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Nuclear orientation following muon capture by spin zero nuclei
pi-1
Nuclear Physics Not to be
A262 (1976) 433 - 443
: @ North-Holland
Publishing
Co., Amsterdam
reproduced by photoprint or micro6lm without written permission from the publisher
NUCLEAR ORIENTATION FOLLOWING MUON CAPTURE BY SPIN ZERO NUCLEI t P. R. SUBRAMANIAN Department
of Nuclear Physics,
University of Madras, Madras 600025, India
R. PARTHASARATHY Institut de Physique Corpusculaire,
tt
Universitk de Louvain, B-1348 Louvain-la-Neuve,
Belgium
and V. DEVANATHAN Institut f$r Theoretische
++t
Physik der Universitgt Frankfurt/Main,
Frankfurt,
W. Germany
Received 17 November I975 Abstract: It is shown that in the proposed experimental set-up of the Louvain-Saclay collaboration, the nuclei recoiling into the forward (or backward) hemisphere in the reaction p- +A(J, = 0) --) B(Jf 2 1) + Y,, will not only be polarized but also aligned. A general expression for the alignment of the recoiling nucleus is derived in terms of the other observables and it is shown to be independent of nuclear structure and the dynamics of the muon capture interaction. Numerical results are presented for muon capture by “C and 160.
1. Introduction In recent years there has been a revival of interest in the study of nuclear polarization ’ - 6, resulting from the capture of polarized muons by a spin zero nucleus since it is now realized that the polarization, because of its insensitivity to nuclear structure, is capable of yielding a better estimate of the coupling constants of the fundamental muon capture interaction. Moreover it is now feasible to retain and measure this polarization by the ion-implantation technique developed by the Louvain-Saclay collaboration ‘). In these experiments, the direction of the recoil nucleus was not Qbserved and hence only the average polarization of the final nucleus could be determined. But there is a proposal 6, to modify this experimental set-up in order to determine simultaneously the longitudinal polarization PL and average polarization PN of the recoil nucleus and indirectly the recoil asymmetry coefficient CI.The effect of the new set-up is to preserve at a given time the orientation of the recoil nucleus recoiling into only one of the hemispheres. The orientation of the nucleus recoiling into the other hemisphere is completely destroyed. The purpose of this article is to t Partly supported by the Gesellschaft fiir Schwerionenforschung (GSI), Germany. tt Permanent address: Department of Physics, Sri Pushpam College, Poondi, Tbanjavur District, 613 503, India. ‘W Permanent address: Department of Nuclear Physics, University of Madras. Madras 600025, India. 433
434
P. R. SUBRAMANTAN
et 01.
show that in the proposed experimental set-up the recoil nucleus will be not only polarized but also aligned if the spin of the final nucleus Jf 2 1. A general expression for the polarization has been derived in ref. “) in terms of P, and P, using an explicit form for the muon capture interaction. Subsequently, Bernabeu 5, has derived the same expression using the rotational invariance and the fixed helicity of the neutrino. In this article we extend the method of Bernabeu 5, to derive an expression for the nuclear alignment following muon capture by a spin zero nucleus. This expression is general and valid for any final nucleus with spin Jf >= 1. Numerical results are presented for the reactions p- +=C(O+) -+ “B(l+)+v,,
(1)
,LL-f160(0+)
(2)
-+ i6N(2-)+v,.
The method consists in choosing a proper frame of reference in which the helicity formalism of Jacob and Wick 7, can be applied. The normal choice for studying the reaction p- +A(Ji = 0) -+ B(J, >= l)+v,,
(3)
is the rest frame of the initial system. This corresponds, in the final state, to the c.m. system, with the recoil momentum p = -v, v being the neutrino momentum. The moving reference system, associated with the direction p^,is obtained from the rest system by rotation through the Euler angles (4,8,0) (see fig. 1). In the latter system, since the entire reaction can be described by two independent helicity amplitudes and their relative phase, there cannot be more than three independent observables and hence one can obtain certain relations among the several observables in the muon capture reaction. Using only the rotational invariance of the interaction Hamiltonian and the definite helicity (- $) of the neutrino, we obtain an expression for the alignment in terms of other observables viz., longitudinal polarization and average polarization of the recoil nucleus.
Fig. 1. The rest frame xyz and the moving
reference frame x’y’z’. The z-axis is pointing in fig. 2.
along fi as shown
435
NUCLEAR ORIENTATION
2. Nuclear spin orientation 2.1. SPIN ORIENTATION
PARAMETERS
The density matrix pr of the final nucleus completely specifies its spin orientation which can be conveniently represented ” ‘, ‘) by a set of parameters (53 defined by (T.?
= Tr U”V,)/Tr
pf,
(4)
where TpK denotes a spherical tensor operator of rank Kin the spin space of the final nucleus and satisfies the normalization condition Tr (7”; 7”g’) = (25, + 1)6,, K’6mX, ,,,+
(5)
subject to the restriction 0 5 K 5 2Jr. The spherical tensor TrK satisfies+ the relations lo): (J,M;IT,mK(JfMf)
= (2K+
(T,o)
l)+C(J,KJ,;
iVQz&f),
= (2J, + l)+G,(J,),
(6)
(7)
where G,(J,) are the statistical tensors introduced by Fano 12’13). They are given by I’) GK(j) = c (- ly-“‘p,,,C(jjK; m, - ml, m where p, is some suitably normalised weight factor giving the population sublevel m. The normalization chosen in ref. 12) is ;Pm
= 1.
(f-3 of the
(9)
The quantities p, are proportional to the diagonal elements of the density matrix pf when it is diagonal. Since in our case Trp, is not normalized to unity, P,,, = (d,,,D’r
ffy
(10)
is a diagonal element of pf in diagonal form. Clearly eq. (10) is consistent with eq. (9). As shown in ref. 12), pm and G,(J) are transforms of each other and it is a matter of simple algebra to obtain the following:
where (P,),,
P,,, = ‘2
CW(J,~J,; mO)p,, 111
= c
[W(J&J,;
mO)lCJflZ-
(11) W)
K=O t For angular momentum coefficients, reduced matrix elements and rotation matrices, we follow the notation and definition of Rose, ref. ‘I).
436
P. R. SUBRAMANTAN et nl.
Here [K] is defined as [K-j = (2K+ l)%
(13)
From eqs. (11) and (12), it is clear that if the population of the sublevels are known, the tensor moments (Ti) can be calculated and vice versa. The quantities (Ti) are the spin orientation parameters we are interested in. When (TF) [or G,(J,)] is not equal to zero, we say the nucleus is “polarized” [ref. I’)]. When (Ti)[or G&J,)] is not equal to zero, the nuclei are said to be “aligned” [ref. “)I. Since (Tg)are given in terms of traces of matrices [see eq. (4)] we can use any convenient representation to evaluate the traces. 2.2. THE CHOICE OF z-AXIS
Since we are interested in the evaluation of tensor polarizations averaged over a hemisphere, the z-axis has to be chosen carefully. Let us choose the z-axis to coincide with the normal drawn to the plane bounded surface of the forward hemisphere
Fig. 2. Forward
hemisphere
(fhs), backward hemisphere (bhs) and the z-axis. The residual muon polarization P, is inclined to fi.
pointing in the forward direction [see fig. 2; also ref. 3)]. Only then the limits of 8 for the forward hemisphere will be 0 and $, and $c and n for the backward hemisphere. Let k^be a unit vector along the z-axis. In ref. ‘), P, = k^and this corresponds to the hemispheres with their plane face at right angles to P,. The choice P,, = E has its own advantages and disadvantages. Thus when P, = &, the calculations become easier. There is no $-dependence for the elements of pr (in the helicity basis) and only (T,"") with mK = 0 exist. In general when the z-axis is along k^
(14) where ths denotes integration over the forward hemisphere. The quantities M+ and M_, are the helicity amplitudes given in ref. “). The x- and y-components of the polarization vector P, of the muon are P, and P,, and & and t?,,are unit vectors
437
NUCLEAR ORIENTATION
along the X- and y-axes (rest frame). The quantities q and PH are given by +
3p
=
H
2fi ReCM@?+)_p IM,I'+
IM_,12
L’
where PL is the longitudinal polarization of the recoil nucleus obtained with unpolarized muons 4, and PH is the average polarization obtained with completely polarized muons. The term $fi Im (M_+A$)(P2$P,,L?,.)does not appear in refs. 3, 5, for the following reasons. In ref. ‘), P, = k and hence P, = P, = 0. The time-reversal invariance implied in the choice of the Hamiltonian in ref. 3, restricts the amplitudes M, and M_*.+to be real. Thus in both cases the above term vanishes. For further calculations we choose the z-axis along k as shown in fig. 2. 2.3. THE DENSITY MATRIX FOR THE FINAL NUCLEUS
In the helicity basis the density matrix pf has only four non-vanishing elements (&, If’ where A, and 2; can take only the two values 0 and - 1. This is due to the fixed helicity for the neutrino which restricts the helicity values of the recoil nucleus to 0 and - 1 in the helicity basis. It is a straightforward calculation to get 47@,),, = ~M~~2[1+PZcos8+sin8(P_e’~+P+e-i~)], (17)
= lM,12(l + P, * R, 47c(p,),, _1 = -M,M~,[P, W&L
l_,cl= -M_%Mf[P,
sin 8+P+e-“(1
-cos d)-P_e”(l
+cos @)I,
sin~+P_e’~(1--cos6)-P+e-‘~(1+cos8)],
(18)
(19)
~M_~~2[1-P,cos6-sin0(P_e’9+P+e-i~)]
47c(p,)_,,_;= =
lM_,12(1-P;
19,
(20)
where P,
= P,fiP,.
(21)
The component of P, along the z-axis is P, and it is easy to see that eqs. (17)-(21) give the elements of pr obtained in ref. ‘) when pfl = k^. 2.4. EXPRESSION FOR <7-F)
Since the density matrix pr has only four non-vanishing elements in the helicity basis it is easy to evaluate Tr(TFKPf) in the helicity basis. The helicity basis is obtained + Eq. (16) is essentially the same as that given by Bernabeu. However our definitions of polarization differ from those given in ref. 5, by a factor Jr Thus P,, of ref. ‘) is equal to JfPav of ref. 5).
P. R. SUBRAMANIAN
438
et al.
from the rest frame by a rotation through Euler angles (4, 0, 0), and the law of transformation for a tensor of rank K is given by 11)
It can be easily seen that Tr(Tpxpf) transforms as a tensor of rank K under rotation and hence Tr(G?W
= c M
c DEL,&%?)&, &L,,
Ip
&,,Af =
D~,O(T~)b,o(~f)d,ofD~~,
+DETE,
1 CT&,-,
-~K1)‘-,,&,)b,
-I
(~f)l1,O+D~,O(TKO)'-1,-1(~f)l-1,-,,
(23)
where of,, M = D&, &,
0,0).
(24)
The prime in the matrices Tf and pf denote that their elements are to be found in the helicity basis. Use has been made of the fact that Lf and if, can take the values 0 and - 1 and eq. (6) yields A4 F &-it,. Using eqs. (4), (6) and (17)-(24), (7’FK) can be found. Since we have to average over the unphysical azimuthal angle 4, we have to take into account the angular dependence of @J;,,1,,. This dependence comes from D$m (4, 13,0) where j = 0 or 1 only. Now 21r of;, ,($, 8, O)o’,,, ,(cb, RO)d4 = 2namK,,, f(Q (25) s0 wheref(8) is a function and this then implies glance at eqs. (17)-(20) to
of 0 only. Thus when mK = 0, only oj,,,,($, 8,O) contributes that only 4 independent terms of pf contribute. A simple shows that of the three components of P,, only P, contributes a complete integration over the recoil direction we obtain 1‘)
Thus full integration yields a,, j and since j = 0, 1, we arrive at our earlier result ‘) that when an integration over the recoil (or neutrino) direction is made, the recoil nucleus can have only vector polarization when Jf 2 1. Thus we get for K 2 2, C@f”,,
= 0.
(27)
K >=2,
(28)
= G)i.h,+<~KO)blls
Hence CT:>,,,, = -(T,o>,,,,
NUCLEAR ORIENTATION
439
where bhs (fhs) denotes that integration over the recoil direction is made in the backward (forward) hemisphere and “full” denotes complete integration. 2.5. ALIGNMENT
OF RECOIL NUCLEUS
In principle all (T,O) can be calculated using eqs. (6) and (17)-(24). What we are primarily interested in is the evaluation of (T,O) in view of the experimental proposals ‘“) to measure them for “B(1’) and 16N(2-). We proceed as follows. Since 11) DA,,&, /I, y) = e-im’ad~~,(~)e-i”Y,
(29)
we get
%,*M(4, 40) = d:,,(e)
=
(-
1)~
(& >kde),
(30)
using table 1 of ref. ‘). The quantity P,,(e) is the unnormalized Legendre polynomial and is defined on page 241 of ref. ‘I). Using eq. (30) we obtain d;, ,(e) = $3 COS' e - i), dg, ,(e) = -d;,
_,(e) = (p
sin
(31) e cos 8.
Using eq. (7) and the table of C-coefficients 15) we get (T,“>b,cl= --YG (T,_l)‘,,,
= -($)%,
(T,‘Y&- 1 = @.d% (32) (T-)‘+1
= (~-VP,
where t = ($(4?j - 3)) -4.
(33)
By direct integration we obtain (T,o),,,
= +-t(q+2qPL+9PH)(P;
r;>.
(34)
The quantities ye,PH and t are given by eqs. (15), (16) and (33) respectively. Note that P, = P, . k. Eq. (34) stems only from the rotational invariance and the defined helicity (-3 of the neutrino. Hence the relation is independent of the nuclear structure, the coupling constants of the muon capture interaction (with or without second class current terms) and the dynamics of the muon capture interaction. Here we have evaluated (Ti),,, directly by evaluating. the traces. Alternative methods of evaluating (Tt) are indicated in the appendix. 3. Numerical results and discussions We have calculated (T:) for the reactions (1) and (2). For muon capture in 12C(Of), we have used our earlier results 3, for PL and PH [= P,/P of ref. 3)] in
440
P. R. SUBRAMANIAN
et al.
TABLE 1 Variation 3 of A with gp/gA for “3(1+)
B&A 30.0 25.0 20.0 17.5 12.5 7.5 5.0 3.5 ? 0.0 -5.0 - 10.0 -20.0 -25.0 - 30.0
Model I
Model II
-0.0601 -0.0197 + 0.0288 + 0.0546 3-0.1059 -1-0.1522 i-O.1723 -i-0.1832 -l-o.1794 + 0.2052 + 0.2285 + 0.2434 -l-o.2550 -l-o.2550 +0.2527
- 0.0768 -0.0409 +0.0044 co.0295 +0.0813 +0.1305 +0.1527 i-O.1650 f0.1614 i-o.1903 i-O.2183 + 0.2372 -t-0.2543 +0.2559 i-O.2547
“) A =
BP/S,4
30.0 25.0 20.0 17.5 12.5 7.5 5.0 3.5 “) 0.0 -5.0 -10.0 -20.0 -30.0
Model II (GV)
-P,.
pH
A
9061 9775 9997 9895 9306 8358 7819 7488 7624 6722 5702 4812 3448 2530
-1744 833 3628 4965 7253 8835 9350 9578 9492 9906 9985 9763 8912 7942
-802 -620 -340 -175 169 488 627 702 672 853 1014 1122 1230 1259
-P, 9126 9808 9992 9872 9256 8298 7758 7428 7560 6667 5655 4776 3427 2518
Pa -1570 1027 3811 5131 7373 8906 9400 9617 9537 9923 9982 9751 8895 7930
A
Model IV (MIGDAL)
Model III (EF)
-P,,
-792 9168 -603 9821 -319 9991 -153 9873 190 9274 505 8342 641 7815 715 7492 686 7625 863 6744 1021 5743 1126 4867 1232 3513 1259 2591
PH
A
-FL
Pu
A
-1453 1106 3835 5128 7332 8857 9356 9579 9495 9903 9991 9787 8968 8020
-786 -596 -316 -153 183 493 628 701 672 849 1008 1116 1227 1259
9209 9858 9982 9827 9145 8141 7588 7253 7376 6488 5484 4620 3307 2428
-1342 1330 4147 5461 7648 9093 9540 9729 9670 9978 9972 9697 8797 7820
-779 -577 -280 -109 238 550 682 754 729 896 1045 1143 1239 1261
3 A = <~;&,sfO’,,.@-
b) Values obtained with gA = -0.986 gv and gp/gA = 3.5 [ref. 23)]. The rest of the values are obtained with gA = - 1.230 gv.
the shell model and the generalized lp shell wave functions of Wirooka et al. 16). For the case of 160, we have used four models - the independent particle model (IPM), Gillet and Vinh Mau wave functions (GV) 17), Elliott and Flowers (EF)
NUCLEAR
ORIENTATION
441
wave functions 18) and the Migdal wave functions I’). Our results are presented in tables 1 and 2. We have used the Fujii-Primakoff effective Hamiltonian 20) and included the momentum dependent terms of order l/M of the muon capture interaction, M being the nucleon mass. Also the complete set of nuclear tensor operators has been taken into account. We have used the same set of values for the coupling constants and oscillator length parameters 1.64 fm for 12C and 1.76 fm for 160, as in refs. 3*21). The following salient features are observed. Polarizations are almost insensitive to nuclear models but sensitive to the gP/gA ratio. Eq. (18) of ref. 5, in our notation reads (4~ + 1)P; + 2(3P, + 2y)P, + 9P; = 0.
(35)
From table 2 one can see that eq. (35) is satisfied, taking into account computational round-off. It is a matter of simple algebra to show that eq. (35) represents an ellipse with its centre at (6, -9 for all J, 2 1 in a coordinate system with its origin at (0,O) and PH and PL chosen along the x- and y-axes respectively. Treating eq. (35) as a quadratic in PH, we obtain the mathematical limits of PL and vice versa. We thus obtain --+Jf 5 P, 5 $Jf+l), -1
SPLSO.
(36)
(37)
Of course the physical limits for PH are - 1 and + 1. For Jf = 2, the mathematical limits of PH obtained from eq. (36) are -g and 1. For Jf = 1, the limits of PH are -f and $. Thus for Jf $ 2, the mathematical limits are more significant than the physical limits. From eq. (37), it is clear that PL is always negative. It must be remembered that the helicity of the neutrino is taken as - f and time-reversal invariance is implied in arriving at eq. (35) [eq. (18) of ref. “)I. Fig. 3 depicts the variation of A = (T~),,,/P, . k^with gP/gA ratio for “B (using the lp generalized wave functions) and 16N (using the Migdal wave functions). For the case of 12C --, “B, the recent experimental value 2, 22) of PH = 0.48ti.y: yields
In conclusion we wish to reiterate the usefulness of the helicity formalism in arriving at interesting relations between various observables independent of nuclear structure, the coupling constants and the dynamics of the muon capture interaction. With the improvements in the experimental techniques, more and more precise experiments to measure nuclear spin orientation parameters will be forthcoming in the near future and these studies will not only serve to check these relations but also yield a better estimate of the coupling constants of the muon capture interaction.
P. R. SUBRAMANIAN
442
0.26
--‘\
0.22 \ 0.1 8
Al&l
I
-30
-20
---
126(,+)
-
16~~2-1
\I T
-\ t A
et al.
\
I
- 0.06-- 0.06&
Fig. 3. Variation of A with gP/gA for “B(l+)
and 16N(2-).
One of us (R.P.) is grateful to Prof. J. P. Deutsch for the hospitality at the Institut de Physique Corpusculaire, Louvain-La-Neuve, Belgium and to Prof. L. Grenacs for useful discussions, and the other (V.D.) thanks Prof. Walter Greiner for the warm hospitality at the Institut fur Theoretische Physik at Frankfurt/M. It is a pleasure to thank Dr. L. Palffy for fruitful communications. Useful discussions with Dr. J. Bernabeu and Mr. K. Iyakutti are also gratefully acknowledged. Appendix
In the text we have evaluated (T;) directly. Here we indicate other methods of evaluating the same. A. 1. METHOD
BASED ON THE POPULATION
OF SUBLEVELS
If the populationp, of the sublevels m are known, then any (Tj)can be calculated using eq. (11). Since p,,, are the diagonal elements of pf when it is in diagonal form, one has to obtain the density matrix in diagonal form from the density matrix in helicity basis using the rotation matrices. Thus pr = D*p;, DT where the asterisk
NUCLEAR ORIENTATION
443
and T denote respectively the complex conjugate and transpose of the matrix D = Dj(cj, 8,0).Once this is diagonalized, (Ti)can be easily found using eq. (11). A.2. DIRECT EVALUATION
USING THE EXPLICIT FORM FOR THE HAMILTONIAN
One can choose tin effective Hamiltonian as in refs. 1’20) and then find out the terms that contribute to Tr(Tipr). Thus when the parity change is (- l)Jr+ I, the terms that contribute to Tr(Tt& are (I’. O)(M, * M,*), (P. @IO * M,12, (0. M2) x (P * AI;), M,(P * Ad;), (P * O)(G. M2)M~, (P . O)(iM, * it x M,*), and iP * (0 x MJ (B . Mz). The quantities MI (I = 1 to 4) are defined by eq. (14) of ref. ‘) and P = P,. By a procedure similar to the one followed in ref. 4), one can prove algebraically that eq. (34) is true. Of course eq. (34) is valid for both changes of parity ) 1. References 1) V. Devanathan, R. Parthasarathy and P. R. Subramanian, Ann. of Phys. 73 (1972) 291 2) A. Possoz, D. Favart, L. Grenacs, J. Lehmann, P. Macq, D. Meda, L. Palffy, J. Julien and C. Samour, Phys. Lett. 50B (1974) 438 3) V. Devanathan and P. R. Subramanian, Phys. Lett. 53B (1974) 21 4) P. R. Subramanian and V. Devanathan, Phys. Rev. Cl1 (1975) 520 5) J. Bernabeu, Phys. L&t. 55B (1975) 313 6) D. Favart, L. Grenacs, J. Lehmann, P. Macq, L. Palffy and A. Possoz, Implantation des noyaux de recul ‘*B issus de la reaction p-+l’C --t vp +“B et measure de leur polarisation, Universite de Louvain report, 1973, unpublished; L. Palffy, private communications 7) M. Jacob and G. C. Wick, Ann. of Phys. 7 (1959) 404 8) W. Lakin, Phys. Rev. 98 (1955) 139 9) G. Ramachandran, Nucl. Phys. B2 (1967) 565 10) P. R. Subramanian and V. Devanathan, J. of Phys. A7 (1974) 1995 11) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 12) M. E. -Rose, in Lectures in theoretical physics, Brandeis Summer Institute, 1961, vol. 2 (Benjamin, New York, 1962) 13) M. E. Rose, Phys. Rev. 108 (1957) 362 14) L. Palffy, private communications 15) E. U. Condon and G. H. Shortley, The theory of atomic spectra (Cambridge University Press, 1970) 16) M. Hirooka, T. Konishi, R. Morita, H. Narumi, M. Soga and M. Morita, Prog. Theor. Phys. 40 (1968) 808 17) V. Gillet and N. Vinh Mau, Nucl. Phys. 54 (1964) 321; Nucl. Phys. 57 (1964) 698 18) J. P. Elliott and B. H. Flowers, Proc. Phys. Soc.*A242 (1957) 57 19) A. B. Migdal, Nucl. Phys. 57 (1966) 29; M. Rho, Phys. Rev. Lett. 18 (1967) 671; V. Devanathan, M. Rho, K. Srinivasa Rao and S. C. K. Nair, Nucl. Phys. B2 (1967) 329 20) V. Devanathan, R. Parthasarathy and G. Ramachandran, Ann. of Phys. 72 (1972) 428 ; A. Fujii and H. Primakoff, Nuovo Cim. 12 (1959) 327 21) V. Devanathan and P. R. Subramanian, Ann. of Phys. 92 (1975) 25 22) L. Grenacs, private communication 23) M. Rho, private communication
Nuclear Physics Not to be
A262 (1976) 433 - 443
: @ North-Holland
Publishing
Co., Amsterdam
reproduced by photoprint or micro6lm without written permission from the publisher
NUCLEAR ORIENTATION FOLLOWING MUON CAPTURE BY SPIN ZERO NUCLEI t P. R. SUBRAMANIAN Department
of Nuclear Physics,
University of Madras, Madras 600025, India
R. PARTHASARATHY Institut de Physique Corpusculaire,
tt
Universitk de Louvain, B-1348 Louvain-la-Neuve,
Belgium
and V. DEVANATHAN Institut f$r Theoretische
++t
Physik der Universitgt Frankfurt/Main,
Frankfurt,
W. Germany
Received 17 November I975 Abstract: It is shown that in the proposed experimental set-up of the Louvain-Saclay collaboration, the nuclei recoiling into the forward (or backward) hemisphere in the reaction p- +A(J, = 0) --) B(Jf 2 1) + Y,, will not only be polarized but also aligned. A general expression for the alignment of the recoiling nucleus is derived in terms of the other observables and it is shown to be independent of nuclear structure and the dynamics of the muon capture interaction. Numerical results are presented for muon capture by “C and 160.
1. Introduction In recent years there has been a revival of interest in the study of nuclear polarization ’ - 6, resulting from the capture of polarized muons by a spin zero nucleus since it is now realized that the polarization, because of its insensitivity to nuclear structure, is capable of yielding a better estimate of the coupling constants of the fundamental muon capture interaction. Moreover it is now feasible to retain and measure this polarization by the ion-implantation technique developed by the Louvain-Saclay collaboration ‘). In these experiments, the direction of the recoil nucleus was not Qbserved and hence only the average polarization of the final nucleus could be determined. But there is a proposal 6, to modify this experimental set-up in order to determine simultaneously the longitudinal polarization PL and average polarization PN of the recoil nucleus and indirectly the recoil asymmetry coefficient CI.The effect of the new set-up is to preserve at a given time the orientation of the recoil nucleus recoiling into only one of the hemispheres. The orientation of the nucleus recoiling into the other hemisphere is completely destroyed. The purpose of this article is to t Partly supported by the Gesellschaft fiir Schwerionenforschung (GSI), Germany. tt Permanent address: Department of Physics, Sri Pushpam College, Poondi, Tbanjavur District, 613 503, India. ‘W Permanent address: Department of Nuclear Physics, University of Madras. Madras 600025, India. 433
434
P. R. SUBRAMANTAN
et 01.
show that in the proposed experimental set-up the recoil nucleus will be not only polarized but also aligned if the spin of the final nucleus Jf 2 1. A general expression for the polarization has been derived in ref. “) in terms of P, and P, using an explicit form for the muon capture interaction. Subsequently, Bernabeu 5, has derived the same expression using the rotational invariance and the fixed helicity of the neutrino. In this article we extend the method of Bernabeu 5, to derive an expression for the nuclear alignment following muon capture by a spin zero nucleus. This expression is general and valid for any final nucleus with spin Jf >= 1. Numerical results are presented for the reactions p- +=C(O+) -+ “B(l+)+v,,
(1)
,LL-f160(0+)
(2)
-+ i6N(2-)+v,.
The method consists in choosing a proper frame of reference in which the helicity formalism of Jacob and Wick 7, can be applied. The normal choice for studying the reaction p- +A(Ji = 0) -+ B(J, >= l)+v,,
(3)
is the rest frame of the initial system. This corresponds, in the final state, to the c.m. system, with the recoil momentum p = -v, v being the neutrino momentum. The moving reference system, associated with the direction p^,is obtained from the rest system by rotation through the Euler angles (4,8,0) (see fig. 1). In the latter system, since the entire reaction can be described by two independent helicity amplitudes and their relative phase, there cannot be more than three independent observables and hence one can obtain certain relations among the several observables in the muon capture reaction. Using only the rotational invariance of the interaction Hamiltonian and the definite helicity (- $) of the neutrino, we obtain an expression for the alignment in terms of other observables viz., longitudinal polarization and average polarization of the recoil nucleus.
Fig. 1. The rest frame xyz and the moving
reference frame x’y’z’. The z-axis is pointing in fig. 2.
along fi as shown
435
NUCLEAR ORIENTATION
2. Nuclear spin orientation 2.1. SPIN ORIENTATION
PARAMETERS
The density matrix pr of the final nucleus completely specifies its spin orientation which can be conveniently represented ” ‘, ‘) by a set of parameters (53 defined by (T.?
= Tr U”V,)/Tr
pf,
(4)
where TpK denotes a spherical tensor operator of rank Kin the spin space of the final nucleus and satisfies the normalization condition Tr (7”; 7”g’) = (25, + 1)6,, K’6mX, ,,,+
(5)
subject to the restriction 0 5 K 5 2Jr. The spherical tensor TrK satisfies+ the relations lo): (J,M;IT,mK(JfMf)
= (2K+
(T,o)
l)+C(J,KJ,;
iVQz&f),
= (2J, + l)+G,(J,),
(6)
(7)
where G,(J,) are the statistical tensors introduced by Fano 12’13). They are given by I’) GK(j) = c (- ly-“‘p,,,C(jjK; m, - ml, m where p, is some suitably normalised weight factor giving the population sublevel m. The normalization chosen in ref. 12) is ;Pm
= 1.
(f-3 of the
(9)
The quantities p, are proportional to the diagonal elements of the density matrix pf when it is diagonal. Since in our case Trp, is not normalized to unity, P,,, = (d,,,D’r
ffy
(10)
is a diagonal element of pf in diagonal form. Clearly eq. (10) is consistent with eq. (9). As shown in ref. 12), pm and G,(J) are transforms of each other and it is a matter of simple algebra to obtain the following:
where (P,),,
P,,, = ‘2
CW(J,~J,; mO)p,, 111
= c
[W(J&J,;
mO)
(11) W)
K=O t For angular momentum coefficients, reduced matrix elements and rotation matrices, we follow the notation and definition of Rose, ref. ‘I).
436
P. R. SUBRAMANTAN et nl.
Here [K] is defined as [K-j = (2K+ l)%
(13)
From eqs. (11) and (12), it is clear that if the population of the sublevels are known, the tensor moments (Ti) can be calculated and vice versa. The quantities (Ti) are the spin orientation parameters we are interested in. When (TF) [or G,(J,)] is not equal to zero, we say the nucleus is “polarized” [ref. I’)]. When (Ti)[or G&J,)] is not equal to zero, the nuclei are said to be “aligned” [ref. “)I. Since (Tg)are given in terms of traces of matrices [see eq. (4)] we can use any convenient representation to evaluate the traces. 2.2. THE CHOICE OF z-AXIS
Since we are interested in the evaluation of tensor polarizations averaged over a hemisphere, the z-axis has to be chosen carefully. Let us choose the z-axis to coincide with the normal drawn to the plane bounded surface of the forward hemisphere
Fig. 2. Forward
hemisphere
(fhs), backward hemisphere (bhs) and the z-axis. The residual muon polarization P, is inclined to fi.
pointing in the forward direction [see fig. 2; also ref. 3)]. Only then the limits of 8 for the forward hemisphere will be 0 and $, and $c and n for the backward hemisphere. Let k^be a unit vector along the z-axis. In ref. ‘), P, = k^and this corresponds to the hemispheres with their plane face at right angles to P,. The choice P,, = E has its own advantages and disadvantages. Thus when P, = &, the calculations become easier. There is no $-dependence for the elements of pr (in the helicity basis) and only (T,"") with mK = 0 exist. In general when the z-axis is along k^
(14) where ths denotes integration over the forward hemisphere. The quantities M+ and M_, are the helicity amplitudes given in ref. “). The x- and y-components of the polarization vector P, of the muon are P, and P,, and & and t?,,are unit vectors
437
NUCLEAR ORIENTATION
along the X- and y-axes (rest frame). The quantities q and PH are given by +
3p
=
H
2fi ReCM@?+)_p IM,I'+
IM_,12
L’
where PL is the longitudinal polarization of the recoil nucleus obtained with unpolarized muons 4, and PH is the average polarization obtained with completely polarized muons. The term $fi Im (M_+A$)(P2$P,,L?,.)does not appear in refs. 3, 5, for the following reasons. In ref. ‘), P, = k and hence P, = P, = 0. The time-reversal invariance implied in the choice of the Hamiltonian in ref. 3, restricts the amplitudes M, and M_*.+to be real. Thus in both cases the above term vanishes. For further calculations we choose the z-axis along k as shown in fig. 2. 2.3. THE DENSITY MATRIX FOR THE FINAL NUCLEUS
In the helicity basis the density matrix pf has only four non-vanishing elements (&, If’ where A, and 2; can take only the two values 0 and - 1. This is due to the fixed helicity for the neutrino which restricts the helicity values of the recoil nucleus to 0 and - 1 in the helicity basis. It is a straightforward calculation to get 47@,),, = ~M~~2[1+PZcos8+sin8(P_e’~+P+e-i~)], (17)
= lM,12(l + P, * R, 47c(p,),, _1 = -M,M~,[P, W&L
l_,cl= -M_%Mf[P,
sin 8+P+e-“(1
-cos d)-P_e”(l
+cos @)I,
sin~+P_e’~(1--cos6)-P+e-‘~(1+cos8)],
(18)
(19)
~M_~~2[1-P,cos6-sin0(P_e’9+P+e-i~)]
47c(p,)_,,_;= =
lM_,12(1-P;
19,
(20)
where P,
= P,fiP,.
(21)
The component of P, along the z-axis is P, and it is easy to see that eqs. (17)-(21) give the elements of pr obtained in ref. ‘) when pfl = k^. 2.4. EXPRESSION FOR <7-F)
Since the density matrix pr has only four non-vanishing elements in the helicity basis it is easy to evaluate Tr(TFKPf) in the helicity basis. The helicity basis is obtained + Eq. (16) is essentially the same as that given by Bernabeu. However our definitions of polarization differ from those given in ref. 5, by a factor Jr Thus P,, of ref. ‘) is equal to JfPav of ref. 5).
P. R. SUBRAMANIAN
438
et al.
from the rest frame by a rotation through Euler angles (4, 0, 0), and the law of transformation for a tensor of rank K is given by 11)
It can be easily seen that Tr(Tpxpf) transforms as a tensor of rank K under rotation and hence Tr(G?W
= c M
c DEL,&%?)&, &L,,
Ip
&,,Af =
D~,O(T~)b,o(~f)d,ofD~~,
+DETE,
1 CT&,-,
-~K1)‘-,,&,)b,
-I
(~f)l1,O+D~,O(TKO)'-1,-1(~f)l-1,-,,
(23)
where of,, M = D&, &,
0,0).
(24)
The prime in the matrices Tf and pf denote that their elements are to be found in the helicity basis. Use has been made of the fact that Lf and if, can take the values 0 and - 1 and eq. (6) yields A4 F &-it,. Using eqs. (4), (6) and (17)-(24), (7’FK) can be found. Since we have to average over the unphysical azimuthal angle 4, we have to take into account the angular dependence of @J;,,1,,. This dependence comes from D$m (4, 13,0) where j = 0 or 1 only. Now 21r of;, ,($, 8, O)o’,,, ,(cb, RO)d4 = 2namK,,, f(Q (25) s0 wheref(8) is a function and this then implies glance at eqs. (17)-(20) to
of 0 only. Thus when mK = 0, only oj,,,,($, 8,O) contributes that only 4 independent terms of pf contribute. A simple shows that of the three components of P,, only P, contributes a complete integration over the recoil direction we obtain 1‘)
Thus full integration yields a,, j and since j = 0, 1, we arrive at our earlier result ‘) that when an integration over the recoil (or neutrino) direction is made, the recoil nucleus can have only vector polarization when Jf 2 1. Thus we get for K 2 2, C@f”,,
= 0.
(27)
K >=2,
(28)
= G)i.h,+<~KO)blls
Hence CT:>,,,, = -(T,o>,,,,
NUCLEAR ORIENTATION
439
where bhs (fhs) denotes that integration over the recoil direction is made in the backward (forward) hemisphere and “full” denotes complete integration. 2.5. ALIGNMENT
OF RECOIL NUCLEUS
In principle all (T,O) can be calculated using eqs. (6) and (17)-(24). What we are primarily interested in is the evaluation of (T,O) in view of the experimental proposals ‘“) to measure them for “B(1’) and 16N(2-). We proceed as follows. Since 11) DA,,&, /I, y) = e-im’ad~~,(~)e-i”Y,
(29)
we get
%,*M(4, 40) = d:,,(e)
=
(-
1)~
(& >kde),
(30)
using table 1 of ref. ‘). The quantity P,,(e) is the unnormalized Legendre polynomial and is defined on page 241 of ref. ‘I). Using eq. (30) we obtain d;, ,(e) = $3 COS' e - i), dg, ,(e) = -d;,
_,(e) = (p
sin
(31) e cos 8.
Using eq. (7) and the table of C-coefficients 15) we get (T,“>b,cl= --YG (T,_l)‘,,,
= -($)%,
(T,‘Y&- 1 = @.d% (32) (T-)‘+1
= (~-VP,
where t = ($(4?j - 3)) -4.
(33)
By direct integration we obtain (T,o),,,
= +-t(q+2qPL+9PH)(P;
r;>.
(34)
The quantities ye,PH and t are given by eqs. (15), (16) and (33) respectively. Note that P, = P, . k. Eq. (34) stems only from the rotational invariance and the defined helicity (-3 of the neutrino. Hence the relation is independent of the nuclear structure, the coupling constants of the muon capture interaction (with or without second class current terms) and the dynamics of the muon capture interaction. Here we have evaluated (Ti),,, directly by evaluating. the traces. Alternative methods of evaluating (Tt) are indicated in the appendix. 3. Numerical results and discussions We have calculated (T:) for the reactions (1) and (2). For muon capture in 12C(Of), we have used our earlier results 3, for PL and PH [= P,/P of ref. 3)] in
440
P. R. SUBRAMANIAN
et al.
TABLE 1 Variation 3 of A with gp/gA for “3(1+)
B&A 30.0 25.0 20.0 17.5 12.5 7.5 5.0 3.5 ? 0.0 -5.0 - 10.0 -20.0 -25.0 - 30.0
Model I
Model II
-0.0601 -0.0197 + 0.0288 + 0.0546 3-0.1059 -1-0.1522 i-O.1723 -i-0.1832 -l-o.1794 + 0.2052 + 0.2285 + 0.2434 -l-o.2550 -l-o.2550 +0.2527
- 0.0768 -0.0409 +0.0044 co.0295 +0.0813 +0.1305 +0.1527 i-O.1650 f0.1614 i-o.1903 i-O.2183 + 0.2372 -t-0.2543 +0.2559 i-O.2547
“) A =
BP/S,4
30.0 25.0 20.0 17.5 12.5 7.5 5.0 3.5 “) 0.0 -5.0 -10.0 -20.0 -30.0
Model II (GV)
-P,.
pH
A
9061 9775 9997 9895 9306 8358 7819 7488 7624 6722 5702 4812 3448 2530
-1744 833 3628 4965 7253 8835 9350 9578 9492 9906 9985 9763 8912 7942
-802 -620 -340 -175 169 488 627 702 672 853 1014 1122 1230 1259
-P, 9126 9808 9992 9872 9256 8298 7758 7428 7560 6667 5655 4776 3427 2518
Pa -1570 1027 3811 5131 7373 8906 9400 9617 9537 9923 9982 9751 8895 7930
A
Model IV (MIGDAL)
Model III (EF)
-P,,
-792 9168 -603 9821 -319 9991 -153 9873 190 9274 505 8342 641 7815 715 7492 686 7625 863 6744 1021 5743 1126 4867 1232 3513 1259 2591
PH
A
-FL
Pu
A
-1453 1106 3835 5128 7332 8857 9356 9579 9495 9903 9991 9787 8968 8020
-786 -596 -316 -153 183 493 628 701 672 849 1008 1116 1227 1259
9209 9858 9982 9827 9145 8141 7588 7253 7376 6488 5484 4620 3307 2428
-1342 1330 4147 5461 7648 9093 9540 9729 9670 9978 9972 9697 8797 7820
-779 -577 -280 -109 238 550 682 754 729 896 1045 1143 1239 1261
3 A = <~;&,sfO’,,.@-
b) Values obtained with gA = -0.986 gv and gp/gA = 3.5 [ref. 23)]. The rest of the values are obtained with gA = - 1.230 gv.
the shell model and the generalized lp shell wave functions of Wirooka et al. 16). For the case of 160, we have used four models - the independent particle model (IPM), Gillet and Vinh Mau wave functions (GV) 17), Elliott and Flowers (EF)
NUCLEAR
ORIENTATION
441
wave functions 18) and the Migdal wave functions I’). Our results are presented in tables 1 and 2. We have used the Fujii-Primakoff effective Hamiltonian 20) and included the momentum dependent terms of order l/M of the muon capture interaction, M being the nucleon mass. Also the complete set of nuclear tensor operators has been taken into account. We have used the same set of values for the coupling constants and oscillator length parameters 1.64 fm for 12C and 1.76 fm for 160, as in refs. 3*21). The following salient features are observed. Polarizations are almost insensitive to nuclear models but sensitive to the gP/gA ratio. Eq. (18) of ref. 5, in our notation reads (4~ + 1)P; + 2(3P, + 2y)P, + 9P; = 0.
(35)
From table 2 one can see that eq. (35) is satisfied, taking into account computational round-off. It is a matter of simple algebra to show that eq. (35) represents an ellipse with its centre at (6, -9 for all J, 2 1 in a coordinate system with its origin at (0,O) and PH and PL chosen along the x- and y-axes respectively. Treating eq. (35) as a quadratic in PH, we obtain the mathematical limits of PL and vice versa. We thus obtain --+Jf 5 P, 5 $Jf+l), -1
SPLSO.
(36)
(37)
Of course the physical limits for PH are - 1 and + 1. For Jf = 2, the mathematical limits of PH obtained from eq. (36) are -g and 1. For Jf = 1, the limits of PH are -f and $. Thus for Jf $ 2, the mathematical limits are more significant than the physical limits. From eq. (37), it is clear that PL is always negative. It must be remembered that the helicity of the neutrino is taken as - f and time-reversal invariance is implied in arriving at eq. (35) [eq. (18) of ref. “)I. Fig. 3 depicts the variation of A = (T~),,,/P, . k^with gP/gA ratio for “B (using the lp generalized wave functions) and 16N (using the Migdal wave functions). For the case of 12C --, “B, the recent experimental value 2, 22) of PH = 0.48ti.y: yields
In conclusion we wish to reiterate the usefulness of the helicity formalism in arriving at interesting relations between various observables independent of nuclear structure, the coupling constants and the dynamics of the muon capture interaction. With the improvements in the experimental techniques, more and more precise experiments to measure nuclear spin orientation parameters will be forthcoming in the near future and these studies will not only serve to check these relations but also yield a better estimate of the coupling constants of the muon capture interaction.
P. R. SUBRAMANIAN
442
0.26
--‘\
0.22 \ 0.1 8
Al&l
I
-30
-20
---
126(,+)
-
16~~2-1
\I T
-\ t A
et al.
\
I
- 0.06-- 0.06&
Fig. 3. Variation of A with gP/gA for “B(l+)
and 16N(2-).
One of us (R.P.) is grateful to Prof. J. P. Deutsch for the hospitality at the Institut de Physique Corpusculaire, Louvain-La-Neuve, Belgium and to Prof. L. Grenacs for useful discussions, and the other (V.D.) thanks Prof. Walter Greiner for the warm hospitality at the Institut fur Theoretische Physik at Frankfurt/M. It is a pleasure to thank Dr. L. Palffy for fruitful communications. Useful discussions with Dr. J. Bernabeu and Mr. K. Iyakutti are also gratefully acknowledged. Appendix
In the text we have evaluated (T;) directly. Here we indicate other methods of evaluating the same. A. 1. METHOD
BASED ON THE POPULATION
OF SUBLEVELS
If the populationp, of the sublevels m are known, then any (Tj)can be calculated using eq. (11). Since p,,, are the diagonal elements of pf when it is in diagonal form, one has to obtain the density matrix in diagonal form from the density matrix in helicity basis using the rotation matrices. Thus pr = D*p;, DT where the asterisk
NUCLEAR ORIENTATION
443
and T denote respectively the complex conjugate and transpose of the matrix D = Dj(cj, 8,0).Once this is diagonalized, (Ti)can be easily found using eq. (11). A.2. DIRECT EVALUATION
USING THE EXPLICIT FORM FOR THE HAMILTONIAN
One can choose tin effective Hamiltonian as in refs. 1’20) and then find out the terms that contribute to Tr(Tipr). Thus when the parity change is (- l)Jr+ I, the terms that contribute to Tr(Tt& are (I’. O)(M, * M,*), (P. @IO * M,12, (0. M2) x (P * AI;), M,(P * Ad;), (P * O)(G. M2)M~, (P . O)(iM, * it x M,*), and iP * (0 x MJ (B . Mz). The quantities MI (I = 1 to 4) are defined by eq. (14) of ref. ‘) and P = P,. By a procedure similar to the one followed in ref. 4), one can prove algebraically that eq. (34) is true. Of course eq. (34) is valid for both changes of parity ) 1. References 1) V. Devanathan, R. Parthasarathy and P. R. Subramanian, Ann. of Phys. 73 (1972) 291 2) A. Possoz, D. Favart, L. Grenacs, J. Lehmann, P. Macq, D. Meda, L. Palffy, J. Julien and C. Samour, Phys. Lett. 50B (1974) 438 3) V. Devanathan and P. R. Subramanian, Phys. Lett. 53B (1974) 21 4) P. R. Subramanian and V. Devanathan, Phys. Rev. Cl1 (1975) 520 5) J. Bernabeu, Phys. L&t. 55B (1975) 313 6) D. Favart, L. Grenacs, J. Lehmann, P. Macq, L. Palffy and A. Possoz, Implantation des noyaux de recul ‘*B issus de la reaction p-+l’C --t vp +“B et measure de leur polarisation, Universite de Louvain report, 1973, unpublished; L. Palffy, private communications 7) M. Jacob and G. C. Wick, Ann. of Phys. 7 (1959) 404 8) W. Lakin, Phys. Rev. 98 (1955) 139 9) G. Ramachandran, Nucl. Phys. B2 (1967) 565 10) P. R. Subramanian and V. Devanathan, J. of Phys. A7 (1974) 1995 11) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 12) M. E. -Rose, in Lectures in theoretical physics, Brandeis Summer Institute, 1961, vol. 2 (Benjamin, New York, 1962) 13) M. E. Rose, Phys. Rev. 108 (1957) 362 14) L. Palffy, private communications 15) E. U. Condon and G. H. Shortley, The theory of atomic spectra (Cambridge University Press, 1970) 16) M. Hirooka, T. Konishi, R. Morita, H. Narumi, M. Soga and M. Morita, Prog. Theor. Phys. 40 (1968) 808 17) V. Gillet and N. Vinh Mau, Nucl. Phys. 54 (1964) 321; Nucl. Phys. 57 (1964) 698 18) J. P. Elliott and B. H. Flowers, Proc. Phys. Soc.*A242 (1957) 57 19) A. B. Migdal, Nucl. Phys. 57 (1966) 29; M. Rho, Phys. Rev. Lett. 18 (1967) 671; V. Devanathan, M. Rho, K. Srinivasa Rao and S. C. K. Nair, Nucl. Phys. B2 (1967) 329 20) V. Devanathan, R. Parthasarathy and G. Ramachandran, Ann. of Phys. 72 (1972) 428 ; A. Fujii and H. Primakoff, Nuovo Cim. 12 (1959) 327 21) V. Devanathan and P. R. Subramanian, Ann. of Phys. 92 (1975) 25 22) L. Grenacs, private communication 23) M. Rho, private communication