[Pw_forum] LO-TO splitting in dynmat.x

xirainbow nkxirainbow at gmail.com
Sun May 1 13:44:41 CEST 2011

Dear Professor Stefano Baroni:
Thank you for you exhaustive explanation. I am deeply benefited from it.
This basic question has been troubling my mind for many years.
Thank you very much :)

On Sun, May 1, 2011 at 6:08 PM, Stefano Baroni <baroni at sissa.it> wrote:

> 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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Hui Wang
School of physics, Fudan University, Shanghai, China
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