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Heat capacity of an electron gas From Wikipedia, the free encyclopedia
In solid state physics the electronic specific heat, sometimes called the electron heat capacity, is the specific heat of an electron gas. Heat is transported by phonons and by free electrons in solids. For pure metals, however, the electronic contributions dominate in the thermal conductivity.[citation needed] In impure metals, the electron mean free path is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.[citation needed]
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Although the Drude model was fairly successful in describing the electron motion within metals, it has some erroneous aspects: it predicts the Hall coefficient with the wrong sign compared to experimental measurements, the assumed additional electronic heat capacity to the lattice heat capacity, namely per electron at elevated temperatures, is also inconsistent with experimental values, since measurements of metals show no deviation from the Dulong–Petit law. The observed electronic contribution of electrons to the heat capacity is usually less than one percent of . This problem seemed insoluble prior to the development of quantum mechanics. This paradox was solved by Arnold Sommerfeld after the discovery of the Pauli exclusion principle, who recognised that the replacement of the Boltzmann distribution with the Fermi–Dirac distribution was required and incorporated it in the free electron model.
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When a metallic system is heated from absolute zero, not every electron gains an energy as equipartition dictates. Only those electrons in atomic orbitals within an energy range of of the Fermi level are thermally excited. Electrons, in contrast to a classical gas, can only move into free states in their energetic neighbourhood. The one-electron energy levels are specified by the wave vector through the relation with the electron mass. This relation separates the occupied energy states from the unoccupied ones and corresponds to the spherical surface in k-space. As the ground state distribution becomes:
where
This implies that the ground state is the only occupied state for electrons in the limit , the takes the Pauli exclusion principle into account. The internal energy of a system within the free electron model is given by the sum over one-electron levels times the mean number of electrons in that level:
where the factor of 2 accounts for the spin up and spin down states of the electron.
Using the approximation that for a sum over a smooth function over all allowed values of for finite large system is given by:
where is the volume of the system.
For the reduced internal energy the expression for can be rewritten as:
and the expression for the electron density can be written as:
The integrals above can be evaluated using the fact that the dependence of the integrals on can be changed to dependence on through the relation for the electronic energy when described as free particles, , which yields for an arbitrary function :
with which is known as the density of levels or density of states per unit volume such that is the total number of states between and . Using the expressions above the integrals can be rewritten as:
These integrals can be evaluated for temperatures that are small compared to the Fermi temperature by applying the Sommerfeld expansion and using the approximation that differs from for by terms of order . The expressions become:
For the ground state configuration the first terms (the integrals) of the expressions above yield the internal energy and electron density of the ground state. The expression for the electron density reduces to . Substituting this into the expression for the internal energy, one finds the following expression:
The contributions of electrons within the free electron model is given by:
Compared to the classical result (), it can be concluded that this result is depressed by a factor of which is at room temperature of order of magnitude . This explains the absence of an electronic contribution to the heat capacity as measured experimentally.
Note that in this derivation is often denoted by which is known as the Fermi energy. In this notation, the electron heat capacity becomes:
For temperatures below both the Debye temperature and the Fermi temperature the heat capacity of metals can be written as a sum of electron and phonon contributions that are linear and cubic respectively: . The coefficient can be calculated and determined experimentally. We report this value below:[1]
Species | Free electron value for in | Experimental value for in |
---|---|---|
Li | 0.749 | 1.63 |
Be | 0.500 | 0.17 |
Na | 1.094 | 1.38 |
Mg | 0.992 | 1.3 |
Al | 0.912 | 1.35 |
K | 1.668 | 2.08 |
Ca | 1.511 | 2.9 |
Cu | 0.505 | 0.695 |
Zn | 0.753 | 0.64 |
Ga | 1.025 | 0.596 |
Rb | 1.911 | 2.41 |
Sr | 1.790 | 3.6 |
Ag | 0.645 | 0.646 |
Cd | 0.948 | 0.688 |
In | 1.233 | 1.69 |
Sn | 1.410 | 1.78 |
Cs | 2.238 | 3.20 |
Ba | 1.937 | 2.7 |
Au | 0.642 | 0.729 |
Hg | 0.952 | 1.79 |
Ti | 1.29 | 1.47 |
Pb | 1.509 | 2.98 |
The free electrons in a metal do not usually lead to a strong deviation from the Dulong–Petit law at high temperatures. Since is linear in and is linear in , at low temperatures the lattice contribution vanishes faster than the electronic contribution and the latter can be measured. The deviation of the approximated and experimentally determined electronic contribution to the heat capacity of a metal is not too large. A few metals deviate significantly from this approximated prediction. Measurements indicate that these errors are associated with the electron mass being somehow changed in the metal, for the calculation of the electron heat capacity the effective mass of an electron should be considered instead. For Fe and Co the large deviations are attributed to the partially filled d-shells of these transition metals, whose d-bands lie at the Fermi energy. The alkali metals are expected to have the best agreement with the free electron model since these metals only one s-electron outside a closed shell. However even sodium, which is considered to be the closest to a free electron metal, is determined to have a more than 25 per cent higher than expected from the theory.
Certain effects influence the deviation from the approximation:
Superconductivity occurs in many metallic elements of the periodic system and also in alloys, intermetallic compounds, and doped semiconductors. This effect occurs upon cooling the material. The entropy decreases on cooling below the critical temperature for superconductivity which indicates that the superconducting state is more ordered than the normal state. The entropy change is small, this must mean that only a very small fraction of electrons participate in the transition to the superconducting state but, the electronic contribution to the heat capacity changes drastically. There is a sharp jump of the heat capacity at the critical temperature while for the temperatures above the critical temperature the heat capacity is linear with temperature.
The calculation of the electron heat capacity for super conductors can be done in the BCS theory. The entropy of a system of fermionic quasiparticles, in this case Cooper pairs, is:
where is the Fermi–Dirac distribution with and
The heat capacity is given by . The last two terms can be calculated:
Substituting this in the expression for the heat capacity and again applying that the sum over in the reciprocal space can be replaced by an integral in multiplied by the density of states this yields:
To examine the typical behaviour of the electron heat capacity for species that can transition to the superconducting state, three regions must be defined:
For it holds that and the electron heat capacity becomes:
This is just the result for a normal metal derived in the section above, as expected since a superconductor behaves as a normal conductor above the critical temperature.
For the electron heat capacity for super conductors exhibits an exponential decay of the form:
At the critical temperature the heat capacity is discontinuous. This discontinuity in the heat capacity indicates that the transition for a material from normal conducting to superconducting is a second order phase transition.
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