theorem as where m is the atomic mass, ν is the frequency, u is the average vibration amplitude, kB is the Boltzmann constant, and T is the absolute temperature. If the threshold value of u2 is c2a2 where c is the Lindemann constant and a is the atomic spacing, then the melting point is estimated as Several other expressions for the estimated melting temperature can be obtained depending on the estimate of the average thermal energy. Another commonly used expression for the Lindemann criterion is From the expression for the Debye frequency for ν, we have where θD is the Debye temperature and h is the Planck constant. Values of c range from 0.15 to 0.3 for most materials. Melting point prediction In February 2011, Alfa Aesar released over 10,000 melting points of compounds from their catalog as open data. This dataset has been used to create a random forest model for melting point prediction which is now freely available. Open melting point data are also available from Nature Precedings. High quality data mined from patents and also models developed with these data were published by Tetko et al. Melting point of the elements See also Congruent melting Hagedorn temperature Highest melting point List of elements by melting point Melting points of the elements (data page) Phase diagram Simon–Glatzel equation Slip melting point Triple point Zone melting References Citations Sources Works cited External links Melting and boiling point tables vol. 1 by Thomas Carnelley (Harrison, London, 1885–1887) Melting and boiling point tables vol. 2 by Thomas Carnelley (Harrison, London, 1885–1887) Patent mined data Over 250,000 freely downloadable melting point data. Also downloadable at figshare Atmospheric thermodynamics Physical quantities Phase transitions Threshold temperatures