Interatomic distances time average

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

Before considering shape principles, we should take a look at the sizes of molecules, particularly their internuclear distances. Electron diffraction studies of molecules in the gas phase are especially useful for the determination of these distances (and the angles that the atom pairs form with one another). An interatomic distance is defined as the average internuclear distance between two atoms bonded together. Because these two atoms vibrate, the distance between them alternately lengthens and shortens in rapid succession but, if the distance is averaged for a period of time, the atoms appear to be separated by some fixed distance. A fourth generalization can be made about these distances. 

There are two interesting regimes of time evolution in the probing/detection of dynamical nonequilibrium structures. In the regime of dynamics, the time evolution of atomic positions is detected on its intrinsic timescale, i.e., femtoseconds. Short X-ray pulses - on the timescale of atomic motion - are required in order to follow the dynamics of the chemical bond. In the regime of kinetics, which has to do with the time evolution of populations - and in the context of time-resolved X-ray diffraction -the time evolution in an ensemble average of different interatomic distances or the structural determination of short-lived chemical species is considered. 

In this section, we discuss briefly a few experimental results. The experimental time resolution has so far not allowed for direct observation of atomic motion (with the exception of a few experiments in the solid phase). However, as discussed in the following, short X-ray probe pulses allow for the detection of the time evolution in an ensemble average of different interatomic distances associated with the recombination of atoms following the interaction with a solvent cage or for the structural determination of short-lived chemical species. 

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