[Pw_forum] LO-TO splitting in dynmat.x

Stefano Baroni baroni at sissa.it
Sun May 1 12:08:16 CEST 2011

Eyvaz, All: I beg to differ, here, though it's probably a matter of terminology.

On Apr 29, 2011, at 9:41 AM, Eyvaz Isaev wrote:

> Dear Wang,
> Let me reiterate: LO-TO splitting takes place ONLY at the Gamma point. 

I would rather say a LO-TO splitting ALWAYS occur at q/=0 in any crystal, simply because there is no reason why it should not. (if two modes are not related by symmetry, their frequencies usually differ, and LO and TO modes are NOT related by any symmetry operations. Weird things only occur at q=0 in polar materials. See below ...

> >For cubic crystals splitting is equal in all directions.
> So, for non-cubic LO-TO splitting  also occurs and the splitting is different for different directions. That is 
> why one can see a discontinuity near the Gamma point in phonon dispersion relations.

the situation is even a bit more messy (also, see below ...)

> >When q is zero, there is no longitude and transverse mode. 
> Really? How about optical modes? Did you pay attention to "O"?  

Strictly speaking, I believe Wang is right. At q=0 it makes no sense to speak about longitudinal or transverse modes, simply because it makes no sense to say that the polarization of the mode is parallel (L) or perpendicular (T) to a vector (q) whose norm is 0 (q=0). The problem is, in a polar *and infinite* crystal q=0 modes do not exist (!!!), because the infinite range of the Coulomb interaction makes the dynamical matrix ill-defined at q=0. So, in this case, not only is it impossible to assign a transverse or longitudinal character to a lattice vibration (which would be true for non polar materials as well), but the very concept of lattice vibration breaks down. What one actually calculates when one calculates the LO or TO modes is the q->0 limit of finite-q modes. When the system is non polar, this limit is well defined and independent of the relative orientation of the polarization and wavevector. In polar materials, instead, this limit depends on this relative orientation, hence it is not defined in the q-> limit. So, the L or T character of a lattice vibration is NOT a property of the vibrations at q=0, but only in the q->0 limit.

So far, so good, if at small but finite q the polarization of normal modes can be chosen to be parallel or perpendicular to the direction of propagation of the vibration. This is indeed the case for phonons propagating along high-simmetry lines in cubic materials. For low-simmetry lines in cubic materials, or any line in non-cubic materials, this may not even be the case and lattice vibrations in the q->0 limit in general are not longitudinal nor transverse. What continues to be true is that the q->0 limit is that vibrational frequencies will depend on the direction of propagation of the phonon, so that, strictly speaking, lattice-perdiodic vibrations are not well defined ...

Hope to have clarified a bit the (admittedly messy) situation ...

Stefano B

Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center - Trieste
http://stefano.baroni.me [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)

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