On circular polirization measurements in γ-γ perturbed angular correlations

3•.A [

Nuclear Physics 86 (1966) 677--680; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON CIRCULAR POLARIZATION M E A S U R E M E N T S IN ~,-y PERTURBED A N G U L A R CORRELATIONS E. G. BELTRAMETTI

lstituto di Fisica dell'Universit3 di Genova LN.F.N. - Sezione di Genova Received 9 May 1966 Abstract: The role of circular polarization measurements in 7-7 perturbed angular correlations is

examined. It is shown, in particular, that g-factors of spin ½- excited nuclear states could be measured by allowing circular polarization detection.

Magnetic dipole moments (g-factors) and electric quadrupole moments of shortlived excited states of nuclei have been extensively measured by use of gamma-gamma perturbed angular correlations (PAC)2). Other methods, e.g. those involving Mdssbauer effect, require very special conditions, and may be used in a small number of cases. Insofar PAC techniques seldom involved measurements of 7-ray polarization (ref. z)) 1. In the present note we consider the role of circular polarization (CP) detection in PAC; experiments of this kind seem practical thanks to the high sensitivity of CP analysers now available 2). It will be shown, e.g., how g-factors of spin ½ excited states could be searched allowing CP measurements; the absence of quadrupole interaction would make this case attractive. Consider a two-step nuclear cascade; let Jl and j2 denote the spin of the initial and final nuclear states which are assumed to be randomly oriented; let j be the spin of the intermediate state whose time evolution operator is denoted by

A(t)=exp

-

K t ' dt'

,

K(t) being the perturbing Hamiltonian. Let ks, k2 be the propagation vectors of the first and second radiation (Yl, Y2). If the physical observations made on 7i are independent of rotations about its propagation vector kl (e.g. for motion direction and CP detection), the double correlation function may be written 3, 4) (aside from J l , J, J2 dependent factors): w(~l, :~2, t)

"~ ~ (--1)L'+Lz+~z[(2VI-t-1)(2Vz-Plf]-~{ j j v l I [ j jvz} (L~L';L~'2) L t IZ, jlttL 2 L'2 J2 × cv,, o(L,L',)cv2, o ( L 2 I J 2 ) ( j [ I L I H J l ) ( j [ [ I . J l l ] j , ) * ( j 2 [ [ L 2 ] ] j ) ( j 2 l l L ' 2 [ l J )

x G~'l"~(t)YO[*(k,)Y~":(k2).

*

(l)

t The problem of finite solid angle corrections, which is particularly relevant when circular polarization is detected, has been extensively considered by Yates 8). 677

678

E. G. BELTRAMETTI

Here ai summarizes the observations made on Yi; use has been done of 6-j symbols; cvi ' o(Li, LI) are the so-called radiation parameters; (j[ILIpJjl) denotes the reduced matrix element for emission of y~ with multipolarity L 1 ; the "perturbation factor" G ~ ( t ) is defined by 4)

rna, mb

ma

--m a

rtl

mb

--rob

t t , x (mb[A(t)[ma)(mblA(t)]rn,~)

n2 (2)

where 3-j symbols are used. The radiation parameters depend upon the informations specified by the experiment under discussion: if the left (LC) or right (RC) circular polarization is observed, we get 5)

LCc~,o(L, E ) = ( - 1 ) L + I [ ( 2 L + 1)(2L'+ 1)(2v+ 1)] ~ (L

_L'I 0) '

RCCv,o(L , J~)~ (--1)L+ 1-v[(2L'~- 1))2L"-~-1)(2v-{-])]~ (L

-/~ 1

0)

'

(3)

with v any integer number; if the CP is not observed, we get

dir'cv,o ( L ,

E)

= 0

dir'cv, o(L,/~) = LCcv, o(L, L') = RCc~,o(L, L')

for v odd, for v even.

The appearence of odd values of v in eqs. (1) and (3) is typical for CP measurements 5). If the interaction in the intermediate state is absent, G,,,2 .... = 6v~,v26..... and the correlation (1) reduces to the one discussed in a previous paper 5). In this case the detection of the CP of both y-rays is required, in order to get a correlation not identical to the unpolarized directional correlation. For perturbed correlations, the CP of only one quantum, say ~,1,is sufficient to modify the unpolarized correlation, providing G nln2 .... ¢ 0 for v~ # v2 (this of course is relevant for experiments since CP analysers introduce a serious loss in the counting rate). The 6-j coefficients appearing in eq. (1) involve the restriction 0 __< vi ~ rain (2j, Li+ L'i). In standard directional correlations, v~ is further restricted to even values so that non-isotropic correlations may be obtained only if j > 1. Allowing CP detection, non-isotropic correlations are obtained also for j = ½, thus providing a possibility for g-factor measurements of spin ½ excited states. Consider the usual technique in which a static external magnetic field B is applied; assuming the quantization axis

679

7"7 PERTURBED A N G U L A R CORRELATIONS

along B, the perturbation factor becomes 4) Gm.Z(t~ = vlv2v,

e

inio~Bt

(~vi, v 2 0 n l , n2 ,

¢d)B - -

(4)

Bp ,

hj

(/~ being the magnetic dipole moment). The CP of both 7-rays is thus required in order to get a correlation not identical to the direction-direction correlation and, in particular, to allow the determination of g-factors f o r j = ½ excited states. Denoting by WLC_Rc(k~, kz) the correlation in which the left CP of 71 and the right CP of 72 are observed, and by use of similar notation if LC (RC) is substituted by RC(LC), we obtain, due to eq. (3), the symmetry relations W L C - L C = W R C - R C , W L C - R C = WRC-LC- If k t and k 2 a r e chosen perpendicular to B, one gets a correlation of the form (q~ is the angle between k 1 and k 2 ) Vrnax

W±(a,, a2, t) ~ Z b.(a,, ~2) cos [n(q~+c0Bt)],

Vma x =

min (2j, L l + E l ,

L2 +/J2),

n=0

showing that the perturbing magnetic field rotates the correlation pattern t by O)Bt.

Due to the presence of odd values of n, eq. (5) exhibits asymmetries around the value ~0 = i n - c o a t; they are typical of CP correlations and are inverted if a L C - L C (or R C - R C ) correlation is replaced by a L C - R C (or R C - L C ) correlation (see eq. (3)). Let us shortly quote the perturbation of a fixed, axially symmetric, electrostatic gradient with the electric quadrupole moment Q of the intermediate nuclear state. The perturbation factor is (with the notation of ref. 4)).

× exp [ - 3i(

-

t].

In general Gn,n ..... # 0 for v I v~ vz (except ifv 1 or v 2 is zero) so that theCP of only one quantum, say Yl, introduces significant modifications in the correlation. We remark that changing Q into - Q introduces in G"'" ..... a factor ( - 1) v'+v2. Thus CP measurement may determine the sign 8) of Q since odd values of vl (or v2) are allowed. Of course, non-vanishing values of G~',',~ for vl ~ v2 also occur in the case of combined static magnetic and electric interaction 4). The detailed role of CP measurement in this case, as for other interactions, may be found in formulae (1)-(3).

t T h e explicit f o r m of the coefficient b~ is obtained by eqs. (1) a n d (4). F o r pure multiple radiations the reduced m a t r i x elements m a y be ignored a n d b n becomes a s u m o f t e r m s like F v ( L I , j l , j ) × Fv(L2, Jz, J), where the functions F v are t a b u l a t e d in refs. ~, 6) for even r a n d in ref. 7) for o d d v.

680

I~. G. BELTRAMETTI

References 1) E. Karlsson, E. Matthias and K. Siegbahn, Perturbed angular correlations (North-Holland. Publ. Co., Amsterdam, 1964) 2) R. M. Steffen and H. Frauenfelder, in Alpha-, beta- and gamma-ray spectroscopy, Vol. 2, ed. by K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1965) p. 1453; M. J. L. Yates, in op. cir. appendix 9 3) L. C. Biedenharn and M. E. Rose, Revs. Mod. Plays. 25 (1953) 729 4) H. Frauenfelder and R. M. Steffen, in Alpha-, beta- and gamma-ray spectroscopy, op cit., p. 997; in Perturbed angular correlations, op. cit., p. 3 5) E. G. Beltrametti, Nuovo Cim. 8 (1958) 445 6) H. Ferentz and N. Rosenzweig, Argonne National Laboratory Report 5324 (1964) 7) K. Alder, B. Stech and A. Winther, Phys. Rev. 107 (1957) 728; A. H. Wapstra, G. J. Nijgh and R. van Lieshout, Nuclear spectroscopy tables (North-Holland Publ. Co., Amsterdam, 1959) 8) S. M. Harris, Nuclear Physics 11 (1959) 387; H. J. Behrend and. D. Budnick, Z. Phys. 168 (1962) 155