Section: 12 | Fermi Energy and Related Properties of Metals |

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FERMI ENERGY AND RELATED PROPERTIES OF METALS

Lev I. Berger

In the classical Drude theory of metals, the Maxwell-Boltzmann velocity distribution of electrons is used. It states that the number of electrons per unit volume with velocities in the range of $d \vec{v}$ about any magnitude $\vec{v}$ at temperature T is

$f_{B} (\vec{v}) d \vec{v} = n (\frac{m}{2 π k_{B} T}) \exp (- \frac{m v^{2}}{2 k_{B} T}) d \vec{v}$

where n is the total number of conduction electrons in a unit volume of a metal, m is the free electron mass, and k_B is the Boltzmann constant. In an attempt to explain a substantial discrepancy between the experimental data on the specific heat of metals and the values calculated on the basis of the Drude model, Sommerfeld suggested a model of the metal in which the Pauli exclusion principle is applied to free electrons. In this case, the Maxwell-Boltzmann distribution is replaced by the Fermi-Dirac distribution:

$f (\vec{v}) d \vec{v} = 2 {(\frac{m}{h})}^{3} d \vec{v} {\exp [(\frac{m v^{2}}{2} - k_{B} T_{0}) / k_{B} T] + 1}^{- 1}$

Here h is the Planck constant and T₀ is a characteristic temperature which is determined by the normalization condition

$n = \int d \vec{v} \cdot f (\vec{v})$

The magnitude of T₀ is quite high; usually, T₀ > 10⁴ K. So, at common temperatures (T < 10³ K), the free electron density of a metal is much smaller than in the case of the Maxwell-Boltzmann distribution. This allows us to explain why the experimental data on specific heat for metals are close to those for insulators.

The maximum kinetic energy the electrons of a metal may possess at T = 0 K is called the Fermi energy, e.g.,

$E_{F} = \frac{ℏ^{2} k_{F}^{2}}{2 m} = (\frac{e^{2}}{2 k_{B}}) {(k_{F} r_{B})}^{2}$

where k_F is the Fermi momentum or the Fermi wave vector

k_F = (3π²n)^1/3

e is the electron charge, and r_B is the Bohr radius

r_B = ħ²/me² = 0.529·10⁻¹⁰ m

Another, more common expression for the Fermi energy is

$E_{F} = \frac{1}{2} m v_{F}^{2}$

where ν_F = ħk_F/m is the Fermi velocity which can be expressed using the concept of the electron radius, r_s. It is equal to radius of a sphere occupied by one free electron. If the total volume of a metal sample is V and the number of conduction electrons in this volume is N, then the volume per electron is equal to

$\frac{V}{N} = \frac{1}{n} = \frac{4}{3} π r_{S}^{3}$

and

$r_{S} = {(\frac{3}{4 π n})}^{1 / 3}$

The following table contains information pertinent to the Sommerfeld model for some metals. The magnitudes of T₀ are calculated using the expression

$T_{0} = \frac{E_{F}}{k_{B}} = \frac{58.2 \cdot 10^{4}}{{(r_{S} / r_{B})}^{2}} K$

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Fermi Energy and Other Ground-State Properties of the Electron Gas in Metals^d

Metal	Name	Valency	n/10²⁸ m^–3	r_S/pm	r_S/r_B	E_F/eV	T₀/10⁴ K	k_F/10¹⁰ m^–1	v_F/10⁶ m s^-1
Continued on next page...
Li^a	Lithium	1	4.70	172	3.25	4.74	5.51	1.12	1.29
Na^b	Sodium	1	2.65	208	3.93	3.24	3.77	0.92	1.07
K^b	Potassium	1	1.40	257	4.86	2.12	2.46	0.75	0.86
Rb^b	Rubidium	1	1.15	275	5.20	1.85	2.15	0.70	0.81
Cs^b	Cesium	1	0.91	298	5.62	1.59	1.84	0.65	0.75
Cu	Copper	1	8.47	141	2.67	7.00	8.16	1.36	1.57
Ag	Silver	1	5.86	160	3.02	5.49	6.38	1.20	1.39
Au	Gold	1	5.90	159	3.01	5.53	6.42	1.21	1.40
Be	Beryllium	2	24.7	99	1.87	14.3	16.6	1.94	2.25
Mg	Magnesium	2	8.61	141	2.66	7.08	8.23	1.36	1.58
Ca	Calcium	2	4.61	173	3.27	4.69	5.44	1.11	1.28
Sr	Strontium	2	3.55	189	3.57	3.93	4.57	1.02	1.18
Ba	Barium	2	3.15	196	3.71	3.84	4.23	0.98	1.13
Nb	Niobium	1	5.56	163	3.07	5.32	6.18	1.18	1.37
Fe	Iron	2	17.0	112	2.12	11.1	13.0	1.71	1.98
Mn^c	Manganese	2	16.5	113	2.14	10.9	12.7	1.70	1.96
Zn	Zinc	2	13.2	122	2.30	9.47	11.0	1.58	1.83
Cd	Cadmium	2	9.27	137	2.59	7.47	8.88	1.40	1.62
Hg^a	Mercury	2	8.65	140	2.65	7.13	8.29	1.37	1.58

^aAt 78 K.
^bAt 5 K.
^cα-phase.
^dThe data in the table are for atmospheric pressure and room temperature unless otherwise noted.

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FERMI ENERGY AND RELATED PROPERTIES OF METALS