User:RedAcer/linearchain

Linear Chain - Classical treatment
In order to simplify the analysis of a 3-dimensional lattice of atoms it is convenient to model a 1-dimensional lattice or linear chain. The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step.(adiabatic approximation)


 * n-1 n  n+1  &larr;  d  &rarr;

$$\cdots$$o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o$$\cdots$$
 * &rarr;&rarr;&rarr;&rarr;&rarr;&rarr;
 * $$u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}$$

Where $$n$$ labels the n'th atom, $$d$$ is the distance between atoms when the chain is in equilibrium and $$u_n$$ the displacement of the n'th atom from it's equilibrium position.

If C is the elastic constant of the spring and m the mass of the atom then the equation of motion of the n'th atom is :


 * $$-2Cu_n + C(u_{n+1} + u_{n-1}) = m{\operatorname{d^2}u_n\over\operatorname{d}t^2}$$

This is a set of coupled equations and since we expect the solutions to be oscillatory, new coordinates can be defined by a discrete Fourier transform, in order to de-couple them.

Put:


 * $$u_n = \sum_{k=1}^N U_k e^{iknd}$$

Here $$nd$$ replaces the usual continuous variable $$x$$. The $$U_n$$ are known as the normal coordinates. Substitution into the equation of motion produces the following decoupled equations.(This requires a significant amount of algebra using the orthonormality and completeness relations of the discrete fourier transform )


 * $$ 2C(cos\,kd-1)U_k = m{\operatorname{d^2}U_k\over\operatorname{d}t^2}$$

These are the equations for harmonic oscillators which have the solution:
 * $$U_k=A_ke^{i\omega_kt};\qquad\quad \omega_k=\surd\Big\{ {2C \over m}(1-coskd)\Big\}$$

Each normal coordinate $$U_k$$ represents an independent vibrational mode of the lattice with wavenumber $$k$$ which is known as a normal mode. The second equation for $$\omega_k$$ is known as the dispersion relation between the angular frequency and the wavenumber.