6.3: Degenerate Electron Gases
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Consider N electrons trapped in a cubic box of dimension a. Let us treat the electrons as essentially non-interacting particles. According to Section [snon], the total energy of a system consisting of many non-interacting particles is simply the sum of the single-particle energies of the individual particles. Furthermore, electrons are subject to the Pauli exclusion principle (see Section [siden]), because they are indistinguishable fermions. The exclusion principle states that no two electrons in our system can occupy the same single-particle energy level. Now, from the previous section, the single-particle energy levels for a particle in a box are characterized by the three quantum numbers, lx, ly, and lz. Thus, we conclude that no two electrons in our system can have the same set of values of lx, ly, and lz. It turns out that this is not quite true, because electrons possess an intrinsic angular momentum called spin. The spin states of an electron are governed by an additional quantum number, which can take one of two different values. (See Chapter [sspin].) Hence, when spin is taken into account, we conclude that a maximum of two electrons (with different spin quantum numbers) can occupy a single-particle energy level corresponding to a particular set of values of lx, ly, and lz. Note, from Equations ([e7.38]) and ([e7.39]), that the associated particle energy is proportional to l2=l2x+l2y+l2z.
Suppose that our electrons are cold: that is, they have comparatively little thermal energy. In this case, we would expect them to fill the lowest single-particle energy levels available to them. We can imagine the single-particle energy levels as existing in a sort of three-dimensional quantum number space whose Cartesian coordinates are lx, ly, and lz. Thus, the energy levels are uniformly distributed in this space on a cubic lattice. Moreover, the distance between nearest neighbor energy levels is unity. This implies that the number of energy levels per unit volume is also unity. Finally, the energy of a given energy level is proportional to its distance, l2=l2x+l2y+l2z, from the origin.
Because we expect cold electrons to occupy the lowest energy levels available to them, but only two electrons can occupy a given energy level, it follows that if the number of electrons, N, is very large then the filled energy levels will be approximately distributed in a sphere centered on the origin of quantum number space. The number of energy levels contained in a sphere of radius l is approximately equal to the volume of the sphere—because the number of energy levels per unit volume is unity. It turns out that this is not quite correct, because we have forgotten that the quantum numbers lx, ly, and lz can only take positive values. Hence, the filled energy levels actually only occupy one octant of a sphere. The radius lF of the octant of filled energy levels in quantum number space can be calculated by equating the number of energy levels it contains to the number of electrons, N. Thus, we can write N=2×18×4π3l3F.
EF=l2Fπ2ℏ22mea2=π2ℏ22ma2(3Nπ)2/3,
The mean energy of the electrons is given by ˉE=EF∫lF0l24πl2dl/43πl5F=35EF,
The total energy of a degenerate electron gas is Etotal=NˉE=35NEF.
In turns out that the conduction (i.e., free) electrons inside metals are highly degenerate (because the number of electrons per unit volume is very large, and EF∝n2/3). Indeed, most metals are hard to compress as a direct consequence of the high degeneracy pressure of their conduction electrons. To be more exact, resistance to compression is usually measured in terms of a quantity known as the bulk modulus, which is defined B=−V∂P∂V
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)