Introduction to the Physics of Matter, Basic atomic, molecular, and solid-state physics Chapter 1

Introductory Concepts

Author: Eze-Odikwa Tochukwu Jed

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College Reg Number: MOUAU/CME/14/18475

1.1 Basic Ingredients

A substantial body of experimental evidence accumulated mainly through the late 19th and early 20th century eventually convinced the community of physicists and chemists that any piece of matter (e.g. pure helium gas in a vessel, a block of solid ice, a metallic screw, a mobile phone, a block of wood, a bee…) is ultimately composed of a (often huge) number of electrons and atomic nuclei.

An electron is a bona-fide elementary point-like particle characterized by a mass = m_e ≃ 0.911 × 10⁻³⁰ kg, and a negative charge = −q_e, where the elementary charge q_e ≃ 1.60 × 10−19 C.

Nuclei come with a complicated inner structure, involving length scales ≃10⁻¹⁵ m and excitation energies ≃10⁻¹³ J. These nuclear properties are largely irrelevant to the “ordinary” properties of matter: to most practical purposes one can describe matter by modeling nuclei as approximately structure less point-like particles. A nucleus containing Z protons and A nucleons (protons and neutrons alike), has positive charge = Zq_e and mass M ≃ A a.m.u. = A × 1.66 × 10⁻²⁷ kg >> m_e.

If we neglect relativistic effects and the interaction of a piece of matter with its surroundings, then all internal microscopic interactions among its components are of simple electromagnetic nature. The non-relativistic motion of nuclei and electrons in the sample is governed by the following (Hamiltonian) energy operator:

H_{tot}=T_{n}+T_{e}+V_{ne}+V_{nn}+V_{ee} (1.1)

where

T_{n}=\frac{1}{2}\sum_α\frac{P^2_{Rα}}{M_{α}} (1.2)

is the kinetic energy of the nuclei (P_Rα is the conjugate momentum to the position R_α),

T_{e}=\frac{1}{2m_{e}}\sum_iP_{ri}^{2} (1.3)

is the kinetic energy of the electrons (P_ri is the conjugate momentum to r_i),

V_{ne}=-\frac{q_{e}^{2}}{4πε_{0}}\sum_α\sum_i\frac{Z_{α}}{\begin{bmatrix}R_{α}- r_{i} \end{bmatrix}}(1.4)

is the potential energy describing the attraction between nuclei and electrons,

V_{nn}=\frac{q_{e}^{2}}{4πε_{0}}\frac{1}{2}\sum_α\sum_{β≠α}\frac{Z_{α}Z_{α}}{\begin{bmatrix}R_{α}- R_{β} \end{bmatrix}}(1.5)

describes the nucleus–nucleus repulsion, and finally

V_{ee}=\frac{q_{e}^{2}}{4πε_{0}}\frac{1}{2}\sum_i\sum_{j≠i}\frac{1}{\begin{bmatrix}R_{i}- r_{j} \end{bmatrix}}(1.6)

represents the electron–electron repulsion. Basically, the distinction between a steel key and a bottle of beer is the result of their different “ingredients”, i.e. the number of electrons and the number and types of nuclei (charge numbers Z_α and masses M_α) involved.

A state ket |ξ ⟩ containing all quantum-mechanical information describing the motion of all nuclei and electrons evolves according to Schrödinger’s equation

i\hbar\frac{d}{dt}|ξ(t) ⟩ =H_{tot}|ξ(t) ⟩(1.7)

This equation, based on Hamiltonian (1.1), is apparently simple and universal. This simplicity and universality indicates that in principle it is possible to understand the observable behavior of any isolated macroscopic object in terms of its microscopic interactions. In practice, however, exact solutions of Eq. (1.7) are available for few simple and idealized cases only. If one attempts an approximate numerical solution of Eq. (1.7), (s)he soon faces the problem that the information contents of a N-particles ket increases exponentially with N, and soon exceeds the capacity of any computer. To describe even a relatively basic material such as a pure rarefied molecular gas, or an elemental solid, nontrivial approximations to the solution of Eq. (1.7) are called for.

The application of smart approximations to Eq. (1.7) to understand observed properties and to predict new properties of material systems is a refined art. These approximations often represent important conceptual tools linking the macroscopic properties of matter to the underlying microscopic interactions. The present textbook proposes a panoramic view of several observed phenomena in the physics of matter, introducing a few standard conceptual tools for their understanding. The proposed schemes of approximation represent a pedagogical selection of rather primitive idealizations: the bibliography at the end suggests directions to expand the reader’s conceptual toolbox to approach today’s state of the art in research. We should be aware of the limitations of state-of-the-art tools: even smart and experienced physicists of matter risk to deliver inaccurate predictions of a basic property such as the electrical conductivity of a pure material of known composition and structure, before actually measuring that property. For more complex systems (e.g. biological matter), quantitative and often even qualitative predictions based on Eq. (1.7) outrun the capability of today’s modeling capability and computing power.

The motions described by Hamiltonian H_tot involve several characteristic dimensional scales, dictated by the physical constants [14] in H_tot, where the absence of the speed of light c is noteworthy. Firstly, observe that in H_tot the elementary charge q_e and electromagnetic constant ε₀ always appear in the fixed combination

e^{2}\equiv\frac{q_{e}^{2}}{4πε_{0}}=2.3071\times10^{-28}Jm

of dimensions energy × length. A unique combination of e², Planck’s constant ℏ, and electron mass m_e yields the characteristic length

a_{0}=\frac{\hbar^{2}}{m_{e}e^{2}}=0.529177\times10^{-10}m(1.8)

named Bohr radius, which sets the typical length scale of electronic motions. Most microscopic structures and patterns of matter arise naturally with spacings of the order of a0. For example, atomic positions can be probed by means of scanning microscopes. These instruments slide a very sharp tip over a solid surface: the atomic force microscope (AFM) maps the forces that the surface atoms exert as they come into contact with the tip; the scanning tunneling microscope (STM) maps an electronic tunneling current between the tip and the surface as they are kept a fraction of nm apart. This (and other) class of experiments provide consistent evidence, e.g. Fig. 1.1, that indeed in materials atoms are typically spaced by a fraction of nm, namely approximately 2 ÷ 10 times a₀.

The interaction energy of two elementary point charges at the typical distance a₀ named Hartree energy, sets a natural energy scale for phenomena involving one electron in ordinary matter. In practice, the eV (≃0.037 E_Ha) is a more commonly used energy unit. The nuclear charge factors Z_α ≤ 10² can scale the e² electron–nucleus coupling constant up by ≤10², the electron–nucleus distance down by ≥10^−2, and therefore increase the binding energies of electrons by up to 4 orders of magnitude (10⁴ E_Ha ≃ 300 keV). On the other hand, delicate balances may occasionally yield electronic excitation energies as small as 1 meV. The motions of the nuclei are usually associated to smaller energies (∼10⁻⁴ ÷ 10⁻³ E_Ha) and velocities than electronic motions, because of the at least 1,836 times larger mass at the denominator of the kinetic term of Eq. (1.2).

E_{Ha}=\frac{e^{2}}{a_{0}}=\frac{m_{e}e^{4}}{\hbar^{2}}=4.35975\times10^{-18}J=27.2114 eV(1.9)

The typical timescale of electronic motions is inversely proportional to its energy
scale: