Clarification on Einstein's model for heat capacity of a solid

TL;DR

This paper clarifies the equivalence of various descriptions of Einstein's model for solid heat capacity, demonstrating they all lead to the same Einstein expression.

Contribution

It provides a clear clarification showing that different interpretations of Einstein's model are equivalent and consistent.

Findings

Different descriptions of Einstein's model are equivalent.

All descriptions lead to the same heat capacity expression.

Clarification resolves confusion in literature.

Abstract

Seemingly different interpretations or descriptions of Einstein's model for the heat capacity of a solid can be found in textbooks and the literature. The purpose of this note is to clarify the equivalence of the different descriptions, which all lead to the same Einstein expression for the heat capacity of a solid of N atoms.