Melting Point - Predicting The Melting Point of Substances (Lindemann's Criterion)

Predicting The Melting Point of Substances (Lindemann's Criterion)

An attempt to predict the bulk melting point of crystalline materials was first made in 1910 by Frederick Lindemann. The idea behind the theory was the observation that the average amplitude of thermal vibrations increases with increasing temperature. Melting initiates when the amplitude of vibration becomes large enough for adjacent atoms to partly occupy the same space. The Lindemann criterion states that melting is expected when the root mean square vibration amplitude exceeds a threshold value.

Assuming that all atoms in a crystal vibrate with the same frequency ν, the average thermal energy can be estimated using the equipartition theorem as

E = 4\pi^2 m \nu^2~u^2 = k_B T

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

T_m = \cfrac{4\pi^2 m \nu^2 c^2 a^2}{k_B} .

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

T_m = \cfrac{m \nu^2 c^2 a^2}{k_B} .

From the expression for the Debye frequency for ν, we have

T_m = \cfrac{2\pi m c^2 a^2 \theta_D^2 k_B}{h^2}

where θD is the Debye temperature and h is the Planck constant. Values of c range from 0.15–0.3 for most materials.

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