Ahmad, I

Madani, J H

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yi,anl Phys. 75A (3). 259-262 (2001)U P  Aan internaiiunal lournalNuclear surface thickness using semi-i^enomenological nucleondensity distributionI Ahmad*, J H Madani and M S Al-GhamdiDepartment of Physics, Faculty ot Science, KinjJ Ahdula/iz University Jeddah, Saudi Arabia [IE mail iahmad(n>kaau cduila R e c e iv e d  I I  A u f fu s t  2 0 0 0 .  a a e p t e d  21 $ e p ie in h e r  2 0 0 0>Aracf : Surface thickness parameters and for proton and neutron densities respectively have been calculated for P-stablc nuclei for mass nber A ;> 12 using the semi-phcnomenological nucleon density distribution It is found that in general, the calculated values for odd proton uirnn) nuclei differ noticeably from those for even proton (neutron) nuclei in a systemalic way Best fit cuives describing dependence of the ,uldtal and on A as well as (N-Z)/A have been obtainedywords Nuclear surface thickness, nucleon density distribution, scmiphenomcnological model r s  Nos, . 21.10 Ft, 21.IO.GvIntroductionme years ago, Gambhir and Pali! [1,2] proposed a simple cbraK : expression for nucleon densities which incorporates Tcctly the asymptotic and central behavior of the density as ju irc d  from some theoretical considerations. This semi- .momenological density has only one free parameter namely half-density radius which can be fixed by demanding the ulcl to reproduce the m easured charge rms radius, ‘plications of the semi-phenomcnological density show that :ivcs quite satisfactory results for the charge forni factor [3] J lor the neutron density distributions [4].Nuclear surface thickness, defined as the distance over which nucleon density falls from 90% to 10% of its central value, is important characteristics of nuclei. It plays an important role Jetermining hcavy-ion scattering at medium energies, which •ensitive mainly to the surface region of the colliding nuclei This is evident from the success of the simple Karol model lor calculating the nuclear transparency function. In this the transparency function is derived using surface- malized Gaussian densities for the colliding nuclei, which c^r greatly from the realistic densities in the interior region of nuclei.•^“responding A uthor.Gambhir and Patil [2] used their scmi-phenomenological density model to calculate also the surface thickness parameters and for proton and neutron density distributions. They showed that the values as predicted by the model are in satisfactory agreement with the experimental ones. For they noted that in general, it increases with the mass number. However, since the main aim of their work was to study the applicability of the scmi-phenomcnological density model, they considered a very limited number of nuclei and did not undertake a quantitali vc study of the dependence of and on some characteristics of nuclei like the mass number A and (N-Z)/A.In this work, we use the scmi-phenomenological density to calculate and for a large number of p-stablc nuclei for A > 12, to study their dependence on mass number A and (N-Z)/A. It turns out that the average behaviour of r , and as a function of A and (N-Z)/A can be described in terms of simple expressions, reasonably well. The present study is expected to be especially useful for calculating the transparency function for hcavy-ion systems involving neutron rich nuclei.1. Theoretical considerationsConsidering general requirements to be satisfied by nuclear densities, Gambhir and Patil [1,2] proposed the following©2001 lACS2603. Results and discussion/  Ahm ad, J  H  M adani and  M  S  A l-G ham dialgebraic fonn for proton and neutron density distributions in nucleip, (r)  = - p :i = (/A «). (»)a, = i -  ip, n) , ,yl2(2nie, )Q I .where is the proton (neutron) separation energy, m is the nucleon mass and0 , for neutrons ^ $7 .- $e ^  , for protonswith Z as the atomic number.The unknown parameters in the expression for proton and neutron densities arc , p)J and R, The first two arc determined from the normalization conditions and the last one, namely R is fixed by requiring that the proton density, after considering finite proton charge distribution, must give the measured charge rms radius. The study of Gambhir and Palil 12] shows that the values of R so determined arc well described by the relation :R ^ r ^ N \ (4)where = 1.31 fm and N is the neutron number. It may be mentioned that a recent study [71 shows that this dependence of R on N, to a very good approximation, is the same as shown by the diffraction radii extracted from the zeros of the charge form factors of the nuxlified Helm's model [8J.To calculate the surface thickness (i = p, n) for the semi- phenomenological density given by eq. (1), wc use the relation(4) and numerically solve the equations.p ,.{q )~ 0 .9 p ,(()) = (),p ,( r2 ) -0 .1 p ,(0 ) - ( )for rj and which respectively arc the radial distances at which p ,( r )  falls to 90% and ]0%  of its central value. The required separation energies are taken from the compilation of Audi and Wapstra [9] and the proton and neutron surface thicknesses are obtained by subtracting from r^The results of our calculation for and as a function oi number A are shown in Figure 1. The filled and open circles Figure 1(a) show the values for odd and even neutron nu respectively, while in Figure 1(b) they show the t^  valucsin^  odd and even proton nuclei. It is seen in Figure i(a) ihm ,where p stands for proton and n for neutrons. The parameter R is the half-density radius to a very good degree of approximation and the quantities « and a ,  arc given by(2)(3)Figure I. Proton and neutron surface thicknc.s.s and \e i\ u \  \  filii,, circles . for odd proton (neulion) nuclei Open circles / joi oui proton (neutron) nucleivalues for odd neutron nuclei arc systematically larjjcr iIlh those for even neutron nuclei. It is further seen that t  ^ ii both the cases, increase with the mass number A. 'fhe daslu, and solid curves in Figure 2(a) show that the average bchavn u! o f f o r  odd as well as even neutron nuclei, as a function tii A can be described by the relation=£/ + /? In i"'where the best-fit parameter values are :1.51 fm, >^ = 0.85 fm, for odd neutron nuitci a = 0.79 fm, /? = 1.015 fm, for even neutron nucleieven p rot on (ne utro n)  nuclei.Nuclear surface thickness using setni'phenonienological nucleon density distribution 261Xhe A-depcndcnce of t^  ^ is shown in Figure 1(b). As expected, we sec again that the values lor odd proton nuclei are larger than those for even proton nuclei. In either case, t^  increases with A up to A -  50 and then starts decreasing slowly, [towever, the rale of decrease of t^  is quite small for even proton fujdci, it may be assumed essentially as constant for this mass range. Ttic values for the two cases come close together near ihc end of the periodic table. We find that the average A- jependence of for odd as well as even proton nuclei can be well described by the relationt r: a ~ h A - c f  A , (6)where the best fit parameter values are :r/ = 2.671fm, /? = 0.002 fm, c = 7.532 fm,for odd proton nuclei; (7 = 2.394 fm, ft = 0.001 fm, c = 6.600 fm,for even proton nuclei.Bdore concluding the discussion of Figure 1, it may be poinied out that the larger for odd proton (neutron) nuclei as compared to the even proton (neutron) nuclei is simply a nllccUon of the fact that the separation energy which dominantly governs the surface diffuseness of the nucleon dcnsiiy through cq. ( 2 ) ,  is smaller for odd proton (neutron) nuclei ih.m lor even proton (neutron) nuclei.In higure 2, we show plots of and as a function of iN -/j /A . In this case also, solid circles show for odd proton (neutron) nuclei and open circles even proton (neutron) nuclei. It IS seen that for both odd and even proton nuclei is» on the a\cragc, a linearly decreasing function of (N“Z)/A, while for hoih odd and even neutron nuclei is a linearly increasing function i>r(N-Z)/A. For , we find that the best fit parameter values for ihc linear relation= r t - f t (A -Z ) /A , (7)a = 2.55 fm, ft = 1.6 fm. for odd proton nuclei;a = 2.30 fm, ft = 0.65 fm, for even proton nuclei.On the other hand, for t^^ , the best fit parameter values for thelinear relation/„ = « 4 -f t(A -Z )/A (8)arc« = 2.50fm, ft = 5.04 fm, for odd neutron nuclei; «=2.13fm, ft=5.05fm, for even neutron nuclei.Coming to the A-dependence of as seen in Figure I, it been already stated that it is a reflection of the A- ^epcndence of the proton and neutron separation energies. To elaborate this point, it is to be noted that in the asymptoticregion, the radial dependence of the density as given by cq. ( I) is of the form :i - i p .  r}). (9)In fact, it was to achieve this theoretically desired asymptotic form that cq. ( I ) that was propo.sed by Clambhir and Pali! 11,2). Now in case of neutron dcnsiiy, a,, = 1, hence the density fall- off is Characteristically governed by the parameter «  ^ which dcpcntls upon the neutron separation energy through eq. (2). Siicc on the average, £„ decreases with A, the parameter 77 ,^ increases as A incrca.scs. In consequence, the rate of fall of P;,(7 )|wilh r decreases as A increases, causing t^  ^ to increase with A| 'Phe difference in the t^  ^ values for even and odd neutron nuclei.iis due to the well known fact that for even neutron nuclei is larger than for odd neutron nuclei.On the other hand for Ppir),  the parameter r/,,, after combining eqs. (3) and (2), is given by = 1 + 0.069(7--1 which increases in importance as A increases. Thus unlike p,, (/ ), the a.symptotic behaviour of pp(r)  is characterised by both cc^  and The exponential factor in eq. (9), as m the case of {/), tends to make Pp(r) to extend to larger r values as A increases; but this is opposed by the factor which grows inimportance with A. In consequence, except for lighter nuclei is essentially con.stant over a large mass region and slightly decreases for very large mass nuclei. Phe even oild effect in has the similar interpretation as for t^  discussed above.The trends seen m Figure 2 arc merely a rcllection ol the trends .seen in Figure 1 because for (i-slablc nuclei, the quantity (N-Z)/A in general, increases with A. Hence, incrca.ses with A while has a decreasing trend However, it should be noted that values have a large scatter and its linear decrease with (N-Z)/A is not so well deilncd as linear increase for t^  values.The present study is expected to be particularly useful for calculating heavy-ion total reaction cross sections using the modified microscopic model applied by Shen etal. (10], which unlike the Karol model |61 makes the provision for proton and neutron densities of the colliding nuclei to be different. In these models, the realistic densities (assumed to be described by the two-paramclcr Fermi distribution) are replaced by the surface normalized Gaussian densities who.se parameters depend upon the half density radius and the surface thickness parameters of the respective Fermi distributions. Shen er a i  | lO], using the half-density radius parameters as given by the droplet model and taking t^  -  -  2.4 fm, find that the interaction radiusparameter deduced from the total reaction cross section measurements is not reproduced, ITiey further show that the interaction radius data could be satisfactorily accounted for if it is assumed that increases linearly with (N-Z). This is in agreement with the trend of t^  found in Figure 2.262 I A h m d, J H Madani and M S Al-Ghamdi4. ConclusionsWe have calculated proton and neutron surface thickness parameters and respectively for a large number of nuclei for /i ^ 12, using Gambhir-Palil scmi-phenomenological density. The calculated values for odd proton (neutron) nuclei are found to be systematically larger than those for even proton (neutron) nuclei. It is also found that the calculated and values show dependence on the mass number A and (N-Z)/A which on the average, can be described by simple formulae reasonably well. These formulae could be useful for more realistic calculation of the nucleus-nucleus transparency function using the modified Karol model [ 10| in which the neutron and proton surface thicknesses are taken to be different.Kcferencefi(1) Y  K Gambhir and S H Patil Z  Phys A321 161 (1925)[21 Y K Gambhir and S H Patil Z  Pliys. A324 9 (1986)[.1] Y  K Gambhir, P Ring and H de Vries Europhys Lett ]q i >,, (1989)[4] G A  Lalazissis, C P Panos, M E  Grypeos and Y  K Gamhhir Phys A3S7 429 (1997)j.5] S Kox el al Phys. Rev. C35 1678 (1987)[6] P J Karol Phy.s Rev C I l  120.3 (197.5)|7] I Ahmad, M A  Abdulmomcn and J H Madani Pme DAE .S\,m, on N u d  Phys. 41B 120 (1998)18] J Friedrich, N  Vocglcr and P G Rcinhard Nucl Ph\.s A4S9 n(1986)[91 G Audi and A  H Wapstra Nud Phys. A56S 66 (1993)[10] W Shen et ul Nud Phys. A491 130 (1989)

Nuclear Surface Thickness Using Semi-Phenomenological Nucleon Density Distribution

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Nuclear Surface Thickness Using Semi-Phenomenological Nucleon Density Distribution

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