Free electrons in a metal

The Schrodinger equation for a free electron in a box is the simplest model describing electrons in a metal. A comprehensive treatment is found in the book by Kittel.  

Looking at a row of atoms along an x-axis of length I (one-dimensional model), the Schrodinger equation is 

The eigen values of the energy E are quadratic functions of the quantum number n 

One can introduce periodic boundary conditions. A wave function satisfying periodic boundary conditions has the form 

In the three-dimensional case a wave vector must be defined with 

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Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependent dielectric constant of 1 - a>2/(co2 - ia>/r), with (op = (47me2/m)l/2 the plasma frequency and r a collision time. For metals like Ag and Au, and with frequencies (o corresponding to visible or ultraviolet light, this simplifies to give a real part... 

The energy of a free electron in a metal is essentially the kinetic energy due to its motion, given by... 

The inverse case, a semiconducting catalyst supported by a metal, termed inverse supported catalyst, has been studied systematically only in the last few years. Here, even more drastic effects can be expected because normally the number of free electrons in a metal is several orders of magnitude higher than in semiconductors. The effects are indeed considerably larger as will be shown below. However, the principles and the theory involved are more complex (6-8). 

Figure 3. Energy diagram for free electrons in a metal. The positive background charge of the core ions leads to a potential energy well with respect to the energy of the electron in vacuum vac- The averaged kinetic energy of the free electrons is indicated with dashed lines 3/2 k%T according to the Drude model, and 3/5 according to the Sommerfeld model. The electrochemical potential of the electrons in the metal [Fermi level] is also indicated.

The Sommerfeld Model for Free Electrons in a Metallic Phase... 

The interaction between a free electron in a metal and the ionic displacements u is frequently given by the deformable potential approximation (E.g.92), p. 128). If the potential felt by the electron in position r is V 0) without the lattice being distorted, then upon distortion of the lattice to an extent of u(r), the potential experienced by the electron is y(r-u). Another approximation views the potential as the sum of separate ionic potentials based on the instantaneous positions X(k) + u(k) of each ion... 

For free electrons in a metal, it can be shown that the E-k relationship in three dimensions is... 

Figure 1 Z.Z The energy versus wave vector curve Tor a free electron in a metal...

The energy of an electron in a quantum well can be calculated using the approach outiined in Section 2.3.6. If it is assumed that the electron is free, and trapped by an infinite boundary potential, the same equations for a free electron in a metal apply. Thus, the energy, E, of a free electron in a rectangular parallelepiped with edges a, b and c is given by Equation (2.15) ... 

Figure 13.27 (a) The density-of-states function, N E), for a free electron in a metal (b) the stqj-like density-of-states function for an electron trapped in a quantum well (c) the density-of-states function of an electron trapped on a quantum wire and (d) the density-of-states function of an electron trapped in a quantum dot... 

As the concentration of free electrons in a metal is much greater than that of eiectroactive compounds in an electrolytic solution, then one can assume that this concentration is constant, and it no longer features in the equation for the apparent rate iaw of the eiementary redox half-reactions at a metal electrolyte interface. This would be different if we wanted to write kinetic iaws for a redox half-reaction at an interface between a semiconductor and an eiectroiyte. However, here the description will be confined to metaiiic eiectrodes. 

To understand why a stable equilibrium state of two metals in contact includes a contact potential, we can consider the chemical potential of the free electrons. The concept of chemical potential (i.e., partial molar Gibbs energy) applies to the free electrons in a metal just as it does to other species. The dependence of the chemical potential of free electrons in metal phase a on the electric potential 0 of the phase is given by the relation of Eq. 10.1.6 on page 287, with the charge number zi set equal to -1 ... 

Hence, we can see that for Ihe free electrons in a metal, the kinetic energy is non-null at the temperature of 0 K this is the consequence of the application of the Fermi-Dirac statistics. Thus, we can no longer define a temperature scale on the basis of the kinetic energy of the free electrons in a metal. 

Cv(st) - contribntion of free electrons in a metal to the molar heat capacity. 

The free electrons in a metal are treated as a collective of zero-frequency oscillators (o>o = 0), i.e., the restoring force is zero. This leads to the well-known Drude relation ... 

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