Editing Phonon (section)

===Dispersion relation===
[[File:Diatomic phonons.png|class=skin-invert-image|thumb|Dispersion curves in linear diatomic chain]]
[[File:Optical & acoustic vibrations-en.svg|class=skin-invert-image|thumb|250px|Optical and acoustic vibrations in a linear diatomic chain.]]
[[File:Diatomic chain.gif|class=skin-invert-image|thumb|Vibrations of the diatomic chain at different frequencies.]]
[[File:Phonon dispersion relations in GaAs.png|class=skin-invert-image|thumb|250px|Dispersion relation ''ω''&nbsp;=&nbsp;''ω''('''k''') for some waves corresponding to lattice vibrations in GaAs.<ref name=Cardona/>]]
For a one-dimensional alternating array of two types of ion or atom of mass ''m''<sub>1</sub>, ''m''<sub>2</sub> repeated periodically at a distance ''a'', connected by springs of spring constant ''K'', two modes of vibration result:<ref name=Misra/>
:<math>\omega_\pm^2 = K\left(\frac{1}{m_1} +\frac{1}{m_2}\right) \pm K \sqrt{\left(\frac{1}{m_1} +\frac{1}{m_2}\right)^2-\frac{4\sin^2\frac{ka}{2}}{m_1 m_2}} ,</math>
where ''k'' is the wavevector of the vibration related to its wavelength by
<math>k = \tfrac{2 \pi}{\lambda}</math>.

The connection between frequency and wavevector, ''ω''&nbsp;=&nbsp;''ω''(''k''), is known as a [[dispersion relation]]. The plus sign results in the so-called ''optical'' mode, and the minus sign to the ''acoustic'' mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

The speed of propagation of an acoustic phonon, which is also the [[speed of sound]] in the lattice, is given by the slope of the acoustic dispersion relation, {{sfrac|∂''ω<sub>k</sub>''|∂''k''}} (see [[group velocity]].) At low values of ''k'' (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ''ωa'', independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of ''k'', i.e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in its [[Wigner-Seitz cell#Primitive cell|primitive cell]], the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the [[wavevector]]. The boundaries at −{{sfrac|{{pi}}|''a''}} and {{sfrac|{{pi}}|''a''}} are those of the first [[Brillouin zone]].<ref name=Misra>{{cite book |title=Physics of Condensed Matter |first=Prasanta Kumar |last=Misra |chapter-url=https://books.google.com/books?id=J6rMISLVCmcC&pg=PA44 |pages=44 |chapter=§2.1.3 Normal modes of a one-dimensional chain with a basis |publisher=Academic Press |isbn=978-0-12-384954-0 |year=2010}}</ref> A crystal with ''N''&nbsp;≥&nbsp;2 different atoms in the [[primitive cell]] exhibits three acoustic modes: one [[Longitudinal wave|longitudinal acoustic mode]] and two [[Transverse wave|transverse acoustic modes]]. The number of optical modes is 3''N''&nbsp;–&nbsp;3. The lower figure shows the dispersion relations for several phonon modes in [[GaAs]] as a function of wavevector '''k''' in the [[Brillouin zone#Critical points|principal directions]] of its Brillouin zone.<ref name=Cardona>
{{cite book |title=Fundamentals of Semiconductors |series=Physics and Materials Properties |first1=Peter Y. |last1=Yu |first2=Manuel |last2=Cardona |chapter-url=https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111 |page=111 |chapter=Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes |isbn=978-3-642-00709-5 |year=2010 |edition=4th |publisher=Springer}}</ref>

The modes are also referred to as the branches of phonon dispersion. In general, if there are p atoms (denoted by N earlier) in the primitive unit cell, there will be 3p branches of phonon dispersion in a 3-dimensional crystal.  Out of these, 3 branches correspond to acoustic modes and the remaining 3p-3 branches will correspond to  optical modes. In some special directions, some branches coincide due to symmetry.  These branches are called degenerate. In acoustic modes, all the p atoms vibrate in phase. So there is no change in the relative displacements of these atoms during the wave propagation.

Study of phonon dispersion is useful for modeling propagation of sound waves in solids, which is characterized by phonons. The energy of each phonon, as given earlier, is ''ħω.'' The velocity of the wave also is given in terms of ''ω'' and k ''.'' The direction of the wave vector is the direction of the wave propagation and the phonon polarization vector gives the direction in which the atoms vibrate. Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic for phonon propagation.

A wave is longitudinal if the atoms vibrate in the same direction as the wave propagation. In a transverse wave, the atoms vibrate perpendicular to the wave propagation.  However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse. For general anisotropic crystals, the phonon waves are longitudinal or transverse only in certain special symmetry directions. In other directions, they can be  nearly longitudinal or nearly transverse. It is only for labeling convenience, that they are often called longitudinal or transverse but are actually quasi-longitudinal or quasi-transverse.   Note that in the  three-dimensional case, there are two directions perpendicular to a straight line at each point on the line. Hence, there are always two (quasi) transverse waves for each (quasi) longitudinal wave.

Many phonon dispersion curves have been measured by [[inelastic neutron scattering]].

The physics of sound in [[fluid]]s differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support [[shear stress]]es (but see [[viscoelastic]] fluids, which only apply to high frequencies).