Modern Physics Unit SS Solid State Physics
Overview: This unit gives an introduction to solid physics. It begins with a discussion of conduction and the distinctions among metals, insulators, and semiconductors. It also introduces the Maxwell-Boltzmann distribution, the equipartition theorem as applied to specific heats, and the exclusion principle and complex atoms, all of which have application to the theory of solids. A second section treats the elementary quantum theory of solids.
Prerequisite: QM5
Section I: Introduction to Solids:
Read:
Kip, Chapt. 7 - The Physics of Conductivity (printout provided), Sec. 7.1-7.4; R. Eisberg and R. Resnick Quantum Mechanics of Atoms, Solids, Nuclei and Particles (2nd Ed.) , Chapt. 13 - Solids, Conductors, and Semiconductors, Sec. 13.4.
After completing this section you should understand::
Questions:
Discussion:
The Maxwell-Boltzmann Distribution
In a classical
collection of similar particles, not all of the particles have the same energy.
In general there is a spread or
The Maxwell-Boltzmann distribution is a useful approximation to the quantum mechanical distributions under certain conditions, such as high temperature. This distribution is described in unit T5 and can be reviewed in Appendix C of Eisberg and Resnick.
The Equipartition Theorem and Classical Specific Heats
A simple but important theorem can be derived from the Maxwell-Boltzmann distribution. The equipartition theorem states that an average energy of (1/2)k_{B} T is associated with each coordinate or momentum component appearing quadratically in the energy, where k_{B} is Boltzmann's constant. The kinetic energy of an ideal gas molecule, for example, contains the momentum terms p_{x}^{2}, p_{y}^{2} and p_{z}^{2}, and the equipartition theorem thus predicts the average energy to be (3/2)k_{B} T. The total energy of a gas of N molecules is thus U = (3/2)N k_{B} T, and the specific heat C_{V} = (∂ U/∂ T)_{V} is (3/2)N k_{B} .
We can also apply the equipartition theorem to solids. A simple model of the atoms in a solid pictures them as vibrating in three dimensions as if interconnected by springs. The total energy of this ensemble of three dimensional oscillators consists of a kinetic energy term with the same squared momentum components as above, but now with an additional potential energy term containing the squares of the coordinates x^{2}, y^{2} and z^{2}. With six quadratic terms the average energy of one atom is now 6 (1/2)k_{B} T = 3 T k_{B}, giving U = 3 N k_{B} T and C_{V} = 3N k_{B}. Classically, then, the solid is expected to have a constant specific heat of 3k_{B}per particle (the Dulong-Petit law) and this is observed at high temperatures. At low temperatures, however, C_{V} -> 0 and the heat capacity of metals and insulators is observed to be about the same. In the above classical picture we would expect that the free electrons in a metal would act as an ideal gas, adding (3/2) k_{B} per free electron for the specific heat of a conductor. These contradictions can be resolved only by the quantum theory of solids.
The equipartition theorem is reviewed on page 14 of Eisberg and Resnick and its application to solids is discussed on pages 421 and 422.
The Exclusion Principle and Atoms
Solids are made up of atoms, each of which when isolated has definite discrete energy levels, certain of which are important to the collective behavior of the solid. An approximate quantum description of atoms is in terms of single electron wave functions and energy levels: each electron is assumed to move in the electrostatic potential of the nucleus and the other electrons. Since the charge of the other electrons is spread out over the atom this potential does not depend upon the radius r as 1/r. Although the states can be labelled by the same quantum numbers n, l, m_{l}, m_{s} as the levels of the hydrogen atom, the energy depends upon both n and l for complex atoms, increasing with both n and l. In the usual notation the letters s, p, d, f, g,... are associated with the angular momentum quantum numbers l = 0, 1, 2, 3, 4,... and the energy levels increase according to 1s, 2s, 2p, 3s, 3p, (4s,3d), 4p,... with the parenthesis indicating that the 4s and 3d levels are nearly the same. The energy levels of the ground state of atoms are strongly influenced by the Pauli exclusion principle, which forbids two electrons to be in the same quantum state. Thus, for example, the ground state of the neutral sodium atom, with 11 electrons, has the configuration 1s^{2}, 2s^{2}, 2p^{6}, 3s^{1}, and its chemical properties are determined mainly by the presence of a single weakly bound electron in the 3s level. More details about the structure of multielectron atoms and the periodic table can be found on pages 359 and 362 of Eisberg and Resnick.
Section II: Quantum Theory of Solids:
Questions:
How does current heat a wire from the point of view of quantum mechanics.
Eisberg and Resnick, Chapter 11: 3,7,20,23,34; Chapter 13: 2,10,11,12
Problems
Kip, Chapter7: 2C,2E,2F