Electromagnetic form factors of3He and3H

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Electromagnetic form factors of 3He and 3H

S S M E H D I a n d V K G U P T A *

Department of Physics, Jamia Millia Islamia, Jamia Nagar, New Delhi, India.

*Department of Physics & Astrophysics, University of Delhi, Delhi 110 007, India.

MS received 28 February 1980

Abstract. The electric and magnetic form factors of 3He and SH are calculated with 3-nucleon wave functions obtained from the solution of Schrt~dinger equation with separable potentials of two different shapes which have already been employed in the coulomb energy calculation. The effect of important meson exchange corrections is evaluated and their dependence on the wave function studied. The form factors can depend rather sensitively on the nucleon form factors as well, and this dependence is studied by using two different parametrisations for the latter.

Keywords. Electromagnetic form factors; meson exchange currents; separable potentials; nucleon form factors.

1. In~oduefion

In the last few years interest in the electromagnetic f o r m factors o f SH a n d 3He has sustained mainly because o f the m i n i m u m in the SHe c h a r g e f o r m factor at a b o u t 11.6 f m -~ ( M c C a r t h y et al 1970; 1977). The occurrence o f the m i n i m u m suggests the existence o f strong correlations in the three-body wave function at short distances.

In the last decade a large n u m b e r o f 'exact' calculations o f the binding energy a n d the wave function o f a three-nucleon system have been carried out, t h a n k s to the availability o f large computers. The electromagnetic f o r m factors have been calculated in the impulse a p p r o x i m a t i o n (IA) with the ' e x a c t ' wave functions as well as with variational wave functions. While these calculations yielded a m i n i m u m in the 8He charge f o r m factor, a precise agreement between t h e o r y a n d experiment could n o t be obtained (Haftel a n d K l o e t 1977). T h e y m a d e a n exhaustive s t u d y o f Various corrections to the I A charge f o r m factors. T h e various m e s o n exchange current corrections turned o u t to be the most i m p o r t a n t ones. Calculation o f the exchange current contribution (ECC) to the magnetic f o r m factors have also been done, the m o s t recent being by Hadjimichael (1978).

An alternative often e m p l o y e d in m a n y three-body a n d o t h e r f e w - b o d y calculations because o f its simplifying character is the use o f separable potentials with excellent results f o r the binding energy a n d doublet n-d scattering length (Mitra 1969; Schrenk et al 1970). A n earlier investigation o f the electromagnetic radii a n d f o r m factors at low m o m e n t u m transfers also gave encouraging results ( G u p t a et al 1967). H o w - ever, when t a k e n to higher m o m e n t u m transfers the results (Mehdi a n d G u p t a 1976) completely disagreed with the experimental findings--no m i n i m u m in the SHe charge f o r m f a c t o r was f o u n d even after the inclusion o f a repulsive t e r m in the singlet N - N potential. Calculations with a different potential shape ( s h a p e - I I o f M e h d i 425 P--1

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426 S S M e h d i and V K Gupta

and Gupta (1976), which is one of a class of potentials proposed by Kharchenko et al (1968), though gave a vast improvement in the overall agreement with experiment in binding energy, various radii and form factors, as well as in the 3He coulomb energy (Mehdi and Gupta 1979), failed to produce the elusive minimum. The inability to produce the required, minimum appears to be a basic drawback of the separable potential model with soft core repulsion. This observation is further supported by the calculation of Haftel and Kloet (1977) in which the only potential that does not yield a minimum, even after the inclusion of repulsion in the singlet and tensor force in the triplet state, is separable potential.

Because of their extreme simplicity, separable potentials still serve as a useful model for many complex few-body systems. For a simple model calculation in- volving these complex systems, one would like to have a simple potential without the formidable complications of a tensor force term, and if possible, even a repulsive term. The shape-II potential seems to be a good candidate for this purpose-- much better than the commonly used shape-I potential. One would however like to see if a reasonable agreement for the electro-magnetic form factors can be obtained with shape-II potentials, once the ECC, the most dominant of the various corrections, is included in the study.

The form factors, and especially the exchange contribution to these, depend rather significantly on the nucleon form factors as well. This comes about because the expressions for the form factors involve terms with opposite signs. Thus, in particular the precise position of the minimum could depend sensitively on the nucleon form factors for which quite a few sets of data exist. It will be of interst to study this sensitivity.

In this paper we report our results on the 3-nucleon form factors obtained with the inclusion of ECC. There are many elementary diagrams that contribute to exchange currents. Of these we, in our calculation, have included I(a) and I(b) of Kloet and Tjon (I974), which make the dominant contributions. The sensitivity of the results to the nucleon form factors is studied by doing the calculation for two different parametrisations of the nucleon form factors (de Vries et al 1964; Janssens et al

1966) which give equally good fit to the electron scattering data.

2. Summary of the formalism

The general form of a rank two separable potential in the singlet state is - - M ( P l Vsl q ) = A~ [ s ( p ) s ( q ) - - h ( p ) h ( q ) ] ,

where h ( p ) provides the necessary repulsion. The traditional forms of s ( p ) and h (p), which we call shape-I, are

s ( p ) - (p~ + fl~)-l; h ( p ) = np 2 (p2 + ff~)-~.

For the triplet state we have, in the absence of tensor force, the single term potential

--M(Pl

vrl q ~ -= )~r c(p) c(q).

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T h e t r a d i t i o n a l f o r m o f c ( p ) , like t h a t o f s ( p ) , is c (p) = ( : +

S h a p e - I I o n t h e o t h e r h a n d c o r r e s p o n d s t o

w i t h s t r e n g t h p a r a m e t e r s c h a n g e d t o n e w v a l u e s ~ a n d ~T-

T h e p a r a m e t e r s o f t h e v a r i o u s sets o f singlet p o t e n t i a l s u s e d i n o u r p r e v i o u ~ s t u d y ( M e h d i a n d G u p t a 1976), a s well as in t h e p r e s e n t one, a n d t h e t w o - b o d y d a t a t o w h i c h t h e y a r e f i t t e d a r e l i s t e d in t a b l e s 1 a n d 2. T h e t r i p l e t p a r a m e t e r s (/~, h T o r /8,, )~T, a s t h e c a s e m a y be) a r e a d j u s t e d t o give the c o r r e c t d e u t e r o n b i n d i n g e n e r g y (2.22 M e V ) a n d s c a t t e r i n g l e n g t h (5.40 fm).

T h e i m p u l s e a p p r o x i m a t i o n e x p r e s s i o n f o r t h e v a r i o u s f o r m f a c t o r s a r e w e l l - k n o w n a n d h a v e b e e n d e r i v e d e a r l i e r ( G u p t a et a l 1967). T h e e x t e n s i o n t o t h e c a s e w h e n t h e p o t e n t i a l i n c l u d e s a r e p u l s i v e t e r m is s t r a i g h t f o r w a r d . T h e c a l c u l a t i o n r e d u c e s t o t h e e v a l u a t i o n o f c e r t a i n m u l t i p l e i n t e g r a l s i n v o l v i n g t h e 3 - b o d y w a v e f u n c t i o n w h i c h is k n o w n f r o m t h e s o l u t i o n o f e n e r g y e i g e n - v a i u e p r o b l e m .

Table 1. The potential parameters fls, ;~s,/3h and n for various potential sets of shape-I and the two-body parameters as and r0m (in Fermis) to which the former are fitted.

a is the deuteron binding energy parameter. S and H represent, respectively, the attractive and repulsive parts of the 1S0 potential and C eft the 'effective' central part of aS1 potential. For meaning of suffixes, Y, N, G~ see Mitra (1969) and Schrenk et al (1970). The triplet parameters, which correspond to at=5"378 fm andr0t = 1"716 fin, are the same for sets I, III and IV (flc=6.255a, A~.=33"36aa). F o r set II these parameters are/3c=5.8a , ~T=22.9a s.

Set Potential /~/a Asia 3 ~hla n --as ros

I ¢~ff + S y 6-255 23.4 - - - - 23.7 2.151

II CN + Shr 5-8 18"6 - - - - 23.7 2-33

llI C~y ff + ( S + H ) N 8.0 62.4 8.0 2.333 23.7 2.355

IV COy ff + (S+H)6'~ 5.7 18.95 7-8 2.74 18.0 2.70

Table 2. The potential parameters [3s, As, flh and ~ for various potential sets of shape-II and the two-body parameters as and r0s (in Fermis) to which the former are fitted.

C, S, H a n d a have the same meaning as in table 1. The triplet parameters (~c:9"0a, AT= 1-083 × 10 +6 a 7) are the same for all the potential sets and correspond to at = 5.405 fro, rot = t.76 fm.

Set Potential ~ / a ~s x lO-Va 7 /~h/a n --a z ros

I" CMO + SMOx 7"411 1"803 - - - - 18"0 2"7 II' CM~ + SM~s 7"16 1"412 - - - - 18"0 2"8

I I I ' CMG + (S+H)MO t 7"70 2.464 12"0 0.0953 18.0 2.7

I V ' CMG + ( S + H ) M o 2 7"40 1"839 10"0 0"0455 18"0 2"8 V" CM~ + ( S + H ) M o 3 7.60 2"260 8"5 0"0320 18"0 2"8

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428 S S M e h d i and V K Gupta

The 3-body wave function, in the absence of D-state, consists of a totally space- symmetric S-state, with a small admixture of S'-state (~-~1 ~o). Thus for the purpose of evaluating the correction due to exchange effects one can safely approximate it by a pure S-state. In that case, following Kloet and Tjon (1971; 1974), the evaluation of meson exchange corrections to the various form factors also becomes fairly straightforward and reduces to the evaluation of just two multiple integrals.

The 3-nucleon electromagnetic form factors are the products of the ' body form factors' with the electromagnetic form factors of the nucleons. The nucleon form factors are generally parametrised in terms of important pole contributions and many such parametrisations exist (de Vries et al 1964; Janssens et al 1966), which give equally good fit to the electron scattering data, especially at low momentum transfers. To study the dependence of the 3-nucleon form factors on the nucleon form factors the calculation has been done for both these parametrisations of the latter.

3. Results and discussion

The form factors in the impulse approximation both for shape-I and shape-II are shown in figures l(a)-(d). The effect of the ECC as well as the nucleon form factors are depicted in figures 2 and 3. The IA results for the form factors (and for the radii as well) were discussed in Mehdi and Gupta (1976) and are included here for the sake of completeness. The main features to note are a marked overall improvement with shape-II potential sets (primed) compared to shape-I potential sets (un- primed) and a reduction in the effect of repulsive core with shape-II compared to its effect with shape-I. This wide difference between the two shapes can be understood from the effect of the shape on the wave function. To illustrate the point we show in figure 4 the deuteron wave function as obtained with the two shapes. The wave function near the origin is much smaller with shape-II than with the other. Thus with shape-II potential any two nucleons in 3H or 3He as well, tend to stay away from each other, thereby increasing the three-body radii and depressing the form factors. Since the wave function with shape-II is already quite hollowed out, the effect of introducing a repulsive core is much smaller than with shape-I, thus explain- ing both features mentioned above.

As for the meson exchange currents, first of all we observe that their effect on magnetic form factors is almost negligible both for 8He and SH. The oscillation in the sign of the ECC has no dynamical significance but is due to the definition; the quantity /~Fmag, where /~ is the static magnetic moment (which is also affected by exchange currents), decreases in a uniform manner on the addition of these effects.

The effect of exchange currents on the charge form factors is quite different for SH and SHe. Whereas for the former it remains small (starting of course with zero at zero momentum transfer), for the latter it is much more appreciable and is maxi- mum at ~ 4 fm -z after which it decreases slowly. Since the form factor itself falls quite rapidly, the relative effect of exchange currents keeps on increasing with increasing momentum transfer.

In figure 2 is also shown the effect of the nucleon charge from factor on the charge form factor of SHe. As has already been mentioned, all the 3-nucleon form factors depend upon the values of the nucleon form factors. Of these, however, the SHe

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100 16 2

t M ¢ -

l d 4

156 100

IcT 2

~E 0 Lt_

161

0

~ , , 3He Charge form factor (a)

_ %

, .",,%

- . : , I v

I'

-- V f "-111'

t I I t I _

N \ 5He

Magnetic form factor (c)

k :Jl

__ " ; * ~ I V

-

...//7::'

-- IIl

& IV'

I I L _ I I

4 8 ~2 /6 20

~ 3H Charge form factor (b)

"-..

_

V'/~'~-~ Ill & IV

I I I I I

' ~ , , ,

5H

Magnetic form fodor

^(d⁾

-- "" " " < ' - . / I I

:'~IV'

) I I ,~ I

0 4 8 12 16 20

[<2 ( fm-~ )

Figure 1. Charge and magnetic form factors of SH and 3He in the impulse approximation for the various potential sets of tables 1 and 2. Solid and dashed curves represent the form factors for potentials with and without the repulsive term, respectively. Data points are from McCarthy et al 1970.

charge f o r m factor which is the difference o f I A a n d E C C terms, b o t h being o f the same order, is the m o s t sensitive one. Indeed, whereas the Janssens et al p a r a - metrisation is able to p r o d u c e a change in the sign o f the f o r m factor, the de Vries et al p a r a m e t r i s a t i o n is not (the thick and thin lines respectively in figure 2). T h e effect on the other f o r m factors (not shown in the figure) is comparatively small.

The dependence o f E C C on the wave function can be seen f r o m curves I - I I / in figure 2. T h e higher the f o r m factor in the IA, the m o r e is the depression b r o u g h t a b o u t by the inclusion o f ECC. Thus the E C C tends to reduce the differences in the f o r m factor due to different 3-body wave functions. In particular, the effect of a soft core t e r m which is already m u c h less for s h a p e - I I than for shape-I potentials, is further reduced when exchange currents are included, so m u c h so that for I A q - E C C f o r m factors are nearly the same with or without the repulsive core. The effect o f the tensor force h o w e v e r m a y not follow the same p a t t e r n because o f the introduction of D-state.

In conclusion, wherever the tensor force effects can be neglected the s h a p e - I I potential gives a better fit to the three-body data including the f o r m factors a n d provides a m o r e a p p r o x i m a t e wave function for use in other calculations t h a n shape-/.

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10 0 ]He Form factors lo-1

lo-Z

v

(+ve) i(-ve) • ~.. (+ re)

i l ... "-... !

,, ,

"\i

t:

(-~e)

~o-SI

, I , I I I I 0 4 8 12 16 20 K2(fm -2) Figure 2. The exchange current contribution for SHef orm factors. The three dashde lines (I-III) represent the ECC to charge form factor for potential sets V', II' and I, respectively. The dotted line represents the ECC for magnetic form factor. The solid lines represent the IA+ECC to charge form factor for the de Vries et al (1964) and Janssens et al (1966) parametrisation to the nucleon form factors.

10 ° 10-1 -- -2 10

lo-31-! .--"

/ ! .~'"~'"-..(÷ve) .,. ... --

/ i-/ .... .. i (-~) ...

~641-/!! • i

[/

^'~"

!" ""

16 s I I I I I

0 4 8 12 16 20 K2,(.fm -2) Figure 3. The ECC to the magnetic form factor (dotted line), charge form factor (dashed line) and IA+ECC to the charge form factor (solid line)for SH.

4~. ¢..,0

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0 8 u(r)

0.4 1.2

o I t 1 I I

0.8 1.6 2.4 3"2 4"0

r(frn)

Figure 4. The deuteron wave function for the two potential shapes, I and II, with fl=6.255a for shape-I and fl=9.0a for shape-II.

Also, the r e p u l s i v e c o r e h a s a m u c h less crucial role to p l a y a n d the wave f u n c t i o n o b t a i n e d w i t h o u t its i n c l u s i o n is n e a r l y as a c c u r a t e as the o n e w i t h it. F o r f i n e r a g r e e m e n t o n t h e p o s i t i o n o f the m i n i m u m a n d the s e c o n d a r y m a x i m u m , a p a r t f r o m t h e effects o f e x c h a n g e c u r r e n t s a n d t h r e e - b o d y forces, etc., the n u c l e o n f o r m f a c t o r s also p l a y a n i m p o r t a n t role.

References

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Gupta V K, Bhakar B S and Mitra A N 1967Phys. Rev. 153 1114

Hadjimichael E 1978 A study of some finer properties of 3H and 3He with non-variational wave functions Nucl. Phys. A294 513

Haftel M I and Kloet W M 1977 Phys. Rev. C15 404 Janssens T et al 1966 Phys. Rev. 142 922

Kharchenko V F, Petrov N M and Storozhenko S A 1963 Nucl. Phys. AI06 464 Kloet W M and Tjon J A 1971 Nucl. Phys. A176 481

Kloet W M and Tjon J A 1974 Phys. Lett. B49 419 McCarthy J S 1970 Phys. Rev. Lett. 25 884

McCarthy J S et al 1977 Phys. Rev. C15 1396

Mehdi S S and Gupta V K 1976 Few body dynamics (eds) A N Mitra et al (Amsterdam: North Holland Pub. Co.)

Mehdi S S and Gupta V K 1979 Pramgma 13 667

Mitra A N 1969 Advances in nuclear physics (eds) M Baranger and E Vogt (New York: Plenum Press)

Shrenk G L, Gupta V K and Mitra A N 1970 Phys. Rev. CI 895