Question

Fermi-Dirac Distribution.

(a) Calculate the magnitude of the Fermi momentum and the Fermi wavelength for 4.2 x10²¹ electrons confined in a box of volume 1 cm³.

(b) Compute the Fermi energy (in eV) for this system.

(c) If the electrons are replaced by neutrons (also spin-1/2), compute the Fermi momentum, the Fermi wavelength, and the Fermi energy.

Answer 1

(a)

Fermi momentum can be found if one knows the electron density:

4.2 x10²¹ electrons confined in a box of volume 1 cm³.

(n) = 4.2 x10²¹ /cm3 = 4.2 x10²⁷ /m3

( 1J = 1 kg·m²·s^-2 )

The fermi momentum P(f ) = h/2 * (3/ * n)^1/3 = 6.626*10^-34 J-s / 2 * (3/ * 4.2 x10²⁷ /cm3)^1/3

The fermi momentum P(f ) = 5.26x10^-25 kg-m/s

fermi wavelength ( λ(f) ) = h/P(f) = 6.626*10^-34 J-s /5.26x10^-25 kg-m/s = 1.26*10^-9 m =1.26 nm

(b) Compute the Fermi energy

E (f) = 1/2* mv(f)^2 = m²v(f)^2/ 2m = P(f )/ 2m                    (P(f) = m*v(f))

m(e) =9.1*10^-31 kg

(5.26x10^-25 kg-m/s)² / 2*9.1*10^-31 kg = 1.52*10^-19 J

1 eV = 1.6*10^-19 J

E(f) = 1.52*10^-19 J /1.6*10^-19 J/eV = 0.95 eV

(c) For neutron :