Phonon dispersion relation

A phonon dispersion relation gives its angular frequency $\omega$ as a function of its wavevector $k$. For an isotropic power law, $\omega =C|k{|}^{\alpha }$.

For a translation-invariant monatomic chain with symmetric pair force constants, $M{\ddot{u}}_p={\sum }_{n>0}{C}_n(u_{p+n}+u_{p-n}-2u_p)$. Substituting a traveling normal mode gives the displayed even, reciprocal-periodic phonon dispersion relation. If $\sum n^2{C}_n$ converges and the chain is stable, its long-wavelength speed is $a\sqrt{\sum n^2{C}_n/M}$. A truly long-range interaction need not have a finite second moment or a finite linear acoustic slope.

Let $\alpha =C_2/C_1>0$ and $x={\sin }^2(ka/2)$. The squared frequency in units $4C_1/M$ is $(1+4\alpha )x-4\alpha x^2$. For $\alpha \le 1/4$ its maximum on $0\le x\le 1$ is at the zone edge and equals one. For $\alpha >1/4$ it moves inside the zone to $x=(1+4\alpha )/(8\alpha )$, with the displayed frequency ratio. The small-wavevector speed instead increases by $\sqrt{1+4\alpha }$; maximizing the frequency and finding the sound speed are different operations.

A continuous family of normal modes labelled by the wavevector and a branch index $s$. For $r$ chemical atoms per three-dimensional primitive unit cell, the dynamical matrix has $3r$ eigenvalues, giving $3r$ branches: three acoustic phonon branches and $3r-3$ optical phonon branches for a mechanically stable ordinary crystal structure.

The crystal momentum of a phonon is defined modulo a reciprocal lattice vector. Extending its dispersion relation periodically beyond the first Brillouin zone labels equivalent excitations by different extended wavevectors. A scattering transfer $Q=G+q$ can be outside that zone while its reduced phonon wavevector $q$ is inside. The measured neutron recoil depends on the actual transfer $Q$, not on the reduced representative alone.

For alternating masses $m,M$ at spacing $d$, equal nearest-neighbour spring constants $K$, and cell length $2d$, a two-component normal mode givesThe minus sign is the acoustic phonon branch and the plus sign is the optical phonon branch. For $M>m$ the edge frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, leaving a gap. At $k=0$ they are zero and $\sqrt{2K(1/m+1/M)}$.

In an alternating-mass chain with nearest-neighbour spacing $a$ and cell length $2a$, the inter-sublattice Bloch coupling is proportional to $\cos qa$. At the first-zone edge $q=\pi /(2a)$ it vanishes. For $M>m$, the acoustic edge mode moves only the heavy sublattice and the optical edge mode only the light one. Their frequency difference is $\sqrt{2\alpha /m}-\sqrt{2\alpha /M}$. At long wavelength the acoustic displacements are in phase, whereas the optical mode has zero centre-of-mass displacement.

For alternating masses $m_A,m_B$, nearest-neighbour spacing $a$ and equal spring constants $\alpha$, the zone-centre optical angular frequency and acoustic group velocity obeyTherefore $\alpha =m_A{m}_B{\omega }_{\mathrm{op}}^2(0)/[2(m_A+m_B)]$ and $a=v_s\sqrt{(m_A+m_B)/(2\alpha )}$. These infer effective model parameters from a measured intercept and initial slope of a phonon dispersion relation. Fitting a real polar crystal to a nearest-neighbour scalar model does not guarantee that every measured point or its structural spacing is reproduced exactly.

For equal masses $m$ at spacing $d$ joined by nearest-neighbour springs of spring constant $K$, Newton's second law gives $m{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. A normal mode $u_j=u_0{e}^{i(kjd-\omega t)}$ givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.

A nearest-neighbour chain with the Lennard-Jones potential $V(x)=4ϵ[(\sigma /x{)}^{12}-(\sigma /x{)}^6]$ has equilibrium spacing $a=2^{1/6}\sigma$ and spring constant $K=V^{\prime \prime }(a)=72ϵ/a^2$. The monatomic-chain dispersion relation gives speed of sound $v=a\sqrt{K/m}=\sqrt{72ϵ/m}$ and maximum angular frequency $2v/a$. The cancellation of $\sigma$ from the sound speed holds in this chain model; it does not remove the spacing from the cutoff frequency.

With real wave number and time convention $e^{-i\omega t}$, an on-site friction force $-\Gamma {\dot{u}}_j$ shifts the frequencies as displayed. Underdamped oscillations have amplitude envelope $e^{-\Gamma t/(2m)}$, while overdamped modes have two nonoscillatory rates. At small wave number a stable acoustic chain has slow decay $\omega \simeq -im{c}_s^2{k}^2/\Gamma$. Exactly uniform displacement remains a zero-restoring-force mode, so a constant displacement is not forced to vanish.

A periodic chain of $N$ sites and length $Na$ has allowed signed wave numbers separated by $2\pi /(Na)$. The continuum state density on the signed axis is $Na/(2\pi )$. Combining opposite propagation directions onto $0<|q|<\pi /a$ doubles it to $Na/\pi$. This counts the same $N$ modes, not two independent acoustic phonon branches.

An acoustic phonon branch has $\omega (k)\to 0$ as $k\to 0$. Its long-wavelength group velocity is the speed of sound.

An optical phonon branch has a nonzero frequency as $k\to 0$, corresponding to relative motion of degrees of freedom within one primitive cell.

For equal masses $m$ separated by $a$ and alternating spring constants $\lambda$ and $\alpha \lambda$, the primitive-cell length is $2a$ and the two branches areThe minus and plus signs give the acoustic phonon branch and optical phonon branch, respectively.

Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg \kappa >0$. The primitive unit cell contains a strong-bonded pair, so its length is $2a$ and the Brillouin zone ends at $\pm \pi /(2a)$. Expanding the exact alternating-spring chain dispersion relation givesThe low-frequency acoustic phonon branch moves almost rigid dimers of mass $2m$ coupled by weak springs; its speed of sound tends to $a\sqrt{2\kappa /m}$. The high-frequency optical phonon branch describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$.

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives