Topic 6

Question 1

Why is the approximation of atoms in a solid vibrating in 3 dimensions not a really good?

Answer 1

Bad approximation for many solids at high temperature

Question 2

What is the assumption made for the equipartition of a solid?

Answer 2

Allowed energy is continuous.

The equipartition prediction has limiting behaviours at high T

Question 3

What is the statistical viewpoint of potential?

Answer 3

The energy between adjoining atoms can be described as

𝑉_𝐿𝐽=𝜖((𝑟_0/𝑟)^12−2(𝑟_0/𝑟)^6 )
where r_0 equilibrium separation and r is the separation

*Note: electrons overlap due to the Pauli exclusion principle

Question 4

How do we go from the LJ potential to the quadratic potential?

Answer 4

By assuming the displacement of the atom is reasonable small so we can extend potential around the equilibrium so V(r-r_0) then using the taylor expansion about r=r_0

LJ potential has a minimum at r=r_0 (the equilibrium separation)

𝑉_𝐿𝐽≈𝜀[36((𝑟−𝑟_0)/𝑟_0 )^2−1]=𝐶+𝐾(𝑟−𝑟_0 )^2
where 𝐾=36𝜀/(𝑟_0^2 ) is the harmonic potential with a spring constant

Question 5

How do we derive the Einstien solid?

Answer 5

V_LJ can be applied to derive Einstien solid

Assume each “spring” joining an atom to its neighbour is independent

A solid consisting of N atoms is represented as 3N simple harmonic oscillators

Schrödinger’s equation with a harmonic potential leads to
E_n=𝜔(n+1/2); 𝑛=0, 1, 2, …

𝜔=√(𝐾∕𝑚) is the classical frequency for a harmonic oscillator of two masses 𝑚 joined by a spring with spring constant 𝐾

• Can use the MCE approach
• CE (which is easier)

Question 6

What is Einstein solid?

Answer 6

3𝑁𝑘_𝐵 (ℏ𝜔/(𝑘_𝐵 𝑇))^2 [𝑒^(ℏ𝜔\/𝑘_𝐵 𝑇)/(𝑒^(ℏ𝜔\/𝑘_𝐵 𝑇)−1)^2 ]

Note that it does deviate from data at low temperatures. To improve using (Debye model(treated as a coupled oscillator) , which recognises that the harmonic oscillators are not independent (phonons))

Question 7

Derive Einsteins solid using the MCE approach

Answer 7

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Question 8

Derive Einsteins using the CE approach?

Answer 8

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