# Metals

Corresponding Wikipedia article: Metal

## Thermal and electrical conductivity

The metals don’t have any distinct energy levels of theirs electrons, and there is an electron gas. The electrons, as the most mobile particles, are the basic effective transmitters of heat and electrical current within the solid metallic bodies. Debye's phonons are absorbed by the material particles. Assuming that the electron gas is an ideal gas, the mean speed $$v_0$$ of the electrons thermal motion can be determined from the following equation: $kT=\frac{mv_0^2}{2}\tag{1}$ When $$T$$ = 300K, then $$v_0$$ = 9,5·104 m/s, and an electron momentum $$p_0$$ = 8,7·10-26 kg·m/s.

Flux of the energy (in case of heat) or the charge (in case of electric current) through an area unit per a time unit
Thermal conduction Electrical conduction
Law $$\mathbf{\overrightarrow{q}}=-\chi\;grad\;T$$
(Fourier's law)
$$\mathbf{\overrightarrow{j}}=-\sigma\;grad\;\varphi=\sigma\mathbf{\overrightarrow{E}}$$
(differential Ohm's law)
Particle energy $$\epsilon$$ $$kT$$ $$e\varphi$$
Particles-carriers all free electrons
Quantity which a particle carries $$C_MT/N_A$$ $$e$$

According to the classical Drude theory, the electrons move from one obstacle (atoms of the crystal lattice, etc.) to another during the relaxation time $$\tau$$, as a ratio of the mean free path $$l_0$$ to the mean velocity $$v_0$$ of thermal motion. The velocity of drift, which is caused by an external force, is a product of the acceleration with the relaxation time. The number of particles, which are drifting through an area unit per a time unit, is determined by multiplying of this velocity by the volumetric concentration of particles $$n$$: $\mathbf{\overrightarrow{s}}=-n\frac{\tau}{m}grad\;\epsilon=-n\frac{l_0}{p_0}grad\;\epsilon\tag{2}$ The stable locations of electrons are in the nodes of the standing phonon waves. Therefore, the mean free path is approximately the average half of wavelength of these phonons, and it is not greater than the crystal lattice period.

The metal conductivity decreases with increase in temperature, when the free path decreases and the momentum increases.

Fourier's law of the thermal conduction: $\mathbf{\overrightarrow{q}}=\frac{C_MT}{N_A}\mathbf{\overrightarrow{s}}=-n\frac{C_Mk}{N_A}\frac{l_0}{p_0}T\;grad\;T\tag{3}$ $\chi=n\frac{C_Mk}{N_A}\frac{l_0}{p_0}T\tag{4}$ The thermal conductivity $$\chi$$ for all metals, except some alloys, decreases with increase in temperature above the Debye temperature.

Assuming that the electrons are the only transmitters of heat, the particles concentration is determined by the atomic number $$Z$$ and by the molar volume $$V_M$$: $n=\frac{N_AZ}{V_M}\tag{5}$ Then the electron mean free path is estimated by the formula: $l_0=\frac{V_Mp_0\epsilon}{ZC_MkT}\tag{6}$

Mean free path for different metals, which are separated into few intervals with gaps
$$l_0$$, pm
(at $$T$$ = 300 К)
Metals
>150 Lightweight metals, where the mean free path is comparable with the atomic radii (lithium, beryllium, sodium, magnesium, aluminium, potassium, calcium).
78...85 Silver, copper (maximal conductivity).
34 Gold.
25...30 Chromium, zinc, molybdenum, strontium.
17...21 Iron, nickel, cobalt (ferromagnetism).
Tungsten (infusibility).
<12 Heavy metals with low conductivity (titanium, manganese, yttrium, palladium, barium, tantalum, platinum, mercury, lead, bismuth etc.).
Electron radius is estimated to be 2...3 pm (see "Mass and momentum").

The strong spatial structures of electrons are strengthening the weak binding of electrons. For example, the atoms of iron, cobalt, nickel and copper, which have the paired nuclear protons arranged in an icosahedron, arrange the electrons into the similar structures. The electron condensation in these materials produces two mutually exclusive phenomena: the high permeability (ferromagnetic) or the high thermal and electrical conductivity (copper).

The magnetostriction is the crystal structure deformation under the external magnetic field influence. This effect is most significant in the ferromagnetic materials, where the electrons attract each other by their magnetic field.

Copper is a diamagnetic, because its electrons are paired and allow the nucleus to respond to an external magnetic field.

Silver is an element of the copper group with maximal conductivity.

Ohm’s law, in its differential form, determines the electric current density by the free electrons concentration $$n_e$$: $\mathbf{\overrightarrow{j}}=e\mathbf{\overrightarrow{s}}=-n_ee^2\frac{l_0}{p_0}grad\;\varphi=n_ee^2\frac{l_0}{p_0}\mathbf{\overrightarrow{E}}\tag{7}$ The electrical conductivity (specific conductance) $$\sigma$$ and the specific resistance $$\rho$$: $\sigma=\frac{1}{\rho}=n_ee^2\frac{l_0}{p_0}\tag{8}$ The electrical conductivity of all metals, except some alloys, decreases with increase in temperature.

The superconductivity is an abnormal increase in the electrical conductivity at the ultralow temperature. This anomaly occurs because the low-energy electrons become totally paired (so-called Cooper pairs), putting the chaotic flow in order. The related Meissner effect is explained by the eddy currents in a superconductor, which are induced by external magnetic field.

The work of motion of an electric charge over any resistive conductor is equal to the produced thermal energy, because the ohmic resistance is a mechanical interaction of particles. For a conductor with a cross-section area $$S$$ and a length $$\Delta l$$ per time $$\Delta t$$, it is: $A=\mathbf{\overrightarrow{F}}\Delta\mathbf{\overrightarrow{l}}=q\mathbf{\overrightarrow{E}}\Delta\mathbf{\overrightarrow{l}}=\mathbf{\overrightarrow{E}}\mathbf{\overrightarrow{j}}S\Delta l\Delta t\tag{9}$ The volumetric power density of heat, which is produced by electric current, is expressed by Joule's first law in a differential form: $w=\mathbf{\overrightarrow{E}}\mathbf{\overrightarrow{j}}=\sigma E^2=\rho j^2\tag{10}$ The higher is electric current, the more directional is the chaotic thermal motion of particles. The electric current actually reduces the thermal energy of electron gas. Therefore the linear Ohm's and Joule’s laws are valid when the work of electrons in electric field does not exceed the thermal energy of electron gas: $\rho j^2<n_ekT\tag{11}$ $j=\sqrt{\sigma n_ekT}\tag{12}$ Exceeding the critical value of the current density produces an increase in electron speed $$v_0$$ by electric field, i.e. the direct increase in temperature of the conductor, which is much greater than Joule’s effect. This effect is used, for example, in the fuses.

Assuming that all electrons in a conductor are free ($$n_e=n$$), the critical current density can be calculated by the approximate formula using (5), where $$R$$ is the ideal gas constant: $j_{MAX}=\sqrt{\frac{\sigma ZRT}{V_M}}\tag{13}$ Substituting the values for $$T=300 K$$, gives the critical current density 7,7·108A/m2 for copper, and 3,4·108A/m2 for aluminum. For example, the diameter of a copper wire for the 1А fuse should be about 0,04 mm, and an aluminum wire about 0,06 mm. For higher currents, a significant heating of the fuse due to its ohmic resistance should be taken in account, according to Joule's first law.

For the metals, a ratio of the thermal conductivity to the electric conductivity is almost directly proportional to the absolute temperature with a coefficient, which is called the Lorenz number $$L$$. This ratio is called the Wiedemann-Franz law, and after substituting of (4) and (8) it has the following form: $\frac{\chi}{\sigma}=\frac{nC_M}{n_eR}\frac{k^2}{e^2}T=L_0\frac{k^2}{e^2}T=LT\tag{14}$ The dimensionless coefficient, which depends on material and temperature, is: $L_0=\frac{nC_M}{n_eR}\approx 3\dotsc 4\tag{15}$

Examples of the common metals
Metal $$\chi$$,
W/(m·K)
$$\sigma$$,
MS/m
$$L_0$$ $$C_M/R$$ $$n/n_e$$ Comment
silver 429 62,5 3,08 3,05 1,01
copper 401 58,8 3,06 2,94 1,04
gold 317 45,5 3,13 3,05 1,02
aluminum 237 36,0 2,96 2,93 1,01
tungsten 163 18,2 4,02 2,92 1,38 Infusible metal with a large fraction of the fixed bonding electrons.
magnesium 156 22,7 3,08 3,00 1,03
zinc 116 16,9 3,08 3,06 1,01
nickel 90,9 11,5 3,55 3,14 1,13 Ferromagnetic materials with the magnetic domains of electrons.
iron 80,4 10,0 3,61 3,02 1,19
platinum 71,6 9,54 3,37 3,11 1,08
tin 66,8 8,33 3,60 3,26 1,10 Stable crystal lattice.
lead 35,3 4,81 3,29 3,21 1,03
mercury 8,3 1,04 3,58 3,37 1,06

The unique metals in the neighbourhood of 80th element in the periodic table:

• Tungsten (74), the highest melting and boiling points.
• Gold (79) and platinum (78) are the widespread noble metals.
• Mercury (80), the lowest melting and boiling points.

## Hall effect

The Hall effect is a production of transverse EMF within a conductor by Lorentz force:

$E=vB=\frac{jB}{n_e e}\tag{16}$ $$E$$ is a field strength of transverse EMF;
$$v$$ is a drift velocity of charge carriers (electrons in metal);
$$B$$ is a magnetic flux density.

The specific Hall resistance is:

$\frac{E}{j}=\frac{B}{n_e e}=\rho_H B\tag{17}$ $$ρ_H$$ is a Hall coefficient.

The flow of current in the Hall effect can be considered 2-dimensional, and 3-dimensional quantities could be replaced by 2-dimensional quantities:

$R_H=\frac{E}{j_2}=\frac{B}{n_2 e}\tag{18}$ $$j_2$$ is a surface current density, A/m;
$$n_2$$ is a surface concentration of charge carriers (electrons in metal), m–2;
$$R_H$$ is a Hall resistance, Ohm.

The motion of an electron with momentum $$p$$ along a circle of radius $$r$$ is described by the equation:

$p=\frac{h}{\lambda}=reB\tag{19}$

The electron pairs of metal are concentrated at the nodes of standing matter waves of length $$\lambda$$. So the electron density is twice the density of 2-dimensional half waves ("Waves", 6):

$n_2=2\frac{\pi}{\lambda^2}\tag{20}$

Substituting of (19) and (20) into (18) gives:

$R_H=\frac{h\lambda}{2\pi re^2}\tag{21}$

The multiplicity of wavelength and circumference causes the resonant quantum Hall effect as well as the quantum effects in atom:

$2\pi r=n\lambda\;\;\;\;\;n=1,2...$

The permitted resistance values are:

$R_H(n)=\frac{h}{ne^2}\tag{22}$

The fractional quantum Hall effect ($$0<n<1$$) occurs within strong magnetic fields, when the nodes of standing waves contain more than two electrons, because their mutual electrostatic repulsion is not strong enough.