LETTERE AL NUOVO CIMENTO VOL. II, N. 14 11 Novembre 1969

Hartree-Fock Charge Distribution and the Form Factor of 4He.

R. S. KAUSHAL a n d Y. R . WAGHMARE

Physics Department, Indian Institute o] Technology - ~Kanp~r

(ricevuto il 1 ° Settembre 1969)

Recently, the Hartrec-Foek (HF) theory along with an effective interaction which is obtained as a result of uni tary transformation, has provided satisfactory results for a large number of nuclei (1-3). The calculations based on this approach are further extended for fight nuclei (A < 16) after including the corrections due to Coulomb effects and centre-of-mass motion in a self-consistent manner (4). The binding energies so obtained are in fairly good agreement with experiments. On the other hand, as a test of the wave functions the calculated charge densities are compared (3) with the electron scattering data for a few light nuclei such as 4He, 3Be, lzC, l°0.

The recent experimental results of FROSCH et al. (6) at large momentum transfers have shown the failure of the Gaussian model for ~He. In order to explain these results, there have been several attempts (~) by modifying the oscillator charge distribution (3) for short-range dynamical correlations. These various attempts differ either in the choice of the correlation function or in the inclusion of the antisymmetrization in the wave function of the alpha-particle. The form of the charge distribution used by CHOU (6) consists of the difference of two Gaussiau shapes with different parame- ters. JAIN (lo)has compared the charge density obtained from the Irving wave functions with these experiments. From these analyses, it seems that the electron- scattering results for large momentum transfers provide an effective tool to test the theories dealing with nucleon-nucleon correlations. The purpose of this note is to c o m -

(1) C. ~ . $HA~I~ r, Y. R. WA(~HMARE and M. H. HULL: Phys. Rev., i6 t , 1006 (1967). (z) C. 1~. SHAKIN, Y. R. WAGHY.ARE, 1~. TO~C-aSELLI and M. H. HULL: Phys. Rev., 161,1015 (1967). (s) M. R. GUN~: Nucl. Phys., A l l8 , 174 (1968). (~) R. M. SINaRU, I. KAKKAR, R. S. KAUSHAL and Y. R. WAGH~RE: I IT/K. Tech. l~eport No.

Phys. 5/68 (1968), to be published. (5) R. S. KAUSHAL: PrOC. 01 Nucl. Phys. Symposium (Madras, 1968), p. 210. (e) R. F. FROSCH, J. S. McC~LRTHr, 1~. E. R~LND and M. 1~. YEARIAN: Phys. 1~¢v., i60, 874

(1967). (7) W. CZY2 and L. LESNIAK: Phys. Left., 25B, 219 (1967); Y. C. TANO and R. C. HERNDON-"

Phys. Left., 25 B, 307 (1967); A. MA~bECKI and P. PICCHI: Phys. Rev. Left., 21, 1395 (1968). (s) R. HOFSTADTER: ~Rn. Re~. Nucl. Sci., 7, 231 (1957). (') T. T. CHOU: Phys. Rev., 168, 1574 (1968).

(lo) S. C. JAIN: Nuovo Cimento, 57B, 135 (1968).

639

640 R . S . KAITSHAL and Y. R. ~VAGH3IARE

pare the H F charge densi ty wi th the results of FRoscH et al. (~) as well as wi th o~ner (7.~o) theoret ical results. We also use a simple model of charge d is t r ibut ion given by

doo(r) (1) £(r) = Oo(r ) ~- er ,

dr

where o0(r ) = ~0 exp [--r2/b2], the usual Gaussian iorm, and e is a parameter . The choice of this d is t r ibut ion is based on the cr i ter ion that , for 4He being a spin-zero nucleus, at large values of m o m e n t u m t ransfer the second: t e rm would provide the monopole vibra- t ions (u). Now for ~o0(r ) as a Gaussian form, (1) gives a charge dis t r ibut ion which is sinfilar in na ture to the dis t r ibut ion of p-shell nuclei (8).

0nee we know the charge densi ty o(r), the form factor if(q) can be calculated f rom

(2) 4~

/:"(q) = q J .'2(") r sin (qr ) d r ,

0

where q is t i le m o m e n t u m t ransferred by the electron to the ta rge t nucleus. Wi th a Gaussian form, (I) leads to the form factor

sq2 b 2 ] F(q) = 1 + 2 (1 - -3e ) J exp[--q~b2/4]

and to the root mean square (r.m.s.) charge radius

5e) [ 2 ( 1 - - 3e)

The charge distr ibut ion of the 4He nucleus obtained from H F theory is given in Fig. 1.

0.08 ~ . \

0.06

'E 0"0~

r..

O.,

0102

r \

1.0 2.0 3.0 r ( f m )

F i g . 1. - C h a r g e d e n s i t i e s of 4 t ie . T h e so l i d l i n e r e p r e s e n t s t h e H F r e s u l t s ; d a s h e d a n d d o t - d a s h e d l i ne s r e p r e s e n t t h e r e s u l t s of TAI,;G o~nd I-IERNDOX (~) c o r r e s p o n d i n g to c o r r e l a t i o n lengths of 0.6 fm and 0.5 fm respectively. Ex- perimental points are from Flaoscll el al. U).

For the details of this theory and tlle calculat ional procedure we refer to ti le previous work (1,2). I t can be not iced tha t H F charge dis t r ibut ion has a somewhat longer ta i l compared to the exper imenta l results. The form factor calculated nmner iea l ly f rom eq. (2) is compared wi th exper iments in Fig. 2. The agreement be tween H F results and the exper imenta l values is excel lent for q2< 10 fin -~, however , i t is somewhat poor for large q2. H F results give a diffraction min imum at q ° - = l l fm -z against the exper imenta l results at q-~ = l0 (fro)-% The results of Cz¥2 and LESNIAK (7) Ycho have calculated the F(q) wi th lowest-order correct ion to the uncorre la ted ground-s ta te charge densi ty are also shown in this Figure. F r o m this Figm'e it can be not iced that , though the agreement of H F results wi th exper iments is a lmost as good as tha t of CzY~ and LES- xIAK, i t is be t t e r than tha t of TANG and HERNDO~-(7), whose results are g iven for a J a s t row correlat ion length of 0.6 fm.

(11) R . ~)~AI~t:L&EL, H. ~BERALL a n d C. WERXTZ: Phys. l?ev., 152, 899 (1966).

H A R T R E E - F O C K C H A R G E D I S T R I B U T I O N A N D T H E F O R M F A C T O R OF 4 H e 641

For a Gaussian form of Co(r), the distribution (1) is shown in Fig. 3 for two sets of parameters (~=- -0 .23 , b = 1.22 fm and e = - - 0 . 2 4 , b = 1.18 fm) and the correspond- ing form factor is given in Fig. 2. In comparison with HF results, the agreement of

10

O- LL

10 - t

10 - z

\ \

\

\

l r i 10"-30 ,4 8 28

12 16 20 24

q Z ( f m - 2 )

F i g . 2. - F o r m f a o t o r o f 4He. T h e s o l i d c u r v e r e p r e s e n t s t h e H F r e s u l t s , t h e d a s h e d c u r v e r e p r e s e n t s the r e s u l t s o f T&I~G a n d lcFERlqDO~ (~) f o r a c o r r e l a t i o n l e n g t h of 0 .6 f ro , t h e d o t t e d c u r v e r e p r e s e n t s the r e s u l t s o f C z ~ a n d LESlCIAK (7) a n d t h o d o t - d a s h e d c u r v e showS t h e r e s u l t s o f t h e s i m p l e m o d e l

f o r 8 = -- 0 .24 , b = 1 .18 f m . E x p e r i m e n t a l p o i n t s o f FROSCB[ et a l . (e) a r e s h o w n w i t h e r r o r s .

these results with experiments is somewhat poor particularly in t he region of large q~. The values of the parameters obtained by fitting the HF charge distribution with this

we summarize the results of various model are e ---- - - 0.22, b = 1.20 fm. Fur ther authors in Table I.

For the HF results it can be noted that the half-charge density radius c is somewhat smaller and the skin thickness t as well as the r.m.s, radius are somewhat larger com- pared to the values obtained from the ex- periments (cf. Table I). In addition to this,

F i g . 3. - C h a r g e d e n s i t y of 4He o b t a i n e d f r o m t h e s i m - p l e m o d e l ( e q . (1)) . R e s u l t s a r e s h o w n f o r t w o s e t s o f p a r a m e t e r s . C u r v e s d e n o t e d b y 1) a n d 2) cor- r e s p o n d t o t h e f i r s t a n d t h e s e c o n d t e r m s of (1) f o r * = -- 0 .23 , b = 1.22 f ro . E x p e r i m e n t a l p o i n t s a r e f r o m FROSCH et al. (e). _ _ - - , s = - - 0 . 2 4 , b = l . 1 8 f m .

0.06

I E 0.04 v

%o.o2

X ,

1.0 2.0 3.0 rCfm)

642 n . s . K A U S H A L a n d Y. 1~. VCAGHMARE

TABLE I . -- Comparison o] the results obtained ]rom present calculations with those o] other authors.

Various methods

Present calculat ions (HF results) Present calculat ions (simple model :

e = - - 0 . 2 3 , b = 1.22fm) Present calculat ions (simple model :

ref. (fm)

1.778

1 . 6 8 4

c t ! i

(fro) ! (fro) i ~%h(O)]2e

i i 1 . 1 8 ! 1 . 7 6 ~ 0 . 0 6 3 6

1.28 i 1.45 0.0587

e - -0 .24 , b = 1.18fro) I rv ing wave funct ions I t a rd core (~) (r~ = 0.5 fro) Hard core (a) (re = 0.6 fro) Gaussian model Expe r imen ta l

(a) TANG a n d HERNDON.

- - 1.635 1.26 (lO) 1.73 1.08 (~) 1.67 1.13 (~) 1.73 1.20

1.68 1.14 1 (.) (6) 1 . 6 7 1 . 3 2 i

1.6 0.063 1.58 0.0767 1.55 0.076 1.56 0.065 1.64 0.0698 1.45 0.0595

H F results show a l i t t le shift of l (fin) -2 of tile diffraction min imum towards large q2 (ef. Fi-'. 2). These discrepancies are not surprisin~ but unders tandable f rom the fol- lowing: The wave funct ions used to calculate ti le charge densi ty are ob ta ined for sI)herical l t a r t ree -Foek . In this case the correlated wave funct ion is expanded in terms of harmonic-osci l la tor basis (el. eq. (12) of ref. (2)) and the smnmat ion is res t r ic ted to the radial quan tum nmnber n. alone (for fixed 1). However , different unoccupied l-orbitals may also cont r ibute and consequent ly improve the results. In o ther words, t he skin par t of the charge dis t r ibut ion is not proper ly t rea ted and is s t re tched towards the tai l side in these calculations. Thus, the reason fin' smaller values of c seems obvious. On the o ther hand. we are includinm' the effects of far separated n-orbitals. This probably causes the hmg tail in the t i p charge dis t r ibut ion and consequent ly increases the r.m.s, radius in comparison wi th the observed vahle.

Though, the simple model (1) does not reproduce the exper imenta l values of /;(q) for large q2 (cf. Fig. 2). the o ther calculated quant i t ies show a nice agreement wi th exper iments (of. Table I). The results obta ined wi th this s imple model are much be t te r than the results of dalX (,o) who has used the h 'v ing wave funct ions to calculate the charge density. F rom Fig. 2 it can be seen tha t the overal l behav iour of the ~'(q) obta ined wi th this model is reasonable in comparison to the results of T~t~G and HERNDON (7) and CzY~ and I,ESNIAK (v), whose results car ry the effects of short-r~nge correlations. Besides, this simple model of 4He assumes a Gaussian form for ~o0(r ). For spherical nuclei heavier than 4He, the va l id i ty of this model wi th corresponding oscil- la tor d is t r ibut ion for ~)0(r) seems cneoura~zing. This is beeausc of the fact tha t the second term in (1) not only enhances the uni form p~rt in the central region, bu t ~lso gives rise to a dip or bump (dependino" upon the sign of e) at the centre (cf. Fig. 3). On the o ther band, the small f la t tening of the ehargc d is t r ibut ion near the centre is sufficient to reproduce a diffraction minimuin in the fornl factor.

One of the authors (R.S.K.) gra teful ly acknowledges the financial help of C.S.I .R. , New Delhi, India.