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Effects of core polarization and exchange currents on the form factors of magnetic transitions in 12C
NuckarPhysics A322 (1979) 361-368 Q North-HoNand Publishing Co ., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permipion from the publisher
EFFECTS OF CORE POLARIZATION AND EXCHANGE CURRENTS ON THE FORM FACTORS OF MAGNETIC TRANSITIONS IN uC H. SAGAWAt, T. SUZUKI, H. HYUGA and A. ARIMA Department ojPhysics, University of Tokyo, Tokyo 113, Japan Received 30 October 1978 Abdrtack We have studied the fotYrt factor of the magnetic dipole excitation to the 15 .1 MeV, 1 + state in t2C, taking into account the effects of con polarization and exchange currents . It is shown by using the Cohen-Kttrath wave functions that the core polarization contributions, from both central and tensor interactions, are important and enhance signifit~rttly the second maximum of the magnetic form factor, while the effects of the exchange currents are shown to be small .
1. Iatfrodactioa A magnetic transition form factor to a state at 15 .1 MeV (JR =1+, T =1) in t2C has been observed by electron scattering t). This form factor, however, has not been explained quantitatively by any theoretical calculations over the whole momentum transfer region observed, 0 fm-t s q s 3 fm-t. The aim of this paper is to investigate the effects of core polarization and exchange currents on the magnetic form factor of this transition. As is well known, light nuclei around t2C are described well by the intermediate t~upling shell model. It has been shown by Hiro-Oka s) that the magnetic transition form factor to the 15 .1 MeV state in t2C is well reproduced by the wave functions given by Cohen and Ktuath 3) when the momentum transfer q is small (0 fm-t s q s 1.0 fm -t ) . On the other hand, there is a large discrepancy between the calculation and the experimental data in the high-q region (q a 1 .8 fm-'). Chemtob and Lumbroso a) calculated the effect of the exchange currents on this form factor. Their results showed that the exchange currents enhance the magnetic form factor, especially in the high-q region . Their results are, however, about four times larger than the experimental data in magnitude partly because they used apure lp-lh (Opl-Opt t) wave function to describe the 1 + state. Moreover, their treatment of the two-body operators due to one-pion exchange is not exact; their approximation becomes worse at high-q region . The effect of the d(1236) resonance was considered by Grecksch et al. s) in this form factor . Their calculation showed that the d -resonance affects the magnetic t Present address : The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark . 361
362
H. SAGAWA et al.
form factor in the high-q region, but did not succeed in explaining quantitatively the experimental data even in the low-q region . Mukhopadhyay and Martorell 6 ) calculated the impulse contribution by using sophisticated wave functions such as wave functions given by the Migdal model and Cohen-Kurath wave functions . They failed, however, to reproduce the second peak of the form factor . Furthermore, in order to investigate the dependence of the impulse contributions on the radial forms of single particle wave functions, they used harmonic oscillator wave functions, Woods axon wave functions and those produced by density-dependent Hartree-Fock calculations . The differences among form factors given by those wave functions are not only very small but also not enough to explain the second peak at all. On this basis, we decided to use harmonic oscillator wave functions. In this paper, we use the Cohen-Kurath wave functions produced by the (8-16) potential interaction s) because these wave functions reproduce very well many experimental datafor 12C (for example, the muon capture rate of 1Z C and the ft value of the ß-decay of 12B)') . We consider the impulse contribution, the effect of the core polarization and that of the exchange currents including picnic, pair and nucleonic currents . By core polarization, we mean the mixing of configurations outside the 0 flm shell model space assumed for the Cohen-Kurath wave functions. 2. Core polarization effects Inelastic electron scattering to the 15.1 MeV, 1 + state has only an isovector magnetic dipole form factor, 1 (A=l,t-1) 2 = 4~ + (A-l,t-1) + rr(q)1 IFM ~ 2l+ll (I -1 ' T - 111fiM (q)11j=0 , r=0)I , where TM`) (q) is the magnetic 11-pole operator given by Tii't) (q)
= J drJa(4r)Y(~i)(~) . ,1(r),
(2)
and l(r) is the sum of the one-body and two-body current operators. The amplitude of the impulse contribution to the magnetic form factor is given as follows : (1+T, = -1 .r-1) 111TM (q)11o+T = 0)imPulse
Ap (j~2111~M1) (q)11112), (3) 6( tr where a(j, j') denotes the sum of the products of the one-body c.f .p.'s of the initial a~ final states and Ap is the nucleon number in the p-shéll, Ao = 8. Here (1~2111fiM 1) (4)11112) is the reduced matrix element of the magnetic dipole operator due to the one-body current. The numerical values of a (j, j') are a (2, 2) _ -0.01453, a (z, 2) = 0.2440, a (~, i) = 0.08476 and a (2, z) _ -0.02693, respectively . The oscillator parameter b for harmonic oscillator wave functions is taken to be b =1.64 fm _ ~ a(1.1~)(-)t-f,
FORM FACTORS
363
IFM I 2
10~
10 5
10-s
Piy. 1 . The aquara of the Ml form factor to the 15 .1 MeV state (J` =1 + , T =1) in ' 2 C. The dashed, dashed-dotted, dashed-two-dotted and solid curves represent the individual contribution of the impute term, the mre polarization produced by the Rasenfeld mixture and the tenwr force, and the exchange currents, respectively. Signs + and - denote the relative sign of the individual contribution to the form factor . The experimental data is taken from the compilation in ref. 1 ).
which is fitted to the elastic and inelastic electron scattering from t2C [ref.')]. The c.m. correction and the proton finite size effect are taken into account by multiplying a factor exp [ - â(ap - zb2 )g 2 ] with ap = 0.658 fm. The calculated result of the impulse contribution is shown in figs . 1 and 4. Main contributions to the impulse form factor come from terms with the coefficients a (~, z) and a(z,?) . They interfere destructively at low q ; a cancellation that is essential to producing a good fit to the experimental data in the q s 1 .5 fm-t region . In the high-q region (q ~ 1 .8 fm-t), the calculated result of the impulse form factor is several times smaller than the experimental data . The core polarization is calct~ated by first order perturbation theory : ~1 + T
= 1 i M(q) Q V + VQ Ti,,t(q)IIO + T = .0
e
N
=E
n
{(i+T
e
/ 1
=111fiM `' (q)Iln ; o+z = o>-(n ; o+T = ol ~Io+T = 0> Eph
+(1 + T=11VIn ; 1+T=1)É~(n ; 1+T=111~M`'(q)Ib+T=O)}~
364
H . SAGAWA et al.
The p-h excited state In ; JT) is given by J where I -1 (aJRTR)) are the Cohen-Kurath wave functions of A =11 systems and (nplPjo ) specifies the quantum numbers of the particle state outside the Os and Op shells#. The first term of eq . (4) can be written by using the shell-model technique as follows:
E (1+T =1111~M `' (q)Illn ; 0+T = 0>~Ph (n ; o+T = ol ~IO+T = 0> _
(U ~ clel,zJ .utIz~tT' uillu~T'Jn
+T
= O{Il -Z~~Tc)(1t12)~)(1+T=1{Il -2(I~~T~)(%ili)~1'T')Ctz
X
~~((l ijv)1TI VI (1i 12)~)({RJ~T~{1i jz )J' T'}1 + T =1111
X
TM t) (4)IIIUx+cTo(1 t lv)JT}~+T - ~) + (It ~-s12 ).
where CtZ equals ZnP(nD -1) for two particles in the p-shell and np n, when a particle is in the p-shell while the other is in the s-shell. The core polarization effects are estimated by . using a residual interaction consisting of central and tensor parts. Since any realistic effective interaction is not available in t2C, we use two phenomenological potentials as the residual interactions . [As stated above, the (8-16) potential of ref. s) is used for p-shell nucleons.] As the central part, we adopt the Rosenfeld interaction and the Ferrell-Vischer interaction with the Gaussian radial dependence . Adimensionless range parameter A = ro/fib is assumed to be 0.694 and the strength of the potential in the triplet-even state is taken to be -60 MeV . These parameters reproduce fairly well the matrix elements in the Op shell obtained by Cohen and Kurath 8). As the tensor part, we use that of the Hamada-Johnston potential multiplied by a cut-off function f (r) =1 for r ~ 0.7 fm and f (r) = 0 for r < 0.7 fm [ref. lb)]. The rms deviations of the matrix elements between Op shell bound states calculated by using (Rosenfeld +tensor) and (FerrellVischer+tensor) interactions from those given by Cohen and Kurath are 1.78 MeV and 1.33 MeV, respectively. The deviations are thus smaller than the large matrix elements which amount to 5 MeV. We take into account intermediate states up to 6tw, excitations. The single particle energies of higher configurations are assumed to be those of the harmonic oscillator shell model and the value ~m is taken as 15 .37 MeV. The numerical results are shown in fig. 1 . In the case of nuclei with simple configurations, the core polarization effect by the central interaction decreases the magnetic dipole form factor as is shown by Arita 9) and Arima et al.'s However this is not the case in the present problem, because ~ Here j stands for (UP3iz, ~Piiz) or (Osliz) .
FORM FACTORS
365
many Otur configurations compete here. The core polarization by the central interaction increases the magnetic form factor of t2C especially in the high-q region . It is also worthwhile mentioning that if we want to introduce an effective operator to simulate the effect of the cere polarization, we need not only one-body but also two-body operators. This complication results from a fact that t2C has eight valence nucleons in the Op shell. The tensor interaction has a considerable effect on the magnetic properties of nuclei. Our result shows that the core polarization by the tensor interaction is somewhat larger than that produced by the central interaction of the Rosenfeld mixture in the low-q side (0.5 fm-' s q s 2 fm') and has a node atq ~ 2.4 fm'. The effects of the central and tensor interactions have opposite signs around the first peak and cancel each other. Near the second peak, however, they become additive and increase the impulse form factor by from three to four times. 3 . E:change carrent effects As for the exchange currents, we consider one-pion exchange currents, i.e., the picnic and pair currents and the nucleonic current (d(1236) resonance effect) tt) . The numerical results of the exchange current contributions also are shown in figs . 1 and 2 . - 0 .04
r
4 FM(4)
-0.02 0.0 0.02 0 .04 0.06 0:08 0 .10 Fig 2 . The form factor of the 1 + state multiplied by e'/q, where y ~ (b~(A-1)/4A+a ;)qZ . The dashed, dashed-dotted and solid curves are for the impulse term, the impulse term plm the picnic current and the impuhe term plus the picnic plus pair currents, respectively. Dotted point ahw indude the effect of the d-resonance.
366
H. SAGAWA er al.
As is shown in fig. 2, the effect of the pionic current decreases the impulse form factor. On the other hand the pair current increases it. The effect of thepair current is about three times larger than that of the pionic current. The net effect of the mesonic currents is small and, at most, 15% of the impulse form factor . These features are qualitatively consistent with the calculation of Dubach et al. t2 ) for 3 He and of Chemtob and Lumbroso a) for tZC. However, our result is smaller than that of ref. 4), particularly on the high-q side. There are two reasons for this discrepancy: (i) we take the intermediate coupling wave functions which are necessary in order to obtain a good fit to the experimental data in the low-q region . The intermediate coupling weakens the contribution of the exchange currents . (ü) An approximation used by ref. a) in the calculation of the pionic and pair current contributions is good only up to q ~ 1 .5 fm t. It is expected to become worse at the high-q region . On the other hand, we treat exactly the two-body currents. For the sake of comparison we also calculated the mesonic current contributions using the j-j limit wave functions which were used in ref. `). The results are shown in fig. 3a. We see from fig. 3a that the cancellation among contributions from various components of the Cohen-Kurath wave function is essential in reducing considerably the exchange current contribution especially around the second peak of the form factor . The difference between our result and that of Chemtob and Lumbroso
10 6
10'
0
1 q(f mi)Z
3
0
1
4(}m~)2
3
Fig. 3 . The calculated results of contributions to We Ml form factor from the pionic plus pair currents and from the nucleonic current are shown in (a) and (b), respectively . The dashed curves are obtained by using the f-j liatit wave functions while the solid curves by using the Cohen-Kurath wave functions .
FORM FACTORS
367
thus comes mainly from the difference in the wave functions used . Our result is ts) consistent with that of Dubach and Haxton where another set of Cohen-Ktuath wave functions ((8-16) 2BME) is used . The effect of the nucleonic current is smaller than that of the picnic and pair currents and in fact negligible in our case (see fig. 2). This feature is also pointed out by Dubach t4) in the magnetic form factor of t 'O . However, our result is quite different from the result in ref. S ), though their result has the same sign of the contribution from the nucleonic current as ours . We show in fig. 3b the dependence of the nucleonic current contribution on wave functions. Making a comparison between the results of the Cohen-Kurath and j! limit wave functions, one sees that the effect is quite sensitive to the wave functions used. Thus we believe that the discrepancy between our result and that of ref. s) mainly results from the difference in the wave functions. The difference may be partially attributed to the difference in methods of the treatment of the d-resonance effect . 4 . Condasion
The magnetic form factor including all contributions is given in fig. 4. We obtain a quite satisfactory agreement with the experimental data by the Rosenfeld interaction . In the case of the Ferrell-Vischer interaction, the agreement is not so good as M12
,2 C (1', T=1) Ex =15.1 MeV
IF
b=1.64 fm
f ~5
f ~6
L 0 Fry . 4 . The squares of the calculated Ml torm .factors to the 15.1 MeV, 1 + state in 1 ~C. The dashed curve shows the impulse contribution . The caltarlated form [actors including all contributions are shown by the solid and dashed-dotted curves . The former is produced by the Rosenfeld interaction and the latter by the Ferrell-Viecher interaction .
368
H. SAGAWA et al.
in the case of the Rosenfeld interaction. When one takes a large oscillator length (b ~ 1.8 fm) in the latter case, the calculated form factor is improved. In the present calculation, the coherent effect of the core polarization is shown to be very important in explaining the second peak in the form factor . The mesonic currents increase the form factor in almost thewhole q-region but are very small compared to the effect of the core polarization . One may expect that relativistic corrections could explain the remaining discrepancy between the experimental data and the theoretical results in the high-q region . We believe that the relativistic correction is not too large even in the high-q region and is probably of the order of q a /8M Z s 0.05 times the impulse contribution as in the case of the charge density ts), The authors would like to thank Professor K. Yazaki for the enlightening discussions. This work was supported in part by the Institute for Nuclear Study and High Energy Laboratory (KEK) at Tsukuba. One of them (H.S.) acknowledges the Soryushi-Shogakukai for the scholarship. The numerical calculations were performed using HITAC 8800/8700 at We Computer Centre of the University of Tokyo. References 1) 2) 3) 4) S) 6) 7) 8) 9) 10) 11) 12) 13) 14) 1S) 16)
A. Yamaguchi, T. Terasawa, K. Nakahara and Y. Torizuka, Phys. Rev. C3 (1971) 1750 M. Hiro-Oka, Prog. Theor. Phys. 43 (1970) 689 S. Cohen and D. Kurath, Nucl . Phys. 73 (1965) 1 M. Chemtob and A. Lumbroso, Nucl . Phys . B17 (1970) 401 E. Gr~ecksch, M. Dillig and M. G. Huber, Phys . Lett . 72B (1977) 11 N. C. Mukhopadhyay and J. Martorell, Nucl . Phys . A296 (1978) 461 M. Hiro-Oka, T. Koniahi, R. Monta, H. Narumi, M. Sofia and M. Monta, Prog. Theor. Phys . 40 (1968) 808 A. Anima and S. Yoshida, Nucl . Phys . A161 (1971) 492 K. Arits, Proc. Meeting on giant resonances and related topics ed. H. Sagawa (INS, University of Tokyo, 1977) p. 119 A. Anima, Y. Horikawa, H. Hyuga and T. Suzuki, Phys . Rev. Lett. 40 (1978) 1001 J. Hockert, D. O. Riska, M. Gari and A. Hufiman, Nucl. Phys. A217 (1973) 14 J. Dubach, J. H. Koch and T. W. Donnelly, Nud. Phys. A271(1976) 279 J. Dubach and W. C. Haxton, preprint J. Dubach, as quoted in M. V. Hynes, H. Miska, B. Norum, W. Bertoui, S. Kowalaki, F. N. Rad, C. P. Sargent, T. Sasanuma, W. Turchinetz and B. L. Berman, preprint J. L. Friar, Ann. of Phys. 81 (1973) 332 K. Shimizu, M. Ichimura and A. Anima, Nucl. Phys. A226 (1974) 282
EFFECTS OF CORE POLARIZATION AND EXCHANGE CURRENTS ON THE FORM FACTORS OF MAGNETIC TRANSITIONS IN uC H. SAGAWAt, T. SUZUKI, H. HYUGA and A. ARIMA Department ojPhysics, University of Tokyo, Tokyo 113, Japan Received 30 October 1978 Abdrtack We have studied the fotYrt factor of the magnetic dipole excitation to the 15 .1 MeV, 1 + state in t2C, taking into account the effects of con polarization and exchange currents . It is shown by using the Cohen-Kttrath wave functions that the core polarization contributions, from both central and tensor interactions, are important and enhance signifit~rttly the second maximum of the magnetic form factor, while the effects of the exchange currents are shown to be small .
1. Iatfrodactioa A magnetic transition form factor to a state at 15 .1 MeV (JR =1+, T =1) in t2C has been observed by electron scattering t). This form factor, however, has not been explained quantitatively by any theoretical calculations over the whole momentum transfer region observed, 0 fm-t s q s 3 fm-t. The aim of this paper is to investigate the effects of core polarization and exchange currents on the magnetic form factor of this transition. As is well known, light nuclei around t2C are described well by the intermediate t~upling shell model. It has been shown by Hiro-Oka s) that the magnetic transition form factor to the 15 .1 MeV state in t2C is well reproduced by the wave functions given by Cohen and Ktuath 3) when the momentum transfer q is small (0 fm-t s q s 1.0 fm -t ) . On the other hand, there is a large discrepancy between the calculation and the experimental data in the high-q region (q a 1 .8 fm-'). Chemtob and Lumbroso a) calculated the effect of the exchange currents on this form factor. Their results showed that the exchange currents enhance the magnetic form factor, especially in the high-q region . Their results are, however, about four times larger than the experimental data in magnitude partly because they used apure lp-lh (Opl-Opt t) wave function to describe the 1 + state. Moreover, their treatment of the two-body operators due to one-pion exchange is not exact; their approximation becomes worse at high-q region . The effect of the d(1236) resonance was considered by Grecksch et al. s) in this form factor . Their calculation showed that the d -resonance affects the magnetic t Present address : The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark . 361
362
H. SAGAWA et al.
form factor in the high-q region, but did not succeed in explaining quantitatively the experimental data even in the low-q region . Mukhopadhyay and Martorell 6 ) calculated the impulse contribution by using sophisticated wave functions such as wave functions given by the Migdal model and Cohen-Kurath wave functions . They failed, however, to reproduce the second peak of the form factor . Furthermore, in order to investigate the dependence of the impulse contributions on the radial forms of single particle wave functions, they used harmonic oscillator wave functions, Woods axon wave functions and those produced by density-dependent Hartree-Fock calculations . The differences among form factors given by those wave functions are not only very small but also not enough to explain the second peak at all. On this basis, we decided to use harmonic oscillator wave functions. In this paper, we use the Cohen-Kurath wave functions produced by the (8-16) potential interaction s) because these wave functions reproduce very well many experimental datafor 12C (for example, the muon capture rate of 1Z C and the ft value of the ß-decay of 12B)') . We consider the impulse contribution, the effect of the core polarization and that of the exchange currents including picnic, pair and nucleonic currents . By core polarization, we mean the mixing of configurations outside the 0 flm shell model space assumed for the Cohen-Kurath wave functions. 2. Core polarization effects Inelastic electron scattering to the 15.1 MeV, 1 + state has only an isovector magnetic dipole form factor, 1 (A=l,t-1) 2 = 4~ + (A-l,t-1) + rr(q)1 IFM ~ 2l+ll (I -1 ' T - 111fiM (q)11j=0 , r=0)I , where TM`) (q) is the magnetic 11-pole operator given by Tii't) (q)
= J drJa(4r)Y(~i)(~) . ,1(r),
(2)
and l(r) is the sum of the one-body and two-body current operators. The amplitude of the impulse contribution to the magnetic form factor is given as follows : (1+T, = -1 .r-1) 111TM (q)11o+T = 0)imPulse
Ap (j~2111~M1) (q)11112), (3) 6( tr where a(j, j') denotes the sum of the products of the one-body c.f .p.'s of the initial a~ final states and Ap is the nucleon number in the p-shéll, Ao = 8. Here (1~2111fiM 1) (4)11112) is the reduced matrix element of the magnetic dipole operator due to the one-body current. The numerical values of a (j, j') are a (2, 2) _ -0.01453, a (z, 2) = 0.2440, a (~, i) = 0.08476 and a (2, z) _ -0.02693, respectively . The oscillator parameter b for harmonic oscillator wave functions is taken to be b =1.64 fm _ ~ a(1.1~)(-)t-f,
FORM FACTORS
363
IFM I 2
10~
10 5
10-s
Piy. 1 . The aquara of the Ml form factor to the 15 .1 MeV state (J` =1 + , T =1) in ' 2 C. The dashed, dashed-dotted, dashed-two-dotted and solid curves represent the individual contribution of the impute term, the mre polarization produced by the Rasenfeld mixture and the tenwr force, and the exchange currents, respectively. Signs + and - denote the relative sign of the individual contribution to the form factor . The experimental data is taken from the compilation in ref. 1 ).
which is fitted to the elastic and inelastic electron scattering from t2C [ref.')]. The c.m. correction and the proton finite size effect are taken into account by multiplying a factor exp [ - â(ap - zb2 )g 2 ] with ap = 0.658 fm. The calculated result of the impulse contribution is shown in figs . 1 and 4. Main contributions to the impulse form factor come from terms with the coefficients a (~, z) and a(z,?) . They interfere destructively at low q ; a cancellation that is essential to producing a good fit to the experimental data in the q s 1 .5 fm-t region . In the high-q region (q ~ 1 .8 fm-t), the calculated result of the impulse form factor is several times smaller than the experimental data . The core polarization is calct~ated by first order perturbation theory : ~1 + T
= 1 i M(q) Q V + VQ Ti,,t(q)IIO + T = .0
e
N
=E
n
{(i+T
e
/ 1
=111fiM `' (q)Iln ; o+z = o>-(n ; o+T = ol ~Io+T = 0> Eph
+(1 + T=11VIn ; 1+T=1)É~(n ; 1+T=111~M`'(q)Ib+T=O)}~
364
H . SAGAWA et al.
The p-h excited state In ; JT) is given by J where I -1 (aJRTR)) are the Cohen-Kurath wave functions of A =11 systems and (nplPjo ) specifies the quantum numbers of the particle state outside the Os and Op shells#. The first term of eq . (4) can be written by using the shell-model technique as follows:
E (1+T =1111~M `' (q)Illn ; 0+T = 0>~Ph (n ; o+T = ol ~IO+T = 0> _
(U ~ clel,zJ .utIz~tT' uillu~T'Jn
+T
= O{Il -Z~~Tc)(1t12)~)(1+T=1{Il -2(I~~T~)(%ili)~1'T')Ctz
X
~~((l ijv)1TI VI (1i 12)~)({RJ~T~{1i jz )J' T'}1 + T =1111
X
TM t) (4)IIIUx+cTo(1 t lv)JT}~+T - ~) + (It ~-s12 ).
where CtZ equals ZnP(nD -1) for two particles in the p-shell and np n, when a particle is in the p-shell while the other is in the s-shell. The core polarization effects are estimated by . using a residual interaction consisting of central and tensor parts. Since any realistic effective interaction is not available in t2C, we use two phenomenological potentials as the residual interactions . [As stated above, the (8-16) potential of ref. s) is used for p-shell nucleons.] As the central part, we adopt the Rosenfeld interaction and the Ferrell-Vischer interaction with the Gaussian radial dependence . Adimensionless range parameter A = ro/fib is assumed to be 0.694 and the strength of the potential in the triplet-even state is taken to be -60 MeV . These parameters reproduce fairly well the matrix elements in the Op shell obtained by Cohen and Kurath 8). As the tensor part, we use that of the Hamada-Johnston potential multiplied by a cut-off function f (r) =1 for r ~ 0.7 fm and f (r) = 0 for r < 0.7 fm [ref. lb)]. The rms deviations of the matrix elements between Op shell bound states calculated by using (Rosenfeld +tensor) and (FerrellVischer+tensor) interactions from those given by Cohen and Kurath are 1.78 MeV and 1.33 MeV, respectively. The deviations are thus smaller than the large matrix elements which amount to 5 MeV. We take into account intermediate states up to 6tw, excitations. The single particle energies of higher configurations are assumed to be those of the harmonic oscillator shell model and the value ~m is taken as 15 .37 MeV. The numerical results are shown in fig. 1 . In the case of nuclei with simple configurations, the core polarization effect by the central interaction decreases the magnetic dipole form factor as is shown by Arita 9) and Arima et al.'s However this is not the case in the present problem, because ~ Here j stands for (UP3iz, ~Piiz) or (Osliz) .
FORM FACTORS
365
many Otur configurations compete here. The core polarization by the central interaction increases the magnetic form factor of t2C especially in the high-q region . It is also worthwhile mentioning that if we want to introduce an effective operator to simulate the effect of the cere polarization, we need not only one-body but also two-body operators. This complication results from a fact that t2C has eight valence nucleons in the Op shell. The tensor interaction has a considerable effect on the magnetic properties of nuclei. Our result shows that the core polarization by the tensor interaction is somewhat larger than that produced by the central interaction of the Rosenfeld mixture in the low-q side (0.5 fm-' s q s 2 fm') and has a node atq ~ 2.4 fm'. The effects of the central and tensor interactions have opposite signs around the first peak and cancel each other. Near the second peak, however, they become additive and increase the impulse form factor by from three to four times. 3 . E:change carrent effects As for the exchange currents, we consider one-pion exchange currents, i.e., the picnic and pair currents and the nucleonic current (d(1236) resonance effect) tt) . The numerical results of the exchange current contributions also are shown in figs . 1 and 2 . - 0 .04
r
4 FM(4)
-0.02 0.0 0.02 0 .04 0.06 0:08 0 .10 Fig 2 . The form factor of the 1 + state multiplied by e'/q, where y ~ (b~(A-1)/4A+a ;)qZ . The dashed, dashed-dotted and solid curves are for the impulse term, the impulse term plm the picnic current and the impuhe term plus the picnic plus pair currents, respectively. Dotted point ahw indude the effect of the d-resonance.
366
H. SAGAWA er al.
As is shown in fig. 2, the effect of the pionic current decreases the impulse form factor. On the other hand the pair current increases it. The effect of thepair current is about three times larger than that of the pionic current. The net effect of the mesonic currents is small and, at most, 15% of the impulse form factor . These features are qualitatively consistent with the calculation of Dubach et al. t2 ) for 3 He and of Chemtob and Lumbroso a) for tZC. However, our result is smaller than that of ref. 4), particularly on the high-q side. There are two reasons for this discrepancy: (i) we take the intermediate coupling wave functions which are necessary in order to obtain a good fit to the experimental data in the low-q region . The intermediate coupling weakens the contribution of the exchange currents . (ü) An approximation used by ref. a) in the calculation of the pionic and pair current contributions is good only up to q ~ 1 .5 fm t. It is expected to become worse at the high-q region . On the other hand, we treat exactly the two-body currents. For the sake of comparison we also calculated the mesonic current contributions using the j-j limit wave functions which were used in ref. `). The results are shown in fig. 3a. We see from fig. 3a that the cancellation among contributions from various components of the Cohen-Kurath wave function is essential in reducing considerably the exchange current contribution especially around the second peak of the form factor . The difference between our result and that of Chemtob and Lumbroso
10 6
10'
0
1 q(f mi)Z
3
0
1
4(}m~)2
3
Fig. 3 . The calculated results of contributions to We Ml form factor from the pionic plus pair currents and from the nucleonic current are shown in (a) and (b), respectively . The dashed curves are obtained by using the f-j liatit wave functions while the solid curves by using the Cohen-Kurath wave functions .
FORM FACTORS
367
thus comes mainly from the difference in the wave functions used . Our result is ts) consistent with that of Dubach and Haxton where another set of Cohen-Ktuath wave functions ((8-16) 2BME) is used . The effect of the nucleonic current is smaller than that of the picnic and pair currents and in fact negligible in our case (see fig. 2). This feature is also pointed out by Dubach t4) in the magnetic form factor of t 'O . However, our result is quite different from the result in ref. S ), though their result has the same sign of the contribution from the nucleonic current as ours . We show in fig. 3b the dependence of the nucleonic current contribution on wave functions. Making a comparison between the results of the Cohen-Kurath and j! limit wave functions, one sees that the effect is quite sensitive to the wave functions used. Thus we believe that the discrepancy between our result and that of ref. s) mainly results from the difference in the wave functions. The difference may be partially attributed to the difference in methods of the treatment of the d-resonance effect . 4 . Condasion
The magnetic form factor including all contributions is given in fig. 4. We obtain a quite satisfactory agreement with the experimental data by the Rosenfeld interaction . In the case of the Ferrell-Vischer interaction, the agreement is not so good as M12
,2 C (1', T=1) Ex =15.1 MeV
IF
b=1.64 fm
f ~5
f ~6
L 0 Fry . 4 . The squares of the calculated Ml torm .factors to the 15.1 MeV, 1 + state in 1 ~C. The dashed curve shows the impulse contribution . The caltarlated form [actors including all contributions are shown by the solid and dashed-dotted curves . The former is produced by the Rosenfeld interaction and the latter by the Ferrell-Viecher interaction .
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in the case of the Rosenfeld interaction. When one takes a large oscillator length (b ~ 1.8 fm) in the latter case, the calculated form factor is improved. In the present calculation, the coherent effect of the core polarization is shown to be very important in explaining the second peak in the form factor . The mesonic currents increase the form factor in almost thewhole q-region but are very small compared to the effect of the core polarization . One may expect that relativistic corrections could explain the remaining discrepancy between the experimental data and the theoretical results in the high-q region . We believe that the relativistic correction is not too large even in the high-q region and is probably of the order of q a /8M Z s 0.05 times the impulse contribution as in the case of the charge density ts), The authors would like to thank Professor K. Yazaki for the enlightening discussions. This work was supported in part by the Institute for Nuclear Study and High Energy Laboratory (KEK) at Tsukuba. One of them (H.S.) acknowledges the Soryushi-Shogakukai for the scholarship. The numerical calculations were performed using HITAC 8800/8700 at We Computer Centre of the University of Tokyo. References 1) 2) 3) 4) S) 6) 7) 8) 9) 10) 11) 12) 13) 14) 1S) 16)
A. Yamaguchi, T. Terasawa, K. Nakahara and Y. Torizuka, Phys. Rev. C3 (1971) 1750 M. Hiro-Oka, Prog. Theor. Phys. 43 (1970) 689 S. Cohen and D. Kurath, Nucl . Phys. 73 (1965) 1 M. Chemtob and A. Lumbroso, Nucl . Phys . B17 (1970) 401 E. Gr~ecksch, M. Dillig and M. G. Huber, Phys . Lett . 72B (1977) 11 N. C. Mukhopadhyay and J. Martorell, Nucl . Phys . A296 (1978) 461 M. Hiro-Oka, T. Koniahi, R. Monta, H. Narumi, M. Sofia and M. Monta, Prog. Theor. Phys . 40 (1968) 808 A. Anima and S. Yoshida, Nucl . Phys . A161 (1971) 492 K. Arits, Proc. Meeting on giant resonances and related topics ed. H. Sagawa (INS, University of Tokyo, 1977) p. 119 A. Anima, Y. Horikawa, H. Hyuga and T. Suzuki, Phys . Rev. Lett. 40 (1978) 1001 J. Hockert, D. O. Riska, M. Gari and A. Hufiman, Nucl. Phys. A217 (1973) 14 J. Dubach, J. H. Koch and T. W. Donnelly, Nud. Phys. A271(1976) 279 J. Dubach and W. C. Haxton, preprint J. Dubach, as quoted in M. V. Hynes, H. Miska, B. Norum, W. Bertoui, S. Kowalaki, F. N. Rad, C. P. Sargent, T. Sasanuma, W. Turchinetz and B. L. Berman, preprint J. L. Friar, Ann. of Phys. 81 (1973) 332 K. Shimizu, M. Ichimura and A. Anima, Nucl. Phys. A226 (1974) 282