TY - JOUR

T1 - Extended Regression Analysis for Debye–Einstein Models Describing Low Temperature Heat Capacity Data of Solids

AU - Gamsjäger, Ernst

AU - Wiessner, Manfred

N1 - Publisher Copyright: © 2024 by the authors.

PY - 2024/5/26

Y1 - 2024/5/26

N2 - Heat capacity data of many crystalline solids can be described in a physically sound manner by Debye–Einstein integrals in the temperature range from (Formula presented.) to (Formula presented.). The parameters of the Debye–Einstein approach are either obtained by a Markov chain Monte Carlo (MCMC) global optimization method or by a Levenberg–Marquardt (LM) local optimization routine. In the case of the MCMC approach the model parameters and the coefficients of a function describing the residuals of the measurement points are simultaneously optimized. Thereby, the Bayesian credible interval for the heat capacity function is obtained. Although both regression tools (LM and MCMC) are completely different approaches, not only the values of the Debye–Einstein parameters, but also their standard errors appear to be similar. The calculated model parameters and their associated standard errors are then used to derive the enthalpy, entropy and Gibbs energy as functions of temperature. By direct insertion of the MCMC parameters of all (Formula presented.) computer runs the distributions of the integral quantities enthalpy, entropy and Gibbs energy are determined.

AB - Heat capacity data of many crystalline solids can be described in a physically sound manner by Debye–Einstein integrals in the temperature range from (Formula presented.) to (Formula presented.). The parameters of the Debye–Einstein approach are either obtained by a Markov chain Monte Carlo (MCMC) global optimization method or by a Levenberg–Marquardt (LM) local optimization routine. In the case of the MCMC approach the model parameters and the coefficients of a function describing the residuals of the measurement points are simultaneously optimized. Thereby, the Bayesian credible interval for the heat capacity function is obtained. Although both regression tools (LM and MCMC) are completely different approaches, not only the values of the Debye–Einstein parameters, but also their standard errors appear to be similar. The calculated model parameters and their associated standard errors are then used to derive the enthalpy, entropy and Gibbs energy as functions of temperature. By direct insertion of the MCMC parameters of all (Formula presented.) computer runs the distributions of the integral quantities enthalpy, entropy and Gibbs energy are determined.

KW - Bayesian framework

KW - Markov chain Monte Carlo (MCMC)

KW - probability density distribution

KW - regression analysis

KW - thermodynamic functions

UR - http://www.scopus.com/inward/record.url?scp=85197884416&partnerID=8YFLogxK

U2 - 10.3390/e26060452

DO - 10.3390/e26060452

M3 - Article

AN - SCOPUS:85197884416

VL - 26.2024

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 6

M1 - 452

ER -