1. Condensed Matter Physics: Quantum Statistics & Electronic Structure in Solids
Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics)
Homework due Wednesday Nov. 12th
Chapter 10: 1, 3, 6, 9, 11, 20, 23, 25, 29, 35

2. Quantum Statistics
In quantum physics, identical particles must be treated as indistinguishable – this affects counting of states
In quantum physics there are two types of particles – fermions (half-integer spin, e.g. electrons) and bosons (integral spin, e.g. photons)
Bosons and Fermions obey different statistics

3. Different statistics Fermions Bosons
Obey the Pauli Exclusion Principle – no two particles can have the exact same quantum numbers
Distribution function

4. Differences fermions bosons
Only one particle per quantum state allowed
Cannot all “condense” to E=0
Must fill up to the “Fermi Energy”
ABE does not depend strongly on temperature
The only states that have any probability at low temperatures are those at E=0 for which the exponential approaches 1
This is Bose-Einstein Condensation

5. Free Electron Gas in Metals
Solid metals are bonded by the metallic bond
One or two of the valence electrons from each atom are free to move throughout the solid
All atoms share all the electrons. A metal is a lattice of positive ions immersed in a gas of electrons. The binding between the electrons and the lattice is what holds the solid together

6. Fermi-Dirac “Filling” Function
Probability of electrons to be found at various energy levels.
For E – EF = 0.05 eV  f(E) = 0.12 For E – EF = 7.5 eV  f(E) = 10 –129
Exponential dependence has HUGE effect!
Temperature dependence of Fermi-Dirac function shown as follows:

7. Free Electron Gas in Metals
The number of electrons in the interval
E to E+dE is therefore
The first term is the Fermi-Dirac distribution
and the second is the density of states g(E)dE

8. Free Electron Gas in Metals
From n(E) dE we can calculate many global characteristics of the electron gas. Here are just a few
The Fermi energy – the maximum energy level occupied by the free electrons at absolute zero
The average energy
The total number of electrons in the electron gas

9. Free Electron Gas in Metals
The total number of electrons N is given by
The average energy of a free electron is given by

10. Free Electron Gas in Metals
At T = 0, the integrals are easy to do.
For example, the total number of electrons is

11. Free Electron Gas in Metals
The average energy of an electron is
This implies
EF = k TF defines
the Fermi
temperature

12. Summary of metallic state
The ions in solids form regular lattices
A metal is a lattice of positive ions immersed in a gas of electrons. All ions share all electrons
The attraction between the electrons and the lattice is called a metallic bond
At T = 0, all energy levels up to the Fermi energy are filled

13. Heat Capacity of Electron Gas
By definition, the heat capacity (at constant
volume) of the electron gas is given by
where U is the total energy of the gas. For a gas
of N electrons, each with average energy <E>,
the total energy is given by

14. Heat Capacity of Electron Gas
Total energy
In general, this integral must be done
numerically. However, for T << TF, we can use
a reasonable approximation.

15. Heat Capacity of Electron Gas
At T= 0, the total energy of the electron gas is
For 0 < T << TF, only a small fraction kT/EF
of the electrons can be excited to higher energy
states
Moreover, the energy of
each is increased by
roughly kT

16. Heat Capacity of Electron Gas
Therefore, the total energy can be written as
where a = p2/4, as first shown by Sommerfeld
The heat capacity of the
electron gas is predicted to be