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Gamma-ray linear polarization distribution
ATOMIC
DATA
AND
NUCLEAR
DATA
GAMMA-RAY
TABLES
LINEAR
37,53-68
POLARIZATION
J. RIKOVSKA Clarendon
(1987)
Laboratory,
DISTRIBUTION
and N. J. STONE Oxford,
United Kingdom
The formalism for describing the linear polarization of gamma radiation from oriented nuclei is reviewed. Coefficients Ak2 are introduced which give the polarization-sensitive terms in the directional distribution of radiation the same form as the polarization-insensitive terms. Expressions for ,4k2 coefficients are given for cases involving admixture of up to three multipole components in the observed transition. Also given are specific expressions appropriate to tests of time-reversal invariance in gamma decay. The coefficients C,(L, L', If, 1;) tabulated herein are to enable the evaluation of .4k2. Values of C,(L, L', if, I,) are listed for the ranges X == 2-6, L, L' = 1, 2, 3, and integer spins 1 < I, < 8 and halfinteger spins 3/2 G 1, < 15/2. Analytical expressions for Legendre polynomials P,(cos 0) and associated Legendre polynomials p2(cos 0) for X < 6 are provided additionally, to facilitate the calculation of polarization distributions. Examples of the use of the tables in specific cases are given. e 1987 Academic Press. Inc
0092-640X/87
$3.00
Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
53
Atomc
Data and Nuclear
Data Tables.
Vol. 37. No. 1, July 1987
J. RIKOVSKA
Gamma-Ray Linear Polarization Distribution
and N. J. STONE
CONTENTS
INTRODUCTION . . .._.__......_......................... Gamma-Ray Linear Polarization Distribution Time-Reversal Invariance in Gamma Decay
54 54 56
EXPLANATION
OF TABLES
58
OF USE OF TABLES
59
EXAMPLES
TABLES I. Analytical Expressionsfor Legendre and Associated Legendre Polynomials II. Linear Polarization Distribution Coefficients
61 62
INTRODUCTION Atomic nuclei oriented with respect to a definite direction in space can produce gamma rays polarized to a high degree. The main interest concerns linear polarization, inasmuch as the directional dependence of the linear polarization of such gamma rays dependsupon the type of the radiation (magnetic/electric) and thus on the relative parity of the nuclear states between which the transition occurs. This important information cannot be obtained from a measurement of the directional distribution of gamma rays from oriented nuclei, performed by polarization-insensitive detectors, which depends only upon the spins of the states and the multipole order of the radiation. Further, for mixed multipole radiation, the frequent occurrence of more than one mixing ratio compatible with the measureddirectional distribution can be resolved with the aid of a measurementof the polarization in a specified direction. Thus linear polarization measurementsperformed with sufficient accuracy on oriented nuclei provide, in combination with gamma-ray directional distribution data, more complete information on excited nuclear statesand the transitions between them. They are also of prime importance in testing such fundamental interaction symmetries in nuclei astime-reversal invariance and parity nonconservation.
Analysis of a linear polarization measurement generally involves comparison between the experimental data and calculation. Formulas for certain special caseshave been published. l-4 In the presentation below, the method outlined is applicable to the most general calculation of the polarization distribution. The approach here is especially convenient when more than two multipole components are present in the transition. Gamma-Ray Linear Polarization Distribution The angular distribution of linearly polarized gamma rays from an axially oriented ensemble of nuclei can be written, following Steffen and Alder,’ as an expansion in Legendre and associatedLegendre polynomials Ph and P::
+ c
B,U,A;zPZ,(cos Qcos 2#.
(1)
h=WeFi
Equation (1) applies to a gamma-ray transition from state 111)to state II,), wherein the state 11;)is populated either directly or through intervening unobserved transitions 54
Atomc
Data
and
Nuclear
Data
Tables.
Vol.
37.
No.
1, July
,987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
from the oriented parent state ]Z). Z, Zi, and Z, are the corresponding spin values. 0 is the angle of emission of a gamma quantum with respect to the orientation axis of the parent nuclear ensemble, and # is the angle between the electric E vector of the quantum and the plane defined by the orientation axis and the direction of gamma emission. The coefficients
for j + 1 unobserved transitions in a direct cascade. For a single transition between ]I,$ and ]Z,) states &(zkb)
=
2 L
aL ~A(~k~lJ%
(5)
where aL is a fraction of a component carrying L units of angular momentum (for example multipole order for gamma radiation, partial wave order for a particle transfer) and, again, c aL = 1. In particular, for the most frequent case of a gamma-ray transition with mixed components L and L’ = L + 1 we have
B,(Z) = [(2X + 1)(21+ 1)]“2 2 (- l)‘+m
u,(zkz[LL’) = [ u,(&z,L) + 62uA(zkJIL’)]/( 1 + 6*),
where 6 is the multipole mixing ratio and S*/( 1 + S2) is thus the fractional L’ contribution to the total intensity. For gamma-ray transitions where internal conversion is a significant decay mode this is included by writing
are tensors describing the degree of orientation of the ground state IZ) of the parent nucleus.5 Multiplying the Wigner 3j symbols are the Boltzmann population parameters p(m), which are dependent on the mechanism of orientation of the state IZ). For example, P(m) = ew(mWT)l
c exP(mAM/T) m=-I
for pure magnetic dipole orientation, splitting (in millikelvin) is
uA(zkzlLL’) = [( 1 + a)UX(zkz&) + (1
(24
i
G’b)
exp(-m2AE/T)
Ar = 1.160{3@‘,1)/[4Z(2Z-
(2~)
(7)
+bp2],
(8)
u,(zkz,L) = (- l)‘k+L’+L+A x [(2& + 1)(2Z,+ I)]“* [;
;
9.
(9)
Specific expressions for &(ZkZl) for alpha and beta decay and electron capture and s-wave neutron capture are given in Ref. 8. The angular distribution coefficients Ax for a y transition between states IZi) and IZr> with multipole components aL and a’L’ (a = E or M) used here are defined as
(24
AA= C FA(LL’z~zih(~L)r*(a ‘L’IIC I$~L)12, ( 10) CL/L CL‘ where y(aL) is the absolute transition amplitude’ of the aL component. The multipole mixing ratio is given by G(a’L’faL) = y(a’L’)/y(uL). The Fx are the functions
where n is the number of branches via which the state ]Z;) is fed, and branching intensities are normalized so that C w, = 1. For each branch, * - - U,(zjZJ
+ b)s*],
where q is the ratio of the EO to E2 reduced transition matrix elements. The coefficient &(ZkZ/L), appearing in Eqs. (5)-(g), is given in terms of the Wigner 6j symbols as
where the electric quadrupole moment Q is in barns and the electric field gradient V,, is in lOI V/cm* (see Vianden6 and Hagn7). The deorientation coefficients U, describe the effect of any transition (radiation or particle) from the oriented parent state and succeeding states which precede the state ]Zi) from which the observed radiation is emitted. In general,8 KU - - * ZJ = 2 w,[ UA(Z- * * Zi)ln, (3)
UA(Z* * * ZJ = u,(zz,)u~(z’z*)
+ (1
Xs2UX(ZkZkL’)]/[bq2~‘+(1 +a)+(1
is l)],
a)
u,(z,&LL ‘) = [bq%* + ( 1 + a) ux(ZkZkL)+ ( 1 + b)
m=-I
and the energy splitting (in millikelvin)
1 +
version mode also contributes to the transition with AZ = 0, for example an EO + Ml + E2 multipolarity mixture, we have
In Eq. (2b), the magnetic dipole moment p of the state II) is in nuclear magnetons and the polarizing hyperfine field B is in tesla (see Krane6 for a table of numerical values). For pure electric quadrupole orientation, p(m) = exp(-m2AE/T)/
+ b)62UA(zkz&‘)]/[(
where a and b are the conversion coefficients of the L and L’ components, respectively. When the EO internal con-
where the energy
AM = 0.366pBJZ.
(6)
Fh(LL’ZfZi) = (- 1)“+“+‘[(2X X (2L’ + 1)(2Zi + l)]“*
(4) 55
Atomic
(f
+ 1)(2L + 1) “;
Data and Nuclear
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Vol. 37. No. 1, July 1997
J. RIKOVSKA
Gamma-Ray
and N. J. STONE
Explicit expressions for Ax for transitions with two and three multipole components can be found in Refs. 3, 4. In Eq. (l), the ,4)X2coefficients in the polarizationsensitive term depend on the type u of the electromagnetic radiation through the phase factor (-l)A’d’, where h(E) = 0 and A(M) = 1. They differ from the AX2 of Ref. 1 in that they include the factor 2[( h - 2)!/( X + 2)!] ‘12, leading to simplification of Eq. (1) and Eq. (20) below. We have A;2 = -
[we,#=o)+
(14)
This formulation gives the linear polarization-sensitive term in Eq. (1) the same form as the directional distribution term, but with the Fx coefficients replaced by the C, coefficients (compare Eqs. (10) and (12)). Then, using Eqs. ( 12)-( 14) the Ai coefficients for transitions with two and three multipole components can be written as
-{(-l)"'"'C~(LLz,-Z,)+[(-l)A'v" ‘~(~)+~~+L'+h]Ch(LL'I/z~)~(u~L~/uL)+ (- I)“‘“”
x C,(L’L’z~z;)62(u’L’/uL)~/[
1 + a2(u’L’/uL)],
(I 5)
and
A;? (uLu'L'u"L")
= -(( - l)*‘“‘C,(LLZ/Z,) + [(- 1)““”
+(-I) *(c)+L+L’+h]C,(LL’Zfz,)qu’L’/uL) +[(-l)*‘“V)+(-l) +1(-l)
A(0
+ (-
A'"'+L+r~+X]C,(LL'Z~Zi)~
(U”L”/UL)
l)A(d)+l.‘+L”+X]CA(LILnlfz~)
x G(u'L'/aL)&r"L"/uL)+(-l)"'"'C,(L'L'zJzi) x G'(u'L'/uL)S(-
1)-""'C,(L"L"Z,-z,)S2(U"L"/UL))/ [ 1 + S’(u’L’/uL) + fi’(u”L”/uL)].
(17)
where Z’(B)is given in Eq. ( 17) and Q is the linear polarization detection efficiency (for details seeRef. 11). The factors Bx, Ux, Ai, and Pi are normalized so that &, = U, = A0 = PO = 1. Angular momentum coupling selection rules cause terms with X > X,,, to vanish as follows: Bh, X,,, = 21; Ah, h,,, = 2L,,, or 2Z,, whichever is smaller: U,, h,,, = 2Zk or 2Z,, whichever is smaller. A;* vanish for h < 2 and for X,, = 2L,, or 2Zi, whichever is smaller. Here L,,, is the largest multipole order present in the transition. For parity conserving and time-reversal invariant radiation only even values of X contribute to lV(0, 4); otherwise X-odd terms must be also considered. In this paper, we give listings of the C, coefficients with a range encompassing the casein which the highest occurring multipole component is octupole. For such a case, all the coefficients in Eq. (1) are required up to X = 6. In Table I. we have given for convenience the analytical expressionsfor the normalized Legendre polynomials and the unnormalized associated Legendre polynomials of Eq. (1). In Table II we list CA(LL’ZfZi) for h = 2-6, L and L' = 1, 2, and 3, and integer and halfinteger spin values between 1 and 8. The coefficients Bx, U,, and Fx have been tabulated by Krane for the range X = l-4 in Refs. 5 and 9 and for X = l-6 in Ref. 10. The coefficients F,+ and U, (X = l-4) can also be found in Appendixes 2 and 3 of Ref. 1.
The C, change sign with interchange of L,L’ as
+(-I)
w(k1C/=d2)1,
where the normalization is such that - 1 < p(0) < + 1. Our ,4;, coefficients do not include linear polarization efficiency, which many authors (for example Ref. 1) include in their coefficients AX2. This has its origin in a very general definition of angular distribution W = Tr(t . p), where t is the efficiency matrix, representing the observation process, and p is the density matrix of the system of oriented nuclei. In practice, efficiency is not usually calculated but is measured in an independent experiment. Thus, without loss of generality, the measured degree of linear polarization a can be related to the calculated value by A = P(W2, (18)
= [(A - 2)!/(X + 2)!]“2
A;2(oLu'L')=
Distribution
P(B) = [ W(B, ys = 0) - lv(l3, $ = 7r/2)]/
where
CA(LL'ZfZ,) = (- l)L+L’+“C~(L’LZj-Zi).
Polarization
of gamma emission. Thus the degree of linear polarization is defined as
2 (-l)“‘d’C~(LL’Z~Zi) oLdL’
C,(LL’Z,Zj)
Linear
(16)
Time-Reversal Invariance in Gamma Decay
The angular dependence of the distribution W(8, 4) is governed by the ordinary Legendre polynomials ~(COS e), the unnormalized associated Legendre polynomials Px(cos 8) and cos 2$. Examination of Eq. (1) showsthat the maximum variation in lV(0, J/) is for $ = 0 and I/ = r/2, that is between the gamma quantum being emitted with E vector parallel or perpendicular to the plane determined by the orientation axis and the direction
Linearly polarized radiation from oriented nuclei can be used in testing time-reversal (T) symmetry in ydecay.r2-14T violation would manifest itself through the mixing ratio of a y transition with two multipole components L, L' such that G(u’L’/uL)
56
= I6(u’L’/uL)l(cos
Atomic
Data and Nuclear
17 + i
Data Tables,
sin II),
(19)
Vol. 37, No. 1, July 1987
J. RIKOVSKA and N. J. STONE
Gamma-Ray Linear Polarization Distribution
where the phase 11differs from 0 or x. The linear polarization of gamma rays from an axially symmetric polarized state of pure, unmixed parity, including T-sensitive terms, is given by
of the degree of linear polarization reversal asymmetry parameter AT.
p(0) and the time-
References =
2 B,lJ,A,P,(cos X=eWn +
c B,&A;* h=l3Wl
+ c
1. R. M. Steffen and K. Alder, in The Electromagnetic Interaction in Nuclear Spectroscopy, edited by W. D. Hamilton (North-Holland, Amsterdam, 1975), p 505
0)
p: (cos 0)cos 2# @sin 2#.
(-i)B,&l&2ff(COS
2. P. J. Twin,
3. T. Aoki, K. Furuno, Y. Tagishi, and J-Z. Ruan ATOMIC DATAANDNUCLEARDATATABLES 23,349 (1979)
(20)
h=odd
Here w,(e) and l+‘,(e, #) denote normal T-even directional and linear polarization distributions in the notation used above. I+‘,(fI, 11/)is the T-violating term. From the 3; symbol in the expression for ,412 (Eq. (12)) it follows that h must be at least 2 and since only odd-X terms contribute to the T violation, the first such term is for X = 3. The coefficients Ak2 for X = odd can be expressed as A;2(aLafL’) = -[(- l)A(ul)-(-l)A(o)tL+L’+h] X CA(LL’IfIf)16(u’L’/aL)li
sin q/[ 1 + I&(u’~‘/uL)~~].
4. J. Rikovska,
++,) - wr
mv,4d+
24-26,963
6. K. S. Krane, Hyperfine Interactions (1983); R. Vianden, ibid., p. 1081 7. E. Hagn, Hyperfine
(1985)
Interactions
15/16,
1069
22, 19 ( 1985)
8. K. S. Krane, in Low Temperature Nuclear Orientation, edited by N. J. Stone and H. Postma (NorthHolland, Amsterdam, 1986), p. 65
(21)
9. K. S. Krane, Los Alamos Scientific Laboratory LA-4677 (1971) 10. K. S. Krane. in Ref. 8, Appendixes
Report
5. 6
11. J. Rikovska, N. J. Stone, and V. R. Green, Nucl. Instrum. Methods A 241,46 1 ( 1985) 12. R. J. Blin-Stoyle, in Fundamental Interactions and the Nucleus (North-Holland, Amsterdam, 1973), Chaps. 1, 10
- em, n + km
13. N. K. Cheung, H. E. Hendriksen, Phys. Rev. C 16, 2381 (1977)
w~-kd+~m)i = w3 tern, 1cldi w (em).
Hyperfine Interactions
5. K. S. Krane, NUCLEAR DATA TABLES 11,407 (1973)
Note the difference in sign inside the bracket in Eq. (2 I) as compared to Eq. (15). This arises because we are dealing with a complex mixing ratio (see Eq. (19)). The X = 3 term in w3(0, +) reaches its maximum for 8, = 54.7” f n* and $, = 45” + na/2. For this choice of angles W2(B,, +,,,) and the X = 2 term in w,(0,) vanish. W3 changes sign under the time-reversal transformation13 0 + r - 8, $ + r + +; hence the time-reversal asymmetry AT may be introduced: A de, +) = [ we,
in Ref. 1, p 701
(22)
and F. Boehm,
14. K. S. Krane, in Ref. 8, p. 249
The tabulated coefficients C,, used in conjunction with Eqs. ( 17) and (22) and the other tabulated quantities Bx, U,, F,,, Px, and Pf , enable calculations to be made
15. D. I. Bradley, N. J. Stone, J. Rikovska, D. Novakova, and J. Ferencei, J. Phys. G 12, 115 (1986)
57
Atomic
Data
and
Nuclear
Data
Tables.
Vol.
37.
No.
1, July
1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
EXPLANATION TABLE I.
OF TABLES
Analytical Expressions for Legendre and Associated Legendre Polynomials Expressions for normalized Legendre polynomials Px(cos 0) and unnormalized associatedLegendre polynomials P?(cos 0) to be used with Eqs. ( 1) and (20) are given for 1 G X < 6.
TABLE II. Linear Polarization Distribution Coefficients The coefficients CA(LL’ZfZj) are listed for X = 2-6; L, L' = 1, 2, and 3; integer spins 1 < Zi < 8; Half-integer spins 3/2 < Z, < 15/2. L L’ 1~ Ii
CA
Multipole order of the first multipole component Multipole order of the second multipole component Spin of the final state Spin of the initial state CA(LL'ZfZi), X = 2-6 (seeEqs. (13) and (14))
58
J. RIKOVSKA
and N. J. STONE:
EXAMPLES
Gamma-Ray Linear Polarization Distribution
OF USE OF TABLES
Example 1 Calculation of the value of linear polarization p(B = 90”) (Eq. (17)) of the 2015-keV (ZT = 4+, Zj = 4’.) transition with mixed multipole Ml and E2 components (6 (E2/MI) = 0.83 (Ref. 15)) in 56Fe:The initial state at 4100.4keV is populated directly in 0 decay (pure L = 1, no parity change) of the ground state (I” = 4+) of 56Co,oriented in the Fe matrix at low temperatures, Observation of the magnetic dipole orientation due to hyperfine interaction of the magnetic dipole moment of the 56Coground state (P = 3.830 nm.) in Fe (Z&r = -28.9 T) at 5.83 mK is considered. (a) Orientation coefficients Bx (Eqs. (2)-(2.b)) are Bz = I .3556,
B4 = 0.5543
for AM= 10.13mK,
AM/T= 1.7378.
(b) Deorientation coefficients U, (Eq. (9)) are U,(44 1) = 0.8500,
U4(44 1) =0.5000.
(c) Angular distribution coefficients Ax (Eqs. (10) and (11)): Using F2( 1144) = -0.4387,
E;( 1244) = -0.3354,
F,(2244) = 0.2646,
and F4( 1144) = 0,
F4( 1244) = 0,
F,(2244) = -0.498 1,
for 6(E2/Ml) = 0.83, we have A2 = -0.48 15 and
A4 = --0.2032.
(d) The linear polarization distribution coefficients A’,* (Eqs. (12)-( 14)): Using C,( 1144) = -0.2 194,
C,( 1244) = -0.0559,
C,(2244) = -0.1323,
and C,( 1144) = 0,
C,(1244) ==0,
C,(2244) = -0.0415,
for 6(E2/Ml) = 0.83, and A(a) = l(0) for Ml (E2) components, respectively, we have from Eq. (15) A& = -0.02 10 and
A& = 0.0 169.
(e) The ordinary Legendre polynomials for x = cos 90” are P&X) = -0.5000
and
P4(x) = +0.375
and associated Legendre polynomials are e(x) = 3.0000
and
59
ti (x) = -7.5000.
Atomic
Data and Nuclear
Data Tables.
Vol. 37, No. 1, July 1997
J. RIKOVSKA
Gamma-Ray
and N. J. STONE
EXAMPLES
Linear
OF USE OF TABLES
Polarization
Distribution
continued
(f) Using Eq. (I), for X = 0, 2, 4 allowed by the selection rules, we obtain in the present case W(0 = 90”, ic,= 0”) = 1 + BzU2A2P2(cos 0) + B4U&Pq(cos
0)
+ B2 lJ2A;2P$(cos 6)cos 2+ + BJIqA&P:(cos
B)cos 2$
= 1.1486 and W(O = 90”,$ = 90”) = 1.3640. Hence P(B = 90”) = -0.0857. Example 2 Calculation of the time-reversal asymmetry parameter AT(/3, $) (Eq. (22)) for the transition considered in Example 1, at critical angles 0, = 54.7” and $J, = 45”: The only X-odd value allowed by the selection rules is X = 3. Therefore we need, together with values of coefficients quoted above for X = 2 and 4, B3 = -0.9680, C,( 1144) = 0,
U3(441) = 0.7000,
C,( 1244) = -0.0802,
C,(2244) = 0.
From Eq. (2 1), we obtain A& = iO.0788 sin 9. For P4(x) = -0.3894,
x= cos 54.7”, we have P2(x) = 0, The expression A,(B,
e(x)
= 5.7735.
for Ar(B, $) will be
= 54.7”, $, = 45”) = -iB,lJgl’&(cos
8,)sin 2$,)/
[ 1 + Bz U~A~P~(cos 8,) + 84 iJ&P4(cos
it?,,,)]
and the resulting value AT(Bm, $,J = -0.3017 sin 7. Comparing AT with a corresponding experimental the time-reversal nonconserving factor sin q.
60
value,4 we can determine
Atomic
Data
and
Nuclear
Data
Tables,
Vol.
37.
No.
1. July
1987
J. RIKOVSKA
TABLE I. Analytical
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
Expressions for Legendre and Associated Legendre Polynomials See page 58 for Explanation of Tables Normalized
Legendre Polynomials
P,(x) = 1 P,(x) =x Pz(x) = i(3x2 - 1) Pj(X) = $(5x3 - 3x) P4(x) = 335x4 - 30x2 + 3) P,(x) = %63x5 - 70x3 + 15x) Ps(x)=+&!31x6-315x4+ Unnormalized
105x2-5)
Associated Legendre Polynomials
P$(x) = 3( 1 - x2) P:(x) = 15( 1 - x2)x Pi(x) = 3 1 - x2)(7x2 - 1) P:(x) = +q 1 - x2)(3x3 - x) P;(x) = $q 1 - x2)(33x4 - 18x2 + 1)
61
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’
Ii
If
1 1 1 2 1 1 1 2 2 3 2 2 3 3
1 1 2 2 1 2 3 2 3 3 2 3 3 3
1 1 2 1 1 1 2 3 1 1 1 2 2 3 2 2 3 3
1 2 2 1 2 3 3 3 1 2 3 2 3 3 2 3 3 3
2 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1
2 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3
0 1 1 1 2 2 2 2 2 2 3 3 3 4 112 112
l/2 312 312 312 312 312 512 512 512 512 512 512 712 712
712 912 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3
Gamma-Ray Linear Polarization Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
C2
c3
1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.3535 -0.1768 -0.1768 0.1768 0.0354 0.0791 0.0707 -0.1768 -0.1581 0.1414 0.0505 0.0945 -0.1768 0.0589
312
312 312 312 312 312 312
0.2500 -0.1443 0.2500 -0.2000 -0.1291 0.0408 -0.1581 0.2000 0.0500 0.0986 0.0667 -0.1786 -0.0845 -0.0500 0.0714 0.1157 -0.1667 0.0833
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
0.2988 0.2092 -0.1559 0.0199 0.1494 -0.1336 0.2390 -0.2092 -0.1021 0.0488 -0.0640 -0.1429 0.0598 0.0598 0.1091 0.0639
0.0000 0.0000 0.1054 -0.0589 0.0000 0.0791 0.0000 0.0000 -0.0690 -0.0722 0.0000 0.0211 0.0000 0.0000 0.0184 0.0472
312 312 312 312
312 312 312 312
312
312 312
c4
c5
c6
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1291 0.0000 0.0000 -0.0577 -0.0913 0.0707 0.0000 0.0000 0.0126 0.0373 0.0000 -0.0567 0.0000 0.0000 0.0173 0.0000 0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0891 0.0000 0.0000 0.0668 0.0594 0.0166 -0.0297 0.0000 0.0000 -0.0273 -0.0255 -0.0133 0.0445 0.0000 0.0000 0.0060
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
62
Atomic
Data and Nuclear
Data Tables.
Vol. 37. No. I. July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
rABLE II. Linear Polarization See page
L
If
L’
2 2 3 2 2 3 3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2
2 3 3 2 3 3 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2
2 3 2 2 3 3
3 3 2 3 3 3
3 2 2 3
3
1 1 1 2 2 3 1 1 1 2 2 3 1
2
2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1
2 2 2 2 2 2 l/2
112
l/2 312 312 312 312 3/2 3/2 512 512 512 512 512 512 712 712 7/2 712 712 712 Q/2 Q/2 Q/2
11/2 0 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4
512 512 512 512 512 512 w 5/2 512 512 512 512 512 5/2 512 512 512 512 5/2 512 512 512 512 512 512 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
-0.1707 -0.0437
Polarization
Distribution
58 for Explanation
c2
Ii
Linear
Distribution
Coefficients
of Tables
c3
c4
0.0000
0.0064 0.0054
0.0000
0.0000
0.0000
0.0000
-0.0604
c5
c6
-0.1096 0.0854 0.1263 -0.1494 0.0996
0.0000 0.0000 0.0249 0.0000 0.0000
-0.0297 -0.0007 -0.0010 0.0099
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0013
0.0000
0.0000
0.2673 -0.0945 0.2673 0.1871 -0.1581
0.0000 0.0646 0.0000 0.0000 0.0926
-0.0514 0.0182 -0.0514 0.0000 0.0000
0.0267 0.0954 -0.1468 0.1470
-0.0685 0.0000 0.0656 0.0000
0.0579 0.0588
-0.2138 -0.0845 0.0525
0.0000 -0.0742 -0.0598
-0.0954 -0.1245 -0.0089 0.0668
0.0000 -0.0031 0.0000 0.0000
0.1157 0.0619 -0.1623 -0.0179
0.0226 0.0528 0.0000
-0.1336 0.0954 0.1324 -0.1336 0.1114
-0.0590 0.0000 0.0000 0.0302 0.0000 0.0000
0.2887
0.0000
0.2474 -0.1157 0.2165 0.1732
0.0000 0.0722 0.0000
-0.1581 0.0309
0.0000 0.0845 -0.0722
0.0619 -0.1479 0.0914 -0.2165 -0.0722 0.0546 -0.1134 -0.1091 -0.0481 0.0722
0.0000 0.0527 0.0000 0.0000 -0.0772 -0.0510 0.0000 -0.0170 0.0000 0.0000
63
0.0054 0.0257 0.0000 0.0000 -0.0337 -0.0331 -0.0133 0.0257 0.0000 0.0000 0.0089 0.0098 0.0074 -0.0343 -0.0012 -0.0017 0.0148 -0.0023 -0.0711 -0.0372 0.0174 -0.0118 0.0000 0.0000 0.0522 0.0558
0.0000
0.0000
-0.0545 0.0000 0.0000 0.0000
0.0000
0.0000 0.0000 0.0326 0.0000
0.0000
0.0000 0.0000 0.0000
0.0000
0.0000 -0.0133 0.0000 0.0000 0.0000
0.0000
0.0000 0.0000 0.0034
0.0000
0.0000 0.0000 -0.0005 0.0000
0.0000
0.0000
0.0000
0.0000 0.0000
0.0435 0.0000
-0.0386 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0326 0.0000 0.0000 0.0000 0.0000
0.0000 0.0352 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0182 0.0000 0.0000
0.0000 0.0355 0.0000 0.0000 -0.0369 -0.0372 -0.0123 0.0118 0.0000
Atomc
Data and Nuclear
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0181 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0073 0.0000
Data Tables.
“ol
37. NO. 1, July 1997
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
TABLE II. Linear Polarization See page
L
L’
1 1 2 2 3 2 2 3 3 3 2 2 3 1 1 1 2 2 3 1 1 1
If
Ii
2 3 2 3 3 2 3 3 3 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3
2 2 3 1 1 1 2
3 1 2 3 2
2 3
3 3
2 2 3 3
2 3 3
3 2 2 3 1 1 1 2 2 3 1 1 1 2
3 2 3 3 1 2 3 2 3 3 1 2 3 2
3
112 312 312 312 512 512 512
c3
3 3 3 3 3 3
0.1203 0.0603 -0.1546 0.0000 -0.1443 0.1031 0.1364
0.0257 0.0564 0.0000 -0.0564 0.0000 0.0000
3 3
-0.1203 0.1203
712
0.2728 0.2338 -0.1263 0.1818 0.1637 -0.1575
512
512 512
712 712 712 712 712
712 912
912 912 912 912 912 1112 1112 1112 1312
712 712 712 712 712 712 712 712 712 712 712
712 712 712
712 712 712 712
712 712
712 712
712 712 712 1 2 2 2 3 3 3 3 3 3 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4
0.0337 0.0390 -0.1458 0.0546 -0.2182 -0.0630 0.0558 -0.1247 -0.0967 -0.0727 0.0764 0.1236 0.0591
0.0340 0.0000 0.0000 0.0000 0.0000 0.0739 0.0000 0.0000 0.0790 -0.0739 0.0000 0.0426 0.0000 0.0000 -0.0790 -0.0445 0.0000 -0.0257 0.0000 0.0000
0.1091 0.1391 -0.1091
0.0282 0.0589 0.0000 -0.0538 0.0000 0.0000 0.0370 0.0000
0.1273
0.0000
0.2612 0.2238 -0.1324 0.1567 0.1567 -0.1567 0.0357 0.0224 -0.1429 0.0285 -0.2194 -0.0559 0.0567 -0.1323
0.0000 0.0000 0.0739 0.0000 0.0000 0.0749 -0.0747 0.0000 0.0349 0.0000 0.0000 -0.0802 -0.0395 0.0000
-0.1481 0.0130 -0.1491
64
Polarization
Distribution
58 for Explanation
c2
3
Linear
Distribution
Coefficients
of Tables
C8
c5
c4
0.0000
0.0000
0.0111 0.0124 0.0087
0.0000 0.0000
0.0055 0.0000 0.0000
-0.0348 -0.0017 -0.0022 0.0183 -0.0032 -0.0570 -0.0298 0.0161 0.0063 0.0000 0.0000 0.0484 0.0531 -0.0031 0.0359 0.0000
0.0000 -0.0003 0.0000
0.0000 0.0000 -0.0303 0.0000
0.0218
0.0000 0.0000 -0.0211
-0.0112 0.0023 0.0000 0.0000 0.0129 0.0145 0.0095 -0.0340 -0.0021 -0.0027
0.0020 0.0000
-0.0008 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0350 0.0000 0.0000 0.0000
0.0000 -0.0389 -0.0398
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0071 0.0000 0.0000 -0.0012
0.0000 0.0000 -0.0290 0.0000 0.0000 0.0000 0.0000 0.0000 0.0218 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0106 0.0000 0.0000 0.0000 0.0000 0.0000 0.0033 0.0000 0.0000 -0.0006
0.0209 -0.0040
0.0000 0.0000
-0.0484 -0.0254
0.0000 0.0000
0.0140 0.0000
-0.0254 0.0000 0.0000 0.0000 0.0000 0.0000 0.0342 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0253 0.0000 0.0000 0.0000 0.0000 0.0000 0.0230 0.0000 0.0000 0.0000 0.0000
0.0150 0.0161 0.0000 0.0000 0.0455 0.0507 -0.0051 0.0337 0.0000 0.0000 -0.0401 -0.0415
Atomic
Data and Nuclear
0.0001
Data Tables.
Vol. 37. NO. 1, July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
TABLE II. Linear
Polarization Distribution Coefficients See page 58 for Explanation of Tables
L
L’
If
Ii
C2
c3
c4
c5
c6
2 3 1 1 1 2 2 3 2 2 3 3
3 3 1 2 3 2 3 3 2 3 3 3
4 4 5 5 5 5 5 5 6 6 6 7
4 4 4 4 4 4 4 4 4 4 4 4
-0.0867 -0.0893 0.0798 0.1261 0.0581 -0.1424 0.0230 -0.1510 0.1140 0.1410 -0.0997 0.1330
-0.0316 0.0000 0.0000 0.0302 0.0608 0.0000 -0.0512 0.0000 0.0000 0.0393 0.0000 0.0000
-0.0102 -0.0044 0.0000 0.0000 0.0142 0.0161 0.0100 -0.0326 -0.0025 -0.0031 0.0229 -0.0047
-0.0231 0.0000 0.0000 0.0000 0.0000 0.0000 0.0085 0.0000 0.0000 -0.0015 0.0000 0.0000
-0.0128 0.0000 0.0000 0.0000 0.0000 0.0000 0.0044 0.0000 0.0000 -0.0009 0.0001
3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 2 2 3 3
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 2 3 3 3
312 5/2 512 512 712 7/2 712 712 712 712 912 912 912 912 912 912 11/2 11/2 11/2 11/2 1112 11/2 13/2 1312 13/2 15/2
912 912 912 912 912 912 912 912 912 912 912 g/2 912 912 912 912 g/2 912 912 912 912 912 912 912 912 912
0.2523 0.2163 -0.1364 0.1376 0.1514 -0.1559 0.0373 0.0098 -0.1398 0.0092 -0.2202 -0.0503 0.0573 -0.1376 -0.0785 -0.1009 0.0826 0.1281 0.0573 -0.1376 0.0308 -0.1514 0.1180 0.1424 -0.0917 0.1376
0.0000 0.0000 0.0733 0.0000 0.0000 0.0718 -0.0751 0.0000 0.0287 0.0000 0.0000 -0.0810 -0.0355 0.0000 -0.0357 0.0000 0.0000 0.0318 0.0622 0.0000 -0.0490 0.0000 0.0000 0.0412 0.0000 0.0000
-0.0427 -0.0224 0.0141 0.0220 0.0000 0.0000 0.0434 0.0488 -0.0064 0.0311 0.0000 0.0000 -0.0410 -0.0427 -0.0094 -0.0093 0.0000 0.0000 0.0154 0.0175 0.0104 -0.0312 -0.0028 -0.0034 0.0245 -0.0054
0.0000 0.0000 -0.0221 0.0000 0.0000 0.0000 0.0000 0.0000 0.0333 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0244 0.0000 0.0000 0.0000 0.0000 0.0000 0.0096 0.0000 0.0000 -0.0018 0.0000 0.0000
0.0103 0.0000 0.0000 -0.0224 0.0000 0.0000 0.0000 0.0000 0.0000 0.0233 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0143 0.0000 0.0000 0.0000 0.0000 0.0000 0.0054 0.0000 0.0000 -0.0011 0.0001
3 2 2 3 1 1 1 2 2 3 1
3 2 3 3 1 2 3 2 3 3 1
5 5 5 5 5 5 5 5 5 5 5
0.2453 0.2103 -0.1391 0.1227 0.1472 -0.1552 0.0385 0.0000 -0.1369 -0.0057 -0.2208
0.0000 0.0000 0.0726 0.0000 0.0000 0.0694 -0.0753 0.0000 0.0238 0.0000 0.0000
-0.0386 -0.0202 0.0134 0.0258 0.0000 0.0000 0.0417 0.0472 -0.0073 0.0284 0.0000
0.0000 0.0000 -0.0198 0.0000 0.0000 0.0000 0.0000 0.0000 0.0324 0.0000 0.0000
0.0081 0.0000 0.0000 -0.0202 0.0000 0.0000 0.0000 0.0000 0.0000 0.0233 0.0000
2 3 3 3 4 4 4 4 4 4 5
65
Atomsc Data and Nuclear
0.0000
Data Tables.
vol. 37. NO. 1, July 1997
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’
If
Ii
1 1 2 2 3 1 1 1 2 2 3 2 2 3 3
2 3 2 3 3 1 2 3 2 3 3 2 3 3 3
5 5 5 5 5 6 6 6 6 6 6 7 7 7 8
3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 2 2 3
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 2 3 3
512 712 712 712 912 912 912 912 912
3 2 2 3 1 1 1 2 2
3 2 3 3 1 2 3 2 3
912
1112 1112 1112 1112 1112 1112 1312 1312 1312 1312 1312 1312 1512 1512 1512
3 4 4 4 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Gamma-Ray Linear Polarization Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
c2
c3
c* 0.0000
-0.0456 0.0577 -0.1415 -0.0716 -0.1095 0.0849 0.1297 0.0566 -0.1334 0.0371 -0.1510 0.1213 0.1435 -0.0849 0.1415
-0.0817 -0.0323 0.0000 -0.0387 0.0000 0.0000 0.0332 0.0633 0.0000 -0.0470 0.0000 0.0000 0.0428 0.0000 0.0000
-0.0417 -0.0436 -0.0086 -0.0129 0.0000 0.0000 0.0163 0.0187 0.0107 -0.0297 -0.0031 -0.0037 0.0258 -0.0059
1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112
0.2397 0.2054 -0.1410 0.1106 0.1438 -0.1545 0.0395 -0.0079 -0.1343 -0.0174 -0.2212 -0.0418 0.0581 -0.1445 -0.0658 -0.1159 0.0869 0.1310 0.0560 -0.1298 0.0422 -0.1501 0.1242 0.1444 -0.0790
0.0000 0.0000 0.0718 0.0000 0.0000 0.0674 -0.0754 0.0000 0.0198 0.0000 0.0000 -0.0821 -0.0296 0.0000 -0.0410 0.0000 0.0000 0.0343 0.0642 0.0000 -0.0452 0.0000 0.0000 0.0441 0.0000
6 6 6 6 6 6 6 6 6
0.2350 0.2015 -0.1424 0.1007 0.1410 -0.1539 0.0403 -0.0144 -0.1319
0.0000 0.0000 0.0711 0.0000 0.0000 0.0658 -0.0754 0.0000 0.0164
66
c5
c6
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0254 0.0000 0.0000 0.0000 0.0000 0.0000 0.0105 0.0000 0.0000 -0.0020 0.0000 0.0000
-0.0155 0.0000 0.0000 0.0000 0.0000 0.0000 0.0062 0.0000 0.0000 -0.0014 0.0001
-0.0356 -0.0186 0.0128 0.0283 0.0000 0.0000 0.0403 0.0459 -0.0080 0.0259 0.0000 0.0000 -0.0421 -0.0442 -0.0080 -0.0156 0.0000 0.0000 0.0172 0.0197 0.0109 -0.0283 -0.0034 -0.0039 0.0268
0.0000 0.0000 -0.0180 0.0000 0.0000 0.0000 0.0000 0.0000 0.0316 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0262 0.0000 0.0000 0.0000 0.0000 0.0000 0.0113 0.0000 0.0000 -0.0023 0.0000
0.0067 0.0000 0.0000 -0.0184 0.0000 0.0000 0.0000 0.0000 0.0000 0.0231 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0164 0.0000 0.0000 0.0000 0.0000 0.0000 0.0069 0.0000 0.0000 -0.0016
-0.0332 -0.0174 0.0123 0.0301 0.0000 0.0000 0.0392 0.0447 -0.0085
0.0000 0.0000 -0.0167 0.0000 0.0000 0.0000 0.0000 0.0000 0.0309
0.0057 0.0000 0.0000 -0.0171 0.0000 0.0000 0.0000 0.0000 0.0000
Atomic
Data and Nuclear
0.0000
Data Tables.
Vol. 37. NO. 1. July 1967
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’ 3
3
1 1
1
1
2 2 3 1 1 1
2 2 3 2 2 3 3 2 2 3 1 1 1
2 2 3 1 1 1
2 2 3 1 1 1
2 2 3
2 3 2 3 3 1 2 3 2 3 3 2 3 3 3 2 3 3 1 2 3 2 3 3 1
2 3 2 3 3 1
2 3 2 3 3
3 2 2 3 1
3 2 3 3
1
2 3 2 3 3 1
1 2 2 3 1
1
Linear
Polarization
Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
Ii
If
Gamma-Ray
c2
c3
5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
-0.0269
0.0000
-0.2216 -0.0386 0.0583 -0.1468 -0.0609 -0.1209 0.0886 0.1321 0.0555 -0.1266 0.0466 -0.1491 0.1266 0.1451 -0.0739
0.0000
712 912 912 9!2 11/2 11/2 11/2 11/2 11/2 11/2 1312 13/2 13/2 1312 1312 1312 15/2 15/2 15/2 15/2 15/2 15/2
13/2 13/2 13/2 1312 13/2 13/2 13/2 1312 13/2 1312 13/2 1312 13/2 1312 1312 1312 1312 1312 13/2 1312 13/2 13/2
4 5 5 5 6 6 6 6 6 6 7
7 7 7 7 7 7 7 7 7 7 7
c4
c5
c6
-0.0825 -0.0273 0.0000 -0.0427 0.0000 0.0000 0.0353 0.0649 0.0000 -0.0436 0.0000 0.0000 0.0452 0.0000
0.0237 0.0000 0.0000 -0.0425 -0.0447 -0.0074 -0.0178 0.0000 0.0000 0.0179 0.0205 0.0111 -0.0271 -0.0036 -0.0041 0.0276
-0.0268 0.0000 0.0000 0.0000 0.0000 0.0000 0.0120 0.0000 0.0000 -0.0025 0.0000
0.0228 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0171 0.0000 0.0000 0.0000 0.0000 0.0000 0.0075 0.0000 0.0000 -0.0018
0.2311 0.1981 -0.1435 0.0924 0.1387 -0.1533 0.0410 -0.0198 -0.1297 -0.0347 -0.2219 -0.0358 0.0585 -0.1486 -0.0567 -0.1248 0.0901 0.1331 0.0551 -0.1238 0.0502 -0.1479
0.0000 0.0000 0.0704 0.0000 0.0000 0.0644 -0.0753 0.0000 0.0136 0.0000 0.0000 -0.0828 -0.0253 0.0000 -0.0441 0.0000 0.0000 0.0362 0.0656 0.0000 -0.0421 0.0000
-0.0313 -0.0164 0.0119 0.0313 0.0000 0.0000 0.0382 0.0438 -0.0090 0.0218 0.0000 0.0000 -0.0428 -0.0451 -0.0069 -0.0195 0.0000 0.0000 0.0185 0.0212 0.0112 -0.0259
0.0000 0.0000 -0.0156 0.0000 0.0000 0.0000 0.0000 0.0000 0.0303 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0272 0.0000 0.0000 0.0000 0.0000 0.0000 0.0126 0.0000
0.0050 0.0000 0.0000 -0.0160 0.0000 0.0000 0.0000 0.0000 0.0000 0.0225 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0176 0.0000 0.0000 0.0000 0.0000 0.0000 0.0081
0.2278 0.1953 -0.1444 0.0854 0.1367 -0.1528 0.0416 -0.0244 -0.1278 -0.0412 -0.2221
0.0000 0.0000 0.0698 0.0000 0.0000 0.0633 -0.0753 0.0000 0.0112 0.0000 0.0000
-0.0298 -0.0156 0.0115 0.0323 0.0000 0.0000 0.0374 0.0429 -0.0093 0.0200 0.0000
0.0000 0.0000 -0.0148 0.0000 0.0000 0.0000 0.0000 0.0000 0.0297 0.0000 0.0000
0.0045 0.0000 0.0000 -0.0151 0.0000 0.0000 0.0000 0.0000 0.0000 0.0222 0.0000
67
0.0000 0.0000 0.0000 0.0000 0.0000
Atomic
Data and Nuclear
Data Tab&.
Vol. 37, NO. 1, July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
TABLE II. Linear Polarization See page
L 1 1 2 2 3 1 1 1 2 2 3 3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3
c3
2 3 2
7 7 7
7 7 7
-0.0334 0.0587 -0.1500
-0.0830 -0.0236 0.0000
3 3 1 2 3
7 7 8 8 8
7 7 7
-0.0530 -0.1280 0.0915
-0.0452 0.0000 0.0000
2 3 3
8 8 8
7 7 7
0.1339 0.0547 -0.1213
7 7
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3
15/2
912 11/2 11/2 1112 1312 13/2 13/2 1312 1312
1512 1512 15/2 15/2 15/2 15/2 15/2 1512
1312 15/2 15/2
1512 15/2 15/2
15/2 1512 1512 1512
15/2 15/2 1512 1512
3 2 2 3
3 2 3 3
1 1
1 2
5 6 6 6 7 7
1 2 2 3
3 2 3 3
7 7 7 7
1 1 1 2 2 3
1 2 3 2 3 3
8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
Polarization
Distribution
58 for Explanation
C2
If
L’
Linear
Distribution
Coefficients
of Tables
c4
c5
c6
0.0000
0.0000
0.0000
-0.0430 -0.0455
0.0000
0.0000
0.0000
0.0000
0.0370 0.0661 0.0000
-0.0065 -0.0209 0.0000 0.0000 0.0190 0.0219
-0.0276 0.0000 0.0000 0.0000 0.0000 0.0000
0.0534 -0.1467
-0.0408 0.0000
0.0113 -0.0248
0.0132 0.0000
0.2249 0.1928 -0.1451
0.0000 0.0000 0.0692
-0.0285
0.0000 0.0000
0.0794 0.1350 -0.1524 0.0421 -0.0284 -0.1260 -0.0468
0.0000
-0.0181 0.0000 0.0000 0.0000 0.0000 0.0000 0.0086 0.0041
0.0000 0.0000 0.0623 -0.0752 0.0000 0.0091
-0.0150 0.0112 0.0330 0.0000 0.0000 0.0367 0.0422 -0.0096
-0.0141 0.0000 0.0000 0.0000 0.0000 0.0000 0.0292
0.0000 0.0000 -0.0143 0.0000 0.0000 0.0000 0.0000 0.0000
-0.2223 -0.0313
0.0000 0.0000 -0.0832
0.0185 0.0000 0.0000
0.0000 0.0000 0.0000
0.0219 0.0000 0.0000
0.0588 -0.1512 -0.0497 -0.1305
-0.0221 0.0000 -0.0462 0.0000
-0.0432 -0.0457 -0.0061 -0.0220
0.0000 0.0000 -0.0279 0.0000
0.0000 0.0000 0.0000 -0.0184
0.2224 0.1907 -0.1456 0.0741 0.1335 -0.1520 0.0425
0.0000 0.0000 0.0686 0.0000
-0.0275 -0.0144 0.0110 0.0336
0.0000 0.0000 -0.0135 0.0000
0.0037 0.0000 0.0000 -0.0137
0.0000 0.0614
0.0000 0.0000 0.0000 0.0000 0.0288 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0281 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0216 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0187
-0.0318 -0.1244 -0.0515
-0.0751 0.0000 0.0073 0.0000
0.0000 0.0000 0.0361 0.0416 -0.0098 0.0171
-0.2224 -0.0295 0.0589 -0.1522 -0.0468 -0.1327
0.0000 -0.0833 -0.0208 0.0000 -0.0469 0.0000
0.0000 0.0000 -0.0434 -0.0459 -0.0057 -0.0230
68
Atomic
Data and Nuclear
Data Tables.
Vol. 37. No. 1. Juiy 1987
DATA
AND
NUCLEAR
DATA
GAMMA-RAY
TABLES
LINEAR
37,53-68
POLARIZATION
J. RIKOVSKA Clarendon
(1987)
Laboratory,
DISTRIBUTION
and N. J. STONE Oxford,
United Kingdom
The formalism for describing the linear polarization of gamma radiation from oriented nuclei is reviewed. Coefficients Ak2 are introduced which give the polarization-sensitive terms in the directional distribution of radiation the same form as the polarization-insensitive terms. Expressions for ,4k2 coefficients are given for cases involving admixture of up to three multipole components in the observed transition. Also given are specific expressions appropriate to tests of time-reversal invariance in gamma decay. The coefficients C,(L, L', If, 1;) tabulated herein are to enable the evaluation of .4k2. Values of C,(L, L', if, I,) are listed for the ranges X == 2-6, L, L' = 1, 2, 3, and integer spins 1 < I, < 8 and halfinteger spins 3/2 G 1, < 15/2. Analytical expressions for Legendre polynomials P,(cos 0) and associated Legendre polynomials p2(cos 0) for X < 6 are provided additionally, to facilitate the calculation of polarization distributions. Examples of the use of the tables in specific cases are given. e 1987 Academic Press. Inc
0092-640X/87
$3.00
Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
53
Atomc
Data and Nuclear
Data Tables.
Vol. 37. No. 1, July 1987
J. RIKOVSKA
Gamma-Ray Linear Polarization Distribution
and N. J. STONE
CONTENTS
INTRODUCTION . . .._.__......_......................... Gamma-Ray Linear Polarization Distribution Time-Reversal Invariance in Gamma Decay
54 54 56
EXPLANATION
OF TABLES
58
OF USE OF TABLES
59
EXAMPLES
TABLES I. Analytical Expressionsfor Legendre and Associated Legendre Polynomials II. Linear Polarization Distribution Coefficients
61 62
INTRODUCTION Atomic nuclei oriented with respect to a definite direction in space can produce gamma rays polarized to a high degree. The main interest concerns linear polarization, inasmuch as the directional dependence of the linear polarization of such gamma rays dependsupon the type of the radiation (magnetic/electric) and thus on the relative parity of the nuclear states between which the transition occurs. This important information cannot be obtained from a measurement of the directional distribution of gamma rays from oriented nuclei, performed by polarization-insensitive detectors, which depends only upon the spins of the states and the multipole order of the radiation. Further, for mixed multipole radiation, the frequent occurrence of more than one mixing ratio compatible with the measureddirectional distribution can be resolved with the aid of a measurementof the polarization in a specified direction. Thus linear polarization measurementsperformed with sufficient accuracy on oriented nuclei provide, in combination with gamma-ray directional distribution data, more complete information on excited nuclear statesand the transitions between them. They are also of prime importance in testing such fundamental interaction symmetries in nuclei astime-reversal invariance and parity nonconservation.
Analysis of a linear polarization measurement generally involves comparison between the experimental data and calculation. Formulas for certain special caseshave been published. l-4 In the presentation below, the method outlined is applicable to the most general calculation of the polarization distribution. The approach here is especially convenient when more than two multipole components are present in the transition. Gamma-Ray Linear Polarization Distribution The angular distribution of linearly polarized gamma rays from an axially oriented ensemble of nuclei can be written, following Steffen and Alder,’ as an expansion in Legendre and associatedLegendre polynomials Ph and P::
+ c
B,U,A;zPZ,(cos Qcos 2#.
(1)
h=WeFi
Equation (1) applies to a gamma-ray transition from state 111)to state II,), wherein the state 11;)is populated either directly or through intervening unobserved transitions 54
Atomc
Data
and
Nuclear
Data
Tables.
Vol.
37.
No.
1, July
,987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
from the oriented parent state ]Z). Z, Zi, and Z, are the corresponding spin values. 0 is the angle of emission of a gamma quantum with respect to the orientation axis of the parent nuclear ensemble, and # is the angle between the electric E vector of the quantum and the plane defined by the orientation axis and the direction of gamma emission. The coefficients
for j + 1 unobserved transitions in a direct cascade. For a single transition between ]I,$ and ]Z,) states &(zkb)
=
2 L
aL ~A(~k~lJ%
(5)
where aL is a fraction of a component carrying L units of angular momentum (for example multipole order for gamma radiation, partial wave order for a particle transfer) and, again, c aL = 1. In particular, for the most frequent case of a gamma-ray transition with mixed components L and L’ = L + 1 we have
B,(Z) = [(2X + 1)(21+ 1)]“2 2 (- l)‘+m
u,(zkz[LL’) = [ u,(&z,L) + 62uA(zkJIL’)]/( 1 + 6*),
where 6 is the multipole mixing ratio and S*/( 1 + S2) is thus the fractional L’ contribution to the total intensity. For gamma-ray transitions where internal conversion is a significant decay mode this is included by writing
are tensors describing the degree of orientation of the ground state IZ) of the parent nucleus.5 Multiplying the Wigner 3j symbols are the Boltzmann population parameters p(m), which are dependent on the mechanism of orientation of the state IZ). For example, P(m) = ew(mWT)l
c exP(mAM/T) m=-I
for pure magnetic dipole orientation, splitting (in millikelvin) is
uA(zkzlLL’) = [( 1 + a)UX(zkz&) + (1
(24
i
G’b)
exp(-m2AE/T)
Ar = 1.160{3@‘,1)/[4Z(2Z-
(2~)
(7)
+bp2],
(8)
u,(zkz,L) = (- l)‘k+L’+L+A x [(2& + 1)(2Z,+ I)]“* [;
;
9.
(9)
Specific expressions for &(ZkZl) for alpha and beta decay and electron capture and s-wave neutron capture are given in Ref. 8. The angular distribution coefficients Ax for a y transition between states IZi) and IZr> with multipole components aL and a’L’ (a = E or M) used here are defined as
(24
AA= C FA(LL’z~zih(~L)r*(a ‘L’IIC I$~L)12, ( 10) CL/L CL‘ where y(aL) is the absolute transition amplitude’ of the aL component. The multipole mixing ratio is given by G(a’L’faL) = y(a’L’)/y(uL). The Fx are the functions
where n is the number of branches via which the state ]Z;) is fed, and branching intensities are normalized so that C w, = 1. For each branch, * - - U,(zjZJ
+ b)s*],
where q is the ratio of the EO to E2 reduced transition matrix elements. The coefficient &(ZkZ/L), appearing in Eqs. (5)-(g), is given in terms of the Wigner 6j symbols as
where the electric quadrupole moment Q is in barns and the electric field gradient V,, is in lOI V/cm* (see Vianden6 and Hagn7). The deorientation coefficients U, describe the effect of any transition (radiation or particle) from the oriented parent state and succeeding states which precede the state ]Zi) from which the observed radiation is emitted. In general,8 KU - - * ZJ = 2 w,[ UA(Z- * * Zi)ln, (3)
UA(Z* * * ZJ = u,(zz,)u~(z’z*)
+ (1
Xs2UX(ZkZkL’)]/[bq2~‘+(1 +a)+(1
is l)],
a)
u,(z,&LL ‘) = [bq%* + ( 1 + a) ux(ZkZkL)+ ( 1 + b)
m=-I
and the energy splitting (in millikelvin)
1 +
version mode also contributes to the transition with AZ = 0, for example an EO + Ml + E2 multipolarity mixture, we have
In Eq. (2b), the magnetic dipole moment p of the state II) is in nuclear magnetons and the polarizing hyperfine field B is in tesla (see Krane6 for a table of numerical values). For pure electric quadrupole orientation, p(m) = exp(-m2AE/T)/
+ b)62UA(zkz&‘)]/[(
where a and b are the conversion coefficients of the L and L’ components, respectively. When the EO internal con-
where the energy
AM = 0.366pBJZ.
(6)
Fh(LL’ZfZi) = (- 1)“+“+‘[(2X X (2L’ + 1)(2Zi + l)]“*
(4) 55
Atomic
(f
+ 1)(2L + 1) “;
Data and Nuclear
;)I; Data Tables.
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;I*
(I l)
Vol. 37. No. 1, July 1997
J. RIKOVSKA
Gamma-Ray
and N. J. STONE
Explicit expressions for Ax for transitions with two and three multipole components can be found in Refs. 3, 4. In Eq. (l), the ,4)X2coefficients in the polarizationsensitive term depend on the type u of the electromagnetic radiation through the phase factor (-l)A’d’, where h(E) = 0 and A(M) = 1. They differ from the AX2 of Ref. 1 in that they include the factor 2[( h - 2)!/( X + 2)!] ‘12, leading to simplification of Eq. (1) and Eq. (20) below. We have A;2 = -
[we,#=o)+
(14)
This formulation gives the linear polarization-sensitive term in Eq. (1) the same form as the directional distribution term, but with the Fx coefficients replaced by the C, coefficients (compare Eqs. (10) and (12)). Then, using Eqs. ( 12)-( 14) the Ai coefficients for transitions with two and three multipole components can be written as
-{(-l)"'"'C~(LLz,-Z,)+[(-l)A'v" ‘~(~)+~~+L'+h]Ch(LL'I/z~)~(u~L~/uL)+ (- I)“‘“”
x C,(L’L’z~z;)62(u’L’/uL)~/[
1 + a2(u’L’/uL)],
(I 5)
and
A;? (uLu'L'u"L")
= -(( - l)*‘“‘C,(LLZ/Z,) + [(- 1)““”
+(-I) *(c)+L+L’+h]C,(LL’Zfz,)qu’L’/uL) +[(-l)*‘“V)+(-l) +1(-l)
A(0
+ (-
A'"'+L+r~+X]C,(LL'Z~Zi)~
(U”L”/UL)
l)A(d)+l.‘+L”+X]CA(LILnlfz~)
x G(u'L'/aL)&r"L"/uL)+(-l)"'"'C,(L'L'zJzi) x G'(u'L'/uL)S(-
1)-""'C,(L"L"Z,-z,)S2(U"L"/UL))/ [ 1 + S’(u’L’/uL) + fi’(u”L”/uL)].
(17)
where Z’(B)is given in Eq. ( 17) and Q is the linear polarization detection efficiency (for details seeRef. 11). The factors Bx, Ux, Ai, and Pi are normalized so that &, = U, = A0 = PO = 1. Angular momentum coupling selection rules cause terms with X > X,,, to vanish as follows: Bh, X,,, = 21; Ah, h,,, = 2L,,, or 2Z,, whichever is smaller: U,, h,,, = 2Zk or 2Z,, whichever is smaller. A;* vanish for h < 2 and for X,, = 2L,, or 2Zi, whichever is smaller. Here L,,, is the largest multipole order present in the transition. For parity conserving and time-reversal invariant radiation only even values of X contribute to lV(0, 4); otherwise X-odd terms must be also considered. In this paper, we give listings of the C, coefficients with a range encompassing the casein which the highest occurring multipole component is octupole. For such a case, all the coefficients in Eq. (1) are required up to X = 6. In Table I. we have given for convenience the analytical expressionsfor the normalized Legendre polynomials and the unnormalized associated Legendre polynomials of Eq. (1). In Table II we list CA(LL’ZfZi) for h = 2-6, L and L' = 1, 2, and 3, and integer and halfinteger spin values between 1 and 8. The coefficients Bx, U,, and Fx have been tabulated by Krane for the range X = l-4 in Refs. 5 and 9 and for X = l-6 in Ref. 10. The coefficients F,+ and U, (X = l-4) can also be found in Appendixes 2 and 3 of Ref. 1.
The C, change sign with interchange of L,L’ as
+(-I)
w(k1C/=d2)1,
where the normalization is such that - 1 < p(0) < + 1. Our ,4;, coefficients do not include linear polarization efficiency, which many authors (for example Ref. 1) include in their coefficients AX2. This has its origin in a very general definition of angular distribution W = Tr(t . p), where t is the efficiency matrix, representing the observation process, and p is the density matrix of the system of oriented nuclei. In practice, efficiency is not usually calculated but is measured in an independent experiment. Thus, without loss of generality, the measured degree of linear polarization a can be related to the calculated value by A = P(W2, (18)
= [(A - 2)!/(X + 2)!]“2
A;2(oLu'L')=
Distribution
P(B) = [ W(B, ys = 0) - lv(l3, $ = 7r/2)]/
where
CA(LL'ZfZ,) = (- l)L+L’+“C~(L’LZj-Zi).
Polarization
of gamma emission. Thus the degree of linear polarization is defined as
2 (-l)“‘d’C~(LL’Z~Zi) oLdL’
C,(LL’Z,Zj)
Linear
(16)
Time-Reversal Invariance in Gamma Decay
The angular dependence of the distribution W(8, 4) is governed by the ordinary Legendre polynomials ~(COS e), the unnormalized associated Legendre polynomials Px(cos 8) and cos 2$. Examination of Eq. (1) showsthat the maximum variation in lV(0, J/) is for $ = 0 and I/ = r/2, that is between the gamma quantum being emitted with E vector parallel or perpendicular to the plane determined by the orientation axis and the direction
Linearly polarized radiation from oriented nuclei can be used in testing time-reversal (T) symmetry in ydecay.r2-14T violation would manifest itself through the mixing ratio of a y transition with two multipole components L, L' such that G(u’L’/uL)
56
= I6(u’L’/uL)l(cos
Atomic
Data and Nuclear
17 + i
Data Tables,
sin II),
(19)
Vol. 37, No. 1, July 1987
J. RIKOVSKA and N. J. STONE
Gamma-Ray Linear Polarization Distribution
where the phase 11differs from 0 or x. The linear polarization of gamma rays from an axially symmetric polarized state of pure, unmixed parity, including T-sensitive terms, is given by
of the degree of linear polarization reversal asymmetry parameter AT.
p(0) and the time-
References =
2 B,lJ,A,P,(cos X=eWn +
c B,&A;* h=l3Wl
+ c
1. R. M. Steffen and K. Alder, in The Electromagnetic Interaction in Nuclear Spectroscopy, edited by W. D. Hamilton (North-Holland, Amsterdam, 1975), p 505
0)
p: (cos 0)cos 2# @sin 2#.
(-i)B,&l&2ff(COS
2. P. J. Twin,
3. T. Aoki, K. Furuno, Y. Tagishi, and J-Z. Ruan ATOMIC DATAANDNUCLEARDATATABLES 23,349 (1979)
(20)
h=odd
Here w,(e) and l+‘,(e, #) denote normal T-even directional and linear polarization distributions in the notation used above. I+‘,(fI, 11/)is the T-violating term. From the 3; symbol in the expression for ,412 (Eq. (12)) it follows that h must be at least 2 and since only odd-X terms contribute to the T violation, the first such term is for X = 3. The coefficients Ak2 for X = odd can be expressed as A;2(aLafL’) = -[(- l)A(ul)-(-l)A(o)tL+L’+h] X CA(LL’IfIf)16(u’L’/aL)li
sin q/[ 1 + I&(u’~‘/uL)~~].
4. J. Rikovska,
++,) - wr
mv,4d+
24-26,963
6. K. S. Krane, Hyperfine Interactions (1983); R. Vianden, ibid., p. 1081 7. E. Hagn, Hyperfine
(1985)
Interactions
15/16,
1069
22, 19 ( 1985)
8. K. S. Krane, in Low Temperature Nuclear Orientation, edited by N. J. Stone and H. Postma (NorthHolland, Amsterdam, 1986), p. 65
(21)
9. K. S. Krane, Los Alamos Scientific Laboratory LA-4677 (1971) 10. K. S. Krane. in Ref. 8, Appendixes
Report
5. 6
11. J. Rikovska, N. J. Stone, and V. R. Green, Nucl. Instrum. Methods A 241,46 1 ( 1985) 12. R. J. Blin-Stoyle, in Fundamental Interactions and the Nucleus (North-Holland, Amsterdam, 1973), Chaps. 1, 10
- em, n + km
13. N. K. Cheung, H. E. Hendriksen, Phys. Rev. C 16, 2381 (1977)
w~-kd+~m)i = w3 tern, 1cldi w (em).
Hyperfine Interactions
5. K. S. Krane, NUCLEAR DATA TABLES 11,407 (1973)
Note the difference in sign inside the bracket in Eq. (2 I) as compared to Eq. (15). This arises because we are dealing with a complex mixing ratio (see Eq. (19)). The X = 3 term in w3(0, +) reaches its maximum for 8, = 54.7” f n* and $, = 45” + na/2. For this choice of angles W2(B,, +,,,) and the X = 2 term in w,(0,) vanish. W3 changes sign under the time-reversal transformation13 0 + r - 8, $ + r + +; hence the time-reversal asymmetry AT may be introduced: A de, +) = [ we,
in Ref. 1, p 701
(22)
and F. Boehm,
14. K. S. Krane, in Ref. 8, p. 249
The tabulated coefficients C,, used in conjunction with Eqs. ( 17) and (22) and the other tabulated quantities Bx, U,, F,,, Px, and Pf , enable calculations to be made
15. D. I. Bradley, N. J. Stone, J. Rikovska, D. Novakova, and J. Ferencei, J. Phys. G 12, 115 (1986)
57
Atomic
Data
and
Nuclear
Data
Tables.
Vol.
37.
No.
1, July
1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
EXPLANATION TABLE I.
OF TABLES
Analytical Expressions for Legendre and Associated Legendre Polynomials Expressions for normalized Legendre polynomials Px(cos 0) and unnormalized associatedLegendre polynomials P?(cos 0) to be used with Eqs. ( 1) and (20) are given for 1 G X < 6.
TABLE II. Linear Polarization Distribution Coefficients The coefficients CA(LL’ZfZj) are listed for X = 2-6; L, L' = 1, 2, and 3; integer spins 1 < Zi < 8; Half-integer spins 3/2 < Z, < 15/2. L L’ 1~ Ii
CA
Multipole order of the first multipole component Multipole order of the second multipole component Spin of the final state Spin of the initial state CA(LL'ZfZi), X = 2-6 (seeEqs. (13) and (14))
58
J. RIKOVSKA
and N. J. STONE:
EXAMPLES
Gamma-Ray Linear Polarization Distribution
OF USE OF TABLES
Example 1 Calculation of the value of linear polarization p(B = 90”) (Eq. (17)) of the 2015-keV (ZT = 4+, Zj = 4’.) transition with mixed multipole Ml and E2 components (6 (E2/MI) = 0.83 (Ref. 15)) in 56Fe:The initial state at 4100.4keV is populated directly in 0 decay (pure L = 1, no parity change) of the ground state (I” = 4+) of 56Co,oriented in the Fe matrix at low temperatures, Observation of the magnetic dipole orientation due to hyperfine interaction of the magnetic dipole moment of the 56Coground state (P = 3.830 nm.) in Fe (Z&r = -28.9 T) at 5.83 mK is considered. (a) Orientation coefficients Bx (Eqs. (2)-(2.b)) are Bz = I .3556,
B4 = 0.5543
for AM= 10.13mK,
AM/T= 1.7378.
(b) Deorientation coefficients U, (Eq. (9)) are U,(44 1) = 0.8500,
U4(44 1) =0.5000.
(c) Angular distribution coefficients Ax (Eqs. (10) and (11)): Using F2( 1144) = -0.4387,
E;( 1244) = -0.3354,
F,(2244) = 0.2646,
and F4( 1144) = 0,
F4( 1244) = 0,
F,(2244) = -0.498 1,
for 6(E2/Ml) = 0.83, we have A2 = -0.48 15 and
A4 = --0.2032.
(d) The linear polarization distribution coefficients A’,* (Eqs. (12)-( 14)): Using C,( 1144) = -0.2 194,
C,( 1244) = -0.0559,
C,(2244) = -0.1323,
and C,( 1144) = 0,
C,(1244) ==0,
C,(2244) = -0.0415,
for 6(E2/Ml) = 0.83, and A(a) = l(0) for Ml (E2) components, respectively, we have from Eq. (15) A& = -0.02 10 and
A& = 0.0 169.
(e) The ordinary Legendre polynomials for x = cos 90” are P&X) = -0.5000
and
P4(x) = +0.375
and associated Legendre polynomials are e(x) = 3.0000
and
59
ti (x) = -7.5000.
Atomic
Data and Nuclear
Data Tables.
Vol. 37, No. 1, July 1997
J. RIKOVSKA
Gamma-Ray
and N. J. STONE
EXAMPLES
Linear
OF USE OF TABLES
Polarization
Distribution
continued
(f) Using Eq. (I), for X = 0, 2, 4 allowed by the selection rules, we obtain in the present case W(0 = 90”, ic,= 0”) = 1 + BzU2A2P2(cos 0) + B4U&Pq(cos
0)
+ B2 lJ2A;2P$(cos 6)cos 2+ + BJIqA&P:(cos
B)cos 2$
= 1.1486 and W(O = 90”,$ = 90”) = 1.3640. Hence P(B = 90”) = -0.0857. Example 2 Calculation of the time-reversal asymmetry parameter AT(/3, $) (Eq. (22)) for the transition considered in Example 1, at critical angles 0, = 54.7” and $J, = 45”: The only X-odd value allowed by the selection rules is X = 3. Therefore we need, together with values of coefficients quoted above for X = 2 and 4, B3 = -0.9680, C,( 1144) = 0,
U3(441) = 0.7000,
C,( 1244) = -0.0802,
C,(2244) = 0.
From Eq. (2 1), we obtain A& = iO.0788 sin 9. For P4(x) = -0.3894,
x= cos 54.7”, we have P2(x) = 0, The expression A,(B,
e(x)
= 5.7735.
for Ar(B, $) will be
= 54.7”, $, = 45”) = -iB,lJgl’&(cos
8,)sin 2$,)/
[ 1 + Bz U~A~P~(cos 8,) + 84 iJ&P4(cos
it?,,,)]
and the resulting value AT(Bm, $,J = -0.3017 sin 7. Comparing AT with a corresponding experimental the time-reversal nonconserving factor sin q.
60
value,4 we can determine
Atomic
Data
and
Nuclear
Data
Tables,
Vol.
37.
No.
1. July
1987
J. RIKOVSKA
TABLE I. Analytical
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
Expressions for Legendre and Associated Legendre Polynomials See page 58 for Explanation of Tables Normalized
Legendre Polynomials
P,(x) = 1 P,(x) =x Pz(x) = i(3x2 - 1) Pj(X) = $(5x3 - 3x) P4(x) = 335x4 - 30x2 + 3) P,(x) = %63x5 - 70x3 + 15x) Ps(x)=+&!31x6-315x4+ Unnormalized
105x2-5)
Associated Legendre Polynomials
P$(x) = 3( 1 - x2) P:(x) = 15( 1 - x2)x Pi(x) = 3 1 - x2)(7x2 - 1) P:(x) = +q 1 - x2)(3x3 - x) P;(x) = $q 1 - x2)(33x4 - 18x2 + 1)
61
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’
Ii
If
1 1 1 2 1 1 1 2 2 3 2 2 3 3
1 1 2 2 1 2 3 2 3 3 2 3 3 3
1 1 2 1 1 1 2 3 1 1 1 2 2 3 2 2 3 3
1 2 2 1 2 3 3 3 1 2 3 2 3 3 2 3 3 3
2 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1
2 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3
0 1 1 1 2 2 2 2 2 2 3 3 3 4 112 112
l/2 312 312 312 312 312 512 512 512 512 512 512 712 712
712 912 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3
Gamma-Ray Linear Polarization Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
C2
c3
1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.3535 -0.1768 -0.1768 0.1768 0.0354 0.0791 0.0707 -0.1768 -0.1581 0.1414 0.0505 0.0945 -0.1768 0.0589
312
312 312 312 312 312 312
0.2500 -0.1443 0.2500 -0.2000 -0.1291 0.0408 -0.1581 0.2000 0.0500 0.0986 0.0667 -0.1786 -0.0845 -0.0500 0.0714 0.1157 -0.1667 0.0833
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
0.2988 0.2092 -0.1559 0.0199 0.1494 -0.1336 0.2390 -0.2092 -0.1021 0.0488 -0.0640 -0.1429 0.0598 0.0598 0.1091 0.0639
0.0000 0.0000 0.1054 -0.0589 0.0000 0.0791 0.0000 0.0000 -0.0690 -0.0722 0.0000 0.0211 0.0000 0.0000 0.0184 0.0472
312 312 312 312
312 312 312 312
312
312 312
c4
c5
c6
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.1291 0.0000 0.0000 -0.0577 -0.0913 0.0707 0.0000 0.0000 0.0126 0.0373 0.0000 -0.0567 0.0000 0.0000 0.0173 0.0000 0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0891 0.0000 0.0000 0.0668 0.0594 0.0166 -0.0297 0.0000 0.0000 -0.0273 -0.0255 -0.0133 0.0445 0.0000 0.0000 0.0060
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
62
Atomic
Data and Nuclear
Data Tables.
Vol. 37. No. I. July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
rABLE II. Linear Polarization See page
L
If
L’
2 2 3 2 2 3 3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2
2 3 3 2 3 3 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2
2 3 2 2 3 3
3 3 2 3 3 3
3 2 2 3
3
1 1 1 2 2 3 1 1 1 2 2 3 1
2
2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1
2 2 2 2 2 2 l/2
112
l/2 312 312 312 312 3/2 3/2 512 512 512 512 512 512 712 712 7/2 712 712 712 Q/2 Q/2 Q/2
11/2 0 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4
512 512 512 512 512 512 w 5/2 512 512 512 512 512 5/2 512 512 512 512 5/2 512 512 512 512 512 512 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
-0.1707 -0.0437
Polarization
Distribution
58 for Explanation
c2
Ii
Linear
Distribution
Coefficients
of Tables
c3
c4
0.0000
0.0064 0.0054
0.0000
0.0000
0.0000
0.0000
-0.0604
c5
c6
-0.1096 0.0854 0.1263 -0.1494 0.0996
0.0000 0.0000 0.0249 0.0000 0.0000
-0.0297 -0.0007 -0.0010 0.0099
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0013
0.0000
0.0000
0.2673 -0.0945 0.2673 0.1871 -0.1581
0.0000 0.0646 0.0000 0.0000 0.0926
-0.0514 0.0182 -0.0514 0.0000 0.0000
0.0267 0.0954 -0.1468 0.1470
-0.0685 0.0000 0.0656 0.0000
0.0579 0.0588
-0.2138 -0.0845 0.0525
0.0000 -0.0742 -0.0598
-0.0954 -0.1245 -0.0089 0.0668
0.0000 -0.0031 0.0000 0.0000
0.1157 0.0619 -0.1623 -0.0179
0.0226 0.0528 0.0000
-0.1336 0.0954 0.1324 -0.1336 0.1114
-0.0590 0.0000 0.0000 0.0302 0.0000 0.0000
0.2887
0.0000
0.2474 -0.1157 0.2165 0.1732
0.0000 0.0722 0.0000
-0.1581 0.0309
0.0000 0.0845 -0.0722
0.0619 -0.1479 0.0914 -0.2165 -0.0722 0.0546 -0.1134 -0.1091 -0.0481 0.0722
0.0000 0.0527 0.0000 0.0000 -0.0772 -0.0510 0.0000 -0.0170 0.0000 0.0000
63
0.0054 0.0257 0.0000 0.0000 -0.0337 -0.0331 -0.0133 0.0257 0.0000 0.0000 0.0089 0.0098 0.0074 -0.0343 -0.0012 -0.0017 0.0148 -0.0023 -0.0711 -0.0372 0.0174 -0.0118 0.0000 0.0000 0.0522 0.0558
0.0000
0.0000
-0.0545 0.0000 0.0000 0.0000
0.0000
0.0000 0.0000 0.0326 0.0000
0.0000
0.0000 0.0000 0.0000
0.0000
0.0000 -0.0133 0.0000 0.0000 0.0000
0.0000
0.0000 0.0000 0.0034
0.0000
0.0000 0.0000 -0.0005 0.0000
0.0000
0.0000
0.0000
0.0000 0.0000
0.0435 0.0000
-0.0386 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0326 0.0000 0.0000 0.0000 0.0000
0.0000 0.0352 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0182 0.0000 0.0000
0.0000 0.0355 0.0000 0.0000 -0.0369 -0.0372 -0.0123 0.0118 0.0000
Atomc
Data and Nuclear
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000
0.0000 0.0000 0.0000
0.0000 0.0181 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0073 0.0000
Data Tables.
“ol
37. NO. 1, July 1997
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
TABLE II. Linear Polarization See page
L
L’
1 1 2 2 3 2 2 3 3 3 2 2 3 1 1 1 2 2 3 1 1 1
If
Ii
2 3 2 3 3 2 3 3 3 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3
2 2 3 1 1 1 2
3 1 2 3 2
2 3
3 3
2 2 3 3
2 3 3
3 2 2 3 1 1 1 2 2 3 1 1 1 2
3 2 3 3 1 2 3 2 3 3 1 2 3 2
3
112 312 312 312 512 512 512
c3
3 3 3 3 3 3
0.1203 0.0603 -0.1546 0.0000 -0.1443 0.1031 0.1364
0.0257 0.0564 0.0000 -0.0564 0.0000 0.0000
3 3
-0.1203 0.1203
712
0.2728 0.2338 -0.1263 0.1818 0.1637 -0.1575
512
512 512
712 712 712 712 712
712 912
912 912 912 912 912 1112 1112 1112 1312
712 712 712 712 712 712 712 712 712 712 712
712 712 712
712 712 712 712
712 712
712 712
712 712 712 1 2 2 2 3 3 3 3 3 3 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4
0.0337 0.0390 -0.1458 0.0546 -0.2182 -0.0630 0.0558 -0.1247 -0.0967 -0.0727 0.0764 0.1236 0.0591
0.0340 0.0000 0.0000 0.0000 0.0000 0.0739 0.0000 0.0000 0.0790 -0.0739 0.0000 0.0426 0.0000 0.0000 -0.0790 -0.0445 0.0000 -0.0257 0.0000 0.0000
0.1091 0.1391 -0.1091
0.0282 0.0589 0.0000 -0.0538 0.0000 0.0000 0.0370 0.0000
0.1273
0.0000
0.2612 0.2238 -0.1324 0.1567 0.1567 -0.1567 0.0357 0.0224 -0.1429 0.0285 -0.2194 -0.0559 0.0567 -0.1323
0.0000 0.0000 0.0739 0.0000 0.0000 0.0749 -0.0747 0.0000 0.0349 0.0000 0.0000 -0.0802 -0.0395 0.0000
-0.1481 0.0130 -0.1491
64
Polarization
Distribution
58 for Explanation
c2
3
Linear
Distribution
Coefficients
of Tables
C8
c5
c4
0.0000
0.0000
0.0111 0.0124 0.0087
0.0000 0.0000
0.0055 0.0000 0.0000
-0.0348 -0.0017 -0.0022 0.0183 -0.0032 -0.0570 -0.0298 0.0161 0.0063 0.0000 0.0000 0.0484 0.0531 -0.0031 0.0359 0.0000
0.0000 -0.0003 0.0000
0.0000 0.0000 -0.0303 0.0000
0.0218
0.0000 0.0000 -0.0211
-0.0112 0.0023 0.0000 0.0000 0.0129 0.0145 0.0095 -0.0340 -0.0021 -0.0027
0.0020 0.0000
-0.0008 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0350 0.0000 0.0000 0.0000
0.0000 -0.0389 -0.0398
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0071 0.0000 0.0000 -0.0012
0.0000 0.0000 -0.0290 0.0000 0.0000 0.0000 0.0000 0.0000 0.0218 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0106 0.0000 0.0000 0.0000 0.0000 0.0000 0.0033 0.0000 0.0000 -0.0006
0.0209 -0.0040
0.0000 0.0000
-0.0484 -0.0254
0.0000 0.0000
0.0140 0.0000
-0.0254 0.0000 0.0000 0.0000 0.0000 0.0000 0.0342 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0253 0.0000 0.0000 0.0000 0.0000 0.0000 0.0230 0.0000 0.0000 0.0000 0.0000
0.0150 0.0161 0.0000 0.0000 0.0455 0.0507 -0.0051 0.0337 0.0000 0.0000 -0.0401 -0.0415
Atomic
Data and Nuclear
0.0001
Data Tables.
Vol. 37. NO. 1, July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray Linear Polarization Distribution
TABLE II. Linear
Polarization Distribution Coefficients See page 58 for Explanation of Tables
L
L’
If
Ii
C2
c3
c4
c5
c6
2 3 1 1 1 2 2 3 2 2 3 3
3 3 1 2 3 2 3 3 2 3 3 3
4 4 5 5 5 5 5 5 6 6 6 7
4 4 4 4 4 4 4 4 4 4 4 4
-0.0867 -0.0893 0.0798 0.1261 0.0581 -0.1424 0.0230 -0.1510 0.1140 0.1410 -0.0997 0.1330
-0.0316 0.0000 0.0000 0.0302 0.0608 0.0000 -0.0512 0.0000 0.0000 0.0393 0.0000 0.0000
-0.0102 -0.0044 0.0000 0.0000 0.0142 0.0161 0.0100 -0.0326 -0.0025 -0.0031 0.0229 -0.0047
-0.0231 0.0000 0.0000 0.0000 0.0000 0.0000 0.0085 0.0000 0.0000 -0.0015 0.0000 0.0000
-0.0128 0.0000 0.0000 0.0000 0.0000 0.0000 0.0044 0.0000 0.0000 -0.0009 0.0001
3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 2 2 3 3
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 2 3 3 3
312 5/2 512 512 712 7/2 712 712 712 712 912 912 912 912 912 912 11/2 11/2 11/2 11/2 1112 11/2 13/2 1312 13/2 15/2
912 912 912 912 912 912 912 912 912 912 912 g/2 912 912 912 912 g/2 912 912 912 912 912 912 912 912 912
0.2523 0.2163 -0.1364 0.1376 0.1514 -0.1559 0.0373 0.0098 -0.1398 0.0092 -0.2202 -0.0503 0.0573 -0.1376 -0.0785 -0.1009 0.0826 0.1281 0.0573 -0.1376 0.0308 -0.1514 0.1180 0.1424 -0.0917 0.1376
0.0000 0.0000 0.0733 0.0000 0.0000 0.0718 -0.0751 0.0000 0.0287 0.0000 0.0000 -0.0810 -0.0355 0.0000 -0.0357 0.0000 0.0000 0.0318 0.0622 0.0000 -0.0490 0.0000 0.0000 0.0412 0.0000 0.0000
-0.0427 -0.0224 0.0141 0.0220 0.0000 0.0000 0.0434 0.0488 -0.0064 0.0311 0.0000 0.0000 -0.0410 -0.0427 -0.0094 -0.0093 0.0000 0.0000 0.0154 0.0175 0.0104 -0.0312 -0.0028 -0.0034 0.0245 -0.0054
0.0000 0.0000 -0.0221 0.0000 0.0000 0.0000 0.0000 0.0000 0.0333 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0244 0.0000 0.0000 0.0000 0.0000 0.0000 0.0096 0.0000 0.0000 -0.0018 0.0000 0.0000
0.0103 0.0000 0.0000 -0.0224 0.0000 0.0000 0.0000 0.0000 0.0000 0.0233 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0143 0.0000 0.0000 0.0000 0.0000 0.0000 0.0054 0.0000 0.0000 -0.0011 0.0001
3 2 2 3 1 1 1 2 2 3 1
3 2 3 3 1 2 3 2 3 3 1
5 5 5 5 5 5 5 5 5 5 5
0.2453 0.2103 -0.1391 0.1227 0.1472 -0.1552 0.0385 0.0000 -0.1369 -0.0057 -0.2208
0.0000 0.0000 0.0726 0.0000 0.0000 0.0694 -0.0753 0.0000 0.0238 0.0000 0.0000
-0.0386 -0.0202 0.0134 0.0258 0.0000 0.0000 0.0417 0.0472 -0.0073 0.0284 0.0000
0.0000 0.0000 -0.0198 0.0000 0.0000 0.0000 0.0000 0.0000 0.0324 0.0000 0.0000
0.0081 0.0000 0.0000 -0.0202 0.0000 0.0000 0.0000 0.0000 0.0000 0.0233 0.0000
2 3 3 3 4 4 4 4 4 4 5
65
Atomsc Data and Nuclear
0.0000
Data Tables.
vol. 37. NO. 1, July 1997
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’
If
Ii
1 1 2 2 3 1 1 1 2 2 3 2 2 3 3
2 3 2 3 3 1 2 3 2 3 3 2 3 3 3
5 5 5 5 5 6 6 6 6 6 6 7 7 7 8
3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3 2 2 3
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3 2 3 3
512 712 712 712 912 912 912 912 912
3 2 2 3 1 1 1 2 2
3 2 3 3 1 2 3 2 3
912
1112 1112 1112 1112 1112 1112 1312 1312 1312 1312 1312 1312 1512 1512 1512
3 4 4 4 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Gamma-Ray Linear Polarization Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
c2
c3
c* 0.0000
-0.0456 0.0577 -0.1415 -0.0716 -0.1095 0.0849 0.1297 0.0566 -0.1334 0.0371 -0.1510 0.1213 0.1435 -0.0849 0.1415
-0.0817 -0.0323 0.0000 -0.0387 0.0000 0.0000 0.0332 0.0633 0.0000 -0.0470 0.0000 0.0000 0.0428 0.0000 0.0000
-0.0417 -0.0436 -0.0086 -0.0129 0.0000 0.0000 0.0163 0.0187 0.0107 -0.0297 -0.0031 -0.0037 0.0258 -0.0059
1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112 1112
0.2397 0.2054 -0.1410 0.1106 0.1438 -0.1545 0.0395 -0.0079 -0.1343 -0.0174 -0.2212 -0.0418 0.0581 -0.1445 -0.0658 -0.1159 0.0869 0.1310 0.0560 -0.1298 0.0422 -0.1501 0.1242 0.1444 -0.0790
0.0000 0.0000 0.0718 0.0000 0.0000 0.0674 -0.0754 0.0000 0.0198 0.0000 0.0000 -0.0821 -0.0296 0.0000 -0.0410 0.0000 0.0000 0.0343 0.0642 0.0000 -0.0452 0.0000 0.0000 0.0441 0.0000
6 6 6 6 6 6 6 6 6
0.2350 0.2015 -0.1424 0.1007 0.1410 -0.1539 0.0403 -0.0144 -0.1319
0.0000 0.0000 0.0711 0.0000 0.0000 0.0658 -0.0754 0.0000 0.0164
66
c5
c6
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
-0.0254 0.0000 0.0000 0.0000 0.0000 0.0000 0.0105 0.0000 0.0000 -0.0020 0.0000 0.0000
-0.0155 0.0000 0.0000 0.0000 0.0000 0.0000 0.0062 0.0000 0.0000 -0.0014 0.0001
-0.0356 -0.0186 0.0128 0.0283 0.0000 0.0000 0.0403 0.0459 -0.0080 0.0259 0.0000 0.0000 -0.0421 -0.0442 -0.0080 -0.0156 0.0000 0.0000 0.0172 0.0197 0.0109 -0.0283 -0.0034 -0.0039 0.0268
0.0000 0.0000 -0.0180 0.0000 0.0000 0.0000 0.0000 0.0000 0.0316 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0262 0.0000 0.0000 0.0000 0.0000 0.0000 0.0113 0.0000 0.0000 -0.0023 0.0000
0.0067 0.0000 0.0000 -0.0184 0.0000 0.0000 0.0000 0.0000 0.0000 0.0231 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0164 0.0000 0.0000 0.0000 0.0000 0.0000 0.0069 0.0000 0.0000 -0.0016
-0.0332 -0.0174 0.0123 0.0301 0.0000 0.0000 0.0392 0.0447 -0.0085
0.0000 0.0000 -0.0167 0.0000 0.0000 0.0000 0.0000 0.0000 0.0309
0.0057 0.0000 0.0000 -0.0171 0.0000 0.0000 0.0000 0.0000 0.0000
Atomic
Data and Nuclear
0.0000
Data Tables.
Vol. 37. NO. 1. July 1967
J. RIKOVSKA
and N. J. STONE
TABLE
L
L’ 3
3
1 1
1
1
2 2 3 1 1 1
2 2 3 2 2 3 3 2 2 3 1 1 1
2 2 3 1 1 1
2 2 3 1 1 1
2 2 3
2 3 2 3 3 1 2 3 2 3 3 2 3 3 3 2 3 3 1 2 3 2 3 3 1
2 3 2 3 3 1
2 3 2 3 3
3 2 2 3 1
3 2 3 3
1
2 3 2 3 3 1
1 2 2 3 1
1
Linear
Polarization
Distribution
II. Linear Polarization Distribution Coefficients See page 58 for Explanation of Tables
Ii
If
Gamma-Ray
c2
c3
5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
-0.0269
0.0000
-0.2216 -0.0386 0.0583 -0.1468 -0.0609 -0.1209 0.0886 0.1321 0.0555 -0.1266 0.0466 -0.1491 0.1266 0.1451 -0.0739
0.0000
712 912 912 9!2 11/2 11/2 11/2 11/2 11/2 11/2 1312 13/2 13/2 1312 1312 1312 15/2 15/2 15/2 15/2 15/2 15/2
13/2 13/2 13/2 1312 13/2 13/2 13/2 1312 13/2 1312 13/2 1312 13/2 1312 1312 1312 1312 1312 13/2 1312 13/2 13/2
4 5 5 5 6 6 6 6 6 6 7
7 7 7 7 7 7 7 7 7 7 7
c4
c5
c6
-0.0825 -0.0273 0.0000 -0.0427 0.0000 0.0000 0.0353 0.0649 0.0000 -0.0436 0.0000 0.0000 0.0452 0.0000
0.0237 0.0000 0.0000 -0.0425 -0.0447 -0.0074 -0.0178 0.0000 0.0000 0.0179 0.0205 0.0111 -0.0271 -0.0036 -0.0041 0.0276
-0.0268 0.0000 0.0000 0.0000 0.0000 0.0000 0.0120 0.0000 0.0000 -0.0025 0.0000
0.0228 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0171 0.0000 0.0000 0.0000 0.0000 0.0000 0.0075 0.0000 0.0000 -0.0018
0.2311 0.1981 -0.1435 0.0924 0.1387 -0.1533 0.0410 -0.0198 -0.1297 -0.0347 -0.2219 -0.0358 0.0585 -0.1486 -0.0567 -0.1248 0.0901 0.1331 0.0551 -0.1238 0.0502 -0.1479
0.0000 0.0000 0.0704 0.0000 0.0000 0.0644 -0.0753 0.0000 0.0136 0.0000 0.0000 -0.0828 -0.0253 0.0000 -0.0441 0.0000 0.0000 0.0362 0.0656 0.0000 -0.0421 0.0000
-0.0313 -0.0164 0.0119 0.0313 0.0000 0.0000 0.0382 0.0438 -0.0090 0.0218 0.0000 0.0000 -0.0428 -0.0451 -0.0069 -0.0195 0.0000 0.0000 0.0185 0.0212 0.0112 -0.0259
0.0000 0.0000 -0.0156 0.0000 0.0000 0.0000 0.0000 0.0000 0.0303 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0272 0.0000 0.0000 0.0000 0.0000 0.0000 0.0126 0.0000
0.0050 0.0000 0.0000 -0.0160 0.0000 0.0000 0.0000 0.0000 0.0000 0.0225 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0176 0.0000 0.0000 0.0000 0.0000 0.0000 0.0081
0.2278 0.1953 -0.1444 0.0854 0.1367 -0.1528 0.0416 -0.0244 -0.1278 -0.0412 -0.2221
0.0000 0.0000 0.0698 0.0000 0.0000 0.0633 -0.0753 0.0000 0.0112 0.0000 0.0000
-0.0298 -0.0156 0.0115 0.0323 0.0000 0.0000 0.0374 0.0429 -0.0093 0.0200 0.0000
0.0000 0.0000 -0.0148 0.0000 0.0000 0.0000 0.0000 0.0000 0.0297 0.0000 0.0000
0.0045 0.0000 0.0000 -0.0151 0.0000 0.0000 0.0000 0.0000 0.0000 0.0222 0.0000
67
0.0000 0.0000 0.0000 0.0000 0.0000
Atomic
Data and Nuclear
Data Tab&.
Vol. 37, NO. 1, July 1987
J. RIKOVSKA
and N. J. STONE
Gamma-Ray
TABLE II. Linear Polarization See page
L 1 1 2 2 3 1 1 1 2 2 3 3 2 2 3 1 1 1 2 2 3 1 1 1 2 2 3
c3
2 3 2
7 7 7
7 7 7
-0.0334 0.0587 -0.1500
-0.0830 -0.0236 0.0000
3 3 1 2 3
7 7 8 8 8
7 7 7
-0.0530 -0.1280 0.0915
-0.0452 0.0000 0.0000
2 3 3
8 8 8
7 7 7
0.1339 0.0547 -0.1213
7 7
3 2 3 3 1 2 3 2 3 3 1 2 3 2 3 3
15/2
912 11/2 11/2 1112 1312 13/2 13/2 1312 1312
1512 1512 15/2 15/2 15/2 15/2 15/2 1512
1312 15/2 15/2
1512 15/2 15/2
15/2 1512 1512 1512
15/2 15/2 1512 1512
3 2 2 3
3 2 3 3
1 1
1 2
5 6 6 6 7 7
1 2 2 3
3 2 3 3
7 7 7 7
1 1 1 2 2 3
1 2 3 2 3 3
8 8 8 8 8 8
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
Polarization
Distribution
58 for Explanation
C2
If
L’
Linear
Distribution
Coefficients
of Tables
c4
c5
c6
0.0000
0.0000
0.0000
-0.0430 -0.0455
0.0000
0.0000
0.0000
0.0000
0.0370 0.0661 0.0000
-0.0065 -0.0209 0.0000 0.0000 0.0190 0.0219
-0.0276 0.0000 0.0000 0.0000 0.0000 0.0000
0.0534 -0.1467
-0.0408 0.0000
0.0113 -0.0248
0.0132 0.0000
0.2249 0.1928 -0.1451
0.0000 0.0000 0.0692
-0.0285
0.0000 0.0000
0.0794 0.1350 -0.1524 0.0421 -0.0284 -0.1260 -0.0468
0.0000
-0.0181 0.0000 0.0000 0.0000 0.0000 0.0000 0.0086 0.0041
0.0000 0.0000 0.0623 -0.0752 0.0000 0.0091
-0.0150 0.0112 0.0330 0.0000 0.0000 0.0367 0.0422 -0.0096
-0.0141 0.0000 0.0000 0.0000 0.0000 0.0000 0.0292
0.0000 0.0000 -0.0143 0.0000 0.0000 0.0000 0.0000 0.0000
-0.2223 -0.0313
0.0000 0.0000 -0.0832
0.0185 0.0000 0.0000
0.0000 0.0000 0.0000
0.0219 0.0000 0.0000
0.0588 -0.1512 -0.0497 -0.1305
-0.0221 0.0000 -0.0462 0.0000
-0.0432 -0.0457 -0.0061 -0.0220
0.0000 0.0000 -0.0279 0.0000
0.0000 0.0000 0.0000 -0.0184
0.2224 0.1907 -0.1456 0.0741 0.1335 -0.1520 0.0425
0.0000 0.0000 0.0686 0.0000
-0.0275 -0.0144 0.0110 0.0336
0.0000 0.0000 -0.0135 0.0000
0.0037 0.0000 0.0000 -0.0137
0.0000 0.0614
0.0000 0.0000 0.0000 0.0000 0.0288 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0281 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0216 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0187
-0.0318 -0.1244 -0.0515
-0.0751 0.0000 0.0073 0.0000
0.0000 0.0000 0.0361 0.0416 -0.0098 0.0171
-0.2224 -0.0295 0.0589 -0.1522 -0.0468 -0.1327
0.0000 -0.0833 -0.0208 0.0000 -0.0469 0.0000
0.0000 0.0000 -0.0434 -0.0459 -0.0057 -0.0230
68
Atomic
Data and Nuclear
Data Tables.
Vol. 37. No. 1. Juiy 1987