Chapter 19 (I)

Chapter 19 (I) - Chapter 19 Fermi-Dirac Gases - The...

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Chapter 19 Fermi-Dirac Gases --- The Electron Gas in Metals
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19.1 Introduction The most familiar example for Fermi-Dirac gases is the conduction-band electrons in metals. There are a large number of conduction electrons confined to the interior of a metal as gas molecules confined in a container. We speak of the electrons as an electron gas . One central issue is why the electronic heat capacity C v e is temperature-dependent and negligible at ordinary temperatures . This cannot be explained by the classical statistical theory. First of all, we should ask: Is an electron gas in a metal similar to an ideal monatomic molecular gas in nature? Will the Maxwell-Boltzmann (M-B) statistics applicable to an electron gas in a metal?
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To answer this question, we should review the condition for using M-B stat, which can be expressed as That is, the average inter-particle distance is far larger than the de Broglie wavelength of the particle. For an electron, m e =9.11x10 -31 kg, its λ dB e at T=300 K is given by . 2 3 1 dB mkT h N V λ π = >> = l . 3 . 4 ) ( 10 3 . 4 300 10 38 . 1 10 11 . 9 14 . 3 2 10 63 . 6 2 9 23 31 34 nm m kT m h e e dB = × = × × × × × × × = =
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In typical metals, there is about one conduction electron per atom . Thus, the average distance between electrons is about the same as the interatomic distance (the lattice constant). The latter is typically 2-5 Ǻ =0.2-0.5 nm. It follows that l e << λ dB e ,( 1 9 . 1 ) that is, the classical limit condition is severely violated. In other words, the M-B stat is NOT valid for the electron gas in metals. Instead, we have to apply quantum statistics, namely Fermi-Dirac statistics, to treat an electron gas, as electron is a Fermion .
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Chapter 19 (I) - Chapter 19 Fermi-Dirac Gases - The...

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