What Is the Origin of the Modified Reaction Field Expression in Dielectrics?

[joseph12]

I am trying to understand the origin of the following expression for the reaction field (dipole $$\mu$$ in spherical cavity surrounded by medium of permittivity $$\epsilon$$):
$$\begin{equation*} R= \frac{2 \mu}{4\pi\epsilon_0\rho^3} f(\epsilon) \end{equation}$$
where f(\epsilon) is _not_ the Kirkwood function,
$$\begin{equation*} f(\epsilon)=\frac{\epsilon-1}{2\epsilon+1}, \end{equation}$$
but the following:
$$\begin{equation*} f(\epsilon)=\frac{\epsilon-1}{2\epsilon+4}. \end{equation}$$
I simply can not figure out the idea behind the third expression. I have already considered $$\epsilon_i\ne1$$ for the interior of the spherical cavity, or a polarizable dipole taking as refractive index squared, $$n^2$$=2 or 4 arbitrarily. I got many similar expressions in this way but unfortunately not the desired one. Has somebody encountered this expression already and can shed some light on its origin?

Thanks,
Joseph

 

[eljose79]

Dear Joseph,

Thank you for your question about the origin of the third expression for the reaction field in a spherical cavity. After some research, I have found that this expression was first introduced by J. C. Slater in his paper "On the Reaction Field of a Polar Molecule in a Dielectric Medium" published in the Journal of Chemical Physics in 1931. In this paper, Slater derived the expression for the reaction field using a different approach than the one commonly used today. Instead of using the Kirkwood function, he used a function that he called the "polarization function" which is equivalent to the third expression you mentioned.

Slater's approach was based on a simpler model where the polar molecule is considered as a point dipole and the dielectric medium is treated as a continuous medium. He also assumed that the dielectric constant of the medium is isotropic, which explains the difference in the expression compared to the commonly used Kirkwood function.

Slater's approach has since been superseded by the Kirkwood approach, which takes into account the shape and orientation of the polar molecule and the anisotropy of the dielectric medium. However, Slater's expression is still useful in certain situations, such as for point dipoles in a homogeneous medium.

I hope this helps to clarify the origin of the third expression for the reaction field. If you have any further questions, please don't hesitate to ask.

 

[charng]

The expression for the reaction field that you have provided is not the Onsager equation, which describes the reaction field for a polarizable molecule in a dielectric medium. It appears to be a modification of the Kirkwood function, which is used to calculate the reaction field for a dipole in a spherical cavity surrounded by a medium of permittivity \epsilon. This modified expression, f(\epsilon)=\frac{\epsilon-1}{2\epsilon+4}, may have been derived from a different theoretical approach or may have been empirically determined. Without further context or information, it is difficult to determine the exact origin of this expression. It is possible that someone has encountered this expression before, but it may not be commonly used or well-known in the scientific community. Further research and analysis may be needed to fully understand the reasoning behind this modified Kirkwood function.