Atomic motion in solids

As in molecules, the starting point of a study of atomic motions in solid is the potential surface on which the atoms move. This potential is obtained in principle from the Born-Oppenheimer approximation (see Section 2.5). Once given, the... 

This linear problem is thus exactly soluble. On the practical level, however, one cannot carry out the diagonalization (4.11) for macroscopic systems without additional considerations, for example, by invoking the lattice periodicity as shown below. The important physical message at this point is that atomic motions in solids can be described, in the harmonic approximation, as motion of independent harmonic oscillators. It is important to note that even though we used a classical mechanics language above, what was actually done is to replace the interatomic potential by its expansion to quadratic order. Therefore, an identical independent harmonic oscillator picture holds also in the quantum regime. 

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

Once least-squares methods came into general use it became standard practice to refine not only atomic positional parameters but also the anisotropic thermal parameters or displacement parameters (ADPs), as they are now called [22]. These quantities are calculated routinely for thousands of crystal structures each year, but they do not always get the attention they merit. It is true that much of the ADP information is of poor quality, but it is also true that ADPs from reasonably careful routine analyses based on modem point-by-point or area diffractometer measurements can yield physically significant information about atomic motions in solids. We may tend to think of crystal structures as static, but in reality the molecules undergo translational and rotational vibrations about their equilibrium positions and orientations, as well as internal motions. Cruickshank taught us in 1956 how analysis of ADPs can yield information about the molecular rigid-body motion [23], and many improvements and modifications have been introduced since then. In particular, various computer programs are available to estimate the amplitudes of simple postulated types of internal molecular motion e.g., torsion-... 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

Molecular motion in solids has been the object of many studies in the field of physical chemistry of polymers , but dynamic processes in molecular crystals of organic and inorganic compounds are less well investigated. In fact, the average chemist is not aware of the fact that processes like internal rotation or ring inversion proceed in solids quite often with barriers which are not very different from those found for these types of internal motion in the liquid state. Thus, for the equatorial axial ring inversion of fluorocyclohexane values of 42.4 and 43.9 kJ mol have been measured in the liquid and the solid, respectively. The familiar thermal ellipsoids of individual atoms obtained from X-ray studies are qualitative indicators of molecular motion in the crystal, but a more quantitative study of such processes is only possible after appropriate solid state NMR techniques are applied. 

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

A. Rahman, A Comparative Study of Atomic Motions in Liquid and Solid Argon, unpublished. 

A classical dynamics model is used to investigate nuclear motion in solids due to bombardment by energetic atoms and ions. Of interest are the mechanisms of ejection and cluster formation both of elemental species such as Nin and Arn and molecular species where we have predicted intact ejection of benzene-CgHg, pyridine-Cs N, napthalene-CigHg, bipheny 1-0 2 10 an[Pg.43]

Important examples of chemical interest include particles that move in the central held on a circular orbit (V constant) particles in a hollow sphere V = 0) spherically oscillating particles (V = kr2), and an electron on a hydrogen atom (V = 1 /47re0r). The circular orbit is used to model molecular rotation, the hollow sphere to study electrons in an atomic valence state and the three-dimensional harmonic oscillator in the analysis of vibrational spectra. Constant potential in a non-central held dehnes the motion of a free particle in a rectangular potential box, used to simulate electronic motion in solids. 

Bradbury and Elliot have reported64 the infrared spectrum of crystalline NMA. Polarized spectra were run with the E vector parallel to each of the crystal axes in turn. The temperature dependence of the spectra was quite marked (see also Ref.9 ) and this was attributed, at least in part, to the previously mentioned solid phase transition. Apparently the atomic motions in NMA are of considerable complexity. The far-infrared spectrum has been reported by Itoh and Shimanouchi65. Schneider and co-workers66-68 have done an extensive study of the vibrational spectra of solid NMA and its deuterated analogues and have done complete normal coordinate analyses of these compounds69. ... 

Einstein s theory of specific heat leads to the same result. This theory connects the molecular motion in solid bodies with Planck s theory of radiation, and has been confirmed in the main by the experimental researches of Nernst and his collaborators in the last few years. Einstein assumes that the heat motion in solid bodies consists of vibrations of the atoms about a point of equihbrium, as distinct from the translational motion of the molecules which we assume for gases. The energy of these vibrations—and this is the characteristic feature of the theory, and also of Planck s theory of radiation—is always an integral multiple of a quantity of energy e, which, in turn, is the product of a universal constant (. e. a constant independent of the nature of the substance) and the frequency i/ (number of vibrations R,... 

Inelastic neutron scattering (INS) is suitable to detect librations, low-energy rotational motions in solids. It was used to follow molecular reorientations as a function of temperature [25]. These reorientations should not be confused with pseudorotation as they involve actual displacements of atoms in the crystal they correspond to an abrupt change in the crystal field [4] and their intensity scales with the crystal field strength. 

Mossbauer effect. A nuclear phenomenon discovered in 1957. Defined as the elastic (recoil-free) emission of a 7-particle by the nucleus of a radioactive isotope and the subsequent absorption (resonance scattering) of the particle by another atomic nucleus. Occurs in crystalline solids and glasses but not in liquids. Examples of y-emitting isotopes are iron-57, nickel-61, zinc-67, tin-119. The Mossbauer effect is used to obtain information on isomer shift, on vibrational properties and atomic motions in a sohd, and on location of atoms within a complex molecule. 

The situation is much the same in liquids, although the molecules are more closely spaced and the molecular interactions are stronger and more frequent. Similarly, in a solid, conduction may be attributed to atomic activity in the form of lattice vibrations. The modern view is to ascribe the energy transfer to lattice waves induced by atomic motions. In a non-conductor, the energy transfer is exclusively via these waves, in a conductor it is also due to the translational motion of the free electrons. 

Atomic Transport in Solids by A. R. Allnatt and A. B. Lidiard, Cambridge University Press, Cambridge England, 1993. This book is an exhaustive modern treatment of point defects and their motion. I turn to this book often. 

Enhanced diffusion has been found in Si. It is featured in alloys. Atomic motion in most metals and solid-solution alloys usually occurs by the interchange between atoms and neighboring vacant sites. For such a diffusion process, the atomic diffusion coefficient is given by... 

Phonons are quasiparticles, which are quantized lattice vibrations of all atoms in a solid material. Oscillating properties of the individual atoms in nonequivalent positions in a compound, however, are not necessarily equivalent. The dynamics of certain atoms in a compound influence characteristics such as the vibration of the impurity or doped atoms in metals and the rare-earth atom oscillations in filled skutterudite antimonides. Therefore, the ability to measure the element-specific phonon density of states is an advantageous feature of the method based on nuclear resonant inelastic scattering. Element-specific studies on the atomic motions in filled skutterudites have been performed (Long et al. 2005 Wille et al. 2007 Tsutsui et al. 2008). 

Sveijensky DA, Sahai N (1996) Theoretical prediction of single-site surface protonation equilibrium constants for oxides and silicates in water. Geochim Cosmochim Acta 60 3773-3797 Tasker PW (1979) Stability of ionic crystal surfaces. J Physics C Solid State Physics 12 4977-4984 Toukan K, Rahman A (1985) Molecular-dynamics study of atomic motions in water. Phys Rev B- Con Mat 31 2643-2648... 

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