Lorentz' Theory

Lorentz combined the ideas of Thomson, Riecke, and Drude[113] into the most complete form.

It supposed that:

• the atoms of the metal are fixed
• in the interstices between them a large number of resinous electrons are in rapid motion.

The mutual collisions of the electrons are disregarded, so that their collisions with the fixed atoms alone come under consideration; these are regarded as analogous to collisions between moving and fixed elastic spheres.

The flow of heat and electricity in the metal is supposed to take place in a direction parallel to the axis of x, so that the metal is in the same condition at all points of any plane perpendicular to this direction; and the flow is supposed to be steady, so that the state of the system is independent of the time.

Consider a slab of thickness dx and of unit area; and suppose that the number of electrons in this slab whose x-components of velocity lie between u and u + du, whose y-components of velocity lie between v and v + du, and whose z-components of velocity lie between w and w + du, is

f(u, v, w, x) dx du dv dw.

One of these electrons, supposing it to escape collision, will in the interval of time dt travel from (x, y, z) to (x + u dt, y + vdt, z + wdt): and its x-component of velocity will at the end of the interval be increased by an amount eEdt/m, if m and e denote its mass and charge, and E denotes the electric force. Suppose that the number of electrons lost to this group by collisions in the interval dt is a dx du dv dw dt, and that the ​number added to the group by collisions in the same interval is b dx du dv dw dt. Then we have

f (u, v, w, x) + (b - a)dt = f(u + eEdt/m, v, w, x + udt),

and therefore

The law of distribution of velocities which Maxwell postulated for the molecules of a perfect gas at rest is expressed by the equation

where N denotes the number of moving corpuscles in unit volume, r denotes the resultant velocity of a corpuscle (so that r2 = u2 + v2 + w2), and a denotes a constant which specifies the average intensity of agitation, and consequently the temperature. It is assumed that the law of distribution of velocities among the electrons in a metal is nearly of this form; but a term must be added in order to represent the general drifting of the electrons parallel to the axis of x. The simplest assumption that can be made regarding this term is that it is of the form

u × a function of r only;

we shall, therefore, write

The value of χ(r) may now be determined from the equation

for on the left-hand side, the Maxwellian term

would give a zero result, since b is equal to a in Maxwell’s system; thus b-a must depend solely on the term uχ(r); and ​an examination of the circumstances of a collision, in the manner of the kinetic theory of gases, shows that (b − a) must have the form -urχ(r)/l, where l denotes a constant which is closely related to the mean free path of the electrons. In the terms on the right-hand side of the equation, on the other hand, Maxwell’s term gives a result different from zero; and in comparison with this we may neglect the terms which arisefrom -uχ(r). Thus we have

and thus the law of distribution of velocities is determined.

The electric current i is determined by the equation

where the integration is extended over all possible values of the components of velocity of the electrons. The Maxwellian term in f(u, v, w) furnishes no contribution to this integral, so we have

When the integration is performed, this formula becomes

or

The coefficient of i in this equation must evidently represent the ohmic specific resistance of the metal; so if γ denote the specific conductivity, we have

Let the equation be next applied to the case of two metals. A and B in contact at the same temperature T, forming an ​open circuit in which there is no conduction of heat or electricity (so that i and da/dx are zero). Integrating the equation

across the junction of the metals, we have

Discontinuity of potential at junction or since

ma2, which represents the average kinetic energy of an electron, is by Drude’s assumption equal to qT, where q denotes a universal constant, we have

Discontinuity of potential at junction

This may be interpreted as the difference of potential connected with the Peltier[114] effect at the junction of two metals; the product of the difference of potential and the current measures the evolution of heat at the junction. The Peltier discontinuity of potential is of the order of a thousandth of a volt, and must be distinguished from Volta’s contact-difference of potential, which is generally much larger, and which, as it presumably depends on the relation of the metals to the medium in which they are immersed, is beyond the scope of the present investigation.

Returning to the general equations, we observe that the flux of energy W is parallel to the axis of x, and is given by the equation

where the integration is again extended over all possible values of the components of velocity; performing the integration, we have

or, substituting for E from the equation already found,

Consider now the case in which there is conduction of heat without conduction of electricity. The flux of energy will in this case be given by the equation

where κ a denotes the thermal conductivity of the metal expressed in suitable units; or

If it be assumed that the conduction of heat in metals is effected by motion of the electrons, this expression may be compared with the preceding; thus we have

and comparing this with the formula already found for the electric conductivity, we have

an equation which shows that the ratio of the thermal to the electric conductivity is of the form T × a constant which is the same for all metals. This result, accords with the law of Wiedemann and Franz,

Moreover, the value of q is known from the kinetic theory of gases; and the value of e has been determined by J.J. Thomson[115] and his followers; substituting these values in the formula for κ/γ, a fair agreement is obtained with the values of κ/γ determined experimentally.

It was remarked by J. J. Thomson that if, as is postulated in the above theory, a metal contains a great number of free electrons in temperature equilibrium with the atoms, the specific heat of the metal must depend largely on the energy required in order to raise the temperature of the electrons.

Thomson considered that the observed specific heats of metals are smaller than is compatible with the theory, and was thus led to investigate[116] the consequences of his original hypothesis[117] regarding the motion of the electrons, which differs from the one just described in much the same way as Grothuss’ theory of electrolysis differs from Arrhenius'.

Each electron was now supposed to be free only for a very short time, from the moment when it is liberated by the dissociation of an atom to the moment when it collides with, and is absorbed by, a different atom. The atoms were conceived to be paired in doublets, one pole of each doublet being negatively, and the other positively, electrified.

Under the influence of an external electric field the doublets orient themselves parallel to the electric force, and the electrons which are ejected from their negative poles give rise to a current predominantly in this direction. The electric conductivity of the metal may thus be calculated.

In order to comprise the conduction of heat in his theory, Thomson assumed that the kinetic energy with which an electron leaves an atom is proportional to the absolute temperature; so that if one part of the metal is hotter than another, the temperature will be equalized by the interchange of corpuscles. This theory, like the other, leads to a rational explanation of the law of Wiedemann and Franz.