c5.3 adam shell

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International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (309-311)

Shell Correction With Strutinskys Model

Adam Romulo Nuclear Physics and Biophysics Research Division,

Department of Physics, Faculty of Mathematics and Natural Sciences,Institut Teknologi Bandung, Jl. Ganesa 10 Bandung 40132 INDONESIA

E-mail: [email protected]

AbstractSHELL CORRECTION WITH STRUTINSKYS MODEL. Besides by experiment, the values of binding

energy per nucleon can be obtained through the semi-empirical approach. One of it is Weizsackers formulawhich is based on the liquid drop model. There are five contributions considered in this approach: volumeenergy, surface effect, symmetry effect, coulomb interaction, and correction related to spin. Liquid drop model,quantitatively, describes the global trend nuclear potential energy; however, the value of nuclear potentialenergy obtained still has deviation from the energy of liquid drop. The most deviation is related to the shelleffect: with closed shell nuclei is bond stronger than an average nuclei, and a mid-shell nuclei is bond weaker.

Based on this overview, it is necessary to make a shell correction formula so that the value of nuclear potential

energy can be obtained more precisely. Strutinskys Method is one of shell correction calculation method by thesingle particle energy in a given deformation. In this method, the shell correction is built by the geometricinterpretation, and the emphasis is more to the single particle than to the level of density. By using Strutinskysmodel, the deviation of nuclear potential energy to the liquid drop energy is expected to be eliminated.

Introduction The shell correction arises because of

fluctuations in the actual distribution of single- particle levels telative to a smooth distributionof levels. Since neutrons and protonsindependently fill the single-particle orbit set,shell correction is given as independent sums of

a term for neutrons and as an analogous term for protons.There are some approaches to calculate

the value of shell correction. One of thoseapproaches is known as Strutinskys method. Inthis method, shell correction is calculated fromthe single-particle energies at a givendeformation. Based on geometric interpretation,the emphasize of this Strutinkys method ismore emhasized to the single-particle energythan to the single-particle level density.

With the aid of Fig.1 as an example, thesolid curve (like staircase) gives single neutronenergy dependence n to the single particle number n. For a macroscopic sistem without single-particleeffects, the whole enery will be lied on the smoothcurve; however, the discretness of the single-

particle sauses some fluctuation about a monotonicincreasing function of n. Descrete energy n can beregarded as staircase function stair (n) which areformed by vertical and horizontal lines through the

points.

Figure 1 . Dependence of single-particle number for a spherical 208Pb nucleus

Local energy fluctuation stair (n) isremoved when retaining its long-renge behavior by

passing a smooth curve (n) to the staircasefunction. Shell correction for N specified particlenumber is the difference between the area under thestaircase curve and smooth curve at an appropriate

N, that is:Although the calclulation for shell

correction will be applied for even particlenumbers, equation (1) is still applied for odd valueof N calculation.

In Strutinskys method, the quantity that is

explicitly given is the invers of (n) function whic

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is known as the average of particle number n () asa function of the single-particle energy. Thisquantity is reached by noting that exact particlenumber is given by

,with be single-particle level density is

Basicly, the summation in eq (2) should extendover only the bound states, with the continuumstates represented by an appropriate continuousexpression. However, because the continuum statesaffect only the final shell correction when

determining the smooth curve (n) for particlenumbers nat and below the fermi surface, they can

be approximated with a good enoughapproximation in terms of descrete states. The nextstep is to separate the exact level density g( ) into a

smoothly varying part g () and a part of g ()that has the local fluctuations, that is:

,

This will be accomplished by expanding the function in a series of Hermite polynomials whichis followed by separating the terms into a smoothlyvarying parts and a fluctuating part. This expansionincludes automatically a Gaussian weightingfunction and leads to:with abbreviation:

,And the coefficients c m are given by:

.

A scaling factor, which has the dimension of energy, is introduced to make the argumentsdimensionless and to control the Gaussianweighting function keeps working at the range thatits values nonzero. The summation over m involveonly even m since the coefficients of every oddHermite polynomials are zero.Since Hermite polynomials of low order oscillateslower than the higer order, some terms fromHermite polynomilas in equation (3) representingsmooth variation to g( ), and the rest terms givefluctuation contribution. Therefore, smooth level

density g () is given by an expression analogousto eq (3), but with the summation over m extendingonly to p(which defines the order of shellcorrection) rather than to infinity. In the fluctuating

terms of g (), the summation begin from p+1 toinfinity.

Now, average particle number can be explicitlyevaluated, that is:

Conceptually, the next step is to invert this equationto get the value of average single-particle energy (n) as function of particle number. This function

could then be inserted to the eq (1) and the resultingintegral evaluated numerically to give the shellcorrection.However, it will be easier if the integration in theeq (1) is evaluated to the energy, that is

.

The upper limit is interpreted as Fermi energyfor the smooth distribution as illustrated in Figure1. The value is explicitly determined by the relation

which is iteratively solved with a given n ( ) fromeq (4). The desired integral can be evaluated to getwith abbreviation:

.

(2)

(4)

(1)

(3)


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The subtitution of this result to the eq (1) will givethe shell correction.

Calculation Result Using the eq (5) and (1), the calculation of

shell correction for several nuclei with various mass

numbers A was done. The value of is taken for 38 MeV and the values of are varying depend onA with relation = 41MeV/A 1/3. Based onCPM{16,14} method, the values of n for every

nuclei involved are generated by using Maslise2007. The harmonic oscilator potential is used as

potential function. In this calculation, the radius r isvarying depend on the valur of A (with relation r =1.2A 1/3). Using 10 -8 tollerance, the CPM {16,14}isevaluated from lower limint -100 to upper limit100. The calculation result can be viewed in fig.2.

below. The whole calculations of eq (5) and (1) has been done under Microsoft Office Excel 2007.

Figure 2 . Shell correction Vs A Curve

Reference 1. M. Bolsterli, New Calculation of Fission

Barrier for Heavy and Superheavy Nuclei, Phys. Rev. C. Vol 5, No. 3, 1972.

2. Verlee Ledoux, Study of Special Algorithms for solving Sturm-Liouvilleand Schrdinger Equations , Ph.D. Thesis,

Dept. of Applied Mathematics andComputer Science, Ghent University,2007.

3. http://en.wikipedia.org/wiki/Fermi_energ

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