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) > > La morale est une logique de l'action comme la logique est une morale de la > pens?e - Jean Piaget > > Please, if possible, don't send me MS Word or PowerPoint attachments > Why? See: http://www.gnu.org/philosophy/no-word-attachments.html > > > > > _______________________________________________ > Pw_forum mailing list > Pw_forum at pwscf.org > http://www.democritos.it/mailman/listinfo/pw_forum > > -- ____________________________________ Hui Wang School of physics, Fudan University, Shanghai, China -------------- next part -------------- An HTML attachment was scrubbed... URL: http://www.democritos.it/pipermail/pw_forum/attachments/20110501/f7cce685/attachment.htm
