The amount of heat needed to raise a unit amount of substance by a unit in temperature

The constant value of the heat capacity of most simple solids

In 1819, Dulong and Petit found experimentally that Cv for most solids at room temperature is 3R = 25 J/Kmol

System treated as N oscillators connected by springs in one dimension

Assumed that all oscillators oscillate at same frequency

Recovers Dulong-Petit law at high temperature

The maximum vibration frequency is determined by Debye frequency

Periodic boundary condition which impose the restriction that a wave function must be periodic on a certain Bravais lattice

The condition often applied to model an ideal crystal

Consist of chains of large number of identical atoms with identical masses, M connected by identical springs of spring constant, K

Force to the right

Force to the left

Total force

Equation of motion for nth atom

The dispersion relation of the monoatomic for 1D lattice

Relationship between frequency and wave number is known as dispersion relation

The result is periodic and the only unique solutions are correspond to values in the range,

Phase velocity, Vp

Group velocity, Vg

The cos forces the group velocity to 0 at the edge of the 1st Brillouin zone

Small value of q is close to the centre of the Brillouin zone

Linear dispersion

1D crystal has only one sound velocity

Lattice that have long wavelengths or small frequencies behaves as a continuum or no dispersion takes place

atoms are equally spaced with separation (1/2) a

Equation of motion for mass M

Equation of motion for mass m

Any possible motion of a mass-spring system can be described in terms of its normal modes

A pattern of motion in which all parts of the system move in a sinusoidal pattern, with the same frequency

At A, the two atoms are oscillating in antiphase with their centre of mass at rest

At B, the lighter mass m is oscillating and M is at rest

At max.acoustical point C, M oscillates and m is at rest

At min.optical point B, m oscillates and M is at rest

The optical branch is a higher energy vibration

The term “optical” comes from how these were discovered - notice that if atom 1 is +ve and atom 2 is -ve, that the charges are moving in opposite directions

These modes can be excited with electromagnetic radiation

If the two masses are equal, the two branches join (become degenerate) at π/a

At optical branch,

At acoustical branch,

The difference between maximum frequency and minimum frequency becomes narrower when both atoms are nearly identical and no longer exist for m=M

When M=m,

Equation of motion in 3D is presented in terms of normal mode,

When k lies in the direction of high symmetry, two of the branches as transverse and one is longitudal

In real crystals, two transverse modes degenerate on only in special high-symmetry directions

For directions of propagation, these two transverse modes and the one longitudinal mode all have different frequencies

Brillouin zones of the reciprocal lattice

Wigner-Seitz cell of the reciprocal lattice is also called the 1st Brillouin zone

Contains all information about the lattice vibrations of the solid

For 3D lattices

Doesn't carry physical momentum because the center of mass of the crystal doesn't change it position under vibrations (except k=0)

In elastic scattering of a crystal is governed by K’ = K +G, where G is a vector in the reciprocal lattice, K is the wavevector of the incident photon and K’ is the wavevector of the scattered photon

Phonon is created, K’+ k = K +G

Phonon is absorbed, k' = k + K + G

1. Electric field applied to a piezoelectric material (quartz, cadmium sulfide, etc..), strain experienced

2. Oscillating electric field generated, the field then swing the piezoelectric transducer at the same frequency

3. Phonon transmitted by the transducer, into the specimen

1. Current flows through metal wires causes the electron's temperature to rise

2. The electrons release energy by emitting phonon and photons into the metal and its surrounding

Above the threshold frequency, only photons are produced

1. Thin layer of insulator is placed between two thin layers of metal to form a barrier for the electrons