Phonon dispersions

Crystalline materials are composed of periodic replicas of unit cells. In our case, the unit cell is a parallelepiped defined by the edge vectors ai, aa and aa- The volume of the unit cell is the absolute value of determinant of the lattice matrix A = [ai, 2, 83], which is nonzero, as the column vectors of the matrix are linearly independent. The smallest unit cell is called the primitive cell. The positions of the atoms in the primitive cell form the basis of the crystal. 

The crystal is built by translating the primitive cell by all the translation vectors 

At finite temperature, atoms vibrate around their equilibrium positions, and their displacement can be described by a small vector u. The actual position of an atom is given by 

The total potential energy (f) of the crystal is a function of the positions of the atoms. The Taylor-expansion of the potential energy is 

Bartok-Paitay, The Gaussian Approximation Potential, Springer Theses, 

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Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The phonon dispersion relations for ( ,0) zigzag tubules have 4 X 3/j = 12/j degrees of freedom with 60 phonon branches, having the symmetry types (for n odd, and D j symmetry) ... 

Fig. 1. Phonon modes in 2D and 3D graphite (a) 3D phonon dispersion, (b) 2D phonon dispersion, (c) 3D Brillouin zone, (d) zone center q = 0 modes for 3D graphite.

However, in graphite, consistent with the weak interlayer interaction, the phonon dispersion parallel to... 

In the following sections, we first show the phonon dispersion relation of CNTs, and then the calculated results for the Raman intensity of a CNT are shown as a function of the polarisation direction. We also show the Raman calculation for a finite length of CNT, which is relevant to the intermediate frequency region. The enhancement of the Raman intensity is observed as a function of laser frequency when the laser excitation frequency is close to a frequency of high optical absorption, and this effect is called the resonant Raman effect. The observed Raman spectra of SWCNTs show resonant-Raman effects [5, 8], which will be given in the last section. 

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

FegNi. Frozen phonon calculations combined with the determination of the electron-phonon matrix in the framework of the theory of Varma and Weber have been carried out for the ferrous alloy. The resulting phonon dispersion for the bet phase was already presented elsewhere . As expected, no softening or anomalous curvatures have been detected. This confirms the existence of a bet ground state for FesNi. 

Figure 4. Phonon dispersion for fee FeaNi (renormalized with the eleetron-phonon eoupling). Foree eonstants have been obtained from ab initio ealeulations. Diamonds mark experimental results. ...

CuZn. We have investigated the phonon dispersion of the B2 phase. Our result compares well with the experimental findings marked as diamonds in Fig. 7. Similar to the fee FcsNi phase, a soft transversal mode is detected in bcc CuZn. This [110]... 

Figure 7. Phonon dispersion including the electron-phonon interaction for bcc CuZn. Force constants have been obtained from ah initio calculations. Dashed line is the phonon dispersion without the V-i contribution. Diamonds mark experimental data. ...

Figure 3 Phonon dispersion curves obtained by inelastic neutron scattering revealing precursor behaviour prior to the 14M transformation in Ni-AI. The dip at q = 1/6 [110] (a) deepens upon cooling and (b) shifts under an external load .

From Eqs. (53), one determines the real and imaginary parts of the Green s functions self-consistently. However, we can disregard the phonons dispersion... 

Fig. 22. Phonon dispersion relations for a (5,5) carbon nanotube. This armchair nanotube would be capped with a Cr,o hemisphere [194],...

Fig. 5. Schematics of the formation of the surface phonon dispersion of a (111) f.c.c. crystal.

Kinematics of surface phonon He spectroscopy. The thick lines correspond to scan curves of a 18 meV He beam. The thin lines display the Rayleigh phonon dispersion curve of Pt(lll) along the f M azimuth. 

Fig. 10. (a) He time-of-flight spectrum taken from a LiF(001) surface along the < 100) azimuth at an incident angle Si = 64.2°. The primary beam energy was 19.2 meV. (After Ref 25.). (b) Measured Rayleigh phonon dispersion curve of LiFfOOl) < 100), including a scan curve (dashed) for the kinematical conditions in (a). (After Ref. 25.)... 

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

Fig. 17. Bulk phonon dispersion of longitudinal modes in Cu and Ni crystals, along the (110) and (100) directions. After Ref. 37.)...

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