Lattice Vibrations

Consider the ID lattice Fig.5.1

Hooke's law is obeyed exactly

If one atom is displaced by x from equilibrium position

Potential energy of the atom is a function of distance from neighboring atoms Displaced position potential energy is changed by

potential energy when distance from neighbour =x

Expanding in Taylor series about a and keying only 1st significant term

potential function of a harmonic oscillator

But this

assumption that only one atom oscillates since atoms are bond to each other we expect that vibration of one atom in lattice will set other atom to vibration

let displacement of and atom from equilibrium

Fig.5.2

l = integer

force acting on l th atom, a distance from adjacent atoms and for two adjustment atoms putting it to themselves.

Force F on l th atom

2 nd law to atom

.............(i)

Then (i) become

wave equation (ii)

Complex oscillation possible

let regular frequency = where q = wave no.

from equation (i).

proper Sign should be chosen to make

Fig.5.3

Classical single harmonic oscillator there is a single frequency

Lattice vibration are characterized by a continum of frequencies with a limiting maximum value But repetition is there for or

So Brillouin zone concept is coming out.

Now We consider a Crystal of finite member of atoms N

Boundary condition for the displacement of lattice atom at the end of crystal may be taken periodic boundary condition to be no displacement.

Periodic boundary condition

The lattice vibration we talked about applicable to material like in which all atoms identical and one atom/unit cell. Material with 2 atoms/unit cell or 2 types of atoms/unit cell=Semiconductors.

Double periodicity

Fig.5.4

Two equations describes the vibrations (2 atom 2 period case)