# Monatomic Linear Chain - Comparison of Numerical and Analytical Results

1. Sep 29, 2011

### sebpinski

Hi all,

I'm having a few problems with crystal dynamics of a simple monatomic chain.

Taking the dispersion relation:

$\omega^2 = \frac{4k}{m}\left(\sin^2 \left( \frac{\kappa a}{2}\right)\right)$

Where k=spring constant, m=mass, $\kappa$=wavevector, a= lattice constant and $\omega$= frequency.

Now assume we have a periodic chain made up of four ions and applying the boundary conditions we get:

$\kappa = \frac{2 \pi n}{a L}$

where n is just an integer and L is the length of the chain (here L=4).
Substituting this into the dispersion relation and assuming mass and springs are unity we can calculate the four frequencies for the four admissible modes as $\omega^2=$0,2,2 and 4.

Now solving numerically, we construct the dynamical matrix using the same values for masses and springs as:

(2, -1, 0, -1; -1, 2, -1, 0; 0, -1, 2, -1; -1, 0, -1, 2)

Diagonalizing this gives the eigenvalues ($\omega^2$) identical to the ones above calculated analytically and the following eigenvectors (extensions from equilibrium):

(0.5, 0.5, 0.5, 0.5)
(0.7071, 0, -0.7071, 0)
(0, 0.7071, 0, -0.7071)
(-0.5, 0.5, -0.5, 0.5)

I've tried analytically to obtain these eigenvectors but I just can't see how. I recall the equation for extension from equilibrium as:

$u_s = U exp[\frac{i 2 \pi n s}{L}]$

Again L=4, U is the amplitude and for normalization it is $L^{-1/2}$, $u_s$ is the extension of site s. If I just substitute the values site numbers 1 to 4 in along with mode numbers n= 1 to 4. I do not reproduce the eigenvectors found from diagonalization.

Thanks for any help in advance.
Sebastian