HW-4

Theory of Solids HW-4 Solution By Qifeng Shan Ashcroft and Mermin’s book 2-4 Insensitivity of the distribution function to small changes in the total number of electrons In deriving the Fermi distribution (page 41) we argued that the probability of a given level being occupied should not change appreciably when the total number of electrons is changed by one. Verify that the Fermi function (2.56) is compatible with this assumption as follows: (a) Show, when k BT  <<  << εF, that when the number of electrons changes by one at fixed temperature, the chemical potential changes by 1 ,      (2.108) V  g  F  Where g  Where g (ε) is the density of levels. (b) Show, as a consequence of this, that the most the probability of any level being occupied can change by is 1  F 1 .    f   (2.108) 6 k BT  N  [Use the free electron evaluation (2.65) of  g (εF).] Although temperatures of 8 milidegrees Kelvin can be reached, at which εF / k B T  ≈ 10 , when N  when  N  is   is of order 22 10  then ∆ f  is  is still neglibly small. Solution: (a) When k BT  <<  << εF, from Eq. (2.78), we have

 1   k  T   2  B       F 1     3   2 F          F

(1)

and  F



3n 2 g  F 

.

(2)

Thus,

    F  Q.E.D. 1

1 V  g  F 

 

(3)

(b) The probability change of the energy level occupied only happens near the Fermi level.

  f   

 f    f    1

1

2           k BT Vg   F     k  T   k  T   1 e e      



     B



1

1

 F

B

1

. 6 k BT  N 

Q.E.D.

2

(4)