face Lecture
120 min.
Fermi 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 Homework
For electrons with an energy \(\varepsilon\gg mc^2\), where \(m\) is the mass of the electron, the energy is given by \(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons in a cube of volume \(V=L^3\) the momentum takes the same values as for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a gas of \(N\) electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}
assignment Homework
face Lecture
120 min.
ideal 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 Homework
Let 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)?
face Lecture
30 min.
thermodynamics statistical mechanics
These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.assignment Homework
Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.
Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.
assignment Homework
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 \(kT\).
assignment Homework
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.)
assignment Homework
Consider 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.
Show 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 \(T\rightarrow 0\).