Equation is the definition of specific heat given in terms of the fermi-dirac distribution function f and the density of states equation g C_e(T_e)=int^infinity_infinity (df(epsilon,, nu, T_e))/(dTe) The first consideration that I make is that the fermi level is a function of temperature

lexi13xoxla

lexi13xoxla

Answered question

2022-08-12

Equation is the definition of specific heat given in terms of the fermi-dirac distribution function f and the density of states equation g
C e ( T e ) = f ( ε , μ , T e ) T e g ( ε ) ε   d ε ,
The first consideration that I make is that the fermi level is a function of temperature, so I use the following conservation equation to solve for the fermi level
N e = f ( ε , μ ( T e ) , T e ) g ( ε ) d ε .
where N e is the number of free electrons. For example in Copper it would be 1 electron/atom.
My question is around how to determine the number of free electrons in a metal as a function of temperature. All the analysis that I have seen just assumes a static amount of free electrons for a material, but we know as T the number of free electrons goes towards the atomic number of the metal.
My question is how can I solve for N e ( T ) if I know N e ( T 0 )?

Answer & Explanation

kilinumad

kilinumad

Beginner2022-08-13Added 21 answers

While your definition of the heat capacity is correct, you are misinterpreting the second equation you write. The sum of the Fermi distribution multiplied by the density of states over all energy is equal to the total number of electrons not free electrons i.e. it says that the number of electrons doesn't change as you heat up the sample. It is necessary to calculate this as the temperature changes (at least for larger temperatures) because the position of the Fermi level will change, due to the uneven shape of the density of states.
Consequently, you don't need to know the number of free electrons to calculate the heat capacity using your equation (1).

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