Vibration of atoms in crystals

In addition to the dynamic disorder caused by temperature-dependent vibration of atoms, protein crystals have static disorder due to the fact that molecules, or parts of molecules, do not occupy exactly the same position or do not have exactly the same orientation in the crystal unit cell. However, unless data are collected at different temperatures, one cannot distinguish between dynamic and static disorder. Because of protein crystal disorder, the diffraction pattern fades away at some diffraction angle 0max. The corresponding lattice distance <7mm is determined by Bragg s law as shown in equation 3.7 ... 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

Heat can be conducted through solids, liquids, and gases. Conduction in solids is the most illustrative since it is the most common heat transfer mechanism in that type of medium. Conduction is the energy transfer between adjacent molecules or atomic particles at motion. The nature of the motion depends on the system and on the molecular and particle state. The motion can range from vibration of atoms in a crystal lattice of solids to the chaotic fluctuations of gas molecules. In metallic solids, movement of free electrons contributes to heat conduction. 

It is generally agreed that thermally induced vibrations of atoms in solids play a major role in melting [2.144]. The simple vibrational model of Linde-mann predicts a lattice instability when the root-mean-square amplitude of the thermal vibrations reaches a certain fraction / of the next neighbor distances. However, the Lindemann constant/varies considerably for different substances because lattice anharmonicity and soft modes are not considered, thus limiting the predictive power of such a law. Furthermore, Born proposed the collapse of the crystal lattice to occur when one of the effective elastic shear moduli vanishes [2.138], Experimentally, it is found instead that the shear modulus as a function of dilatation is not reduced to zero at Tm and would vanish at temperatures far above Tm for a wide range of different substances [2.145]... 

Finally, there are possible also vibrations of the atoms within the molecule, which are apparently to be treated just like the vibrations of atoms in the crystal. These explain the occurrence of molecular heats higher than R for diatomic, or than for polyatomic gases. 

Some interesting and important conclusions were drawn by separating the phonon spectrum in accordance with the polarization of the oscillations [15]. The whole spectrum was divided into six branches, each of which has an almost Gaussian form of the distribution curve g( ). For cubic crystals, these six branches consist of three acoustical branches (one branch of longitudinal and two branches of transverse waves) and three optical branches (one longitudinal and two transverse waves). The acoustical vibrations can be compared with the vibrations of atoms in a unit cell, and the optical vibrations with mutual oscillations of the sublattices in relation to one another. The curves of the density distribution of oscillations in each [Pg.180]

Heat is essentially the vibration of atoms in a material. Consequently thermal properties reflect the type and strength of interatomic bonding and the crystal structure. The important thermal properties of any material are... 

Obviously, thermal vibrations of atoms in a solid are strongest on the verge of melting. Sutherland was the first (1891) to suggest that melting occurs when the amplitude of vibrations reaches a certain fraction (equal for all the elements) of the atomic size [13]. In 1910, Lindemann [14] developed this idea and related the critical amplitude to the temperature of melting (Tm) and atomic oscillation frequency v proportional to the characteristic Debye temperature ( ). In its modern form [15] the Lindemann s rule states that a material melts at the temperature at which the amplitude of thermal vibration exceeds a certain critical fraction of the interatomic distance, and this fraction depends somewhat on the crystal structure, position in the Periodic Table, and perhaps other unspecified physical quantities. These works initiated numerous... 

The thermal vibration of atoms in the crystal lattice is strongly temperature dependent and is less effective in assisting dislocation motion at low temperatures. The interaction of dislocations with thermal vibrations is complicated, but it is nonetheless satisfying to find that ductility usually decreases somewhat with a decrease in temperature. 

Fig.2.9. Oscillator and phonon description of the vibrations of atoms in a crystal. The index s stands for (qj), where q is the wave number and j specifies the branch...

The main method of crystal stmctnre determination is X-ray diffraction analysis (refer to Section 6.3.5). Experiments that nse small single crystals and/or polycrystalline samples allow one to determine the mutual location of atoms in crystals of rather complex chemical componnds (for instance, a thonsand or more atoms in the unit cell) together with the nature and parameters of atomic thermal vibrations (their root mean square displacement values). For investigation of special qnestions, methods of neutron and electron diffraction are also used (refer to Chapter 7.2). 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

The vibrational motion of atoms in diatomic molecules and, by extension, in crystals cannot be fully assimilated to harmonic oscillators, because the potential well is asymmetric with respect to Xq. This asymmetry is due to the fact that the short-range repulsive potential increases exponentially with the decrease of interionic distances, while coulombic terms vary with 1/Z (see, for instance, figures 1.13 and 3.2). To simulate adequately the asymmetry of the potential well, empirical asymmetry terms such as the Morse potential are introduced ... 

The extension of the quantum-mechanic interpretation of the vibrational motion of atoms to a crystal lattice is obtained by extrapolating the properties of the diatomic molecule. In this case there are 3 ( independent harmonic oscillators (9l is here the number of atoms in the primitive unit cell—e.g., fayalite has four... 

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