Lattice Vibration III
Monoatomic chain
1D Diatomic chain of 2N atoms
Acoustical/Optical branches
Normal mode
Total number of eq. of motions 
Definition: pattern of motion in which all parts of the system move in a sinusoidal fashion, with the same frequency.
Normal mode frequencies of a chain of two types of atom
-At A, two atoms are oscillating in antiphase with their centre of mass at rest
-At B, lighter mass m is oscillating and M is at rest
-At C, M is oscillating and m is at rest
join the end of crystal to form a ring
satisfied periodic boundary condition (set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell)
Acoustical (vibrate in long wavelength limit)
Optical (vibrate in long wavelength limit)
two atoms in the unit cell move opposite to each other (one +ve and another -ve) and the light mass amplitude is greater
displacement of both atoms has the same amplitude, direction and phase.
excite these mode with EM radiation
Dispersion relation
Maximum frequency of optical branch (at k=0);
Minimum frequency of acoustical branch;
For ka= π (at maximum frequency);
For ka= π (at minimum frequency) ;
N allowed values of k in range;
Displacement for a crystal that contains single atom;
Simplest crystal is the one dimensional chain of identical atom with identical masses.
Connected by identical springs of spring constant K.
- Atoms only move in direction parallel to the chain.
- Atoms seperated by distance of 'a'.
Equation of motion for nth atom
The dispersion relation of the monoatomic 1D chain is
PHASE AND GROUP VELOCITY.
- Longitudinal wave the particles are displaced parallel to the direction to the wave travels
- Energy is transported by the wave at a generally slower speed than group velocity.
w-k relation : Dispersion Relation.
- Transverse wave is particles are displaced perpendicular to the direction the wav travels.
- Longitudinal wave the particle are displaced parallel to the direction the wave travel.
Force to the right :
Force to the left :
Total force = Force to the right-Force to the left
One dimensional linear chain of N atoms of mass m and M.
Each mass m has two near neighbours of mass M and vice versa.
The repeat distance between M and M is a.
The force constant that coupled each atom to their nearest neighbour is K.
(mass) (acceleration)
= restoring force
Only the first neighbour interaction is considered
Two equations of motion, M and m, equation of motion :
Two different types of atoms move in opposite direction, thus further complicating the model
Eq. of M :
Eq. of m :
a pair of algebraic equations for α and ω as a function of k.
As n does not appear in above equation indicates that our assumed solution is of the correct form and may be rewritten in the form:
a quadratic equation for ω2 can be obtained by cross-multiplication and the two roots are:
ω versus k relation for diatomic chain
:
two values of ω for each value of k, thus two branches (corresponds to positive and negative signs of dispersion relation);
The dispersion relation is periodic in k with a period 2 π /a = 2 π /(unit cell length).
This result remains valid for a chain of containing an arbitrary number of atoms per unit cell.