Atomic and Molecular Displacements

This predicted intensity falloff was observed experimentally by W. H. Bragg. He studied rock salt at two temperatures, 15 and 37°C, and found that the intensities of Bragg reflections, particularly those at high 20-values, were smaller at the higher temperature. This effect was described in Chapter 7, where it was shown that the falloff in intensity as a function of scattering angle (20) is routinely used to obtain an average B value for the entire crystal structure by way of a Wilson plot.  

Atomic vibrations are displacements from equilibrium positions, with frequencies of such vibration typically of the order of 10 per second. The frequencies of X rays (velocity of light/wavelength of X rays) are much faster, of the order of (3 x 10 cm/sec)/(1.5 x 10 cm) = 2 x 10 per second. As a result, the atom may vibrate and be viewed by X rays 

Because the diffraction experiment involves the average of a very large number of unit cells (of the order of 10 in a crystal used for X-ray diffraction analysis), minor static displacements of atoms closely simulate the effects of vibrations on the scattering power of the average atom. In addition, if an atom moves from one disordered position to another, it will be frozen in time during the X-ray diffraction experiment. This means that atomic motion and spatial disorder are difficult to separate from each other by simple experimental measurements of intensity falloff as a function of sm6/X. For this reason, atomic displacement parameter is considered a more suitable term than the terms that have been used historically, such as temperature factor, thermal parameter, or vibration parameter for each of the correction factors included in the structure factor equation. A displacement parameter may be isotropic (with equal displacements in all directions) or anisotropic (with different values in different directions in the crystal). 

Now that fairly precise measures of electron density can be made, atomic displacement parameters can be refined so that the best possible fit to the experimental electron-density profiles of each atom is obtained. This is done by the introduction of additional atomic parameters, one parameter if the displacements are isotropic, six if they are anisotropic. When this least-squares refinement of displacement parameters is completed, the crystallographer is then left with the problem of explaining the atomic displacement parameters so obtained in terms of vibration, static disorder, dynamic disorder, or a combination of these. 

FIGURE 13.2. Diagrams of potential energy curves for an atomic position showing (a) perfect order, (b) static order, (c) dynamic disorder, and (d) a mixture of static and dynamic disorder of the position of one atom. The percentage occupancy of each site is listed. Vertical axes = energy, horizontal axes = position. 

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Atomic and molecular displacement under constraint. Thermal expansion and compressibility are large and anisotropic. Sometimes structural data have been extrapolated from the room temperature (RT) down to low temperature (LT) simply by considering changes in lattice dimensions. This has led to disappointing results since, even in the absence of a phase transition, molecular shapes and orientations may change substantially. Similarly, if we find an isostatic pressure at room temperature whose effect is equivalent to a given temperature decrease at ambient pressure for, say, the chain contraction, the equivalence will not usually match for, say, the... 

Although the pulse sequences used to study phase transitions are usually quite simple in the examples presented in this review (one to maximum four pulses), the interpretation may be subtle. Solid-state NMR nevertheless remains a difficult technique since quantitative interpretation of the spectra rely on a profound knowledge of the chemical composition and structure of the sample analysis of NMR results also requires a model to relate the observed NMR spectral shapes or relaxation behavior to hypothesis concerning the structure and dynamics of the atoms or molecules carrying spins. That NMR motionally average the atomic and molecular displacements that occur on a time-scale faster than 10—8 10—9s is an important point that should be considered in the interpretation of data. In particular, the difference in perception between NMR and X-ray diffraction with regard to fast and slow dynamical disorder in molecular crystals undergoing phase transitions between different polymorphs was illustrated. In fact, the interpretation of NMR data almost always needs the support of other data obtained by different techniques. Therefore, we emphasized the different complementarities with X-ray (or neutron) diffraction, IQNS and other spectroscopic methods to provide, by cross-correlation of the different data, consistent picture of the phase transition. 

Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

A minimum of potential energy such as that at r, in Fig. X-2, separated by a maximum from a still lower region, is often met in atomic and molecular problems and is called a position of metastable equilibrium, or a metastable state. It is stable as far as small displacements are concerned, but a large displacement can push the system over the maximum, after which it does not return to the original position but to the entirely different configuration of really lowest potential energy. In all such... 

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

The reason is that such a PES contour plot describes a dissociative chemisorption process for a particular location and orientation of the molecule above a particular geometry of the surface atoms. In real systems, all of the PES features will vary with orientation of the molecule f, position of the molecule in the surface unit cell (X, T), and displacement of the surface atoms (Yj —These other variables may complicate the problem tremendously, and may even control the particular dynamical process. For example, the atomically and molecularly chemisorbed species may be stable at different positions in the surface unit cell. Spectroscopies probes of these species would then determine two completely different regions of the PES without any means for connecting them. 

At the atomic and molecular levels, some of the charge displacements will reach saturation at high E-field strength levels (Section 8.4.1). The alignment of dipoles in a polar dielectric will reach a maximum when the field energy is of the same order of magnitude as the Boltzmann factor kT. 

The dielectric polarization, or simply polarization, within dielectric materials is a vector physical quantity, denoted by P, and its module is expressed in C.m Electric polarization arises due to the existence of atomic and molecular forces and appears whenever electric charges in a material are displaced with respect to one another under the influence of an apphed external electric field strength, E. On the other hand, the electric polarization represents the total electric dipole moment contained per unit volume of the material averaged over the volume of a crystal cell lattice, V, expressed in cubic meters (m ) ... 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

The force microscope is also well suited for atomic and molecular manipulation as it allows the measurement and control of forces involved in the manipulation process. In fact, the force needed to move a Co atom or a CO molecule across a Cu(lll) surface has been quantified in a combined NC-AFM/STM experiment [238]. This experiment and other NC-AFM manipulation experiments have initially been performed at cryogenic temperatures in analogy to procedures known from STM manipulation. However, sophisticated experimental methods of atom tracking and feed-forward techniques also allow imaging, manipulation, and spectroscopy with atomic precision at room temperature [239-242]. Controlled vertical manipulation has been demonstrated by displacement of individual silicon atoms on a Si(lll)7x7 surface by soft nanoindentation [243] and lateral manipulation for adsorbates on a Ge(lll)-c(2x8) surface [244]. The concept of lateral manipulation has further been developed to create atomic structures on semiconductor surfaces at room temperature by using sophisticated manipulation protocols [245, 246]. Room-temperature, atomic-scale manipulation has also been achieved on insulating surfaces [247, 248] however, the processes involved are more complicated and the degree of control is lower in this case. 

Since the index of refraction is known to be a function of wavenumber, the dielectric constant, e, also depends on wavenumber. This follows from the atomic and molecular composition of matter. The material consists of molecules or atoms in which the charges are bound, and therefore acts as a collection of oscillators. Consider the case of charges in the molecules and atoms subjected to polarized radiation traveling in the x-direction with electric field E = Eq e Each molecule experiences a force due to the radiation field of the form F = gEo cos oat, where q is the electric charge. This force causes small displacements, r, of the charges, and thereby gives rise to a polarization of the material. The total polarization, P, is the volume sum of all individual dipole moments p = qi r generated by the field. 

The time dependence is in this case due to the oscillation of the 4tn electric field as given by Eq. (80), as well as the displacement of the charged particles, electrons and nuclei within the atomic or molecular system. 

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