Show that P(E), the occupancy probability in Eq. 41-6, is symmetrical about the value of the Fermi energy; that is, show that .
It is shown that that is the occupancy probability is symmetrical about the value of the Fermi energy.
The probability of a state to be occupied by and electron is referred to as the occupancy probability. At 0 K temperature, the states below the Fermi level have occupancy probability equal to and for the states above the Fermi level, its value is .
The occupancy probability of the state with energy E is-
( i )
Here is the Fermi energy, and T is the absolute temperature.
Upon expansion in view of equation (i), the LHS value of the given equation can be solved as follows:
On further solving,
Hence, the given condition is proved and this implies the symmetrical condition for occupancy probability.
Assume that the total volume of a metal sample is the sum of the volume occupied by the metal ions making up the lattice and the (separate) volume occupied by the conduction electrons. The density and molar mass of sodium (a metal) are and , respectively; assume the radius of the Na+ ion is . (a) What percent of the volume of a sample of metallic sodium is occupied by its conduction electrons? (b) Carry out the same calculation for copper, which has density, molar mass, and ionic radius of 8960, 63.5g/mol, and 135 pm, respectively. (c) For which of these metals do you think the conduction electrons behave more like a free-electron gas?
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