Neglecting the vibrational motion, a diatomic molecule may be treated as a three-dimensional rigid rotator. The Hamil tonian Hm of the molecule is written as a sum of a translational,Htr, plus a rotational,Hrot, term (that is,Hm =Htr +Hrot).
Consider a system of N very weakly interacting molecules of this kind, in a container of volume V , at a given temperature T .
(a)Obtain an expression for Hrot in spherical coordinates. Show that there is a factorization of the canonical partition function of this system. Obtain an expression for the specific heat at constant volume.
(B) Now suppose that each molecule has a permanent electric dipole moment μ and that the system is in the presence of an external electric field E (with the dipole μ along the axis of the rotor). What is the form of the new rotational part of the Hamiltonian? Obtain an expression for the polarization of the molecule as a function of field and temperature. Calculate the electric susceptibility of this system.
Physics
The Lagrangian of a free rotator (two atoms of mass m and a fixed interatomic distance a) is given by
from which we have the Hamiltonian
Therefore,
In the presence on an electric field (taken along the x direction), and with the magnetic moment along the axis of the rotator, we have
The associated partition function is given by
Thus, we have
where L (x) is known as the Langevin function.
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