CORE POLARIZATION EFFECTS OF SOME ODD SD-SHELL NUCLEI USING M3Y EFFECTIVE NUCLEON-NUCLEON INTERACTION

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1 NUCLEAR PHYSICS CORE POLARIZATION EFFECTS OF SOME ODD SD-SHELL NUCLEI USING M3Y EFFECTIVE NUCLEON-NUCLEON INTERACTION KHALID S. JASSIM, and HASSAN ABU KASSIM Department o Physics, College o Education or Pure Science, University o Babylon, PO Box 4, Hillah-Babylon, Iraq Khalid_ik74@yahoo.com Department o Physics, Faculty o Science, University o Malaya, Kuala Lumpur, Malaysia Received July 8, 0 Inelastic electron scattering orm actors in some odd-a sd-shell nuclei ( 7 O, 7 Al and 39 K) have been calculated taking into account higher energy conigurations outside sdshell model space, which are called core-polarization (CP) eects. The two-body wildenthal interaction is used or the sd-shell model space. The two-body Michigan three Yukawas (M3Y) interactions are used or the core-polarization CP matrix elements. This interaction was given in LS-coupling. A transormation between LS and jj must perormed to get the relation between the two-body shell model matrix elements and the relative and center o mass coordinates, using the harmonic oscillator radial wave unctions with Talmi-Moshinsky transormation. The sd-shell model calculations ail to describe the data in both the transition strength and the orm actor; the inclusion o CP eects modiies the orm actors markedly and describes the experimental data very well in both the absolute strength and the momentum transer dependence. Key words: Inelastic electron scattering, orm actor, Shell model. PACS B 5.30.Dh.60.Cs 7.0.+n (6 A 9) t (0 A 38). INTRODUCTION The central problem in nuclear physics has long been the nucleon-nucleon interaction. As it is well known, this interaction is usually described by a potential based on a meson exchange. There are a number o realistic models or potentials. They all describe the nucleon-nucleon scattering data very well []. Several theoretical models have been introduced or this purpose, shell-model is one o them and it leads to a very successul description. The sd-shell is considered in the present work, this model deals with only the distribution and coupling o the valence nucleons within a ew model-space orbits. According to this model the 6 O is considered as an inert core. The active orbits o the model-space are labeled by the quantum numbers: nlj=d 5/, s /, d 3/. The nucleons o the core ( 6 O) are 8 protons and 8 neutrons which are inert in the (s /,p 3/,p / ) J=0,T=0 coniguration and the remaining A-6 nucleons are distributed over all possible combinations o the d 5/,s / and d 3/ orbits according to Pauli Exclusion Principle. To investigate Rom. Journ. Phys., Vol. 58, Nos. 3 4, P , Bucharest, 03

2 30 Khalid S. Jassim, Hassan Abu Kassim the eects o the higher conigurations outside the sd-shell model space on the closed core, the core-polarization eects are introduced. Core-polarization eects are taken into account through irst order perturbation theory, which allows particle-hole excitation rom the sd-shell core orbits and also rom the valence sd-shells to the higher allowed orbits with 6 ω excitations. Several shell-model calculations or sd-shell nuclei are available. The work o Wildenthal [] is adopted here. Coulomb orm actors o C4 transitions in even-even nuclei are discussed [3] taking into account core-polarization eects. Higher conigurations are taken into account through a microscopic theory, which allow particle-hole excitations rom the s and p shells core orbits and also rom the sd-shell orbits to the higher allowed orbits with excitations up to 4 ω. The core-polarization is ound essential in both the transition strengths and momentum transer dependence o orm actors, and gives a remarkably good agreement with the measured data with no adjustable parameters. The calculations are based on the wildenthal interaction or the sd-shell model space and the Modiied Surace Delta Interaction (MSDI) or the core-polarization eects. A microscopic model has been recently used [4] in order to study the irst order CP eects on C orm actor o p-shell nuclei. Those calculations depend on the realistic two-body eective interaction (M3Y) as a residual interaction to generate the core-polarization matrix elements that be added to the model-space matrix elements. The results are quite successul and describe the data very well in both the transition strength and momentum transer dependence. The longitudinal orm actors or electron scattering have been calculated or p-shell nuclei [5] using enlarged model space includes all orbits in p and s-d shells. The aim o the present work is to consider the particle-hole excitations o the core and model space to calculate the longitudinal orm actors or electron scattering rom some sd-shell odd-odd nuclei. A more realistic nucleon-nucleon interaction is adopted in the present work or core-polarization calculation which is called the Michigan three Yukawas (M3Y) realistic two-body interactions [6] where its parameters are adjust rom the nucleon-nucleon scattering data. So, we do not adjust any parameters in the calculations o the various matrix elements.. THEORY The reduced matrix elements o the electron scattering operator T ˆ η consist o two parts, one is or the Model space matrix elements, and the other is or the Core-polarization matrix elements Γ Tˆ η Γ = Γ Tˆ η Γ + Γ δtˆ η Γ i i i MS CP ()

3 3 Core Polarization eects o some odd sd-shell nuclei 3 Where the state Γ i and Γ are described by the model-space wave unctions. Greek symbols are used to denote quantum numbers in coordinate space and isospin, i.e. Γ i J iti, Γ J T and JT. The Model Space (MS) matrix elements are expressed as the sum o the product o the one-body density matrix elements (OBDM) times the single-particle matrix elements, which is given by: ˆη T OBDM(, ) Tˆη Γ Γ = αβ α β i MS αβ, where α and β denote the inal and initial single particle states respectively (isospin is included) or the model space. Similarly, the core-polarization matrix element in equation () can be written as ollows: ˆη T OBDM(, ) Tˆη Γ δ Γ = αβ αδ β i cp αβ, According to the irst order perturbation theory, the single-particle matrix element or the higher-energy conigurations is given by [7]: ˆη Q ˆη ˆη αδt β=αv T β+αt Q V β (4) J res (0) J J (0) res E H E H i The operator Q is the projection operator onto the space outside the model space. For the residual interaction, V res, we adopt the M3Y [6]. E i and E are the energies o the initial and inal states, respectively. H (0) is the unperturbed Hamiltonian. Equation (4) is written as [7] αδˆt β= β+α +Γ ( ) e e e + e α, α, Γ β α α α Γ MS cp α β (Γ+ ) α α Γ ˆ η ααv βα α T α ( +δ )( +δ ) + terms with α res αα αβ and α exchanged with an overall minus sign, (5) where the index α runs over particle states and α over hole states and e is the single particle energy. The core-polarization parts are allowing particle-hole excitations rom the s-, p- and sd-shell orbits into higher orbits. These excitations are taken up to 4 ω. The reduced single particle matrix element becomes: () (3)

4 3 Khalid S. Jassim, Hassan Abu Kassim 4 where: ˆ T + α β = α ˆ α η η TJT IT ( t ) z TJt z t z I T or T = 0 ( t ) = z t z or T = ( ) (6) (7) where t z =/ or proton and -/ or a neutron. Electron scattering orm actor involving angular momentum J and momentum transer q, between the initial and inal nuclear shell model states o spin J i, and isospin T i, are [8] T 4 T Ti η π T T ( ) ( ) z η J = Γ J, T( ) Γi Z (J ) T = 0, i + T M Z T T Zi F q T q cm. ( ). s( ) F q F q (8) where T z is the projection along the z-axis o the initial and inal isospin states and is given by t z = (z-n)/. The nucleon inite size (s) orm actor is F s (q) = = exp (-0.43q /4) and F cm (q) = exp (q b /4a) is the correction or the lack o translation invariance in the shell model. A is the mass number and b is the harmonic oscillator size parameter. The single-particle energies are calculated according to [7]: with: ( l+ ) ( r) nl or j= l e = ( n+ l ) ω+ nlj l () r or nl j= l+ () r 0A /3MeV nl ω= 45A /3 5A /3 (9) (0)

5 5 Core Polarization eects o some odd sd-shell nuclei 33 For the two-body matrix elements o the residual interaction αα Vres βα, which appear in equations (5), the Michigan three Yukawas Γ (M3Y) interaction o Bertch et al. [6] is adopted. The interaction is taken between a nucleon in any core-orbits and nucleon that is excited to higher orbits with the same parity and with the required multipolarity ( ), and also between a nucleon in any sd orbits and that is excited to higher orbits with the same parity and with the required multipolarity. The orm o the potential is deined in equations ()-(3) in re. [6]. The parameters o Elliot are used which are given in Table o the mentioned reerence. This interaction was given in LS-coupling. A transormation between LS and jj must be perormed to get the relation between the two-body shell model matrix elements and the relative and center o mass coordinates, using the harmonic oscillator radial wave unctions with Talmi-Moshinsky transormation. 3. RESULT AND DISCUSSION In the present work, the core-polarization eects are included through microscopic theory to discuss electron scattering or the light doubly odd N=Z sdshell nuclei 7 O, 7 Al and 39 K. Core-polarization eects are taken into account through the irst-order perturbation theory, which allows particle-hole and two particles-two holes excitation respectively, rom the s and p shell core orbits to the higher allowed orbits with ω excitation. The one-body density matrix elements (OBDM) values or even sd-shell nuclei under consideration in the present work are taken rom re. [9]. The core-polarization single-particle matrix elements are calculated according to equation (5). The many-particle matrix element that includes both the model space and the core-polarization eects are calculated according to equation (). Finally, the nuclear orm actor can be obtained rom equations (8). The single-particle wave unctions are those o the HO potential with size parameters b, chosen to reproduce the root mean square charge radius. Oxygen is one o the important elements in our lie. The shell model calculation or 7 O assumed 6 O as an inert closed core and the remainder nucleon can occupy s / or d 5/ or d 3/ o the model space. In this transition the nucleus is π + excited rom the ground state ( J i T i = 5/ /) to the excited state π + ( J i T i = / /) by the electron with an excitation energy o 0.87 MeV. The single-particle wave unction or this transition is that o HO potential with size parameter b r.m.s =.763 m [0]. The result o sd-shell model calculations with

6 34 Khalid S. Jassim, Hassan Abu Kassim 6 inclusion o CP eect predicts the B(C) value to be.97 e m 4 in comparison with the measured value.8±0.6 e ƒm 4 [0], which is more close to the measured value. The longitudinal C electron scattering orm actor with core-polarization eect or this transition is shown in Figure (as a solid curve) and compare with the experimental data o re. []. The results estimates the experimental data in the irst and the second maximum region; these results with core-polarization eect are very good especially at the second maximum region with the large bar errors and it is generally acceptable. This indicates that the M3Y suitable to describe the data at the high region o momentum transer upon disagreement o MSDI which is ails to describe the experimental data at the high momentum transer [3]. The one-body density matrix elements (OBDM) values or this transition are tabulated in Table. E-3 7 O C: 0.87 MeV (/ + /) E-4 F(q) E-5 E-6 E q(m - ) Fig. The inelastic longitudinal (C) orm actors or the / + / state at 0.78 MeV in 7 O, the solid curve is the single-particle model space calculations with core-polarization eects. The experimental data are taken rom Re. [3].

7 7 Core Polarization eects o some odd sd-shell nuclei 35 E-3 E-4 39 K C:.53 MeV(/ + ) F(q) E-5 E-6 E q(m - ) Fig. The inelastic longitudinal (C) orm actors or the / + / state at.53 MeV in 39 K, the solid and dashed curves are the single-particle model space calculations with and without core-polarization eects, respectively. The experimental data are taken rom Re. [4]. Table The one-body density matrix elements or 0.87 MeV (5/ + / / + /) and.53 MeV (3/ + / / + /) transition in 7 O and 39 K respectively Nucleus j a j b OBDM ( T=0) OBDM ( T=) 7 O 5/ / K 3/ / The results or the nucleus o 39 K are presented or the transition C (3/ + / / + /).53 MeV. The size parameter or the single-particle wave unction is taken to be: b r.m.s =.950 m [0]. In this transition, the nucleus is excited i T i J π T rom the ground state ( J π = 3/ + /) to the state ( =/ + /) with an

8 36 Khalid S. Jassim, Hassan Abu Kassim 8 excitation energy o.53 MeV due to electron scattering. The values o the OBDM elements or this transition are given in Table. Figure shows the calculations o the transverse C electron scattering orm actor with and without CP eect as a solid curve and dashed curve, respectively. The sd-shell model ails to describe the data in both the transition strength B(C) and the orm actor. The calculated B(C ) value without including core-polarization eects is.5 e m 4 which is low in comparison with the measured value 8.90±.8 e m 4 [0]. When the CP eect is included, the B(C ) value becomes 0.5 e m 4 which is closer to the measured value than that calculated with (0++4) ω extended model space [], where their calculated value is underestimated by a actor o. The inclusion o core-polarization eect gives an excellent agreement in the irst maximum and the second maximum region up to.8 m -, than overestimates the data beyond that. E- E-3 7 Al C: MeV (/ + ) E-4 F(q) E-5 E-6 E-7 E q(m - ) Fig. 3 The inelastic longitudinal (C) orm actors or the / + / state at MeV in 7 Al, the solid and dashed curves are the single-particle model space calculations with and without core-polarization eects, respectively. The experimental data are taken rom Re. [5].

9 9 Core Polarization eects o some odd sd-shell nuclei 37 E- E- E-3 7 Al C+C4:. MeV (7/ + ) F(q) E-4 E-5 E-6 E-7 E q(m - ) Fig. 4 The inelastic longitudinal (C+C4) orm actors or the 7/ + / state at. MeV in 7 Al, the solid and dashed curves are the single-particle model space calculations with and without core-polarization eects, respectively. The experimental data are taken rom Re. [5] Table The values o the OBDM elements or the longitudinal C transition to the / + / at E =0.844 MeV or 7 Al x j a j b OBDM ( T=0) OBDM ( T=) 5/ 5/ / / / 3/ / 5/ / 3/ / 5/ / / / 3/

10 38 Khalid S. Jassim, Hassan Abu Kassim 0 Table 3 The values o the OBDM elements or the longitudinal C transition to the 7/ + / at E =. MeV or 7 Al x j a j b OBDM ( T=0) OBDM ( T=) 5/ 5/ / / / 3/ / 5/ / 3/ / 5/ / / / 3/ Table 4 The values o the OBDM elements or the longitudinal C4 transition to the 7/ + / at E =. MeV or 7 Al x j a j b OBDM ( T=0) OBDM ( T=) 5/ 5/ / 3/ / 5/ The nucleus o the 7 Al has been studied both theoretically and experimentally. It is considered as 6 O and eleven nucleons are occupying the sd-shell space according to Pauli Exclusion Principle. Calculation are presented or the transitions rom the ground state (5/ + /) to the / + / state and 7/ + / at MeV and. MeV, respectively. The size parameter or the single-particle radial wave unction is taken to be b r.m.s =.804 m [0]. The experimental reduced transition strength B(C ) or / + / state at MeV is equal to.79±0.5 e m 4 [4] and the calculated one is 7 e m 4, which is higher than the experimental value. The OBDM values or this transition are listed in table (,3,4) and are taken rom re. [3]. The sd-shell model calculation o C orm actor including CP eects in / + / state at MeV gives a good result in the irst maximum up to q=.8 m - and the result is gradually shited rom the experimental data in which the curve underestimates this data, the deviation can be interpreted within the large bar errors as show in Figure 3. The experimental orm actor or. MeV (7/ + /) in 7 Al doublet o the C and C4. The theoretical results o (C+C4) orm actor or this state, which shown in Figure 4 give a good results in the irst maximum region up to q=.6 m -, but at the second maximum region it overestimates the data ater.8 m -.

11 Core Polarization eects o some odd sd-shell nuclei 39 Acknowledgements. The author would like to express his thank to Pro. Dr. R. A. Radhi, (Physics Department/ Science College/ Baghdad University/IRAQ) or his valuable notes. I thank University o Malaya, Faculty o Science as well as Department o Physics, University o Babylon, College o Education or Pure Science, Department o Physics or supporting this work. REFERENCE. T.T.S. Kuo, Scott Bogner and L. Coraggio: Nucl. Phys. A704, 07c (00).. B. M. Preedom and B. H. Wildenthal: Phys. Rev. C6, 633 (97). 3. R. A. Radhi: Eur. Phys. J. A6, 387 (003). 4. K. S. Jassim Ph.D. Thesis, College o Science, University o Baghdad (007). 5. R. A. Radhi, A. K. Hamoudi and K. S. Jassim: Indian Journal o Physics, 8, 7 (007) 6. G. J. Bertsch, Borysowicz and McManus, H.: Nucl. Phys. A84, (977) 7. P. J. Brussard and P. W. M. Glademans, Shell-model Application in Nuclear Spectroscopy, North-Holland Publishing Company Amsterdam (977). 8. T. William Donnelly and I. Sick: Review-Mod. Phys. 56, 46(984). 9. B. A. Brown R. A. Radhi and B. H. Wildenthal: Phys. Rep. 0, 33 (983). 0. B.A. Brown, W. Chung and B. H. Wildenthal: Phys. Rev. C, 774 (980).. D. M. Manley, D. J. Millener, B. L. Berman, W. Bertozzi, T. N. Buti, J. M Finn, F. W. Hersman, C.E. Hyde-Wright, M. V. Hynes, J.J. Kelly, M. A. KoVash, S. Kowalski, R. W. Lourie, B. Murdock, B. E. Norum, B. Pngh, and C.P. Sargent: Phys. Rev. C4, 448 (990).. S. Karataklidis, B.A. Brown, K. Amos, P.J Dortmans: Phys. Rev. C55 86 (997). 3. D. M. Manley, D. J. Millener, B. L. Berman, W. Bertozzi, T. N. Buti, J. M. Finn, F. W. Hersman, C.E. Hyde-Wright, M. V. Hynes, J.J. Kelly, M. A. KoVash, S. Kowalski, R. W. Lourie, B. Murdock, B. E Norum, B. Pngh, and C.P. Sargent: Phys. Rev. C36, 700 (987). 4. Th., G. Richter, A. G. Schrieder, E. Spamer, W. Stock: Nuclear Physics, A35, 769 (987). 5. P.J. Ryan, R.S. Hicks, A. Hotta, J. Dubach, G.A Peterson and D.V. Webb: Phys. Rev. C7, 55 (983).