Let's say we have a cubic lattice of identical masses $m$, each connected to its 6 nearest neighbors by identical spring constants $k$. Essentially, the problem is I get an eigenvalue problem with three equal eigenvalues $\omega^2$, but I've heard that longitudinal waves should have higher velocities than transverse waves, and I'm trying to figure out if I made a math mistake or if this rule doesn't apply to this particular system.

Let the coordinates of an atom be $(x, y, z)$, and the coordinates of the surrounding atoms be $\mathbf{r}_i = (x_i, y_i, z_i)$, with $i=1,2,...,6$ for the 6 nearest neighbors. Since we're looking for plane wave solutions, we want $\mathbf{r} = (x,y,z)$ = $(A_x,A_y,A_z) e^{i \omega t} e^{i (K_x x + K_y y + K_z z)}$. (The phonon amplitude is $(A_x, A_y, A_z)$, and $(K_x, K_y, K_z)$ is the phonon wavevector.) For the nearest neighbors,

$\mathbf{r}_1 = e^{-i K_x a} \mathbf{r}$

$\mathbf{r}_2 = e^{i K_x a} \mathbf{r}$

$\mathbf{r}_3 = e^{-i K_y a} \mathbf{r}$

$\mathbf{r}_4 = e^{i K_y a} \mathbf{r}$

$\mathbf{r}_5 = e^{-i K_z a} \mathbf{r}$

$\mathbf{r}_6 = e^{i K_z a} \mathbf{r}$

(The lattice constant is $a$.) For the equations of motion, in the $x$ direction I get

$U = \frac{1}{2} k (\mathbf{r} - \mathbf{r}_1)^2 + \frac{1}{2} k (\mathbf{r}-\mathbf{r}_2)^2 + ... = \frac{1}{2} k [(x - x_1)^2 + (y-y_1)^2 + (z-z_1)^2 + (x-x_2)^2 + ...$

$\implies \frac{dU}{dx} = k [(x - x_1) + (x-x_2) + ... + (x-x_6)] = k [6x - x_1 - x_2 - ... - x_6]$

$= k A_x [6 - e^{-i K_x a} - e^{i K_x a} - e^{-i K_y a} - ...]$

When I try and solve this, I get this equation of motion:

$-\omega^2 m A_x = k A_x (6 - e^{-i K_x a} - e^{i K_x a} - e^{-i K_y a} - e^{i K_y a} - e^{-i K_z a} - e^{i K_z a})$

and similar ones for the $y$ and $z$ directions. Presumably, this means I have three equal eigenvalues, but this means transverse and longitudinal phonons would have the same velocity, and I've read a number of arguments saying that longitudinal velocities should be larger than transverse velocities. (For example, see Ch7, section 7.7 of the classical mechanics notes posted here; granted, that assumes an isotropic material.)

Is there a problem with my math, or does the idea that longitudinal velocities are greater than transverse velocities not apply here?


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