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Book Chapter

Lattice dynamics: Theory and application to minerals

By
Carlo M. Gramaccioli
Carlo M. Gramaccioli
Dipartimento di Scienze della Terra, Università degli Studi di Milano Via Botticelli 23, I–20133 Milano, Italy
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Published:
January 01, 2002

Abstract

Atoms are continuously moving, their displacements from the “equilibrium” position being contrasted by a “restoring” force field. Arough (although useful) approximation is that of considering such a force field to derive from resistance to altering “ideal” values of bond lengths and angles concerning the atom k in question with its neighbours k′, or also from electrostatic interaction with the other charged atoms in the molecule or – more in general – in the crystal structure.

A well-known ideal case for a molecule (see also below) occurs when the force F is proportional to the atomic displacement Δri, (i = 1, 2, 3 if the displacement is along x, y, or z), i.e., Fi = −ΦiiΔri, where Φii is a “force constant”. In this case the following equation can be written: 
formula
where m(k) is the atomic mass and t is time. More in general, if the interaction with many atoms k′ and the dependence of Fi on the other components of the shift are considered as well, we can write a summation of all these effects: 
formula
where F(k) is the force acting upon the atom k, and Φ(kk′) and Δr(k′,t) are the corresponding “force constant matrices” and “displacement vectors”, respectively. It is useful considering that the elements of Φ(kk′) are the second derivatives of the potential energy E with respect to the atomic shifts, and such derivatives are considered to be time-independent, i.e.

Φ(kk′)ij = 

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Contents

European Mineralogical Union Notes in Mineralogy

Energy Modelling in Minerals

Carlo Maria Gramaccioli
Carlo Maria Gramaccioli
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Mineralogical Society of Great Britain and Ireland
Volume
4
ISBN electronic:
9780903056397
Publication date:
January 01, 2002

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