Order 1252715: Condensed Matter physics

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Lattice vibrations – 1D

Consider one dimensional system of atoms in a line

Recall: The potential between two neighbouring atoms has the

form above

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In region of minimum, Taylor expansion:

Lattice vibrations – 1D

At finite temperature T the atoms can oscillate between and

Since potential is asymmetric away from minimum, this leads to an

Average position greater than

- Thermal Expansion (though not all systems behave like this)

Handout 6

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Lattice vibrations – 1D

Compressibility/elasticity

Hooke’s Law – quadratic potential about minimum

Applying a force to compress system

- reduces distance between atoms

Compressibility: (assuming )

In one-dimension, with L the length:

(taking = )

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Lattice vibrations – 1D

In an isotropic compressible fluid

sound waves with velocity:

For the 1-D solid take the density as where is the mass of atom

then

bulk modulus

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Lattice vibrations -1D chain Handout 6

Ch 22

Ashcroft &

Mermin

Let the position of the atom be

And the equilibrium position be

Allowing motion of atoms:

Can write total potential energy as:

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Lattice vibrations -1D chain

The force on the mass

Ansatz Solution:

Substitute solution into (1)

(1)

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Dispersion of one-dimensional chain

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in general we have that ω depends on k.

ω(k) is called the dispersion relation. Periodic in

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The first Brillouin zone

Recall: The first Brillouin zone is the region of reciprocal

space which is closer to one reciprocal lattice point than to

any other (Wigner-Seitz cell in reciprocal space).

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Dispersion of one-dimensional chain

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sound

wave

dispersion relation

For small k the sin is equal to its argument

but for k very small (lambda very long) the crystalline structure is

unimportant and we get sound waves.

.

Dispersion of one-dimensional chain

sound

velocity

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Dispersion of one-dimensional chain

We expect periodicity since:

In general for integer p,

The set of points in k-space which are equivalent to k=0 is known

as the reciprocal lattice (seen this before!)

belongs to the reciprocal lattice if:

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Dispersion of one-dimensional chain

At shorter wavelength (larger k) we define:

speed at which a wave packet moves

speed at which maxima and minima move

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examples of dispersion relations

vibrations in a 1D chain

a quantum mechanical particle

k

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examples of dispersion relations

light in vaccum

k

in vacuum the dispersion relation of light is linear.

Light travels with c independent of the frequency.

light in matter

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Counting normal modes

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Periodic boundary conditions

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N

A finite chain with no end!

Max Born and Theodore von Karman (1912)

chain with N atoms:

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Finite chain with 10 unit cells and

one atom per unit cell

• N atoms give N so-called normal modes of vibration. • For long but finite chains, the points are very dense.

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counting normal modes.....

boundary conditions

chain with 1 atom / unit cell and

N unit cells N x 1 modes

(since we have N degrees of freedom)

# k-points

# eigenvalues per k-point

# k-points

# eigenvalues per k-point

chain with 2 atom / unit cell and

N unit cells N x 2 modes

(since we have 2xN degrees of freedom)

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Single harmonic oscillator: quantum model

The energy levels are

quantized

image source: wikimedia, author AllenMcC.

Quantum Modes: Phonons

In our chain, harmonic oscillator can be a collective normal mode,

not just motion of a single particle

Correspondence: for classical harmonic system with normal

oscillation mode at frequency , corresponding quantum system

will have eigenstates with energy:

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Quantum Modes: Phonons

The ground state being n=0 eigenstate, and has zero-point energy

The lowest energy excitation is of energy greater than the

ground state, corresponding to n=1 eigenstate.

Each excitation of this normal mode by a step up

(increasing quantum number n) is known as a phonon

A phonon is a discrete quantum of vibration

Adv.: In Handout 6, read on about effect of temperature

(Bose occupation factor) and how the heat capacity can be obtained.