Caesium chloride (CsCl) has a simple cubic Bravais lattice and there are two atoms in the basis. The primitive lattice vectors are,

\begin{equation} \vec{a}_1=a\,\hat{x},\quad \vec{a}_2=a\,\hat{y},\quad\vec{a}_3=a\,\hat{z}, \end{equation}

and the basis is Cs (0,0,0) and Cl ($\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{2}$). Each Cs atom has 8 nearest neighbor Cl atoms and 6 next-nearest neighbor Cs atoms, and vice versa. Let $u^x_{lmn}$ the displacement of Cs 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 $C_{\text{Cs-Cl}}$, $C_{\text{Cs-Cs}}$, and $C_{\text{Cl-Cl}}$, then the equations of motion for the atoms are,

\begin{equation} \begin{split} M_{\text{Cs}} \frac {d^{2}u^{x}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cs-Cl}}}{3}\Big[+(v^{x}_{lmn} - u^{x}_{lmn}) + (v^{y}_{lmn} - u^{y}_{lmn}) + (v^{z}_{lmn} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)mn} - u^{x}_{lmn}) - (v^{y}_{(l-1)mn} - u^{y}_{lmn}) - (v^{z}_{(l-1)mn} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)(m-1)n} - u^{x}_{lmn}) + (v^{y}_{(l-1)(m-1)n} - u^{y}_{lmn}) - (v^{z}_{(l-1)(m-1)n} - u^{z}_{lmn})\\ &+(v^{x}_{l(m-1)n} - u^{x}_{lmn}) - (v^{y}_{l(m-1)n} - u^{y}_{lmn}) + (v^{z}_{l(m-1)n} - u^{z}_{lmn})\\ &+(v^{x}_{lm(n-1)} - u^{x}_{lmn}) + (v^{y}_{lm(n-1)} - u^{y}_{lmn}) - (v^{z}_{lm(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)m(n-1)} - u^{x}_{lmn}) - (v^{y}_{(l-1)m(n-1)} - u^{y}_{lmn}) + (v^{z}_{(l-1)m(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)(m-1)(n-1)} - u^{x}_{lmn}) + (v^{y}_{(l-1)(m-1)(n-1)} - u^{y}_{lmn}) + (v^{z}_{(l-1)(m-1)(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{l(m-1)(n-1)} - u^{x}_{lmn}) - (v^{y}_{l(m-1)(n-1)} - u^{y}_{lmn}) - (v^{z}_{l(m-1)(n-1)} - u^{z}_{lmn})\Big] \\ &+C_{\text{Cs-Cs}}\left( -2 u^{x}_{lmn}+u^{x}_{(l-1)mn}+u^{x}_{(l+1)mn}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cl}} \frac {d^{2}v^{x}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cs-Cl}}}{3}\Big[+(u^{x}_{(l+1)(m+1)(n+1)} - v^{x}_{lmn}) + (u^{y}_{(l+1)(m+1)(n+1)} - v^{y}_{lmn}) + (u^{z}_{(l+1)(m+1)(n+1)}- v^{z}_{lmn})\\ &+(u^{x}_{l(m+1)(n+1)} - v^{x}_{lmn}) - (u^{y}_{l(m+1)(n+1)} - v^{y}_{lmn}) - (u^{z}_{l(m+1)(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{lm(n+1)} - v^{x}_{lmn}) + (u^{y}_{lm(n+1)} - v^{y}_{lmn}) - (u^{z}_{lm(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{(l+1)m(n+1)} - v^{x}_{lmn}) - (u^{y}_{(l+1)m(n+1)} - v^{y}_{lmn}) + (u^{z}_{(l+1)m(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{(l+1)(m+1)n} - v^{x}_{lmn}) + (u^{y}_{(l+1)(m+1)n} - v^{y}_{lmn}) - (u^{z}_{(l+1)(m+1)n} - v^{z}_{lmn})\\ &+(u^{x}_{l(m+1)n} - v^{x}_{lmn}) - (u^{y}_{l(m+1)n} - v^{y}_{lmn}) + (u^{z}_{l(m+1)n} - v^{z}_{lmn})\\ &+(u^{x}_{lmn} - v^{x}_{lmn}) + (u^{y}_{lmn} - v^{y}_{lmn}) + (u^{z}_{lmn} - v^{z}_{lmn})\\ &+(u^{x}_{lm(n+1)} - v^{x}_{lmn}) - (u^{y}_{lm(n+1)} - v^{y}_{lmn}) - (u^{z}_{lm(n+1)} - v^{z}_{lmn})\Big] \\ &+C_{\text{Cl-Cl}}\left( -2 v^{x}_{lmn}+v^{x}_{(l-1)mn}+v^{x}_{(l+1)mn}\right), \end{split} \end{equation}

There are similar terms for the $y-$ and $z-$displacements.

Show the equations for the $y-$ and $z-$displacements.

\begin{equation} \begin{split} M_{\text{Cs}} \frac {d^{2}u^{y}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cs-Cl}}}{3}\Big[+(v^{x}_{lmn} - u^{x}_{lmn}) + (v^{y}_{lmn} - u^{y}_{lmn}) + (v^{z}_{lmn} - u^{z}_{lmn})\\ &-(v^{x}_{(l-1)mn} - u^{x}_{lmn}) + (v^{y}_{(l-1)mn} - u^{y}_{lmn}) + (v^{z}_{(l-1)mn} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)(m-1)n} - u^{x}_{lmn}) + (v^{y}_{(l-1)(m-1)n} - u^{y}_{lmn}) - (v^{z}_{(l-1)(m-1)n} - u^{z}_{lmn})\\ &-(v^{x}_{l(m-1)n} - u^{x}_{lmn}) + (v^{y}_{l(m-1)n} - u^{y}_{lmn}) - (v^{z}_{l(m-1)n} - u^{z}_{lmn})\\ &+(v^{x}_{lm(n-1)} - u^{x}_{lmn}) + (v^{y}_{lm(n-1)} - u^{y}_{lmn}) - (v^{z}_{lm(n-1)} - u^{z}_{lmn})\\ &-(v^{x}_{(l-1)m(n-1)} - u^{x}_{lmn}) + (v^{y}_{(l-1)m(n-1)} - u^{y}_{lmn}) - (v^{z}_{(l-1)m(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)(m-1)(n-1)} - u^{x}_{lmn}) + (v^{y}_{(l-1)(m-1)(n-1)} - u^{y}_{lmn}) + (v^{z}_{(l-1)(m-1)(n-1)} - u^{z}_{lmn})\\ &-(v^{x}_{l(m-1)(n-1)} - u^{x}_{lmn}) + (v^{y}_{l(m-1)(n-1)} - u^{y}_{lmn}) + (v^{z}_{l(m-1)(n-1)} - u^{z}_{lmn})\Big] \\ &+C_{\text{Cs-Cs}}\left( -2 u^{y}_{lmn}+u^{y}_{l(m-1)n}+u^{y}_{l(m+1)n}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cl}} \frac {d^{2}v^{y}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cs-Cl}}}{3}\Big[+(u^{x}_{(l+1)mn} - v^{x}_{lmn}) + (u^{y}_{(l+1)mn} - v^{y}_{lmn}) + (u^{z}_{(l+1)mn} - v^{z}_{lmn})\\ &-(u^{x}_{l(m+1)(n+1)} - v^{x}_{lmn}) + (u^{y}_{l(m+1)(n+1)} - v^{y}_{lmn}) + (u^{z}_{l(m+1)(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{lm(n+1)} - v^{x}_{lmn}) + (u^{y}_{lm(n+1)} - v^{y}_{lmn}) - (u^{z}_{lm(n+1)} - v^{z}_{lmn})\\ &-(u^{x}_{(l+1)m(n+1)} - v^{x}_{lmn}) + (u^{y}_{(l+1)m(n+1)} - v^{y}_{lmn}) - (u^{z}_{(l+1)m(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{(l+1)(m+1)n} - v^{x}_{lmn}) + (u^{y}_{(l+1)(m+1)n} - v^{y}_{lmn}) - (u^{z}_{(l+1)(m+1)n} - v^{z}_{lmn})\\ &-(u^{x}_{l(m+1)n} - v^{x}_{lmn}) + (u^{y}_{l(m+1)n} - v^{y}_{lmn}) - (u^{z}_{l(m+1)n} - v^{z}_{lmn})\\ &+(u^{x}_{lmn} - v^{x}_{lmn}) + (u^{y}_{lmn} - v^{y}_{lmn}) + (u^{z}_{lmn} - v^{z}_{lmn})\\ &-(u^{x}_{(l+1)mn} - v^{x}_{lmn}) + (u^{y}_{(l+1)mn} - v^{y}_{lmn}) + (u^{z}_{(l+1)mn} - v^{z}_{lmn})\Big] \\ &+C_{\text{Cl-Cl}}\left( -2 v^{y}_{lmn}+v^{y}_{l(m-1)n}+v^{y}_{l(m+1)n}\right), \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Cs}} \frac {d^{2}u^{z}_{lmn}}{dt^{2}} & = \frac{C_{\text{Cs-Cl}}}{3}\Big[+(v^{x}_{lmn} - u^{x}_{lmn}) + (v^{y}_{lmn} - u^{y}_{lmn}) + (v^{z}_{lmn} - u^{z}_{lmn})\\ &-(v^{x}_{(l-1)mn} - u^{x}_{lmn}) + (v^{y}_{(l-1)mn} - u^{y}_{lmn}) + (v^{z}_{(l-1)mn} - u^{z}_{lmn})\\ &-(v^{x}_{(l-1)(m-1)n} - u^{x}_{lmn}) - (v^{y}_{(l-1)(m-1)n} - u^{y}_{lmn}) + (v^{z}_{(l-1)(m-1)n} - u^{z}_{lmn})\\ &+(v^{x}_{l(m-1)n} - u^{x}_{lmn}) - (v^{y}_{l(m-1)n} - u^{y}_{lmn}) + (v^{z}_{l(m-1)n} - u^{z}_{lmn})\\ &-(v^{x}_{lm(n-1)} - u^{x}_{lmn}) - (v^{y}_{lm(n-1)} - u^{y}_{lmn}) + (v^{z}_{lm(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)m(n-1)} - u^{x}_{lmn}) - (v^{y}_{(l-1)m(n-1)} - u^{y}_{lmn}) + (v^{z}_{(l-1)m(n-1)} - u^{z}_{lmn})\\ &+(v^{x}_{(l-1)(m-1)(n-1)} - u^{x}_{lmn}) + (v^{y}_{(l-1)(m-1)(n-1)} - u^{y}_{lmn}) + (v^{z}_{(l-1)(m-1)(n-1)} - u^{z}_{lmn})\\ &-(v^{x}_{l(m-1)(n-1)} - u^{x}_{lmn}) + (v^{y}_{l(m-1)(n-1)} - u^{y}_{lmn}) + (v^{z}_{l(m-1)(n-1)} - u^{z}_{lmn})\Big]\\ &+C_{\text{Cs-Cs}}\left( -2 u^{z}_{lmn}+u^{z}_{lm(n-1)}+u^{z}_{lm(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{Cs-Cl}}}{3}\Big[+(u^{x}_{(l+1)mn} - v^{x}_{lmn}) + (u^{y}_{(l+1)mn} - v^{y}_{lmn}) + (u^{z}_{(l+1)mn} - v^{z}_{lmn})\\ &-(u^{x}_{l(m+1)(n+1)} - v^{x}_{lmn}) + (u^{y}_{l(m+1)(n+1)} - v^{y}_{lmn}) + (u^{z}_{l(m+1)(n+1)} - v^{z}_{lmn})\\ &-(u^{x}_{lm(n+1)} - v^{x}_{lmn}) - (u^{y}_{lm(n+1)} - v^{y}_{lmn}) + (u^{z}_{lm(n+1)} - v^{z}_{lmn})\\ &+(u^{x}_{(l+1)m(n+1)} - v^{x}_{lmn}) - (u^{y}_{(l+1)m(n+1)} - v^{y}_{lmn}) + (u^{z}_{(l+1)m(n+1)} - v^{z}_{lmn})\\ &-(u^{x}_{(l+1)(m+1)n} - v^{x}_{lmn}) - (u^{y}_{(l+1)(m+1)n} - v^{y}_{lmn}) + (u^{z}_{(l+1)(m+1)n} - v^{z}_{lmn})\\ &+(u^{x}_{l(m+1)n} - v^{x}_{lmn}) - (u^{y}_{l(m+1)n} - v^{y}_{lmn}) + (u^{z}_{l(m+1)n} - v^{z}_{lmn})\\ &+(u^{x}_{lmn} - v^{x}_{lmn}) + (u^{y}_{lmn} - v^{y}_{lmn}) + (u^{z}_{lmn} - v^{z}_{lmn})\\ &-(u^{x}_{(l+1)mn} - v^{x}_{lmn}) + (u^{y}_{(l+1)mn} - v^{y}_{lmn}) + (u^{z}_{(l+1)mn} - v^{z}_{lmn})\Big] \\ &+C_{\text{Cl-Cl}}\left( -2 v^{z}_{lmn}+v^{z}_{(lm(n-1)}+y^{z}_{lm(n+1)}\right), \end{split} \end{equation}

The general form for the normal mode solutions for crystals with two atoms in the basis are,

$$\vec{u}_{lmn} = \vec{u}_{\vec{k}}\exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right),\qquad\vec{v}_{lmn} = \vec{v}_{\vec{k}}\exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right).$$

For a cubic crystal with a lattice constant $a$ these solutions are,

$$\vec{u}_{lmn} = \vec{u}_{\vec{k}}\exp\left(i\left(lk_xa+mk_ya+nk_za-\omega t\right)\right),\qquad\vec{v}_{lmn} = \vec{v}_{\vec{k}}\exp\left(i\left(lk_xa+mk_ya+nk_za-\omega t\right)\right).$$

Substituting this solution into the differential equations above results in six algebraic equations.