# 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    d

${\displaystyle \cdots }$o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o${\displaystyle \cdots }$

→→  →→→
${\displaystyle u_{n-1}\qquad \quad u_{n}\qquad \quad u_{n+1}}$

Where ${\displaystyle n}$ labels the n'th atom, ${\displaystyle d}$ is the distance between atoms when the chain is in equilibrium and ${\displaystyle u_{n}}$ 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 :

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

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.[1]

Put:

${\displaystyle u_{n}=\sum _{k=1}^{N}U_{k}e^{iknd}}$

Here ${\displaystyle nd}$ replaces the usual continuous variable ${\displaystyle x}$. The ${\displaystyle U_{n}}$ 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 [2])

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

These are the equations for harmonic oscillators which have the solution:

${\displaystyle U_{k}=A_{k}e^{i\omega _{k}t};\qquad \quad \omega _{k}=\surd {\Big \{}{2C \over m}(1-coskd){\Big \}}}$

Each normal coordinate ${\displaystyle U_{k}}$ represents an independent vibrational mode of the lattice with wavenumber ${\displaystyle k}$ which is known as a normal mode. The second equation for ${\displaystyle \omega _{k}}$ is known as the dispersion relation between the angular frequency and the wavenumber.[3]

1. ^ Mattuck R. A guide to Feynman Diagrams in the many-body problem
2. ^ Greiner & Reinhardt. Field Quantisation
3. ^ Donovan B. & Angress J.; Lattice Vibrations