REFERENCEIC/68/42

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

MICROSCOPIC THEORY OF EFFECTIVE

OPERATORS OF ELECTROMAGNETIC

INTERACTIONS IN NUCLEI

M. GMITRO

A. RIMINI

J. SAWICKI

AND

T. WEBER

1968

MIRAMARE - TRIESTE

IC/68/42

OTTSHHATIOKAL ATdllC EEERGT AGMCT

CBHTES FOR THEORETICAL PETSICS

MICROSCOPIC THSORT OP EFFECTIVE OPERATORS

OP ELECTROMAGNETIC INTERACTIONS IS HUCLEI * t

11. Gmitro **

A. Rimini ***

J. Sawioki ****

and

T. Weber *****

TRIESTS

June 1968

* To be submitted for publication.

** International Centre for Theoretical Physics, Trieste^On leave of absence from Nuclear Research Institute, Rea (Prague),Czechoslovakia.

*** Istituti di Fisica Teorica e di T'eccanica dell'Universita diTrieste, Italy.

**** International Centre for Theoretical Physics, Trieste, and

Istituto di Fisica Teorica dell'TJniversita di Trieste, Italy.

***** Iatituto di Fisica Teorica dell'Universita di Trieste, Italy,

t Supported in part by Istituto Nazionale di Fiaica Nucleare.

ABSTRACT

Effective operators for valence (open-shell) nucleons

only are constructed, for electromagnetic interactions in nuclei.

Such operators are calculated in terms of excitations and de-

excitations of particle-hole pairs of the oore nucleons. The

nuclear force involved is a realistic; nucleon-nucleon potential

(Yale-Shakin). The theory is applied to the Sn region described in

terms of five valence neutron subshells and of all the core and

open subshells of both the neutrons and protons between the magic

numbers 8 and 126. The effective quadrupole charge matrix has all

the elements of about the same order of magnitude,thus giving a

first partial support to the concept of a oonstant neutron effective

charge. Calculations of observables are performed with the states

of the even tin isotopes described in terms of the two- and four-

quasiparticle Tamm-Dancoff theories involving explicitly only

the valence neutrons. Results are presented for the B(E2, 2, - * 0 . ) t

the quadrupole moment Q ( 2 ) , the gyromagnetic factor g - and

the inelastic electron-scattering form factors for the 2+ and 3r116 — XI

states of Sn . Except for the last (3-* ) form factor, a good

seraiquantitative agreement with all the corresponding experimental

data is obtained. The reported calculations involve no ad hoc

adjustable parameter.

- 1 -

MICBOSCOPIC TH30RY OP EFFECTIVE OPERATORS

OP ELECTROMAGNETIC INTERACTIONS

IN NUCLEI

1. INTRODUCTION

The enormous complexity and the completely prohibitive dimensions

of the many-body problem of a finite nucleus force us, in the shell-

model description of the nuclear spectra and of the nuclear ground state,

to an elimination from our explicit treatment of the so-called oore.

The oore which constitutes the main bulk of the nuoleus is the supposedly

inert part of the nucleus: usually it is the ensemble of all the

protons and neutrons of all the closed subshells (possibly only of

all the closed major shells) in the ground state. These lie deep

inside the Fermi sea and are less important for the properties of

all the low-lying states than are the open (valence) shell nucleons.

This picture is, however, a fair approximation only when one works with

effective phenomenological nuclear forces which are supposed to

contain implicitly all the effects of the excited configurations of the

core nucleons. In fact, it has been shown that the effects

of such configurations, i.e., the so-called effects of the core polarization,

are extremely important in a description in terms of realistic nucleon—

nucleon potentials. These have to be drastically renormalized if they

are to "be used for mixing configurations of only the valence nucleons.

Similarly, in a phenomenological description of electromagnetic

properties and interactions of nuclei^the nucleonic charges of the

valence protons and neutrons are supposed to be renormalized for all the

contributions of the core nucleons. This is the concept of the effective

charge which is,in a phenomenological theory, an adjustable parameter. This

leads to an uncomfortable freedom as the effective charge is different

for the neutrons and the protons and for various multipoles.

In fact, in addition to the philosophical difficulty of mixing purely

phenomenological and microscopic concepts, one also usually has too

many adjustable parameters in the theory. Since,

in contrast to the situation of our knowledge of the actual nuclear

forces, the electromagnetic interactions with nucleons are well-known,

we have even less excuse for a phenomenological approach to these

interactions than for such an approach to the nucleon-nucleon

interactions between the valence protons and neutrons. It is clear that

- 2 -

a fully microscopic theory of nuolear properties and of nuclear spectra

in terms of realistic nucleon-nucleon potentials should be free of

the concept of a phenomenological adjustable parameter.

7)-9)Several authors ' have attempted microscopic derivations

and estimates of the effective charge using the picture of virtual

excitations of core nucleons. Por example, a neutron-effective charge

could arise from eecond-order processes in which a virtual or a real

photon is absorbed by a core proton creating a particle-hole pair which

is subsequently annihilated (de-excited) in a collision with a valence

neutron. Unfortunately, the description of Refs. 7) - 9) has been based

only on phenomenological nuclear forces and involved many crude,

schematic approximations thus giving only qualitative or, at best,

semiquantitative estimates.

It is clear that only realistic nucleon-nucleon potentials, avoid-

ing the introduction of new adjustable parameters, are to be used in this

kind of calculation when a quantitative comparison with experimental data

ia wanted.

It is our aim now to study the problem in detail and in a

quantitative way in relation to a realistic nucleon-nucleon potential.

Our numerical analysis is performed on the example of the even tin

isotopes, which are quite representative of an important region of the

periodic table: of the so-called vibrational nuclei. We derive formulae

for the effective electric (or magnetic) multipole operator, O\ ,

which may then be treated with all the retardation effects (no approximation

for the radial integrals involved) or in the long wave-length limit

approximation.

In particular, we examine the question to what extent

J -operator be replaced "by e ,, O\^ when

g ,r ie a unique over-all constant effective charge (independent of

the transition configuration).

We apply our computed CK £ to the study of some of the

reduced E2-transition probabilities, B(E2, I. ->I,,), of the quadrupole

moment of the first excited 2, - state, Q(2-,), of the gyromagnetic factorL •L/form factors corresponding to thej

gc- of the 5, - state and of the inelastic ere~c" rbn scatterirfgTfiJaalstates 2-, and 37* We compare our results with the recent experimental

""" 116data on these observables, in particular for the nucleus Sn

-3-

II. EFFECTIVE SINGLE-PARTICLE OPERATORS

In the case of a realistic nucleon—nucleon potential containing

a strongly repulsive part at small separation distances (hard core or

at least a "soft" core for a local, static potential) the effective nuclear

interaction Hamiltonian is defined in the sense of the Brueckner theory,

i.e., the two-body potential V(i,p is replaced by the appropriate

Brueckner reaction matrix G(v,-j) in the original expression for the

potential energy operator. Any perturbation theory calculation is

based on such an effective Ramiltonian (an expansion in terms of

G(i>j) (of., e.g., Refs. l) -3)). Only if one uses a strongly non-local

or velocity-dependent potential such as, e.g., that of Tabakin ,

can one use the standard perturbation theory in terms of Vflj]) itself.•j

For the sake of definiteness let us now consider the problem

of a nucleus with a doubly magic core possessing neutrons only in the

open (valence) shells, such as a tin isotope with the 50-50 core. Vfe

have now to calculate the effective operator of a multipole Gy ,> for

an extra core single neutron. Clearly it is the neutron-proton

two-body potential V ™ (or fCp) which is responsible for the transmission

of the electromagnetic interaction from the core protons to the neutrons.

To lowest order,this effective interaction can be represented loy the

two diagrams of Pig. 1. These correspond to the second-order perturbation

theory.

In fact, since there is no first-order contribution (exceptA N

for O^nbeing a magnetic interaction with the neutron spin;, we obtain

in this case the following expression for the single-neutron matrix

element:

s -1

(1)

where the first term in the sum on the r.h.s, of eq..(l) corresponds to

the diagram(a) of Fig.l and the second to the diagram(b). Here q = +1

if 3T is a (proton) particle and y^ a (proton) hole, and ~ 0 otherwise,

and the energy denominators are C. ± - E v ~ E"^ ± ^Evi - Ejj).

-4-

In the following we shall limit ourselves to the spherical symmetry

with the degeneracy in the magnetic subata,tes. Consequently, we utilize

the Wigner-Eckart theorem and only reduoed matrix elements of the

operators are involved. We choose the notation with latin subscripts

for the single-particle states with the exclusion of the j-projeotion

quantum number m (a as corresponding to (X = (a,m^)t etc.). It is

convenient to introduce the particle-hole coupled reduced matrix elements

FMp(abod, X ) of the neutron-proton potential V as defined by:

= -^ L

The Wigner-Eckart theorem for the element (<S' | Oy^ I <3 ) can be written as:

It follows then immediately from eq. (l) that for an electric 2'-pole •

FF W PL

(4)

In Ref. 5) are employed reduced F-elements in the isotopic spin formalism.

Our Fw.p(n1'nppl, J ) occurring in eq. (4) are connected with these F-elements

through the formula

where T is the isospin of a pair (ab and cd) and the corresponding

matrix element < <X 0 |V(l"P/( ^ Y /* i s antisymmetrized in the two

interacting nucleons.

One can also consider higher-order terms as corrections

to O\ J of eq. (l). For example, one can easily include

iterations of the bubble diagram of Fig. 1, which is the simplest

correction (cf. Fig. 2).

One can easily calculate the correspondingly corrected^1 ]j 0^ + A 0 A I11 /

even including all the appropriate • exchange diagrams (not indicated

in Fig. 2). In fact, one then obtains a formula of eq. (4) merely with

F(abcd, "X ) replaced by appropriately renormalized <r-elements of Ref. 5)•

-5-

However, the corresponding corrections are, in general, quite email• In

the following we snail give a discussion of these corrections in connection

•with our numerical examples.

Let us also consider the case when we have valence protons in

addition to valence neutrons in a given nucleus. The following formulae

will also "be available to the magnetio multipoles of interaction via

the core neutron spin. We calculate the effective 0\-operator for a

valence proton. This contains the direct interaction as well as that

of the diagrams similar to those of Fig. 1 with V«p replaced by Vpp(l-P)

where P is the exchange operator of the two protons involved. For the

reduced matrix element we find:

fj FPPQ.

tA.

itwhere q^ and q.v are appropriate projectors of the

proton and neutron (ph)-pairs, respectively.

The last sura in eq. (6) which runs over all the neutron particle-hole

pairs is actually present only if O\ is a magnetic multipole operator

of interaction with the neutron spin. The quantity Fpp(abcd,^.) is

expressed in terms of F with definite isotopic spin T, i.e., we have:

The reduced matrix elements Fpp correspond to antisymraetrized elements

of the potential V p p, i.e., to elements of U p p = Vpp(l-P12) where ? 1 2

exchanges all the co-ordinates 1 and 2. The exchange terms of Fpp in

eq. (6) are connected with the elementary prooesses represented by the

diagrams of Fig. 3« "^~

(7)

Obviously, the formula of eq. (6) obtains for neutrons in the place of that ofA

o<l.(4) In a oaso where Ox is * non-vanieking photon-neutron interaotion

operator (e.g., for Ml-transitions) ; here Ppp is replaced "by F ^

again of the form of eq. (7)•

Our eqs. (4) and (6) are essentially similar to the formulae

given in Ref. 7) i*1 terms of Slater integrals. Refs. 8) and 9) omit

terms corresponding to our diagrams ("b) and (b1) of Figs. 1 and 31respectively

In the present paper we do not attempt to analyse any more

complicated diagrams (microscopio processes) whioh may give additional

contributions to OyJJ . However, the diagrams of Figs. 1 and 3 are

the most important.

III. ELSCTRGEAGNETIC EFFECTIVE TRANSITION KATRIX ELEMENTS

Since the nuclear many-body states constructed explicitly with

the valence nucleons only, with which we are to compute the matrix

elements of the operators ^i > . are usually given in the second

quantized form, it is useful to express these operators in terms of the

creation ( C y ) and annihilation (c^) operators. Using the Wigner-Eckart

theorem one can write:

A.where </ nv|| 0 ^ || ft is a reduoed matrix element and

H Z. C-) k + f m" tfcU jrvaj-wi,, U^)ck, (9)

creates an (n^n ) pair of spin A . In practice,in medium heavy nuolei

one deals with many quasiparticle states and it is useful to perform a

quasipartiole transformation of eq. (9)* An explicit expression for

this form of X is given in eq. (ll) of Ref. 12).

E The matrix element for a ^-transition from the nuclear state

I X jj^ / to the state j x ' , / o a n n o w ^e' expressed as"

(10)

where \ £ J 11 A i\£ J / is a reduced matrix element of X between the two

many-body s ta tes in question. If one uses for /a j | (X | a ^ i n ©Q.*

-7-> = •

the expressions of eqs. (4) or (6), one introduoes an approximation which

is not identioal with the standard many-body perturbation theory-

appropriate to obtain the nuolear many-body states with which to calculate

the matrix elements of O\ . In our numerical analysis we shall

examine in detail the goodness of this approximation. Strictly speaking,

the formulae of eqs. (4) and (6) are most appropriate and directly

applicable to the independent particle model, e.g., to describing creations

of pure single particle-hole pairs in a shell model without residual

interactions.

It is interesting to oompare the above prescription of using

eqs. (4) and (6) with the corresponding formula following from the

standard perturbation theory. Let us consider the Hamiltonian H ™ of

the interaction of the valence neutrons with the core protons, a

perturbation field for many-body system described by [j \ |0)> \ where

1 < X M / refers to the valence neutrons only and 0 A is the spin

zero ground state of the proton core. By perturbation theory we calculate

the perturbed set involving the proton particle-hole (ph)-pair excited

configurations \ ! \ l//— \ (X) piljA >. ) T M 7 (a vector product).

On the perturbed set we now calculate the matrix elements of O\,i 1

the operator which acts only on the proton co-ordinates. Again one

obtains the formula of eq. (io)j only the s.p. reduced matrix element

has to be replaced with (cf. Appendix i)

(11)

The only difference in relation to eq. (4) is the dependence on the

energies E and E( of the two states involved of the valence neutron

system in place of the dependence on the unperturbed s.p. neutron

energies E and E , respectively. In operator form^ CX «' of eq. (ll)

can be written as

which differs from eq.. (8) by the dependence on E and on the effective

total Hamiltonian of th e valence neutrons, H*,. Similarly to 0^^

of eq. (8) j Q\VL (-£•*) o f e<l* ^12^ i s a nonlinear neutron

operator; the operator of eq. (12) is a many-body operator. The formulae

of eqs. (ll)and(l2) are quite general (i.e. they are valid for any ntany-

toody theory of the valence nuclear system).

-S-

Quite similar formulae are obtained for the case where core

nucleons are involved of the same nucleonic charge aa that of the valence

nucleons. For example, if pr and p1 ' are two single valence proton

states, we obtain ^p (j O \ / (E , H Vcjj) {jb \ exactly of

the form of eq. (6) where E ,, - E ,.. in e , e is replaced ~by H ,"E;p p *r •— vax

H . is the effective total Hamiltonian of the valence nucleon system and

E is its initial state eigenvalue. In our application we use a similar

formula for the valence neutrons in calculating magnetic moments of

excited states of the even tin isotopes.

Explicit formulae for the reduced matrix elements E'l'\\)\(^'] I^JU^ /

where I ( V J M / J are zero-, two (QTD)- or four (QSTD)-quasiparticle

eigenvectors are given in Refs. .12) - 14).

The Feynman diagrams appropriate to the processes of eq. (ll)

are those of Pig. 1 but the' lines \>, V' should f be

replaced by the "phonon" lines of j YJ-JV, )> and j V'j'Yi'/ » respectively.

The dependence on E and ET of < [? 0/^' (E, E') 1] )> of eq. (ll)

is an extra complication in relation to the simple / f: O\"* \: / of

eqs. (4), (6) and (8). It is dear that the latter version of our

theory has, in practioal calculations, the great advantage of universality,

i.e., of the independence of the many-body eigenstates involved in the

applications. Still another complication arises from the following

difference between the symmetry properties of ^hrJ a nd ^X/x Iy y pp XM- AM

of eq. (ll). With the . usual phase conventions ' which

we use in the present work it is immediately verified that, while

<aH6^1(a) - (-)^~^' <o.H6>ll<X<) } the same symmetry

relation holds for the effective operator of eqs. (4), (6) and (8):

Instead, for the operator of eq. (ll) we find:

The fact of having to interchange E and E1 together with n arid n1 in the

latter case renders the calculation of <f E j'[; O_^ ^ (Ej£)|jEJy> more

complicated than that of <( t" j' |j 6 "' | ^/> for a given E7 -* E' J'

transition. Only for diagonal (E = E1) matrix elements no such extra

trouble arises (calculations of the quadrupole moments, magnetic moments,

etc.; in this case calculations with 0^ (E (£ E £) are equivalent to

those with (X \ " of eqs. (4) and (6) neglecting E^, - E^ in e + ) .

- 9 -

It follows from our numerical results presented "below that the

two different definitions of the effective electromagnetic operators

lead to only very small differences in the respective values of the

calculated observables. This then seems to justify extensive use

of the simpler version ^\JJ whioh is independent of E and E 1.

As for a critical evaluation of the usefulness of the

extremely crude effective charge approach we calculate the "effective

charge matrix" (ECM) defined as

where <CY\' j O\ 11/ p is the "reference matrix" defined in the

usual way for "direct" n-^n* transitions and e ' = 1. ECM gives

the actual -theoretical effeotive charge for each individual n-> n1

transition.

IV. APPLICATION TO ELECTROMAGNETIC PROPERTIES AND TO INELASTICSCATTERING OP ELECTRONS FROM THE EVEN ISOTOPES OP TI3T

The spectra of the even isotopes of tin have been generally

successfully described by two— and four-quasiparticle RPA and Tamm—

Dancoff theories 5)» 6), l3)-lB)^ jn particular, realistic

micleon-nucleon potentials of Tabakin '. and of Yale-Shakin ''» °'

including the core polarization renormalization have been applied

in the quasiparticle Tamm-Dancoff (QTD) and second Tarara—Dancoff

(QSTD) theories *'t °' . It is our aim here to calculate the

effective electromagnetic operators appropriate to our equations

of Sec. 3 a-nd corresponding to one such realistic effective nuclear

force, and apply them subsequently to calculatiM some electromagnetic

properties of the even tin isotopes. The five valence neutron

subshells are: 2 a 5 M» 1^/ 2» ^

a\ fo» 2d3/2 a n d lhll/2* F o r t h e

particle-hole excitations of the core nucleons we consider the subshells

up to all those between the magic numbers 8 and 126.

- 10 -

Unfortunately, the laok of direct experimental information

on the shell-model s.p. energies on the one hand, and of any self-

consistent Hartree-Fock (HF) or Hartree-Fock-Bogolubov (HPB)

calculations of such energies on the other forces us to use other

methods for determining them which are much less satisfactory. On

the other hand, many observables and even some energy levels not of

a collective character are well known to be sensitive to the input

values of the s.p. energies. A procedure which seems to be the most

appropriate in the ciroumstances is that of deriving the s.p.

energies from the observed energy levels of the odd isotopes by the

inverse gap equation (iGE) method * • This procedure is

now being applied by the present authors in collaboration with21)

R. Alzetta and T.K. Gambhir . Since the validity of the main

conclusions of the present paper does not hinge on a detailed

quantitative fit to any particular experimental data,we are in

our present numerical applications limiting ourselves to a less

well justified choice of the set of the s.p. energies involved.

To do so,we have employed the values of the s.p. binding energies

obtained by the Bonn group " " with a reasonable Woods-Saxonpotential. The energies (in MeV) for the five valencesubshells are: -10.52(2d|), -9-36(lg|), -8.45(3s|), - 7 . |

—7.16(1115—); for the eight important proton core (hole) subshells

we have: -3O.O9(ld|) , -27.93(ldf), -27.07(2s|), -22.9l(lf§),

-19.07(lff), -I8.82(2p|), -17.28(2p|), -15-24(lgf); in additionto the five valence subshells we consider six higher proton particle

subshells: -2.26(21^), - I . 1 4 ( 3 P | ) , -o.23(3?i), +1.01(21^), +1.04(1*5?-),+1.07(lh~). These energies are most appropriate to the isotope116

Sn . Any other particle or hole subshells give only negligible

contributions. The Woods-Saxon radial wave functions are reasonably

approximated with those of the harmonic osoillator (h.o.) with

b~ = yv" = O.46 F~ . It appears that the (e, e'p) data on some

light nuclei and other similar information suggest generally considerably

larger binding energies for the deep—lying subshells than those

corresponding to a "reasonable" Woods—Saxon potential. Correcting

for some many-body effeots, this would imply more spread-out s.p.

HF energies than in our present calculation (a smaller s.p. level

- 11 -

density)* This oould lead to a reduction of the contributions

to < 11 Ov ^ 1/ of the deep-lying core nucleons. On the other hand,

the most important contributions (suoh as that of the lg£ subshell)

may be even greater with an actual HP basis.

Let us consider first the most important E2 transitions.

With all the above-mentioned s.p. states there are in all 29 non-

vanishing (E2 allowed transitions) proton matrix elements. The

nine non-vanishing distinct (n . n1) elements e-(n.n') of eq. (15)

for the valence neutrons are given in Table I. O, ..'' is

that of eq* (4). The nuolear force

Yale-Shakin reaction matrix.

F ™ elements) is the "bare"

Table I

3sJL

2 d |

3 B 2

0

(0

0

.6143

.6154)

.6459

0.6757

(0.6096)

0.6989

(0.6398)

O.6521

7 11

- -

1.1636

(1.0516)

1.1132

(1.1054)

1.0844

0.6535

The corresponding values of &?^ ~* (n,n') with <^T\ |[ O (£ £lr

of eq. (ll) are given in parenthesis for comparison (clearly the

diagonal elements of the two variants of our theory are identical

for E' = E). We observe only very small differences between the two

respective table entries for each off-diagnonal n* -^ n element in Table I.

- 12 -

A feff)The differences between the elements <^||^ II11/^ °f ®q.« (9)

and the elements < 1 l| Oj*V(E,E* = E-V1. 2<j tjeVll^are preciselyof the same order. The values of G (n,n?) are of the desired

sign and the same order of magnitude. They are actually grouped

in two clusters: those somewhat higher than unity and those somewhat smaller

than 0.7. The composition of each one of the nine elements of Table I

in terms of the partial contributions of the subshells of the (ph)-

pairs involved is given in Table II. The entire If2p major shell,9

plus Xgr of the core with all their transitions to the five lowest-

lying partiole (p) subshells, contribute on the average slightly

more than about 50$ of all the e (n,nf). Transitions from the same

to the six s.p. levels of the upper major shell (2f3p» li^ • lbr)

contribute the surprisingly large amount of 30-40$ of all the

e_(n,n')« The Id2s major shell of the core is of little importance.

The numbers of Table II refer to O, " as defined in eq. (4).

A similar distribution of the (ph) contribution obtains for Oz " (E^Erw-

The two terms on the r.h.s. of eq. (4) (or of eq. (ll)) e a c h

gives contributions of the same order of magnitude (equal for n = n*

and for E1 = E). Some authors '* suggested schematic

models in whioh they were considering only one of the two terms.

This is clearly invalid for a quantitative analysis in view of the

symmetry properties of these terms leading to eq. (l3) or (l4).

In order to examine the relative importance of our individual

e (n,n') we compute with the numbers of Table I the observables

B(E2, 2n -» 0- ) and quadrupole moment of the 21 state, Q(21). The

corresponding eigenvectors JO-^ and j 2 w of Sn are those of

Ref. .23) computed in the QTD and QSTD approximation with the Yale-

Shakin foroe renormalized for the core polarization. The core

polarisation renormalization of all the proton and neutron subshells

mentioned is taken to second order which is a good approximation '' '*t

the valence neutron subshells are assumed to be, on the average,

exactly half occupied; no other approximation of the propagators

of the core-polarization terms is made. The s.p. energies and

wave functions are exactly those of our e?(n,n') calculation. In

QTD \ 0j_/ is qp-vacuum itself and 2^ \ a nine-component veotor.

The corresponding QSTD vectors of Ref. 23) have 56 and 94 components/

respectively. These are free of all the basic spurions due to the

nuoleon-number non-conservation (such kets are projected out).

The QSTD 0, eigenvalue lies by -0.363 MeV lower than the qp-vacuum,

- 13 -

J

j.

i

oroVOONCO

o

roON

©

OHP»

o

o

&VJ1VD

O

CJo-p*ro

omt j jON

oCoCOro

o*-P»o

—Iro

o

«ro

- jVO- J

o

b|—i

VO

O

b-p-M

o*o-pi

- J

o

bON—3

O

b—jro

o

oo

o

oCO

o*oCO-piVJ1

o

o-p.

bd

o*oON

o*

"^

o•oCAON

o•Mroro

O

bO N

OaJ I

Oro

o

bCTN

o

oi

ro-P=>

o*oCo

- 4

Ot

MCO-P»o

o*MON

VJ1

o•roro

o

C?N

o

—JONo

o

VJ1^o

o#COM

u»

o*-p i- JON

o•MCO

O J—3

ro

p .rof-o

ro

Wro***

roj^

roll—1

p j

.NHJI

ro

r o M

ro

roTji

pi.r o M

roro [ o

ro

rokjiM

roj-j

C3

^ ^

piroll-1

rolM

(0

H

and the QSTD 2- - eigenvalue is 1.153 MeV; the QTD 2^ energy l i e s at

1.259 MeV; the observed z\ energy i s 1.291 MeV. In QSTD we distinguish

for j O j . / between the case ( i ) , in which we define the four-quasiparticle

spurious kets j V c t ^ ^u e *° *^e nucleon number non-conservation so

t h a t they have no vaouum component

in which<.0| \±O

O \ y .y , and the oase (ll)

n the latter case we project out

here If

operator, and JL its correct eigenvalue.)

exactly the fluctuation of N -N. where I is the nucleon number

o

Table III

^v theory

observable >v

B(E2, 2*-»O^)

(in e¥)

Q<2+)(in barn)

e(n) _ ,2

QTD

317.0

+0.036

QSTD(II)

273.7

+0.1251

e (n,n*) of Table I

of eq. (4)

QTD

232.3

+0.021

QSTD(I)

259.4

QSTD(II)

202.2

+0.094

of eq. (ll)

QTD

229.5

+0.022

QSTD

-

+0.091

In Table III we compare the calculated (QTD and QSTD) values

of B(E2, 2^ -3> 01) and Q(2Z) both theoretical (computed with the e (n,n')

of Table i) and computed with the neutron effective charge, e? = 1 . The

reported observed value of B(E2, 2^ -* 0^) varies between about 200

and about 500 e F . In comparing this with our theoretioal values of

Table III we should keep in mind that our predicted results were obtained

with no adjustable parameter involved. It is clear that by varying

the s.p. parameters a better agreement with experiment could be obtained.. T l ^ 2/i

The observed value of Q ^ ) of Sn is " +0.4+0.3 barn. Our

QSTD values lie around the lower limit of the experimental error.

Q(2i) is generally a "delicate" quantity sensitive to the detailed

structure of the 2-, \ vector. The QSTD predictions are much better

than those of QTD because of the most important enhancement due to the

- 15 -

large two-qp-four-qp interference -terms even in the oase of quite

small four-qp components. It should "be stressed that our theory

is based on the purely spherical shell model; we feel that the

assumption

premature,

assumption of a stable deformation in the 2 state in Sn ie probably

An interesting observable is the static magnetic moment

u(57) of the 5? state (observed at 2.35 MeV in Sn 1 1 6 and at 2.29 MeV^120120 -

in Sn - this latter may be almost degenerate with the 3^ state120 — 120

in Sn and thus any data for 5i in Sn have to be taken with25")

oaution). For /4(5"£) Bodenstaedt e.t al. give the followingrespective values of the g-factors:

gc-(A = 116) = -O.O65 + 0.005

g_-(A = 120) = -O.O58 + 0.007

The g-factor is defined as

^ i ={TCT*-t)Ca.j+o}"s<T»Awn'> (16)in the usual notation, where the magnetic dipole operator is

In Table IV we give the computed "effective magnetic reduction matrix"

(EMRM) analogous to ECM of Table I, The name suggests that the

"hare" s.p. matrix elements '('Wl/Vo^/Y a r e • Provid-ed ^0, generally

greater than the computed theoretical values based on eq. (6)

(here specified for neutrons). yThisis indeed the oase and it

goes in the direction of a better agreement with experiment. The

"bare" matrix corresponds in this case to pure neutronic matrix

elements of the valence neutrons only. We find that the contributions

of the (ph)-pairs (both neutrons and protons) are of opposite sign

to those of the valence neutron, and lead to an over-all reduction, i .e . ,

0<ufa,h')< 1 where u(n,n ' ) is our EMRM s X" f jao) I'' j . ((pit)- P<H."£) ' 9 H . ' '

The allowed til—transitions/ are : lg^r <•—» lgr (both protons1 1 Q

and neutrons) and lh—«-» lb*- (neutrons only). The inclusion of the

latter transitions (with the upper major shell) leads to up to 40$

reductions of u(n,n'). The off-diagonal /x(2d- 2d|-) is given also

for M^fC (EJE'sE) of the type of eq. (l2) (the number in

parenthesis). The symbol OJ for /U.(3s^2d^) and yU(2d-Jlgj) in

Ke choose here to use the neutron (ph)-projector q • %• for all

the (ph)-paire involved, whioh corresponds to an average one-half

oooupatio

have u/32

oooupation of the neutron valence subshells; indeed in Sn we

Table IV

34

4

24

^ 13 B 2

0.6377

4

0.6972

24

-

0.3900Co.5280)

0.6333

_

-

cO

0.4275

-

_

-

0.5935

Table V

theory

5]_

with <H [\

QTD

-0.1382

&.> '> bare

QSTD

-0.1504

with/fe^of eq. (6)

QTD

-0.0579

QSTD

-0.0648

QTD

-0.0536

QSTD

-0.0611

- 17 -

Table IV means that the "bare" elements <-n |JJU.(|, |[ii'V vanish

(are X-forbidden) in these cases, while the corresponding

theoretical "effeotive" elements <^|/^YT [[n') a r e ^° (the £-selection

rule is relaxed through the formula of eq. (6)\ The most important

reduction in u(n,n') comes from the (ph)-neutron spin parts of

/YifOv " §\\\ * less important negative contributions are those

of the core proton spin terms; the core proton current (electric)

terms are roughly by an order of magnitude smaller.

In Table V we give our computed QTD and QSTD "bare" and

theoretical values of the g-factor gK- of Sn . The QTD and QSTD

eigenvectors 5T / refer to the same Yale—Shakin force renormalized

for oore polarization which was used for our computations of Table I I I .

We note that while our four theoretical values differ from each

other only a l i t t l e , they are by a factor of the order of l/3

smaller than the "bare" values. The theoretical results compare

very nicely with the observed value for Sn . We may mention

that a calculation by Lombard based on the 5~ J QTD

eigenvector of Ref. 15) and QTD and QSTD calculations of Hef. 23)

using <Vi ]j kX, j1)llil/viar l e d "fco a sharp disagreement with experiment

similar to what we find with the same <T) [| a, (vjj^' / ,

Recently, Barreau and Bellicard 2 " published the first

experimental data on the inelastic electron scattering from the

even tin isotopes 116, 120 and 124 with the excitation of the 2, and

the 37 states. The bombarding electron energy was 150 MeV, and the

scattering angle varied between 45° and 80°. The electric quadrupole

and octupole form factors square^ P. (Q)| » have been extracted

from the differential oross-seotions as ]Fin| = ^ 0 ^ /2 c M o t t ^ Z " l

Both the absolute values of j F. j and their angular distributions

should, particularly when combined with the static EX moments and

the B(EJX), serve as a good test of any microscopic or other nuclear

wave functions of the excited states in question.

In a letter by three of us 3' the corresponding squares\ 12

of theoretical form factors I F. I have been presented calculated

2n / a n d 3 7 / eigenvectors of Ref, 23)11)corresponding to the two-body nuclear potential of Tabakin '

renormalized for core polarization. The ooncept of a constant

- 18 -

effective charge, e\ ' was applied, and the calculations were

"based on the "reference" s.p. matrix elements, <*V\ || 0 ^ I!'ft1/ f

of the appropriate 0^. The Coulomb and the transverse electric

parts (spin and current terms) of | F. 1 are calculated according

to the definitions in eq. (3«64) of De Forest and Walecka »

Although the calculations were done essentially in Born approximation,

corrections for the distortion effects as proposed by Czyz and

Gottfried30"^ Jot. also eq. (8.13) of Hef. Z<$ )J were included.

The transverse electric terms are found to he negligible as compared

with the Coulomb parts. With the numerical values of the effective

constants e^ f = 1.23 and e^eff^= 2.19, good f i ts to the data of

Ref. 27) were obtained.

In the present work we have redone the calculations of

Ref. 28) with our present(i t e . , those ofRe£23J)QTD and QSTD eigenvectors

| 2. ^ and | 37 / appropriate to the core-renormalized Yale-Shakin

two-body force. The constant effective charge results we then

oompare with those obtained with the theoretical effective operators

calculated according to eqs. (4) and (6).

In Fig. 4 we compare with the data of Ref. 2?} our

theoretical ]p. i2(0* -* 2*) calculated with the QTD, ^STD(l)

and QSTD(II) eigenvectors ) 0 ^ and 2 - ^ . Except for large

angles (large momentum transfer)j agreement with the

data is rather good both for the angular distribution and for the

absolute values. The calculated angular distribution of

| F . \2(OT -> 3") is consistent with the data of Ref. 27).

Unfortunately, the absolute values of the same quantity are too small,

as are those of Ref. 28). If the },~ cross-section data of Ref. 2f)

are indeed based on a sufficiently precise resolution discriminating W

neighbouring 57 an<^ other states, the explanation of the lat ter

discrepancy is to be sought probably in the inadequacy of our

treatment of some exoited configurations of the core nucleons.

In Table VI we give the (e,e f) angular distribution of

the ratio of the theoretical effective form factor I F. (theor)(ft*) i n

to the referenoe form factor j F. (ref., e^ ' = l) j, i.e., to

the one computed with a constant effective charge (=l). This ratio

can be interpreted as an effective 2 -pole charge which depends

on the (e,ef) scattering angle $. This dependence measures the

-19-

Table VI

2 +

3"

QTD

QSTD(I)

QSTD(II)

QTD

QSTD(I)

QSTD(II)

45°

0.861

0.860

0.863

0.937

0.935

0.938

50°

O.858

O.856

0,858

0.954

0.953

0.955

55°

0.850

O.848

O.85O

0.975

0.974

0.976

60°

0.832

0.831

0.832

. 1.001

1.000

1.001

t

ti

0.795 j

0.793

0.794

1.032 I

1.032 \i

\

1.032

-20-

inadequacy of a oonstant effective charge theory of the (e,e')

cross-sections. Prom Table VI we realize that the ratio I F. (theor)'/

Pin(ref., e^ ; = l)J varies between about 0.79 and 0.86

for the 0+-^ 2t excitation and between about O.94 and 1.03

for 0 -> 37 in "the region 45,$ ^,^5°* Although the theoretical

effective charge is larger in the 37 case, it is still much too

small (by a factor of the order of 3) "to explain the experimental

data on the absolute values of the (e,e') cross-section of Ref. 2/0

(cf. the results of Ref. 2 3)).

In general, the differences between the respective QTD,

QSTD(l) and QSTD(ll) results (cf. Table Vl) are quite small. By

increasing the (e,e') scattering angle d the theoretical-to-

referenoe ratio of Table VI decreases slightly in the 2, case

and increases slightly in the 3-. case.

V. CONCLUSIONS

Our calculations of the effective operators of electro-

magnetic interactions with nuclei are fully microscopic and they

involve essentially no adjustable parameters, i.e., the only

parameters of the theory are the same single-particle parameters

which are involved in the corresponding shell-model spectra. The

two—body nuclear force is derived (reaction matrix) from a realistic

nucleon-nucleon potential. The theoretical construction is based on

a perturbation theory treatment of the particle-hole excitations of

the core nucleons. In this sense it is similar to the Kuo—Brown 1)**4)

core polarization of the two—body nuolear force (cf. als.o Refs.5),6))

and to the double Multiple) scattering terms of the Brueckner theory.

Our numerical calculations for even Sn-isotopes provide

a partial justification (for an otherwise completely arbitrary)

concept and approximation of a constant effective charge. The

computations are based on the Yale-Shakin realistic nucleon—nucleon

force and on a set of s.p. parameters of a reasonable Woods-Saxon

potential. The over-all agreement with the observed B(B2) ,

0,(2,) , /*(57) values and the inelastic electron scattering form

factors (except for the 0. -* 37 transition) is even surprisingly

good in view of the lack of any ad hoc adjustable parameter involved

and in view of the crudeness of some of the theoretioal

-21-

ingredients assumed. The results rsehem to be encouraging, and future

oaloulations for other nuclei andiwith s.p.basis which ara determined

in a possibly self-consistent (HF) way are moat desirable,

A direct proof of the validity of our perturbation-type

procedure by performing direct calculations with shell-model wave

functions explicitly involving configurations of the core nuoleons

treated in a more exact way is not possible as yet because of the enormcu;;

dimensions of such shell-model problems. Approximate treatments (cf.AppK.I]

based on simple QTD and QRPA calculations seem to be quantitatively

inconsistent (e.g. they leave out energetically equivalent four-q.p

or two-particle-two-hole excited configurations), and cannot provide aA(eff)

valid criterion for our theoretical 0\ . We may point out that, in

oontrast to a suggestion by Bando" " ' , we find that the bare nuclear

force is a sufficient approximation in the second-order calculation

(Kabcdjj") in eqs. (4), (6), (ll)) of the effective electromagnetic

interaction 0 ^

.Vfter the present work was oompleted an independent work by

namamoto and I'olinari ' oame to our attention. Their letter ' is

concerned with the quadrupole effective oharge of nuclei around Z - 28,

and conoluaions are reached similar to those of Ref. 10).

ACKNOWLEDGMENTS

We are indebted to Dr. M. Beiner for communicating to us

details of the eigenstates of Ref. .22). All our computations were

performed on the IBM 7044 computer of the University of Trieste.

Two of us (M.G. and J.S.) express their thanks to Professor Abdus

Salam and the IAEA for kind hospitality at the International Centre for

Theoretical Physios, Trieste. Financial support from UiNiCSCO to

one of us (M.G.) is gratefully acknowledged.

- 22 -

DERIVATION OF THE FORMULA OF EQ.(ll)

The Hamiltonian of interaction between the valence neutrons

and the core protons, H,™, generates proton particle-hole pairs (ph) out

of a given unperturbed state J V J M / I ^ where j j j ^ i s the eigenvector

of the valence neutron system and ( O ^ i o the ground state of the closed

shell proton oore, The many-body perturbation theory then gives for

the state perturbed to first order the expression

where

(A2)

and where ( | ^*- /®j(p''M/l. is a vector-coupled product of the unperturbed

valence neutron eigenvector of energy B, spin J and a proton (ph)-pair

of spin X •

The Hamiltonian H-_p can be put in the form

where A ^ j A / . O is defined in eq..(9) and

CJ)VV(a,CLl) ^

(M).

is a proton (aa1) partiole-hole creation operator. A straightforward

algebra gives for the expansion ooeffioient (A.2) the resul t :

=2.2.

-23-

(AS).

We now prooeed to calculate the lowest-order non-vanishing

contributions to a matrix element of the electric 2 - pole,E' A r E \

®XL*- M /• T ^ e r e ^xe *vo first-order terms, the first of which is

A simple evaluation of this expression leads to

'JM(A6)

T

(A7)where

r

-. r. (m.V&te.X)r

A quite similar formula is easily obtained for the second first-orderterra. Combining the two terms, we finally obtain eq. ( l l ) .

\ A

Let us now take a 2 -pole magnetic operator O\ . In ,addition to the non-vanishing zero—order (bare) matrix elements < ,..,

j ^ 1 ^we find in this case the first-order terms of the virtual excitationof the core neutrons corresponding to the terms of eq.(ll) with F,™replaced by the ant isymmetri zed elements R™ of the valence neutron-coreneutron interaction. In deriving the latter formula, in the same way a3that of eq.(6)» all the contractions between the neutron creation andannihilation operatoiB(c^ and c) are to be made in the matrix elements

i.e., 'JT is oontraoted with and \T while is contracted with V and V

(")( and 3r are distinct as a hole and a particle, respectively).

- 2 4 -

APPENDIX II

REMARKS OET EXPLICIT TREATMENT OF CORE CONFIGURATIONS ITT QTD

In the QTD approximation it is still possible explicitly to

include at least some of the excited configurations of the core

nucleons. Even the limitation to a relatively small number of extra

subshells in Sn. (in addition to those of the valence neutrons) does

not render such a model very reasonable.

First, an explicit treatment of the core nucleons means that,

to avoid double-counting, no corresponding core polarization corrections

must be included. It is readily found that for bare realistic two-

body forces (K-matrix), one finds generally only a weak BCS pairing

effect even for the subshells close to the Fermi level, i.e,, the

corresponding energy gaps and single-qp energies are small •?'* '.

Then it is clear that, in the circumstances, the quasiparticle

approach is bound to fail and an exact shell-model approach is

necessary. On the other hand, we know that the Cooper pair-elements

of the core polarization corrections to the effective nuclear force

are large. Consequently, they mustVbexreated on the same footing

with the corresponding bare elements, i.e., by the BCS method. The

remaining (residual) parts of the qp-transformed Hamiitonian are

then relatively weak and can be treated by a Tamm-Dancoff type

approximation.

As an illustration we give here the numerical prediction of

B(E2, 2*^>0*) in such a simple extended QTD. We have considered 10

neutron and 10 proton subshells in Sn, i.e., the subshell lg -s and

the entire major shell (pf) in addition to the five valence neutron

subshells. The numerical values of all the single-particle parameters

involved were exactly those of the main text of the present paper,

i.e., those of Ref,23 and the same Yale-Shakin nuclear force was

employed.

The calculated energy spectrum of Sn is in marked

disagreement with experiment. For B{E2, 2+-*0!J") we find only the

very small value of 5.03 e F (with the neutron effective charge

exactly => 0). One extra reason for this bad failure of the model

to reproduce even the right order of magnitude

-25-

of the observed value is the fact that the six subshells of the upper

major shell should he most important for a collective effect of all the

single-proton transitions in B(E2, 2*->0 ), and the (sd) major shell

should be included. In fact, one must not oompare the last result with

the results of Table III "but, if at a l l , rather with the corresponding

model where only the five core subshells (lg-7 a n ^ Ef) a r e included both

in calculating the QI!D eigenvectors and the effective charge matrix

(ECU). In "this oase we obtain B(E2, 2*-*0*) - 50.19 e2F4 which is about

five times smaller than the value calculated in our theory with all the

single-particle levels between the magic numbers 8 and 126.

-26-

REFERENCES AUD FOOTNOTES

l ) T.T.S. KUO and G.E. BROW, Nuol. Phys. 8£, 40 (1966) ;

A92. 481 (1967).

2) E.P. LYXCH and T.T.S. KUO, Huol. Phys. Aj£, 561 (1967).

3) T.T.S. KUO, Nuol. Phys. A£2, 199 (1967).

4) G. SARTOHIS and L. ZAMICK, Phys. Le t t e r s ,2^3, 5 (1967).

5) M. GMITRO, J . HENDEKOVIC and J . SAtflCKI, Fnys. Rev., 16£, 983

and Phys. L e t t e r s 263, 252 (1968).

6) M.. GMITRO and J . SAWICKI, Phys. Le t t e r s 2j>B, 493 (1966);

T.T.S. KUO, i b i d . 26B. 63 (1968).

7) A. EE-SKALIT, Phys. Hev. 1 1 ^ , 547 (1959).

8) B. MOTTELSOM", Rendioonti Scuola Internaz ionale di Fisioa

"E. Fermi" , Varenna (i960), p . 44.

9) Historically, the ea r l i e s t considerations in terms of

configuration mixing are : R.B. AMADO and B.J, BLIN-ST0YL3,

Proc. Phys. Soo. (London) AJO, 532 (l957); A. ARIMA and

H. H0Vl6, Progx. Theoret. Phys. (Kyoto) 11, 509 (1954)

and 12, 622 (1954); R. BLIN-STOYLSt Proc. Phys. Soc. (London)

A66. 729 (1953) and R.J. BLIN-STOYLS and M.A. PERKS, iTjid.

A67. 885 (1954).

10) E. GMITRO, A. En^INI, J. SAWICKI and T. 1VSB3R, Phys. Rev. Letters

2.01 1185 (1968); by a misprint the definitions of the symbols e..

and e 2 of eq.(l) of Ref.10) are interchanged; all the results

of Ref .10)corrospond to its eq.(l) with, the correct definitions

of e., and e«»

11) F. TABAKUr, Ann. Fhya. (HY)30, 51 (1964).

12) J, HENDEKOVltf, P.L. OTTATIANI, M. SAVOIA and J. SAWICKI,

Uuovo Cimento $$3, 80 (1968).

13) P . L . 0TTAVIA1H, H. SAVOIA, J . SAWICKI and A. TOMASOTC, Phys .

Rev. l^i, 1138 (1967).

-27-

14) A. RIMINI, J. SAWICKI and T. WEBER, Phya. Rev. 168, 1401 (1968).

15) R. ARVIEU, Ann. Phys. (Paris) 8., 407 (1963); R. ARVIEU,

E. BARANGER, M. BARAKGER, M. VENEROITI and V. GILLET, Phys.

Letters 4, 119 (1963).

16) P.L. OTTAVIANT, M. SAVOIA and J . SAWICKI, Phys. Let ters 24B.

353 (1967).

17) K.E. LASSILA et a l . . Phys. Rev. 126, 881 (1962).

18) CM. SHAKIN et a l . . Phys. Rev. 161,, 1006 (1967).

19) V. GILLET and il. RHO, Phys. Let ters 21., 82 C1966).

20) Y.K. GAI-3HIR, ICTP, Tr i e s t e , preprint IC/68/3 2 and Fnys. Let ters 2 63,

695 (1968).

21) R. ALZETTA ejb al., to be submitted for publication.

22) K. BLEUL3R, M. BEIHER and R. DE TOUREIL, Uuovo Cimento 52B.

45f 149 (1967) and private communication from M. Beiner.

23) M. OMITHO, A. RIMIHI, J. SAWICKI and T. WEBER, ICTP, Trieste,

preprint IC/68/29.

24) J. BE BOER, Proceedings of the International Conference on

Nuclear Structure, Tokyo 1967» P* 203.

25) .E. BODEHSTAEDT et al.. Z. Phys. ljS8_, 370 (1962); Co-operation

of the Angular Correlation Groups of Bonn and Hamburg,

Nucl. Phys. 8£, 305 (l$66).

26) R.J. LOMBARD, Huol. Phys. JXt 348 (1965).

27) P. BARREAU and J.B. BELLICARD, Phys. Rev. Letters 1 ., 1444 (196?)

A. HIMIffI, J. SAWICKI and T. WEBER, Phys. Rev. Letters 20,

676 (1968).

29) T. DE FOREST and J.D. WALECKA,.Advan. Phys. 1£, 1 (1966).

- 28 -

30) w. C3YZ and K. GOTTFRIED, Ann. Phys. (NY) 21, 47 (1963).

32) H. BAUDO, Progr. Theoret. Phys. (Kyoto) 8, 1285 (1967).

32) I. HJtt! 'OTO and A. KOLDTAHI, PhyB. Letters 26B, 649 (1968)

-29-

TABLE CAPTIONS

TABLE I . Matrix of the effective quadrupole charge of eq. (15)

for E2 t rans i t ions for the five valence-neutron sub-

shel ls in Sn. The numbers without parenthesis refer

to Cr® ^ of eq. (4)» and those in parenthesis to

the 0£g '(E = E ' ) of eq. ( l l ) . The s .p . and

other parameter values are explained in the t e x t .

TABLE II. Partial contributions to the elements e (n,n() of

Table I coming from the four groups of the h<*-»p

transitions: A = (2p, lf -, lg§0 <H* (3S, 2d, lg-L lh—) 5 (a),

B = C = = (2plf) *

(2f , 3P» l i g , lh|") = (upper major s h e l l ) .

TA3LE III. Values of the reduoed transition rate B(E2, 2, •» 0. )

^ calculated with e^ 1 a n dand of of Sn calculated withwith e (n,n<) of Table I . The QTD and Q3TD eigen-

vectors ( l and I I explained in the text) refer to the

renormalized Yale-Shakin foroe of Ref. 23).

TABLE IV. Matrix of {n,n*) = < ^ H Mdefined in the text for Ml transitions for the five

The numbers without

in

E1 = E) of the analogue of eq. (ll).

valence-neutron subshells in Sn.

parenthesis refer top- ®\ ' of eq. (6) and those iA(eff) l}

parenthesis to

TABLE V. The gyromagnetic factor g,.- of the 5~ state of Sn

calculated with ^ I ^ f , ) W / - ^ and with fi-%) givenin the t ex t . The QTD and QSTD eigenvectors referto the renormaliaed Yale-Shakin force of Ref. 23)•

-30-

TABLE VI. Ra t io s |F i n ( t heo r ) | / | F i n ( ey f f ^= l ) | of the theoretical

and the "reference" (e x ' = l ) inelast ic electron

form f a c t o r s f o r t h e r e a o t i o n s Sn ( e f e ' J Sn

(2 , , 3n)» The theoretical effective 0\ ' operators

are calculated as from eqj. (4) and (6) for each

(e,e () scattering angle 6. The QTD and QSTD (i and II)

eigenvectors refer to tlie renormalized Tale-Shakin

force of fief. .23).

- 3 1 -

FIGUES CAPTIOETS

Pig. 1 Lowest-order diagrams for processes contributing to

with neutrons only in the valence shells.matrix elements <'K\I || 0 i n o a s e o f a nucleus

Pig. 2

Pig. 3

Simplest diagram contributing to the higher-ordercorrections to matrix elements <*ft' jj O t, li

Diagrams contributing to the exchange terms of Fpp ineq..(6).

Fig. 4 Theoretical inelastic scattering form factor

Pin(theor)j2 for the reaotion Sn 1 1 6 Ce,e')

Sn (2,) at 150 MeV (incoming electron energy).

Q is the momentum transfer. The QTD (the dashed

line ) , QSTD(l) (the solid line) and

QSTD(ll) (the dotted line ) results refer

to the eigenvector of Ref. 23) obtained for the

renormalized Yale-Shakin force. The experimental

data (bars) are those of Ref. 27).

- 3 2 -

V.NP

x \ NP

(a) ib)

Fin. 2

- 3 3 -

It

(a') Vb')

-34-

0.6 0.7 u. 6 0. 9

. TiC.

- 3 5 -

Available from the Office of the Scientific Information and DocumentationOfficer, International Centre for Theoretical Physics, 34100 TRIESTE, Italy

1678

• 1i ;•" -