[Pw_forum] problems with lattice dynamical principle

degironc degironc at sissa.it
Thu Nov 23 14:09:42 CET 2006

MY explanation of what happens is the following:

Just take a simple model of nearest neighbhor interaction... I know
coulomb interaction is long range but in
TA(X) mode you are moving back and forth laterally (100) planes of NaCl,
these planes are neutral on average
so let us forget about what happens beyond the first shell.
Let's have a coulomb attractive part -Z^2/R between (any) atom of a
(100) plane and its nearest neighbhor in the adjacent
plane (which happens to be of opposite charge) and a short range
repulsive part A/R^n again only between nearest neighbhors.
Then calculate the restoring force for this nearest neighbhor
interaction: you move atom (1) in the central plane and calculate the
negative of the force induced on an atom (2) in the nearby plane:
Phi (1,alpha;2,beta) = - Z^2/R^3 (delta_{alpha,beta} - 3 R_alpha R_beta
/ R^2) +
n A/ R^n+2 (delta_{alpha,beta} - (n+2) R_alpha R_beta / R^2)

with symmetry arguments and sum rules you get Phi(1,alpha;1beta) = -2 *
Phi(1,alpha;2,beta) and similarly for the other
At X motion of nearest neighbor planes are out of phase so the total
restoring force for such a phonon is -4 * Phi(1,alpha;2,beta) or
Phi(X)_{alpha,beta} = +4 [ Z^2/R^3 (delta_{alpha,beta} - 3 R_alpha
R_beta / R^2) -
n A/ R^n+2 (delta_{alpha,beta} - (n+2) R_alpha R_beta / R^2) ]
Now for TA modes and just nearest neighbhor interaction displacement is
orthogonal to R so forget about R_alpha R_beta term.
You are left with Phi( X, TA) = +4 [ Z^2/R^3 - n A/ R^n+2 ]
As you apply pressure, the negative (unstable) term coming from the
short range repulsive interaction grows more rapidly
(because n>>1) than the positive (stabilizing) Couomb term thus leading
to instability. The higher the ionicity (the larger Z) of
the system the larger will be the pressure needed to destabilize it
(assuming similar short range term) ....

At least this is the way I saw the situation when I discussed with Alex
Zunger the results of their calculations in Aspen back in 1998...

Again, if you want genuine author's explanation you should ask them.


luch001 wrote:

> Dear all,
> I have read the paper titled ¡°Theory of systematic absence of
> NaCl-type (¦Â-Sn-type) high pressure phases in covalent (ionic)
> semiconductors¡± (Phys Rev Lett. *82*, 767, 1999). The phonon behaviors
> are dominated by the competition between the attractive electrostatic
> Madelung energy ¨CZ^2 /R (where Z is the excess charge within the
> atomic sphere) and the repulsive short-range energy. I can not
> understand well the formula of the force constant given in the last
> part of page 769, which reads:
> ¦µ_¦Á¦Â = nA¦Ä_¦Á¦Â /R^n+2 -Z^2 ¦Ä_¦Á¦Â /R^3 -¦ËR_¦Á R_¦Â / R^5
> How does it come from? Does it apply to all the systems or just apply
> to some specific system? The values of A and ¦Ë are determined by what
> factors?
> The authors claimed that the transverse acoustic mode at X point
> lowers the repulsive energy and raises the Madelung energy, but why?
> I have pondered hardly and consulted with others. But I have reached
> no conclusion. I wrote letter to Alex Zunger, one of the two authors.
> He did not write back yet.
> Any reply will be appreciated!
> With all my best regards!
> Yours sincerely
> Jingyun Zhang

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