...CoPlot.

``CoPlot," computer plotting software, ©1988 and 1990 by CoHort Software, Berkeley, California, USA

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...program.

emTeX, TeX processing software, written by Eberhard Mattes and available free of charge on the Internet.

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...``crystals''.

Many textbooks, including Kittel [2], Ashcroft and Mermin [1], and Yu and Cardona [6] provide introductions to crystal symmetries. We refer the reader to those books for descriptions of three-dimensional crystals. Here we will use two-dimensional examples to illustrate the concepts.

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...table.

An introduction to basic group theory is given by Yu and Cardona [6] pp. 21-43 or Harrison [5] pp. 16-20.

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...crystallographers.

See Ashcroft and Mermin [1] pp. 122-126.

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...down).

LaCuO72#72 is an antiferromagnetic insulator. High temperature superconductivity was discovered in a closely related compound, La73#73Ba2#2CuO72#72. See J.G. Bednorz and K.A. Müller, Z. Physik B 64, 189 (1986).

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...Å;

Descriptions of the hcp structure are given by Harrison [5] pp. 2-4, Ashcroft and Mermin [1] pp. 76-79, Ibach and Lüth [4] p. 23, and Kittel [2] pp. 23-24.

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...forms:

See Problem 1.6.

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...electrons.

See Ashcroft and Mermin [1] p. 410.

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...zero.

The compressibility of solids is defined as the ratio between the volume change in response to the change in pressure: 167#167. The thermal expansion is 168#168, where T is the temperature. These concepts are discussed in detail in Chapter .

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...pointlike.

Note that the Madelung sum converges very slowly, and make sure that your computer procedure works on NaCl, or other structures, where the Madelung constant is known.

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...spectrometer

See, for example, Ibach and Lüth [4] p. 45.

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...isotropic?

The phonon spectrum in Ref. [4] p. 57 is incomplete; for the (1 1 0) direction one of the acoustic modes is missing.

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...by

This Ansatz is suggested by Ibach and Lüth [4] p. 61.

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...approximation.

see Problem .

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...approximation.

see Problem .

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...HREF="node62.html#2ddos1">4.3.

This expression is a straightforward generalization of the one-dimensional dispersion formula. It is also the solution of an equation of motion, as discussed by Kittel [2] Problem 4.1, p. 122. The atomic displacements are perpendicular to the plane. Interestingly, the interaction required to obtain this result does not correspond to ``springs." Rather, it is like the bending of elastic ``rods" between the atoms.

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...280#280.

The band structure of a one-dimensional electron gas in the presence of weak, periodic potential has been discussed in problems Problems . and .

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...spin.

The Holstein-Primakoff transformation is discussed, for example, by Mattis [8] p. 154.

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...342#342.)

See Mattis [8] p. 76.

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...371#371.

This density of states is encountered in the BCS theory of superconductivity (see Tinkham [20] pp. 41-44), or in quasi-one-dimensional charge density wave systems (see Problem ).

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...materials.

The data were measured by F. Pawlek and K. Riecher, Z. Metallkunde, 47, 347 (1956) and discussed in detail by J. Friedel, Can. J. Phys., 34 1190 (1956).

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...phonons.

These modes are easily observed in Raman or neutron spectroscopy.

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...exceeds

See Problems 2.16 and 9.1.

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...system.

This Hamiltonian is used in the Stoner model of conduction electron ferromagnetism. (See Ziman [3] p. 339). In terms of the more general, wavenumber dependent potential 486#486, the second term in the Hamiltonian corresponds to 487#487. A Fourier transformation reveals that this represents an ``infinite range" interaction between the electrons.

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...system.

This Hamiltonian is equivalent to the one in Problem 9.4.

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...511#511.

See Solution .

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...impurity.

This is a Friedel oscillation, as discussed by Ziman [3] p. 159, and others.

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...532#532.

As discussed in problem , the divergence of 529#529 indicates divergent susceptibility. At these wavenumbers Kohn anomalies occur in the phonon spectrum, or a weak electron-electron interaction will cause an instability of the metallic state.

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...Hamiltonian

P.W. Anderson, Phys. Rev., 124, 41 (1961).

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...band.

The first three terms of Eq. 9.11 describe a system similar to that investigated in Problem 5.2.

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