Sodium chloride (NaCl) has an fcc Bravais lattice and there are two atoms in the basis. The primitive lattice vectors are,

\begin{equation} \vec{a}_1=\frac{a}{2}(\hat{x}+\hat{z}),\quad \vec{a}_2=\frac{a}{2}(\hat{x}+\hat{y}),\quad\vec{a}_3=\frac{a}{2}(\hat{y}+\hat{z}), \end{equation}

and the basis is Na (0,0,0) and Cl (0.5,0,0). Each Na atom has 6 nearest neighbor Cl atoms and 12 next-nearest neighbor Na atoms. Let $u^x_{lmn}$ the displacement of Na atom in the unit cell at position $\vec{r}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ in the $x$-direction from its equilibrium position and $v^x_{lmn}$ the displacement of Cl atom in the unit cell at position $\vec{r}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ in the $x$-direction from its equilibrium position. If the restoring force that pushes the atoms back to their equilibrium positions is modelled with linear springs with spring constants $C_{\text{Na-Cl}}$, $C_{\text{Na-Na}}$, and $C_{\text{Cl-Cl}}$. The simplest approximation is to connect only nearest neighbor atoms with linear springs. The equations of motion in this case are,

$$M_{\text{Na}} \frac {d^{2}u^{x}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 u^{x}_{lmn}+v^{x}_{lmn}+v^{x}_{(l-1)(m-1)(n+1)}\right),$$ $$M_{\text{Na}} \frac {d^{2}u^{y}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 u^{y}_{lmn}+v^{y}_{(l-1)m(n+1)}+v^{y}_{l(m-1)n}\right),$$ $$M_{\text{Na}} \frac {d^{2}u^{z}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 u^{z}_{lmn}+v^{z}_{(l-1)mn}+v^{z}_{l(m-1)(n+1)}\right),$$ $$M_{\text{Cl}} \frac {d^{2}v^{y}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 v^{x}_{lmn}+u^{x}_{lmn}+u^{x}_{(l+1)(m+1)(n-1)}\right),$$ $$M_{\text{Cl}} \frac {d^{2}v^{y}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 v^{y}_{lmn}+u^{y}_{l(m+1)n}+u^{y}_{(l+1)m(n-1)}\right),$$ $$M_{\text{Cl}} \frac {d^{2}v^{z}_{lmn}}{dt^{2}} = C_{\text{Na-Cl}}\left( -2 v^{z}_{lmn}+u^{z}_{l(m+1)(n-1)}+u^{z}_{(l+1)mn}\right).$$

These equations decouple into three cases of 1-d chains of atoms with two different masses. The resulting density of states contains van Hove singularities that are characteristic of one-dimensional systems but that is not expected for a three dimensional problem. For a better calculation of the phonon dispersion, the next nearest neighbors can be taken into account. The equations are,

\begin{equation} \begin{split} M_{\text{Na}} \frac {d^{2}u^{x}_{lmn}}{dt^{2}} & = \frac{C_{\text{Na-Na}}}{2}\Big[ (u^{x}_{(l+1)mn} - u^{x}_{lmn})+ (u^{x}_{(l-1)mn} - u^{x}_{lmn})+ (u^{x}_{l(m+1)n} - u^{x}_{lmn}) + (u^{x}_{l(m-1)n} - u^{x}_{lmn})\\ & + (u^{x}_{(l+1)m(n-1)} - u^{x}_{lmn})+ (u^{x}_{(l-1)m(n+1)} - u^{x}_{lmn})+ (u^{x}_{l(m+1)(n-1)} - u^{x}_{lmn})+ (u^{x}_{l(m-1)(n+1)}- u^{x}_{lmn})\\ & + (u^{y}_{l(m+1)n} - u^{y}_{lmn}) + (u^{y}_{l(m-1)n} - u^{y}_{lmn}) - (u^{y}_{(l+1)m(n-1)} - u^{y}_{lmn}) - (u^{y}_{(l-1)m(n+1)} - u^{y}_{lmn})\\ &+ (u^{z}_{(l+1)mn}- u^{z}_{lmn})+ (u^{z}_{(l-1)mn} - u^{z}_{lmn})-(u^{z}_{l(m+1)(n-1)} - u^{z}_{lmn})- (u^{z}_{l(m-1)(n+1)} - u^{z}_{lmn})\Big] \\ &+C_{\text{Na-Cl}}\left( -2 u^{x}_{lmn}+v^{x}_{lmn}+v^{x}_{(l-1)(m-1)(n+1)}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Na}} \frac {d^{2}u^{y}_{lmn}}{dt^{2}} &= \frac{C_{\text{Na-Na}}}{2}\Big[ (u^{y}_{l(m+1)n} - u^{y}_{lmn}) + (u^{y}_{l(m-1)n} - u^{y}_{lmn}) + (u^{y}_{(l+1)m(n-1)} - u^{y}_{lmn})+ (u^{y}_{(l-1)m(n+1)} - u^{y}_{lmn}) \\ &+ (u^{y}_{(l-1)(m+1)n} - u^{y}_{lmn}) + (u^{y}_{lm(n-1)} - u^{y}_{lmn}) + (u^{y}_{(l+1)(m-1)n} - u^{y}_{lmn}) + (u^{y}_{lm(n+1)} - u^{y}_{lmn})\\ &+ (u^{x}_{l(m+1)n} - u^{x}_{lmn}) + (u^{x}_{l(m-1)n} - u^{x}_{lmn}) - (u^{x}_{(l-1)m(n+1)} - u^{x}_{lmn}) - (u^{x}_{(l+1)m(n-1)} - u^{x}_{lmn}) \\ &+ (u^{z}_{lm(n-1)} - u^{z}_{lmn})+(u^{z}_{lm(n+1)} - u^{z}_{lmn}) - (u^{z}_{(l+1)(m-1)n} - u^{z}_{lmn})- (u^{z}_{(l-1)(m+1)n} - u^{z}_{lmn}) \Big]\\ & +C_{\text{Na-Cl}}\left( -2 u^{y}_{lmn}+v^{y}_{(l-1)m(n+1)}+v^{y}_{l(m-1)n}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Na}} \frac {d^{2}u^{z}_{lmn}}{dt^{2}} &= \frac{C_{\text{Na-Na}}}{2}\Big[ (u^{z}_{(l-1)mn} - u^{z}_{lmn})+ (u^{z}_{(l-1)l(m+1)n} - u^{z}_{lmn})+ (u^{z}_{l(m+1)(n-1)} - u^{z}_{lmn}) + (u^{z}_{lm(n-1)} - u^{z}_{lmn}) \\ &+ (u^{z}_{lm(n+1)} - u^{z}_{lmn})+ (u^{z}_{(l+1)mn} - u^{z}_{lmn})+ (u^{z}_{(l+1)(m-1)n} - u^{z}_{lmn})+ (u^{z}_{l(m-1)(n+1)} - u^{z}_{lmn})\\ &+ (u^{x}_{(l+1)mn} - u^{x}_{lmn}) + (u^{x}_{(l-1)mn} - u^{x}_{lmn})-(u^{x}_{l(m-1)(n+1)} - u^{x}_{lmn}) - (u^{x}_{l(m+1)(n-1)} - u^{x}_{lmn})\\ &+(u^{y}_{lm(n+1)} - u^{y}_{lmn}) + (u^{y}_{lm(n-1)} - u^{y}_{lmn}) - (u^{y}_{(l-1)(m+1)n} - u^{y}_{lmn})- (u^{y}_{(l+1)(m-1)n} - u^{y}_{lmn})\Big]\\ &+ C_{\text{Na-Cl}}\left( -2 u^{z}_{lmn}+v^{z}_{(l-1)mn}+v^{z}_{l(m-1)(n+1)}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cl}} \frac {d^{2}v^{x}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cl-Cl}}}{2}\Big[ (v^{x}_{(l+1)mn} - v^{x}_{lmn})+ (v^{x}_{(l-1)mn} - v^{x}_{lmn})+ (v^{x}_{l(m+1)n} - v^{x}_{lmn}) + (v^{x}_{l(m-1)n} - v^{x}_{lmn})\\ & + (v^{x}_{(l+1)m(n-1)} - v^{x}_{lmn})+ (v^{x}_{(l-1)m(n+1)} - v^{x}_{lmn})+ (v^{x}_{l(m+1)(n-1)} - v^{x}_{lmn})+ (v^{x}_{l(m-1)(n+1)}- v^{x}_{lmn})\\ & + (v^{y}_{l(m+1)n} - v^{y}_{lmn}) + (v^{y}_{l(m-1)n} - v^{y}_{lmn}) - (v^{y}_{(l+1)m(n-1)} - v^{y}_{lmn}) - (v^{y}_{(l-1)m(n+1)} - v^{y}_{lmn})\\ &+ (v^{z}_{(l+1)mn}- v^{z}_{lmn})+ (v^{z}_{(l-1)mn} - v^{z}_{lmn})-(v^{z}_{l(m+1)(n-1)} - v^{z}_{lmn})- (v^{z}_{l(m-1)(n+1)} - v^{z}_{lmn})\Big] \\ &+C_{\text{Na-Cl}}\left( -2 v^{x}_{lmn}+u^{x}_{lmn}+u^{x}_{(l+1)(m+1)(n-1)}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cl}} \frac {d^{2}v^{y}_{lmn}}{dt^{2}} &= \frac{C_{\text{Cl-Cl}}}{2}\Big[ (v^{y}_{l(m+1)n} - v^{y}_{lmn}) + (v^{y}_{l(m-1)n} - v^{y}_{lmn}) + (v^{y}_{(l+1)m(n-1)} - v^{y}_{lmn})+ (v^{y}_{(l-1)m(n+1)} - v^{y}_{lmn}) \\ &+ (v^{y}_{(l-1)(m+1)n} - v^{y}_{lmn}) + (v^{y}_{lm(n-1)} - v^{y}_{lmn}) + (v^{y}_{(l+1)(m-1)n} - v^{y}_{lmn}) + (v^{y}_{lm(n+1)} - v^{y}_{lmn})\\ &+ (v^{x}_{l(m+1)n} - v^{x}_{lmn}) + (v^{x}_{l(m-1)n} - v^{x}_{lmn}) - (v^{x}_{(l-1)m(n+1)} - v^{x}_{lmn}) - (v^{x}_{(l+1)m(n-1)} - v^{x}_{lmn}) \\ &+ (v^{z}_{lm(n-1)} - v^{z}_{lmn})+(v^{z}_{lm(n+1)} - v^{z}_{lmn}) - (v^{z}_{(l+1)(m-1)n} - v^{z}_{lmn})- (v^{z}_{(l-1)(m+1)n} - v^{z}_{lmn}) \Big]\\ &+ C_{\text{Na-Cl}}\left( -2 v^{y}_{lmn}+u^{y}_{l(m+1)n}+u^{y}_{(l+1)m(n-1)}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cl}} \frac {d^{2}v^{z}_{lmn}}{dt^{2}} &= \frac{C_{\text{Cl-Cl}}}{2}\Big[ (v^{z}_{(l-1)mn} - v^{z}_{lmn})+ (v^{z}_{(l-1)l(m+1)n} - v^{z}_{lmn})+ (v^{z}_{l(m+1)(n-1)} - v^{z}_{lmn}) + (v^{z}_{lm(n-1)} - v^{z}_{lmn}) \\ &+ (v^{z}_{lm(n+1)} - v^{z}_{lmn})+ (v^{z}_{(l+1)mn} - v^{z}_{lmn})+ (v^{z}_{(l+1)(m-1)n} - v^{z}_{lmn})+ (v^{z}_{l(m-1)(n+1)} - v^{z}_{lmn})\\ &+ (v^{x}_{(l+1)mn} - v^{x}_{lmn}) + (v^{x}_{(l-1)mn} - v^{x}_{lmn})-(v^{x}_{l(m-1)(n+1)} - v^{x}_{lmn}) - (v^{x}_{l(m+1)(n-1)} - v^{x}_{lmn})\\ &+(v^{y}_{lm(n+1)} - v^{y}_{lmn}) + (v^{y}_{lm(n-1)} - v^{y}_{lmn}) - (v^{y}_{(l-1)(m+1)n} - v^{y}_{lmn})- (v^{y}_{(l+1)(m-1)n} - v^{y}_{lmn})\Big]\\ &+ C_{\text{Na-Cl}}\left( -2 v^{z}_{lmn}+u^{z}_{l(m+1)(n-1)}+u^{z}_{(l+1)mn}\right). \end{split} \end{equation}

To Do:

• Substitute the general normal mode solution for phonons into these equations. The general solution is, $$\vec{u}_{lmn} = \vec{u}_{\vec{k}}\exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right),\qquad\vec{v}_{lmn} = \vec{v}_{\vec{k}}\exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right).$$ This converts the six second order differential equations into a set of six algebraic equations.
• Solve the resulting eigenvalue problem. There will be 6 frequencies $\omega$ for every $k$ value. These correspond to 3 acoustic modes and 3 optical modes.