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The first theory explaining mechanism of melting in the bulk
was proposed by Lindemann , who used
vibration of atoms in the crystal to explain the melting transition.
The average amplitude of thermal vibrations increases when
the temperature of the solid increases. At some point the amplitude of vibration
becomes so large that the atoms start to invade the space of their nearest neighbors and
disturb them and the melting process initiates.
Quantitative calculations based on the model are not easy, hence Lindemann offered a simple criterion:
melting might be expected when the root mean vibration amplitude exceeds a certain
threshold value (namely when the amplitude reaches at least of the nearest neighbor
Assuming that all atoms vibrate about their equilibrium positions with the
same frequency (the Einstein approximation) the average thermal vibration energy
can be estimated relying on the equipartition theorem as:
where is the atomic mass, is the Einstein frequency,
is the mean square thermal average amplitude of vibration,
and is absolute temperature. Using the Lindemann criterion for the threshold
, where is Lindemann's constant one can estimate the melting point
Lindemann's constant was assumed to be the same for crystals with similar structure,
hence it could be calculated from the melting temperature of one particular crystal.
A detailed experimental examination showed that is not strictly a constant
and the correlation is only fair (See Fig. 1.2).
Recently, the validity of the Lindemann instability criterion have been tested in computer simulations
of bulk melting of Lennard-Jones fcc crystals . It has been found that
melting occurs when a sufficiently large number spatially correlated destabilized atoms
of the crystal ( e.g. cluster of quasiliquid) are generated (See Fig. 1.3).
These clusters are distributed homogeneously thruout
the solid. The Lindemann criterion of the lattice instability is found to be valid for these clusters.
The accumulation, growth and coalescence of the clusters of the liquid phase constitute, according to Jin et al. ,
the mechanism of homogeneous bulk melting.
The measured melting temperature versus the
melting temperature estimated using the Lindemann rule, from ref. 
It should be stressed that the original Lindemann model for vibrational melting, like many of its
more sophisticated successors, refers only to a crystal with the simplest possible structure, i.e.
assemblies of closed packed atoms. Crystals containing more complex molecules as unit of structure
exhibit a vibrational complexity which rules out any simple rule of lattice stability, determined merely
by vibrational amplitudes of the molecular centers of mass. Futhermore, the Lindemann model is based on harmonic
forces, which never give way, whereas melting must involve bond breaking. This is another serious defect of the model.
Furthermore numerous experiments carried out at high pressures indicate that the Lindemann model
does not estimate adequately the pressure dependence of the melting temperature .
The most serious defect of the model is that melting is described in term individual atomic property, i.e.
mean square amplitude of vibration, while a phase transition is a cooperative process. In addition,
the Lindemann model describes melting in terms of the solid alone, although the melting transition
must involve both solid and liquid phases. Nevertheless the predictive success of the Lindemann melting criterion
lent support to the belief that melting could be a gradual process, beginning within the solid at temperatures
below the melting point. Subsequent theories and numerous experiments helped to bolster the idea.
3D visualization of the collective appearance of the Lindemann particles at .
(a) a few clusters with 20-200 particles (larger black circles) against other Lindemann particles
(smaller gray circles) which do not form such clusters (b) four large clusters with 219, 214, 187, and
117 particles colored with red, blue, black and gray, respectively. From ref. 
Next: The Born criterion
Up: Bulk melting