrn. The thickness of the source used in this experiment
was 0.79 rnrn. Scattering in sources of Iess than critical thickness
reduces the ani.sotropy by less than a factor of 0. !$ (2], p. 150)'
55
The syrnrnetry of the source was confirrned by observing the sin-
gles counting rates a.t r.arious angles.
The 2.3 year CrI34 sources used in this experirnent were
prepared frorn rnater:i.al obtained from the Oak Ridge National
Laboratorv. This was in the forrn of high specific activity CsCl
dissolved in HCi. A srnaIl arnount of this rnaterial was dropped
into
a. srnal1
cavity in the lucite source holder, the liquid was al-
lowed to evaporate and the process was repeated until a source of
the desired activity was obtained. The cavity was then sealed with
a lucite cap to contain the source rnaterial and to rninirnize evaporation.
Figure 3 shows the garnrna ray pulse height distribution in
the region of interest for the
C
"134.
Cornrnercial sea-led Co60 sources of 0. I and I. I rnillicu-
ries were used. fo:: testrng the spectrorneter by perforrning a
directional corr elati on.
Ni60
and 563 Kev
Backscatter
80
/r05,569,
peak
70
60
d50
0)
a
O
797 and 802 Kev
Sno
@
N
o.r 30
d
a
-20
o
k
#0)
.3
ro
bo
0)
+t
r0
z0
30
40
50
60
70
Pulse Height (Volts)
Figure 3. PULSE HEIGHT SPECTRUM OF C"t34.
(n
6
57
EXPERIMENTA L
A
PPARA TUS
The arrangernent of the electronic equiprnent used in this
investigation was sirnilar to the conventional fast-slow delayed
coincidence system in which the functions of tirning and energy
selection were perforrned separately and then cornbined in a triple
coincidence circuit.
A pair of scintillation counters was used to detect the nu-
clear garrlrrra rays. The voltage pulses generated by each counter
were proportional to the energy of the garrrma rays detected in that
counter. These output pulses were fed into pre-amplifiers and
then into linear amplifiers, after which the signals were separated
into tirning and energy selection channels. Constant arnplitude output pulses frorn each amplifier were generated by selected ampli-
fied counter pulses and fed into a fast coincidence circuit. Addi-
tionally, unselected arnpiified counter pulses were fed into single
channel differential pulse height analyzers. Each pulse height
anaLyzer generated a constant amplitude output pulse whenever the
size of an input pulse fell between two voltage levels, V and
V + dV, thus acting as an energy selector. The output pulses
from the analyzers were fed into a slow coincidence circuit.
The
output of the fast coincidence circuit was delayed before being fed
58
to the triple coincidence stage. This was necessaryinorder to compensate
for the delay introduced by the pulse height analyzers. By
this technique only those triple coincidences were forrned corresponding to signals frorn
detectors I and 2 whose amplitudes
fell within the respective w[ndows of the analyzers. Figure
4
shows a block diagram of the scintillation spectrometer.
The garnma-ray detectors consisted of NaI(Tl) crystals
I.5 inches in diarneter and 1.0 inches thick optically coupled to
Dumont K-1719 photornultiplier tubes. This assembly was positioned in a steel cylinder which acted as a rnechanical support and
a partial electrornagnetic shield. Cylindrical rnu-meta1 magnetic
shields were installed aroun{ the steel cylinders to elirninate ef-
fects produced by the earthrs rnagnetic field.
The preamplifiers were stacked cathode followers which
provided a fast rising, negative output pu1se. The arnfli,fiers were
R-C coupled non-overloading types des,igned for a rise tirne of.0.2
microsecond. The pulse height selector output pulse used for the
fast coincidence was ,a 30 volt negative pulse of 0.7 rnicrosecond
duration. Each differential pulse height analyzer was fed by
a
positive output pulse of I.5 rnicroseconds duration followed by
negative undershoot with an exponential decay.
The differential pulse height anaLyzers were of
a
a
(Fixed)
Linear
Amp
Scaler
1
Scaler
D. P, H,
Anal.
E
Figure 4. BLOCK DIAGRAM OF THE SCINTILLATION
SPECTROMETER.
1
50
conventional design. Each unit produced a negative output pulse
only when a positive input signal had an amplitude falling within
two preset voltage levels. The instrurnent deterrnined whether or
not the height of the input pulse exceeded the lower level of the
channel and then whether or not the pulse size exceeded the upper
level.
When the input pulse size exceeded the lower level a nega-
tive output pulse was generated which would then be vetoed by
an
anticoincidence circuit if the pulse also exceeded the upper level.
In this way output pulses were generated only for those pulses
whose sizes fe11 between the upper and lower levels set on the instrument.
The coincidence circuit consisted of two fast and two slow
channels. The fast pulse height selector pulses frorn the arnplifiers were fed directly to the fast coincidence circuit. If
these
two pulses arrived within the resolving tirne of the fast coincidence
circuit, a coincidence occurred producing a large pulse at
a corrrrrron plate junction. This pulse, after shaping, was fed to
the triple coincidence bus after having been delayed about 5.5
rnicroseconds in order to cornpensate for the extra tirne needed
to carry out the energy selection in the slow channel pulse height
ar.aLyzexs. Output pulses frorn the analyzers were fed to the two
slow channels of the coincidence circuit.
The outputs of the two
6l
fast and the two slow channels, all positive pulses, were fed to the
junction of four diodes and a load resistor on the triple coincidence
bus. The bias across each diode was set so that any one of the diodes could hold the voltage down on the load
one
resistor even though
or rnore of the other diodes was cut off. This condition could
be attained by applying a positive pulse to the cathode of the diode
or by opening the circuit through the diode by a rnanual switch.
When a
triple coincidence occurred (fast and two slows) a positive
pulse rvas generated on the triple coincidence bus" This pulse was
fed to the output stage which in turn fed a scaler of the convention-
aI type.
The photornultiplier high voltage power supplies were designed to provide a very stable (0.02 percent per day) d-c voltage
source for precision scintillation counting. The output voltage
changes by less than 0.35 percent
for a current increase frorn
zero to rnaxirnurn load (1 rnilliarnpere) and by less than 0.00035
percent per volt change of line frorn 100-130 volts. The ripple
was less than 0.01 percent of the output voltage. Each unit con-
sisted of an input regulating transforrner followed by a high voltage
transforrner., a rectifier, and an R-C filter network. The filtered
d-c voltage was applied to a bank of 17 cold cathode tubes frorn
which the stabilized output voltage was obtained.
6z
One detector was fixed on the spectrorneter table while the
other could be rotated about the central axis, on which the source
was placed, and set at various angular positi.ons to within 0.25 de-
gree. The distance of each detector frorn the central axis was adjustable. The alignrnent of the counters and the source was done
with the aid of a telescope.
63
EXPERIMEN TAL MEASUREMENTS
Before fiIeasurernents on Bul34 were begun an angular
correlation rneasurement was perforrned with the ganuna rays of
Ni60. The data were taken at five angles using a 0. I rnillicurie
Co50
source. rr,. Ni50 correlation is weII known and serves
as
a convenient rneans of testing the experirnental rnethods and equip-
rnent. The experirnent was done in the sarne way as is described
below in the casd.of 8"134. rhe Ni60 rneasurerrrent yielded re-
sults in agreernent with the published (50, p. 553) directional cor-
relation coefficients for Ni60 and with the theory. With this assurance that the equipment was functioning properly and that the
rnethod of data treatrnent was satisfactory the directional correla-
tion rneasurernents with g^134 were undertaken.
_ 134
The Cs - source was prepared as described in the section on source preparation and was rnounted in the source holder
on the angular correlation table where
it rernained for the duration
of the entire experirnent. This served to eliminate fatigue effects
in the photornultiplier tubes that are observed when the tubes are
first exposed to a source. The source strength was about 80
rnicrocuries. This gave a ratio of true to chance coincidence
of
about two. A11 runs were rnade at a source to detector distance
64
of 7.0 crn.
Two different angular correlations were measured in the
gamrna decay of
B.134. The decay scherne is quite cornplicated
and includes sorne garrlrna rays of nearly the sarne
energy.
The
energy resolution of the detecting system was rneasured to be
about nine percent in the geornetry used, so that garrrrrra rays dif -
fering in energy by less than this amount could not be resolved.
The
first correlation that was rneasured was an t'overallr' correla-
tion which is the cornposite correlation rnade up of the 797-605
Kev, the 802-563 Kev, the
802-[
569-7 97 Kev
correlations, and the
sfi] -605 Kev cross-over correlation. Because of the srnall
energy differences the low energy photopeak will be cornposed of
the 605 Kev, 569 Kev, and 563 Kev gamma rays. The higher
energy photopeak will be rnade up of the 797 Kev and 802 Kev garnrna rays. For sirnplicity the lower peak will be terrned the 605
Kev peak and the upper one the 797 Kev peak.
In order to measure the rroverallrr correlation the gains of
the arnplifiers were adjusted so that the pulses in the 797 Kev gamrrra ray peak
in channel 2 had the sarne arnplitude as the pulses
frorn the 605 Kev garnma ray peak in channel 1. The arnplifier
and analyzer discriminators of each channel were then set at
point L in Figure 5. The windows of the analyzers were
then
65
set so as to bracket the peaks. The discrirninator on arnplifier I
was then varied slightly to optimize t}:.e delay and pulse shape and
rnaxirnize the coincidence counting rate, The background level
with no source in place was found to be negligible. The counting
rates in each channel were kept to about I03 per second so that
counting losses were negligible.
In order to be able to correct the total coincidence counting rate for chance coincidences the effective resolving time of
the fast-slow coincidence circuit had to be determined. It was
rneasured by the incoherent source rnethod during which the set-
tings of the electronic equiprnent were identical to those used in
a run. The directional correlation source was left in position
where it was viewed by the fixed detector and the rnovable detec-
tor was rernoved frorn the table and inserted into a thick-walled
lead cave. A second source of the rnaterial under investigation
was placed in the cave and
its distance frorn the counter was
varied until the singles counting rate of that channel was identical
to its rate during a correlation run. The two detectors and their
sources were cornpletely shielded frorn one another so that the
coincidences that were recorded were due only to chance. From
this rneasurement the resolving time was given by
66
N
zT- Nl ch *,aux
.,r*
"rr*
where N aux', N-Z aux', and Nch, aux are the analyzed singles of
I
channels I and Z and the coincidence rate, respectively, of this
auxilliary experirnent. The resolving time was rrleasured during
each run and was found to be about
lZx lO-8 ,u. rernaining essen-
tially constant frorn run to run.
The second correlation rneasurernent that was atternpted,
the I'separatedt' correlation, was that cornposed of two cascades
only; the 797-605 Kev and the 802-[sfi] -605 Kev correlations.
With an energy resolution of nine percent the half widths at half
rnaxirnurn of the photopeaks of the 569 Kev and 563 Kev gamrna
rays would be about 25 Kev. The arnplifier gains were readjusted
and the arnplifier and analyzer discrirninators were set as shown
in Figure 6 at Lt with the an.alyzer windows, 'W, bracketing
the
peaks as shown. In this way, setting the discrirninator levels of
channel I at about 605 Kev, the contributions of the 563 Kev and
559 Kev garrurla rays were
largely elirninated. Frorn this point
the sarne procedures were followed as in rneasuring the "overall"
correlation.
The
triple coincidence and slow singles rates were deter-
mined at 19 angles between t0 degrees and 270 degrees in ten
L
I40
67
605 Kev
,t,
lz0
r00
80
Channel
\r
60
0)
+J
d
d
40
I
bo
H
.a
+) ZO
U
Channel
797 K ev
d
o
0
L6
i8
z0 zz
24
/z
26 z8
Pulse Height (Volts)
30 32
34
Figure 5. PULSE HETGHT SeECTRUM oF C"134 poR CHANNELS
AND Z IN THE ''OVERALL" CORRELATION.
LZO
Lr
605 Kev
{,
797 Kev
100
,f
Channel I
80
w
60
o
H+o
d
h0
'i( z0
Channel
2
d
o
,
uo
6z 64
66 68
70
7Z
74
76
78
80
Pulse Height (VoIts)
Figure'6. PULSE HEIGHT SeECTRUM oF C"l34 l,.oR CHANNELs
AND Z IN THE "SEPARATED'' CORRELATION.
68
degree incrernents for each correlation. The fast singles were
observed as a stability rnonitor. In each case the data were col-
lected in a series of half-hour runs at each angle alternating with
half-hour runs at 90 degrees (or 270 degrees) for norrnalization
purposes. In the overall correlation at least
105
true coincidence
counts were accurnulated at each ang1e, except for 90 (or 270) de-
grees, where approximately 106 were accurnulated. For the
(797 + 802)-505 Kev (rrseparatedr')
correlation about 4xI04 true
coincidences were obtained at each angle except 90 (or 270) de-
grees where approxirnately 4x105 counts were obtained. A slow
electronic drift was apparent during an eight or ten hour run
so
the equiprnent was reset whenever the singles counting rate in
either channel changed by rnore than one percent. The data were
reduced according to the treatrnent given in the appendix. The re-
sults for the two correlation rrreasurertents are given in Figures
and 8 and
7
in Table 4. In the figures the solid curve is the theoret-
ical directional correlation function for the case 4lzlzl2)0 rnodified to correct for the finite solid angles subtended by the two de-
tectors. The experimental coefficients presented in Table 4 have
been corrected
for solid angle. The uncertainties indicated
the experirnental points are due to statistical effects only.
on
1.160
L. L4O
t.120
1.100
o
6
1.080
1.060
(D
F
1.040
t.o20
1.000
€l
0.9801 I
90
_-/L
t
100
l
I
'
110
'
120
'
t40
130
e
Figure z. 4"I34 "oVERALL" CoRRELATIoN.
150
160
180
o
\o
1.160
n
llllllrlf
1.140
I
1.
t20
1.100
T
6./,/[
1
150
160
,/t
E( r.oao
F
9 r.ooo
,/[,
,/f,
I
T,/
o
I
B
1.040
1.
-L
020
1. OOO
0.980
90
100
110
t20
130
e
Figure B. B"134 "SEeARATED" CoRRELATIoN.
140
t70
180
\t
o
7l
Table
4.
DIRECTIONAL CORRELATION COEFFICIENTS AND
ANISOTROPY Or 8a134.
Ao
Theory
tL?-l?!?l-o-
I
O+
oz
+
0'
10
20
+
0'
0091
Anisotropy
+0'
1667
8"134
least
-:gYlI::
"Overall"
correlation
I +0.0904 +.0026 +0.0097 +.0052
'rseparatedrr l
correlation
+o.o96g
+0,L478 +.0061
+.0043 +0.0142 *.00g6 +0.r6ro +.orr4
72
CONCLUSIONS
The results of the garnma ray directional correlation
rneasurements in g^134 are surnrn arized. in Figures ? and 8 and
in Table 4.
In the decay of a state with angular rnornenturn j to a
state with angular rnornenturn jt by the emission of radiation with
angular rnornentum L, the possible values of L are deterrnined
by the conservation law l: - :'J < L S j + j', so that angular rnornentum sequences colnmensurate with this selection rule rnust be
considered. Any garnrna ray to the ground state in Bal34 *rr"t b.
a pure rnultipole since the ground state angular rnornenturn
On the basis of reported work the 505 Kev and
taken to be 2* and 4+
is zero.
l40l Kev states are
with the 797 Kev and 605 Kev
"."puctively
radiations both electric quadrupole.
The "separatedil correlation will be cornposed of two components, the 797 Kev-505 Kev cascade, and the 802 Kev-[S6: Xev] 505 Kev cascaile
in which the interrnediate 563 Kev radiation is un-
observed. The expected correlation function can be calculated
on
the basis of the decay scheme by weighting the contributions ac-
cording to the garrurra ray intensities. On the basis of lifetirne
considerations and reported internal conversion coefficients some
73
combinations of angular rnornenta for the 1970 Kev and Il68 Kev
levels can be ruled out. The remaining reasonable pairs of values
of the angular rnornenta of the 1970 Kev and 1168 Kev states, re-
spectivelyr ?r€ 3 and l, 3 and 2, and 4 and 2. Table 5 contains
the calculated directional correlation coefficients for the 'rseparatedrr correlation for the possible pure rnultipole orders of the
radiations for these angular rnornenta. The contributions have
been weighted using the experirnentally observed gamrra ray inten-
sities given in Figure I and the weights are 72.5/81.7 and 9. ?./8L.7
respectively in the order in which the contributions appear in Table
5.
None of these
possibilities can be ruled out on the basis of the
experimental coefficients, particularly since rnultipole rnixing is
possible and suitable rnixing can bring the calculations into agreernent with the experirnent within the experirnental error.
The rroverall" correlation is a composite of four cascades,
the ?97 Kev-605 Kev 172.5/LoZ.4l, the 802 Kev-563 Kev (9. z/toa.4\
the 569 Kev-797 Kev lll.5/102.4l', and the 802 Kev-[S5: fev] -605
Kev (9. Z/lOZ.4) in which the 553 Kev radiation is again unobserved.
Expected coefficients have again been calculated for the possible
angular rnornenta and multipolarities and are shown in Table
6
where rneasured intensities have again been used to deterrnine the
weights of the contributions. The weight of each cascade is given
TABLE
5.
CALCULATED COEFFICIENTS FOR TtlE "SEPARATED" CORRELATION IN ge134
I
T
1970
1
Kev
Kev
168
Combined
E2t2(2lo
412)2(2)0
+. oo91
+.0091
\
A4
o
0
+.0945
+. 0865
+. oo81
+. oo81
o
+.0865
+.0081
1020
0
+.05,14
3(212D) 2(2lo
+.0437
-.0233
+.@22
+.0790
+.0954
+. oo81
+. Ot42
+. oo55
4142lLl 2l2lo
+.0510
-.0060
-.o2t9
+. 0026
+. 0963
+. 0880
+.OO74
E2)2tA 42ro
3(2)L
w
z?)o
+. 0357
3Q)LA a?)o
-.0357
3<tl2 [1] 42to
-. 0357
+. 0153
\[rzLA
z(2)o
3G)2.r1) 2(2'to
412)421O
rr
-.
Separatedrr Expe rimental Coefficients
A2
A4
+.0084
= +0.0968 + 0.0043
= +0. 0142 + 0. m86
-t
A
TABLE
6.
CALCULq,TED COEFFICIENTS FOR THE .OVERALL', CORRELATION IN Ba134
II
1970
Kev
1168
Combined
569-797
Kev
3<2lt(tlz
3<4t t1)42!o
3lLl412l2
3l2lL(212
3(2ltL2) 42',,o
3(1141212
3(2lt(tlz
312lL(2)2
3{tlz(L)2
312)2(L)2
3ltl2(212
3{212(2)2
3{\Lt41r2
34212(t'tz
3lt)2(2r2
342)2(2t2
E2)41)2
412)2(2)2
41212(tlz
41212(2)2
-.0500
-.t429
+.0153
+.0437
-.0500
-.t429
+.0153
+.0437
+.O7L4
-.O2L9
+.0714
-.o2t9
0
0
O
-.0233
3{2ltlt) 42lo
3{zllLd 42lo
312)41212
3(L)2 U) 2(2lo
3(1141212
342t2IlJ 42to
3(Ll2 L2) 2(2lo
3<1141212
3(2t2L2l42p
Lrl42lf.
3{.2)412)2
3{L)2D) 42ri0.
3{21412)2
3(2)2L2142yo
312)412)2
qzv
A)2Q)o
411)412)2
E2)2L2)4210
411)412)2
412)2lL) 42)o
412)2t2) 42)o
\21412)2
O
-.0233
O
+. 0026
O
+.0026
312't2
rr
412)E2)2
OveralIr Experimental
Az
A4
+.0590 +.O064
+.O500 +.0064
+.O77O -.0144
+. O68O -.OLu
3{1144212
42',,o
3(rl2w
A+
312144212
3tLlEzlz
q2tqz12
0
0
A2
C
O
-;1403
-.1403 -O...
0
-.1403
t&3
0
-.
+.020O -. 1853
+. O2OO -. 1853
+. O20O -. 1853
+.O2OO -.1853
+.0488
+.0345
+.0592
+.0643
+. O668
+.0525
+.0772
+. 0823
+.0O64
+. 0113
+.1964
+. t964
-. Lt84
-. lt84
+.1053
+.0903
+.0699
+.0550
+.0059
oef fic ients
0
0
+. 1516
+.
1516
Az=
44=
+.0064
+. O@3
-.0144
-.
OO95
-.0144
-.0186
+. O069
+.0229
+.0239
+ o. m26
+O.O@7 + 0.0052
+0.0904
{'
(rl
76
above in parentheses after the cascade. The constant contribution
of the 797 Kev-505 Kev 4(ZlZl2)0 cascade has been ornitted frorn
the table for simplicity but its effect is included in the calculated
coefficients. The coefficients for the
cascade have not been listed
802 Kev-[S5: Xev] -605 Kev
explicitly since they are the same
as
those given in Table 5. In this case rnultipole rnixing is possible
in at least two of the radiations for each pair of angular rnomenta
and by the choice of proper mixing ratios the calculations
in each
case can be brought into agreement with the experirnental data
within their errors. None of the three possible angular rnornentum sequences can be ruled out on the basis of these experirnental
results alone. Since it is possible to include rnixing in at least
two radiations the values of the mixing ratios cannot be assigned
unarnbiguously.
The interpretation of the results of directional correlation
measurements between the 1365 Kev and the 605 Kev radiations
are not cornplicated by interfering radiations on a more cornplete
decay scherne and indicate the assignment of angular rnomenturn
4 to the 1970 Kev level l3Z, p, 66l', (I8, p. 9551, (54, p. 691).
Taking the 1970 Kev leveI to have angular rnornentum 4 and, from
the allowed nature of the beta branch, to have positive parity
leaves angular rnomenturn Z as the probable assignrnent for the
77
I158 Kev Ievel. This angular momenturn sequence is supported
by the recent work of Segaert, et aI. (48,
this state is taken to
p. 87).
The
parity
be positive so that the 802 Kev garnma
of
ray is
electric quadrupole. Otherwise the 802 Kev radiation would be
rnagnetic quadrupole, electric octupole, or a higher order rnulti-
pole and would be very weak. Cornparing Tables 5 and 5 for these
angular rnornenta it appears that the experirnental O, of the
rrseparatedil correlation is best fitted by taking the 563 Kev radia-
tion to be rnagnetic dipole whereas the coefficient for the "overall'l
correlation is best fitted by taking the 563 Kev radiation to be
electric quadrupole and the
569 Kev gamma
ray to be rnagnetic
dipole. This apparent inconsistency can be rernoved by considering both the 563 Kev and the 569 Kev radiations to be magnetic
dipole-electric quadrupole rnixtures. The other possible rnixtures
would involve rnagnetic octupole or higher order rnultipoles and
can be neglected. Figure 9
is a plot of the experirnentally allowed
versus those of 6-,^, where the rnixing ratio
values of 6-,^
-
563
569'
6
is, in each case, the ratio of the electric quadrupole to rnagnetic
dipole intensity in the radiation. In the 'rseparatedrr correlation
only 6S5: appears. By requiring the theoretical rnixed O, for
this case to equal the experirnental O, within its standard error,
the restriction 0 .
IUUO, I
a 0.9? is obtained. The horizontal
78
lines labeled I at 6553 = + 0. 92 then define the boundaries of the
allowed values of the mixing ratio to fit the separated correlation.
The experimental On for the rrseparatedrr case places no restric-
tions on 6S5r. The same procedure was applied to the overall
correlation in which both mixing ratios appear. The solid rrverti-
cal" lines labeled
2
in Figure t represent the boundaries of the
region of allowed rnixing in order for the theoretical rnixed
On
of the rroverall'r correlation to agree with the experirnental rrover-
all'r An within its standard error. Finally, the areas within
the
pairs of curves labeled 3 and 4 represent the allowed values of
mixing so that the theoretical mixed OZ for the I'overallrr correlation agrees with the experimental rroverall" O, within its standard error.
The areas of overlap of these three regions are shaded
of 6S5g and 6 S6g on
the basis of these experiments, The arrow at 65b3 = 0, 6559= -0.38
and represent the allowed range of values
indicates the closest fit to the rrseparatedrr and I'overaIl'r experimen-
tal Ars. At this point the calculated 'rseparatedtr O, is *0. 0963
and the calculated rroverall'r O, is *0.0904. tr-rom these curves it
is
seen that
either or both of the 563 Kev or
569 Kev
radiations can
be at most 50 percent electric quadrupole, corresponding to
6 = t 1. This is in disagreernent with the results reported by
Stewart et al. 154, p. 694) who report the 559 Kev radiation to be
+o
<^
"/
?^ \ ---
_A
/1
-2.O -1.6
-1.2
-0.8 -O.4
0
+0.4
+0.8 +L.2
+
6sog
Figure 9. EXPERIMENTALLY ALLO\MED 6553 VERSUS
EXPERIMENTA LLY ALLO1ME D 6
s6g'
1.6
+2.0
80
t4 percent electric quadrupole. If the condition on the rroverall
An (curves 2l is ignored in Figure 9r however, it is seen that
large positive
6
569,
a
corresponding to that radiation being pre-
dominantly electric quadrupole in agreement with Stewart et al.
,
is not in disagreement with the Or'r of the present work provided that the 553 Kev radiation is taken to be a nearly pure mag-
netic dipole. Generally, the results of the present experiment
support the assignment of large rnagnetic dipole admixtures to
both the 569 Kev and the 553 Kev garnrna rays.
If the angular momenta and parities of the leveIs being considered are assumed to be 2+, zl, 4*, 4* in order of ascending
energy then the asymmetric rotor rnodel of Mallmann is able to
correctly predict the energies of these levels. One result of the
rnodel is that magnetic dipole transitions are forbidden. Tables
of reduced electric quadrupole transition probabilities have been
prepared by Day and Mallrnann (15, p. t). Assurning the experirnental beta branching ratios and neglecting the presence of the
three etates of unknown angular momenta, the garnma ray intensi-
ties were calculated using these reduced transition probabilities.
The probability per unit tirne
for an electric quadrupole transition
is proportional to E5 (E is the gamrna ray energy) tirnes the reduced transition
probability. The results are given in Table
7.
8I
Table
7.
THEORETICAL AND MEASURED GAMMA RAY
INTENSITIES IN Ba134.
Arymmetric Rotor
Gamma
Energy
Reduced Transition
(Kev)
Probability
Gamma Rays per 100 Decays of cr134
Model
563
s69
505
797
802
r 158
1365
0.3196
0.0780
0.2553
0.3535
0. 1145
0.0r08
0.0029
Measured
8.6
14.
z.z
11.5
0
78.6
98.
63. z
72.5
18.2
11.1
6.6
I0.
0
5
2.0
5.0
The comparison for the radiations frorn the 1970 Kev level
is most meaningful and is not good. On the basis of the model all
these radiations must be electric quadrupole or higher order rnul-
tipoles so rnixing can be neglected. Using this fact and the calculated intensities, the expected correlation coefficients were corrr-
puted. The predictions for the rrseparatedil correlation are
A^ - +0.0882, A. - +0.0078, and those for the 'roverall'r correla2-'4
tion are A^Z+
= *0. 0718, A. - +0.0I17. These coefficients are not
in agreement with the experimental results.
In summary, the results of these directional correlation
rrreasurements are in agreement with the rracceptedrr assignments
of angular morrrenta and parities of 2+, Z*, 4*, 4* to the 505 Kev,
1158
Kev, l40l Kev, and 1970 Kev excited states respectively,
but they are not in disagreernent with angular momenta
8Z
assignrnents of Z, L,
4, 3 ot 2, Z, 4, 3 to these states resPectively.
For the assignments ?+, 2t, 4*, 4* the experimental results indicate, however, the predorninant magnetic dipole character of two of
the radiations in contradiction to the selection rule irnposed by the
asymmetric rotor model, a rnodel which successfully predicts the
energies of these excited states in B"134.
83
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85
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28. Heitler, W. On the radiation ernitted by a multipole and its
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Physical Review
UUC""n.
87:67 6-677 . 1952.
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6 134m. Physical Review 97 :451-456.
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1955.
"
3?,. Klerna, E. D. Garnrna-garrrrrra angular correlation in g.
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"rr4
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37. Mansfield, J. A. and L. Schecter. Gamrna-ray attenuation
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r96?.
38.
F. and M. Deutsch. A study of the polarizationdirection correlation of successive garnrrra-ray quanta.
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4L, Preston, M. A. Physics of the nucleus. Reading, AddisonWesley, 1962. 55I p.
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New York, Wiley, 1957. ?48 p.
rnornenturn.
45. Rose, M. E. Internal conversion coefficients. Arnsterdarn,
North-Ho11and,' 1958. 173 p.
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87
50. Siegbahn, K.
) Beta and garnma-ray spectroscopy.
Arnsterdam, North-Holland, L955, 959 p.
(ed.
5I. Siegbahn, K. and M. Deutsch. The disintegration scheme of
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"116
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APPENDICES
88
APPENDIX I
Geornetrical Correction Factors to
W(Q)
The results of an angular correlation experiment are rnost
conveniently expressed in terrns of an expansion in Legendre poly-
nornials. That is, the corrected true coincidence counting rate
per unit solid angle is proportional to
\w(e)
= ) a P (cosO)
'exPLnn
(l)
n
where n takes the values 0, Z, and 4, and the
frorn a least-squares fit to the data.
W(e)^___
exP
"r,
are determined
is related to the theo-
retical correlation function W(+). Since the scintillation crystals
subtend
finite solid angles at the source, the theoretical correlation
function rnust be rnodified. This srneared (or averaged) correlation
function W(6I is then compared with the W(0)e*p deterrnined from
the measurements (47, p. 610). The object here is to find the relationship between Wiil and W(e)-___.
The geornetry under considexp
eration is shown in Figure I0, where
*(p,
'1
0.1
) are the interaction paths for the garnrrra rays in the
crystals,
are the azimuth angles of the radiations measured
with respect to the crystal axes,
is the angle between the crystal axes,
89
T
t
ll
r _>r
kr
.L
Figure
10.
GEOMETRY Or" THE CRYSTALS FOR THE
FINITE DETEC TOR STZE CORREC TION.
90
+
is the angle between the directions of propagation
of
the cascade garnma rays,
is the angle between the axis of crystal I and the
propagation direction of radiation Z,
is the distance frorn the source to the face of either
crystal,
is the thickness of each crystal,
and
is the radius of each crystal.
The crystals are identical right circular cylinders with the
bases oriented toward the source. The source is situated at the
origin of the coordinate systern at the intersection of the crystal
axes. In this case, as will be seen, each coefficient in the expansion will be multiplied by an easily determined attenuation factor.
=(pi) is the interaction path of garnma i at an angle p., so that
'
-r.(E.)x(9.)
'
1
I
the absorption is proportional to (l - e
' ), where ,r(Ei)
is the energy-dependent absorption coefficient. The srneared out,
or measured, correlation function is given
by
-.-'t*(Pt',,, - "-'z*(92,.)dw, dw,
J*,*,(t
wfol =
^ ^)(t-" u ')dwrdw,
J{t-"
where the theoretical function is given by
W(0) =
\-
)o,.",.,cosg)
with v = 0, Z, 4. Substituting this into Equation (2) it can be
(zl
91
seen that the integrals of irrterest are of the forrn
-rr*(Pr) -rz*(!t)
(,,r=JPr.(cos0)(I-e ' ')(I-" o 'l"irrpldpld+l
x sin PZilPZig.Z
(3)
with
-(pi) = tsec p.
_l
for 0$iS tan -
r/(tr+t1 = p'
and
*(pi)
=
tcscP.-h secp. for p' .Pi <tan'
I
r/h = y
The evaluation of Equation (3) is accornplished by using the addi-
tion theorem for spherical harmonics
p--(cos +r
it
v'
4'
= ?v+l
\
*
,.**'(pl+r) yf(a+z).
\
A double application of this theorern and the realization that *(0t)
and *(pZ) are independent of +l and QZ, respectively, and that
0 is independent of +I and +Z so that the integration over these
angles can be perforrned easily leads to
Prr(cos +) = Pv(co" 9l)Prr(cos
c)
c) = Pr.(cos Pr)P,.(cos
0).
and
P,.(cos
Recalling Equation (2), the use of these expressions in Equation
(3) gives
_w(0) = r
Jv(r) J
L ouru(cos 0) rya
(2)
(4)
9Z
where
-rt*(p;)
r.Y
Pr.(cos Fi)(I - e ^'
J,.(i) =
) sin PidP,
J
o
and again
v - 0,2,4. Since Equations (I)and (4) are supposed
to
represent the sarrle thing, then
J (r) J
(2\
(1) J
vJoo
(zl
a=-A-v-v
n
relates the experirnental a and the theoretical A coefficients.
nv
The Jv integrals have been evaluated by Stanford and Rivers for
crystals of various r and t for values of h frorn 7 to 50 crn.
using the absorption coefficients for NaI corresponding to energies
frorn 0.05 Mev to 5.0 Mev (52, p.
719!..
The Jv(i) integrals can be approximated
by
rY
pi) €(pi) sin p. dp.
=
v-o \ P,.(cos
.1."(i)
where € (pi) is the experirnentally deterrnined angular resolution,
J0, J Z, and Jn can then be evaluated by graphical integration of
this integral. The Jr. values have been experirnentally deterrnined
in this way for the detectors used in this experirnent by Glasgow
et aI. (25, p. 68:) and by Mansfield and Schecter (37, p. 574).
The results support the use of Stanford and Riversr calculated
values for the correction factors.
93
APPENDIX II
Treatment of the Data
The dependence of the coincidence counting rate on the an-
gular position of the movable detector corresponds to the theoret-
ical correlation function only under assurrrptions of point detectors, centered point sources, and the absence of scattered or disturbing radiations. The cornparison of the experirnental and theo-
retical results then follows after corrections have been rnade to
the experirnental data for deviations from such an ideal arrange-
rnent. The source and its container were rnade srnall (active
source volume about 7x10-n
so that no corrections were nec-
"*')
essary for the source size or scattering in the source or its container. The experirnents were performed with a scintillation
spectrorneter with its energy selection protection against scattered and other disturbing
radiations. In addition the sides of the
scintillation crystals were shielded with lead for further protection against radiation scattered frorn one crystal into the other.
The true coincidence
rate Ni,(et ) of the ith run at the
angle 0X was deterrnined in each case by subtracting the chance
(or accidental) coincidence rate Nr.fr(Ot) = ZT Nr,(OO)NiZ(et
)
from the total coincidence rate Nrror(0U). NiI (0k) and *rr(rO)
94
are the ar,alyzed singles rates of channels one and two and 2T is
the resolving tirne of the systern.
The true coincidence rate was then divided by the product
of the slow singles rates of channels one and two. Ttriis division
by the singles rates will correct to first order srnall errors in the
centering of the source (50, p. 597), and should also be independ-
ent, to first order, of any changes in the efficiencies of the two
channels. The result is the true coincidence ratio
Nir(eo)
D.
(0O)
=
-
Nir (er.)Niz(er)
*r.or(ao) - zTN. r (or.)Ni
z(er.)
Nii (er.)Ntz(er.)
The standard error in a function tr.(x.) due to errors in
1
the variables x. is siven bv
t"
or{*r)
t/z
=
(8, p. 313). Then the standard error associated with Di(et ) i.s
1r
'o.{eu) -=
D'(e')l-
wi
where N.cfr
aux
P
N-2
itot(0t )
+
nteo)
I
+ negligibie terrns
ch
aux
t/z
I
J
isthe coincidence
- rate of the auxiliary two source
experiment.
The true coincidence
ratio at 0U was divided by the true
coincidence ratio at 90 degrees to give a norrnalized ratio
R.(ek) = D.(ek)/D.(90) independent to first ord.er of instrurnental
95
drift. The error associated with this norrnalized ratio is
t/z
r-Z
hteur =R(ok)
Lfi*
+
"'"ry
oitgot
The average weighted ratio for aII the i runs at angle tO will be
)
KI*
R(0.)-l
*.*.,ro,
11
i1
where the weigh, *i associated with the ith run is the ratio of
a
norrnalizing factor b, chosen for convenience, and the square of
1a'
with the ith ratio, R.(ek). The
the standard. error associated
standard error of the weighted
.g1
average)rratio is given by
\,
/
ar/z
Fl*
and
t/z
-L
L*rd,
11
(n-1)
for internal and external consistency respectively. n is the nurnber of pieces of data and each. d, is the deviation of each R.(ek)
frorn R(ek). No data were rejected.
Th-ese two expressions were
found to give nearly the sarne value for the standard error On{eU)
of R(ek), though the larger of the two values was used in
each
case. The weight assigned to R(ek) in the least- squares analysis
was taken as the ratio of the norrnalizi.ng factor br and. OilaOl.
96
?
Table A-I shows a sarnple calculation of w.R.(0O), di, d.-, and
-Z
As an exarnple, at 0k = I40 degrees, I4 experirnental de*idi-.
terrninations of R.(140) were made for the case of the "overa11"
correlation, yielding a weighted rnean value of 1.0693 +0.0059.
Table A-1. Sarnple calculation of *iRi(140), di, d?, and
oi
R (r40)+ oR.
w.
1
1
1
b
=-=--
uz
RR.
.Z
w. d.
11
t0-?- w.R.(I40)
oz
11
d.11= R( 140) -R. ( I 40)
z.30Lt
2. r645
.000462
1.063I+0.0215
-2
w. d.
11
d?
1
. 0062.
00008 2
0000 38
The total reduction of the 14 pieces of data gave2
t. The rnean weighted norrnalized ratio
)
R(140)
*.*.
=
noy
31.07r6
?%
)
i
Z.
t,
oi7 = l' 0693+o' 0059
w.
1
The standard error associated with R(140)
O.=
1n
J/z
l. t6axto-2
___=_=
(zt. oof / z
P-1,,'
= 0. 0059
97
O=
ex
H
t/z
t/z
10973xi0-6
3. The I east- square s weight of
I 0-4
0.0000348
br
i l. s.
-
o
Z
= 0.0054
13x29.06
R(140)
=
2.87
Z7
in
The identical procedure was carried ou.t at each angle.
Least Squares Analysis of the Data
A least-squares analysis was perforrned to deterrnine the
coefficients describing W(0)e*p. A series of Legendre polynornials was used to fit the R(ek) as closely as possible (47, p. 613):
4
w(0, ) = R(0. ) = t a.P.(cos 0, ) = f".a, .
Kexp
r( L J KJ
K lr J J
j
,'"'3.
The rnost probable coefficients
"j
for the given data were ob-
tained by rninirniztng the function
\[\1'.
*"
*
F,ro,
! ^i "r.:
J
yielding the norrnal equations
\lLu*o
L*,ro,
-
\
!
I
.
^,oo: Jou,
--
o
98
A symmetric square matrix is now defined
so that
c=AwA
with elements
\.
C.. - ) w.A..A..
- C..rJ
I KJ l(1
lJ
JI
k
where W is a diagonal rnatrix with elerrrents *k, A is a rnatrix
of Legendre polynomials, and A is its transpose. Now define
g=AwR
where R is a matrix with elements R(ek). From the normal
equations
it follows that Ca = B, or multiplying
by the inverse of
C, a = C-t-8.
In detail,
..JLJlI
= Tcl.te.
i
gives the desired coefficients.
a is a linear homogenous function of the counting rates
R(ek). Hence, due to the existence of a variation in
R(OO) ex-
2
pressed by 0-n1eO) there will be corresponding rrrean square de-
viations in the coefficients. Writing out the expression for
f-r
^j = LwnR(oo)clj
Ar.l
k,I
and
g?=
a.
J
I,F'; ^,.')'
*kzz
unleo)
^i,
99
This can be written
t,
=lr,,"u' "u'ArrAmrl
: ,,,! "u' ";,'
=
Now
let i =j
o'I ";t
!ri
oi.leol
oi"reol
on, Aki *k
"u' "r,
and the sum
0z = blC..
a.
JJ
I
br
over !.
and
j is perforrned giving
t
J
The square of the standard error in the coefficients is
given by the diagonal elernents of C-1 rnultiplied by b'. After
I
deterrnining the coefficients and their errors they were corrected
for the finite size of the detectors using the JO, J r: ar.d Jn discussed in Appendixl, assurning that the errors in the Jr swere
negligible. Tatrles A-2 and A-3 show the calculation of the ,j,
their errors, and \tr(0).*p for the two cases rneasured-
TABLE
A-2.
ot
CALCULATION SIiEET FOR THE TEAST SQUARES ANALY$S OF TIiE EXIERIMENTAL COEITFICIENTS FOR THE "O\iIERALL"
CORRELATION OF 8a134
R(ek)
k
90
o.9998
39.0625
1@
o.9979
3.4294
110
120
1.0164
2.0,rc8
1. 0230
130
140
150
160
170
180
*tAt4
+t4.il84
+ 0.9119
-
0.0078
0.5899
1.3633
o.9165
+ O.O538
+ O.9693
+ 2.37Ot
+ 2.1626
*kAkz
Axz
w
"1,
-19. 5313
- 1.5596
2.0,()8
-o. so@o
-o.45478
-o.32453
-0.12500
0542
3. 1888
+O.11977
+
1. 0693
+O.38O23
0889
2.8727
2.2957
1.1139
2.O&8
+O.82453
t.1270
1.14t6
2.7778
2. t626
+O.95418
+ 2.6522
+ 2.1626
+O.91161
+1. O000O
1.
1.
"?n
+0.14063
+O.O7O7l
+0. Ofi)O15
+O.08356
+O. t8278
*ooin
+5.4932
+4.2425
+O. OO0O3
+1. Ofi)OO
*tAt
zAt+
O.3819
+O. O1435
+ l.@23
+ 1.4348
+ 1.6827
+O. 14458
0.6623
wnR(eo)
-o.4147
+0.(X25
+
+0. fiD55
+0.0O13
+O.0336
+0.22558
+0.4604
+2.0222
+2.1626
+O.7992
+2.26?9
+1. finOO
o.2551
+O.7@3
+O.2149
+0.0319
+O.0457
+0.4153
+0.8968
+39.0547
+O.0737
+O.728OO
+9.7656
-7.3242
+0. 1705
+0.5829
+O.2924
+O.lOl79
Z50OO
+O.20683
+O.10532
+O.01563
-
+0.625O0
-0.1633
-o- 3485
+2.1626
.z
*k^kz
3.4222
+ 2.0743
+ 2.0878
+ 3.3616
+ 3.4718
+ 2.4998
+ 2.2733
+ 3.1306
+ 2.4589
+0.
+0.39O63
+0.67985
-I9.5773
+
+
+
1. ft6EO
L.5624
L,8744
+ 2.989O
+ 2.4589
+0,37500
+0.26592
-0. 00383
-0. 28906
-o.42753
-o. 31904
+O.02344
+O.47495
+0. 85323
+1.3874
+2.5322
+2.1626
*totzR(ot)
- 1.5563
- 0.6732
- o.26to
- o.&x
oon
+1. fiXX)O
*rAraR(or)
+ 14.
+
6455
O.91OO
- o.oo79
-
0.9800
+
0. 0586
+
1.0797
0.6035
t.4372
+ 2.67L|
+ 2.4689
c()
t0l
Table A- 2 (Continued)
I
)
*-
\
= *5I. eIIe
!
\
/,-l(Kw. R(0, )
*ooo, = -rz.6ot7
p *ooi, -
+18.
k
\
I6re
f
\
p*uoon = *18.2385
\2
f
*ooin -
\
?*u
cl= )* o
r kkz
!
a
^z
^4
f
= +63. 4448
*ut r*,ru) = -tr. 5526
*o\n*(0k) = +18.805I
*xoxz
p
\?
*ooi,
I *oou
\
f
+0.0343
+0. 0l
+0.
-0. 0507
2. 9L 6r
!
f
*ooo n
01 56
-
+l I .4?.80
T
\
*oourAk4 =
,zAk
*ooo.oon
55
-uoon
\?
f *uoi
'-4
63. +448
-0. 0507
+
+0.0545
-0.0085
-1i.
5526
0085
+0. I 663
+i8.
80 51
- 0.
TAZ
Table A-2 (Continued)
a rs norrnalized to
n
aoo=L.0402
a =I.0000
aZZ= 0.0869
a =0.0076
44
a- = 0.0835
a =0.0073
Error in the a n
oZ
to
=
btC.l= r o-4 * 3. 4Z'?xLO-Z
iJ
Oa
=
0.0019
=
0.0025
Ou4 =
0.0041
o
O
^z
Error in the an norrnalized
Oa
and cornpounded.
=
0.0026
=
0.0024
=
0.0039
o
O
^z
O
^4
Solid angle correction of the ..,
Jo(l
)Jo(z)
o. 0835
AZ=^2ry=ffi=0.0904
A
-
Ooz = 0.0025
n
1
103
Table A-2 (Continued)
Jo(i
)Jo(z)
n4- d.4ffi
A_
oo
,,4 =
w(0)exp
=
I*
=m
o. oo?3
o' 0052
(0.0904 +0.0026)Pr(cos 0)
+(0- 0097 +0.0052)Pn(cos 0)
Aexp= Y-(I89=l;=Y(90)
W(90)
= 0.
I 4tB +0. 0067
=
0.0097
TABLE
A-3.
CALCULATION SHEET FOR THE IEAST SQUARES ANALYSIS OF TT{E EXPERIMSNTAL COEFFICIENTS FOR THE IISEPARAIED''
CoRRELATION OF 8a134.
ot
R(e. )
w
K
90
o.9992
100
0.9939
110
1.0011
16.0000
t. t3L7
0.9070
r20
1.0265
1. 1080
130
150
t, o46t
t. 07 t4
r. rt25
1.0000
1.1815
o.7432
160
L. L2t2
t70
1. 1256
1. 1450
140
180
wA
kk4
Atz
k
-0. s0000
-o.45478
-o.32453
-8.0m0
+4.0000
+O.2341
+0. 1198
+0.38023
+0.62500
+O.4492
+O, 14458
+0. 1708
-0.31904
+0.4645
+0. 2903
+O.O23M
1. 1080
+0. 82453
+0.7533
+O,47495
L, L3L7
+O.95478
+0. 9 116
+L, O3L7
+0. 85323
0.5102
+ 1. OOOOO
+O.9 136
+ 1. 0806
+O.5102
+0.39063
+0.67985
+O
+ 1. OOOOO
-0.
12500
+O. L1977
.z
*k^k4
"Ln
*tAtzAt4
+1
L5,9872
-7.9936
-o.5116
-o.2947
+0. 1,top3
+2.2500
-3.
OOOO
+
+O. O7O7
+0.0800
-0.
1369
+ L.t248
-0.0035
-0. 3203
+0.
-0.4275
-o.3769
+O. L8278
+O. LOUg
+O. Ot74
+0.00055
+0. 22558
+O.72800
OOOO15
+O. O8356
+ 1. OO0OO
+0.0OOO1
+O. OO11
+
+O,@26
+0. 1828
+0. 1203
+O. O4OO
+
+
+
+0. 0OO4
+O,2499
+0. O1O9
+O.4339
+0. 8239
+0.5102
+O.9220
+0.51o2
-0.0512
-0.
1433
0.9080
L.1374
-0. 28906
-o.42753
S1O2
*tAtzR(ot
3O1O
+0. 37500
+O.26592
-0. 00383
+0. 0955
+O. Ot73
+0. 0143
woR (0u)
OOO0
L
1
- OOOOO
+0.
+0.9656
+0.5102
ooa
+0.25000
+0.20683
+0. 10532
+0.01563
+0.01435
-o.5147
-o.2944
-0.1385
+6.
+0. 5263
*ooi,
of,,
*kAkz
)
*tAt+R(or.)
+5.9952
+O,2991
-0. 0035
-0. 3288
-0.4472
1.2658
-o.1422
+0, 1253
+0. 4813
+ 0.8268
+O,5t67
+0.0194
+ L,2423
+ 1.2852
+L. 0243
+1.227 L
+0. 5842
+0. 5900
+
1, o461
O.5842
-o. 4039
+1.0966
+0. 5842
A
105
Table A-3 (Continued)
) *
kk
\'
|
\2
!
\
f
= *24.82t4
= *7.117
'= -
L
" 4t33
p *o*,ru) = *25. 407 g
*xAxz = -5.4097
*ooi,
*oourAk4
|
6
*oourR(ek)
)
\
*ooon = t7. r9zz
=
-4. ea3r
*ooonR(ek) = t7.
!
4011
\,
( *ooin = t4. 3I o l
t
\
Lw,
kK
lc
\
Q
*oou,
a
a
o
z
4
|
\2
p *ooorou*
*oAi,
\,
\ *'rorzoo+
! *uoin
f
!*ooun
a
!**.Axz
!
\
*ooon
0402
+0.0876
+0.
+0.0402
+0. I 687
_0. r3zg
-0.
01
17
-0. I3z9
+25"407 9
-0.
-
01
1
7
+0. 4500
4.9831
+ 7"4011
r06
Table A-3 (Continued)
an rs norrnalized to I
ao
= 1. 0407
ao = 1. 0000
a^Z = 0.0931
aZ = 0.0894
a4 = 0.0IIi
a = 0.0I05
4
Error in the
a
n
2_1-4-?
o"AJJ = brc..^=10 ^x9.756xi0
o
Oa
Oa
= 0.0030
o
= 0.004I
z
O
= 0.0067
^4
Error in the a norrnalized and cornpounded.
n
Oa
= 0.004I
o
O
= 0.0039
^z
Ou4 = 0.0064
Solid angle correction of
Jo(r)rolz)
.
"Z
the
*zJ^(1)J^(2)
Z'Z
O.
A =
Z
0.0043
a
n
0.0894
0.9610x0"9614
107
Table A-3 (Continued)
o+=u4ffi=#=o.or4z
Oo4 =
W(0)u*p =
I*
0.0085
(0. 0958+ 0. 0043)Pr(cos 0)
+ (0. 01
o"*o
=
Yg%,@
42*0.0085)Pn(cos 0)
=
o' 1510 *o' oI14