Consider a Fermi gas at low temperatures k T ≪μ(0), where μ(0) is the chemical potential at T=0.

Consider a Fermi gas at low temperatures k T ≪μ(0), where...

Key Concepts

Low-Temperature Thermodynamic Corrections

At low temperatures, only a fraction ~ kT/? of the fermions near the Fermi surface contribute to thermal properties. For example, the energy increases by an amount proportional to T^2 as these excited particles contribute extra energy, while the heat capacity and entropy, being temperature derivatives of energy, are linear in T. Similarly, the Helmholtz free energy and the chemical potential obtain their leading temperature-dependent corrections from these few thermally excited states, with the free energy displaying T^2 corrections and the chemical potential shifting typically by an amount proportional to T^2.

Density of States at the Fermi Level

The density of states (DOS) provides the number of available quantum states at a particular energy level per unit energy. Because the thermal excitations in a degenerate Fermi gas are confined to a narrow region around the Fermi energy, the behavior of the DOS near the Fermi level is crucial in determining the magnitude and temperature dependence of thermodynamic quantities. The form of the DOS directly influences whether the leading temperature corrections are proportional to T^2, T, or another power of T.

Sommerfeld Expansion

The Sommerfeld expansion is a mathematical technique used to approximate integrals that involve the Fermi–Dirac distribution at low temperatures. Because only states within a narrow window about the Fermi energy contribute significantly, one can expand the integrals in powers of temperature. Typically, this leads to corrections of order T^2 in quantities like energy and chemical potential, and first derivatives of these quantities (such as heat capacity and entropy) then inherit a linear T dependence.

Fermi–Dirac Distribution

The Fermi–Dirac distribution gives the probability that an energy state is occupied by a fermion. At very low temperatures, the distribution is nearly a step function, with a sharp cutoff at the Fermi energy. The modifications of the step function due to finite temperature lead to thermally activated excitations that occur only in an energy band of order kT around the Fermi surface. This underpins the temperature-dependent corrections in energy, heat capacity, entropy, free energy, and chemical potential.

Fermi Gas and Degenerate Fermion Systems

A Fermi gas is a system of non?interacting fermions that obey Fermi–Dirac statistics. In the degenerate limit, at temperatures much lower than the Fermi energy, the vast majority of fermions occupy states well below the Fermi level. Only those near the Fermi surface have energy available to be thermally excited. This restricted set of excitations is the basis for the characteristic low-temperature behavior of various thermodynamic quantities.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that for a system of fermions at room temperature, we need to compute the probability of a single particle state being occupied.

00:13 If this energy is given by some values in the options in here.

00:19 Now, the first thing that we are going to do is to know how is the probability.

00:24 And the probability of a state being occupied is given by the fermi -dirac -di -rac.

00:30 Distribution function, that is one overt the exponential of the energy minus the potential, the chemical potential, divided by boltzman constant times a temperature, and this plus one.

00:51 At room temperature, we know that the product between boltzman constant and the temperature is equal to 0 .026.

01:02 Electron balls.

01:04 So the probabilities for for example for part a that we are given that the difference between the energy and the chemical potential is equal to one electron balls.

01:18 When we substitute that into the probability, the distribution function of probability, we are going to have that that is one over the exponential of minus, 1 over 0 .026 plus 1.

01:38 So from this, we obtain that this is equal to 1 plus 2 times times 10 to the minus 17.

01:52 That elevated to the minus 2.

01:55 So we can approximate this to just simply 1.

02:01 So that's a solution for part a.

02:05 Now for part b, we are told that the difference between the energy and the chemical potemption in this case is equal to minus 0 .01 electron balls.

02:20 And then the probability in this case is going to be equal to 1 over the exponential of minus 0 .01 .1.

02:32 Divided by 0 .026 plus 1...

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