COMPARATIVE STUDY OF PHASE SHIFTS FOR HCB MODEL AND HS MODEL

SINHA N.K.; CHOUDHARY B.K.

COMPARATIVE STUDY OF PHASE SHIFTS FOR HCB MODEL AND HS MODEL

SINHA N.K.¹, CHOUDHARY B.K.²*
¹Department of Applied Physics, Cambridge Institute of Technology, Tatisilwai, Ranchi-835103, Jharkhand, India.
²Department of Applied Physics, Cambridge Institute of Technology, Tatisilwai, Ranchi-835103, Jharkhand, India.
* Corresponding Author : binodvlsi@gmail.com

Received : 22-10-2012 Accepted : 20-11-2012 Published : 27-11-2012
Volume : 3 Issue : 1 Pages : 41 - 44
World Res J Appl Phys 3.1 (2012):41-44

Conflict of Interest : None declared

Cite - MLA : SINHA N.K. and CHOUDHARY B.K. "COMPARATIVE STUDY OF PHASE SHIFTS FOR HCB MODEL AND HS MODEL." World Research Journal of Applied Physics 3.1 (2012):41-44.

Cite - APA : SINHA N.K. , CHOUDHARY B.K. (2012). COMPARATIVE STUDY OF PHASE SHIFTS FOR HCB MODEL AND HS MODEL. World Research Journal of Applied Physics, 3 (1), 41-44.

Cite - Chicago : SINHA N.K. and CHOUDHARY B.K. "COMPARATIVE STUDY OF PHASE SHIFTS FOR HCB MODEL AND HS MODEL." World Research Journal of Applied Physics 3, no. 1 (2012):41-44.

Copyright : © 2012, SINHA N.K. and CHOUDHARY B.K., Published by Bioinfo Publications. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Abstract

The intermolecular potential for HCBâ€™s (hard convex bodies) reduces to hard sphere (HS) intermolecular potential for major and minor axis ratio equal to one and has exactly the same surface-to-surface distribution as for HCBâ€™S. The radial wave equation reduces to HS Model expression in terms of the surface-to-surface co-ordinate representation. The phase shifts for HS Model have also been calculated.

Keywords

Phase shifts, HCB & HS Model, Co-ordinate System, Support function h(x), Intermolecular potential, radial wave equation, Constant parameters.

Introduction

The phase shifts is the solution of the radial wave equation. The expression for the radial wave equation of a HCB Model co-ordinate system has been described first and expressed for the pair intermolecular potential specified in terms of the support function h(x) and surface-to-surface co-ordinate representation. The phase shifts have been calculated for many values of the quantum number l and J* for the orientation along major and minor axis of HCB Model. The properties of hard convex bodies (HCBâ€™s) necessary for our analysis are due to Kihara [1] and others [2] .
The phase shifts expression for HCBâ€™s model reduces to hard sphere (HS) Model expressions in terms of surface-to-surface co-ordinate representation for the isotropic-symmetric orientation and for major and minor axis ratio equal to one. The phase shifts for HS Model also have been calculated.

Methodology

The essential task is determing the explict form of is that of determining the scale factors.
The condition for this transformation is that the jacobian
$J=det\begin{vmatrix} \frac{\partial x}{\partial u_{1}} & \frac{\partial x}{\partial u_{2}} & \frac{\partial x}{\partial u_{3}}\\ \frac{\partial y}{\partial u_{1}} & \frac{\partial y}{\partial u_{1}} & \frac{\partial y}{\partial u_{1}}\\ \frac{\partial z}{\partial u_{1}} & \frac{\partial z}{\partial u_{1}} & \frac{\partial z}{\partial u_{1}} \end{vmatrix}$ is non zero.
In terms of HCBâ€™S co-ordinates (K,Î¸,Ã˜)
u₁=K,u₂=Î¸,u₃=Ã˜
and h1 =1
h2 = h(x) + K
h3 = {h(x) + K} sinq
The expression for in terms of HCBâ€™S co-ordinate system is

$\left [ \frac{1}{(h(x)+K)^{2}}\frac{\partial }{\partial K}\left \{ (h(x)+K)^{2}\frac{\partial }{\partial K} \right \}+\frac{1}{(h(x)+K)^{2}\mathrm{sin}\: \theta }\frac{\partial }{\partial \theta }$

$\left ( \mathrm{sin}\: \theta \: \frac{\partial }{\partial \theta } \right )+\frac{1}{(h(x)+K)^{2}\mathrm{sin}^{2}\: \theta }\frac{\partial }{\partial \theta ^{2}}\: ]\Psi =\triangledown ^{2}\Psi \: \: \: \: \: \: \: \: \: \: (1)$

The support function h (x) for a/b = 1 reduces to a = b and the intermolecular potential for HCBâ€™S reduces to HS (Hard Sphere) intermolecular potential in terms of surface-to-surface co-ordinate representation.
If h(x) + k is replaced by a + k, this will be corresponds to the co-ordinate r or r = a + k which is a centre-to-centre co-ordinate representation. The phase shift calculated for the major axis for HCB model will corresponds to the phase shift for HS model with radius parameter a.

Result and Discussion

The asymptotic solution of the radial wave equation for real(interacting) and ideal (non-interacting) pairs of molecules are sinusoidal and differ only in the phase of the sine functions, the difference being the phase shifts,hl (J*). The phase shift depends upon the angular momentum quantum number l and the wave number of relative motion.
It is in general not possible to give an exact solution of the radial wave equation for the phases. The expression for the phase shifts has been given by N.F. Mott [3] and applied by Hulthen to calculate hl for different potentials which gives very satisfactory results [4,5] .
The integral expression for phase shifts for the HCBâ€™S model may be written as

$\mathrm{sin}\: \eta \: \varphi =\left ( \frac{\pi }{2\mathrm{J}^{*}} \right )^{1/2}\int_{0}^{\infty }(1+K^{*})^{\frac{1}{2}}\mathrm{J}\: \varphi +\frac{1}{2}(J^{*}(1+K^{*}))\frac{16\pi ^{2}}{\Delta ^{*2}}$

$.\left ( \frac{1}{K^{*12}}-\frac{1}{K^{*6}} \right ).\frac{1}{J^{*}}\mathrm{sin}(J^{*}.(1+K^{*})-\frac{\varphi \pi }{2}+\eta \varphi )dK^{*}\: \: \: \: \: \: \: \: \: \: (2)$

where,

$J_{\varphi +\frac{1}{2}}(J^{*}(1+K^{*}))$ is a Besselâ€™s function [10] given by the expression.

$J_{\varphi +\frac{1}{2}}(J^{*}(1+K^{*})=\left [ \frac{2}{\pi J^{*}(1+K^{*})} \right ]^{\frac{1}{2}}.$

$\left [ \mathrm{sin}\left ( J^{*}(1+K^{*})-\frac{\varphi \pi }{2} \right ).\sum_{n=0}^{n\leq \frac{1}{2}}\frac{(-1)^{n}(\varphi +2n)\: !}{(2n)\: !\: (\varphi -2n)\: !\: (2(k^{*}+1)J^{*})^{2n}}+$

$\mathrm{cos}\left ( J^{*}(1+K^{*})-\frac{\varphi \pi }{2} \right )\sum_{n=0}^{n\leq \frac{1}{2}(\varphi -1)}\frac{(-1)^{n}(\varphi +1+2n)\: !}{(2n+1)\: !\: (\varphi -2n-1)\: !\: (2(k^{*}+1)J^{*})^{2n+1}}\: ]$ (3)

the expression (2) may be written as

$\frac{16\pi ^{\frac{5}{2}}}{\sqrt{2}.\: J*\: ^{3}/_{2}\: \Delta ^{*^{2}}}\int_{0}^{\infty }(1+K^{*})^{\frac{1}{2}}\: .\: J_{\varphi +\frac{1}{2}}(J^{*}(1+K^{*})).$

$\eta \: \varphi =\mathrm{tan}^{-1}\frac{\left ( \frac{1}{K^{*12}}-\frac{1}{K^{*6}} \right )\mathrm{sin}\left ( J^{*}(1+K^{*})-\frac{\varphi \pi }{2} \right )}{1-\frac{16\pi ^{\frac{2}{5}}}{\sqrt{2}J^{*}\: ^{3}/_{2}\Delta ^{*2}}\int_{0}^{\infty }(1+K^{*})^{\frac{1}{2}}.\: J_{\varphi +\frac{1}{2}}(J^{*}(1+K^{*})).}$ (4)

$\left ( \frac{1}{K^{*12}}-\frac{1}{K^{*6}} \right )\mathrm{cos}(J^{*}(1+K^{*})-\frac{\varphi \pi }{2})$

The expression (4) gives the phase shifts corresponding to the angular momentum quantum number l. The expressions for phase shifts corresponding to l = 0,1,2,3,......................... have been given in terms of atan {A (J*) }, atan {B (J*) },............... for the calculation with the help of computer.
The expression (4), for example for l = 4, can be written as

$\int_{a}^{b}z\frac{1}{J^{*2}}\left ( \frac{1}{K^{*12}}-\frac{1}{K^{*6}} \right ).\left [ \mathrm{sin}(J^{*}(1+K^{*})^{2}-2\pi ).\left ( 1-\frac{45}{J^{*2}(1+K^{*})^{2}}+$

$\frac{105}{J^{*4}(1+K^{*})^{4}})+\mathrm{cos}(J^{*}(1+K^{*})-2\pi ).\left ( \frac{10}{J^{*}(1+K^{*})}- \frac{105}{J^{*3}(1+K^{*})^{3}}\right )\: ].$

h4 = a tan {E (J*)} =

$-2\pi ).\left ( 1-\frac{45}{J^{*2}(1+K^{*})^{2}}+ \frac{105}{J^{*4}(1+K^{*})^{4}}\right )+$

$\mathrm{cos}(J^{*}(1+K^{*})-2\pi ).\left ( \frac{10}{J^{*}(1+K^{*})}+ \frac{105}{J^{*3}(1+K^{*})^{3}}\right ).$

$\mathrm{cos}(J^{*}(1+K^{*})-2\pi )dK$

Where
$J_{4+\frac{1}{2}}(J^{*}(1+K^{*}))=\left ( \frac{2}{\pi J^{*}(1+K^{*})}\right )^{\frac{1}{2}}.$

$\left [ \mathrm{sin}(J^{*}(1+K^{*})-2\pi ).\left ( 1-\frac{45}{J^{*2}(1+K^{*})^{2}}+\frac{105}{J^{*4}(1+K^{*})^{4}}\right ).$

$+\: \mathrm{cos}(J^{*}(1+K^{*})-2\pi ).\left ( \frac{10}{J^{*}(1+K^{*})}-\frac{105}{J^{*3}(1+K^{*})^{3}}\right )\: ]$

and

$Z=\frac{16\pi ^{2}}{\left [ \frac{h}{S.(mE)^{1/_{2}}} \right ]^{2}}$

The constant parameter a, b and E of the pair potential are known [4] . These values are a = 2.792 x 10-8 cm b = 1.396 x 10-8cm and E =1.414 x 10-15 erg. for ratio a/b = 2. These values have been used here for the calculation of the phase shifts.
This expression for phase shift may be used for the calculation of phase shift along any orientation, this has been used only for major and minor axis orientation here. S is a minor or major axis.
The expression (4) has been used for the calculation of phase shifts for HCBâ€™S model along the major and minor axis for He4 and He3. The phase shift with even l (l = 0,2,4,6,.......) have been calculated for He4, since He4 has an even number of nucleons and electrons and zero spin. For He3 which has an odd number of nucleons and elecrons and spin. The phase shift with both for odd l (l = 0,1,3,5,7,......) and even l (l = 0,2,4,.......) have been calculated. The calculation of phase shifts have been done with the help of computer for J = 0.1,0.2,................ 2 for each value of l.
The phase shifts have been plotted against the collision energy J* for both orientation in [Fig-1] , [Fig-2] , [Fig-3] , [Fig-4] , [Fig-5] , [Fig-6] .

References

[1] Kihara T. (1963) Adv. Chem. Phys., 5, 147.
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[2] Kumar B., James C. and Evans G.T. (1988) J. Chem. Phys., 88, 11.
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[3] Mott N.F. and Massey H.S.W. (1949) The Theory of Atomic Collisions.
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[4] She R.S.C. and Evans G.T. (1986) J. Chem. Phys., 85, 1513.
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[5] De Boer J. and Cohen E.G.D. (1951) Physica., 11-12, 99.
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Images

	Fig. 1- Phase shift of He³ along major axis for HCB Model
	Fig. 2- Phase shift of He⁴ along major axis for HCB Model
	Fig. 3- Phase shift of He³ along minor axis for HCB Model
	Fig. 4- Phase shift of He⁴ along minor axis for HCB Model
	Fig. 5- Phase shift of He³ for HS Model
	Fig. 6- Phase shift of He⁴ for HS Model

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