Spectroscopic quadrupole moments from muonic atoms: Nuclear polarizability correction

Nuclear Phyatca A274 (1976) 413-427 ;,© Norrh-XoAad P~bllahinp Co ., dntrrerdlont Not to be reproduced by photoprlnt or micxo8lm without written permiraton tiom the publisher

SPECTROSCOPIC QUADRUPOLE MOINENTS FROM MUONIC ATOMS : NUCLEAR POLARIZABILITY CORRECTION J . MARTORELL and F . SCHECK SIN, CH 523

f

Villtgen, Switzerland

Received I8 August 1976 Abstract : In . extracting spectroscopic quadrupole moments from the quadrupole hyperfne structure of intermediate orbits of muonic atoms nuclear polarizability is an important theoretical correction . Whilst the electric quadrupole polarizability is fairly well understood, this is not so for the electric dipole polarizability . we show that the latter yields a center-0f-gravity shift ö e of the quadrupole multiplet and a shift SA= of the quadrupole hyperfine constant. The quantity ó o is evaluated using nuclear sum rules and is found to be in good agreement with existing numerical studies. The shift. SA z is very small due to strong cancellations in the nuclear matrix elements . This cancellation is studied and proven in a number of nuclear models .

1.

I~ÍIrOdOCÍiO~

Nuclear spectroscopic quadrupole moments can be determined very precisely and in a model-free manner from the quadrupole hyperfine structure of intermediate orbits in muonic atoms. This has been demonstrated recently in the example of the 4f-3d transitions in muonic lutetium t) from which the quadrupole moment could be determined to. less than one percent accuracy. About half of this error is purely statistical, whilst the remainder is due to the uncertainty in the theoretical corrections to the simple picture of â point-like deformed nucleus. Numerically important are the nuclear finite size correction, the nonspherical vacuum polarization, and the nuclear polarizability correction . Whilst the finite size and the vacuum polarization correction can be calculated in a straightforward and very accurate way, the variation of the quadrupole hf constant due to nuclear polarizability is more difficult to obtain and must be discussed iti somewhat more detail. As usual one subdivides this correction into multipole contributions, among which the electric monopole, dipole and quadrupole terms are the most important ones. Evidently, the monopole polarizability only shifts the center-of-gravity of the. multiplet but does not contribute to the quadrupole hf constant . As far as the quadrupole polarizability is concerned there are well-known and fairly reliable methods of calculation based on the rotator model. Here, it is well-known that the main contribution comes from the ground state rotational band Z). The effects, as T

Present address : Institut für Physik, Johannes Gutenberg-Universitilt, Postfach 3980, D-65 Mainz. 413

4l4

J. MARTORELL AND F. SCHECK

calculated for instance by the method developed by McKinley 3 ), turn out to be nonnegligible, of the order of 2 to 3 ~ in the quadrupole moment t). This leaves the polarizability correction due to nuclear dipole transitions whose magnitude is less obvious. In this note we are concerned with the influence of this dipole polarizability on the quadrupole hyperfine constant. We calculate the center-of-gravity shift and give limits for the change of the hyperfine constant through this of%ct, by making use of sum rule techniques . In the case of the intermediate, hydrogen-like orbits we are considering here, penetration ef%cts are small, of the order of five percent or less in a correction which itself is small. As a consequence, muonic and nuclear degrees of freedom separate and the relevant matrix elements factorize into a muonic and a nuclear part. While the muonic part can be calculated in a straightforward and direct manner, the nuclear part can be treated either within the framework of nuclear models or, in a rather model-independent way, by means of sum rules for nuclear dipole transitions t . In sect. 2 we summarize the relevant formulae and reduce the polarizability shifts to two purely nuclear matrix elements which are discussed and evaluated in the subsequent sections. Sect. 3 deals with the calculation of the center-of-gravity shift of the multiplet as a whole, by means of sum rule techniques . In sect. 4 we discuss and estimate the polarizability shift of the gi~adrupole hyperfine constant in the framework of various nuclear models. The concluding sect. 5 summarizes only results. 2. Dipole polarizabílity shifts The energy of an individual state of the quadrupole hyperfine multiplet is given by the well-known formula ~~ = E"~+A z(nj, nC(1,l, ~~

Here E~ is the energy of the muonic state (n, j) without electric quadrupole interaction, I is the nuclear spin, F the total angular momentum, i.e. the vector sum of nuclear spin Iand muonic angular momentum j, Az is the quadrupole hyperfine constant, whilst C(I, j, F) is the well-known angular momentum factor C(1,j, F) -

3X(X -1)-47(I + 1)j(j+ 1) 21j(21-1x2j-1) '

where Methods of analysis and practical calculations of the hyperfine constant A z are described in detail in ref. t), and we do not go into details here . We note only that Az, for intermediate hydrogen-like orbits, is very close to its value for a point-like f Sum rule techniques have also been applied to particle polari7abilities by Ericson and Hfffner `).

QUADRUPOLE MOMENTS

415

nucleus, where it is proportional to the spectroscopic quadrupole moment and to the expectation value <1lr 3 ~ I in the muonic orbit (n, j). The dipole polarizability shift is given by dE EI (F) _

~ 1 I<(Ij)FIHEiI(l'Î~>I Z, LL*II EIIF-ELLF

where HE 1 is the electric dipole Coulomb interaction, HEi

= -

4n z A r e 3 t~ r~

*

~ Yi~,(P,~Yi~,(Pr)

1+~3~ 2

The well-known technical complications in calculating nuclear polarizability shifts stem primarily from the penetration of muonic orbits into the nucleus . In particular, penetration effects are important for the 1 s and 2p orbits in heavy muonic atoms, where there does not seem to be any way of avoiding a tedious numerical evaluation ofthe polarizability shifts . In the case of intermediate muonic orbits, however, the muon moves well outside the nucleus and penetration effects may safely be neglected. We then have, approximately, HE~ ^- -

43

ez

r

~ Y~,(P,~D~

with D~, _ ~; 1 r, Yl,~(Pr}}(1 +T3'ß, the nuclear electric dipole operator. The summation over I', ~" in eq . (3) implies a summation over a complete set of intermediate nuclear end muonic states. We first observe that, in general, the expression on the right-hand side of eq . (3) yields a shift of the center-of-gravity EI (eq. (1)) of the multiplet as well as a modification of the quadrupole constant A2, In other terms the shift eq . (3) assumes the form dEEi (F) = So +bA z C(1, j, ~,

(6)

where C(I, j, ~ is the same angular momentum factor as before, eq . (2). This can be seen, for instance, by applying the closure approximation over nuclear and muonic states to eq. (3). Suppose the energy denominator is replaced by an average excitation energy d. Then dE~, is given by dEEI (F~

~ - d {UIj)FI(HEi) Z I(IJ)F> - IUIj)FIHE,I(Ij)F>IZ }.

The second term in the brackets vanishes due to parity ; the first term contains (Hsi)2 which, in the approximation underlying eq . (5), can be written as Z - 4n z ea

(II EI) -

~3~

ra N

3 4~ {-

1 12n

~oo+

is ~ Ys~(P~z~~,

416

when

J . MARTORELL AND F . SCHECK

~ (lm 1m 2 1L11~D~,D, .~._ are tensor operators of rank L in the space of the nuclear variables. The first term in the curly brackets on the right-hand side of eq . (7) is spherically symmetric and, therefore only gives a shift of the multiplet as a whole, see the term denoted So in eq. (~. The second term has the structure of the first order, diagonal, electric quadrupole interaction. Therefore it gives rise to a modification SA z of the quadrupole hyperfine constant. In this approximation we then obtain I~L, __

One`

1

So ^ 3d 21 -1 3~d
(10)

The expectation value of r~4 in the muonic orbit (n, j) can be calculated, to a good approximation, with Coulomb wave functions for a point-like nucleus
(11)

with a = 1/Zam the Bohr radius. The quantities So and SAZ are now expressed in terms of purely nuclear matrix elements
The shift So of the multiplet as a whole can be calculated by means of sum rules for nuclear dipole transitions, thus avoiding the approximation underlying eq . (9) and the evaluation of the nuclear matrix element
417

QUADRUPOLE MOMENTS

the monopole part So ofdEEl (F~, eq . (3), in terms of the nuclear dipole polarizability : A

aEl

-

~2

~

I
wing

t=1

riz( 1 +T3)In>I

Z

EA - Eo

(12)

Here, dD .stands for the energy of the giant resonance ; d~, represents the average muonic excitation energy . Since d~ is small compared to dD, the whole correction factor is close to one:.Then aEl can be determined using the well-known sum rule derived by Migdal 6), aEl

~ A(rZ ) x 1 .89 x 10_aAs~s fm3, K

__ _e2

(14)

where
(15)

We remark that this result is in very good agreement with detailed numerical calculations by Cole') . As an alternative we can obtain anotherestimate for this same quantity ifwe recast eq. (3) in a form such that the double oommutatorpf HEl with the total Hamiltonian H can be introduced. This leads us to a generalization of the Thomeas-Kuhn-Reiche sum rule and is useful also for the estimate of bAZ, sect . 4. For this purpose we write dEEi(F~ x

1 (dD+d~2 ~ (Er~B-Er"~"F)I<(L1)FIHEiI(I~1)FiI2 1 <(IJyFI (Hei, (H, HEiIII(I1)F>, 2(dD+d~Z _ -

(1~)

where the double commutator has to be taken with the total Hamiltonian of the muon-nucleus system, A 1 H = Tp +TN +VN - ~

e2

+~~

,

(17)

where TM .is the muonic kinetic energy, while TN +VN is the nuclear Hamiltonian.

41 8

J. MARTORELL AND F. SCHECK

After some algebra this double commutator is found to be (in the factorization limit), [Hev [H, Hsi~l = [HEi,

[(TN+ VN)~

4a z 90e' - ( 3 ) mNr~

Hei~~

1 L 4n ~ {1

2 1

2 2 2 1~0 0

L 0) ~`.n~ Yr er(P,a'

(18)

The, first term on the right-hand side leads to the ordinary nuclear dipole sum rule and contributes to So, the monopole shift, only . The second term stems from the commutator [T~, Het] and contributes to both the monopole and the quadrupole polarizability shift. Thus f [Hev [(Tx+

VN~

e° 1 NZ Het~~ = rN mN A '

(19)

where So is given, from eq . (16), by eq . (19) and the term with L = 0 of the expression on the right-hand side of eq . (18). The latter contribution is found to be small, of the order of 1 ~ of what is obtained from eq. (19) and may therefore be neglected. Thus

For the case of lutetium, as an example, this gives - 26 eV, which is very close to the estimate given above and the numerical result of ref. '). 4. Models and esthnates for SA=, the shüt of the hyperfine oonetaet In order to estimate SAz, the modification of the quadrupole hyperfine constant due to nuclear dipole polarizability, we can either start from the closure expression (10) or from the extended Thomas-Kuhn-Reiche sum rule, eq. (18) . In either case the calculation of SAz is reduced to a calculation of the nuclear matrix element
(21)

Even though the closure approximation in both nuclear and muonic variables is a rough approximation, it turns out to be sufficient for our purposes. Indeed, we shall see that the nuclear matrix element (21), and hence also SA z from oqs. (10) or (18), is small so that we obtain rather an upper limit for that quantity . First we notice that >lrzo can be written as the sum of one body and two body operators ~zo = ßió+ßió.

depending on whether i = j or i ~ j in the double sum over nucleon coordinates t Up to the usual oorrediom due to exchange forces . Those can of course be inserted into e9. (19) .and into the final expttiasion for ó o.

QUADRUPOLE MOMENTS

419

eq. (8). From eq . (8) and the definition of D,~ we have " i=i

~ió = ~ (17r7,1-mI20) ~ r, Yi~(Pt)Yi -~(P~) which is easily seen to be

~~~

"_

1 + t(i)

i-i

1 + Tl'1 s , 2 (23)

From this it follows that the nuclear matrix element (21), taken for ßió alone, is proportional to the spectroscopic quadrupole moment, 1 <1II ~/rióhl ~ = 4n

.

(24)

This result is independent of any nuclear model. The two body part of the operator ~so is given by " 1 + k il l + t N (25) ßió = ~ (1~,1-7n~20) ~ rrYi~(P~r1Yi-~(P~) 2 3 2 3 . m

!*1

We now show using different nuclear models, that the matrix element ~II~~ZSOIlI) nearly cancels the matrix element of the one body part, eq . (24) . In some simplified situations this cancellation is complete, in more realistic situation it is not complete but the resulting value for the matrix element of the sum ~so = ~iö+~ZÓ is still rather small. It was Chen s) who noticed first that the exchange matrix elements of the two~Y Pam ßïó cancel to a large. extent the contribution of the direct term of ~~ó and of the one-body part .~rh. In our case where the direct term of ßió vanishes due to parity (D,~ is odd under parity) this cancellation is almost complete . As deformed nuclei are more difficult to describe microscopically than spherical ones we start with the discussion of nuclei close to spherical (closed-shell) nuclei (models I to III). The results are then generalized to open-shell and strongly deformed nuclei subsequently (models I to III - deformed). 4.1 . NUCLEI WITH SPHERICAL CORE

We start with the simplest, though somewhat unrealistic model. Model 1: One odd nucleon outside a doubly closed shell core . In this model the nuclear spin and the quadrupole moment are carriod by the odd nucleon. If we further assume that the orbits can be described as harmonic oscillator states, then it is easy to show that


(26)

420

J. MARTORELL AND F. SCHECK

The proofgoes as follows : First we express the expectation value of ßsó in tenors of two body matrix elements. The direct terms vanish due to parity, and we obtain z «ió)i = ~( - ~ +i (1~ 1-m120) ~ I(ilrY~~li)I Z, (2~ t.~=i

where i,j denote single particle states. The sum over . these can be simplified by remarking that the closed shell core has spin zero, so that ~~zoi~e = 0, and thus only particle-core matrix elements will contribute to (2~. Furthermore, for harmonic oscillator wave functions, the only non-vanishing matrix elements are those for which i, j belong to consecutive major harmonic oscillator shells, so that only the core states of the last occupied major shell will give a contribution to (2~. Finally, a complete set of these states can be constructed using a Cartesian coordinate basis, which allows (2~ to be rewritten as

«iói = 2 ~ ( - ~+ '(lm, l -m120)

~

I(n~,n=ITYi~In~+;~s)I 2,

(28)

("z+"y+"~~N-1)

where for defmiteness we assume that the odd nucleon wave function, In~, n~, n:) with nx +n~+n= = N, is also of this type . Expressing the spherical harmonic operator also in Cartesian coordinates and using standard properties of harmonic oscillator functions, we find

where b = 1 /mco is the harmonic oscillator parameter. Finally, remarking that the quadrupole moment of the odd nucleon is given by ~~,z Q"s +ry"s - -(ns+ny-~sl~ we have (31) This result can easily be extended to any single particle wave function which is a linear combination of states I nx, nr n=) belonging to the same N-shell. This completes the proofof eq . (26) . Before going on to more complicated models we would like to comment on this result . Firstly, when the orbits of the particles are taken to be Hartree-Fock states, with a more realistic radial behaviour, then the cancellation is not complete any more. For example, using Hartree-Fork states pertaining to the Skyrme SIII force ' e) and the expression (16) we find, typically, óAz

- 1 eV.

(32)

QUADRUPOLE MOMENTS

421

Secondly, the model can immediately be generalized, without altering these conclusions, to the following cases Model La: One nucleon outside a closed subshell core whose last filled subshells have süigle particle angular moments differing by more than one unit from the odd nucleon one : This extension comes out trivially from the property which ensures that the discussion following (2~ still applies. Model Lb : One nucleon outside a closed subshell core whose last filled subshells have the same principal quantum number as the odd nucleon : This generalization is again a consequence of the vanishing of the matrix element (33), now due to conservation of parity . We note that these two extensions imply in practice that SA 2 in eq . (6) is zero in almost all nuclei around closed shells . A typical example where simultaneous application of both La and Lb is required is that of Z°9Bi ground state. This cancellation however is not specific for such nuclei, and the same results can be seen to apply also to other nearly spherical nuclei with partially filled shells under somewhat less restrictive conditions, as discussed in the following model. Model II: Here there are n nucleons (n odd) in the lowest seniority state, filling incompletely a subshell, outside a closed shell core. Using the straightforward techniques of ref. 9) part III, extended to tensor operators of arbitrary degree 1 ~, the multinucleon matrix elements of ~Z° can be reexpressed in terms of the ones appearing in model I (a detailed proof given in the appendix). This gives for the one body term, ßió, the expected result proportional to the quadrupole moment, and an analogous expression is also obtained for ßzó. Both can be summarized by


+

~,~

+

Thus ifthe r.h.s. of (34) are equal and opposite in sign for s = 1, 2 as model I predicts for harmonic oscillator functions, again the expectation values of the one and two body parts of ~Z° cancel, and SAZ = 0. Furthermore, since the ratio (2j+ 1-2n)/(2j-1) is always of the order of 1, the estimate (32) applies also in this model. The two preceeding models rely heavily on the validity of pure shell-model configurations in describing nuclear states. We discuss here an alternate approach, inspired by the Thomas-Ferrai model. This suggests that the cancellation of SAZ is not related to shell structure, but to much simpler bulk properties of the nuclear one and two body densities.

422

J. MARTORELL AND F. SCHECK

Motel III:, Slater local density approximation (SLDA). This model assumes that the two body density in nuclei can be approximated by the factorized expression (exact in nuclear matter) P~z~(ri~ rz) ^ Pst(ske(r))P(r~ (35) with s = rz - rl, r = ~}(r 1 + rz) and where ps,, p are the Slater and one body densities respectively. The former is given by Pst(x) = 31i(x)/x,

(36)

kF(r) _ [3~zP(r)]~~

(3~

wherejl is a spherical Beseel function ; the local Fenmi momentum kF{r) is related to the one body density through (we consider only protons) No assumptions are made for the one body density. The validity of SLDA and its improvement through a systematic expansion of plz~ in a complete set of Beseel functions of argument skF, has been discussed extensively in refs. i i . iz) . pf special relevance for our problem are the results discussed in ref. 13) where SLDA is used to evaluate the exchange part of the total Coulomb energy, since expanding in this case the Coulomb interaction in multipoles leads to expressions quite similar to our eq . (2~. In a systematic survey of s-d shell nuclei, it is shown in ref. 's) that the SLDA values differ by about 6 ~ from the Hartree-Fork results, and furthermore, that the quality of agrcement improves with the mass number A, making SLDA specially adequate for heavier nuclei . Given the smallness ofthe effects we are interested in, the quoted errors are quite acceptable for our purposes and no improvement on (35) seems necessary in our case. The evaluation of «io~i in this model is again very simple . We summarize the steps Usingc.m. and relative coordinates, and using the fact that ~ ió is spin independent, we have ~~zói = - 2 where

J

drps(r)

J

dsP~(skr~r))~ió~(s~ r~

(38)

~ió~(s+r) = 4n i(r= ás=) -[(rs+r7)-~Sx+sy)]~+ (39) performing the integration overthe svariable, and using standard properties ofBeseel functions (40) ~~iói = - 4k 2 drp(rX2r=-r~-ry), J which is again (26) . Thus also in this model there is complete cancellation between the one and two body parts of <~zoi .

QUADRUPOLE MOMENTS

423

4.2 . NUCLEI WITH DEFORMED CORE

Up to this point, we have been considering only nearly spherical nuclei. This restriction is not essential, and the results for models I and III can be extended to deformed nuclei without difficulty Using standard strong coupling wave functions [notation as in ref. 14)] the expectation value of our operator is written as ~IMK~~ zo IIMK~ =

2I+1 {(D~rxIDóol~~rx) 8nz

x (K~~ío~~+8a .t(-~-~ ~ (n`~ xIDóiII~`a.-c~I~íol - ~}~ x

(41)

The second term in the r.h.s. is present only when the intrinsic state ~K) has K = ~. However, since J = K for ground ,states, this term contributes only in spin-} nuclei . Since for these the expectation value of ~zo is zero, and thus bAz = 0 trivially, this second term never loccurs in any case of interest . The first term is proportional to the expectation value of ßío in the intrinsic state. To evaluate this, we now extend models I and III to intrinsically deformed states . Model l-deformed: One odd nucleon outside a closed shell core . All single particle states are eigenfunetions of an axially symmetric oscillator well (bx = by $ b=). The evaluation of ßío goes as in (2~, (28) ; we note in particular that since parity is conserved the contribution of the direct term still vanishes. However due to the axial symmetry (29) becomes (42) It is easily seen that which again implies that the one and two body contributions to <~zoi associated to the odd nucleon cancel . In contradistinction to the spherical case, however, there could in principle also be contributions to «rzo) coming form the core itself, given the fact that its spin is no longer zero . These contributions can be shown to vanish by remarking that (42), (43) are valid also for any nucleon in the core, so that the one body contribution of each shell is compensated by the two body contribution due to the terms connecting this shell with the one with same quantum number " sinus one.. This leaves apparently the N = 0 shell uncompensated, but for this shell the expression (42) vanishes, so that cancellation between one and two body parts of ~~zoi ~ oomPlete. Model III - deformed: The extension of the results (38) to (40) obtained in the spherical case is immediate noticing that no assumption was made on the shape and symmetries of the one body density p(r), when we formulated this model, and thus the results remain the same in the defonmed case .

42 4

J . MARTORELL AND F. SCHECK

We complete our discussion of the nuclear matrix elements by considering the inclusion of effects of c.m . motion, using effective charges. As is well known, shellmodel wave functions are not translationally invariant, and, in particular, for harmonic oscillator functions the c.m . wave functions is a harmonic oscillator instead of a plane wave as it should . To avoid introducing spurious c.m . excitations, and in order to guarantee that only intrinsic excitations are taken into account, it is necessary to decompose our operators D~, eq . (5) in two parts acting respectively on c.m. and intrinsic coordinates and keep only the latter. This gives D~ _ ~ ~ nYi~(P~+ 1 ~ éinYi~(P~ i where we introduced of%ctive charges é equal to NlA for protons and -ZlA for neutrons. Thus, replacing D~, by its intrinsic part amounts to introduce et%ctive charges é, in all our expressions, and consequently the neutrons as well as the protons contribute to ss In determing the spectroscopic quadrupole moment from the hyperfine structure of muonic intermediate (i.e. hydrogen-like) orbits, the greatest theoretical uncertainty stems from the polariTability of the nucleus . In this paper we study specifically the case of the electric dipole polarizability. The shift of the center-ofgravity of the hyperfine multiplet as a whole is non-negligible and can be calculated reliably and in a simple manner by means of nuclear dipole sum rules. Our result for the case of lutocium is. in good agreement with the result of detailod numerical calculations of Cole'). As to the modification áA2 of the quadrupole hyperfine constant, we find a strong cancellation between nuclear direct and exchange matrix elements leading to a very small value of áA2, of the order of 1 eV. This cancellation is studied and proven in a variety of nuclear models, giving some confidence as to the general validity ofthe result . These conclusions do not depend on the approximations used in the summation over intermediate states. Closure over muonic states, «l. (18), or closure over muonic and nuclear states, eq . (10), give essentially the same results. One of us (J. M.) thanks GIFT (Spain) for financial support.

QUADRUPOLE MOMENTS

425

Appendix We give here the main steps in the derivation of (34). The evaluation of the expectation value of the one body term (s = 1) is identical to that of the quadrupole moment, since they are related through eq . (24). The corresponding expression can be found e.g. in ref. 9) p. 315. From this and our assumption of lowest seniority states the result in eq . (34) for s = 1 follows. The evaluation of «rZO) is done using eq . (24) from ref. '~ restricted to only one kind ofnucleons. The reduced matrix element corresponding to the interaction ofthe outer nucleons (total angular momentum of one nucleon : jo) with a subshell of the core (total angular momentum of one nucleon : j~) is given by zJ~+i zJ~+i+A +i --_ A~ V~1° +i(~)>ó(Jo~ I = joll ~ ~ ~ió~ri, rJ)IIj~J` (~)Jó(jo~ I = joi +=i

=

1=21~+z

n L,(~JOVO){~- 1V1)~)Z 210 + 1 1~

2i0+ 1 U(Jo 2J1J0~J0J0) ~ m.

~~+ 1(-~+w'

x U(cv'2j°,jo; cojo)
(A.1)

where the U are the normalized Rasah coefficients, as defined in ref. 's) ; and the functions ~io~ri, rJ) are obtained from eq. (8): ~ió~ri, rz) _ ~(1m,1-m120kiYi~(Pi)rzYl-~(Pz) "

(A.2)

The matrix elements ~(.n{~-1(j')i are the standard fractional parentage coefficients (c.f.p.). Since our objective is to relate the multinucleon expectation value of ßïû to the single odd-nucleon case, we particularize(A.1)to n = 1, and, comparing the resulting expression with the one for A, obtain A. _ -Ai9(n~

(A .3)

where we have defined 2 jo ~J,o (A.4) 9(n) = n~(~(Uo){lÏó-lVi)i)z(-~'(2.10+1) Uo .li jo~ . 1~ This reduces our problem to the evaluation of g(n). First we note that writing explicitly the value of the 6j symbol we obtain a socond degree polynomial in x --- j,(ji + 1). The summation overj, can be then performed separately for each of its tenors. For the term of zero degree in x the contribution of the c.f.p. is unity, since they satisfy the orthononmality condition [ref. 9~ p. 273] The result for the terms of first degree caa be easily obtained also from the general

426

J. MARTORELL AND F. SCHECK

relation [ref. ~, p. 280] J'

J'(J'+ lx
n

n

2 J(J+ 1)+,lÏj+ 1).

(A.6):

For the term ofsecond degree explicit properties ofthe c.f p. of an odd n, seniority one, configuration are needed . In this particular case, we note that the states ~-t (J~i can have only seniority zero or two . For seniority zero states : J' = 0 and thus there is no contribution to the summation over J' --_ jt . The c.f.p. corresponding to seniority two states are given in ref. ~, p. 307, un_ 1x2J, +1) (A .7) (~v = 1(J =~~{I,~-tv, = uJ~)i)Z = n(2j+ lx2j _ 1) , which reduces the evaluation of the degree two term to the calculation of zJo -t

which after some algebra is found to be n = (2.1o -1 )2jo(2jo+lxjo+lx`~.1~ó+3jo -1)~ Putting together the results for the three terms, the value for g(n) is

(A .9)

and when this is inserted back into (A .3) and the reduced matrix elements substituted by the non-reduced ones with Ml = I, the eq . (34) for the s = 2 case is obtained . RefereBCes 1) H . J. Leisi, W. Dey, P. Ebersold, R. Engfer, F. Scheck and H. K. Walter, J. Phys . Soc. Jap. Suppl. 34 (1973) 355; W. Dey, Thesis No . 5473 (1975) ETH Zûrich ; W. Dey, P. Ebersold, H. J. Leisi, F. Scheck, H. K. Walter and A. Zehnder, Nucl . Phys., to be published 2) M . Y. Chen, Phys . Rev. Cl (1970) 1176 3) J. M. McKinley, Phys. Rev . 183 (1969) 106 4) T. E. O. Ericson and J. Hüfner, IQucl. Phys. B~7 (1972) 205 5) F. Scheck, Z. Phys. 172 (1963) 239 6) A. B. Migdal, J. Phys. (USSR) 8 (1944) 331 ; J. S. Levinger, Nuclear photo-disintegration (Oxford University Press, London, 1960); A. B. Migdal et al., Nucl. Phys . 66 (1965) 193 7) R. K. Cole Jr., Phys. Lett . 25B (1967) 178 ; Phys . Rev. 177 (1969) 164 8) M. Y. Chen, Phys. Rev. Cl (1970) 1167 9) A. de Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) 10) J. M. Gomez, A. Poves and J. Bernebeu, preprint Univ . Autónoma Madrid, FTUAM/II/75/1, to be published in Ann. R. Soc. EsparSola Fis. Quinn 11) J. W. Negele and D. Vautherin, Phys. Rev. CS (1972) 1472 ; Cll (1975) 1031

QUADRUPOLE MOMENTS

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12) D. W. L. Sprung, M. Vallieren, X. Campi and C. M. Ko, Nucl . Phyn. Á2S3 (197 1 13) P. Quentin, Thesis, Université Paris-Sud, Orsay (1975) ; C. Titin-Schneider and P. Quentin, Phys . Lett . "9B (1974) 397 14) M. A. Preston, Physics of the nucleus (Addison-Wesley, NY, 1965) 15) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ . Press, Princeton, NJ, 1957) 16) M. Reiner, H. Flocard, Nguyen Van Gisi and P. Quentin, Nucl . Phys. A238 (1975) 29