Magnetic electron scattering from p-shell nuclei

Nuclear Physics A569 (1994) 510-522 North-Holland

NUCLEAR PHYSICS A

Magnetic electron scattering from p-shell nuclei J.G.L. Booten, A.G.M. vanHees R.J. Van de Graaff iaboratorium, Rijksuniuersiteit te Utrecht, P.O. Box 80000, 3508 TA Utrecht, The Netherlands

Received 1 September 1993

Abstract

In this paper we report on shell-model calculations of transverse electron scattering form factors of states in several selected p-shell nuclei. Results will be presented for calculations in the p-shell model space and in the extended (0 + 2)&0 model space. The influence of meson-exchange currents on the calculated form factors will be examined. We find that in the beginning of the p-she11 the description of the transverse form factors is improved by extending the model space. However, towards the end of the p-shell the situation deteriorates, indicating deficiencies in the wave functions.

1. Intr~uction In a previous paper we investigated the electron scattering form factors of ‘Li in the framework of the nuclear shell model. This nucleus is pa~~cularly interesting to study, because both longitudinal as well as transverse form factors have been measured for the transitions to three individual states up to very large momentum transfer. The shell-model calculations presented in ref. [l] were performed in the p-shell model space and in the complete (0 + 2)&w model space. For the transverse form factors we included the effects due to pion exchange currents. It was demonstrated that in order to obtain a satisfying description of the experimental data, both the extension of the model space as well as the inclusion of exchange currents were necessary ingredients. In this paper we examine if the conclusions given in ref. [l] also hold for other p-shell nuclei. We restrict our study to the transverse form factors, since these are more affected by meson exchange currents (MEC) than the longitudinal form factors (see e.g. ref. [21). Results are presented for the nuclei 6Li, *‘B, “B, t4N and “N. Many more examples could be studied, but these form factors would all have similar characteristics as the cases examined here. For more complete treatments on magnetic electron scattering we refer to the review articles by Donnelly and Sick 131and by Peterson 141and to the recent review paper by Raman, Fagg and Hicks 151. 0375-9474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0515-A

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In the next section we will present the results of calculated transverse form factors of several p-shell nuclei. Calculations have been performed in the p-shell model space as well as in the extended 2fiw model space. The effective interaction used in the former model space is taken from the work of Cohen and Kurath [6] (the interaction labeled (6-16~2BME), the one used in the latter model space is taken from the work of Wolters et al. [7]. In the large model space the contributions from the purely isovector pion-exchange currents have been added according to the formalism outlined in detail in ref. [I]. Bare-nucleon charges and magnetic moments have been employed for all cases considered. The single-particle wave functions used are of harmonic-oscillator shape. The size parameter b is taken to be the same (1.75 fm) for all form factors calculated in the 2ttw mode1 space, except for the 15N form factors as will be discussed. The value was determined from a least squares fit of calculated charge radii of p-shell nuclei to the experimental values [7]. For a more direct comparison of the Oltio results with those obtained in the extended 2Aw model space, we adopted the same value for the size parameter in the small model space. It should be emphasized that for all cases considered below, the radial shape of the single-particle wave functions was not determined from the form factors. For an overview of some longitudinal form factors of the nuclei to be discussed below, the reader is referred to ref. [8]. In general, the longitudina1 form factors are fairly well reproduced in the extended model space. A clear improvement as compared to the small model space results was observed. The MEC effects on the low-q magnetic observables of p-shell nuclei have been reported in refs. 19,101(for magnetic dipole moments) and ref. [ll] (for Ml transition rates).

2. Results 21. The nucleuf 6Li As a first exampie we will discuss the transverse form factors of a few low-lying states in ‘Li. Since the ground state is a J” = l+, T = 0 state, MEC wili not affect the calculated transverse form factor (a purely isoscalar Ml) in our approach. The results are shown in Fig. 1, together with the data from refs. [12,13]. The first peak is correctly reproduced in the extended model space. However, the diffraction minimum is predicted at a too low momentum transfer q compared to the experimental location q = 1.4 fm-‘. Furthermore, the second maximum is clearly overpredicted. Better results are obtained in the small model space. The location of the minimum is exactly reproduced and the amplitude of the second maximum agrees better with experiment as compared to the large-basis calculation. Even better agreement with experiment is achieved in a phenomenological shell-model treatment as described by Donnelly and Sick 131. In such an approach the wave

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1

2 o (in

3

4

fm-‘)

Fig. 1. Elastic magnetic form factor of 6Li. Results of calculations in the p-shell model space (dashed line) and in the 2hw model space (solid line) are shown. The data are from refs. [12,13].

function amplitudes are determined in a fitting procedure from a variety of observables, such as electron scattering form factors and ground-state moments. A perfect reproduction of the elastic magnetic form factor of 6Li is also recently obtained in terms of cluster model descriptions [14,15], but only after the wave functions were fully antisymmetrized. The role of MEC is investigated in the purely isovector Ml transition from the ground state to the J” = Of, T = 1 state at E, = 3.56 MeV. The calculated transverse form factor is displayed in Fig. 2. In contrast to the elastic magnetic

0

1

2

q

3 (in

4

5

fm-I)

Fig. 2. Transverse form factor for electroexcitation of the J” = Oc, T = 1 state in 6Li. The calculations in the p-shell model space are indicated by the dashed line (without MEC) and the dash-dotted line (MEC included), those in the 2hw model space by the dotted line (without MEC) and the solid line (MEC included). The data are from refs. [16-181.

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form factor, the 21iw calculation predicts the location of the minimum at the correct momentum transfer, while the p-shell calculation overestimates this location. Note that MEC shift the minimum to a slightly higher momentum transfer. The exchange currents provide only a minor correction to the form factor below q = 3 fm-‘. At higher momentum transfer MEC enhance the form factor compared to the one-body prediction. However, no specific MEC-induced structures were observed below q = 5.0 fm-’ in the 2Ro calculation. This is in severe disagreement with the results obtained by Dubach et al. [2] in the p-shell model space. They applied the phenomenological shell-model approach to this transition and observed an MEC-induced second minimum at q = 3.2 fm-‘. For comparison we also plotted our results obtained in the p-shell model space after inclusion of MEC. Note that we used a size parameter b = 1.75 fm, which is smaller than the value used by Dubach (b = 2.03 fm>. As mentioned before, we employed the Cohen and Kurath 161 (616)2BME shell model amplitudes in the small model space instead of the phenomenological wave functions used by Dubach. Furthermore, we also added the MEC contributions coming form the interaction of valence nucleons (occupying a Op-shell orbit) with core nucleons (occupying a Os-shell orbit). Similarly to the result of Dubach we observe a second minimum due to MEC in the small model space. The location of this minimum (at q 2: 4 fm-‘> is shifted to a higher momentum transfer compared to Dubach’s result. This is due to the differences in both calculations pointed out above. Fig. 3 shows the transverse form factor for the purely isovector transition to the J” = 2+, T = 1 state in 6Li at E, = 5.36 MeV. The ~ntributing mu~tipoles in this transition are the Ml, E2 and M3 multipoles. The form factor is rather poorly reproduced in both model spaces. MEC effects only show up at high momentum transfer, leading to the usual enhancement compared to the one-body calculation.

0

2

1

q Fig. 3. Transverse

(in

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4

fm-I)

factor for electroexcitation of the J” = Zf, T = 1 state in 6Li. The same labeling as in Fig. 2 is used. The data are from refs. [16,18,191.

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structures were observed in our calculations below q = 5.0

2.2. i%e nucleus l”B The ground state of l”B is a T = 0 state, as it is for 6Li. Therefore, MEC will not influence the elastic magnetic form factor in the present approach. The form factor is still of particular interest due to the large spin J” = 3+ of the ground state. This implies that the magnetic multipolarities Ml, M3 and M5 are allowed. Shell model calculations restricted to the p-shell allow for a maximum multipolarity M3 in case of one-body operators. Due to the extension of the model space an M5 form factor could show up. Our results for the elastic magnetic form factor of l”B is plotted in Fig. 4. For the 2ko calculation the individual multipole contributions are also shown. No M5 multipole was observed in the 2hw calculation. The M5 contribution turned out to be less than 1O-9 over the entire momentum transfer region studied here. The theoretical curves underestimate the experimental data below q = 2 fm-’ in both model spaces. In Fig. 5 the transverse form factor for the transition from the ground state to the J” = O+, T = 1 state at E, = 1.74 MeV is shown. Note that this is a pure M3 transition. Since the transition is isovector, MEC will yield corrections to the calculated form factor. MEC corrections are rather small below q = 3 fm- ‘. At higher momentum transfer they enhance the form factor compared to the one-body calculation, bringing theory a bit closer to the data. The transverse form factor for electroexcitation of the J” = 2+, T = 1 state at 5.17 MeV is shown in Fig. 6. Five multipole components contribute to this form

J”=3+,T=O

E

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2 q (in

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Fig. 4. Elastic magnetic form factor of *‘B . Results of calculations in the p-shell model space (dashed line) and in the 2hw model space (solid line) are shown. The dotted lines indicate the individual multipole contributions in the 2hw calculation. The data are from refs. [12,20,21].

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3 2 q (in fm-‘)

4

5

Fig. 5. Transverse form factor for electroexcitation of the J” = O+, T = 1 state in “B. The same labeling as in Fig. 2 is used. The data are from refs. [21,22].

factor: the Ml, E2, M3, E4 and M5 multipoles. The high multipoies (E4 and M5) are absent in the small model space calculation (without MEC). At large momentum transfer (4 = 4 fm-‘1 MEC dominate the calculated form factor in the 2hw mode1 space and improve the agreement with experiment. This is entirely due to the high multipole components E4 and M5, as can be seen in the multipole decomposition displayed in Fig. 7. These multipoles are mainly induced by MEC already at moderate momentum transfer (q = 2 fm-‘1. Note that the E2 and Ml multipoles are strongly affected by the introduction of MEC at low q, since the one-body contributions to these multipoles are strongly inhibited, see e.g. ref. [2].

0

1

2

3

4

5

q (in fm-‘) Fig. 6. Transverse form factor for electroexcitation of the f” = 2+, T= 1 state in toB. The same labeling as in Fig. 2 is used. The data are from refs. [21-231.

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2 q (in

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Fig. 7. Multipole decomposition of the transverse form factor of the .I” = 2+, T = 1 state at E, = 5.17 MeV in “B. Dotted lines denote the contributions obtained in the 2ho model space without MEC, full lines the contributions with MEC included.

This, however, does not show up in the total form factor due to the dominating M3 multipole. The calculated transverse form factors of l”B presented here, reproduce the data rather poorly. However, as mentioned before, the value for the harmonicoscillator size parameter was assumed to be the same as obtained from a fit to the charge radii of the p-shell nuclei [7], namely b = 1.75 fm. A somewhat smaller value would certainly improve the agreement with experiment. 2.3. The nucleus “B The nucleus “B has a J” = $-, T = : ground state. The contributing multipoles to the elastic magnetic form factor are the Ml and M3 multipoles. The description is similar to the elastic magnetic form factor of 7Li, discussed in detail in ref. [l]. The form factor is shown in Fig. 8, together with the data from various experiments [12,20,213. The data are reasonably well reproduced up to momentum transfer q E 3 fm-‘, although the main peak of the Ml (at q = 0.7 fm-‘) is somewhat below the data. The same holds for the peak of the M3 multipole (at q = 1.7 fm-‘1. At high momentum transfer the calculations fall off too rapidly compared to the data. The 2frw calculation improves the description compared to the p-shell model result. Much better agreement with experiment, however, can be obtained by determining the radial shape of the single-particle wave functions from a fit to the magnetic scattering data at large q. Hicks et al. [21] obtained a good correspondence of the form factor calculated in the Op-shell model space, employing the Cohen and Kurath amplitudes [6], with all existing data by using fitted Woods-Saxon radial wave functions.

J.G.L. Booten, A.G.M. uan Hees / Electron scattering

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10-6 IO -8

0

1

2

3

4

5

q (in fm-‘) Fig. 8. Elastic magnetic form factor of ‘*B. The same labeling as in Fig. 2 is used. The data are from refs. [12,20,21].

The introduction of MEC leads to a minor enhancement of the calculated form factor below 4 = 4 fm- ‘, improving the agreement with experiment. Similarly to the elastic magnetic form factor of 7Li (see ref. [l]) MEC yield a diffraction minimum around 4 = 4 fm-’ in both the Ml and M3 multipoles (not shown explicitly in the figure). Consequently also the total calculated form factor exhibits this diffractive structure induced by MEC. However, no such structures have been observed experimentally below a momentum transfer of 4 fm-‘. 2.4. The nucleus “N The description of 14N parallels the one given before for 6Li. Transverse electron scattering data [24] exist for the purely isoscalar elastic scattering from the J” = l+, T = 0 ground state and for the purely isovector transition to the J” = O+, T = 1 state at Ex = 2.313 MeV. 14N in terms of a Op-shell model description has two holes in the closed shell I60 nucleus, whereas 6Li has two nucleons above the cIosed shell 4He core. Due to the simplicity of the wave functions, a fully phenomenological shell-model treatment is possible, similarly to the 6Li case. This procedure was followed by Huffman et al. [24] and very recently by Genz et al. [25]. Both calculations were able to reproduce the elastic as well as the inelastic Ml form factor up to a momentum transfer of 2 fm-‘. At larger momentum transfer the calculated form factors feil off much more rapidly than the data. This turned out to hold in particular for the inelastic transition. In Figs. 9 and 10 our results are displayed. Whereas the elastic Ml form factor is reasonably reproduced up to q = 2 fm- ‘, our calculations fail entirely to describe the inelastic transition. Especially in the large model space our calculations underpredict the high-q data by orders of magnitude. The large-basis calcula-

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1o-4

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’'N

-

J”= 1 +,T=O

I.,.,,,,,., 0

,,,,,,,,,

, .\

1

2

q Fig. 9. Elastic

magnetic

form factor

(in

3

.:\...,

.,

I

4

fm-‘)

of 14N. The same labeling ref. [24].

as in Fig. 1 is used. The data are from

tion introduces a minimum near q = 3 fm-’ in the inelastic Ml form factor. MEC shift this minimum to a somewhat higher momentum transfer. The p-shell model calculation does not show any diffractive structure. 2.5. The nucleus 15N In the extreme single-particle model the “N ground state (J” = i-, T = i) is considered to be a pure Op,,, p roton-hole state in the closed shell 160 nucleus. The first excited state (J” = $ -, T = ;I at E, = 6.32 MeV is considered to be a pure OP~,~ proton hole state in this model. The elastic magnetic form factor is determined by a single multipole (Ml), whereas the inelastic transition has two

P’

0

1

2

q Fig. 10. Transverse

(in

3

4

fm-‘)

form factor for electroexcitation of the J” = O+, T = 1 state labeling as in Fig. 2 is used. The data are from ref. [24].

in 14N. The same

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lo--4 10-5-

z

lo+-

lo-'10-8

10-g

-

c

0

1

3 2 q (in fm-‘)

4

5

Fig. 11. Elastic magnetic form factor of “N. The same labeling as in Fig. 2 is used. The data are from ref. [I&].

contributing multipoles (Ml and E2). The calculated form factors in the proton-hole model and in the 2ho model space are displayed in Figs. 11 and 12. The data shown are taken from the work of Singhal et al. 126-J.The harmonic-osciilator size parameter is chosen to be b = 1.65 fm, in order to reproduce the location of the second maximum of the elastic Ml form factor in the 212~ model space. The same size is taken in the small model space. The simple proton-hole model calculation is somewhat improved by extending the model space in case of the inelastic transition. For the elastic magnetic form factor, however, the description deteriorates. The calculated second maximum clearly overpredicts the data in both model spaces, while at large q the calculated form factors fall off much too rapidly

0

1

2

3

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q (in fm-‘) Fig. 12. Transverse form factor for eiectroexcitation of the f” = $- state in “N. The same labeling as in Fig. 2 is used. The data are from ref. 1261.

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compared to the data. Especially in the extended model space the calculation with the one-body Ml operator underestimates the data by orders of magnitude. A similar behavior is observed for the elastic magnetic form factor (a pure Ml as well) of 13C, calculated in the 2hw model space 171.Despite the fact that the 15N wave functions of ref. [7] are unable to fit the elastic Ml form factor data, they provide an improved description of pion-nuclear reactions on 15N (see ref. [27]). Introduction of MEC in the elastic magnetic form factor calculated in the (0 + 2)hw model space (solid curve) enhances the form factor in the region of the second maximum, bringing theory even further away from experiment. This agrees with the results obtained by Singhal et al. [26] in the proton-hole model. At higher values of momentum transfer (q > 3 fm-‘1 MEC quench the total form factor. At some point (q = 3.3 fm-‘> the MEC contribution cancels out the one-body contribution and thus introduces a minimum in the Ml form factor, which has not been observed experimentally. At large q the MEC contributions are dominant. Suzuki et al. [28,29] have shown that first-order core polarization effects, taking into account excitations beyond 2hw, are able to reduce the peak of the second maximum in the elastic Ml form factors of 13C and “N. Inclusion of second-order core polarization effects yields a further quenching of the peak, as shown by Blunden and Caste1 [30], and more recently by Gijkalp and Yilmaz [31]. Blunden and Caste1 showed that these second-order effects are partially cancelled by MEC and they find a good agreement with experiment in the low-q region. At large momentum transfer all calculations mentioned above exhibit the same deficiencies: the calculated elastic magnetic “N form factor falls off much faster than the experimentally observed form factor. There are several reasons for this rapid fall off: (i) harmonic-oscillator wave functions are not realistic at such high q-values; (ii) lack of consistency between the nuclear wave functions and the exchange currents; (iii) quark exchange currents might play an important role (see e.g. ref. [32]); (iv) relativistic corrections should be included. Especially the last point seems to be crucial. Blunden and Kim [33] recently calculated the one-pion exchange current effects on the elastic Ml form factor of 15N in a fully consistent manner within a relativistic framework. Their results agree qualitatively with the non-relativistic treatments mentioned above, with the exception that at high momentum transfer the Ml form factor falls off less drastically.

3. Conclusions The most important conclusions of the present paper can be briefly summarized as follows: (i> Transverse electron scattering form factors are fairly well reproduced in the extended model space in the beginning of the p-shell. In general, an improvement is seen compared to results obtained in the restricted model space.

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(ii) Towards the end of the p-shell, transverse form factors are poorly reproduced in the 2zZw model space. The description worsened compared to the Oho results. MEC could only partly cure the deficiencies. This suggests the need for components in the wave functions other than only 0 + 2hw. (iii) At high momentum transfer MEC contributions enhance the calculated transverse form factors. However, even after inclusion of MEC, the form factors fall off too fast compared to the data. All calculations presented in this paper have been performed with radial wave functions of the harmonic-oscillator shape. The use of Woods-Saxon wave functions would enable us to obtain a better high-q description of the form factors (see e.g. ref. [31). This, however, would complicate the computation of MEC effects considerably.

This work was performed as part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). Numerical calculations were performed on the Cray YMP of the Amsterdam Computer Centre (SARA), with financial support from the Stichting Nationale Computer Faciliteiten (NCF).

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