Charge Form Factor and Cluster Structure of 6 Li Nucleus

Transcription

1 Charge Form Factor an Cluster Structure of Nucleus G. Z. Krumova 1, E. Tomasi-Gustafsson 2, an A. N. Antonov 3 1 University of Rousse, 7017 Rousse, Bulgaria 2 DAPNIA/SPhN, CEA/Saclay, F Gif-sur-Yvette Ceex, France 3 Institute of Nuclear Research an Nuclear Energy, Bulgarian Acaemy of Sciences, 1784 Sofia, Bulgaria Abstract. The charge form factor of nucleus is consiere on the basis of its cluster structure. The charge ensity of is presente as a superposition of two terms. One of them is a fole ensity an the secon one is a sum of an the euteron ensities. Using the available experimental ata for an euteron charge form factors, a goo agreement of the calculations with the experimental ata for the charge form factor of is obtaine, incluing those in the region of large transferre momenta. 1 Introuction It is known that the structure of the nucleus has some peculiarities compare to the other 1p-shell nuclei (see e.g. [1 7]). The electron elastic scattering ata [8] on the charge form factor an the rms raius of cannot be explaine in the framework of the shell-moel by means of an oscillator parameter ω = MeV, the latter proviing a goo escription of these ata for the other 1p-shell nuclei. The usage of another value of ω, the same for the s- an p- nucleons, as well as of two ifferent oscillator parameters for the s- an p-shells is also not successful. The situation is similar in the case of the inelastic form factors. The wave functions of the low-lying states of are significantly ifferent from the commonly accepte an use shell moel wave functions. This fact is important for the analysis of the (p, 2p), (p, p), (p, pα) reactions on, the photonuclear reactions, the (, ), (, α) reactions an others. It has been estimate that has a well pronounce cluster structure an is consiere generally as a system consisting of α an euteron clusters in a mutual motion exchanging nucleons. The small value of the ecay threshol α +, the large nuclear raius, etc. give evience that the α an clusters in are quite isolate. In one of the cluster moels, the Moel of Nucleon Associations (MNA) (e.g. [6]), the problem of the role of the exchange has been stuie by analyzing the elastic an inelastic form factors of the Coulomb electron scattering. The antisymmetrization effect turns out to be substantial only at large values of the isolation parameter x 1, where the parameter x = b/a is the ratio between the relative motion function parameter b an the α-particle function parameter a. At the real value x = , the exchange effects are alreay of no importance. 259

2 260 G. Z. Krumova, E. Tomasi-Gustafsson, A. N. Antonov In MNA the value x =1correspons to the shell-moel structure of, while x =0correspons to the cluster moel (α structure). It has been foun that the isolation parameter x has ifferent values for nuclei with cluster structure. For instance, x = for 9 Be, x =0.7 for 12 Canx =0.8 for 16 O [9, 10]. The elastic scattering charge form factor, although being sensitive to the value of x, can be escribe by ifferent moels ue to the fact that it is obtaine on the base of the charge ensity istribution that is average over the angular variables. The form factor of the inelastic quarupole scattering, however, strongly epens on x an cannot be escribe within the shell moel. The MNA provies a goo rms raius of [11] an with the above values of the isolation parameter (x = ) allows a proper simultaneous escription of the electron elastic an inelastic scattering but only up to transferre momentum values q 2 fm 1. The aim of the present work is to suggest an approach in which the α cluster structure of to be checke by calculations of the charge ensity an the corresponing charge form factor. We construct a scheme in which the charge ensities of an the euteron are inclue an the available experimental ata for them can be use to calculate the charge ensity, the charge form factor an the latter to be compare with the experiment. In this sense, our work has a meaning of a theoretical experiment in the cluster structure analysis of. In Section 2 the theoretical scheme, the results of the calculations an a iscussion are presente. The conclusions are given in Section 3. 2 Charge Density an Form Factor of in Relation to Those of an Deuteron Consiering the problems in the escription of the charge ensity an form factor briefly mentione above, we mae an attempt to stuy these quantities on the base of the corresponing ones for an the euteron within the framework of the α cluster structure of nucleus. Our first attempt was to escribe the charge ensity of by means of foling of the charge ensities of an the euteron: ρ ch ( r) = 3 2 r ρ ch ( r r ) ρ ch ( r ). (1) The charge ensities in Eq. (1) are normalize to the number of protons Z (Z =3, 2 an 1 for, an the euteron, corresponingly). Substituting charge ensity (1) in the efinition of the charge form factor F ch ( q )= 1 r e i q. r ρ ch ( r ) (2) Z we obtain F ch (q) =F ch (q) F ch (q) e q2 /(4A 2/3), (3)

3 Charge Form Factor an Cluster Structure of Nucleus 261 Table 1. The values of the parameters α, β, γ an δ in the parametrizations I an II, obtaine from the global best fit in [16] (the values of α are erive from Eq. (8)). Set α β γ δ I 5.75 ± ± ± ± 0.03 II 5.50 ± ± ± ± 0.03 in which the exponential factor approximately accounts for the centre-of-mass (c.m.) corrections accoring to [12]. In our calculations of the charge form factor of (Eq. (3)) we use the available experimental ata for the charge form factor of (see e.g. [13] an references therein), as well as the experimental ata for the charge form factor of the euteron. The latter are those from the Thomas Jefferson Laboratory experiments in which the euteron charge form factor was measure for a first time to a transferre momentum value up to q =6.64 fm 1 an the noe of the form factor was observe (Abbott et al. [14, 15]). In our calculations for the euteron charge form factor we use a best fit parametrization obtaine in [16]. It is represente by Eqs (4)-(8) [16]: F ch (q2 )=g(q 2 )F ch (q2 ), (4) F ch (q2 )=1 α β + α m2 ω m 2 ω + + β q2 m 2, (5) Φ + q2 where m ω an m Φ are the meson masses (m ω = GeV an m Φ = GeV ). For any values of the two real parameters α an β The factor g in Eq. (4) has the form m 2 Φ F ch (0) = 1. (6) g(q 2 )= 1 (1 + γq 2 ) δ (7) an γ an δ are also real parameters. The requirement of a noe for q0 2 between the parameters α an β: 0.7 GeV 2 gives the following relation α = m2 ω + q 2 0 q 2 0 β m2 ω + q0 2 m 2 Φ +. (8) q2 0 The values of two sets of the parameters α, β, γ an δ obtaine in [16] by a best fit to the experimental ata, which are use in the calculations of the present work, are given in Table 1. Recent large-scale shell-moel (LSSM) calculations of [17] an the analysis of the electron an proton elastic an inelastic scattering ata from 6,7 Li have prove the clustering behavior of these systems. In our work [18] proton, neutron, charge,

4 262 G. Z. Krumova, E. Tomasi-Gustafsson, A. N. Antonov 1 F ch (q) 2 1 exp DW BA 800 MeV PW BA exp DW BA 600 MeV PW BA q [fm -1 ] Figure 1. Charge form factors of the stable isotopes an obtaine in [18] using LSSM ensities in PWBA an in DWBA calculations in comparison with the experimental ata [13, 19 25]. an matter ensities of a wie range of exotic nuclei obtaine in the Hartree-Fock- Bogolyubov metho an in the LSSM (for He an Li isotopes) have been use for Plane-Wave Born Approximation (PWBA) an Distorte-Wave Born Approximation (DWBA) calculations of the relate form factors. This makes it possible to analyze the influence of the increasing number of neutrons on the proton an charge istributions in a given isotopic chain. The obtaine in [18] theoretical preictions for the charge form factors of exotic nuclei are a challenge to their measurements in the future experiments on the electron-raioactive beam colliers in GSI an RIKEN in orer to get etaile information on the charge istributions of such nuclei. Our results for an charge form factors obtaine in [18], compare to the available experimental ata [13, 19 25], are given in Figure 1. One can see a goo agreement with the ata up to q 3 fm 1.

5 Charge Form Factor an Cluster Structure of Nucleus F ch q,fm -1 Figure 2. The charge form factor of the euteron calculate using two sets of parameters α, β, γ an δ (with values given in Table 1) an compare with the experimental ata [14, 15] up to q 3.8 fm 1. In Figure 2 are presente the experimental ata for the euteron charge form factor [14, 15] an the result of the parametrization from [16] up to q 3.8 fm 1 (with parameter sets I an II from Table 1). In Figure 3 are given our results for the square charge form factor of calculate by using of Eq. (3) (taking account of the c.m. correction) an the experimental ata for the charge form factors of an the euteron. For the latter we use the same parametrization from [16] (Eqs (4)-(8)) with two sets of parameters I an II from Table 1. A goo agreement with the experimental ata in the interval of transferre momentum 0 <q 2.7fm 1 can be seen an a isagreement with the values of the form factor for larger q s that are relate to small values of r s, i.e. to the central part of the nuclear ensity. In other wors, the central ensity can be ifferent from the assumption for the foling ensity (Eq. (1)). We note the similarity of the results (compare with the ata) of the calculate charge form factor of for q 2.7 fm 1 from the present work (shown in Figure 3) with those from [18] (shown in the own panel of Figure 1). The results shown in Figure 3 were the reason to look for an extension of the approach. Our secon suggestion is to consier the charge ensity of as a superposition of a foling term an a sum of the charge ensities of an the euteron with weight coefficients c 1 an c 2 : ρ ch ( r )= 3 2 c 1 r ρ ch ( r r ) ρ ch ( r )+c 2 [ ρ ch ( r )+ρ ch ( r )]. (9) The normalization of the ensities in Eq. (9) to Z leas to the conition for the coefficients

6 264 G. Z. Krumova, E. Tomasi-Gustafsson, A. N. Antonov IF ch (q)i q,fm -1 Figure 3. The charge form factor of calculate by using Eq. (3) an the experimental ata for the charge form factors of an the euteron in comparison with the experimental ata ( [13, 19 21, 24, 25]). c 1 + c 2 =1. (10) The charge ensity (Eq. (9)) leas to the following expression of the charge form factor of (with the account for the c.m. correction): { F6 ch Li (q) = c 1 F4 ch He (q) F ch (q)+ c } 2 ch [2F4 3 He (q)+f ch (q)] e q2 /(4A 2/3). (11) For q =0 F6 ch Li (0) = 1. (12) The square charge form factor can be written as: F ch (q) 2 = A + B + C, (13) where A, B an C represent the contributions to the charge ensity of of the foling term (A), of the sum of the charge ensities of an the euteron (B) an the interference term (C). Their explicit expressions are: A = c 1 2 F ch (q) 2 F ch (q) 2 e q2 /(2A 2/3), (14) B = c 2 2 ch [4 F4 9 He (q) 2 + F ch (q) 2 +4 F ch ch (q) F (q) ] e q2 /(2A 2/3), (15) 4He

7 Charge Form Factor an Cluster Structure of Nucleus 265 C = 2 3 c 1c 2 F4 ch He (q) F ch (q) [2 F4 ch He (q) + F ch (q) ] e q2 /(2A 2/3). (16) In the following three Figures 4-6 are presente the results for the square charge form factor of calculate using Eqs (13)-(16) with the experimental ata for the charge form factor of an the euteron an for ifferent sets of the values of the weight coefficients c 1 an c 2. The fit of Eq. (13) to the experimental ata reveals an interval of values of c 1 = an, corresponingly, of c 2 = , for which the results reasonably agree with the experimental ata [13, 19 21, 24, 25] within the limits of the experimental errors. We also show the contributions of the three terms A, B an C experim ent 10-1 A 10-3 B C A+B+C IF ch (q)i q,fm -1 Figure 4. The charge form factor of (Eqs (11)-(16)) calculate for c 1 =0.975 an c 2 = 0.025, in comparison with the experimental ata from [13, 19 21, 24, 25]. The results presente in Figures 4-6 prove that the contribution of the foling ensity to the charge ensity of is about %. This correspons to the weight of the contribution of the sum of an the euteron ensities of about %. It is seen that the term A (Eq. (14)) escribes well the square charge form factor of in the interval 0 <q 2.7 fm 1, while the shell-moel cluster ensity (relate to the term B, Eq. (15)) is important for the escription of the charge form factor of for the large values of q (q 3 fm 1 ), relate to the central nuclear ensity. The interference term C (Eq. (16)) has a contribution to the charge form factor of for q 3 fm 1. The increase of c 1 within the above interval leas to a better escription of the ata for q = fm 1,butat

8 266 G. Z. Krumova, E. Tomasi-Gustafsson, A. N. Antonov 10 0 experim ent 10-1 A 10-3 B C A+B+C IF ch (q)i q,fm -1 Figure 5. The same as in Figure 4 but with c 1 =0.979 an c 2 = experim ent 10-1 A 10-3 B C A+B+C IF ch (q)i q,fm -1 Figure 6. The same as in Figure 4 but with c 1 =0.985 an c 2 =0.015.

9 Charge Form Factor an Cluster Structure of Nucleus 267 the same time to a ecrease of the values of the square charge form factor for q 3 fm 1, unerestimating the ata. Our calculations show that most reasonable results are obtaine for 97.9% contribution of the foling term that correspons to 2.1% weight of the contribution of the sum of an the euteron charge ensities to the charge ensity. As known, the value of the obtaine rms raius is a test for the consistency of any approach to the escription of the nuclear system structure. The charge rms raius of is given by the expression: r6 2 Li = 1 rr 2 ρ ch 6 3 Li ( r). (17) Substituting the expression for the charge ensity of (Eq. (9)) in Eq. (17), we obtain: r6 2 Li = c 1 [ r4 2 He + r 2 ]+ c 2 3 [2 r2 + r 2 ]. (18) The usage of the experimental ata for the rms raii of an the euteron [20,26]: r 2 1/2 =1.676(8) fm, r 2 1/2 =2.116(6) fm in Eq. (18) (with c 1 =0.979 an c 2 =0.021) leas to the following value for the charge rms raius: r6 2 Li 1/2 =2.684 fm, which is in accorance with the experimental estimations for the charge rms raius of [20, 26]: r6 2 Li 1/2 =2.57(10) fm. This coul be expecte ue to the use of the experimental charge ensities of the euteron an, being combine in a realistic theoretical scheme that gives a goo agreement with the experimental ata for the charge form factor of. 3 Conclusions In the present work we suggest a theoretical scheme for calculations of the charge ensity istribution an form factor of in the framework of the α cluster moel of this nucleus. The obtaine results can be summarize as follows: 1. Our calculations show a reasonable escription of the charge form factor of on the basis of a superposition of two ensity istributions: i) a foling ensity obtaine from an the euteron charge ensities, an ii) a sum of the an euteron charge ensities. Provie corresponing experimental ata for both ensities are use, the calculations show that a reasonable agreement with the ata can be obtaine when the weight of the foling ensity contribution is about %

10 268 G. Z. Krumova, E. Tomasi-Gustafsson, A. N. Antonov an the weight of the contribution from the sum of both ensities is about %. 2. The scheme has only one free parameter (c 1 or c 2 ) with a clear physical meaning, namely, it is the weight of the one of the contributions to the ensity of. 3. The behavior of the charge form factor of for 0 < q 2.7 fm 1 is etermine mainly by the foling contribution (of an the euteron ensities) to the charge ensity of (the weight is about %). 4. The shell-moel α cluster ensity of (i.e. the sum of an the euteron charge ensities) is important (though with a small weight of about %) in the central nuclear region an, corresponingly, it is responsible for the values of the charge form factor of at large values of q (q 3 fm 1 ). 5. The calculate within the suggeste scheme charge rms raius of agrees with the experimental estimations of this quantity. 6. We woul like to pay attention to the following facts: i) the minimum of the experimental charge form factor of the euteron is at q 4.2fm 1 [14,15], ii) the minimum of the experimental charge form factor of is at q 3.2fm 1 [13], an iii) the minimum of the experimental charge form factor of is at q 2.9 fm 1 [13, 19 21, 24, 25]. Base on points i) an ii), our estimations show that the latter is etermine mainly by the contribution of the charge ensity an the corresponing form factor of. Acknowlegments We woul like to thank Professor Stephane Platchkov an Professor Pero Sarriguren for the valuable iscussion. This work was partly supporte by the Bulgarian National Science Fun uner the Contracts No. Φ-1416 an Φ References 1. I. S. Gul karov,issleovaniya yaer electronami (Atomizat,Moscow, 1977). 2. Yu. A. Kueyarov, I. V. Kuryumov, V. G. Neuatchin, an Yu. F. Smirnov, Nucl. Phys. A 126, 36 (1969). 3. T. I. Kopaleyshvili, I. Z. Machabeli, Sov. J. Nucl. Phys. (in Russian) 4, 37 (1966). 4. Yu. A. Kueyarov, Yu. F. Smirnov, an M. A. Chebotarev, Sov. J. Nucl. Phys. (in Russian) 4, 1048 (1966). 5. Yu. A. Kueyarov, V. G. Neuatchin, S. G. Serebryakov, an Yu. F. Smirnov, Sov. J. Nucl. Phys. (in Russian) 6, 1203 (1967).

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