Polarisation effects in δ〈r2〉 measurements

Volume 31B, n u m b e r 8

PHYSICS

POLARISATION

EFFECTS

13 Ai)ril 197,)

LETTERS

IN

5(r2)

MEASUREMENTS

J. S P E T H

Physik-Department of the Technische Hochschule M~nchen. Theoretisches TeilinstitM Miinchen , Germany Received 1 F e b r u a r y 1970 The changes of the m e a n square charge radius and the quadrupole m o m e n t of the ground state of 152Sm and 160Dy due to a muon and to a change of the s - e l e c t r o n density respectively, are calculated. It turns out that in M b s s b a u e r m e a s u r e m e n t s the e s t i m a t e d ratio of the energy shift due to this polarization to the usual i s o m e t r i c shift is 2.6 t i m e s as large as in muon e x p e r i m e n t s .

B y m e a s u r i n g t h e i s o m e t r i c s h i f t E i s of r e c o i l l e s s y - r a y s in d i f f e r e n t c h e m i c a l c o m p o u n d s , o r b y m e a s u r i n g t h e s h i f t in n u c l e a r g a m m a - r a y e n e r g i e s in t h e p r e s e n c e of a b o u n d m u o n , t h e d i f f e r e n c e of t h e n u c l e a r c h a r g e r a d i i in t h e g r o u n d s t a t e a n d in a n e x c i t e d s t a t e c a n b e d e t e r m i n e d . P r o v i d e d t h e d i f f e r e n c e of e l e c t r o n d e n s i t i e s a t t h e n u c l e u s A l,l'e(0)[2 i s k n o w n 5 ( r 2) i s d e r i v e d i n t h e c a s e of t h e M ~ s s bauer effect from the well-known formula for the isomeric shift,

Eis : ~ ~Ze 2 A }~I,e(0)l2 5 (r2)

(I)

From an analogous formula in the case of the muonic measurements, one derives the difference of the charge density in the ground state and an excitated state, from which 6(r 2 ) can be calculated. In both cases the total measured energy shift is interpreted as a change of the coulomb energy, which is different in the ground and the excited states of the nucleus. The influence of the change of the electromagnetic field on nuclear structure is neglected. In the case of deformed nuclei, however, one expects that a change of the electromagnetic field will also change the quadrupole moment and the moment of inertia. In the Bohr-Mottelson theory of rotational states, such a change of the moment of inertia will also change the excitation energy of the first excited 2+ state. For muons, Schiffer and De Voe [I] estimated this effect in a crude model to be - 50 • i00 eV. We have calculated, in the framework of the theory of finite Fermi systems [2] the change of the mean square charge radius and of the quadrupole moment of 152Sm and 160Dy both for muons and for a change of the s-electron density in the case of the M~ssbauer technique. The starting point of our calculation is the formula for the change of density matrix SpyI v2 [0] due to a static external field. In linear - response approximation we get E/J1

5Pvlv2(O) =

2EvlEv2(Evl

+Ev2)

~1 v2

V3v4 VlV3V2V4

E~,1 Ev2+ ~v2Evl ~ F~_ ~u3Ev4+ Eu3£v4 2EvlEv 2 (Evl+ Ev 2) v3v4 VlV2V3v4 EV3Av4+ Av3Ev4

~Pv4 v3

T h e i n d i c e s v d e n o t e t h e q u a n t u m n u m b e r s of s i n g l e p a r t i c l e s t a t e , e v a r e t h e s i n g l e p a r t i c l e e n e r 2 1/2 g i e s , Av t h e e n e r g y g a p s a n d E v = (E2v+ Av ) t h e BCS q u a s i - p a r t i c l e e n e r g i e s . In c o n t r a s t to r e f . 3 w e r e q u i r e eq. (2) to c o n s e r v e s e p a r a t e l y t h e p r o t o n a n d n e u t r o n n u m b e r s . T h i s r e q u i r e m e n t , w h i c h f o l l o w s d i r e c t l y f r o m t h e B S C - l i k e e q u a t i o n of L a r k i n a n d M i g d a l [4] f o r t h e q u a n t i t i e s Av, r e d u c e s t h e c a l c u l a t e d e f f e c t s v e r y s t r o n g l y , U n d e r t h i s c o n d i t i o n t h e q u a n t i t i e s VVlV2, / ~ v 3 v2v4 a n d ~P~v2 ~3 v4 have the following forms :

513

Volume 3lB. numl)er 8

PHYSICS

LETTERS

13 A p r i l 1970

A2 protons

o _ Sp1 ~VD VO Vvlv2 vv vEv3 5- ~ l V 2

#o VlV2

.>co IneutrOns v o

1 ~

-

vo

vv _ 3 ~ v Ev

VlV2 Sn Vn

Iprotons

F~

/',2 a

v-

~12~3~

1 ~

'

v

12

@ v ~ E,a,

VlV2V3v4 =

~3~4

5

F~_

neutrons

Fw

~r;b ~El. F'v v

co

3 vv~t 5v 1 112

VlV3V2~l

1 ~

A2

'

- =--~.J ~Fco vv 4 5nVn Ev Vv3 5l~1V2

~2

~

,

1 ~ cv, AVF_~ _ 5 ~IV2 v'3v4 S n V n E~ vq]v3v'4 ~i~2

[neutrons

A2 v

_ 1

Fvlv3v2v4 ~

:

VlV3V2v4

¢ A v VF-~-

co

protons

_

S~ = ~

A 2 / E 3,

Vp

"/' v

S

=~

n

~2v / E

Vn

/

3

v

F ~ v3 v2 v4 and F~lv2 v3 u4 are the usual particle-hole and particle-particle residual interactions. All parameters and numerical details are the same as in ref. 5. The quantities ~V O I]2 are matrix elements of the electromagnetic fields V~ (r) and Ve(r) produced by the muon lq~ (r)i 2 and by the s-electron density difference A]k~e (~-)I2, respectively. The most important fact is theft these two charge densities are not only different in magnitude but also in the radial dependence. The change of the electron density is, in a good approximation, constant over the whole nucleus, in contrast to the muon density. Therefore the solutions 5p~ v2 and 5pe v2 of eq. (2) do not differ merely by a constant factor. 1 1 Table I shows the change of the mean square radius and the quadrupole moment of the ground state due to the electromagnetic field of the muon and the change of the electrons. To estimate the change in the excitation energy of the first 2+ state, we need a relation between the moments of inertia and the quadrupole moments. The relation @ ~ Q2 is in good agreement with experimental results [6]. The change of energy due to this polarisation and the ratio of this energy to the usual experimental isometric shift is shown in table 2. It is very difficult to compare these calculations with the experimental results. In order to extract 5~r2} from a Mbssbauer experiment, we need the s-electron density difference which is known only within 30% and 50% [7]. On the other hand, the change of the density which we need for a calculation of 5 r 2~ from a muon experiment is not unique. However mean square radii derived from Mbssbauer experiments seem to be systematically larger than those derived from muon experiments, as known from the 128W. 184W and 186W isotopes [8,9]. Table 3 shows a comparison of results for 152Sm and 154Gd [7]. The theoretical prediction that the energy shift due to the polarisation is positive and that in MSssbauer measurements the ratio of this shift to the usual isomeric shift is 2.6 times as large as in muon experiments provides a very simple explanation of the experimental Table 1 Change of the mean s q u a r e radius and the quadrupole m o m e n t of the ground state, m e a s u r e d in fm 2. The i n d i c e s /.~ and v' denote muon and e l e c t r o n , r e s p e c t i v e l y .

ZS/?-2\P

Zb(r 2}e

5(Q}~

5(Q)e

5
152Sm

-7.4 ~ 10 -2

-1.58 x I0 -10

-2.62 x 10 -1

-6.09 x l 0 -10

2.32 x 10 -9

160Dy

- S . l x 10 2

-1.82 × 10-10

-3.26 x 10 -1

-7.69 × 10 -10

2.37 × 1 0 -9

Table 2 Change of the energy and the ratio of this energy to the ususal isometric shift. [Energy in eV].

AEV152Sm+110

AE" ~2.49×10 - 7

e,E~/E~×100AE~s×100 14.4

Table 3 Comparison of results measured by the Mbssbauer effect and derived from ~i- mesie atoms for A t =0 taken from ref. 7.

e/
514

au/<~2>× 104

37.4

6.0 8.5

2.5 3.5

4.8 5.9

0.4 0.8