On the Lifshitz formula of dispersion interaction

TL;DR

This paper reviews the Lifshitz formula for dispersion interactions, critiques its derivation, and compares methods for calculating dispersion forces, providing new calculations for specific structures and analyzing force behavior at nanometer scales.

Contribution

It clarifies the derivation of the Lifshitz formula, compares different approaches, and calculates dispersion forces for various structures, highlighting their dependence on distance and thickness.

Findings

Dispersion force density changes to inverse fourth power at small gaps.

Force becomes independent of distance below 1 nm.

Force density for thin plates scales with the square of thickness.

Abstract

The Lifshitz formula and methods of its preparation in the literature are considered. It is shown that in Lifshitz's work itself, this formula is given without a consistent conclusion. Moreover, the approach to the conclusion proposed in this work does not allow us to obtain it. The most general conclusion of this formula can be the method proposed by Levin and Rytov, the variation method of Schwinger and the method proposed by Van Kampen and co-authors. The Levin and Rytov approach is applicable in principle to bodies of arbitrary shape if the diffraction loss fields for electric and magnetic dipoles are determined, while the Van Kampen approach is applicable to any plane-layered structure and is quite simple. It is enough to write down the dispersion equations of the plasmon-polaritone structure. The specific dispersion force for a number of structures is calculated based on the Van Kampen method. It is shown that at small gaps, the force (pressure) density changes the inverse fourth-degree dependence on the distance and practically ceases to depend on it at distances less than 1 nm. For thin identical plates, this density is proportional to the square of their thickness at such distances, but the dependence quickly becomes saturated and already at thicknesses of the order of 10 nm practically ceases to depend on it.