Force constants three-dimensional crystals

For three-dimensional crystals the lattice summation converges only slowly, and in any brute-force computational scheme one must make sure that, as one sums outward from the "zeroth unit cell" at the center of the crystal, the ions included at any stage should have as close to zero net charge as possible. For some crystals the Madelung constants a have been evaluated (Table 8.4), using component potentials obtained by summing certain infinite series. 

In Chap.3, the theory is generalized to three-dimensional crystals with a basis. Chapter 3 also contains a section dealing with the connection of lattice dynamics and the theory of elasticity. The force constants, dynamical matrix and dispersion relation are illustrated with the help of monoatomic crystals with fee structure. 

We have already evaluated the Griineisen parameter y for the linear mono-atomic chain with nearest neighbour interactions, (5.60), with the result that y is proportional to the enharmonic force constant g of the cubic term in the potential energy (5.31,32). If for a general three-dimensional crystal,... 

Sommerfeld suggested that the potential in a metal crystal could be assumed constant. This assumption implies that the forces acting on an electron cancel to zero and that the electrons in a metal can be described like a non-interacting gas of electrons, confined to a box that represents the metal. The only restriction on electronic motion would be the Pauli principle. The electronic energy in a three-dimensional rectangular box is known as... 

We start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

Although much work has been done on the three reference substances described here, there are still needs for refinement of heat capacity measurements and frequency spectra fit. Most emjdiasis has been placed on obtaining a picture of the vibrations in the ideal crystal lattice. The questions about vibrations in small and defect crystals has at pre nt only been opened. Historically, the universal Dulong Petit-Rule of heat capacity which fits only well at elevated temperatures was replaced between 1900 and 1920 mainly throi h the work of Einstein, Debye. Nemst, and Lindemaim by a theory with one characteristic constant for each substance, the 0-temperature. The Tarasov treatment of the 1950 s shows that in case of strong anisotropy of forces, as in one or two dimensionally strongly bonded crystals, a second constant... 

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