- Email: [email protected]
Calculations of elastic and inelastic electron scattering on 19F using large-basis no core-shell model wave functions
Nuclear Physics A 798 (2008) 16–28 www.elsevier.com/locate/nuclphysa
Calculations of elastic and inelastic electron scattering on 19F using large-basis no core-shell model wave functions R.A. Radhi ∗ , A.A. Abdullah, A.H. Raheem Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq Received 4 July 2007; received in revised form 10 October 2007; accepted 15 October 2007 Available online 28 October 2007
Abstract Large basis no core shell model calculations are performed to study the elastic and inelastic electron scattering on 19 F. All major shells s, p, sd and pf are considered with (0 + 2)hω ¯ truncations. Excitations out of major shell space are taken into account through a microscopic theory which allows particle–hole excitations from the sd and pf shell orbits to all higher orbits with 2hω ¯ excitations. Calculations are presented for the transitions from J π T = 1/2+ 1/2 to J π T = 1/2+ 1/2, 3/2+ 1/2, 5/2+ 1/2, 7/2+ 1/2 and 9/2+ 1/2 in 19 F. Excitations out the no core shell model space are essential in obtaining a reasonable description of the longitudinal and transverse electron scattering form factors. © 2007 Elsevier B.V. All rights reserved. PACS: 25.30.Dh; 21.60.Cs; 27.20.+n Keywords: 19 F: Elastic and inelastic electron scattering form factors calculated with large basis no core model space including higher energy configurations
1. Introduction Large-basis no core-shell model calculations have been performed [1,2] for p-shell nuclei using six major shells (from 1s to 3p–2f –1h). In these calculations all nucleons are active. However, constrained by computer capabilities, one can use a truncated no-core calculation, where only those configurations are retained from the full no-core case in which there are up and includ* Corresponding author.
E-mail address: [email protected] (R.A. Radhi). 0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.10.010
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
17
ing few h¯ ω excitations of the lowest unperturbed configuration. As the number of hω ¯ increases, the result will converge and approach those of the full no-core calculations. It was observed [3] that the E2 transition rates obtained in the 4h¯ ω calculations for 6 Li are weaker than those calculated in the 6h¯ ω space. Shell model structure of low-lying excited states in 6,7 Li have been studied [4] using a multi-h¯ ω excitations. However, it was found that the result of the quadrupole moments were far from the experimental values, even with (0 + 2 + 4 + 6)h¯ ω wave functions. A clear improvement in most observables was evident for the calculation of the 10 C → 10 B Fermi matrix elements [5], where the size of the large model space was increased from 2h¯ ω to 4h¯ ω. Calculations of E2 transitions and quadrupole moments for A = 7–11 were underestimated [6] and there were still need for effective charges despite the large model space, 6h¯ ω for A = 7 nuclei, and 4h¯ ω for A = 8–10 nuclei. Convergent results were obtained for A = 3 and A = 4 with 50h¯ ω and 16h¯ ω, respectively [7]. Calculations were also presented for 12 C [7] for model space up to 5h¯ ω. In the present work, we will adopt a large no core harmonic oscillator (HO) model space considering the major shells 1s, 1p, 2s–1d, 2p–1f , to study elastic and inelastic electron scattering from 19 F. We will consider a (0 + 2)h¯ ω truncated no core calculation. The 0h¯ ω configuration is [(1s)4 (1p)12 (2s1d)3 ], while the 2h¯ ω configurations are [(1s)3 (1p)12 (2s1d)4 ] and [(1s)4 (1p)11 (2s1d)3 (2p1f )1 ] for one particle-one hole (1p–1h) excitations. Also, the configurations [(1s)4 (1p)10 (2s1d)5 ] and [(1s)4 (1p)12 (2s1d)1 (2p1f )21 ] are allowed for two particle– two hole (2p–2h) excitations. These excitations form the no-core model space. Excitations out of this model space will be taken into consideration through first-order perturbation theory, where a (1p–1h) excitation is allowed from 2s–1d and 2p–1f shells orbits to all higher orbits with 2h¯ ω excitations. So, the higher shells 3s–2d and 3p–2f are included. Calculations are presented for the transitions from J π T = 1/2+ 1/2 ground state to J π T = 1/2+ 1/2, 3/2+ 1/2, 5/2+ 1/2, 7/2+ 1/2 and 9/2+ 1/2 states in 19 F. 2. Theory The reduced matrix element of the electron scattering operator TˆΛ is expressed as the sum of the product of the elements of the one-body density matrix (OBDM) XΓΛf Γi (α, β) times the single-particle matrix elements, and is given by (Γf |TˆΛ |Γi ) = XΓΛf Γi (α, β)(α|TˆΛ |β), (1) αβ
where α and β label single-particle states (isospin is included) for the shell model space. The states |Γi and |Γf are described by the model space wave functions. Greek symbols are used to denote quantum numbers in coordinate space and isospace, i.e., Γi ≡ Ji Ti , Γf ≡ Jf Tf and Λ ≡ J T . According to the first-order perturbation theory, the single-particle matrix element of the one-body operator is given by [8] Q Vres β (α|TˆΛ |β) = α|TˆΛ |β + αTˆΛ Ei − H0 Q (2) + αVres TˆΛ β. E −H f
0
The first term is the zero-order contribution. The second and third terms are the first-order contributions which give the higher-energy configurations (hec). The operator Q is the projection operator onto the space outside the model space.
18
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Table 1 Excitation energies, reduced transition probabilities and effective charges. The experimental values are from Ref. [11] Jfπ
B(wL) (e2 fm2L )
Ex (MeV) Exp.
Theory
wL
Exp.
1/2+ 3/2+
0.0 1.554
0.0 1.514
5/2+ 7/2+ 9/2+
0.197 4.378 2.780
0.558 5.610 4.099
M1 M1 E2 E2 E4 E4
0.0547 – – 62.8(7) – –
The hec terms given in Eq. (2) are written as [8] (−1)β+α2 +Γ α β (2Γ + 1) α2 α1 eβ − eα − eα1 + eα2 α1 α2 Γ
Effective charges
Theory
proton
neutron
0.0639 0.191 × 10−4 38.53 60.03 75.39 42.10
– 1.233 1.217 1.479 1.231
– – 0.481 0.486 0.965 0.515
Λ (1 + δα1 α )(1 + δα2 β ) Γ
× αα1 |Vres |βα2 Γ α2 |TˆΛ |α1 + terms with α1 and α2 exchanged with an over all minus sign,
(3)
where the index α1 runs over particle states and α2 over hole states and e is the single-particle energy, and is calculated according to [8] −1/2( + 1)f (r)n for j = − 1/2, (4) en j = (2n + − 1/2)h¯ ω + 1/2 f (r)n for j = + 1/2, with f (r)n ≈ −20A−2/3 and h¯ ω = 45A−1/3 − 25A−2/3 . The single particle matrix elements reduced in both spin and isospin, are written in terms of the single-particle matrix elements reduced in spin only [8] 2T + 1 IT (tz )j2 Tˆj tz j1 , (5) α2 |TˆΛ |α1 = 2 tz 1 for T = 0, with IT (tz ) = (6) 1/2−t z (−1) for T = 1, where tz = 1/2 for a proton and −1/2 for a neutron. Higher energy configurations are taken into consideration through 1p–1h excitations from the model space orbits into higher orbits. All excitations are considered with 2h¯ ω excitations. Since we consider a truncated (0 + 2)h¯ ω spsdpf model space, so the 1p–1h excitations are considered through the sd and pf shells. For the residual two-body interaction Vres , the M3Y interaction of Bertsch et al. [9] is adopted. The form of the potential is defined in Eqs. (1)–(3) in Ref. [9]. The parameters of ‘Elliot’ are used which are given in Table 1 of the mentioned reference. A transformation between LS and jj is used to get the relation between the two-body shell model matrix elements and the relative and center of mass coordinates, using the harmonic oscillator radial wave functions with Talmi–Moshinsky transformation. Electron scattering form factor involving angular momentum J and momentum transfer q, between initial and final nuclear shell model states of spin Ji,f and isospin Ti,f are [10] 2 η 2 η 2 Tf T Ti 4π F (q) = ˆ (q)Ffs2 (q) (7) (Jf Tf TJ T Ji Ti ) Fcm J Z 2 (2Ji + 1) −T 0 T z z T =0,1
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
19
with η selecting the longitudinal (L), transverse electric (E) and transverse magnetic (M) form factors, respectively. Tz is given by Tz = (Z − N )/2. The nucleon finite size (fs) form factor is Ffs (q) = Exp(−0.43q 2 /4) and Fcm (q) = Exp(q 2 b2 /4A) is the correction for the lack of translational invariance in the shell model (center of mass correction), where A is the mass number and b is the harmonic oscillator size parameter. The reduced electric transition strength is given by
Z 2 (2J + 1)!! 2 L 2 FJ (k) , B(EJ ) = (8) 4π kJ where k = Ex /h¯ c. η The single-particle matrix elements j2 TˆJ tz j1 for the required electron scattering operators used in this work are those of Brown et al. [11]. The proton and neutron effective charges can be determined microscopically by comparing the form factor result calculated at the photon point including first-order correction with that of the zero-order correction. 3. Results and discussion Calculations of the form factors are presented for the Jfπ Tf (Ex MeV) = 1/2+ (0.0), 3/2+ (1.554), 5/2+ (0.197), 7/2+ (4.378) and 9/2+ (2.78) states. The measured electron scattering form factors to these states are available from Ref. [11], where the data cover a wide range of effective momentum transfer. The radial wave functions for the single-particle matrix elements were calculated with the harmonic oscillator (HO) potential. The oscillator length parameter b = 1.833 fm was chosen to reproduce the measured root mean square charge radius [11]. Large-basis no core model space is used in this study. This space covered the four shells 1s, 1p, 2s–1d and 2p–1f with (0 + 2)h¯ ω truncations. Shell model interactions encompassing the four oscillator shells have been constructed by Warburton and Brown [12]. These interactions are based interactions for the 1p2s1d shells determined by a least square fit to 216 energy levels in the A = 10–22 region assuming no mixing of nh¯ ω and (0 + 2)h¯ ω configurations. The 1p2s1d part of the interaction (cited in Ref. [12] as WBP) results from a fit to two-body matrix elements and single-particle energies for the p shell and a potential representation of the 1p–2s1d crossshell interaction. The WBP model space was expanded to include the 1s and 2p1f major shells by adding the appropriate 2p1f and cross-shell 2s1d–2p1f two-body matrix element of the Warburton–Becker–Milliner–Brown (WBMB) interaction [13] and all the other necessary matrix elements from the bare G-matrix potential of Hosaka, Kubo and Toki [14]. The 2s1d shell interaction of Wildenthal [15] used in WBP interaction is replaced in this study by a new interaction referred as USDB (Universal sd-shell B) [16], where the derivation of the USD Hamiltonian [15] has been refined with an up dated and complete set of energy data. The new Hamiltonian USDB leads to a new level of precision for realistic shell-model wave functions. Shell model calculations were performed with the shell-model code OXBASH [17], where the OBDM elements given in Eq. (1) were obtained. The first term in Eq. (2) is the zero-order contribution, which gives the single-particle matrix element for the large-basis no core model space (ms) contribution. The second and third terms are the first-order contributions which account for the higher energy configurations (hec). These configurations are taken through 1p–1h excitations from the 2s1d and 2p1f shells orbits into higher orbits with 2h¯ ω excitations. For the
20
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 1. Electroexcitation spectra of 19 F at q = 1.95 fm−1 . The solid lines are calculated with the truncated (0 + 2)h¯ ω no core model space and dashed lines are those of Ref. [11].
residual interaction Vres , the M3Y interaction of Bertsch et al. [9] is adopted as described in the theory. In Fig. 1 we present distributions of theoretical longitudinal and transverse form factor values which correspond to a momentum transfer value of 1.95 fm−1 , using truncated (0 + 2)h¯ ω no core model space (solid lines) and 0h¯ ω sd shell model space using USD interaction [15] (dashed lines). The longitudinal and transverse strengths for both model spaces are close to each other. The form factors calculated with the spsdpf model space matrix elements will be displayed as dashed curves and those include higher energy configurations as solid curves. The calculated and measured excitation energies and transition rates are given in Table 1. The calculated effective charges are also given in this table. 3.1. 0.0 MeV, 1/2+ 1/2 state Elastic Coulomb C0 form factor calculated with the model space wave functions is displayed in Fig. 2(a) in comparison with the experimental data of Ref. [11]. An over all agreement is
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
21
Fig. 2. Elastic form factors for the 1/2+ 1/2 ground state in 19 F. The upper panel (a) represents the C0 longitudinal component. The lower panel (b) represents the magnetic M1 form factors calculated with the spsdpf truncated (0 + 2)h¯ ω (dashed curve). The solid curve represents the calculation that includes higher energy configurations. The data are taken from Refs. [11] and [18] for the C0 and M1 form factors, respectively.
obtained through all the experimental momentum transfer values. Elastic magnetic M1 form factors are displayed in Fig. 2(b). The no core-model space result is shown by the dashed curve in comparison with the experimental data of Ref. [18]. The form factor fails to describe the data in the region of 1 < q < 1.5 fm−1 , where the experimental data exhibit a second maximum. Extending the model space to include the shells 3s–2d–1g and 3p–2f –1h through 1p–1h excitations with 2h¯ ω predicts an intermediate maximum and describes the q data very well and slightly over predicts the high q data for the third maximum as shown by the solid curve.
22
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 3. Inelastic form factors for the 3/2+ 1/2 (1.554 MeV) state in 19 F. The upper panel (a) represents the C2 longitudinal component. The lower panel (b) represents the M1 + E2 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
3.2. 1.554 MeV, 3/2+ 1/2 state The longitudinal C2 form factor for this state is shown in Fig. 3(a). The no core-model space result (dashed curve) under predicts the data by a bout a factor of five at the first maximum. The calculated B(C2↑) value is 7.624 e2 fm4 . When higher energy configurations are included, the form factor explains the data very well as shown by the solid curve in Fig. 3(a). The predicted transition rate B(C2↑) becomes 38.53 e2 fm4 . The calculated effective charges at the photon point are 1.233e and 0.481e for the proton and neutron, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
23
Fig. 4. Inelastic form factors for the 3/2+ 1/2 (1.554 MeV) state in 19 F calculated with the sd shell model space (0h¯ ω) (plus symbols) in comparison with the spsdpf truncated (0 + 2)h¯ ω model space (dashed curves). The data are taken from Ref. [11].
The transverse M1 + E2 form factors are given in Fig. 3(b). Extending the model space to include higher energy configurations reduces the form factors as shown by the solid curve. The experimental data are explained better by the no-core model space result (dashed curve). To compare our model space results with those of Ref. [11], we display both results in Fig. 4. The results of the 0h¯ ω calculations of Ref. [11] (sd model space only) are displayed as plus symbols in comparison with those of (0 + 2)h¯ ω calculations (dashed curves). Both results are close to each other and the longitudinal form factors under predict the data by about a factor of five at the maximum. In the 2h¯ ω truncated calculations, one cannot claim to have performed a full
24
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 5. Inelastic form factors for the 5/2+ 1/2 (0.197 MeV) state in 19 F. The upper panel (a) represents the C2 longitudinal component. The lower panel (b) represents the M3 + E2 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
no core calculations since not all nucleons are active, and there still is a partially inert core. So, in this case, when the contribution of the higher shells are taken into consideration, the longitudinal form factor enhanced appreciably and agree very well with the data as shown in Fig. 3(a). In the case of the 0h¯ ω calculations, the sd shell model space results agree with those of the truncated (0 + 2)h¯ ω. When the sd shell model space transition density is combined with Tassie collective model space transition density [19], the results of Ref. [11] agree with our results (ms + hec) and with the experimental data. The B(C2) in Ref. [11] are calculated using effective charges 1.35e and 0.35e, for the protons and neutrons, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
25
Fig. 6. Inelastic form factors for the 7/2+ 1/2 (4.378 MeV) state in 19 F. The upper panel (a) represents the C4 longitudinal component. The lower panel (b) represents the M3 + E4 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
3.3. 0.197 MeV, 5/2+ 1/2 state The no-core spsdpf model space fails to describe the C2 form factor data for this state as shown in Fig. 5(a), by the dashed curve. The predicted B(C2↑) value is 12.72 e2 fm4 which is a factor of about one fifth the measured value (62.8 e2 fm4 ) [11]. Extending the no-core model space to include hec explains the C2 form factor data remarkably well, as shown by the solid curve in Fig. 5(a). The predicted B(C2↑) value is 60.03 e2 fm4 which is very close to the experimental value. The calculated effective charges are 1.217e and 0.486e for the proton and
26
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 7. Inelastic form factors for the 9/2+ 1/2 ( 2.78 MeV) state in 19 F. The upper panel (a) represents the C4 longitudinal component. The lower panel (b) represents the M5 + E4 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
neutron, respectively. The transverse form factor (E2 + M3) is less affected by including hec, and quenched by a small factor from that of the spsdpf model space. 3.4. 4.378 MeV, 7/2+ 1/2 state The longitudinal C4 and transverse M3 + E4 form factors for this state are well described by including hec, as shown by the solid curve in Figs. 6(a) and 6(b), respectively. The calculated effective charges are 1.479e and 0.965e for the proton and neutron, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
27
3.5. 2.78 MeV, 9/2+ 1/2 state The C4 spsdpf form factor under predicts the experimental data at the maximum by about a factor of three, as shown in Fig. 7(a) by the dashed curve. The experimental data are reasonably well explained by including hec as shown by the solid curve. The calculated effective charges are 1.231e and 0.515e for the proton and neutron, respectively. The transverse E4 + M5 form factor is less affected by including hec and both models under predict the data. 4. Conclusions The effect of higher energy configurations out of the large basis no core spsdpf model space with (0 + 2)h¯ ω on electron scattering from 19 F is investigated. In this 2h¯ ω truncated calculation, one cannot claim to have performed a full no core calculations since not all nucleons are active, and there still is a partially inert core. As the size of the model space is increased, the results will approach those of the full no core calculations (indication towards convergence). However, such enlargement is constrained by computer capabilities. The results with the (0 + 2)h¯ ω truncated calculations are still far from the experimental values, especially those of the electric transition rates and longitudinal form factors. Mixing with states that lie outside the model space is taking into account through one particle–one hole 2h¯ ω excitations. These mixings improve the agreement with the experimental data significantly in the absolute strength and the q dependence form factors. Acknowledgements The authors would like to express their thanks to Professor B.A. Brown of the National superconducting cyclotron laboratory, Michigan State University for providing them the computer code OXBASH. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
D.C. Zheng, J.P. Vary, B.R. Barrett, Phys. Rev. C 50 (1994) 2841. D.C. Zheng, B.R. Barrett, J.P. Vary, W.C. Haxton, Phys. Rev. C 52 (1995) 2488. P. Navratil, M. Thoresen, B.R. Barrett, Phys. Rev. C 55 (1997) R573. S. Karataklidis, B.A. Brown, K. Amos, P.J. Dortmass, Phys. Rev. C 55 (1997) 2826. P. Navratil, B.R. Barrett, W.E. Ormand, Phys. Rev. C 56 (1997) 2542. P. Navratil, B.R. Barrett, Phys. Rev. C 57 (1998) 3119. B.R. Barrett, P. Navratil, P.J. Vary, Nucl. Phys. A 704 (2002) 254. P.J. Brussaard, P.W.M. Glaudemans, Shell Model Applications in Nuclear Spectroscopy, North-Holland, Amsterdam, 1977. G. Berstch, J. Borysowicz, H. McManus, W.G. Love, Nucl. Phys. A 284 (1977) 399. T.W. Donnely, I. Sick, Rev. Mod. Phys. 56 (1984) 461. B.A. Brown, B.H. Wildenthal, C.F. Williamson, F.N. Rad, S. Kowalski, H. Crannell, J.T. O’Brien, Phys. Rev. C 32 (1985) 1127. E.K. Warburton, B.A. Brown, Phys. Rev. C 46 (1992) 923. E.K. Warburton, J.A. Becker, B.A. Brown, Phys. Rev. C 41 (1990) 1147. A. Hosaka, K.-I. Kubo, H. Toki, Nucl. Phys. A 244 (1985) 76. B.A. Brown, B.H. Wildental, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. B.A. Brown, W.A. Richter, Phys. Rev. C 74 (2006) 034315.
28
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
[17] B.A. Brown, A. Etchegoyen, N.S. Godwin, W.D.M. Rae, W.A. Richter, W.E. Ormand, E.K. Warburton, J.S. Winfield, L. Zhao, C.H. Zimmerman, MSU-NSCL report number 1289 2005 version. [18] A.J.H. Donee, G. Van Middelkoop, L. Lapikas, T. Suzuli, P.W.M. Glaudemanns, D. Zwartz, Nucl. Phys. A 455 (1986) 453. [19] L.J. Tassie, Aust. J. Phys. 9 (1956) 407.
Calculations of elastic and inelastic electron scattering on 19F using large-basis no core-shell model wave functions R.A. Radhi ∗ , A.A. Abdullah, A.H. Raheem Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq Received 4 July 2007; received in revised form 10 October 2007; accepted 15 October 2007 Available online 28 October 2007
Abstract Large basis no core shell model calculations are performed to study the elastic and inelastic electron scattering on 19 F. All major shells s, p, sd and pf are considered with (0 + 2)hω ¯ truncations. Excitations out of major shell space are taken into account through a microscopic theory which allows particle–hole excitations from the sd and pf shell orbits to all higher orbits with 2hω ¯ excitations. Calculations are presented for the transitions from J π T = 1/2+ 1/2 to J π T = 1/2+ 1/2, 3/2+ 1/2, 5/2+ 1/2, 7/2+ 1/2 and 9/2+ 1/2 in 19 F. Excitations out the no core shell model space are essential in obtaining a reasonable description of the longitudinal and transverse electron scattering form factors. © 2007 Elsevier B.V. All rights reserved. PACS: 25.30.Dh; 21.60.Cs; 27.20.+n Keywords: 19 F: Elastic and inelastic electron scattering form factors calculated with large basis no core model space including higher energy configurations
1. Introduction Large-basis no core-shell model calculations have been performed [1,2] for p-shell nuclei using six major shells (from 1s to 3p–2f –1h). In these calculations all nucleons are active. However, constrained by computer capabilities, one can use a truncated no-core calculation, where only those configurations are retained from the full no-core case in which there are up and includ* Corresponding author.
E-mail address: [email protected] (R.A. Radhi). 0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.10.010
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
17
ing few h¯ ω excitations of the lowest unperturbed configuration. As the number of hω ¯ increases, the result will converge and approach those of the full no-core calculations. It was observed [3] that the E2 transition rates obtained in the 4h¯ ω calculations for 6 Li are weaker than those calculated in the 6h¯ ω space. Shell model structure of low-lying excited states in 6,7 Li have been studied [4] using a multi-h¯ ω excitations. However, it was found that the result of the quadrupole moments were far from the experimental values, even with (0 + 2 + 4 + 6)h¯ ω wave functions. A clear improvement in most observables was evident for the calculation of the 10 C → 10 B Fermi matrix elements [5], where the size of the large model space was increased from 2h¯ ω to 4h¯ ω. Calculations of E2 transitions and quadrupole moments for A = 7–11 were underestimated [6] and there were still need for effective charges despite the large model space, 6h¯ ω for A = 7 nuclei, and 4h¯ ω for A = 8–10 nuclei. Convergent results were obtained for A = 3 and A = 4 with 50h¯ ω and 16h¯ ω, respectively [7]. Calculations were also presented for 12 C [7] for model space up to 5h¯ ω. In the present work, we will adopt a large no core harmonic oscillator (HO) model space considering the major shells 1s, 1p, 2s–1d, 2p–1f , to study elastic and inelastic electron scattering from 19 F. We will consider a (0 + 2)h¯ ω truncated no core calculation. The 0h¯ ω configuration is [(1s)4 (1p)12 (2s1d)3 ], while the 2h¯ ω configurations are [(1s)3 (1p)12 (2s1d)4 ] and [(1s)4 (1p)11 (2s1d)3 (2p1f )1 ] for one particle-one hole (1p–1h) excitations. Also, the configurations [(1s)4 (1p)10 (2s1d)5 ] and [(1s)4 (1p)12 (2s1d)1 (2p1f )21 ] are allowed for two particle– two hole (2p–2h) excitations. These excitations form the no-core model space. Excitations out of this model space will be taken into consideration through first-order perturbation theory, where a (1p–1h) excitation is allowed from 2s–1d and 2p–1f shells orbits to all higher orbits with 2h¯ ω excitations. So, the higher shells 3s–2d and 3p–2f are included. Calculations are presented for the transitions from J π T = 1/2+ 1/2 ground state to J π T = 1/2+ 1/2, 3/2+ 1/2, 5/2+ 1/2, 7/2+ 1/2 and 9/2+ 1/2 states in 19 F. 2. Theory The reduced matrix element of the electron scattering operator TˆΛ is expressed as the sum of the product of the elements of the one-body density matrix (OBDM) XΓΛf Γi (α, β) times the single-particle matrix elements, and is given by (Γf |TˆΛ |Γi ) = XΓΛf Γi (α, β)(α|TˆΛ |β), (1) αβ
where α and β label single-particle states (isospin is included) for the shell model space. The states |Γi and |Γf are described by the model space wave functions. Greek symbols are used to denote quantum numbers in coordinate space and isospace, i.e., Γi ≡ Ji Ti , Γf ≡ Jf Tf and Λ ≡ J T . According to the first-order perturbation theory, the single-particle matrix element of the one-body operator is given by [8] Q Vres β (α|TˆΛ |β) = α|TˆΛ |β + αTˆΛ Ei − H0 Q (2) + αVres TˆΛ β. E −H f
0
The first term is the zero-order contribution. The second and third terms are the first-order contributions which give the higher-energy configurations (hec). The operator Q is the projection operator onto the space outside the model space.
18
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Table 1 Excitation energies, reduced transition probabilities and effective charges. The experimental values are from Ref. [11] Jfπ
B(wL) (e2 fm2L )
Ex (MeV) Exp.
Theory
wL
Exp.
1/2+ 3/2+
0.0 1.554
0.0 1.514
5/2+ 7/2+ 9/2+
0.197 4.378 2.780
0.558 5.610 4.099
M1 M1 E2 E2 E4 E4
0.0547 – – 62.8(7) – –
The hec terms given in Eq. (2) are written as [8] (−1)β+α2 +Γ α β (2Γ + 1) α2 α1 eβ − eα − eα1 + eα2 α1 α2 Γ
Effective charges
Theory
proton
neutron
0.0639 0.191 × 10−4 38.53 60.03 75.39 42.10
– 1.233 1.217 1.479 1.231
– – 0.481 0.486 0.965 0.515
Λ (1 + δα1 α )(1 + δα2 β ) Γ
× αα1 |Vres |βα2 Γ α2 |TˆΛ |α1 + terms with α1 and α2 exchanged with an over all minus sign,
(3)
where the index α1 runs over particle states and α2 over hole states and e is the single-particle energy, and is calculated according to [8] −1/2( + 1)f (r)n for j = − 1/2, (4) en j = (2n + − 1/2)h¯ ω + 1/2 f (r)n for j = + 1/2, with f (r)n ≈ −20A−2/3 and h¯ ω = 45A−1/3 − 25A−2/3 . The single particle matrix elements reduced in both spin and isospin, are written in terms of the single-particle matrix elements reduced in spin only [8] 2T + 1 IT (tz )j2 Tˆj tz j1 , (5) α2 |TˆΛ |α1 = 2 tz 1 for T = 0, with IT (tz ) = (6) 1/2−t z (−1) for T = 1, where tz = 1/2 for a proton and −1/2 for a neutron. Higher energy configurations are taken into consideration through 1p–1h excitations from the model space orbits into higher orbits. All excitations are considered with 2h¯ ω excitations. Since we consider a truncated (0 + 2)h¯ ω spsdpf model space, so the 1p–1h excitations are considered through the sd and pf shells. For the residual two-body interaction Vres , the M3Y interaction of Bertsch et al. [9] is adopted. The form of the potential is defined in Eqs. (1)–(3) in Ref. [9]. The parameters of ‘Elliot’ are used which are given in Table 1 of the mentioned reference. A transformation between LS and jj is used to get the relation between the two-body shell model matrix elements and the relative and center of mass coordinates, using the harmonic oscillator radial wave functions with Talmi–Moshinsky transformation. Electron scattering form factor involving angular momentum J and momentum transfer q, between initial and final nuclear shell model states of spin Ji,f and isospin Ti,f are [10] 2 η 2 η 2 Tf T Ti 4π F (q) = ˆ (q)Ffs2 (q) (7) (Jf Tf TJ T Ji Ti ) Fcm J Z 2 (2Ji + 1) −T 0 T z z T =0,1
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
19
with η selecting the longitudinal (L), transverse electric (E) and transverse magnetic (M) form factors, respectively. Tz is given by Tz = (Z − N )/2. The nucleon finite size (fs) form factor is Ffs (q) = Exp(−0.43q 2 /4) and Fcm (q) = Exp(q 2 b2 /4A) is the correction for the lack of translational invariance in the shell model (center of mass correction), where A is the mass number and b is the harmonic oscillator size parameter. The reduced electric transition strength is given by
Z 2 (2J + 1)!! 2 L 2 FJ (k) , B(EJ ) = (8) 4π kJ where k = Ex /h¯ c. η The single-particle matrix elements j2 TˆJ tz j1 for the required electron scattering operators used in this work are those of Brown et al. [11]. The proton and neutron effective charges can be determined microscopically by comparing the form factor result calculated at the photon point including first-order correction with that of the zero-order correction. 3. Results and discussion Calculations of the form factors are presented for the Jfπ Tf (Ex MeV) = 1/2+ (0.0), 3/2+ (1.554), 5/2+ (0.197), 7/2+ (4.378) and 9/2+ (2.78) states. The measured electron scattering form factors to these states are available from Ref. [11], where the data cover a wide range of effective momentum transfer. The radial wave functions for the single-particle matrix elements were calculated with the harmonic oscillator (HO) potential. The oscillator length parameter b = 1.833 fm was chosen to reproduce the measured root mean square charge radius [11]. Large-basis no core model space is used in this study. This space covered the four shells 1s, 1p, 2s–1d and 2p–1f with (0 + 2)h¯ ω truncations. Shell model interactions encompassing the four oscillator shells have been constructed by Warburton and Brown [12]. These interactions are based interactions for the 1p2s1d shells determined by a least square fit to 216 energy levels in the A = 10–22 region assuming no mixing of nh¯ ω and (0 + 2)h¯ ω configurations. The 1p2s1d part of the interaction (cited in Ref. [12] as WBP) results from a fit to two-body matrix elements and single-particle energies for the p shell and a potential representation of the 1p–2s1d crossshell interaction. The WBP model space was expanded to include the 1s and 2p1f major shells by adding the appropriate 2p1f and cross-shell 2s1d–2p1f two-body matrix element of the Warburton–Becker–Milliner–Brown (WBMB) interaction [13] and all the other necessary matrix elements from the bare G-matrix potential of Hosaka, Kubo and Toki [14]. The 2s1d shell interaction of Wildenthal [15] used in WBP interaction is replaced in this study by a new interaction referred as USDB (Universal sd-shell B) [16], where the derivation of the USD Hamiltonian [15] has been refined with an up dated and complete set of energy data. The new Hamiltonian USDB leads to a new level of precision for realistic shell-model wave functions. Shell model calculations were performed with the shell-model code OXBASH [17], where the OBDM elements given in Eq. (1) were obtained. The first term in Eq. (2) is the zero-order contribution, which gives the single-particle matrix element for the large-basis no core model space (ms) contribution. The second and third terms are the first-order contributions which account for the higher energy configurations (hec). These configurations are taken through 1p–1h excitations from the 2s1d and 2p1f shells orbits into higher orbits with 2h¯ ω excitations. For the
20
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 1. Electroexcitation spectra of 19 F at q = 1.95 fm−1 . The solid lines are calculated with the truncated (0 + 2)h¯ ω no core model space and dashed lines are those of Ref. [11].
residual interaction Vres , the M3Y interaction of Bertsch et al. [9] is adopted as described in the theory. In Fig. 1 we present distributions of theoretical longitudinal and transverse form factor values which correspond to a momentum transfer value of 1.95 fm−1 , using truncated (0 + 2)h¯ ω no core model space (solid lines) and 0h¯ ω sd shell model space using USD interaction [15] (dashed lines). The longitudinal and transverse strengths for both model spaces are close to each other. The form factors calculated with the spsdpf model space matrix elements will be displayed as dashed curves and those include higher energy configurations as solid curves. The calculated and measured excitation energies and transition rates are given in Table 1. The calculated effective charges are also given in this table. 3.1. 0.0 MeV, 1/2+ 1/2 state Elastic Coulomb C0 form factor calculated with the model space wave functions is displayed in Fig. 2(a) in comparison with the experimental data of Ref. [11]. An over all agreement is
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
21
Fig. 2. Elastic form factors for the 1/2+ 1/2 ground state in 19 F. The upper panel (a) represents the C0 longitudinal component. The lower panel (b) represents the magnetic M1 form factors calculated with the spsdpf truncated (0 + 2)h¯ ω (dashed curve). The solid curve represents the calculation that includes higher energy configurations. The data are taken from Refs. [11] and [18] for the C0 and M1 form factors, respectively.
obtained through all the experimental momentum transfer values. Elastic magnetic M1 form factors are displayed in Fig. 2(b). The no core-model space result is shown by the dashed curve in comparison with the experimental data of Ref. [18]. The form factor fails to describe the data in the region of 1 < q < 1.5 fm−1 , where the experimental data exhibit a second maximum. Extending the model space to include the shells 3s–2d–1g and 3p–2f –1h through 1p–1h excitations with 2h¯ ω predicts an intermediate maximum and describes the q data very well and slightly over predicts the high q data for the third maximum as shown by the solid curve.
22
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 3. Inelastic form factors for the 3/2+ 1/2 (1.554 MeV) state in 19 F. The upper panel (a) represents the C2 longitudinal component. The lower panel (b) represents the M1 + E2 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
3.2. 1.554 MeV, 3/2+ 1/2 state The longitudinal C2 form factor for this state is shown in Fig. 3(a). The no core-model space result (dashed curve) under predicts the data by a bout a factor of five at the first maximum. The calculated B(C2↑) value is 7.624 e2 fm4 . When higher energy configurations are included, the form factor explains the data very well as shown by the solid curve in Fig. 3(a). The predicted transition rate B(C2↑) becomes 38.53 e2 fm4 . The calculated effective charges at the photon point are 1.233e and 0.481e for the proton and neutron, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
23
Fig. 4. Inelastic form factors for the 3/2+ 1/2 (1.554 MeV) state in 19 F calculated with the sd shell model space (0h¯ ω) (plus symbols) in comparison with the spsdpf truncated (0 + 2)h¯ ω model space (dashed curves). The data are taken from Ref. [11].
The transverse M1 + E2 form factors are given in Fig. 3(b). Extending the model space to include higher energy configurations reduces the form factors as shown by the solid curve. The experimental data are explained better by the no-core model space result (dashed curve). To compare our model space results with those of Ref. [11], we display both results in Fig. 4. The results of the 0h¯ ω calculations of Ref. [11] (sd model space only) are displayed as plus symbols in comparison with those of (0 + 2)h¯ ω calculations (dashed curves). Both results are close to each other and the longitudinal form factors under predict the data by about a factor of five at the maximum. In the 2h¯ ω truncated calculations, one cannot claim to have performed a full
24
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 5. Inelastic form factors for the 5/2+ 1/2 (0.197 MeV) state in 19 F. The upper panel (a) represents the C2 longitudinal component. The lower panel (b) represents the M3 + E2 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
no core calculations since not all nucleons are active, and there still is a partially inert core. So, in this case, when the contribution of the higher shells are taken into consideration, the longitudinal form factor enhanced appreciably and agree very well with the data as shown in Fig. 3(a). In the case of the 0h¯ ω calculations, the sd shell model space results agree with those of the truncated (0 + 2)h¯ ω. When the sd shell model space transition density is combined with Tassie collective model space transition density [19], the results of Ref. [11] agree with our results (ms + hec) and with the experimental data. The B(C2) in Ref. [11] are calculated using effective charges 1.35e and 0.35e, for the protons and neutrons, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
25
Fig. 6. Inelastic form factors for the 7/2+ 1/2 (4.378 MeV) state in 19 F. The upper panel (a) represents the C4 longitudinal component. The lower panel (b) represents the M3 + E4 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
3.3. 0.197 MeV, 5/2+ 1/2 state The no-core spsdpf model space fails to describe the C2 form factor data for this state as shown in Fig. 5(a), by the dashed curve. The predicted B(C2↑) value is 12.72 e2 fm4 which is a factor of about one fifth the measured value (62.8 e2 fm4 ) [11]. Extending the no-core model space to include hec explains the C2 form factor data remarkably well, as shown by the solid curve in Fig. 5(a). The predicted B(C2↑) value is 60.03 e2 fm4 which is very close to the experimental value. The calculated effective charges are 1.217e and 0.486e for the proton and
26
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
Fig. 7. Inelastic form factors for the 9/2+ 1/2 ( 2.78 MeV) state in 19 F. The upper panel (a) represents the C4 longitudinal component. The lower panel (b) represents the M5 + E4 transverse component. The calculations are performed with the spsdpf truncated (0 + 2)h¯ ω (dashed curves) and with those including higher energy configurations (solid curves). The data are taken from Ref. [11].
neutron, respectively. The transverse form factor (E2 + M3) is less affected by including hec, and quenched by a small factor from that of the spsdpf model space. 3.4. 4.378 MeV, 7/2+ 1/2 state The longitudinal C4 and transverse M3 + E4 form factors for this state are well described by including hec, as shown by the solid curve in Figs. 6(a) and 6(b), respectively. The calculated effective charges are 1.479e and 0.965e for the proton and neutron, respectively.
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
27
3.5. 2.78 MeV, 9/2+ 1/2 state The C4 spsdpf form factor under predicts the experimental data at the maximum by about a factor of three, as shown in Fig. 7(a) by the dashed curve. The experimental data are reasonably well explained by including hec as shown by the solid curve. The calculated effective charges are 1.231e and 0.515e for the proton and neutron, respectively. The transverse E4 + M5 form factor is less affected by including hec and both models under predict the data. 4. Conclusions The effect of higher energy configurations out of the large basis no core spsdpf model space with (0 + 2)h¯ ω on electron scattering from 19 F is investigated. In this 2h¯ ω truncated calculation, one cannot claim to have performed a full no core calculations since not all nucleons are active, and there still is a partially inert core. As the size of the model space is increased, the results will approach those of the full no core calculations (indication towards convergence). However, such enlargement is constrained by computer capabilities. The results with the (0 + 2)h¯ ω truncated calculations are still far from the experimental values, especially those of the electric transition rates and longitudinal form factors. Mixing with states that lie outside the model space is taking into account through one particle–one hole 2h¯ ω excitations. These mixings improve the agreement with the experimental data significantly in the absolute strength and the q dependence form factors. Acknowledgements The authors would like to express their thanks to Professor B.A. Brown of the National superconducting cyclotron laboratory, Michigan State University for providing them the computer code OXBASH. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
D.C. Zheng, J.P. Vary, B.R. Barrett, Phys. Rev. C 50 (1994) 2841. D.C. Zheng, B.R. Barrett, J.P. Vary, W.C. Haxton, Phys. Rev. C 52 (1995) 2488. P. Navratil, M. Thoresen, B.R. Barrett, Phys. Rev. C 55 (1997) R573. S. Karataklidis, B.A. Brown, K. Amos, P.J. Dortmass, Phys. Rev. C 55 (1997) 2826. P. Navratil, B.R. Barrett, W.E. Ormand, Phys. Rev. C 56 (1997) 2542. P. Navratil, B.R. Barrett, Phys. Rev. C 57 (1998) 3119. B.R. Barrett, P. Navratil, P.J. Vary, Nucl. Phys. A 704 (2002) 254. P.J. Brussaard, P.W.M. Glaudemans, Shell Model Applications in Nuclear Spectroscopy, North-Holland, Amsterdam, 1977. G. Berstch, J. Borysowicz, H. McManus, W.G. Love, Nucl. Phys. A 284 (1977) 399. T.W. Donnely, I. Sick, Rev. Mod. Phys. 56 (1984) 461. B.A. Brown, B.H. Wildenthal, C.F. Williamson, F.N. Rad, S. Kowalski, H. Crannell, J.T. O’Brien, Phys. Rev. C 32 (1985) 1127. E.K. Warburton, B.A. Brown, Phys. Rev. C 46 (1992) 923. E.K. Warburton, J.A. Becker, B.A. Brown, Phys. Rev. C 41 (1990) 1147. A. Hosaka, K.-I. Kubo, H. Toki, Nucl. Phys. A 244 (1985) 76. B.A. Brown, B.H. Wildental, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. B.A. Brown, W.A. Richter, Phys. Rev. C 74 (2006) 034315.
28
R.A. Radhi et al. / Nuclear Physics A 798 (2008) 16–28
[17] B.A. Brown, A. Etchegoyen, N.S. Godwin, W.D.M. Rae, W.A. Richter, W.E. Ormand, E.K. Warburton, J.S. Winfield, L. Zhao, C.H. Zimmerman, MSU-NSCL report number 1289 2005 version. [18] A.J.H. Donee, G. Van Middelkoop, L. Lapikas, T. Suzuli, P.W.M. Glaudemanns, D. Zwartz, Nucl. Phys. A 455 (1986) 453. [19] L.J. Tassie, Aust. J. Phys. 9 (1956) 407.