Atomic force constants

The coefficients 0ap in (12.14), which represent the second derivatives of the potential energy with respect to the atomic displacements determined at the equilibrium points, are called atomic force constants. By definition, they have an ex-pUcit physical meaning. The coefficient 4>afi(lk I k ) is equal to the minus force which acts on the atom (Ik) in the direction a, when the other atom (Vk ) deviates per unit distance in the direction /3. The Born-von Karman model implies that all other atoms stay at their equihbrium positions. 

In recent years there has been significant progress in the quantum mechanical calculation of energies and wave functions of polyatomic molecules. Atomic force constants are related in a simple manner to molecular electron density functions (51). A significant beginning has been made in the calculation of atomic force constants (52) and their application to isotope chemistry (53). 

Atomic Force Constants. For applications of this concept to OF2, see [46, 47]. 

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-... 

The coefficients ( i) re called atomic force constants. In the harmonic... 

This equation relates the elastic constants to the atomic force constants via (3.116). Due to the occurrence of the products r r in the sum of... 

Using (3.128) and (3.116), it is possible to express the elastic constants in terms of the atomic force constants. For the case of the fee lattice with nearest-neighbour interactions, we obtain from Table F.l (putting 3 = - F,... 

It is possible to derive relations between the elastic constants and the atomic force constants for crystals with several atoms per unit cell. For a general crystal structure, the expression corresponding to (3.128) is then quite complicated [3.20]. However, if every atom is at a center of inversion, the relation (3.128) still applies but the quantities C, defined in (3.116) are now replaced by the more general expression... 

Because of anharmonicity, which we discuss in Chap.5, the elastic constants and the atomic force constants are not independent of temperature. For the elastic constants, an increase of 10% or more in going from ordinary temperatures to 0 K is fairly typical. 

In Sect.3.6, we established relations between the elastic constants and the atomic force constants (3.116,128,137). In Sect.5.2.2, we obtained the result that due to anharmonicity, the atomic force constants depend on temperature and pressure, (5.54,55). We therefore expect that the elastic con-... 

F) Restriction on Atomic Force Constants Which Follow from the Space Group Symmetry of the Crystal... 

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