Fermi energy of fermion systems Masatsugu Sei Suzuki, Department of Physics (Date: October 14, 2016)

The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi gas, the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of the conduction band. https://en.wikipedia.org/wiki/Fermi_energy

Here we discuss the Fermi energy and the Fermi temperature for some typical systems are evaluated: liquid 3He, white dwarf, neutron star, and conduction electron in metal, and proton and neutron in nucleus.

Using the Fermi energy  F , the Fermi temperature TF is defined by

 F TF  kB

Fermi momentum and Fermi wave number is defined by

ℏ pF  kF  2m F

The Fermi velocity is defined by

p v  F . F m

1. Liquid 3He Liquid 3He as a Fermi gas

1 spin I  (fermion) 2 Density   .0 081 g/cm 3

The Fermi energy

2 ℏ 2 3/2  F  3(  n) 2m0

3 where m0 is the mass of He atom,

.3 016g m   5.0 1023  0.5 1024 g 0 6.0221023

The number density

N N M  .0 081 22 3 n     24  62.1 10 / cm . V M V m0 0.5 10

Then the Fermi energy is

16 4  F  .6 81510 erg  .4 25410 eV.

The Fermi temperature

 F TF   4.94 K. kB

2. White dwarf

Suppose that the sun consists of hydrogen atoms. A hydrogen atom has 1 electron. When mP is the mass of proton, there are N hydrogen atoms in the sun,

M N  sun  .1 188881057 . mp

The Fermi energy of electrons in the white dwarf where we use N electrons for the sun

ℏ2 3 2 N ℏ2 3 2 N ℏ2 9N   ( ) 3/2  ( ) 3/2  ( ) 3/2 F 2m V 2m 4R3 2m 4R3 3

4 where V  R3 and R is the radius of white dwarf; R  2109 cm. 3

8 4  F  .6 30810 erg = 94.3 10 eV

The Fermi temperature is

8 TF  4.569 x 10 K

3. Neutron star The Fermi energy of the neutron star is given by

3 2 N   cℏ( ) 3/1 (relativistic limit) F V

Suppose that R = 10 km, and  = 4.0 x 10 14 g/cm 3

N    2.39 x 10 38/cm 3 V mn

The Fermi energy

-4 8  F 6.07 x 10 erg = 79.3 10 eV.

The Fermi temperature is

12 TF  40.4 10 K.

4. Conduction electron in metal Suppose that there is one free electron per atom. There is one atom per unit cell. We consider the simple cubic crystal with the lattice constant a  3 Å. So the number density of electrons is

1 n   70.3 1022 /cm 3. a3

The Fermi energy

ℏ2   3(  2n) 3/2 = 6.488 x 10 -12 erg = 4.04 eV. F 2m

The Fermi temperature is

4 TF  7.4 10 K.

The Fermi velocity is

8 vF  1.19 x 10 cm/s.

5. Fermi energy of nucleon Model a heavy nucleus of mass number A (= Z + N) as a free Fermi gas of an equal number of protons and neutrons, containing in a sphere of radius

3/1 R  0 Ar (1)

13 where r0  4.1 10 cm. We calculate the Fermi energy and the average energy per nucleon in MeV. The stable nucleus has approximately a constant density and therefore the nuclear radius R can be approximated by the following formula (1), where A = Atomic mass number (= Z + N, Z: the number of protons, N: the number of neutrons N) and

13 r0  4.1 10 cm.

The volume V0 is

4 4 V  R3  r 3 A 0 3 3 0

After assuming that the proton and neutron potential wells have the same radius, we find for a nucleus with

A N  Z  N  0 2

Since the Fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the Fermi energy of the protons in the nucleus, and vice versa. The Fermi momentum is given by

2V 4 N  0 p 3 0 2( ℏ)3 3 F

Then the Fermi momentum pF for proton and neutron can be rewritten as

3/1 3/1  2 A  2  3  3/1 n p  3 N  ℏ  9  ℏ 0  ℏ 2  p F  pF  pF        = 214.7 MeV/c V  4 3  r 8  0   r A  0    3 0 

The Fermi energy is

1 p 1 n  F  pF  pF =24.564 MeV 2mp 2mn

using the mass of proton and neutron ( mp and mn ). The average energy is

E 3  F = 14.738 MeV N0 5

(( Mathematica )) Clear "Global` " ; rule1 kB 1.3806504 10 16 , c 2.99792 10 10 , 1.054571628 10 27 , me 9.10938215 10 28 , mp 1.672621637 10 24 , mn 1.674927211 10 24 , qe 4.8032068 10 10 , eV 1.602176487 10 12 , meV 1.602176487 10 15 , keV 1.602176487 10 9, MeV 1.602176487 10 6, 10 8 ; r0 1.4 10 13 ;

9 1 3 pF . rule1 r0 8 1.1474 10 14

pF c . rule1 MeV 214.697

1 pF 2 EF1 . rule1 2 mp MeV 24.5637

3 EF1 5 14.7382

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