Pion exchange currents and nuclear charge form factors

Nuduar PJrysicr A274 (1976) 428-442 ; © Nords-HoflandPuNirhhp Co., AnutwdamNot to be reproduced by photoprint or microfilm without wrhtm permission from the publisher

PION EXCHANGE CURRENTSAND NUCLEAR CHARGE FORM FACTORS t

MARK RADOMSKI and D . O . RISKADepartment ofPhysics and Cyclotron Laboratory, Michigan State University, East Lansing,

Michigan 48824

Received 9 April 1976

Abetrset : The effect ofthe pion exchange current due to the Born term in the photoproduction amplitudeon the charge form factors and distributions of'He, ` 60and"Caiscalculated in the simpleharmonicoscillator shell model . The exchange current contribution ismost notable in the case ofthe a-particlewhile it is rather unimportant in ' 60 and "Ca . The contribution to the nuclear mean square radiusis small (z 0.03 fin') and roughly independent of the mass number.

1. IntroductionIt has recently been noted that pion exchange currents have a large effect on the

charge form factors of the lightest nuclei (A S 4) t-s) at relatively small values ofmomentum transfer (~_ 3 fm- ') . This observation raises several questions, as forexample how well one can quantitatively calculate these exchange current contribu-tions and how important such effects are in the case of heavier nuclei . The first ofthese two questions will eventually be decided by comparing experimental data forthe charge form factors with the predictions obtained with the present models forthe exchange current operators in conjunction with realistic wave functions for thefew-nucleon bound states . We note in passing that a definite theoretical analysis ispresently hindered by the absence of data on the deuteron charge form factor . Thesecond question above is relatively simpler to answer, at least preliminarily, byexplicit calculation of the exchange current contributions to the charge form factorsof heavier nuclei . In this paper we present the results of such calculations .The pion exchange current contributions to the charge form factors of the doubly

closed shell nuclei 'He, 160and 4°Ca are calculated using simple harmonic oscillatorwave functions . The results indicate that, while the exchange currents are relativelyunimportant at lowmomentum transfer, their effects rapidly increase with momen-tum transfer so as to give rise to corrections of the order 25 %at 10 fm -Z . In the caseof the a-particle, the effect ofthe pion exchange current considered here is even moreimportant, in agreement with the findings of Borysowicz and Risks').

f Research supported by the National Science Foundation .428

We find that the exchange currents give rise to a small positive contribution to thenuclear mean square radius which is roughly independent of nuclear mass numberand of the order 0.03 fm' . Since this correction to the mean square radius is muchsmaller than e.g . the correction due to the finite nucleon size it is oflittle significance .The effect on the nuclear charge densities due to the exchange current is most im-portant in the central region of the nuclei . In the case ofthe a-particle the calculatedcentral density is reduced by 7% because of the exchange current.

This paper falls into five sections, the first of which is this introduction . In thesecond section we define the exchange current operator and find expressions forits matrix elements for doubly closed shell nuclei . In sect . 3 we present numericalresults for the charge form factors and densities for 'He, 160and 4°Ca. In the fourthsection we calculate the exchange current correction to the mean square radius usingthe Fermi gas model in order to find the dependence on the nuclear mass number.Finally sect . 5 contains a concluding discussion.

2. Exchange currents and charge form factorsThe pion exchange currents are caused by those parts of the photo- or electro-

production amplitudes for virtual pions which cannot be described by the single-nucleon currents and nuclear wave functions alone. The importance of taking into .account such exchange currents in the consideration of the magnetic transitionrates and form factors of the few nucleon systems has by now been well demon-strated 4-6). The question of the influence of pion exchange currents on nuclear .charge form factors is less certain since they can be expected to be of appreciablemagnitude only at relatively large values of momentum transfer where other uncer-tainties associated with the nuclear wave functions themselves and the neutron formfactors complicate the analysis. Nonetheless Kloet and Tjon 1) noted that the ex-change current operator associated with the isoscalar part of the Born term in thepion electroproduction amplitude off a nucleon gives rise to large corrections tothe charge form factors of 3H and 'He at moderately small values of momentumtransfer (34 fm -1). Similar large effects have since been predicted .for the chargeform factors of 2H and 'He [refs.The charge component ofthe two-nucleon pion exchange current in question has

the form

t We use unita wherein h = c = 1 .

PION EXCHANGE CURRENTS

429

2Jo(1, 2) =

33 GUg 2)& - qa2 - I ~Z -T22 +(1 +-> 2).

Here G. is the isoscalar ~nucleon magnetic form factor [G�(0) = 0.88], g thepseudoscalar pion-nucleon coupling constant (g2/4n x 14.5), m the nucleon andp the pion mass t. The spin matrices of the two nucleons involved are denoted at,

430

M. RADOMSKI AND D. O . RISKA

a2 and the isospin matrices st, s2 . In (2.1) q is the spatial momentum transfer to thetwo nucleons and I the momentum transferred from nucleon 1 to nucleon 2 by theexchanged pion.

��,f______

Fig. 1 . Diagrammatic representation of pion exchange current in (a) the pseudoscalar coupling modeland (b) the pseudovector coupling model .

The form of the exchange current operator (2.1) is the lowest order relativisticcorrection to the adiabatic limit of the pion photoproduction current. In the pseudo-scalar pion-nucleon coupling model this expression is obtained by considering theadiabatic limit of the "pair-current" diagram in fig . la. In the pseudovector pion-nucleon coupling model the pair diagram gives no contribution and the exchangecurrent operator arises as a relativistic correction to the part ofthe pion photoproduc-tion amplitude with a positive energy nucleon intermediate state t. In either casewe may represent the current operator (2.1) by the point-interaction type diagramin fig . l b.Another pion exchange current ofthe same order in g, called pion "recoil" current

in ref. e), was considered in refs . "). The model for that current is obtained in acoupled channel development ofthe nuclear interaction but lacks a clear interpreta-tion when phenomenological wave functions are used . We shall not consider thatexchange current here except for the observation that its effect is smaller than, andof opposite sign to, that of the pair culrent (2.1) .Other pion exchange currents that may contribute to the charge form factors are

associated with excitations of isospin4 pion-nucleon resonances in the intermediatestate. We assume that they are of much less importance than the one we considerbecause ofthe large masses of those resonances . Thepa exchange current wasshownin ref. 3) to be ofvery little significance for momentum transfer values S 4.5 fm-1 .The contribution to the charge form factor of a nucleus due to a two-body current

of the form (2.1) is

P"~~q2)

- A(2Z 1) fd3rt . . . d3rAgl +(r t . . . rA) fdi2- ir? f3 â

2 )s elk' ."elk''"

X(2,X)3 6(k, +ks-q)Jo(kt, k2)7(rl . . . rA) .

(2.2)

Here kt and k2 are the parts of the momentum transfer q delivered to nucleons 1

t The definition of the exchange current operator has been discussed by Friar 7 ) .


431

and 2 respectively . We consider in this section the harmonic oscillator model for thenuclear wave function T. For this model the c.m . part of the wave function factorsout as a Gaussian and may be replaced by a proper plane wave . This replacement isachieved, as in the case of the impulse approximation, by multiplying the formfactor (2.2) by the factor exp (q'l4Aa2), in which a is the radial scale parametermina in the oscillator wave function 9) .Upon substitution ofthe exchange current expression (2.1) into (2 .2), with account

of the c.m . motion correction, we obtain

P"'-"(q2) = i.1(g 2 ) 1 d3rl . . . d3rAT+ (r l . . . rA)ti . T261 - 4(r 2. P

Here we have used the notation

q2) = A(A-1) 92 _p2

q ZG

g2)e°2/4Aa28Z

4a qm (m)

'

Y,(x) =(1

+l~e-.x x

x Yj(pr)e 4

T(rl . . . rA) .

(2.3)

The vector r is defined as r2-r1 .To evaluate the matrix element in eq. (2.3) we use a Slater determinant of harmonic

oscillator single-particle wave functions as amodel for the nuclear wave function IF .For closed shell nuclei the spin-isospin matrix elements can be calculated separately,and are found to be 0 for the direct term and 12 ¢ " P for the exchange term . Forthe exchange term we thus obtain the results

F'h(g2) _ -12 i i.(g2)

<nvn214 " ~e`'""Yi(hr)1n2, n,>,A(A-1)

6 2NYfs,(r11Y' *2(r2»ss(rl»w,( r2) = aa

E AN(~a2r2)~(Za2R2Y`e-a2(2R

~"51 .02

n z#-0

(2.4)

(2.5)

where n stands for a complete set of orbital quantum numbers for single-particlestates, i.e . {nbn) or, {n,,n n,). The sum in (2.5) is to be taken over all occupied states .To carry out the sum over the closed shell occupations we need the quantity

(2.6)

Here R = Yr,+r2) is the c.m . coordinate, A,, are numerical coefficients and Nis the principal quantum number of the highest closed shell. The independence ofthis expression of the directions of r and R is notable and greatly facilitates theevaluation of IF" . This independence is a special feature of the closed-shell sum

432

M.

RADOMSKI AND D

.

O

.

RISKA

for

the exchange term in the harmonic oscillator model t

.

The coefficients Ax, for

N

5 2 ('He,

.

160 and "Ca) are given in table 1

.

The

coefficients A,,N occurring in the closed-shell sum of the exchange term for the two-body matrix

element

in the harmonic oscillator model (2

.6)

For

the closed-shell ground states under consideration here, expression (2

.6)provides

a more streamlined evaluation of the two-body matrix elements than does

the

more conventional method using the Brody-Moshinsky transformation to)

.

In

the

latter method, the full reflection-rotation symmetry of (2

.6)

is not exploited

.The

angular integrations of expression (2

.6)

are trivial and we find the final result

:

TABLE

1

and

Y1(x) = (1 +x)e-z/x

.

The confluent hypergeometric function 1F1 in (2

.8b)

is

a

polynomial of degree p in this case

.t

The overall rotational symmetry of the closed-shell configuration limits the functional dependence

of

(2

.6)

to the scalars r=, R2 and f - k

.

If the sum is written using a Cartesian basis with real factorized

wave

functions, a further symmetry becomes evident

:

that ofthe exchange of any of the Cartesian com-

ponents

of r, and r=

.

This amounts to symmetry under reflection of r (for fixed R) in any coordinate

plane.

These reflections can be combined with overall rotations to yield reflection of r in an arbitrary

plane.

No dependence on t - R is then possible

.

R=0 1 2 3 4

A1M(40ca)x=0 4i 0 0 i

I -10 -5 -2 -l 02 _1~ 4 a 0 03 -2 -1 0 0 04 4 0 0 0 0

A,1,(160)1=0 1 2 1

1 -2 -2 02 1 0 0

AO

r(4He)

A=0 l

2

2

F`

°h(92) 4 g

(ß)

q

Grr(9z

214Aa= ,q

.1

AN xp(q ,(qZ)~

(2 .7)Z

4n m m10-o

Here

we have used the definitions

°°Ix(92)

2

_

Jo«e-r= CJo (2) +J2 (i2)] Yt (~J 2+Za~

(2.8a)0

l

2IR(9s)=

fODdte-4j( 9

o

4t) Zs+2p _ I'(Fr+ ~-42/ea= t F t (_p

.1'8a2 (2.8b)o

o,2

as


433

Thecorrection to the nuclear mean square radius is readily obtained from eq . (2 .7)

~r2>eu>. =

24

g2

Gs O)1

,!KFz(0)I~(0) .

(2.9)Zm2 4n

m

7 lu -oFor reference we also give the expression for the charge form factor in the impulseapproximation

(F~P(4 2) =

2ZG g2

=/°Aa=

J drr2(2l+ 1)R,.Ar)JO(gr) .2et1~N

Here GE is the isoscalar nucleon charge form factor [GE(O) = 1] and the R�Xr) arethe normalized radial wave functions in the harmonic oscillator model.

It is interesting to study the relative importance of the exchange current and im-pulse approximationcontributions to the charge form factor at very large momentumtransfer . For this purpose we derive an approximate expression for F" valid inthe limit oflarge q. This is most conveniently done by rewriting the expression (2.8a)for the structure function Ix in the form

2

3a

f'Od

2z

g

d3x

Q . (x+Q)

_,n.~z(4) = UQ2J2 Jo

d

.e_J (2n)3 [(x+Q)2+2p2/a2]e

3(X3

Ê2~ 0[2)]3 2

A2_

2.r(I2) _ (A2+1

'

(2.10)

Here we have defined Q = q/(aJ2). If in (2 .11) the ~ integration is first performed,the result contains a Gaussian of the form exp (-4x 2) which limits SKI to smallvalues . It is therefore justifiable to take the limit Q -" oo inside the integrand toobtain

(2.12)

Note that the limit (2.12) renders F", eq. (2 .7), independent of the exchangedmassy.

If hadronic form factors, at the pion vertices in fig. la, are included in definingthe exchange current operator, the asymptotic behaviour is modified. A vertexform factor f(I2) will :result in multiplication of the integrand in eq . (2.11) byf2(}(x+Q)2a2) . Thus, for example, if

and il >- (ju, a), formula (2.12) is only valid in the range (�2, a2) -c q2 < A2,whereas one has an extra factor 4A2/q2 in the asymptotic range q2 >- A2 . Sincesuch hadronic form factors damp the short-range behavior of the exchange currentoperator, their effect is similar to that of short-range (r - 1/il) repulsive two-bodycorrelations in the nuclear wave function. Such correlations will, hence, also alterthe large-g2 behavior of F`=°h . Furthermore, diagrams other than fig. 1 will be im-

434

M. RADOMSKI AND D. O. RISKA

portant at large momentum transfer. The expression for F" which follows from(2.12) therefore lacks physical relevance as a q2 -" oo limit, but it is still interestingas an approximation for intermediate momentum transfer, (p2, a2) -c q2 C A2,

and for the light it throws on the mathematics of the present model.The truly asymptotic limit of the present model (2.7) for FD=°ti, is obtained by

replacing, I, with its leading term,

and Ix with eq . (2.12) to get

1R

a,ri- ~- 8a2 1~e-q2/@&2 / 1+O

C32q?a2/l

(2.13)

~~

2

,� - 3J2 92 Aó.2N a3 Gs

2

2/4Ad=

q 2

Ziv _q_/~_

(2 .14)v

b(4 ) _'D

Jn 47r Z m3 w(g

C8a2) e

which is valid for q2 >- 32N2a2 . Ifone uses the exactF [eq. (2.8b)] and the asymp-totic Ix [eq. (2.12)], the result still has an analytic form, but is valid in the lowerrange q2 >. 2 2, p2.

Since the exchange current contribution falls off as exp (-q2/8a2) for large q,whereas the impulse approximation falls off as exp (-q2/4a2), it is obvious that theexchange contribution will dominate over the impulse approximation contribution.The physical explanation is the sharing ofthe momentum q between the two nucleons( jq to each nucleon) so that the momenta involved in the structure functions aresmaller 2.11) . Indeed one finds in the limit of large q that, except for the c.m . correc-tion factor exp (q2/(4Aa2)),

F``~q2) ,., [F4m°(âg2)]2. (2.15)

This conclusion is modified, as discussed above, by hadronic form factors and short-range nuclear correlations .The nuclear charge distributions are obtained from the form factors F(q 2) as

P(r) Zn2 dgg2F(q2)Jo(qr).

(2.l6)0

In the numerical evaluation of the exchange current contribution to the chargedistribution p the numerical error caused by truncating the integral at a finite valueof q may be bounded by using the asymptotic limit (2.14) .

3. Numerical results for the form factorsWe have calculated the exchange current contributions to the charge form factors

and densities of 'He, 160 and "Ca numerically using the expression (2.7) . Whilethe integrals (2.8b) are obtained in analytic form 12) the integrals over the relativecoordinate (2.8a) must be performed numerically. These integrals were performedusing Simpson's rule with an upper cut-off ~ = (15 fm)a . The results ofthese calcula-


435

tions are shown in figs . 2-4. In the calculations the oscillator wave function parametera was adjusted so that the form factor in the impulse approximation alone andwhenthe exchange current contribution is added lead to the same value ofthe mean squareradius . The experimental rms radii are for 'He 1 .7 fm [ref. t 3)], for t60 2.73 fm[ref. ")] and for"Ca 3.49 fm [ref. ' s)] . For the nucleon form factors we have usedthe parametrization of Iachello, Jackson and Lande ' 6) .

In fig. 2 we have plotted the charge form factor for the a-particle as a function ofsquared (invariant) momentum transfer . The impulse approximation curve (IMP)was obtained with the value a = 0.697 fm -1 and the impulse + exchange currentresult using a = 0.702 fm-1 . The data are taken from ref. t s). The results show thatfor the a-particle the pion exchange current has a large effect already in the regionq2 S 15 fm-2 in agreement with the results obtained in ref. 3). It is striking that,even without the introduction of any short range correlations in the four-nucleonwave function model, one obtains a zero in F at 13.5 fm-2 . This finding supportsthe conclusion ofref. 3) that the minimum in the a-particle form factor is only partlydue to short range correlations in the wave function and partly due to the exchangecurrent effect . It is useful to re-emphasize the fact that the exchange current contribu-tion is less sensitive to the magnitude and form of the short range correlations than

.01F

.001

1q2/fmq

1

Fig. 2. Charge form factor of 4He. The curve IMP represents the impulse approximation result witha = 0.697 fm- ' and the curve IMP+EXCH the result when the pion exchange current is added and a

readjustedto 0.702 fm- ' to keep the same rms radius, l .7 fm . Thedata points are taken from ref. ") .

436

q2/fm-1Fig. 3 . Charge form factor of "0 . The notationis the same as in fig. 2. The data points areobtained from ref. ") . For both curves, a = 0.561

fm - ` .

M. RADOMSKI AND D. 0. RISKA

Fig. 4. Charge form factor of 4°Ca. The notationis the same as in fig. 2 . The "data" points areobtained in the Born approximation from thecross-section data in ref. "°) . The dashed curve isthe Fourier transform ofthe charge distributiongiven in ref. ") . For both theoretical curves,

a = 0.505 fm - ' .

is the impulse-approximation form factor 3) . The correction to the mean squareradius due to the exchange current was found to be <rz>°xch = 0.032 fmz for thea-particle (a = 0.697 fm- ').

In fig . 3 we have plotted the charge form factor o160. These results were obtainedusing a = 0.561 fm - ' . The form factor data are taken from ref. ") . One notes thatthe relative importance of the exchange current correction is smaller in the case of'60 than in the case of 'He, at small values of momentum transfer . In addition, atlarge values of momentum transfer (q Z 4 fm - '), the exchange current contribu-tion has the same sign as the impulse approximation contribution and consequentlyno additional minimum is introduced in the charge form factor . The addition to themean square radius for '60 due to the pion exchange current is 0.031 fine.The calculatéd form factors for "Ca are plotted in fig. 4. These results were

obtained with a = 0.505 fm- ' . Theform factor "data" were obtained from the cross-section data in ref. ts) using the Horn approximation expression . The dashed curverepresents the Fourier transform of the .charge distribution given in ref. t a) and

presumably gives a better description ofthe real form factor than theBomapproxima-tion results . It is noted that the effect ofthe pion exchange current is again relativelysmaller than in the case of the a-particle, but since at large momentum transfer theexchange current contribution has a sign opposite to that of the impulse approxima-tion an additional minimum in the form factor is produced at q2 x 16.5 fin -Z . Thecorrection to the squared rms radius due to the exchange current is 0.032 fm'.

In order to study the validity of the asymptotic limit formula (2.14), we havecalculated the form factors to q2 = 100 fm-2 . For "He the asymptotic expressionleads to results too large by 30 % at 20 frn-2 and by only 4 % at 100 fm-2 . For 160the overestimate given by the asymptotic expression is 60 % at 50 fm -2 and 25at 100 ftn -2 . In the case of "Ca the asymptotic formula still overestimates the formfactor by 50 % at 100 fm -2 .The asymptotic approximation is improved if the exact eq. (2.8b) is retained for

IN while using the approximate eq. (2.12) for Ix. Agreement with the numerical resultsis greatly extended in 160and 4°Caby this improvement. (There is exactly no changein 'He.) Again, one overestimates the exact numerical results, but by less than 75in both 160 and "Ca even at 10 frn-2 , and by 10-12 % at 20 fin-2 and only 3-4at 100 fm-2 . Zero crossings of F"cn, found at 3 .0 and 13.5 fm -2 in 160 and 1 .7and4.1 fm- 2 in "Ca, in the numerical results, appear nearby in theimproved large-q2


437

a 4 sr/fm

Fig. 5. The charge density for `He as calculated from the form factors in the impulse approximation(IMP) and with the inclusion of the pion exchange carrent (EXCH), normalized to unit total . charge. .

[a - 0.697 fm-1] .

438

M. RADOMSKI AND D. O . RISKA

approximation, at 3.2, 14.2, 2.2 and 4.4 fm-2 , respectively . This agreement is notsurprising inasmuch as the condition for validity of(2.12) is q2 >.l�

2, a2) S0.5 fin-2 .The exact numerical results for "He, '60 and "Ca show that inclusion of the

exchange current will not improve by much the results for the form factors obtainedwith the harmonic oscillator model - in fact only in the case of the a-particle is therereally a notable beneficial effect . Themagnitude ofthe exchange current contributionat large momentum transfer is, however, in no case negligible, although in largernuclei it appears that inclusion of the pion exchange current does not change thequalitative features of the form factor as calculated with harmonic oscillator wavefunctions .We have also calculated the charge distributions corresponding to the charge form

factors above by carrying out the Hankel transformation (2.15) numerically. Theintegral was performed using Simpson's rule with an upper limit q = 10 fm- ' .The asymptotic limit (2.13) for the exchange current contribution and the similarlimit for the impulse approximation result ensure that this truncation causes anegligible error The charge densities thus obtained are plotted in figs . 5-7. Thevalues for the oscillator parameter a were those used for the impulse plus exchangeresults quoted above. The exchange current contribution is most notable near thecenter ofthe nuclei and, as might have been expected on the basis ofthe form factor

Fig. 6. Thecharge density for' 60calculated from

Fig. 7. The charge density for 4°Ca calculatedthe form factors in fig. 3 . The notation is thesame

fromthe form factors in fig. 4. The notation is theas in fig. 5 .

same as in fig. 6 .


439

results, most evident in the case of 'He. For 'He the exchange current serves toreduce the central density by 7 %, while only by 2 % for "Ca. For "'0 the effect ofthe exchange current is to increase the central density by 2 %.The effect of two-body correlations should not change our results for P111(q 1)

appreciably. We may estimate such effects by inserting a factor (1-exp (- yr 2))INin the relative coordinate integrals Px (2.8a) . Here "y is a correlation length and Na normalization constant . Even if y is taken to be as small as 1 fm - 1,

<r2)e=ch isreduced by less than 30 % and P"' by less than 55 % at q2 = 20 fm-2 , the changebeing smooth over the range of momentum values from 0 to 20 fin-2 .

4, Exchange current contribution in the Fermi gas modelIn order to study the exchange current contribution to the nuclear mean square

radius as a function of nuclear mass number A we employ the simple Fermi gasmodel l'). In this model the single-particle wave functions are running waves nor-malized in a box of volume r) with periodic boundary conditions.The impulse approximation result for the charge form factor in the Fermi gas

model has the formF'r(g2) = Gsj~(g2)Llo(gRo)+J2(gRo) ],

(3.1)where Ro is the nuclear radius Ro = 1 .12 A} fm. This result is obtained by approx-imating the cubical Fermi gas nucleus by a spherical form.To calculate the matrix element of the exchange current operator one needs the

two-body density in spatial coordinates for the Fermi gas nucleus (withN = Z = JA)which is

P( 2)(rl, r2) = Pü)

(rl, r2)+PÉZ(rl+ r2)

A

9A

,ji(kFr) 1 +Q1 " 72 1 +s l

s2

- (A-1)fì2 - (A-1p2 (kFr) 2

2

2

,

2kFRo

2

f

dxj1

C1- 2kFRO]2

4kFR0] = 1 .0

(3 .2)

with r = r2-r1 and kF z 1 .36 fm -1 . The form of the second exchange part of thetwo-body density is obtained by approximating the sum over occupied momentumstates as an integral overmomenta l') . The normalization condition for the two-bodydensity requires

4rè2 f dsrldsr2 Ji(kF2) = 1,

(3.3)n (kFr)

the integrals being taken over the nuclear volume dì . By changing to relative andc.m . variables this normalization requirement can be cast into the simpler form

(3.4)

440


The polynomial factors multiplying ji arise from the restriction rl, r2 < Ro in(3.3) . Because of the change from the original cubic nuclear geometry to a sphericalone and the approximation used to evaluate the sum over allowed momenta as anintegral the l.h .s. of (3.4) will not equal 1 for smallA. To illustrate this deficiency wehave plotted the l.h .s . of (3.4) in fig . 8.

0.05

<r 2 ) = 3(92)2~

(1)2 s (O)4 47r m m

Fig. 8. The normalization of the exchange part of the two-body density in the Fermi gas model as afunction of A [l.h.s . of eq. (3 .3)] .

. fma <ra>46M

8

A

50

100 AFig. 9 . The pionexchange current correction tothe squared nuclear rms radius as afunction ofA . CurveAshows the result obtained using the Fermi gas model result [eq. (3 .5)] and curve B shows the result after

division by the normalization integral shown in fig. 8.

Having established the two-nucleon density in the Fermi gas model (3 .2) it becomesa straightforward task to calculate the contribution to the mean square radius causedby the two-body current (2.3) . The result is

2kpRa

2

.ii(x)Yi (k~1-2k R ]

Cl+ 4k

(3.5)0

FJL

F 0

FRO}FRO}

This result may be compared to the corresponding result for closed shell nuclei (2.9) .The expression (3.5) for the contribution to the mean square radius is rather

insensitive to the nuclear mass number as illustrated in fig . 9 where <r 2> is plottedas a function of A . Since the normalization integral for the two-nucleon density isunderestimated as demonstrated in fig . 8, we also show <r 2) as a function ofA afterdivision by the normalization integral [l .h .s. of (3.4)] . This latter result shows thatthe radius correction due to the pion exchange current is very insensitive to A.Comparing the results in fig . 9 to the radius corrections obtained for 'He, t60 and"Cain the previous section it is seen that the Fermi gas model result is slightly larger


441

than the results obtained for the closed-shell nuclei- in the harmonic oscillator model,i.e. <r2) ;t~ 0.05 fm' as compared to <r') x 0.03 fin' . These results indicate thatthe pion exchange current correction to the mean square radius is much smaller thanthe correction due to the finite nucleon size ( 0.6 fm') and in general thereforeunimportant.These Fermi gas model arguments can also be extended to nuclei which do not

have isospin 0 by letting the protons and neutrons have different Fermi momenta.In such nuclei the isovector part ofthe pair current also contributes. We have foundthat the size ofthe exchange current correction to <r') does not show any appreciablechange when going e.g . from "Ca to "Ca in this model 19).

5. Conclusions

The main conclusion of this work is that the relative importance of the pionexchange current effect on nuclear charge form factors is smaller in the case of heavynuclei than in the case of the few-nucleon systems (A 5 4) . Even so at large enoughvalues of momentum transfer (> 15 fm-2) the exchange current contributions willeventually dominate over the impulse approximation contribution . The numericalresults obtained are necessarily of only qualitative value at high momenta sincethe deficiencies of the harmonic oscillator shell-model wave functions used mustnecessarily become evident at large momentum transfer . It would of course beinteresting to see results obtained with more realistic wave functions when the ex-change current contribution is included . The complicated structure of the two-bodycurrent however makes calculations with more realistic wave functions very labo-rious .After the completion of this work a calculation by Cheon of the exchange current

effects on the form factor of"Caappeared in print "). Unfortunately we find Cheonsresults to be much too large and of the wrong sign . We believe this can be traced tounrealistic features of Cheon's nuclear model which puts all nucleon pairs in relativeS-states and leads to quantum mechanically impossible spin matrix elements .Another recent calculation of the meson exchange effects on the charge from factorsof the doubly closed shell nuclei has been carried out by Gari, Hyuga and Zabolitzkyusing Brueckner-Hartree-Fuck and Faddeev-Brueckner-Hartree-Fuck wave func-tions including three-body forces We note that our results for F"°h are in qualita-tive agreement with those of ref. Z ~.

Themany helpful suggestions ofProf. J. Borysowicz greatly facilitated the carryingout of this work .

442


References

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Fisica, Mexico, 1960)11) E. Moniz, M. Chemtob and M. Rho, Phys . Rev. C10 (1974) 34412) T. de Forest and J. D. Walecka, Adv. in Phys. 15 (1966) 113) R. F. Frosch et al., Phys . Rev. 160 (1967) 87414) I . Sick and J. S. McCarthy, Nucl . Phys. A150 (1970) 63115) R. F. Frosch et al., Phys . Rev. 174 (19óS) 138016) F. Iachello, A. D. Jackson and A. Lande, Phys . Lett . 43B (1973) 19117) A. de-Shalitland H. Feshbach, Theoretical nuclear physics, vol . 1 (Wiley, New York, 1974)18) 1. T. Cheon, Phys. Lea. 60B (1976) 40919) D. O. Riska, to be published20) M. Gari, H. Hyuga and J. G. Zabolitzky, Preprint RUBTP11/145, Ruhr Universitlit, Bochum

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Pion exchange currents and nuclear charge form factors