Electron scattering from 20Ne and 24Mg in a microscopic boson model

Nuclear Physics A483 (1988) 92-108 North-Holland, Amsterdam

ELECTRON

SCA’ITERING

FROM **Ne AND BOSON

‘*Mg IN A MICROSCOPIC

MODEL

R. KUCHTA’ Laboratory of Theoretical Physics, JINR, Dubna, Head Post O@ce, POB 79, Moscow, USSR Received (Revised

2 November 1987 18 January 1988)

It is shown that a mean-field approximation applied to the microscopically derived boson hamiltonian yields a reasonable description of the form factors for both elastic and inelastic electron scattering from some sd-shell nuclei (aoNe, 24Mg). The results agree well with experimental data for the 0’ + 0’ and O++ 2+ transitions but much less so for the Oc + 4+ transitions. Possible sources of the observed discrepancies are suggested. Comparison with other approaches is also given.

Abstract:

1. Introduction

Form factors of electron scattering are known to provide a deeper insight into the structure of nuclear states than the energy spectra and electromagnetic transition probabilities alone ‘). This is so because the momentum-transfer dependence of the associated nuclear matrix elements contains information about the spatial structure of nuclear states, thereby yielding a more stringent test on the reliability of the model wave functions. Various methods have been proposed to study the electron scattering form factors, both in heavy *“) and in light 4-6) nuclei. In this paper we present a new approach which consists in applying the mean-field (MF) approximation ‘) to a boson hamiltonian derived microscopically from the underlying shellmodel hamiltonian by means of the Belyaev-Zelevinsky-Marshalek (BZM) mapping procedure “). Due to the obvious simplicity, the search for a bosonic description of nuclear spectra has been a long one 9, and it has even more intensified lo) after the success of the phenomenologi~al interacting boson model (IBM) of Arima and Iachello I’). Recently, there have also appeared some attempts toward a boson treatment of the electron scattering form factors 3,5*6).These attempts are based either on the IBM 3,5) or on a specific realization of the symplectic sp(3, R) algebra in terms of the harmonic oscillator boson operators 6). Our approach is conceptually closer to that using the IBM, in the sense that it treats the bosons as counterparts of nucleon pairs “), However, we go beyond the conventional IBM ‘l,‘*) because the MF approximation enables us to take into account the bosons with all possible angular momenta as well as those corresponding to proton-neutron pairs. ’ Permanent

address:

Institute

of Mechanical

Engineering,

0375-9474/88/$03.50 @ Elseviet Science Publishers (North-Holland Physics Publishing Division)

B.V.

360 00 Plzeii, Czechoslovakia.

93

R. Kuchta / Electron scattering The

organization

relevant

of the present

formalism.

(Coulomb)

Sect.

form factors

24Mg. A summary

paper

3 contains

is as follows.

its application

for elastic and inelastic

In sect. 2 we describe to investigating

electron

scattering

the

the charge from “Ne

and

is given in sect. 4.

2. The model 2.1. DERIVATION

We consider nuclear

OF THE BOSON

HAMILTONIAN

a system of n, protons

hamiltonian

and nV neutrons

is taken to have the general &=

C T,pc& @

+t

(n, = n, = ZV, N even). The

form (1)

C Va,+:c;csc,, 4-Y~

where the indices (Y,/I, -y, 6 run through a suitable complete set of single-particle states and cz(c,) are the corresponding creation (annihilation) operators of nucleons. The quantities Tap stand for the matrix elements of a one-body operator, matrix such as the kinetic energy, while the Vapvs represent the antisymmetrized elements of an effective two-body interaction (1) is assumed to have the usual hermitian relation

between the nucleons. The hamiltonian and time-reversal properties. Using the

c~c;c~c, =spsc~cy - c~csc;c,, we can rewrite

(2)

(1) as

where

If we choose the single-particle states to form the Hartree-Fock one-body term (3b) is clearly diagonal because

(HF)

basis,

the

by definition of the HF basis. However, when describing realistic light nuclei, in which the single-particle states can be restricted to those taken from neighbouring shells only, &, may be diagonal in the harmonic oscillator basis as well 13). We make this choice here, so that fi, acquires the form I& = c E,C& a

-a

1 vo~vsc~c&;c~ nPr8

(5)

94

R. Ku&a

/ Elecrron scattering

and the single-particle states (Y,/3, . . . are characterized by the oscillator quantum numbers in the isospin formalism 14), a?= (n,, I,, ja,& m,, T,), where T, =+4(-f) for neutrons (protons). We will also use the letter a to denote the same set except m, and 7,. Now, we introduce the boson creation and annihilation operators b& and hop, which correspond to fermion pair operators czci and cBc,, respectively. These boson operators are assumed to satisfy the following antisymmetry and commutation relations b& = -b& IIbe,,

,

(64

bawl = [ b:p , bbpl = 0 ,

[hap, b&y] = &&I,,,-

6,,3,,,.

(6b) (6~)

We also define the boson vacuum IO), by the condition b,, 1% = 0.

(7)

In the BZM approach ‘), the boson images of the bifermion operators czc$, cBc, and czc, are expressed as power series in the boson operators b$, b,, and the unknown coefficients of these expansions are determined by the requirement that the original fermion commutation relations remain valid in the boson space. In the case of the particle-hole operators c& this leads to a very simple finite form (c;tc,),

= C K&&y, Y

(8)

where the suffix B on the left-hand side denotes the boson image of the corresponding fermion operator. On the other hand, the operators (czci), and (c$c,)~ are in general represented by infinite expansions, which naturally give rise to the problem of convergence. Nevertheless, these complicated operators are not needed to find the boson image of the fermion hamiltonian if the latter is conveniently expressed in the particle-hole form (5). In this case only the simple boson operators (8) are relevant. Replacing czc, in (5) by (czca), according to (8) and arranging the boson operators in the interaction term into the normal order with respect to the boson vacuum IO),, we obtain an exact boson image fiB of the fermion hamiltonian fir in the form ci,=

C e,&;tPb,&

C C &&&b&&J+, ww

MY6

where the quantities eaprs are given by

[

(9)

P”

P 1 9

Ea@yfi= &7, ecJ,y -a c v,,,,

+4 V&S

(10)

The bosons associated with the dynamics of the underlying fermion system can be expressed as functions (in general rather complicated) of the operators b$ , bnp.

R. Kuchta / Electron scattering

In this paper we assume that the corresponding

95

transformation

is linear and unitary,

i.e.

JTah

+(iMI)

(iM7) JJ-~~l/f~f~.~‘bf= SJJ,S,,{S,,,~,,,+(-~)‘,+‘~+~+~S,,,S,,.)

4 c

.

(llc)

The quantities (. . . .I. .) in (11 a ) are the usual Clebsch-Gordan coefficients for the angular momentum and isospin coupling. We characterize the dynamical bosons BTMrby the isospin projection T( r = -1, 0, + 1 for the proton-proton (TUT), protonneutron (TV) and neutron-neutron ( VV) bosons, respectively), the angular momentum projection M and the label i which distinguishes between independent configurations with the same T and M. Using (11) and a standard angular momentum algebra, one gets the boson hamiltonian fi, in the form fin = C J%B:Bz+ where

l=(i,M,~,),

(12)

C W,D&B:&&,

12

1234

2~(i*M~7~) ,...

and (13a)

W 1234

=

C J,J,J1J4

x

The functions

C T, TzTxT,

C ahcdef

W~k$$z~,T3J,T,(M171,

M2r2,

M373,

(i M T )+cI::‘m;c;‘)+~p~)+~~(%;p). (CIJ,‘T,L’

M474)

(13b)

E yThcd’and W$$$zqzJ3T3J4T4(Ml

T, , kt2~2,

M3~3, M4~4) are the angular

momentumand isospin-coupled analogues of the quantities E+,~, VaPrs appearing in (9) and their explicit expressions can be found in ref. 15).

2.2. APPROXIMATION

Up to this point,

METHOD

no approximations

have been made.

Provided

the sums in (12)

extend over all possible values of the labels 1~ ( il MI T1), 2 3 ( i2M2r2), . . . , the boson hamiltonian (12) is a precise image of the original fermion hamiltonian (5). However, the diagonalization of (12) in the entire boson space is practicable only for very special types of interaction and for a substantially limited number of bosons. In addition, such an exact diagonalization will also produce unphysical (spurious) states which have nothing to do with the underlying fermion system because of the overcompleteness of the boson basis. These spurious states must be identified and removed, which has long been a difficult problem 16). Fortunately, it has recently been shown “) that

96

R. Kuchta / Electron seatterhg

(i) this identification and removal of spurious states can be accomplished in a rigorous and systematic way; (ii) the remaining boson states, identified as not spurious, emerge unharmed by the admixtures of unphysical components, and consequently, they have the correct energies associated with the original fermion system. It is rather reassuring to see this, but actually one has not achieved much, because the complications of calculating the fermion matrix elements have only been transferred to other place, namely to the procedure for identifying the spurious boson states. It therefore appears to make more sense not to insist on the exact diagonalization of (12) but rather to look for an approximate solution which would be free from spurious states. This amounts to truncating the hamiltonian to a few bosons which must be chosen so that the truncated part of fiB is only weakly coupled to the neglected rest. No general consensus seems to exist as to how one should find the optimal bosons ‘*). In our opinion, the mean field approach is well suited for this purpose because ‘,14) (i) it is based on a variational criterion which ensures that the states built by the desired bosons are maximally separated from all other states, (ii) it is able to treat important physical properties in terms of few relevant parameters, (iii} it may include many different degrees of freedom in a single intrinsic state, (iv) it has a largely established power to deal successfully with many-body systems. The mean field method starts from the assumption that the transformation coefficients +$$’ in (11) are determined so as to minimize the ground-state expectation value of fis. In the present case of rr,_,protons and n, neutrons (n, = ny = N = even) we can take the ground-state wave function /GSfB to be a condensate consisting of only one kind of bosons (i = g; M = 0; r = 0) (14) where IO), is the boson vacuum defined by eq. (7). Note that IGS)n is properly normalized because [B,,, , B&J = 1 as a consequence of (llb). In writing (14) we have assumed that the ground state is axially symmetric (M = 0) and all the 2N = n, + n, nucleons are paired so as to give rise to N gv-bosons (T = 0). This concept of building the GS wave function is to be compared with the usual approach i9) in which rather the identical nucleons are paired to form the +zr=-and vu-bosons. The latter approach is certainly well justified in heavy nuclei where protons and neutrons occupy very different single-particle orbits. However, for applications to lighter nuclei with protons and neutrons filling the same shell, the explicit inclusion of rrv-bosons is expected to be important. With the assumed form (14) of IGS&, the variational requirement &(GS/I;l,/GS)a

=0

(1%

R. Kuchta / Electron scattering

under

97

the restriction

(16) leads to the following

system

~(@x

-6

JTt,b,J’T’a’b’-

of non-linear

JJ’ 6 ‘“-

eigenvalue

equations

E$‘$'b"+2(N-1)86,b 1

for $$jjy:

c J+@$j

J, T, J2 T2 ted

(17) We use standard iterative techniques to solve (17) and together with (ci%i’, corresponding to the lowest eigenvalue A,, we also obtain the solutions 1,5:‘:zi orthogonal to $9;;’a ,

(18) JTab

Once the structure of the ground state has been fixed, we turn our attention to the excited states, which can conveniently be generated in the framework of the equations-of-motion operator

formalism

*“). In this method

one attempts

to find an excitation

Qz satisfying

Q&#d=o,

(19)

b#dCQo,,& Q:lh) = ~&o~[Qc+,Q%#d

(20)

,

where Q, = (Qii)‘, I& hamiltonian excitation

re P resents a suitable reference state, fi stands for the of the system, the subscript (+ denotes a particular solution of (20) with

energy

w, and the symmetrized [A, B,

It is also useful

double

commutator

Cl = ;{[A, [4 Cll+ [[A, Bl, Cl) .

(but not necessary)

to demand

state the boson

as

(21)

that

[Q,,, Q:] = 6,,, . Here we take as reference to verify that the operator

is defined

ground

(22) state (14). It is then not difficult

Q:, = (I+ ~~o&oo)-“‘%m~,oo satisfies (19), (22) and its action on ]GS), produces in which one of the Bc,,,, bosons in the condensate boson BkO, i.e.

(23)

a normalized boson state (UC), (14) is replaced by an excited

(24)

R. Kuchra

98

/ Electron scattering

By successive application of Q& on /GS), one can generate higher (two-boson, three-boson, . . .) excitations but in this paper we will restrict our discussion to the lowest excited states of the form (24). Substituting (23), (14), (12), (13) into (20) and making use of (22), (llc), we obtain

ytw1 JTQh,J’T’a’b’ = 6,J&&$yh”

--~d*~l?*l&)+2(~--1)

x wz;?;::*,*T(~o,

0% 00, ~Owg~!l*+

JITs*T2; 1CI!I$% cfi!!$~L1~

(25)

Solutions of (2.5) give the structure coefficients $JTah tiKO) 0 f the excited boson I?&,, as well as the excitation energies wrK of the states (24). Since the matrix I%C$$&,~~~~~, is fully specified in terms of the quantities arising from the ground-state calculation (4ti$+tT, h ) eqs. (25) are linear with respect to the amplitudes I,!&$, Note that the lowest (fL’1) solutions of (25) with K = 0 and I( = 2 correspond to the so called p.. and y-bands, respectively ‘). We can also convince ourselves that the K = 0 solutions coincide with the J/$7$ obtained from (17), and consequently, they satisfy the relation (18). In the following we will restrict our discussion only to the K =I0 and K = 2 states of the type (24), and therefore, we will not encounter the problem of identifying and removing the one-boson excitations associated with the rotation of the nucleus as a whole j9>. To proceed further, we must restore the spherical and isospin symmetries which are obviously broken in the boson wave functions of the form (14) and (24). This is accomplished by angular momentum and isospin projection according to /GS; J’I’; MAc&-)~= ~;“+‘,,i;;t;,O]GS)B, (26)

/I; JT; ~~~)~

= P&B”,,&&.,lZK>, ,

where PZ~‘,*Eb are the normalization constants and 3”,,+,,,,, @;;rb, stand for the projection operators onto states with definite angular momentum and isospin, respectively 14). The energies of different nuclear states IJT) are then obtained by simply taking the expectation value of I?, in the states (26).

2.3. PHYSICAL

BOSON

STATES

For the procedure described in the preceding section to be reliable, we must be sure that the basic boson states (14) and (24) are indeed physical, i.e. that they are in one-to-one correspondence with actual states of the underiying fermion system. We therefore construct the following fermion analogues (not images!) of the states (14), (24):

/GSb= &~G’;oo)~ 1%t

(27)

Ii%=

(28)

J&~~O(~&)N-‘lO)F,

R. Kuchta / Electron scattering

where NGs, .N[, are the normalization (c,IO&= 0) and I%-,, = ,g,

,C,

99

constants,

IO), denotes

~~~~‘(i,m~~mpIJM)(~7,~7pI

T+&.~_c&,

the fermion

,

vacuum

i = g, 1

(29)

T@TP that with the same $y$$’ as ’ in (11). At the present stage, it should be emphasized the fermion states (27) and (28) cannor, in general, be considered as counterparts of the boson states (14) and (24), respectively. This is because the former may be linearly dependent (as a consequence of the Pauli principle), while the latter are always linearly independent. It is thus necessary to check explicitly whether or not the linear dependence among fermion states occurs. This is done by diagonalizing the norm matrix F(GS (GS),; ~=

F(GS 1IK),;

F( I’K 1GS)F . . . . . * : . . . . . . . . . . . . . . :

(30)

(K)

HI’,/ ,...............

with H.$’ = ,(1’K 1IK), and looking at the eigenvalues. For the lowest excitations with K = 0,2 and for the parameters described in sect. 3 we have found that none of the eigenvalues is zero, which implies linear independence of the fermion states IGS),, IIK),. Consequently, the boson states (14) and (24) can be put into one-to-one correspondence with the fermion states (27) and (28), thereby showing that they indeed represent certain physical states of the fermion system considered. 2.4. THE ELECTRON

SCATTERING

FORM FACTORS

Our aim is to calculate the charge (Coulomb) electron scattering form factor for a transition from an initial nuclear state characterized by angular momentum Ji and isospin Ti to a final state with Jr and Tf. In the plane-wave Born approximation (PWBA)

this form factor

squared

at momentum

transfer

q is given by ‘)

(31) where 2 is the atomic number of the target nucleus, has the second quantized form gL( q) = e

I

mdr 5 (aIjL(qr)YLM(t-

the transition

t3)b)chp

operator

g=(q)

(32)

( . . . 111. .I/.. . j” means that the matrix element is reduced both in the ordinary and isospin spaces. In (32), e represents the charge of a nucleon, jL(qr) and YLM stand for the spherical Bessel functions and the spherical harmonics, respectively, and t, is the operator of the third isospin component, t,)n) = -b/m), &J)= +$Iv). and the symbol

100

R. Kuchta / Electron scattering

In the present boson formalism, the states JJi~i), IJfTr) are taken to be those given in (26) and the boson image of the transition operator (32) is obtained by replacing czcB with (c~c,), according to (8). 2.5. THE SHELL-MODEL

HAMILTONIAN

In order to carry out explicit calculations we have to specify the characteristics of the nucleon system considered, namely the single-particle states and the two-body interaction. As already remarked in sect. 2.1.) instead of performing a self-consistent HF procedure, we choose as the single-particle basis the three-dimensional harmonic oscillator basis restricted to neighbouring shells only (the Op and Odls shells in the present case) and observe that the HF hamiltonian (3b) is diagonal in this limited single-particle basis. As for the nucleon-nucleon interaction, we combine a phenomenological form 2*) Y,(i)*

V,,,(U) = v0 ‘(riiir,)x

Y,~){l-~+~u(i)~o(j))(l-~i+~“l(i)~7(j)}

L

(33) with a two-body spin-orbit

interaction 22)

to get the correct spin-orbit splitting of the single-particle levels. As is well known r4), any calculation in a space larger than one major oscillator shell introduces the problem of identifyng and removing the centre-of-mass (c.m.) excitations, associated with the oscillations of the nucleus as a whole in the shellmodel potential. In the case of an approximate treatment, these (spurious) c.m. excitations can mix with the actual intrinsic excitations and their mutual separation is often very difficult or even impossible. In this paper we try to diminish the coupling between the c.m. and intrinsic excitations by performing a unitary transformation of the shell-model hamiltonian

A = i$l g+f f {V,,,(i,j)+ V,.,.(i,j)l

(35)

i#j

to

13)

2 = UHU-’

=

(36)

where A stands for the nucleon number, E? are the harmonic-oscillator energies and V,,,,( i, j) =

1

1 -A

Gpi

I

*

pj

+$t?WJ2F~ * Fj

>

.

eigen-

(37)

R. Kuchta / Electron scattering

This transformation among

various

states becomes

13,23) to distribute

the strength

states in such a way that the amount enhanced

task we are then (c.m. excitations) evaluating

is expected

while in others

it turns

101

of c.m. excitations

of c.m. components

in certain

out to be suppressed.

The only

left with is to identify which states belong to the first category and which to the second (intrinsic excitations). This is done by

the quantities %.nl.(JT) = (-WL.IJ~>

3

(38)

where fl km.

is the operator the respective

=

c k=-1

(39)

a:4

that counts the number of c.m. motion quanta 23) and IJT) represents nuclear state. The creation operator u: in (39) is defined as (40)

where

Fk and &

are the c.m. momentum

and coordinate,

respectively.

3. Results The procedure described in sect. 2 was applied to the study of the electron scattering form factors in “Ne and 24Mg. The calculations were performed using the oscillator model single-particle space consisting of the 0~312, Op,,,, Od,,,, l~i,~ and Od3,* shells with energies 24) ~(0~312)

=

-21.8

MeV, ~(lsi,~)

e(Op,,,) = -3.28

= -15.65

MeV,

MeV,

e(Od&

= -4.15

MeV,

e(Od3,2) = 0.93 MeV.

The oscillator length parameter b = mw/ h, as well as the nucleon-nucleon interaction parameters V,, r], p, 5,. have been determined in such a way that the diagonal elements of the HF matrix hiI = T,, + $ & VaPaP [cf. eq. (3b)] provide the best fit to the above single-particle energies and the ground-state energy calculated in the present model reproduces the experimental binding energy of a given nucleus. The resulting values are listed in table 1. In fig. 1 we compare the experimental excitation spectra with those obtained in the present approach. An overall agreement is observed, both for 20Ne and 24Mg. TABLE

Calculated

parameters

Nucleus

V, (MeV . fm3)

“Ne

13.6 74.1

=Mg

1

characterizing the nucleon-nucleon oscillator wave functions 17 0.254 0.257

interaction (33), (34) and the single-particle (b = mo/ h) b (fm)

P 0.509 0.490

-1.26 -1.26

0.16 0.16

1.65 1.68

102

R. Kuchtn

/ Electron

scattering

MNe (T=O)

(T=O) 2’

__-__;: -____

-2'

O'-___ O'--

Z4MCJ

--0' --__

-0'

2*-__

-_2*

o+_-_-t'-__

-m

0'

3+-__

__-2’

4’ ---

EXP

Fig. 1. Positive-parity

3+ 2+

2'_____&a--

2'

2'-___-

--

1'

--__L'

2' ______2+

CALC

EXP

T = 0 levels of “‘Ne and 24Mg. Experimental

CALC

values are taken from ref. ‘*).

ELASTIC

10-l

. . . . . . SP(3,R)

0.5

1.0

1.5

2.;

4 IfmT Fig. 2. Calculated form factor for the elastic electron scattering by ” Ne , compared with experiment. The solid curve shows the result obtained in the present boson mean-field (MF) approach. The dashed curve represents the result of the shell-model calculation in the restricted sd-shell subspace. For comparison, we also show the earlier result obtained by Vassanji and Rowe [ref.6)] in their microscopic boson sp (3, R) model (dotted curve). Experimental data are taken from ref. 25).

R. Kuchta / Electron scattering

103

*'Ne

-

MF BOSON

. ..**

SP(3.R)

-.-._

IBM

0.5

15

1.0

2.0

q[fm-'1 Fig. 3. Calculated form factor for inelastic electron scattering to the first 2+ state in “Ne. The solid, dashed and dotted curves have the same meaning as explained in the caption to fig. 2. The dash-dotted curve is taken from ref. ‘) and it has been added to compare our results with those obtained in the IBM. Experimental data are taken from ref. 25).

With the corresponding wave functions, we have calculated the electron scattering form factors for various transitions in the above nuclei. The results for 20Ne are displayed

(as solid lines) in figs. 2-5. As is seen, the agreement with experimental (fig. 5). For comparison, we data is quite good, except for the 0: +4: transitions also show the shell-model results obtained in the restricted sd-shell single-particle subspace (dashed lines). While for the elastic scattering (fig. 2) as well as for the excitation of 2: and 2: states (figs. 3 and 4a, respectively) the shell-model results do not differ substantially from the mean field (MF) boson ones and both compare well with experimental data, the same is not true for the excitation of the 2: state (fig. 4b) and the 4: state (fig. 5). Nevertheless, it is worthwhile to notice that in both cases the MF boson approach provides a better agreement with experimental data than the shell-model calculation, though for the 4: state the discrepancies are still large. This improvement is most likely due to the fact that the MF boson calculation takes into account the excitations of the I60 core (though in an approximate way), whereas the present shell-model calculation does not. In fact, there is some evidence 26) that the O:, 2: and 2: states in *‘Ne are built mainly by four nucleons outside the 160 core while the 2:, 4: states contain non-negligible excitations from the Op-shell.

scattering

R. Kuchta f Electron

104

r

1o-z-

al

20N~ 0; -2;

10-3_ Nd + 1o-4_ Lb A A //

lci5.,!I 0.5

-

MFBO.SON

----

SHELL

\

1.0

MODEL

\

1.5

21;

\

9~fd'I

20-l

I

b1

/

'*Ne

\ \

I lo+.

\\

/’

1G7_

I

0.5

2

,/--

-.

5.

\-

I’

SHELL MODEL --=i 1 \

It I

1.0

1.5

2.0

stfm-'1 Fig. 4. Electron scattering form factors for the excitation of the 2: (a) and 2: (b) states in *‘Ne. The solid and dashed curves represent the results of the MF boson and shell-model calculations, respectively. For experimental data see the caption to fig. 2.

It is also interesting to compare our results with those obtained in other approaches. In figs. 2 and 3 we therefore reproduce (as dotted curves) the recent results of Vassanji and Rowe 6), and in fig. 3 we also show the IBM results of Park and Elliott “) (dash-dotted curve). Needless to say that this comparison is only qualitative because of the conceptual and computational differences between various approaches. Much work is still required before quantitative conclusions can be drawn. In fig. 6 we present our results for the elastic and inelastic electron scattering form factors of 24Mg. We again observe that the MF boson form factors for the O+ + O+ and 0’ + 2+ transitions (figs. 6a, b, c) reproduce the experimental data quite well (in general better than the shell-model calculations) but those for the 0:+4:

105

R. Kuchta / Electron scaitering

I

"Ne

0.5

10

1.5

2.0

2.5

q [fm-’ 1 Fig. 5. Calculated

form

factor

for the excitation of the first 4+ state in *‘Ne. For further caption to fig. 4.

details

see

transitions (fig. 6d) disagree with the data both in shape and magnitude. Possible sources of the observed discrepancies are the following. First of all, we have performed the variation before projection in this work. It is well known 14) that such an approach does not allow for changes of the self-consistent internal field within a rotational band. Since there exists no a priori decoupling between rotational and intrinsic motion, the method of variation before projection can be expected to work well only in cases when the coupling terms are relatively small, which occurs most likely for not-too-large angular momenta. How large they may be depends in an essential way on the choice of the intrinsic system. It is indeed possible that in the present case the values J = 0, J = 2 (for which we have obtained satisfactory results) still permit a sufficiently good decoupling between intrinsic and rotational motion, whereas

the value J = 4 is already

too large to allow the variation

before projection

to be correct. The second reason for the failure of our approach in reproducing 0: + 4: form factors can be traced back to the occurrence of c.m. components

the in

the 4: state. In table 2 we give the quantities (38), which measure the amount of c.m. excitations in a given state, for the relevant cases (i.e. the O:, 2:, 2:, 4: states). It is seen that the values of these quantities for the 4: states are by two orders of magnitude larger than those for the 0: and 2: (i = 1,2) states, irrespective of the fact that the former are still very small (-10m4). It remains to be investigated to what extent such an admixture of c.m. excitations destroys the structure of a given state. When looking at the spectra displayed in fig. 1, we cannot say that the energies of the 4+ states are reproduced qualitatively worse than those of the 2+ states. However, the electron scattering form factors are known to be more sensitive to the actual form of the wave function than the energy spectrum, and therefore, even a

R. Kuchta

106

f

b a)

10”

2.94

\o

/ Electron

b)

9

,(j2

%

“= ,(j2

\

cr

,/ko‘*,,

I

:$i3

%

b-

24Mg

%1/ a\\

ELASTIC

t

scattering

'$0-3

,,b-

-

16”

Iti ,

d

0.5

1.0

1.5 2.0 slfm-ll

0.5

2.5

IO

d)

o;-2;

‘P

:+ 4 in

-\

‘\

I/

If

‘\\

1.5 2.0 qlfm-‘1

25

24Mg 0;-4;

Ii”.

0.5

1.0

1.5

2.0

25

0.5

1.0

4[fm-‘I

1.5

2.0

2!

qlfm-‘I

Fig. 6. Electron scattering form factors of a4Mg calculated with the MF boson (solid curves) and with the sd-shell model (dashed curves) wave functions. In (a) is the elastic (0: + 0:) form factor, in (b), (c) and (d) are the inelastic form factors for the excitation of the 2:, 2: and 4: states, respectively. Experimental data are taken from ref. 29).

TABLE 2 Expectation

values

of the c.m. quanta counting operator states of *‘Ne and *4Mg

(39) in some I&‘, T = 0)

3f

0:

2+ 1

2’*

4:

‘“Ne *4Mg

0.000 001 0.000 001

0.000 001 0.000 000

0.000 002 0.000 001

0.000 135 0.000 172

R. Kuchta

small

admixture

of the

c.m.

/ Eleciron

excitation

107

scaftering

may

have

large

influence

on the

final

result. We have between

thus

suggested

the experimental

two possible and calculated

sources

of the discrepancies

form factors

for the excitation

observed of the 4:

states in *‘Ne and 24Mg. While the implementation of the first point (i.e. projection before variation) is rather straightforward (though laborious), an answer to the question of how to remove the c.m. excitations from the physical states is still far from clear. Of course, we could resort to an exact treatment in a restricted singleparticle space and use the special techniques for isolating the c.m. excitations, which are available in this case 27). However, the main advantage of our approximate method, namely the possibility of including a larger single-particle space without serious difficulties would then be lost. We are therefore currently engaged in an effort to develop a sound and reliable method for dealing with this problem. 4. Summary In this paper we have studied the electron scattering form factors in some sd-shell nuclei using a second quantized boson representation of the relevant operators. Starting from the BZM mapping procedure, an exact boson image of the microscopic shell-model framework

hamiltonian has been obtained and then solved approximately in the of the MF approach and the equations-of-motion method. The resulting

boson states were shown to be free from spurious components due to the violation of the Pauli principle in the boson space. The angular momentum and isospin eigenstates were projected out of them and used to calculating the electron scattering form factors, both for the elastic scattering (0: + 0:) and for the excitation of the 2t(i = 1,2) and 4: states. The results obtained in this way suggest the potential power and utility of the present boson approach for describing such complicated and model-sensitive physical quantities as the electron scattering form factors. At the same time, however, they indicate the urgent need for a careful and detailed treatment of the effects associated with the symmetry restoration and the interplay between the intrinsic and c.m. excitations. In this article, we have restricted our discussion to nuclei with an equal number of protons and neutrons. An extension of the formalism to nuclei with nrr Z nV, together with an application to “0, is currently under way and some preliminary results have been reported in ref. 30). The author

thanks

Dr. M. Gmitro

for his stimulating

interest

in this work.

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