- Ideal gas Entropy Chemical potential
*face*Fermi and Bose gases*face*Lecture120 min.

##### Fermi and Bose gases

Thermal and Statistical Physics 2020Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.*assignment*Distribution function for double occupancy statistics*assignment*Homework##### Distribution function for double occupancy statistics

Orbitals Distribution function Thermal and Statistical Physics 2020Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

*assignment*Derivative of Fermi-Dirac function*assignment*Homework##### Derivative of Fermi-Dirac function

Fermi-Dirac function Thermal and Statistical Physics 2020**Derivative of Fermi-Dirac function**Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.*assignment*Pressure and entropy of a degenerate Fermi gas*assignment*Homework##### Pressure and entropy of a degenerate Fermi gas

Fermi gas Pressure Entropy Thermal and Statistical Physics 2020Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).

Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).

*face*Ideal Gas*face*Lecture120 min.

##### Ideal Gas

Thermal and Statistical Physics 2020ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.*assignment*Gibbs sum for a two level system*assignment*Homework##### Gibbs sum for a two level system

Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

Solve for the thermal average occupancy of the system in terms of \(\lambda\).

Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

Find an expression for the thermal average energy of the system.

Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.

*face*Review of Thermal Physics*face*Lecture30 min.

##### Review of Thermal Physics

Thermal and Statistical Physics 2020thermodynamics statistical mechanics

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.*assignment*Nucleus in a Magnetic Field*assignment*Homework##### Nucleus in a Magnetic Field

Energy and Entropy 2021 (2 years)Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

- Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

*assignment*Symmetry of filled and vacant orbitals*assignment*Homework##### Symmetry of filled and vacant orbitals

Symmetry Orbitals Thermal and Statistical Physics 2020 Show that \begin{align} f(\mu+\delta) &= 1 - f(\mu-\delta) \end{align} This means that the probability that an orbital above the Fermi level is occupied is equal to the probability an orbital the same distance below the Fermi level*being empty*. These unoccupied orbitals are called*holes*.*face*Thermal radiation and Planck distribution*face*Lecture120 min.

##### Thermal radiation and Planck distribution

Thermal and Statistical Physics 2020Planck distribution blackbody radiation photon statistical mechanics

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.-
Thermal and Statistical Physics 2020
Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

Find an expression for the energy \(U\) of the gas.

Find an expression for the entropy \(\sigma\). The temperature is \(kT\).