Z. Phys. A 357, 429–432 (1997) ZEITSCHRIFTFUR PHYSIK Ac© Springer-Verlag 1997

Semi-phenomenological neutron density distributions

G.A. Lalazissis1, C.P. Panos1, M.E. Grypeos1, Y.K. Gambhir 2

1 Department of Theoretical Physics, Aristotle University of Thessaloniki, GR-54006 Thesssaloniki, Greece2 Physics Department, IIT. Powai, Bombay 400076, India

Received: 1 July 1996Communicated by W. Weise

Abstract. The simple algebraic form for the nuclear densi-ties designed to incorporate correctly, the two physical re-quirements, namely, the asymptotic behaviour and the be-haviour near the centre, is used to calculate the neutrondistributions in several nuclei. Sample neutron densities for58,64Ni, 116Sn and 208Pb are presented. The calculation re-veals an excellent agreement with the experiment as wellas with the relativistic and nonrelativistic microscopic meanfield calculations. The use of this algebraic form of the den-sities in the analytic studies is strongly advocated.

PACS: 21.10.-k; 21.10.Ft

1 Introduction

The nucleon distributions in nuclei is of fundamental impor-tance in our understanding of nuclear properties. The mostreliable information about the proton (or charge) distributionin the nucleus is derived from the electron-nucleus scatter-ing. The experimental data has been analysed [1] both us-ing the model independent and model dependent proceduresyielding the corresponding model independent charge den-sities either in terms of Fourier Bessel series or in termsof sum of Gaussians and the model dependent densities interms of two or three parameter Fermi or three parameterGaussian forms. The information about the neutron distribu-tions is comparatively less reliable and is rather scarce. Thisis because, for this purpose, the strongly interacting probes(e.g. protons,α, pions) are required.

On the theoretical front, due to the developements in themany body techniques, the situation is more comfortableand reliable. For example the density dependent Hartree-Fock (DDHF) calculations with Skyrme or Gogny type in-teractions or the relativistic mean field (RMF) calculations[2] yield fairly reliable ground state nuclear properties in-cluding the densities. The RMF is also able to reproducethe recent isotope shift measurements correctly particularlyfor the Kr-, Sr- and the Pb- isotopes [3, 4] where the con-ventional DDHF failed. These mean field methods yield thedensity distributions in terms of explicit numerical numbers.

Therefore, it is important and meaningful to develop alsoa simple functional form for the densities which is essen-tial for the analytic studies especially in the nuclear scat-tering, reactions and other related processes. Bredicheviskyand Mosel [5] based on their DDHF studies attempted toparameterize the nuclear densities in terms of generalizederror functions. Friedrich et al. [6, 7] describe the nuclearcharge density in the Helm model where the three parametersof the model are determined from the positions of the firsttwo zeroes and from the amplitude of the second maximumof the observed electron scattering form factor. Recently, asemi-phenomenological simple algebraic expression for thenuclear densities [8, 9] has also been proposed which isdesigned to incorporate correctly the physical requirementsnamely the asymptotic and the central behaviour and it au-tomatically yields separately the proton densities and theneutron densities. It is to be mentioned that the celebratedWoods - Saxon form does not satisfy these physical require-ments. The reliability of the semi-phenomenological protondensities has been tested in detail by calculating and com-paring with the experiment the form factors for the electronscattering. Remarkable agreement obtained between the cal-culation and the experiment establishes [10] the reliabilityof these proton densities. In the present note, we attempt totest the corresponding neutron densities with the twin aimsnamely, to establish the reliability and to generate confidencein the semi-phenomenological densities.

2 Semi-phenomenological density distributions

The semi-phenomenological density distributions proposedby Gambhir and Patil [8, 9] has the form:

ρi(r) =ρ0i

1 + [12 + 1

2( rR )2]αi [e(r−R)ai + e

−(r+R)ai ]

. (1)

The index i= n (for neutrons) or p (for protons), R is ameasure of the size of the nucleus andai andαi are given interms of the separation energyεi of the last particle (neutronor proton) through:

430

2.0 4.0 6.0 8.0r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12ρ n(

r) (

fm-3

)

SP-DHF+BCSRMFexpt.

58Ni

Fig. 1. The neutron density distributionρn(r) for 58Ni of the present work(SP-D), the DDHF with SkM* interaction (HF+BCS) and the relativisticmean field theory with the NL-SH parameters (RMF). The experimentalρn(r) (expt.) is taken from [12]

2.0 4.0 6.0 8.0r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

ρ n(r)

(fm

-3)

SP-DHF+BCSRMFexpt.

64Ni

Fig. 2. The neutron density distributionρn(r) for 64Ni. For details seecaption of Fig. 1

ai =~

2√

2mεi; (2)

αi =q

~

√m

2εi+ 1 (3)

Herem is the nucleon mass and q=0 for neutrons and q=Z-1for protons.

The aboveρi(r) correctly incorporates the following twoimportant physical requirements:1) The small r (r→ 0) behaviour which implies that thedensity contains only even powers of r.2) The asymptotic behaviour

ρi(r) → r−2αiexp(−r/ai) , (r →∞) . (4)

There are three unknowns:ρ0n, ρ0

p and R in (1). Two ofthese (the central densitiesρ0

n andρ0p) are determined from

the normalization:

2.0 4.0 6.0 8.0r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

ρ n(r)

(fm

-3)

SP-DHF+BCSRMFexpt.

116Sn

Fig. 3. The neutron density distributionρn(r) for 116Sn. For details seecaption of Fig. 1

2.0 4.0 6.0 8.0 10.0 12.0r (fm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

ρ n(r)

(fm

-3)

SP-DHF+BCSRMFexpt.

208Pb

Fig. 4. The neutron density distributionρn(r) for 208Pb. For details seecaption of Fig. 1

4π∫

ρn(r)r2dr = N , (5)

4π∫

ρp(r)r2dr = Z ; (6)

where N and Z are the total number of neutrons and protonsin the nucleus respectively. The remaining lone parameter R(assumed to be the same for neutrons or protons) is deter-mined eithera) through the requirement that the density reproduces theexperimental root mean square (rms) charge radius

rc ' (r2p − 0.6)1/2 , (7)

whererp is the rms radius for the proton density calculatedfrom (1) and the factor 0.6 accounts for the small correctionarising due to the finite size of the proton and the neutron.or b) through:

R = 1.310N1/3 . (8)

The knowledge ofρ0n, ρ0

p and R then is sufficient to deter-mine the proton and neutron densities (1).

431

Table 1. Comparison of neutron rms values,< r2 >1/2n , of experiment (expt.) taken from [12], the present work (SP-D), relativistic mean field theory with

the NL-SH parametrization (RMF), DDHF with the SkM* Skyrme interaction (HF+BCS) and Hartree Fock Bogoliubov with the SkP interaction (HFB)

Nucleus rp R rnexpt. SP-D RMF HF+BCS HFB

40Ca (3.450) 3.395 3.491 3.203 3.311 3.377 3.39948Ca (3.451) 3.727 3.625 3.658 3.583 3.600 3.65758Ni (3.689) 3.920 3.700 3.678 3.727 3.695 3.72664Ni (3.771) 4.229 3.912 4.015 3.970 3.902 3.91490Zr (4.203) 4.722 4.289 4.241 4.296 4.285 4.302116Sn (4.562) 5.294 4.692 4.752 4.717 4.654 4.676124Sn (4.608) 5.442 4.851 4.921 4.870 4.787 4.798208Pb (5.448) 6.551 5.593 5.778 5.713 5.620 5.611

Table 2. Comparison of neutron rms values,< r2 >1/2n , of the present work (SP-D) with the average values obtained from a study of pionic atoms (P.A)

taken from [16]

24Mg 27Al 28Si 32S 40Ar 40Ca 56Fe 63Cu 75As 195Pt 197Au 208PbSP-D 2.89 2.82 2.89 3.02 3.55 3.20 3.74 3.82 4.20 5.77 5.49 5.78P.A. 3.12 3.11 3.21 3.31 3.54 3.52 3.89 3.99 4.20 5.56 5.59 5.69

0.10 0.15 0.20(N - Z)/A

0.00

0.02

0.04

0.06

0.08

(rn-

r p)/r

av (

fm)

SP-DRMFHF+BCSHFBexpt.

90Zr

64Ni

116Sn

48Ca

124Sn 208

Pb

Fig. 5. The fractional deviation (rn - rp)/ rav , (rav = (rn + rp)/2 ) versus(N-Z/A). The notations used are the same as in Table 1

Analytic expressions for the normalization integral andthe integration for the rms radius required in the explicitcalculations have been obtained [8, 9], in the following form:

4π∫

ρi(r)r2dr ' 4πR3

3ρ0i (1 +x2

i ) , (9)

< r2i >' R2(0.6 + 1.4x2

i ) , (10)

where

xi =π

Rai

+ 2αiR2

(R+ai)2+R2

' πaiR + aiαi

. (11)

3 Results and discussion

Explicit calculations have been carried out for several nuclei.The input information namely the single proton and singleneutron seperation energies are obtained from the availableexperimental mass tables [11]. The central densitiesρ0

n andρ0p are fixed through (5, 6) while R is determined by using the

experimental [1] charge radiusrc through the relation (7).It turns out thatRN−1/3 is roughly a constant for nucleiaround the stability line and for N≥ 14; viz:

RN−1/3 = 1.310± 0.021. (12)

This means that for our purpose one can fix R from this ob-servation,thus making the densities parameter free. As statedearlier the calculated proton densities yield the electron scat-tering form factors in excellent agreement with the experi-ment indicating the reliability of the semi-phenomenologicaldensities. Here we concentrate on the neutron densities.

The calculated neutron distributions (SP-D) for58,64Ni,116Sn and208Pb are displayed in Figs. 1–4. The correspond-ing experimental (expt) [12], those of the relativistic meanfield calculations (RMF) with NL-SH set of lagrangian pa-rameters [13] and of the DDHF with SkM* [14] Skyrme in-teraction (HF+BCS) are also shown for comparison. The ex-perimental densities taken from [12] are shown by two solidlines. The separation between them indicates the experimen-tal uncertainties. Clearly, the present semi-phenomenologicalneutron distributions closely follow the experiment, evenbetter than the mean field results in most of the cases, par-ticularly, the slope around the half density radius. However,in these discussions, the reliability of the experimentally de-duced neutron densities should also be kept in mind. Here,our purpose is not to stress these finer details, it is sufficientto point out that the present neutron densities are not infe-rior to those obtained with the mean field models and alsodo agree with the experiment.

Next we compare the calculated neutron rms radii (rn)in Table 1 with the experiment [12] and also with thoseof RMF, HF+BCS. The recent results of the Hartree-Fock-Bogoliubov (HFB) with the SkP Skyrme interaction [15] arealso listed in the table for comparison. The rms proton radiideduced from the observed charge radii (rc) along with theextracted value of R are also included in the table. It is no-ticed that all the corresponding values ofrn listed are closeto each other. At finer level departure does exist specially for40Ca (208Pb) where the experimental value is slightly larger(smaller) compared to all the rest. Again one should keepin mind the uncertainties and the model dependence of theextracted experimental values ofrn. It is therefore, reason-able to state that the neutron rms radii of the correspondingproposed densities (1) do agree with the experiment and alsowith those of the microscopic mean field models.

432

In Table 2 we compare the calculated neutron rms radiirn with those deduced from the analysis of the data of thepionic atoms (P.A). These are the average of the observedvalues obtained with the different prescriptions, listed in [16]On the average, reasonable agreement is obtained betweenthe calculated and the P.A values [16]. In general, the ob-served values are relatively larger (smaller) for the lighter(heavier) mass nuclei. Due to the model dependence and theuncertainties we do not intend to discuss the comparison atthe finer level.

In Fig. 5 we plot the fractional deviation (rn - rp)/ rav,(rav = (rn + rp)/2 ) versus (N-Z)/A, A being the mass num-ber for the cases where the corresponding experimental in-formation is available [12]. Clearly, the results are as rea-sonable and satisfactory as one would expect.

In conclusion we wish to state that the semi-phenomeno-logical algebraic form of the densities are shown to agreeremarkably well with the experiment and are also consistentwith those obtained in the microscopic mean field models.The form of the nuclear (neutrons and protons) densitiesis so designed so as to incorporate correctly two importantphysical requirements of asymptotic behaviour and the be-haviour near the origin. This, thus lends support and faith inthese functional form of the densities. It is however stressedthat the aim is not to replace the densities obtained in theexplicit microscopic calculations. The main motivation hereis to advocate the use of this simple algebraic form of thedensities with confidence in future analytic studies especiallyin the nuclear scattering, reactions and related processes.

The authors are thankful to S.H. Patil for useful discussions and his interestin the work.

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