ABSTRACT
For infinite objects in 3D with a large number of states to be filled, the choice of periodic or fixed boundary conditions is free because it leads to the same evaluation for the density of states g(k) and for g(E). It is not the same for the first possible states of objects limited along one or several directions, for instance surfaces of a metal bounded by vacuum, a thin layer with parallel sides and of thickness of a few lattice distances (Ex. 7) or aggregates (clusters) of atoms (Exs. 5 and 6). The discrete values of nx, ny, nz correspond to the quantum numbers that a free electron (excluding spin) can take, as n, l, m, represent the quantum numbers of an electron subject to a central potential of an ion in atomic physics. 3. Electron Distribution and Density of States at 0 K: Fermi Energy and Fermi Surface in 3DAs for an atom, the Schrödinger equation only allows the determination of all the possible states of electrons. The states
that are occupied will be obtained by filling each state (2p/Lx)· (2p/Ly)· (2p/Lz) in k-space with two electrons of anti-parallel spin (≠Ø). The electronic states thus differ from one another by at least different quantum numbers nx, ny, nz or spin s (from the Pauli exclusion principle) starting with the lowest energy levels. When N free electrons of the volume Lx, Ly, Lz are thus distributed, we obtain a sphere in k-space limiting the occupied and the empty states at 0 K. The radius kF of this Fermi sphere is 43 2 23 3p pk L L L Nx y zF/ ( ) = , where k nF = ( ) /3 2 1 3p and n = N/V. The electron energy is uniquely kinetic and the velocity v is proportional to k ( v k)m h= . The Fermi sphere visualizes the velocity vectors of the free electrons in a metal. The maximum kinetic energy is given by E k
m m nF F= = ◊h h2 2 2 2 2 32 2 3( ) /p . For alkali metals, the orders of magnitude are n = 5 ¥ 1022 e.cm-3,
kF = 1.2 ¥ 108 cm-1, vF ≈ 1.3 ¥ 108 cm/sec, EF = 5 eV. ∑ Density of states: In k-space, the density of states g(k) between
k and k + dk is the same as that evaluated previously for lattice vibrations (Chapter III). However, g(E) must take into account the particular dispersion of free electrons with two electrons of opposite spin per state. We thus obtain g(E)◊dE = 2g(k)◊dk. In 3D, this results in g(E) = (V/2p2)◊(2m/h2)3/2E1/2. The Fermi energy at 0 K can also be deduced from g E dE N
E ( ) =Ú0 F . 4. Influence of Temperature on the Electron Distribution: Electron-Specific HeatElectrons obey Fermi-Dirac statistics: These are fermions because of their non-integer spin leading the Pauli exclusion principle (in contrast to phonons, Chapter III that are bosons from the Bose-Einstein statistics). As a function of temperature, the occupation probability of an electronic state is given by f(E) = [e(E – EF)/kBT +1]–1.The distribution obtained for T ≠ 0 differs from that at 0 K (where f(E) = 1 for E < EF) only for energies that are very close (several kBT) to the Fermi energy.
