Consistent empirical description of magnetic moments and M1 transitions

Nuclear Physics A457 (1986) 292-300 North-Holland, Amsterdam

CONSISTENT

EMPIRICAL

DESCRIPTION

OF MAGNETIC MOMENTS AND Ml TRANSITIONS Quenching of the diagonal and off-diagonal spin matrix elements* W. KNOPFER’

and

W. MILLER’

Insiitut fiir Thedretische Physik der Universitiit Erlangen-Niimberg,

8250 Erlangen, W. Germany

B.C. METSCH Znstitut firTheoretische Kernphysik der Universitiit Bonn, 5300 Bonn, W. Germany A. RICHTER Institut firKemphysik der Technischen Hochschule Darmstadt, 6100 Darmstadt, W. Germany Received (Revised

10 January 1986 20 March 1986)

Abstract: The spin and tensor parts of an effective Ml operator for nuclei around 90Zr and “‘Pb are empirically determined by minimizing simultaneously the deviation of measured magnetic moments and Ml transitions from standard shell model calculations performed within lowest order shellmodel spaces. The results are linked to a previous analysis of sd-shell nuclei and show approximately the mass dependence of the combined influence of high-lying many-particle-many-hole excitations admixed via the tensor force and of A-h admixtures.

1. Introduction Investigations

of nuclear

spin-isospin

excitations

have

become

an important

branch of current low and intermediate energy nuclear physics activities. The exploration of Gamow-Teller (GT) strength distributions in (p, n) reactions ‘) and /3’ decay *), and the investigation of magnetic dipole transitions with high resolution inelastic

electron

scattering3)

and proton

scattering4)

have revealed

a systematic

quenching of the strength up to a factor of two as compared to standard shell model calculations. Such calculations include either configuration mixing within one major shell or ground state correlations in the RPA. At present two mechanisms are held to be dominant ‘) in the understanding of this quenching effect: (i) the admixtures via the tensor force of high-lying many-particle-many-hole configurations outside the lowest order shell model space via the tensor force (“tensor correlations”) and (ii) the retardation through virtual A-hole excitations (“A-hole effect”). It should be noted, however, that also meson exchange currents contribute and tend to interfere such as to weaken the effect of tensor correlations ‘). In fact this l

Work supported

’ Present address:

by the Deutsche Forschungsgemeinschaft. Siemens AC, Erlangen, W. Germany.

0375-9474/86/%03.50 @ Elsevier Science publishers (North-Holland Physics Publishing Division)

B.V.

293

W. Kniipfer et al. / Consistent description

weakening measured

might

be partly

responsible

in (e, e’) show a clear gradual

for the observation weakening

that

Ml

of the quenching

nuclei around A = 40 towards the lightest nuclei “) in contrast reactions. Indeed, for the latter reactions exchange currents

transitions

in going from

to (p, p’) and (p, n) do not contribute.

Calculations within a microscopic model approach which combine all these mechanisms are very extensive and hence have been performed only for some light nuclei near closed shells by Towner and Khanna ‘). Similar calculations for heavy nuclei are presently beyond feasibility. In this article, we therefore describe an attempt to approach the problem of the structure of the magnetic dipole operator in heavy nuclei in an empirical way. The combined study of Ml transitions and magnetic moments provides a valuable tool to investigate phenomenologically the dominant mechanisms for quenching mentioned above. The reason is the following: The calculations of ref. ‘) have demonstrated a strong effect of tensor correlations due to high lying configurations on the magnetic dipole observables. Thus, apart from the fact that the quenching mechanisms (i) and (ii) produce a renormalization of the spin- and orbital g-factor of the Ml operator, they also lead to an additional term arising from these tensor correlations. As an effect of the mechanisms (i) and (ii) we will see that the retardation of Ml strengths and the quantitative descripton of magnetic moments is controlled by the same mechanism but in a different manner. This gives us the possibility to disentangle the contribution of the spin and the tensor parts, respectively. In a very extensive study along this line, Wildenthal and collaborators lo) determined the effective Ml operator for nuclei of the sd shell. We proceed here in a different way for nuclei around 90Zr and 208Pb and derive empirically the spin and tensor parts of an effective Ml operator which describes simultaneously magnetic moments and Ml transitions. In the following section the conventional considered when calculations are performed

corrections to Ml observables are in a lowest-order shell-model space.

The remaining deviations between experiment and theory are then taken to determine empirically the combined influence of the above mentioned mechanisms (i) and (ii). This is done in the third section.

2. Magnetic moments and retardation of Ml transitions The quenching of Ml transitions and moments has turned out to be a rather intriguing and universal phenomenon. In order to demonstrate this we have plotted in the upper part of fig. 1 the relative deviation of some measured representative Ml transition strengths (12C [ref. *‘)I, 28Si [ref. i2)], 40Ca [ref. 13)], 48Ca [ref. 14], “Sr [ref. 15)], 90Zr [ref. ‘“)I and 208Pb [ref. “)I) f rom RPA and shell model calculations within a restricted model space. On the average this deviation increases from almost zero to about one hundred percent when one goes from the very light to the heavy nuclei.

W Kniipfer et al. / Consistent description

294

I

I

I

I

M 1 Strengths

0

M 1 Strengths effective

200-

0

50

100

150

operator

200

Mass Number A Fig. 1. Relative deviation of selected calculated and experimental Ml spin-flip transitions. Upper part: with the free magnetic dipole operator. Lower part: with the effective Ml operator of eq. (3).

In a similar comparison for the magnetic moments one observes (upper part of fig. 2) that the deviation between calculated and experimental values is much smaller than in the case of Ml transitions. For this plot the calculated moments were taken from different sources: (i) In the sd-shell results from full major shell model calculations i8) were used together with the experimental values quoted there. (ii) For the nuclei around 9oZr and “*Pb only representative selected states with strong single particle or single hole components have been chosen. These are the moments of 89Y($-), 89Sr(z+), “Zr($‘), and ‘lMo(q+) with the experimental values quoted in ref. 19)and magnetic moments of “‘Pb($“), 207Pb(s-), 207Tl(~+),*“‘Pb(~-), 209Pb(;+), ‘“Bi(p-) and 2”At(yf), listed again in refs. 1e,2o).Finally, the moment of ‘i’Sn(f+) has been taken into account 2’). In the theoretical description of the moments the g-factor including first order core polarization is writen as gcp = (@Ilj)lj

(I)

29.5

W. Kniipfer et al. / Consistent description

IMagnetic

Moments]

effective

L 0

I

50 Mass

operator

I

I

100

150

Number

I

I

200

A

Fig. 2. Relative deviation of selected calculated and experimental magnetic moments. Upper part: with an Ml operator of eq. (2). Lower part: with the effective Ml operator of eq. (3).

with

gcp=g~*(g,+~gsp-gt)/(2~+1)

(2)

forj = If $ states has been calculated. In eq. (2) the experimentally derived 22)orbital gl factors gy = 1.1 and g; = -0.03 have been taken, which reflect effects of meson exchange currents and higher order core polarization. The values are in fair agreement with the calculations of refs. 9*19). The free spin factors g, with gt: = 5.58 and g! = -3.82 are corrected by core polarization effects GgCp.Within the framework of lowest-order shell-model calculations these effects are (i) the first order core polarization (the so called Arima-Horie effect 23) and (ii) in the nomenclature of ref. 24) that part of second order corrections 25) which is responsible for the single particle vertex renormalization. However, in view of the retardation of the Ml spin-flip transitions the calculated corrections 6gzp taken from RPA calculations in 9oZr [ref. 19)] and in “‘Pb [ref. 2”)] have been reduced by an factor of l/J2. In this way we correct for the fact that such calculations overestimate the square of the Ml off-diagonal matrix element by a factor of two. This correction seems to be an essential point an is in the spirit of ref. 23). From an empirical analysis of the magnetic moments of the proton h9,2 shell around *‘*Pb it has been found that the effect of the core polarization 6gzp found by older

296

W Kniipfer et al. / Consistent description

calculations g-factor

must be reduced

g(h&

by a factor

of about

= 0.805 of ‘09Bi is in agreement

0.8. In fact our result

with Arima’s

estimate

for the

24) of 0.82.

From the observed deviations in figs. 1 and 2 it follows that the quenching of the Ml strengths is larger than that for the magnetic moments. We therefore conclude that the quenching mechanisms (i) and (ii) act differently in the two observables. This important result was pointed out recently 9,27) and shall be discussed below. Apart from the fact that the quenching mechanisms produce a renormalization of the spin and orbital g-factors they also lead to an additional term arising from the tensor interaction “). One does write Mlefi =

(3)

J

where again g, is the orbital factor discussed above and g, is the sum of the free-spin and the lowest-order core-polarization correction 6gzp. For the case g-factor g? of Ml spin-flip transitions the effect of Sgzp is taken into account in RPA by correlations in the wavefunction 16,17). For magnetic moments 6g:p is determined as described above. The additional terms Sg, and 6g, (the strength of the tensor term) stand for all the many-body corrections, i.e. tensor correlations and A-h effects. Calculating the diagonal matrix elements of this operator (magnetic moments) one finds for the correction

6Ml=

MleR- Ml

(j16Ml ti> - Sgs + sg,, whereas

for off-diagonal

matrix

elements

(4)

of 6Ml (Ml transitions)

one gets

(j,=I+#Mllj<=I-;)--6g,+Sg,.

(5)

Comparing eq. (4) with eq. (5) one finds that the contributions of 6g, and 6g, have opposite signs and thus lead to small corrections to the dipole moment, whereas for the off-diagonal matrix elements between spin-orbit partners (spin-flip Ml transitions) both terms interfere constructively and thus lead to a larger quenching. Consequently effects of A-h admixtures as well as effects of tensor correlations can therefore be better studied for Ml spin-flip transitions than for magnetic moments. However,

in order to disentangle

the tensor

contribution

from the renormalization

of the spin part one has to consider both observables. In the next section we therefore determine empirically the contributions and 6g, of eq. (3) by fitting simultaneously moments and Ml transitions experiental data.

of Sg, to the

3. Empirical determination of the effective Ml operator With the Ml operator given in eq. (3) we performed a least squares linear regression on 6g, and Sg, to minimize the sum of the squares of the deviations 6= (IMl1),,,-(IMl,,I). Here (IMlI) stands both for the magnetic moments and for the square root of the Ml transition strengths around 90Zr and 208Pb. The expectation

297

W Kniipfeer et al. / Consistent description TABLE 1 Magnetic

moments

near 90Zr and Ml transitions in “Zr used for the least-squares and Sg, (see text)

A

i

9’Mo s9Y 9’Zr ‘%r

=g9/2 nP;,‘z vd S/2

lzEXP 1.36 -0.28 -0.52 -0.24

“I%9/2

A

transition

90Zr

0++1+

x B(M1),xrC)

gCP“1

& = g,x, - P

1.471 -0.35 -0.67 -0.335

-0.111 0.07 0.15 0.095

z B(Ml)sPAc)

3.5 t&I

12 I&

determination

sg = g,,p-

of 6g,

P Y

0 0 -0.016 0.008

AB d,

AB,,, “)

8.5 &

2.8 t&

“) Calculated with eq. (2) and Sgsp (corrected by 4 is taken from ref. 19). Since these values do not contain the correction of the single-particle vertex renormalization an additional factor of 0.8 [see ref. “)I has been applied. b, gs’ calculated with eq. (3) and 6g,(n) =0.996, Sg,(p) = -1.17, Sg,(n) = -0.58 and Sg,(p) =0.48 obtained by the fit (see text). ‘) See ref. r6) ? AB =C B(Ml)sr.~-C ‘) Wi,=z: B(Ml)m~,ar-1 of b).

HMl),,, NMl),,,

taking the effective Ml operator

of eq. (3) and the parameters

TABLE 2 Magnetic

A 209gi

‘“At 207Tl ‘09Pb “‘Pb 207Pb “‘Pb A “‘Pb

moments

i

rhwz

near “‘Pb

and Ml transitions in 2“*Pb used for least-squares and Sg, (see text)

“&/2 VP;& uf;;z vi &

0.91 1.31 3.6 -0.3 1.18 0.32 -0.16

transition

1 B(Ml),x,

T’l3/2 ns;;*

0++1+

gCP“)

&XP

17 I&

0.80 1.39 4.96 -0.4 1.16 0.44 -0.27 ‘)

IL WMl),,,‘) 38 &

4s = g,,,

- P

0.11 -0.08 -1.36 0.1 0.02 -0.12 0.11

taking the effective

&Y=&XP-g

AB d,

A&, ‘7

21 I&

1 ILL

Ml operator

of Sg,

litb 1

0.02 0.01 -0.04 -0.02 0 0 0.029

“) Calculated with eq. (2) and sg:p (multiplied with 4) is taken from ref. 26) b, g”’ calculated with eq. (3) and Sg,(n)= 1.44, 6g,(p)=-1.32, 6g,(n) =-0.75 obtained by the fit (see text). ‘) See ref. I’) ‘? AB=X B(Ml),,,-C NMl),,, ‘) A&,=X B(Ml)arA.st-C B(Ml),,, of b).

determination

and

6g,(p)=O.56

of eq. (3) and the parameters

W. Kniipfer et al. / Consistent

298

values

([Ml,&

shell model

have been

calculations

calculated

exberimental

with the wave functions

as outlined

display the bare experimental

description

in the previous

section.

data, the core polarization

data used for the determination

from the standard In tables

corrections

of 6g, and

1 and 2 we

and the residual

6g, in the fit. The Ml

transition strengths and the magnetic dipole moments computed with the effective Ml operator are given in the lower parts of figs. 1 and 2, respectively. In figs. 3 and 4 the resulting values for fig, and Sg, (hatched areas) are plotted separately for protons r and neutrons v as a function of the mass number together with the result of ref. lo), which was obtained in the region of the sd-shell nuclei. In this analysis Brown and Wildenthal lo) selected roughly 200 Ml transitions whose strengths are experimentally determined with good precision 28) and fixed the parameters 6g, and 6g, of eq. (3) from a least squares fit. The isosclar parts of 6g, and Sg,, were obtained from a fit to magnetic moments only. The results are compared with a recent prediction of Khanna and Towner9), denoted as KT in the figure. From fig. 2 (6g,) and fig. 4 (Sg,) it is evident that both terms show a characteristic mass dependence. The absolute value of Sg, increases with increasing mass number, whereas Sg, shows no strong mass dependence. Of course, these curves represent an average trend. State dependent effects as e.g. treated in ref. 29) could lead to fluctuations around the displayed curves.

-ISpin OT 1,

10 >

_

/

/

/

a_____853

B

/-

Neutron

N EE&#+-------

A

0 41;

d

co

-----w--P

- iBKT ‘\ ‘.

-1.

Proton B-

e

t 0

-------=

1

1

50 Mass

100

150

Number

200 A

Fig. 3. Mass dependence of the empirical spin correction 6g, for neutrons (upper part) and protons (lower part). The shaded area for the sd-shell is taken from ref. lo). For the other shaded areas, see the main text. The symbol KT denotes a value obtained in ref. 9). The triangles represent a calculation of the A-h effect only *‘).

In figs. 3 and 4 the adjusted calculated values obtained that the A-h effect accounts

values

of Sg, and

6g, are also compared

to the

when including the A-h mechanism only 27). It follows for less than half of the quenching. This result, however,

W. Kniipfer et al. / Consistent description

0

200

100 Mass

299

Number

A

Fig. 4. Mass dependence of the empirical tensor correction Sg, for protons (upper part) and neutrons (lower part). For the meaning of the various symbols, see caption to fig. 3.

depends very sensitively on the short-range behaviour of the A-N interaction which is presently still a source of large uncertainty 30-32). The empirically determined 6g, value of the proton in the lead region (Sg, = -1.32) is smaller than the recent estimate of Arima 24) with Sg, = -0.94. This can be explained by the fact that in Arima’s estimate the effect of the tensor part 6g, which acts in the opposite direction was not included. On the other hand, our value of the total spin renormalization 6g’,“’= Sgzp+ Sg, for the proton h9,* particle, namely %Stota’= -2.27, is smaller than the value -1.62 obtained in ref. *‘). This is thought to be due to the influence of the A-h effect. Given that the tensor correlations are fully taken into account in ref. 29) we obtain that the spin renormalization due to the A-h effect amounts to about -0.6. This is only slightly smaller than the result of Lawson 27). Finally, we like to remark that with the effective operator of eq. (3) and the lowest order shell model wave function of ref. 33) we obtain a strength for the excitation for the J” = l+ state in 48Ca at E, = 10.2 MeV of B(Ml)T = 4.5 pk and a total strength between 9 and 12.5 MeV of C B(Ml)T = 5.3 )(L;. These values are in agreement with the experimental values 14) of B(Ml)T = 3.9kO.3 pk and C B(Ml)T = 5.5kO.4 pi, respectively. References 1) C. Gaarde, Nucl. Phys. A396 (1983) 127 c 2) T. Bjornstad, M.J.G. Borge, P. Dessagne, R.-D. von Dincklage, G.T. Ewan, P.G. Hansen, A. Huck, B. Jonson, G. Klotz, A. Knipper, P.O. Larsson, G. Nyman, H.L. Ravn, C. Richard-Serre, K. Riisager, D. Schardt and G. Walter, Nucl. Phys. A443 (1985) 283

W. Kniipfer

300 3) A. Richter,

4) 5)

6) 7) 8) 9) 10) 11) 12)

13) 14) 15) 16) 17)

18) 19) 20) 21) 22) 23. 24) 25) 26) 27) 28) 29) 30) 31) 32) 33)

et al. / Consistent

description

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