Lattice Hamiltonian is developed to second order around the nominal equilibrium position, given by the experimental crystal structure. This corresponds to a simple implementation of the quasi-harmonic approximation (the positions of the atom at rest are choosen at the minimum of the free energy, in thermodynamic equilibrium) instead of the harmonic approximation (the position of the atom at rest are choose at the minimum of the potential energy, in mechanical equilibrium)
The Hamiltonian is given by the sum of the kinetic part plus the sum of interatomic two-body central potentials. The Hamiltonian degrees of freedom are the atomic cores and atomic shells coordinates. The atomic shells have zero mass and does not contribute to the kinetic part.
The potentials implemented up to now in OpenPhonon are:
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(1) |
and
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(2) |
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(3) |
where
and
is the charge of the
ion.
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(4) |
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(5) |
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(6) |
Finally, in condensed matter (in particular for metals)
the Coulomb potential is renormalized
by the screening effect of the surrounding charges.
This effect can be calculated in many ways,
we have implemented the simplest approach: the
Thomas-Fermi one, i.e.
replacing the Coulomb potential by the
Yukawa potential:
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(7) |
The motion of the atom labelled
in the
th cell is described by the following equations:
where equation (8) describes the motion of the centre of mass
of each atom while equation (9) refers to the motion of the
shell associated to each atom, and its mass
.
The parameter
is given by
are the spring constants which tie the shells to the cores. Symbols
,
,
,
denote non-coulombian interaction.
The
represent interactions between the cores.
Although coulombian interactions act between the cores too,
they need a special treatment because of their long range, and
have thus their own writing
.
are the elastic interactions between a shell and a core on another
site,
between a core and another site shell and
are the
interactions between two different shells.
These terms are the sum of analytical potentials ( Born-Mayer or Van de Waals )
and Born-von-Karman potentials.
In the case of analytical potentials the
's are derived from the
analogous of equation (10) while in the Born-von-Karman
case they are derived from the longitudinal and transverse second
derivatives
and
, namely :
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(11) |
being the unit vector going from
to
One can cast together
(core charge) and
(shell charge) in equations
(8) and (9) to get
(ion charge=core+shell) and
by using a different set of displacement variables (
and
) where
.
By performing such substitution, by summing together equations
(8) and (9) and considering the limit
, one obtains :
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(12) | ||
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and for the zero-mass shells
Phonon eigenvectors can be written as Bloch waves:
where and
are now
dimensional vectors ( N being the number of atoms in a cell),
Y,Z,K are
dimensional diagonal matrices whose diagonals are equal to the shell charges, ionic charge and
spring constants respectively. The matrices C (coulombian), and the V's are obtained substituting equation 14
into 15 and summing over all the cells up to convergence. The subscript ``
'' in
and in
's denotes the so said ``self-forces''.
The dynamical matrix is found from equation (15) by elimination of
.
The summation of
and
over the cells needs a special treatment because of the Coulomb
long range.