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An Accurate Alternative Method to Introduce Mixed Einstein–Debye Model for Molar Heat Capacity and the Exact Analytical Integral-Free Solution to Its Resulting Integration

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Abstract

In this research, a new different method has been proposed to combine the Einstein and Debye models for molar heat capacity at constant volume (cv) in one (mixed) model in order to give results closer to experimental measurements and better than using each model separately, where Debye's notion was adopted for the acoustic branches of the dispersion relation and Einstein's notion for its optical branches, and, so, the integration resulting from this mixed model was solved analytically obtaining an exact integral-free expression by using polylogarithm functions. For evaluating this method, it was applied to some binary semiconductors (monatomic primitive cell ones are included automatically) (C (diamond), Si. Ge, SiC, ZnS, ZnTe, CdS and CdTe) in addition to NaCl and Cu after removing the electronic contribution to cv, as an attempt to generalize the scientific base of this method on all solid materials to reach a comprehensive model that expresses the molar heat capacity at constant volume. With the help of computer programming by MatLab, the best values compatible with this method for the characteristic acoustic temperature and the ratio between the two characteristic optical and acoustic temperatures for the sample materials were obtained by choosing their values that gave the minimum deviations of the theoretical values from the experimental ones. These values, distinct from the widely recognized standard Einstein and Debye temperatures, are systematically determined by specialized computer algorithms tailored to this model. These algorithms are designed to optimize molar heat capacity (\({c}_{V}\)) calculations, striving to achieve the highest degree of concordance with experimental measurements. Notably, this model effectively rectifies discrepancies in the Debye function curve within the intermediate temperature range, aligning it more closely with experimental data points.

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The authors did not receive support from any organization for the submitted work. No funding was received to assist with the preparation of this manuscript. No funding was received for conducting this study. No funds, grants, or other support was received.

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  1. LPAIS Department of Physics, Faculty of Science DM, USMBA, BP 1796, Fez-Atlas, 30000, Fez, Morocco

    Mohammed Joghlaf, Yahya Ababou & Salaheddine Sayouri

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  1. Mohammed Joghlaf
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Contributions

Mohammed Joghlaf : Writing – original draft, Formal analysis, planned and carried out the computer programming, wrote the paper. Yahya Ababou : Supervision, Formal analysis, conceived and supervised the project, analyzed and interpreted the results Salaheddine Sayouri : analyzed and interpreted the results. All authors contribute to the discussion of the results and reviewed the manuscript.

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Correspondence to Yahya Ababou.

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Joghlaf, M., Ababou, Y. & Sayouri, S. An Accurate Alternative Method to Introduce Mixed Einstein–Debye Model for Molar Heat Capacity and the Exact Analytical Integral-Free Solution to Its Resulting Integration. Int. J. Appl. Comput. Math 9, 137 (2023). https://doi.org/10.1007/s40819-023-01618-z

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