Table Of ContentTABLES OF COEFFICIENTS
for the Analysis of Triple Angular
Correlations of Gamma-rays from
Aligned Nuclei
BY
G. KAYE AND E.J.C. READ
Lecturers in Physics, University of Liverpool
AND
J.C WILLMOTT
Professor of Physics,
University of Manchester
TO INOWTTM II··
PERGAMON PRESS
OXFORD · LONDON · EDINBURGH · NEW YORK
TORONTO SYDNEY · PARIS · BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.l
Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1
Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101
Pergamon of Canada, Ltd., 6 Adelaide Street East, Toronto, Ontario
Pergamon Press (Aust.) Pty. Ltd., Rushcutters Bay, Sydney, N.S.W.
Pergamon Press S.A.R.L., 24 rue des Ιcoles, Paris 5*
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1968
Pergamon Press Ltd.
First edition 1968
Library of Congress Catalog Card No. 67-28102
Printed in Poland
08 003167 6
INTRODUCTION
THE measurement of the angular correlation between the y-rays emitted from
an excited state of a nucleus is one of the most powerful methods of determining
the spins of excited states of nuclei. In this process a state of well-defined (but
usually unknown) spin and parity is formed by some process, and the correla
tion between the subsequently emitted y-rays is measured. The general theory
of this process, together with a detailed discussion of methods of solution, is
presented in the recent monograph by A. J. Ferguson.<i> In certain cases, e.g.
resonance reactions, radioactive decay, the formation process provides valuable
information about the initial nuclear state involved and considerably simplifies
its description, but in the more general case of states formed by particle
reactions, these reactions are too complex to be of help, and the quantities
describing the formation of the initial state must be regarded as adjustable
parameters to be determined from the measured correlation, along with the
spins.
The quantities describing the initial state of spin a are the elements
of the density matrix <a a| ń\a a'> or alternatively the statistical tensors p,^J^a a),
where
p,,(a a)= Σ (-l)"""'(aaa-a'|/cK)<aa|p|aa'>
β a'
and a, a' refer to the magnetic.substates.
These latter quantities are the expansion coefficients of the density operator
in terms of certain moment operators which transform under rotations like
spherical harmonics. Hence the statistical tensors themselves transform accord
ing to the relation
ń,Ëá fl)= ôĽ\\(0Lßy)p,Aa a),
ic'
where the new frame is obtained from the old by a rotation through Euler
angles (a â y) and DΝ'^(a â y) is an element of the rotation matrix.
In general there are (2 α 4-1)^ such elements. If, however, the initial state
is formed in an axially symmetric way and we choose this axis as the z-axis,
then the rotation property above indicates that only the ń^,^{á a) are non-zero.
Further, it can be shown (Litherland and Ferguson) ^2) that if the formation
process is performed with an unpolarized target and beam of incident particles,
and any outgoing particles are either undetected, or detected in an axially
vii
vni INTRODUCTION
symmetric way, and without reference to their state of polarization, then only
even values of k occur. The corresponding condition for the elements of the
density matrix is that only the diagonal terms occur and that (a a| ρ |φf a> =
= <α-α|ρ|α-α>. These are the populations of the magnetic substates.
In any case, if the experiment is performed in such a way as to cause the
formation of odd statistical tensors, their presence will only be detected if a
measurement is made of the circular polarization of the emitted y-rays.
Direction—Direction Correlation—No Polarization
If a state of spin a is formed in an axially symmetric way and no measure
ment is made of the polarization of subsequently emitted y-rays, the correlation
between y ă and y2 illustrated in Fig. 1 is given by
yi is detected at polar angle and is assumed to be a mixture of multipoles
¿1 and L'l, and 72 is detected at polar angle 02 and is a mixture of multipoles
¿2 and L'2; ö is the azimuthal angle between y ă and ^2· The summation is
over K, M, Λ^, Li, L'l, ¿2, L'2, and k, with , L2 ^L'2 and N^O.
'(b)
— (c)
FIG. 1
The functions A^¡^m can be expressed in terms of tabulated functions as
follows:
Alj^iabcLyL, L2 L'2 fc) = (-1/> 2" A ' ¿1 L\ 4 {L,\L\-\\KO)
x(L2lL'2-l MO)(K-NM N\kO)
b Li a
b L'l a W{bL2bL2:cM)
Ě Ę k
and
INTRODUCTION IX
where α, etc., means {2a-\- \γ'^. The term 2" accounts for the dupHcations due
to the negative Ν values and the opposite order of coupling Lj, L'l and
An alternative expression for ^4^^ ^ is
b Li a
^ , ( K - N MN kO)
Ë'κΜ- i - i Y ^ r -— —- c. b a Z , { L 2 b L 2 b : c M ).
k KM Μ Κ k
where Gy and Ć÷ are defined and tabulated by Ferguson.<i)
f2 = a - h c ^ j (K + M - k) + \N\.
The angle functions ě (Φ^ ,62, φ) are defined by Ferguson and Rutledge<^> as
1/2
×Éě(Č,.Č2.Ö) = ĘĚ PIJ (cos Φ1) PJ^ (cos Θ2) cos ΛΤ φ.
(K + N)\{M + N)l
÷ and >^ are the multipole mixing ratios. For the practical case of dipoie quadru-
pole mixtures Pi and P2 take the values 0, 1 and 2 for dipoie, interference and
quadrupole terms, when the multipole mixing ratios are defined as
<b\\L,-^l\\a} <C|IL,-FL||fc>
<Hl^i||^> <c\\L2\\by
where the reduced matrix elements are those for the emission of radiation,
with the operators phased according to the upper sign in the article by Devons
and Goldfarb.<4) In this we agree with Ferguson and differ from Huby.<5)
Huby gives a general discussion of the phases used by various authors. With
this convention a given sign of mixing ratio gives the same sign to the inter
ference term whatever tRe parity of the radiation.
The quantities Qj^ and Qj^ are solid angle correction factors discussed by
Rose,(6) Gove and Rutledge,<7) Rutledge,^») West,<9) Twin and Willmott,<io>
Yates<ii> and Gossett and Davidson.<i2)
If either or 72 is not observed the formula simplifies considerably.
\í{θι)=Σρκο(αα)Α^Ę÷'^α^,Ρ^, (cos Θ),
κ
^ (Θ2) =ΣρΜο(αα)Α^οΜΜ x'' y'' QmPm (cos Θ).
Ě
POLARIZATION
Although direction-direction triple correlation measurements can usually
determine the spins involved, they give no indication of the parity of the emitted
χ INTRODUCTION
radiation. To do this it is necessary to measure the linear polarization of the
emitted y-rays.
The full expression involving polarization sensitivity in both detectors
involves the rotation matrix elements corresponding to the two sets of Euler
angles describing the position and orientation of the two counters. Aside from
the complexity of the expression, such an arrangement represents a very
difficult experimental situation and provides no information that cannot be
obtained more simply.
The polarization-direction correlation for (ji not observed) is given by
κ
where
(φl, Φ) = Pj^(cos+ ( -1( LL ) S ^ S ö' PÍ(cosč^)ďď$2ö
π' = 0 for L\ electric and 1 for L[ magnetic radiation.
The quantities Sq and 5^ are the Stokes parameters of the detector, is
the polar angle of the detector and ö is the angle between the reaction plane
and the plane defined by the outgoing beam direction and Si.
'1^ {LlLl\K2)
FALL):
(/C + 2)!J (L1L-1\K0)'
No attenuation factors have been included as the necessary integrals have
not been performed.
A similar expression exists for (úé not observed).
Af
The Main Tables (Tables 4 and 5)
The tables presented here are of the function A^j^ for L'j Z.2 A equal
to dipole and quadrupole radiation. The other parameters specifying the
function are the three spin values of a, b and c and the rank of the statistical
tensor k. Due to the limitation of the multipolarities to quadrupole and dipole,
k has a maximum value of 8. It would have been possible to present the tables
in terms of the populations of the magnetic substates, as has been done by
Smith.<i8) The use of statistical tensors, however, substantially reduces the
number of entries in the tables. A further advantage is that for spin values
greater than 4 a full description of the state is not required when multipoles
INTRODUCTION XI
higher than quadrupole are not considered, as the correlation function gives no
information on the statistical tensors with k greater than 8. Consequently a
unique description in terms of population parameters is not possible, any solution
corresponding to the correct set of statistical tensors being equally acceptable.
A disadvantage of this method is that when a solution has been found it is
usually necessary to transform the statistical tensors to population parameters
to ensure the solution does not correspond to one with a negative population
parameter. This is not a difficult operation and a preliminary check is provided
by the condition:
Σ ρ Ι ο ^Ι (a integral),
k
1
(a half-integral).
k ^
The population parameters are certainly non-negative if
Σρίο ^ « (a integral),
k
or Σ p^o ^ Φ — ^ half-integral).
k
Subsidiary Tables (Tables 1, 2 and 3)
Tables are also presented of the Clebsch Gordon coefficients
(-1)«-« (aaa-a|fcO) required to convert the p^^o to population parameters
through the relation
<a α|ρ|α a> = Σ (- \ T \a α α - a|k 0) p^ o-
A final table gives the quantities (L L') needed for polarization measure
ments.
Application
For a specified spin sequence the angular correlation is, in general, a func
tion of up to five statistical tensors (four ratios) and two mixing ratios. On the
other hand, there are nineteen possible combinations of K, Μ and Λ^, and thus
eighteen independent ratios. Thus it is possible to over determine the parameters
and determine whether or not a certain set of data fit the proposed spin sequence.
In practice this number of determinations is seldom performed.
If one of the counters is fixed at 90° only fourteen of the nineteen coefficients
occur giving thirteen ratios. If the triple correlation is measured over the edges
of an octant defined by the forward direction and the plane containing the
fixed counter (see Fig. 2) each correlation determines six ratios. However the
0 = 90» 0 variable sector is common to both measurements and leaves a net
Xll INTRODUCTION
number of nine independent parameters. Examples of the use of this type of
correlation are given in ref 13. If these measurements do not yield a unique
Fixed Y-ray
Beam
PATH OF MOVING Y-RAY
FIG. 2
spin assignment it is probably more rewarding to do a measurement with the
fixed counter at some other angle rather than try to get more information over
the octant already mentioned. Experiments using 126° as the fixed angle have
been found very useful.^17)
LAYOUT OF TABLES 4 AND 5
The entries in the table are defined by the following quantum numbers:
fl, b, c, Li, L'l, ¿2 y ^2y ^> ^ The tables are divided into two sections,
the first for integral values of the spins a, b and c and the second for half-integral
values (page 125 onwards). In each section the functions are tabulated with J,¿
and c as the principal division. These three numbers are printed on the extreme
right of each page at the top, and whenever they change. The table is then
divided into subtables labelled by the quantum numbers Li, L'j, L2, Iii, which
are printed under a, b and c and slightly indented. In each subtable the rows
are labelled by KMN, and the columns by k. Throughout the tables the
quantum numbers are arranged in the ^*^speedometer" ordering.
The correlation function is tabulated for all values of a, b and c up to a
maximum value of 6 for each spin. Only dipoie and quadrupole transitions have
been considered.
METHOD OF CALCULATION
All calculations were performed using a Mercury computer. Subroutines
were written for each of the coefficients occurring in the triple angular correla
tion coefficients, and thoroughly tested by checking results against existing
tables. <i4. is. ιω
INTRODUCTION Xlll
These were then combined with a programme which selected the appropri
ate values of the various parameters and organized the page layout of the tables.
As far as possible, consistent with simplicity in organization, redundant
calculation of coefficients was avoided.
Accuracy of the Tables
Errors could have occurred in several ways: purely numerical errors, e.g.
built up of rounding errors, etc., undetected machine failures, mispunching of
tape or misprinting by the teleprinter, programming errors.
From an examination of the tables and processes used for calculation all
entries would appear to be correct to the four figures given.
The computer used was at all times in good working order and during long
periods of calculating, detectable failure never occurred; all internal transfers
in the machine are parity checked, and the probability of transient machine
faults producing incorrect answers appears to be almost zero.
The output tape was coded using a parity checked code and the probability
of one digit changing to another digit is smaller than that of one digit changing
to some other character. The latter was not observed in a visual check of the
tables and it is assumed therefore that the probability of the former having
occurred in either the punching or the printing stage is negligible.
The correctness of the programme is more difficult to check. However, the
following relationship was used connecting the values given in ref. 18 and the
tabulated values.
CLi (a) = Σ (- IR-'(a oc α - a|/c 0 ) ( k ),
k
where Cjjj^ are the functions tabulated by Smith.^is) Several sections of the
tables were read back into the computer and this transformation applied, the
results being checked with those given in ref. 18. No discrepancies were found.
References
1. A. J, FERGUSON, Angular Correlation Methods in Gamma-ray Spectroscopy, North
Holland Publishing Co., Amsterdam, 1965.
2. A. E. LiTHERLAND and A. J. FERGUSON, Can. J. Phys. 39 (1961) 788.
3. A. J. FERGUSON and A. RUTLEDGE, C.R.P.—615, 1957, revised 1962 (unpublished).
4. S. DEVONS and L. J. B. GOLDFARB, Handbuch der Physik, vol. 42, Springer-Verlag,
Berlin, 1957.
5. R. HuBY, Paris Conference on Nuclear Structure, 895 (1958).
6. M. ROSE, Phys. Rev. 91 (1953) 610.
7. H. GOVE and A. RUTLEDGE, C.R.P.—755 (unpublished).
8. A. RUTLEDGE, C.R.P.—851 (unpublished).
9. H. I. WEST, JR., U.C.R.L.—5451 (unpublished).
XiV INTRODUCTION
10. p. J. TWIN and J. C. WILLMOTT, Nuc. Inst. Methods 22 (1963) 109.
11. H. J. L. YATES, Perturbed Angular Correlations, Appendix 4, North Holland Publi
shing Co., Amsterdam, 1964.
12. C. R. GossETT and C. M. DAVIDSON, Naval Research Lab. Report (in preparation).
13. H. GovE and C. BROUDE, Ann. Phys. 23 (1963) 71. G. KAVE and J. C. WILLMOTT,
Nuclear Phys. 71 (1963) 561.
14. A. SIMON, Numerical Table of Clebsh Gordon Coefficients, O.R.N.L.—1718.
15. A. SIMON, S. H. VAN DER SLUIS and L. C. BIEDENHARN, Tables of the Racah Coeffi
cients, O. R. N. L.—1679.
16. W. R. SHARP, J. M. KENNEDY, B. J. SEARS and M. G. HOYLE, Tables of Coefficients
for Angular Distribution Analysis, C.R.P.—556.
17. P. J. TWIN and J. C. WILLMOTTT, Nuclear Phys. 78 (1966) 177.
18. P. M. ENDT and P. Β. SMITH, Nuclear Reactions, vol. II, North Holland Publishing
Co., Amsterdam, 1962.
