Charge Form Factor and Cluster Structure of 6 Li Nucleus
aa r X i v : . [ nu c l - t h ] S e p Charge Form Factor and Cluster Structure of Li Nucleus
G. Z. Krumova
University of Rousse, 7017 Rousse, Bulgaria
E. Tomasi-Gustafsson
DAPNIA/SPhN, CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, France
A. N. Antonov
Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
Abstract
The charge form factor of Li nucleus is considered on the basis of its cluster structure. Thecharge density of Li is presented as a superposition of two terms. One of them is a folded densityand the second one is a sum of He and the deuteron densities. Using the available experimentaldata for He and deuteron charge form factors, a good agreement of the calculations within thesuggested scheme is obtained with the experimental data for the charge form factor of Li, includingthose in the region of large transferred momenta.
PACS numbers: α − cluster structure have started since the for-ties and different theoretical models have been developed till nowadays. Among various α − particle models (APM) that must be noted are, for example, the single APM (with ready α − particles inside the nucleus, e.g. [1, 2, 3]), the dynamical APM of point-like α − particlesinteracting by α − α potentials with solving the Schroedinger or Faddeev equations (e.g.[4]),the microscopic APM of Brink, Bloch and Margenau (e.g.[5]), and others, including morerecent approaches (e.g.[6]). However, though a great number of works within the APMhave been devoted to the structure and interactions of such nuclei, many questions remainopen and deserve further work. This concerns even properties of long-time investigated nu-clei, such as the Li nucleus. It is known that the structure of the Li nucleus has somepeculiarities compared to the other 1p-shell nuclei (see e.g. [6, 7, 8, 9, 10, 11, 12]). Theelastic electron scattering data [13] on the charge form factor and the rms radius of Li can-not be explained in the framework of the shell-model by means of an oscillator parameter¯ hω = 15 ÷ M eV , the latter providing a good description of these data for the other1p-shell nuclei. The usage of another value of ¯ hω , the same for the s- and p- nucleons, aswell as of two different oscillator parameters for the s- and p-shells is also not successful.The situation is similar in the case of the inelastic form factors. The wave functions of thelow-lying states of Li are significantly different from the commonly accepted and used shellmodel wave functions. This fact is important for the analysis of the ( p, p ), ( p, pd ), ( p, pα )reactions on Li, the photonuclear reactions, the ( Li, d), ( Li, α ) reactions and others.It has been estimated that Li has a well pronounced cluster structure and is consideredgenerally as a system consisting of α − and deuteron clusters in a mutual motion exchangingnucleons. The small value of the decay threshold Li → α + d , the large nuclear radius,etc. give evidence that the α − and d − clusters in Li are quite isolated. In another of thecluster models, the Model of Nucleon Associations (MNA) (e.g. [12]), the problem of therole of the exchange has been studied by analyzing the elastic and inelastic form factors ofthe Coulomb electron scattering. The antisymmetrization effect turns out to be substantialonly at large values of the isolation parameter x ≈
1, where x = b/a is the ratio betweenthe relative motion function parameter b and the α -particle function parameter a . At thereal value x = 0 . ÷ .
4, the exchange effects are already of no importance. In MNA thevalue x = 1 corresponds to the shell-model structure of Li, while x = 0 corresponds tothe cluster model ( α − d structure). It has been found that the isolation parameter x has2ifferent values for nuclei with cluster structure. For instance, x = 0 . ÷ . Be, x = 0 . C and x = 0 . O [14, 15]. The elastic scattering charge form factor, althoughbeing sensitive to the value of x , can be described by different models due to the fact that itis obtained on the base of the charge density distribution that is averaged over the angularvariables. The form factor of the inelastic quadrupole scattering, however, strongly dependson x and cannot be described within the shell model. The MNA provides a good rms radiusof Li [16] and with the above values of the isolation parameter ( x = 0 . ÷ .
4) allows aproper simultaneous description of the electron elastic and inelastic scattering but only upto transferred momentum values q ∼ f m − .The aim of the present work is to suggest an approach in which the α − d cluster structureof Li to be checked by calculations of the charge density and the corresponding charge formfactor. We construct a scheme in which the charge densities of He and the deuteron areincluded and the available experimental data for them can be used to calculate the Licharge density, the charge form factor and the latter to be compared with the experiment.In this sense, our work has a meaning of a ’theoretical experiment’ to check the particularcluster structure of Li by a comparison of the results of two different suggestions with theempirical data for this nucleus.In Section I the theoretical scheme, the results of the calculations and a discussion arepresented. The conclusions are given in Section II.
I. CHARGE DENSITY AND FORM FACTOR OF LI IN RELATION TO THOSEOF HE AND DEUTERON
Considering the problems in the description of the Li charge density and form factorbriefly mentioned above, we made an attempt to study these quantities on the base of thecorresponding ones for He and the deuteron within the framework of the α − d clusterstructure of Li nucleus.Our first attempt was to describe the charge density of Li within the framework of theoften used folding procedure. In our case it is a folding of the charge densities of He andthe deuteron: ρ ch Li ( ~r ) = 32 Z d ~r ′ ρ ch He ( ~r − ~r ′ ) ρ chd ( ~r ′ ) . (1)3he charge densities in Eq. (1) are normalized to the number of protons Z ( Z = 3, 2 and 1for Li, He and the deuteron, correspondingly). Substituting Li charge density Eq. (1) inthe definition of the charge form factor F ch ( ~q ) = 1 Z Z d ~r e i~q.~r ρ ch ( ~r ) (2)we obtain F ch Li ( q ) = F ch He ( q ) F chd ( q ) e q / (4 A / ) , (3)in which the exponential factor approximately accounts for the centre-of-mass (c.m.) cor-rections according to [17].In our calculations of the charge form factor of Li (Eq. (3)) we use the available ex-perimental data for the charge form factor of He (see e.g. [18] and references therein),as well as the experimental data for the charge form factor of the deuteron. The latter arethose from the Thomas Jefferson Laboratory experiments in which the deuteron charge formfactor was measured for a first time to a transferred momentum value up to q = 6 . f m − and the node of the form factor was observed (Abbott et al. [19, 20]). In our calculationsfor the deuteron charge form factor we use a best fit parametrization obtained in [21]. It isrepresented by Eq. (4) - Eq. (8) [21]: F chd ( q ) = g ( q ) F chd ( q ) , (4) F chd ( q ) = 1 − α − β + α m ω m ω + q + β m m + q , (5)where m ω and m Φ are the meson masses ( m ω = 0 . GeV and m Φ = 1 . GeV ). For anyvalues of the two real parameters α and βF chd (0) = 1 . (6)The factor g in Eq. (4) has the form g ( q ) = 1(1 + γq ) δ (7)and γ and δ are also real parameters.The requirement of a node for q ≈ . GeV gives the following relation between theparameters α and β : α = m ω + q q − β m ω + q m + q . (8)4he values of two sets of the parameters α , β , γ and δ obtained in [21] by a best fit tothe experimental data, which are used in the calculations of the present work, are given inTable 1. TABLE I: The values of the parameters α , β , γ and δ in the parametrizations I and II, obtainedfrom the global best fit in [21] (the values of α are derived from Eq. (8)).Set α β γ δ I 5 . ± . − . ± .
09 12 . ± . . ± . . ± . − . ± .
08 12 . ± . . ± . Recent Large-Scale Shell-Model (LSSM) calculations of [22] and the analysis of the elasticand inelastic electron and proton scattering data from , Li have proved the ’clustering’behavior of these systems. In our work [23] proton, neutron, charge, and matter densities ofa wide range of exotic nuclei obtained in the Hartree-Fock-Bogolyubov method and in theLSSM (for He and Li isotopes) have been used for Plane-Wave Born Approximation (PWBA)and Distorted-Wave Born Approximation (DWBA) calculations of the related form factors.This makes it possible to analyze the influence of the increasing number of neutrons on theproton and charge distributions in a given isotopic chain. The obtained in [23] theoreticalpredictions for the charge form factors of exotic nuclei are a challenge to their measurementsin the future experiments on the electron-radioactive beam colliders in GSI and RIKEN inorder to get detailed information on the charge distributions of such nuclei.The available experimental data [18, 24, 25, 26, 27, 28, 29, 30]for He and Li charge formfactors are presented in Fig. 1 in comparison with the results of our PWBA and DWBAcalculations from [23]. One can see a good agreement with the data up to q ∼ f m − .In Fig. 2 are presented the experimental data for the deuteron charge form factor [19, 20]and the result of the parametrization from [21] up to q ≈ . f m − (with parameter sets Iand II from Table 1).In Fig. 3 are given our results for the squared charge form factor of Li calculated by usingof Eq. (3) (taking account of the c.m. correction) and the experimental data for the chargeform factors of He and the deuteron. For the latter we used the same parametrization from[21] Eq. (4) - Eq. (8) with two sets of parameters I and II from Table 1. A good agreementwith the experimental data in the interval of transferred momentum 0 < q ≤ . f m − can5 -8 -6 -4 -2 -6 -4 -2 exp DWBA 800 MeV PWBA | F c h ( q ) | He q (fm -1 ) exp DWBA 600 MeV PWBA Li FIG. 1: Charge form factors of the stable isotopes He and Li obtained in [23] using LSSMdensities in PWBA and in DWBA calculations in comparison with the experimental data [18, 24,25, 26, 27, 28, 29, 30]. be seen and a disagreement with the values of the form factor for larger q ’s that are relatedto small values of r ’s, i.e. to the central part of the nuclear density. In other words, thecentral density can be different from the assumption for the folding density (Eq. (1)). Wenote the similarity of the results (compared with the data) of the calculated charge formfactor of Li for q < ∼ . f m − from the present work (shown in Fig. 3) with those from [23](shown in the down panel of Fig. 1).The results shown in Fig. 3 were the reason to look for an extension of the approach. Oursecond suggestion was to consider the charge density of Li as a superposition of a foldingterm and a sum of the charge densities of He and the deuteron with weight coefficients c F c h q (fm -1 ) d d FIG. 2: The charge form factor of the deuteron calculated using two sets of parameters α , β , γ and δ (with values given in Table 1) and compared with the experimental data [19, 20] up to q ≈ . f m − . and c : ρ ch Li ( ~r ) = 32 c Z d ~r ′ ρ ch He ( ~r − ~r ′ ) ρ chd ( ~r ′ ) + c [ ρ ch He ( ~r ) + ρ chd ( ~r )] . (9)The normalization of the densities in Eq. (9) to Z leads to the condition for the coefficients c + c = 1 . (10)Using the charge density (Eq. (9)), the following expression for the charge form factor of Li (with the account for the c.m. correction) is obtained: F ch Li ( q ) = n c F ch He ( q ) F chd ( q ) + c F ch He ( q ) + F chd ( q )] o e q / (4 A / ) . (11)For q = 0 F ch Li (0) = 1 . (12)The squared charge form factor can be written as: | F ch Li ( q ) | = A + B + C , (13)where A , B and C represent the contributions to the charge density of Li of the folding term( A ), of the sum of the charge densities of He and the deuteron ( B ) and the interference7 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.510 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I F c h ( q ) I q (fm -1 ) Li FIG. 3: The charge form factor of Li calculated by using Eq. (3) and the experimental datafor the charge form factors of He and the deuteron in comparison with the experimental data([18, 24, 25, 26, 29, 30]). term ( C ). Their explicit expressions are: A = c | F ch He ( q ) | | F chd ( q ) | e q / (2 A / ) , (14) B = c | F ch He ( q ) | + | F chd ( q ) | + 4 | F ch He ( q ) || F chd ( q ) | ] e q / (2 A / ) , (15) C = 23 c c | F ch He ( q ) || F chd ( q ) | [ 2 | F ch He ( q ) | + | F chd ( q ) | ] e q / (2 A / ) . (16)In the following Fig. 4 are presented the results for the squared charge form factor of Licalculated using Eq. (13) - Eq. (16) and the experimental data for the charge form factor of He, of the deuteron and with different sets of the values of the weight coefficients c and c .The fit of Eq. (13) to the experimental data reveals an interval of values of c = 0 . ÷ . c = 0 . ÷ . c = 0 .
979 and c = 0 . A , B and C .The results presented in Fig. 4 and Fig. 5 prove that the contribution of the foldingdensity to the charge density of Li is about 97 . ÷ . He and the deuteron densities of about 2 . ÷ . .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.510 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I F c h ( q ) I q (fm -1 ) Li experiment FIG. 4: The charge form factor of Li (Eqs. (11) - (16)) calculated for c = 0 . c = 0 . c = 0 . c = 0 .
021 (solid line), and c = 0 . c = 0 .
015 (dashed line). Theexperimental data are taken from [18, 24, 25, 26, 29, 30]. seen that the term A (Eq. (14)) describes well the squared charge form factor of Li in theinterval 0 < q < ∼ . f m − , while the shell-model cluster density (related to the term B , Eq.(15)) is important for the description of the charge form factor of Li for the large values of q ( q > ∼ f m − ), related to the central nuclear density. The interference term C (Eq. (16))has a contribution to the charge form factor of Li for q > ∼ f m − . The increase of c withinthe above interval leads to a better description of the data for q = 1 . ÷ . f m − , but at thesame time to a decrease of the values of the squared Li charge form factor for q > ∼ f m − ,underestimating the data.As known, the value of the obtained rms radius is a test for the consistency of anyapproach to the description of the nuclear system structure. The charge rms radius of Liis given by the expression: h r Li i = 13 Z d ~r r ρ ch Li ( ~r ) . (17)Substituting the expression for the charge density of Li (Eq. (9)) in Eq. (17), we obtain: h r Li i = c [ h r He i + h r d i ] + c h r He i + h r d i ] . (18)9 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.510 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 A B C A+B+C I F c h ( q ) I q (fm -1 ) Li experiment FIG. 5: The same as in Fig. 4 for c = 0 . c = 0 . A , B and C termsare presented. The usage of the experimental data for the rms radii of He and the deuteron [25, 31]: h r He i / = 1 . f m , h r d i / = 2 . f m in Eq. (18) (with c = 0 .
979 and c = 0 . Li chargerms radius: h r Li i / = 2 . f m , which is in accordance with the experimental estimations for the charge rms radius of Li[25, 31]: h r Li i / = 2 . f m. This could be expected due to the use of the experimental charge densities of the deuteronand He, being combined in a realistic theoretical scheme that gives a good agreement withthe experimental data for the charge form factor of Li.
II. CONCLUSIONS
In the present work we suggest a theoretical scheme for calculations of the charge densitydistribution and form factor of Li in the framework of the α − d cluster model of this10ucleus. The obtained results can be summarized as follows: • Our calculations show a reasonable description of the charge form factor of Li on thebasis of a superposition of two density distributions: (a) a folding density obtained from He and the deuteron charge densities, and (b) a sum of the He and deuteron charge densities.Provided corresponding experimental data for both densities are used, the calculationsshow that a reasonable agreement with the data can be obtained when the weightof the folding density contribution is about 97 . ÷ .
5% and the weight of thecontribution from the sum of both densities is about 2 . ÷ . • The scheme has only one free parameter ( c or c ) with a clear physical meaning,namely, it is the weight of the one of the contributions to the density of Li. • The behavior of the charge form factor of Li for 0 < q < ∼ . f m − is determinedmainly by the folding contribution of He and the deuteron densities to the chargedensity of Li (the weight of this contribution is about 97 . ÷ . • The shell-model α − d cluster density of Li (i.e. the sum of He and the deuteroncharge densities) is important (though with a small weight of about 2 . ÷ . Li at large values of q ( q > ∼ f m − ). • The calculated within the suggested scheme charge rms radius of Li agrees with theexperimental estimations of this quantity. • We would like to pay attention to the following facts: (a) the minimum of the experimental charge form factor of the deuteron is at q ≈ . f m − [19, 20], 11 b) the minimum of the experimental charge form factor of He is at q ≈ . f m − [18], (c) the minimum of the experimental charge form factor of Li is at q ≈ . f m − [18, 24, 25, 26, 29, 30]. Based on points (a) and (b), our estimations show that thelatter minimum is determined mainly by the contribution of the charge densityand the corresponding form factor of He.
Acknowledgments
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