Monatomic Linear Chain - Comparison of Numerical and Analytical Results

[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

 

[eljose79]

Hi Sebastian,

I can see why you are having trouble reproducing the eigenvectors from diagonalization using the equation for extension from equilibrium. The equation you are using is for a simple harmonic oscillator, which assumes that the forces between the atoms are linear and the potential energy is quadratic. However, in your case, you are dealing with a chain of atoms connected by springs, which introduces nonlinearity and changes the potential energy.

To solve for the eigenvectors, you will need to use a more advanced approach, such as the method of normal modes. This involves solving a system of equations using the dynamical matrix and the eigenvectors as variables. This will give you a set of equations that you can solve to obtain the eigenvectors.

Alternatively, you can also use a computer program to solve for the eigenvectors. There are many programs available that can solve for the normal modes of a crystal, such as Phonopy or VASP. These programs use advanced algorithms to calculate the eigenvectors and frequencies of the crystal.

I hope this helps and good luck with your research!