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The Drude theory of metals


The Drude theory of metals

We overview the Drude theory of metals and its postulates, while setting the stage for its detailed explorations of certain important physical phenomena.

Solid State physics, being the study of rigid objects, unsurprisingly starts with an attempt to understand the structure and properties of metals. In other words, why are metals as they are and what gives them their properties. This is used as a springboard to comparatively study all other solids, whether non-metallic, crystalline, amorphous or the like. The Drude theory was one of the first successful attempts at this.

Paul Drude theory of metals

Paul Drude

Solid State physics began in the late 1800s with J.J. Thomson’s discovery of the electron, although the field itself was recognised separately by 1900 when the German physicist Paul Drude used these newly-discovered electrons to explain metals, an idea freely — and rather vaguely (over)used in chemistry up to that point, but one which physicists had struggled to properly understand at a fundamental level. The theory was extended by H.A. Lorentz and is also called the Drude–Lorentz theory.

As we shall see over the course of our study, solid state physics, as a branch, makes extensive use of various other fields in physics, particularly quantum theory (for which you can refer to our excellent collection of articles) alongside crystallography and electromagnetism.

The kinetic theory

By the 1900s, electric conduction was generally explained using the movement of electrons. The kinetic theory of gasses had explained the macroscopic, observed properties of gasses. Drude used the two to give a foundation of the understanding of metals, an excellent approximation but ultimately incorrect idea that would only be rectified with the later developments of quantum physics by Arnold Sommerfeld and Hans Bethe.

The Drude theory treats metals as a gas of electrons. This was akin to the gas model except we knew metals were electrically neutral, so there must be a second, positively charged particle, which Drude considered to be relatively heavier when compared to free electrons, and hence immovable. Although the exact nature of this free movement requires a quantum mechanical idea, we will safely proceed with the assumption that valence electrons can freely move through a metal. This idea has since been understood to be valid and useful as an approximation.

The term used for such electrons was conduction electrons, which is still used today in the guise of conduction bands etc. The atom itself, although we had a rather weak idea of it then, was divided into three sections.

Drude theory of metals

Although a single atom has a nucleus of charge eZ_a, for its atomic number Z_a, and electrons of charge -eZ_a, some of these, say Z, are bound weakly enough that they can be considered freely moving valence, or conduction, electrons as shown above. The three regions would then be a nucleus, some (Z_a-Z) electrons comprising of the core and the remaining forming a region of free conduction electrons.

Postulates of the Drude theory

The Drude theory of metals, also sometimes called the Drude model of electrical conduction, itself has no specific formulations, but rather gives a framework with postulates and assumptions which will later help in understanding various properties of metals. Once again, it is important to understand that it is, strictly speaking, incorrect, but is an extremely valid approximation. The basis, as we have discussed, is that in the neighbourhood of valence potentials of other atoms, the conduction electrons effectively get detached due to their already weak attraction towards the nucleus, and they can move freely.

Optional mathematics:

For a metal with N_A = 6.022 \times 10^{23} atoms per mole, and a density of \rho_m/A mol cm^{-3}, A the atomic mass. The number of free electrons per cm^{-3}, under our assumption of Z electrons being contributed per atom, is,

    \begin{align*} n = \frac{Z N_A \rho_m}{A} \end{align*}

This conduction electron density turns out to be of the order of 10^{22} cm^{-3}, by and large, in metals. A helpful visualisation of this comes in the form of what is called the conduction electron sphere — a sphere with radius r_s such that its volume is equal to the volume per conduction electron:

    \begin{align*} \frac{V}{N_A} = \frac{1}{n} = \frac{4 \pi r_s^3}{3} \Longrightarrow r_s = \left( \frac{3}{4 \pi n} \right)^{\frac{1}{3}} \end{align*}

The takeaway here is that these densities are much, much larger than gasses and hence directly applying the kinetic theory of gasses would be incorrect. The only difference being that in the Drude theory, we see free electrons moving against a background of immovable, heavy positive charge centres.

The postulates of the Drude theory are fairly simple:

  1. Particle interactions between collisions are largely neglected. Electrons, realistically, have long range forces of attraction or repulsion with other ions or electrons, but these are neglected. The conduction electrons are therefore considered to be moving in straight lines under an external electric field, \vec{E}, or as determined by classical laws under the application of both \vec{E} and \vec{B} (magnetic) fields.
    When electron–electron interactions are neglected, we call it the independent electron approximation, and when electron–ion interactions are neglected, we call it the free electron approximation. While the former is actually fairly accurate, the latter is most certainly not.
  2. Collisions are instantaneous. By the independent electron approximation, which suggests that the only possible interaction between electrons is actual collision, Drude suggested that these collisions impulsively alter the velocity of electrons — since collisions are supposedly with massive ion cores. Luckily it was later found that electron-electron scattering is indeed negligible, but electron movement as attributed simply to electron-ion collisions is incorrect.
  3. Relaxation time is independent of electron position and velocity. If we were to assume that the probability of an electron colliding is some \frac{1}{\tau} in unit time, then the number of collisions over the interval dt would be \frac{dt}{\tau}. The factor \tau is called either as the relaxation time, the mean free time, or the collision time. Clearly, this is independent of the position or velocity of the electron as we assume that, picking any electron at random, it will travel for the same time factor, \tau, before colliding. Experimentally, this has been found to be a fair approximation in a majority of cases.
  4. Collisions alone are responsible for thermal equilibrium. This one is actually fairly obvious given the first assumption: due to the independent electron approximation, energy exchange only occurs via collisions — on colliding with another particle, an electron’s post-collision energy is conveniently valued and directed in such a way as to maintain the local thermodynamic equilibrium at that point. In relation to temperature too, this makes sense: hotter regions of the metal see faster collisions and vice versa.

These ideas form the basis of the Drude–Lorentz theory of metals. They are found to be remarkably accurate assumptions used till this day to gain a good, macroscopic understanding and — to some extent — to approximately explain microscopic phenomena as well, because the quantum physical picture is understandably more complex for everyday, offhand uses.

In the next set of articles on this subject we will examine how electrical conductivity, thermal conductivity and the Hall effect are explained by the Drude model.

Cover image: Flickr/J.D. Hancock.

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V.H. Belvadi is an Assistant Professor of Physics. He teaches postgraduate courses in advanced classical mechanics, astrophysics and general relativity. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin. He currently serves as the Editor-in-Chief of Physics Capsule.

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