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E2 contributions to the 7Be(p,γ)8B cross section
Nuclear Physics A 660 Ž1999. 249–254
E2 contributions to the 7 Be žp,g /8 B cross section F.C. Barker a
a
Department of Theoretical Physics, School of Physical Sciences and Engineering, The Australian National UniÕersity, Canberra, ACT 0200, Australia Received 16 August 1999; accepted 8 September 1999
Abstract A calculation of the E2 component of the 7 BeŽp,g . 8 B cross section atlow energies, using R-matrix formulae that include channel contributions,finds the resonant strength due to the 1q first excited state of 8 B to be appreciably smaller than was obtained in previous calculations, butgives a nonresonant strength slightly higher than before. q 1999 Elsevier Science B.V. All rights reserved. PACS: 24.10.yi; 25.40.Lw; 27.20.qn Keywords: 7 BeŽp, g . 8 B E2 cross section; R-matrix formulae
1. Introduction The measured cross section for the 7 BeŽp,g . 8 B reaction at lowenergies has a pronounced peak at Ecm f 0.63 MeV, which may be attributed to M1 transitions from the 1q first excited state of 8 B to the 2q ground state. Underlying the region is a nonresonant E1 contribution. Small resonant and nonresonant E2 contributions are alsoexpected. In a recent Coulomb dissociation measurement, Davids et al. w1x determined the ratio of the E2 and E1 strengths near the peak energy. They compared their results with the first-order perturbation theory calculations of Esbensen and Bertsch w2,3x, who used a potential model that gave S E2 rSE1 s 9.5 = 10y4 at Ecm s 0.6 MeV Žand 10.4 = 10y4 at the peak at 0.63 MeV w4x.. Best agreement was obtained with an E2 .8 y4 strength 0.7 times as large as the model value, leading to SE2 rSE1 s 6.7q2 at y1.9 = 10 0.6 MeV w1x. Other potential-model calculations w5–8x gave varying values of S E2 rSE1; e.g., as Davids et al. w1x pointed out, Kim et al. w5x found a value about twice that of Esbensen and Bertsch w2,3x. In their calculation of the E2 S factor, Esbensen and Bertsch found a resonant contribution due to p-wave protons superimposed on an appreciable nonresonant background due about equally to p- and f-waveprotons. As the resonant contribution is coherent with part of thebackground, the shape of the resonant peak is somewhat distorted.The nonresonant E2 strength, S E2 Žnr., in all these calculations w2,3,5–8x has a 0375-9474r99r$15.00 q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 3 8 9 - 9
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F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
similar value, particularly when expressed as a fraction ofthe E1 strength Žand when one allows for the omission of the f-wavecontribution by Typel and Baur w7x.. The peak-energy resonant E2 strengths, however, differ widely in the different calculations. We define the resonant E2 strength as S E2 Žr. s S E2 y SE2 Žnr. evaluated at the peak energy. Esbensen and Bertschw2,3x found S E2 Žr.rSE2 Žnr. f 0.3. They used a model in which both the 1q state and the 2q ground state are assumed to have a 7 BeŽg.s.. q 1p 3r2 proton single-particle structure. Kim et al. w5x and Krauss et al. w6x Žin their favoured calculation with spin-orbit coupling in the entrance channel only. appeared to use the same model,but both found S E2 Žr.rSE2 Žnr. f 1.4. The potentials used in these latter calculations, however, are independent of the total J value of the initial capturing state, so differing from the potentials used by Esbensen and Bertsch,and the resonance erroneously contains 0q, 2q and 3q contributions as well as 1q w9x, so accounting for the larger values of the ratio. Typel and Baur w7x apparently assumed a 7 BeŽg.s.. q 1p proton single-particle structure for both 2q and 1q states, with coupling to channel spin s s 2 only, and they obtained a ratio of about 1.0 Žwhere, however, the nonresonant strength contains only the p-wave part.. Typel et al. w8x used essentially the same model but included the f-wave contribution, and found a ratio of about 0.5. All of these potential-model calculations give widths of the 1q state of 8 B appreciably greater than the experimental value of G Ž1q. s 37 " 5 keV w10x; they are about 70 keV w2,3,5x, 100 keV w6x and 78 keV w7,8x, giving evidence that the descriptions of the 1q state are not adequate. These models assume a spectroscopic factor S Ž1q. s 1, although shell model calculations give much smaller values; e.g., Cohen and Kurath w11x give S 3r2 Ž1q. s 0.3215 and S 1r2 Ž1q. s 0.1240 for p 3r2 and p 1r2 protons, respectively. This defect was previously pointed out w9x for the model of Kim et al. w5x. In a three-cluster GCM calculation, Descouvemont and Baye w12x found a resonant E2 strength appreciably greater than that of Esbensen and Bertsch w2,3x, but as a fraction of the E1 strength it is almost the same. Theirwidth of the 1q state is about 54 keV. In a more recent paper w13x, they gave more details of their results, from which one finds S E2 Žr.rSE2 Žnr. f 0.4 at the peak. In contrast with the above results, Bennaceur et al. w14x, from calculations using the shell model embedded in the continuum, gaveseveral values for the width of the 1q level, all being less than theexperimental value. Their largest value is 25.9 keV, and corresponding to this they gave S E2 rSE1 s 7.72 = 10y4 at the peak, which isconsistent with the experimental value w1x. They also found S E2 Žr.rSE2 Žnr. s 0.187 w15x. In view of the differences between the results of these modelcalculations, in particular for S E2 Žr., and the disagreements with experiment, particularly for the width of the 1q level, we here use adifferent approach, which permits a realistic description of the 1q state, in order to calculate the E2 contributions to the 7 BeŽp, g . 8 Bcross section.
2. R-matrix formulae and procedure In Ref. w16x, a potential-model approach was used to calculate thenonresonant E1 contribution to the 7 BeŽp, g . 8 B cross section, while the resonant M1 contribution was
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
251
fitted using R-matrix formulae. InRef. w17x, the E1 component was also calculated using formulae fromR-matrix theory w18x, extended for photon reactions by the inclusion ofchannel contributions w19x. Here such R-matrix formulae are used to calculate the E2 part of the 7 BeŽp, g . 8 B cross section. The main difference from the formulae given in Ref. w17x is that Eq. Ž6. there is replaced by UsJe il i s yie iŽ v iy f i . 2 P l1r2 kg5r2 i
J glJs Ý Alm i
lm
i e li
½
gmgJ i Ž int . q
(
7
53 Mn1r2 e
15 128
=a 3 Nf1r2 Ý i lq1gmJci u f a s Ž l 200 <10 . U Ž 21 Ji s, l 2 . J2 c Ž l ,1 . c
(
q
7
53 Mn1r2 e
15 128
"k
"
5
a2 Nf1r2 i l iq1u f a e s eŽ l i 200 <10 . U Ž 21 Ji s e , l i 2 . Fl i Ž a .
=G l i Ž a . J2X Ž l i ,1 . ,
Ž 1.
where the labels J f s 2 and l f s 1 are omitted. Also the formulae corresponding to Eqs. Ž11. – Ž13. in Ref. w17x have an extra factor rra ineach integrand. We include p- and f-wave contributions Ž l i s 1,3; Ji s 0–4.. For simplicity, we assume gmJci s 0 for l s 3, and allow for up to two levels for each Ji s 0–3 Žand for J f s 2.. We also allow for up to four proton channels, corresponding to the first four states of 7 Be. The level energies and spectroscopic amplitudes are taken initially from the shell model calculation of Barker w20x, with the energies of the lowest 2q, 1q and 3q levels adjusted to agree with the experimental values w10x. For simplicity, we choose Bc s Sc Ž E1J i . for each Ji value. We also calculate g 11g Žint. from the wavefunctions of Ref. w20x, but take other gmgJ i Žint. s 0. This is because we are primarily interested in the E2 cross section in the neighbourhood of the 1q first excited state of 8 B. For the same reason, we study the effect of changing the spectroscopic amplitudes only for this 1q state and for the 2q ground state, both for the 7 Be ground-state channels. We also consider a range of values of the channel radius a, and of the parameters R and a WS of theWoods–Saxon potentials used for calculating reduced-width amplitudes from the spectroscopic amplitudes. Some calculations are also made in the one-level, one-channel approximation for each Ji and J f value. As ‘standard’ values of these adjustable parameters, we use a s 5.0 fm,
R s 2.391 fm,
g 11g Ž int . s 0.76 MeV 1r2 fm5r2 , 1r2 q Sss 2 Ž 11 . s 0.5431 ,
a WS s 0.65 fm , 1r2 Sss1 Ž 1q 1 . s 0.3988 ,
1r2 Sss1 Ž 2q1 . s 0.5012 ,
together with the two-level, four-channel approximation.
1r2 Sss2 Ž 2q1 . s 0.8746 ,
Ž 2.
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
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3. Results Initially we consider calculations in which the parameter valuesare chosen to agree as well as possible with those used in thecalculation of Esbensen and Bertsch w2,3x. Thus we assume a one-level,one-channel approximation with R s 2.391 fm, a WS s 0.52 fm, g 11g Ž int . s 0.35 MeV 1r2 fm5r2 , 1 5 1r2 q 1r2 Sss Sss2 Ž 1q 1 Ž 11 . s 6 s 0.9129 , 1 .s '6 s 0.4082 , 1 1 1r2 q 1r2 Sss s 0.7071 , Sss2 2q Ž 3. Ž 1 Ž 21 . s 1 .s '2 '2 s 0.7071 . These values of the spectroscopic amplitudes correspond to p 3r2 protonsin each case. The value of g 11g Žint. is somewhat uncertain, asinternal wave functions are required and the assumed 7 BeŽg.s.. q 1p 3r2 proton structure is not fully antisymmetrized, because the 7 Be ground state also contains 1p 3r2 protons. With these values and a s 5.0 fm, we find SE2 Žr.rSE2 Žnr. s 0.32. ŽFor a s 6.0 fm, the ratio is 0.27; for a s 5.0 fm and g 11g Žint. s 0, the ratio is 0.14.. There is goodagreement with the ratio found by Esbensen and Bertsch w2,3x. The energy dependence of the background, as measured say by SE2 Ž1.0 MeV.r SE2 Ž0.2 MeV., also agrees reasonably with Esbensen and Bertsch Ž4.9 compared with 5.2..Using S E1 calculated as in Ref. w15x with the parameter values Ž3., we find S E2 rSE1 s 10.4 = 10y4 at the peak energy, the same as predicted bythe Esbensen and Bertsch model w4x. Also the calculated width of the1q level is about 64 keV. Thus there seems to be good agreement withthe results of Esbensen and Bertsch, when we use their parameter values Žas far as possible.. We now consider the use of the standard parameter values Ž2., in the two-level, four-channel approximation. We find S E2 Žr.rSE2 Žnr. s 0.03, SE2 Ž1.0.rSE2 Ž0.2. s 5.3, SE2 rSE1 s 8.7 = 10y4 at the peak, and G Ž1q. s 32 keV. The main changes are the
(
Table 1 Effect on the 7 BeŽp,g . 8 B cross section of moderate changes in the standard parameter values Ž2. Parameter modified
Change in parameter
G Ž1q . ŽkeV.
SE2 Žr.r SE2 Žnr.
S E2 Ž1.0.r S E2 Ž0.2.
SE2 r SE1 Ž10y4 .
SE2 Žnr.r SE1 Ž10y4 .
32
0.03
5.3
8.7
8.5
29 34 33 29 37 36
0.00 0.05 0.05 0.06 0.04 0.02
5.5 5.2 5.3 5.3 5.3 5.3
8.0 9.2 8.7 8.9 8.8 8.7
8.0 8.8 8.3 8.4 8.4 8.5
32
0.09
5.4
9.3
8.5
32 39
0.05 0.17
5.3 5.1
8.9 9.4
8.4 8.0
Standard parameter set
™ 4.0 ™ 6.0 ™ 2.5 ™ 0.52 ™ 0.5 ™ 0.6 0.5012 ™ 0.7071 0.8746 ™ 0.7071 5 0.76 ™ 0.0 2,4 ™ 1,1
Modified parameter sets a Žfm. 5.0 5.0 R Žfm. 2.391 a WS Žfm. 0.65 1r2 Ž q . 0.3988 Sss 1 11 1r2 Ž q . Sss 0.5431 2 11 1r2 q Sss 1 Ž2 1 . 1r 2 q Sss 2 Ž2 1 . g 11g Žint. m,n1 1
m-level, n-channel approximation.
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
253
reductions in the first ratio and in the width. These values and others showing the effect of moderate changes in the parameter values Ž2. are given in Table 1. The calculated widths are mostly consistent with the experimental value of 37 " 5 keV. The ratios S E2 Ž1.0.rSE2 Ž0.2. and SE2rSE1 are close to the values of Esbensen and Bertsch. All the values of the ratio SE2 Žr.rSE2 Žnr. are, however, appreciably smaller than the value of Esbensen and Bertsch.
4. Discussion The shape of the nonresonant contribution found here agrees wellwith that of Esbensen and Bertsch w2,3x. Essentially the same shape, as measured by SE2 Ž1.0.rSE2 Ž0.2., was found by Krauss et al. w6x and by Typel et al. w8x, but a smaller ratio near 4 was obtained by Descouvemont and Baye w13x and by Bennaceur et al. w14x. When we use the parameter values Ž2. or moderate variations of them, we find values of S E2 Žr.rSE2 Žnr. that are smaller than that found by Esbensen and Bertsch w2,3x. The basic reason for this reduction can be seen most simply in the one-level, one-channel approximation. From Eq. Ž1., the resonant channel contribution is then proportional to
Ý Ss1r2 Ž 1q1 .
Ss1r2 Ž 2q 1 . U Ž 211s,12 . ,
Ž 4.
s
with a positive coefficient since i 1q 1 Ž1200 <10. s yŽy 2r5 . ) 0. With the values Ž3., the quantity Ž4. is 0.5590 y 0.1118 s 0.4472; with the values Ž2. it is 0.1731 y 0.1840 s y0.0109. In the first case, the resonant contributions from the channel and internal regions addconstructively; in the second case, they have opposite signs, leading toa 1r2 Ž q. reduced resonant contribution. The reduction is due mainly to thevalue of Sss 1 11 being smaller in Ž2. than in Ž3.. There is experimental evidence to support the value in Ž2.. The width of the 1q level obtained from Ž2. is closer to the experimental value than that from Ž3.. The total spectroscopic factor for the 1q level is 0.454 from Ž2. and 1.0 from Ž3., compared with the experimental value of 0.48 obtained fromstripping for the . Sss1Ž1q . is 1.85 from mirror state in 8 Li w10x. Also the channel spin ratio Sss 2 Ž1q 1 rS 1 Ž2. and 0.20 from Ž3., while the measured ratio for the analog 17.64 MeV level in 8 Be is 3.2 " 0.5 w10x Žalthough this value is affected somewhat by isospin mixing w20x.. Thus we think that a value of SE2 Žr. appreciably smaller than that given by Esbensen and Bertsch w2,3x is more justifiable. Further improvement might be obtainedby normalizing the spectroscopic amplitudes Ž2. to fit either the measuredwidth of the 1q level of 8 B, or the measured spectroscopic factors for 8 Li, S Ž1q. s 0.48 and S Ž2q. s 0.87 w10x. In view of the differences in the calculated values of G Ž1q. and of S E2 Žr., and because the measurement of Davids et al. w1x was not very sensitive to the resonant E2 strength but rather to the nonresonant strength spread over a range of energies w4x, it would seem better to use the experimental data w1x to give a value of SE2 Žnr.rSE1 rather than SE2 rSE1. The final column in Table 1 gives values of SE2 Žnr.rSE1 at the peak energy obtained with the present model. Renormalization of the spectroscopic amplitudes would not change these values. The Esbensen and Bertsch model w2,3x gives
'
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
254
S E2 Žnr.rSE1 s 7.8 = 10y4 at 0.63 MeV w4x. For the other calculations, our estimates of this ratio Žin units 10y4 . are 7.4 w5x, 7.7 w6x, 6.2 w8x, 6.1 w13x and 6.5 w14,15x. Our calculated values in Table 1 are higher than these, and appreciably higher than that obtained from the experimental data of Davids et al. w1x, which require a ratio of .3 y4 w x 5.5q2 4 ; however, as pointed out by Davids et al.w1,4x, this experimental y1 .6 = 10 value is to be considered as a lower limit ifhigher-order dynamical effects are significant in the Coulomb dissociation. In summary, the resonant E2 strength that is calculated here using R-matrix formulae and shell model spectroscopic factors is much smaller than that found in previous calculations, while the nonresonant E2 strength as a fraction of the E1 strength is slightly larger than that found in the potential-model calculation of Esbensen and Bertsch w2,3x, and appreciably larger than that obtained from fitting the experimental data of Davids et al. w1x using first-order perturbation theory.
Acknowledgements The author is grateful to B. Davids and M. Ploszajczak for communicating additional information about their results.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
B. Davids et al., Phys. Rev. Lett. 81 Ž1998. 2209. H. Esbensen, G.F. Bertsch, Phys. Lett. B 359 Ž1995. 13. H. Esbensen, G.F. Bertsch, Nucl. Phys. A 600 Ž1996. 37. B. Davids, private communication. K.H. Kim, M.H. Park, B.T. Kim, Phys. Rev. C 35 Ž1987. 363. H. Krauss, K. Grun, ¨ T. Rauscher, H. Oberhummer, Ann. Phys. 2 Ž1993. 258. S. Typel, G. Baur, Phys. Rev. C 50 Ž1994. 2104. S. Typel, H.H. Wolter, G. Baur, Nucl. Phys. A 613 Ž1997. 147. F.C. Barker, Phys. Rev. C 37 Ž1988. 2920. F. Ajzenberg-Selove, Nucl. Phys. A 490 Ž1988. 1. S. Cohen, D. Kurath, Nucl. Phys. A 101 Ž1967. 1. P. Descouvemont, D. Baye, Nucl. Phys. A 567 Ž1994. 341. P. Descouvemont, D. Baye, Phys. Rev. C 60 Ž1999. 015803. K. Bennaceur, F. Nowacki, J. Okolowicz, M. Ploszajczak, Nucl. Phys. A 651 Ž1999. 289. M. Ploszajczak, private communication. F.C. Barker, Aust. J. Phys. 33 Ž1980. 177. F.C. Barker, Nucl. Phys. A 588 Ž1995. 693. A.M. Lane, R.G. Thomas, Rev. Mod. Phys. 30 Ž1958. 257. F.C. Barker, T. Kajino, Aust. J. Phys. 44 Ž1991. 369. F.C. Barker, Nucl. Phys. 83 Ž1966. 418.
E2 contributions to the 7 Be žp,g /8 B cross section F.C. Barker a
a
Department of Theoretical Physics, School of Physical Sciences and Engineering, The Australian National UniÕersity, Canberra, ACT 0200, Australia Received 16 August 1999; accepted 8 September 1999
Abstract A calculation of the E2 component of the 7 BeŽp,g . 8 B cross section atlow energies, using R-matrix formulae that include channel contributions,finds the resonant strength due to the 1q first excited state of 8 B to be appreciably smaller than was obtained in previous calculations, butgives a nonresonant strength slightly higher than before. q 1999 Elsevier Science B.V. All rights reserved. PACS: 24.10.yi; 25.40.Lw; 27.20.qn Keywords: 7 BeŽp, g . 8 B E2 cross section; R-matrix formulae
1. Introduction The measured cross section for the 7 BeŽp,g . 8 B reaction at lowenergies has a pronounced peak at Ecm f 0.63 MeV, which may be attributed to M1 transitions from the 1q first excited state of 8 B to the 2q ground state. Underlying the region is a nonresonant E1 contribution. Small resonant and nonresonant E2 contributions are alsoexpected. In a recent Coulomb dissociation measurement, Davids et al. w1x determined the ratio of the E2 and E1 strengths near the peak energy. They compared their results with the first-order perturbation theory calculations of Esbensen and Bertsch w2,3x, who used a potential model that gave S E2 rSE1 s 9.5 = 10y4 at Ecm s 0.6 MeV Žand 10.4 = 10y4 at the peak at 0.63 MeV w4x.. Best agreement was obtained with an E2 .8 y4 strength 0.7 times as large as the model value, leading to SE2 rSE1 s 6.7q2 at y1.9 = 10 0.6 MeV w1x. Other potential-model calculations w5–8x gave varying values of S E2 rSE1; e.g., as Davids et al. w1x pointed out, Kim et al. w5x found a value about twice that of Esbensen and Bertsch w2,3x. In their calculation of the E2 S factor, Esbensen and Bertsch found a resonant contribution due to p-wave protons superimposed on an appreciable nonresonant background due about equally to p- and f-waveprotons. As the resonant contribution is coherent with part of thebackground, the shape of the resonant peak is somewhat distorted.The nonresonant E2 strength, S E2 Žnr., in all these calculations w2,3,5–8x has a 0375-9474r99r$15.00 q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 3 8 9 - 9
250
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
similar value, particularly when expressed as a fraction ofthe E1 strength Žand when one allows for the omission of the f-wavecontribution by Typel and Baur w7x.. The peak-energy resonant E2 strengths, however, differ widely in the different calculations. We define the resonant E2 strength as S E2 Žr. s S E2 y SE2 Žnr. evaluated at the peak energy. Esbensen and Bertschw2,3x found S E2 Žr.rSE2 Žnr. f 0.3. They used a model in which both the 1q state and the 2q ground state are assumed to have a 7 BeŽg.s.. q 1p 3r2 proton single-particle structure. Kim et al. w5x and Krauss et al. w6x Žin their favoured calculation with spin-orbit coupling in the entrance channel only. appeared to use the same model,but both found S E2 Žr.rSE2 Žnr. f 1.4. The potentials used in these latter calculations, however, are independent of the total J value of the initial capturing state, so differing from the potentials used by Esbensen and Bertsch,and the resonance erroneously contains 0q, 2q and 3q contributions as well as 1q w9x, so accounting for the larger values of the ratio. Typel and Baur w7x apparently assumed a 7 BeŽg.s.. q 1p proton single-particle structure for both 2q and 1q states, with coupling to channel spin s s 2 only, and they obtained a ratio of about 1.0 Žwhere, however, the nonresonant strength contains only the p-wave part.. Typel et al. w8x used essentially the same model but included the f-wave contribution, and found a ratio of about 0.5. All of these potential-model calculations give widths of the 1q state of 8 B appreciably greater than the experimental value of G Ž1q. s 37 " 5 keV w10x; they are about 70 keV w2,3,5x, 100 keV w6x and 78 keV w7,8x, giving evidence that the descriptions of the 1q state are not adequate. These models assume a spectroscopic factor S Ž1q. s 1, although shell model calculations give much smaller values; e.g., Cohen and Kurath w11x give S 3r2 Ž1q. s 0.3215 and S 1r2 Ž1q. s 0.1240 for p 3r2 and p 1r2 protons, respectively. This defect was previously pointed out w9x for the model of Kim et al. w5x. In a three-cluster GCM calculation, Descouvemont and Baye w12x found a resonant E2 strength appreciably greater than that of Esbensen and Bertsch w2,3x, but as a fraction of the E1 strength it is almost the same. Theirwidth of the 1q state is about 54 keV. In a more recent paper w13x, they gave more details of their results, from which one finds S E2 Žr.rSE2 Žnr. f 0.4 at the peak. In contrast with the above results, Bennaceur et al. w14x, from calculations using the shell model embedded in the continuum, gaveseveral values for the width of the 1q level, all being less than theexperimental value. Their largest value is 25.9 keV, and corresponding to this they gave S E2 rSE1 s 7.72 = 10y4 at the peak, which isconsistent with the experimental value w1x. They also found S E2 Žr.rSE2 Žnr. s 0.187 w15x. In view of the differences between the results of these modelcalculations, in particular for S E2 Žr., and the disagreements with experiment, particularly for the width of the 1q level, we here use adifferent approach, which permits a realistic description of the 1q state, in order to calculate the E2 contributions to the 7 BeŽp, g . 8 Bcross section.
2. R-matrix formulae and procedure In Ref. w16x, a potential-model approach was used to calculate thenonresonant E1 contribution to the 7 BeŽp, g . 8 B cross section, while the resonant M1 contribution was
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
251
fitted using R-matrix formulae. InRef. w17x, the E1 component was also calculated using formulae fromR-matrix theory w18x, extended for photon reactions by the inclusion ofchannel contributions w19x. Here such R-matrix formulae are used to calculate the E2 part of the 7 BeŽp, g . 8 B cross section. The main difference from the formulae given in Ref. w17x is that Eq. Ž6. there is replaced by UsJe il i s yie iŽ v iy f i . 2 P l1r2 kg5r2 i
J glJs Ý Alm i
lm
i e li
½
gmgJ i Ž int . q
(
7
53 Mn1r2 e
15 128
=a 3 Nf1r2 Ý i lq1gmJci u f a s Ž l 200 <10 . U Ž 21 Ji s, l 2 . J2 c Ž l ,1 . c
(
q
7
53 Mn1r2 e
15 128
"k
"
5
a2 Nf1r2 i l iq1u f a e s eŽ l i 200 <10 . U Ž 21 Ji s e , l i 2 . Fl i Ž a .
=G l i Ž a . J2X Ž l i ,1 . ,
Ž 1.
where the labels J f s 2 and l f s 1 are omitted. Also the formulae corresponding to Eqs. Ž11. – Ž13. in Ref. w17x have an extra factor rra ineach integrand. We include p- and f-wave contributions Ž l i s 1,3; Ji s 0–4.. For simplicity, we assume gmJci s 0 for l s 3, and allow for up to two levels for each Ji s 0–3 Žand for J f s 2.. We also allow for up to four proton channels, corresponding to the first four states of 7 Be. The level energies and spectroscopic amplitudes are taken initially from the shell model calculation of Barker w20x, with the energies of the lowest 2q, 1q and 3q levels adjusted to agree with the experimental values w10x. For simplicity, we choose Bc s Sc Ž E1J i . for each Ji value. We also calculate g 11g Žint. from the wavefunctions of Ref. w20x, but take other gmgJ i Žint. s 0. This is because we are primarily interested in the E2 cross section in the neighbourhood of the 1q first excited state of 8 B. For the same reason, we study the effect of changing the spectroscopic amplitudes only for this 1q state and for the 2q ground state, both for the 7 Be ground-state channels. We also consider a range of values of the channel radius a, and of the parameters R and a WS of theWoods–Saxon potentials used for calculating reduced-width amplitudes from the spectroscopic amplitudes. Some calculations are also made in the one-level, one-channel approximation for each Ji and J f value. As ‘standard’ values of these adjustable parameters, we use a s 5.0 fm,
R s 2.391 fm,
g 11g Ž int . s 0.76 MeV 1r2 fm5r2 , 1r2 q Sss 2 Ž 11 . s 0.5431 ,
a WS s 0.65 fm , 1r2 Sss1 Ž 1q 1 . s 0.3988 ,
1r2 Sss1 Ž 2q1 . s 0.5012 ,
together with the two-level, four-channel approximation.
1r2 Sss2 Ž 2q1 . s 0.8746 ,
Ž 2.
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
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3. Results Initially we consider calculations in which the parameter valuesare chosen to agree as well as possible with those used in thecalculation of Esbensen and Bertsch w2,3x. Thus we assume a one-level,one-channel approximation with R s 2.391 fm, a WS s 0.52 fm, g 11g Ž int . s 0.35 MeV 1r2 fm5r2 , 1 5 1r2 q 1r2 Sss Sss2 Ž 1q 1 Ž 11 . s 6 s 0.9129 , 1 .s '6 s 0.4082 , 1 1 1r2 q 1r2 Sss s 0.7071 , Sss2 2q Ž 3. Ž 1 Ž 21 . s 1 .s '2 '2 s 0.7071 . These values of the spectroscopic amplitudes correspond to p 3r2 protonsin each case. The value of g 11g Žint. is somewhat uncertain, asinternal wave functions are required and the assumed 7 BeŽg.s.. q 1p 3r2 proton structure is not fully antisymmetrized, because the 7 Be ground state also contains 1p 3r2 protons. With these values and a s 5.0 fm, we find SE2 Žr.rSE2 Žnr. s 0.32. ŽFor a s 6.0 fm, the ratio is 0.27; for a s 5.0 fm and g 11g Žint. s 0, the ratio is 0.14.. There is goodagreement with the ratio found by Esbensen and Bertsch w2,3x. The energy dependence of the background, as measured say by SE2 Ž1.0 MeV.r SE2 Ž0.2 MeV., also agrees reasonably with Esbensen and Bertsch Ž4.9 compared with 5.2..Using S E1 calculated as in Ref. w15x with the parameter values Ž3., we find S E2 rSE1 s 10.4 = 10y4 at the peak energy, the same as predicted bythe Esbensen and Bertsch model w4x. Also the calculated width of the1q level is about 64 keV. Thus there seems to be good agreement withthe results of Esbensen and Bertsch, when we use their parameter values Žas far as possible.. We now consider the use of the standard parameter values Ž2., in the two-level, four-channel approximation. We find S E2 Žr.rSE2 Žnr. s 0.03, SE2 Ž1.0.rSE2 Ž0.2. s 5.3, SE2 rSE1 s 8.7 = 10y4 at the peak, and G Ž1q. s 32 keV. The main changes are the
(
Table 1 Effect on the 7 BeŽp,g . 8 B cross section of moderate changes in the standard parameter values Ž2. Parameter modified
Change in parameter
G Ž1q . ŽkeV.
SE2 Žr.r SE2 Žnr.
S E2 Ž1.0.r S E2 Ž0.2.
SE2 r SE1 Ž10y4 .
SE2 Žnr.r SE1 Ž10y4 .
32
0.03
5.3
8.7
8.5
29 34 33 29 37 36
0.00 0.05 0.05 0.06 0.04 0.02
5.5 5.2 5.3 5.3 5.3 5.3
8.0 9.2 8.7 8.9 8.8 8.7
8.0 8.8 8.3 8.4 8.4 8.5
32
0.09
5.4
9.3
8.5
32 39
0.05 0.17
5.3 5.1
8.9 9.4
8.4 8.0
Standard parameter set
™ 4.0 ™ 6.0 ™ 2.5 ™ 0.52 ™ 0.5 ™ 0.6 0.5012 ™ 0.7071 0.8746 ™ 0.7071 5 0.76 ™ 0.0 2,4 ™ 1,1
Modified parameter sets a Žfm. 5.0 5.0 R Žfm. 2.391 a WS Žfm. 0.65 1r2 Ž q . 0.3988 Sss 1 11 1r2 Ž q . Sss 0.5431 2 11 1r2 q Sss 1 Ž2 1 . 1r 2 q Sss 2 Ž2 1 . g 11g Žint. m,n1 1
m-level, n-channel approximation.
F.C. Barkerr Nuclear Physics A 660 (1999) 249–254
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reductions in the first ratio and in the width. These values and others showing the effect of moderate changes in the parameter values Ž2. are given in Table 1. The calculated widths are mostly consistent with the experimental value of 37 " 5 keV. The ratios S E2 Ž1.0.rSE2 Ž0.2. and SE2rSE1 are close to the values of Esbensen and Bertsch. All the values of the ratio SE2 Žr.rSE2 Žnr. are, however, appreciably smaller than the value of Esbensen and Bertsch.
4. Discussion The shape of the nonresonant contribution found here agrees wellwith that of Esbensen and Bertsch w2,3x. Essentially the same shape, as measured by SE2 Ž1.0.rSE2 Ž0.2., was found by Krauss et al. w6x and by Typel et al. w8x, but a smaller ratio near 4 was obtained by Descouvemont and Baye w13x and by Bennaceur et al. w14x. When we use the parameter values Ž2. or moderate variations of them, we find values of S E2 Žr.rSE2 Žnr. that are smaller than that found by Esbensen and Bertsch w2,3x. The basic reason for this reduction can be seen most simply in the one-level, one-channel approximation. From Eq. Ž1., the resonant channel contribution is then proportional to
Ý Ss1r2 Ž 1q1 .
Ss1r2 Ž 2q 1 . U Ž 211s,12 . ,
Ž 4.
s
with a positive coefficient since i 1q 1 Ž1200 <10. s yŽy 2r5 . ) 0. With the values Ž3., the quantity Ž4. is 0.5590 y 0.1118 s 0.4472; with the values Ž2. it is 0.1731 y 0.1840 s y0.0109. In the first case, the resonant contributions from the channel and internal regions addconstructively; in the second case, they have opposite signs, leading toa 1r2 Ž q. reduced resonant contribution. The reduction is due mainly to thevalue of Sss 1 11 being smaller in Ž2. than in Ž3.. There is experimental evidence to support the value in Ž2.. The width of the 1q level obtained from Ž2. is closer to the experimental value than that from Ž3.. The total spectroscopic factor for the 1q level is 0.454 from Ž2. and 1.0 from Ž3., compared with the experimental value of 0.48 obtained fromstripping for the . Sss1Ž1q . is 1.85 from mirror state in 8 Li w10x. Also the channel spin ratio Sss 2 Ž1q 1 rS 1 Ž2. and 0.20 from Ž3., while the measured ratio for the analog 17.64 MeV level in 8 Be is 3.2 " 0.5 w10x Žalthough this value is affected somewhat by isospin mixing w20x.. Thus we think that a value of SE2 Žr. appreciably smaller than that given by Esbensen and Bertsch w2,3x is more justifiable. Further improvement might be obtainedby normalizing the spectroscopic amplitudes Ž2. to fit either the measuredwidth of the 1q level of 8 B, or the measured spectroscopic factors for 8 Li, S Ž1q. s 0.48 and S Ž2q. s 0.87 w10x. In view of the differences in the calculated values of G Ž1q. and of S E2 Žr., and because the measurement of Davids et al. w1x was not very sensitive to the resonant E2 strength but rather to the nonresonant strength spread over a range of energies w4x, it would seem better to use the experimental data w1x to give a value of SE2 Žnr.rSE1 rather than SE2 rSE1. The final column in Table 1 gives values of SE2 Žnr.rSE1 at the peak energy obtained with the present model. Renormalization of the spectroscopic amplitudes would not change these values. The Esbensen and Bertsch model w2,3x gives
'
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S E2 Žnr.rSE1 s 7.8 = 10y4 at 0.63 MeV w4x. For the other calculations, our estimates of this ratio Žin units 10y4 . are 7.4 w5x, 7.7 w6x, 6.2 w8x, 6.1 w13x and 6.5 w14,15x. Our calculated values in Table 1 are higher than these, and appreciably higher than that obtained from the experimental data of Davids et al. w1x, which require a ratio of .3 y4 w x 5.5q2 4 ; however, as pointed out by Davids et al.w1,4x, this experimental y1 .6 = 10 value is to be considered as a lower limit ifhigher-order dynamical effects are significant in the Coulomb dissociation. In summary, the resonant E2 strength that is calculated here using R-matrix formulae and shell model spectroscopic factors is much smaller than that found in previous calculations, while the nonresonant E2 strength as a fraction of the E1 strength is slightly larger than that found in the potential-model calculation of Esbensen and Bertsch w2,3x, and appreciably larger than that obtained from fitting the experimental data of Davids et al. w1x using first-order perturbation theory.
Acknowledgements The author is grateful to B. Davids and M. Ploszajczak for communicating additional information about their results.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
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