Electron scattering form factors with core-polarization effects on 6 Li nucleus. Khalid S. Jassim and Fouad A. MAJEED

Transcription

1 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 Electron scatterng orm actors wth core-polarzaton eects on 6 L nucleus. Khald S. Jassm and Fouad A. MAJEED,Department o Physcs, College o Educaton/Ibn Hayyan, Babylon Unversty, P.O.Box 4, Hlla-Babel, Iraq. E-mal: kald974@googlemal.com, E-mal: ouadalajeel@yahoo.com Jun 00 Abstract: Nucleon-Nucleon (NN) nteracton derved rom nuclear realstc potental s used to study the elastc and nelastc longtudnal and transverse electron scatterng orm actors n 6 L nucle, where a mcroscopc theory s employed to nclude the eects o hgh conguraton outsde the p-shell model space, whch s called corepolarzaton (cp) eects. The Cohen-Kurath nteracton or p-shell model space s used wth the realstc Mchgan three Yukawas (M3Y) eectve NN nteractons as a resdual nteracton or the core-polarzaton (cp) matrx elements calculatons. These nteractons are produced n terms o LS-couplng, and then transormed n terms o the couplng o the total angular momentum (jj-couplng) to make them sutable or use n electron scatterng orm actors calculatons. The radal wave unctons or the sngle-partcle matrx elements have been calculated wth the harmonc oscllator (HO) potental. The eect o core-polarzaton s ound essental or the transton strengths (B(C)) and the q-dependent orm actors, and mproves the agreement wth the expermental data remarkably well, especally or Coulomb scatterng. Keywords: p-shell nucle, (e,e) nelastc longtudnal orm actors, calculaton wth model space ncludng core-polarzaton eects. - Introducton: The rst dea about the nature o the nucleon-nucleon nteracton came rom Yukawa n 935 []. He assumed that the strong nteracton between the nucleons s carred out by an exchange o a partcle o a medum heavy mass o about 00 MeV [], called meson. It s called π -meson whch has ound to be the carrer o the strong nuclear orce. But durng the tes t was dscovered other mesons whch contrbuted to the nucleon-nucleon nteracton. One o the hgh ponts o ths development was the suggeston o Gregory Bret n 958 [3] that the short range repulson should be due to a vector, soscalar meson, the omega meson wth a mass o about 800 MeV. It was a bg success o the meson exchange theory o the nucleon-nucleon nteracton when ths meson was ound also expermentally []. Electron scatterng rom nucle s regarded as a successul and powerull tool n the nvestgaton o nuclear structure. Varous mcroscopc and macroscopc theores have been perormed to study exctaton n nucle. Shell model wthn a restrcted model space s one o the models, whch succeeded n descrbng the statc propertes o nucle, when eectve charges are used. In spte o the success o the p-shell model on statc propertes o nucle n ths regon, t als to descrbe electron scatterng data at hgh momentum transer. Extendng the model space to nclude the ћω conguratons mproves the agreement wth the transverse orm actor n the begnnng o the p-shell but towards the end o the p-shell, the stuaton deterorates [4]. The eectve nteracton s used to understand nuclear propertes mcroscopcally,

2 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 startng wth realstc NN nteracton and usng mechancal many-body theory. For lght nucle, there are several standard eectve nteractons such as the Cohen- Kurath [5] nteractons or p-shell. The concept o the core-polarzaton eects has been ntroduced n order to account or the partcpaton o conguratons rom outsde o the model space n the transton. Core-polarzaton eects are taken nto account through rst order perturbaton theory, whch allows partcle-hole exctaton rom the s-shell core orbts and also rom the valence p-shells to the hgher allowed orbts wth ћω exctatons. The goal o study s to use the realstc eectve nucleon-nucleon (NN) nteracton as a resdual nteracton to calculate the core-polarzaton (cp) eects through a mcroscopc theory whch combnes shell model wave unctons and hghly excted states. We wll dscuss the core-polarzaton eects on the elastc and nelastc electron scatterng orm actors or the low lyng states o 6 L nucle. The CK wave unctons wll be adopted as the zeroth-order wave unctons. The NN nteracton adopted n the present study s a realstc nteracton between two nucleons; t s expressed as a sum o three parts, central potental part, spn-orbt potental part, and tensor potental part. Ths nteracton s used as the resdual nteracton between nucleons n the core, and nucleons excted to hgher orbts. The theoretcal part o the present work ncludes the ormulatons o the elastc and the nelastc electron scatterng and wll be perormed n chapter two. The dervaton o cp eects wth hgher conguraton n the rst order perturbaton theory and the two-body matrx elements o three part o the realstc nteracton: central, spn orbt and strong tensor orce wll be ntroduced n chapter three, where the two-body matrx elements calculaton n harmonc oscllator sngle-partcle bass usng Moshnsky transormaton. The results, dscussons and concluson wll be demonstrated n chapter our. -Theory: The reduced matrx elements o the electron scatterng operator Tˆ consst o two parts, one s or the "Model space" matrx elements, and the other s or the "Corepolarzaton" matrx elements [6]. Tˆ Where the state Tˆ and MS. Tˆ () CP are descrbed by the model-space wave unctons. Greek symbols are used to denote quantum numbers n coordnate space and sospn,.e. J T, J T and JT. The model space (MS) matrx elements are expressed as the sum o the product o the one-body densty matrx elements (OBDM) tmes the sngle-partcle matrx elements, whch s gven by: Tˆ OBDM (, ) Tˆ () MS, where and denote the nal and ntal sngle partcle states respectvely (sospn s ncluded) or the model space. MS

3 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 Smlarly, the core-polarzaton matrx element n equaton () can be wrtten as ollows: Tˆ cp OBDM (, ) Tˆ, cp (3) Accordng to the rst order perturbaton theory, the sngle-partcle matrx element or the hgher-energy conguratons s gven by [7]: Q Q Tˆ V Tˆ Tˆ V (4) J res (0) J J (0) res EH EH The operator Q s the projecton operator onto the space outsde the model space. For the resdual nteracton, V res, we adopt the M3Y [G. Bertsch, J. Borysowcz and H. McManus]. E and E are the energes o the ntal and nal states, respectvely. Equaton (4) s wrtten as [7] Tˆ e ( ) e e e,, ( ) ˆ V res T ( )( ) + terms wth and exchanged wth an overall mnus sgn, (5) where the ndex runs over partcle states and over hole states and e s the sngle partcle energy. The core-polarzaton parts are allowng partcle-hole exctatons rom the s-, p- and sd-shell orbts nto hgher orbts. These exctatons are taken up to 4. The reduced sngle partcle matrx element becomes: ˆ T ˆ (6) TJT IT ( t ) TJt z z tz where: or T 0 (7) IT ( tz) t z or T ( ) where t z / or proton and -/ or a neutron. Electron scatterng orm actor nvolvng angular momentum J and momentum transer q, between the ntal and nal nuclear shell model states o spn J, and sospn T, are [8] F T T 4 T T T ( ) ( ) z J q TJ, T ( q) Z (J ) T 0, T M Z T T Z c. m( q) F. s( q) F (8) 3

4 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 where T z s the projecton along the z-axs o the ntal and nal sospn states and s gven by t z = (z-n)/. The nucleon nte sze (s) orm actor s F s (q) = exp (-o.43q /4) and F cm (q) = exp (q b /4a) s the correcton or the lack o translaton nvarance n the shell model. A s the mass number and b s the harmonc oscllator sze parameter. The sngle-partcle energes are calculated accordng to [7]: e nlj ( nl ) (l) l ( r ) ( r ) nl nl or jl or jl (9) wth: ( r ) 0A / 3MeV (0) nl 45A / 3 5A / 3 For the two-body matrx elements o the resdual nteracton V, res whch appear n equatons (5), the Mchgan three Yukawas (M3Y) nteracton o Bertch et. al [9] s adopted. The nteracton s taken between a nucleon n any coreorbts and nucleon that s excted to hgher orbts wth the same party and wth the requred multpolarty ( ), and also between a nucleon n any sd orbts and that s excted to hgher orbts wth the same party and wth the requred multpolarty. The orm o the potental s dened n equatons ()-(3) n re. [9]. The parameters o Ellot are used whch are gven n table o the mentoned reerence. Ths nteracton was gven n LS-couplng. A transormaton between LS and jj must be perormed to get the relaton between the two-body shell model matrx elements and the relatve and center o mass coordnates, usng the harmonc oscllator radal wave unctons wth Talm-Moshnsky transormaton. 3- Result and dscusson The 6 L nucleus s especally nterestng nucle because s the lghtest state nucle that contan p-shell nucleons. The structure and propertes o 6 L nucleus are expermentally and theoretcally well studed (especally orm actors are o partcular nterest). For the conventonal many partcles shell model, the 6 L nucleus s essentally a three-body system, two valence nucleons dstrbuted over p 3/ -p / shell and presumably nert 4 He core. The HO sngle-partcle wave uncton s employed wth sze parameter b=.88 m [0]. Ths nucleus s mportant to be studed because both longtudnal and transverse orm actors have been measured or the transtons to ve states: M, 0.0 MeV ( + 0) state C0+C, 0.0 MeV ( + 0) state C,.8 MeV (3 + 0) state M, 3.56 MeV (0 + ) state M+E+M3, 5.37 MeV ( + ) state 4

5 F(q) Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October The Elastc Magnetc Form Factor or + 0 State The elastc magnetc orm actor M s a purely soscalar, whch s dened n ths case practcally by a sngle M multpole, and s descrbed n terms o two nucleons outsde a closed s shell. The magnetc orm actor carres normaton about the dstrbutons o magnetzaton current and o convecton current over nuclear volume []. Fg. (4--) shows the transverse M electron scatterng orm actor as a uncton o momentum transer q. The dashed curve represents the results o the orm actor or p-shell model, and the sold curve represents the contrbuton o p-shell model wth core-polarzaton eect. It s notced rom Fg. (), that the ncluson o the core-polarzaton doesn t aect sgncantly the calculaton o the orm actors especally at the rst maxmum regon (q <.5) m -, and expermental data [, 3] are well descrbed n ths regon. Whle, n the second maxmum regon, we notced that the cp aectng toward reducng the orm actor wth the rate o meager and not sucent to predcate the expermental data. The OBDM elements are gven n table (). Table (): The values o the OBDM elements or the transverse M transton o the 0 ground state o 6 L. J J OBDM ( T=0)) / / E-0 / 3/ E-0 3/ / E-0 3/ 3/ E-0.0E-3 6 L M: 0.0 MeV ( + 0).0E-4 p+cp p.0e-5.0e-6.0e q(m - ) Fg.() Elastc M transverse orm actor or the + 0 state n 6 L, wth and wthout core-polarzaton eects. The expermental data are taken rom res. [3] (Squares) [] (crcles). 5

6 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October The Elastc Longtudnal Form Factor For + 0 state The elastc charge orm actors or the p-shell model calculaton or ths transton are llustrated n Fg. (). The total orm actors or ths transton are consdered as the sum o the C0 and C (as a sold curve). The C has not been studed expermentally because o ts smallness, although t deserves undoubtedly to be studed n detal because t yelds straghtorward normaton about the small D- component o 6 L ground state []. And also the multpoles C0 and C cannot be separated expermentally, so only the sum o these squares FL ( q) FC 0 ( q) FC ( q) s measured. Fg. (3) shows a comparson between the calculated orm actors wth the ncluson o cp eect (as a sold curve) and those o p-shell model calculaton (as a dashed curve). The cp eects enhance the C orm actor apprecably by a actor around 3 over the p-shell calculaton. The results o the p-shell model calculaton (wthout cp eect) gve a good agreement wth the expermental data [85] up to the momentum transer q =.35 m -. The ncluson o cp eects enhances the C orm actor apprecably, but due to ts small contrbuton to the total orm actor n the regon o q < m -, t does not aect the total orm actor. The OBDM elements or these cases are shown n tables () and (3). Table (): The values o the OBDM elements or the longtudnal transton C0 o the 0 ground state o 6 L. 6 L C0 Intal State Fnal State OBDM ( T=0) s / s / p / p / E- p 3/ p 3/ E- Table (3): The values o the OBDM elements or the longtudnal transton C o the 0 ground state o 6 L. 6 L C J J OBDM ( T=0)) / 3/ E- 3/ / E- 3/ 3/ E- 6

7 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 E+0 E- E- F(q) E-3 6 L C0+C: 0.0 MeV ( +, 0) p C0 C F(q) E-4 E-5 E-6 E q(m - ) Fg. () Coulomb C0+C orm actor or the transton to the + 0 ground state n 6 L, wthout core-polarzaton eects. The expermental data are taken rom re. []. E+0 E- E- E-3 6 L C0+C: 0.0 MeV ( + 0) p+cp p C (p+cp) C (p) E-4 E-5 E-6 E q(m - ) Fg. (3) Coulomb C0+C orm actor or the transton to the + 0 ground state n 6 L, wth and wthout core-polarzaton eects. The expermental data are taken rom re. []. 7

8 F(q) Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October The.8 MeV (3 + 0) State The nucleus s excted by the ncdent electron rom the ground state ( J T = + 0) to the state J T =3 + 0 wth an exctaton energy o.8 MeV. The OBDM elements values or the C transton are lsted n table (4). The longtudnal C orm actor s o soscalar character. The expermental data or ths transton are avalable [4,5]. In p-shell model, the calculated orm actors underpredct the data n all regons o the momentum transers q, as shown n Fg. (4), as a dashed curve. In ths model only the model space wave unctons are consdered. The p-shell model als to descrbe the data n both the transton strength and the orm actors. The calculated B(C ) value s 6.0 e m 4 whch s low n comparson wth the measured value e m 4 [6]. Core-polarzaton eects enhance the orm actor and reproduce the measured orm actor up to q= m -, as shown by the sold curve o Fg (4). In ths case the calculated B(C ) value s 6.68 e m 4, whch s nearly close to the measured value. A smlar result s obtaned by usng MSDI potental [7,8]. Cluster model calculaton [9] reproduced the data or q < m - and overestmated the data or hgher q values. Good agreement s obtaned by the varatonal Monte Carlo calculaton [0]. Table (4): The values o the OBDM elements or the longtudnal C transton to the 3 0 at E x =.8 MeV or 6 L. J J OBDM ( T=0)) / 3/ E-04 3/ / E-0 3/ 3/ E-0 E- E- 6 L C:.8 MeV (3 + 0) p+cp p cp E-3 E-4 E-5 E q(m - ) Fg. (4) The longtudnal C orm actor or the.8 MeV state n 6 L, wth and wthout corepolarzaton eects. The expermental data are taken rom re. [4] (crcles) and re. [5] (squares). 8

9 F(q) Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October The 3.56 MeV (0 + ) State In ths transton, the nucleus s excted rom the ground state ( state J T J T = + 0) to the =0 + wth an exctaton energy o 3.56 MeV due to electron scatterng. The values o the OBDM elements or ths transton are gven n table (5). The M transton to ths state s a purely sovector. It s known that nelastc magnetc orm actor s dcult to reproduce by a theoretcal calculaton. Extended space calculaton o re. [] predcated the locaton o mnmum at the correct momentum transer, but ther result overestmated the data at the rst and the second maxmum o the orm actor. Fg. (5) shows the calculatons o the transverse M electron scatterng orm actor wth and wthout cp eect as a sold curve and dashed curve, respectvely. Also the cp contrbuton s shown by the plus-symbols. The result o the p-shell model calculaton (wthout cp eect) over predcts the data or the rst lobe, beyond the rst maxmum, and descrbes the data or the second lobe up to q.0 m -, but or hgher momentum transer t als to descrbe the data. Whle the ncluson o cp eect gves an excellent agreement up to the dracton mnmum and overestmates the data beyond that. It has been ound [,,3,4] that the eect o MEC on the orm actor yelds only mnor correctons and thus the dscrepancy between the calculated and measured orm actors may not be attrbuted to ths eect. E- Table (5 ): The values o the OBDM elements or the transverse M transton to the 0 at E x = 3.56 MeV or 6 L. J J OBDM ( T=)) / / / 3/ / / / 3/ L M: 3.56 MeV + E-3 E-4 p+cp p cp exp. exp E-5 E q(m - ) Fg. (5) The transverse M orm actor or the transton to the 0 + (3.56 MeV) state n 6 L, wth and wthout core-polarzaton eects. The expermental data are taken rom re. [5] (squares) and re. [4] (crcles). 9

10 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 state 4..5 The 5.37 MeV ( + ) State. In ths transton, the nucleus s excted rom the ground state ( J T J T = + 0) to the = + wth an exctaton energy o 5.37 MeV due to electron scatterng. The OBDM elements or ths case are shown n the tables (6), (7) and (8). The expermental data are taken rom re. [4, 5]. The transverse orm actor or ths state s o mxed multpolartes, M+E+M3. The transverse orm actors calculated wth the model space wave unctons and core polarzaton eect are shown n Fg. (6) as a dashed curve and a sold curve, respectvely. The data are very well explaned when core-polarzaton eects are ncluded, throughout the momentum transer regons. The suppresson o the transverse orm actors by ncludng core-polarzaton eect agrees wth the prevous studes n the p-shell model space where eectve g-actors reduced rom the ree nucleon values were used. The calculaton o re. [9] usng a cluster model showed that MEC played only a mnor role, but reducng the calculaton by a small percentage. So, n our work, ncludng MEC may brng the calculated orm actor even more closely to the data. The derent ndvdual multpoles, calculated wthout core-polarzaton eects are shown n Fg. (7). The man contrbuton n the most regon o q, comes rom E and M3, where M has the domnant contrbuton n the regon o small values o q < 0.5 m -. The M orm actor s reduced by cp eect or 0.5 q m -, as shown n Fg. (8). The E orm actor s reduced by a actor o as shown n Fg. (9), and M3 s also reduced wth the ncluson o cp, as shown n Fg. (0). Table (6): The values o the OBDM elements or the transverse transton M to the state at E = 5.37 MeV or 6 L. x 6 L M J J OBDM ( T=) / / / 3/ / / / 3/ Table (7): The values o the OBDM elements or the transverse transton E to the state at E x = 5.37 MeV or 6 L. 6 L E J J OBDM ( T=) / 3/ / / / 3/ Table (8): The values o the OBDM elements or the transverse transton M3 to the state at E x = 5.37 MeV or 6 L. 6 L M3 J J OBDM ( T=) 3/ 3/

11 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 E- E-3 6 L (M+E+M3): 5.37 MeV ( + ) p+cp p cp E-4 E-5 F(q) F(q) E q(m - ) Fg. (6) The transverse M+E+M3 orm actor or the 5.37 MeV ( + ) state n 6 L, wth and wthout core-polarzaton eects. The expermental data are taken rom re. [4] (crcles) and re. [5] (squares). E- E-3 6 L (M+E+M3): 5.37 MeV ( + ) p M E M3 E-4 E-5 E q(m - ) Fg. (7) The transverse M+E+M3 orm actor or the 5.37 MeV ( + ) state n 6 L, wthout corepolarzaton eects. The expermental data are taken rom re. [4] (crcles) and re. [5] (squares).

12 Kua s Frst Conerence or physcs, specal Issue (ISSN ), 6-7 October 00 Reerence - H. Yukawa, Proc Math. Phys. Soc. Japan 7, 48 (935). - A. Fassler, ACTA Physca Polonca B, vol.9 (998). 3- Bockmann, C. Hahant, O. Krehl, S. Krewald, and J. Speth, Phys. Rev. C, V60, (999). 4- J.G.L. Booten and A.G.M. Van Hees, Nucl.Phys. A569, 50 (994). 5- S. Cohen and P. Kurath; Nucl. Phys.73, (965). 6- R.A. Radh, A. Bouchebak, (003), Nuclear Physcs, A76, P.J. Brussard and P. W. M. Glademans, "Shell-model Applcaton n Nuclear Spectroscopy", North-Holland Publshng Company, Amsterdam (977). 8- T. Wllam Donnelly and I. Sck, (984), Revew-Mod. Physcs. 56, G. Bertsch, J. Borysowcz and H. McManus, (977), Nuclear. Physcs, A84, F. A. Bumllar, F. R. Buskrk, J. N. Dyer and W. A. Mansen; phys. Rev. C5, 39 (97). - V.I. Kukuln, V.T. Voroncher, T.D. Kapor, R.A. Eramzhyam, Nucl. -.C. Bergstrom and S. B. Kowalsk, Phys. Rev. C5, 56 (98). 3-. Lapkas, G. Box and H. DeVres, Nucl. Phys. A53, 34 (975). 4-.C. Bergstrom, Nucl. Phys. A37, 439 (979). 5- C. Bergstron and E. L. Tomusak, Nucl. Phys. A6, 96 (976). 6- F. Ajzenberg-Selove, Nucl. Phys. A490, (988). 7- R.A. Radh, A.A. Abdullah, Z.A. Dakhl, N.M. Adeeb, Nucl. Phys. A696, 44 (00) 8- R.A Radh, Eur. Phys.J., A6, 387 (003). 9- S. Weber and M. Kachelress, Phys. Rev. C50, 49 (994). 0- R.B. Wrnga, R. Schavlla, Phys. Rev. Lett. 8, 437 (998). - J.G.L. Booten and A.G.M. Van Hees, Nucl.Phys. A569, 50 (994). - Z. A. Dakhl, Ph.D. Thess, Unversty o Baghdad (998). 3- K. Ara and Y. Suzuk, Phys. Rev. C5, No.5, 488 (995). 4- N. Adeeb, Ph.D. Thess, Unversty o Baghdad (00). 5- J.C. Bergstron, I.P. Aure and R.S. Hcks, Nucl. Phys. A 5, 40 (975).