Viscosity for isotopes of Helium with HCB model
Binay Prakash Akhouri1*, Sumit Kaur2
1Department
of Physics, Birsa College, Khunti-835210, Jharkhand (India)
2Department
of Physics, Nirmala College, Ranchi-834002, Jharkhand (India)
*Corresponding Author E-mail: binayakhouri@yahoo.in
The
calculation of viscosities for the fluids of Helium isotopes, He3
and He4, has been made by using HCB models. The phase shifts
required for its calculation of viscosities have been obtained by numerical
integration of the radial wave equation expressed by taking a simple form of
Leenard-Jones intermolecular potential specified in terms of the support
function 
and
surface to surface coordinate representation K. For both He3 and He4
the phase shifts resemble the rigid sphere behavior for high collision energies
of the two colliding helium molecules of the same species. The cross-sections
required for the calculation of viscosities below temperature 5K for the fluids
of helium isotopes has also been discussed.
KEYWORDS: Support function, HCB coordinate system, Phase
shifts, Viscosity.
1. INTRODUCTION:
In
theoretical physics, the theory of molecular scattering, has received much
attention today. This is because of their vast applications fall within the
framework of molecular scattering theory. In the present work we shall deal
with the calculation of the phase shifts for the one-dimensional Schrodinger
radial wave equation and for the orientation of HCB’s along the semi-major axis
and along the semi-minor axis. However, the phase shifts for other orientations
may also be calculated. In this work the phase shifts have been obtained by
solving the radial wave equation for different values of the angular momentum 
and
collision energy
of
the two colliding molecules. The present work has been carried out with a new
form off L-J potential. Once a potential function of the L-J form is chosen,
the evaluated phase shifts can be used to calculate the cross-sections and then
by substituting the corresponding collision integral values in the viscosity
equation, the viscosities of helium isotopes can be determined. The paper has
been organized with different sections and subsections to represent our work.
Points represent the experimental value due to Keller
[20] and Je de Boer [21].
2
CONCLUSIONS:
The
most extensive low-temperature calculations have been made using the Leenard
–Jones (6-12) potential. In general, an intermolecular potential describing the
interaction between two bodies depends not only on their separation distance
but also on their relative orientation. The later is especially significant for
molecules with permanent dipole moments. Nevertheless, we presume for
convenience an intermolecular potential,
, that
depends only on the surface to surface distance,
,
separating two bodies. The two considered orientations in our work is actually
represents separation
in
terms of support function,, and
surface to surface separation,. The
phase shift calculations of our work for both the isotopes are found to be
negative at moderate low energy collision values and positive at high energy
collision values. Expectedly, the results of the phase shifts for these isotopes
agree well with the values of the phase shifts of rigid sphere model [17-22].
For both He3 and He4 the phase shifts have much
resemblance for the rigid sphere behavior at high energies as compared with the
results for both the HCB models. The values of the obtained phase shifts from
these models and hence the evaluated cross-sections for a fixed orientation of
the two HCB bodies have been used for the calculation for viscosity for Helium
isotopes. Our results for the effective He3-He3 and He4-He4
cross-sections are summarized in figure2-3, which represent the diffraction
effects encountered in transport phenomena. Actually, the attractive and
repulsive forces of the L-J 6-12 potential together with the model create an
opportunity for improved predictions of viscosities. Hence, for this simple
potential it was found that the obtained results of viscosities have remarkably
good agreement with the experimental values.
6.
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