Displacement tensor

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U"j, are given by... 

Treating the positional parameters and the elements of the thermal displacement tensor as variables, a best fit of the observed to the calculated structure factors in a least squares sense is determined. As this is a non-linear procedure, it is essential to over determine the problem. For a routine structure... 

After comparing expressions (9.76) and (9.79), we obtain the components of the recoverable displacement tensor... 

Finally, the reader should appreciate a significant difference between the way in which a and r were introduced. In Section 1.2.1 the strain tensor was defined on purely mathematical grounds, whereas the conjugate stress tensor was introduced by purely physical reasoning (i.e., force balance). However, both elastic body arc explicitly excluded. As far as rr is concerned, this is effected by introducing the displacement of mass elements (r) as a vector field and by defining stress tensor r, the symmetry property stated in Eki. (1-14) serves to eliminate rotations. 

The sum over all the individual mode tensors gives the total atomic displacement tensor, due to the internal modes, Aiat-... 

Then, using the hydrogen displacements in the x component of the bending vibration at 1235 cm", as an example, the displacement tensors and their trace values, see Eq. (2.38), are calculated. 

Table 2.3 The trace values, A, of the vibrational displacement tensors for each of the vibrations of the bifluoride ion.

The mode-independent total displacement tensor value for the hydrogen atom is ... 

The are the mean square displacements of the scattering atom, still labelled /, caused by the external vibrations of individual molecules and we have used j to label the 6Amoi different phonons. Again we can define a total displacement tensor for the atom but now related to the external vibrations. 

The INS intensity, 5 (g,fo), as calculated from the Scattering Law, Eq. (2.41), is related to the mean square atomic displacements, weighted by the incoherent scattering cross sections. What is required to calculate this quantity is the mean square atomic displacement tensor, Bi, and this can be obtained from the crystalline equivalent of L/ " ( A2.3), the normalised atomic displacements in a single molecule Eq. (4.20). This is and was introduced above, in Eq. (4.55). We have seen how... 

Following a number of manual iterations in the program to best match the intensities the external modes displacement tensor y ext is extracted, see Fig. 5.6. 

This, Eq. (5.21), is the external displacement tensor of benzene adsorbed on zeolite, as measured by diffraction. It is about twice that measured by INS, Eq. (5.18), which is a difference too great to be explained by simple experimental errors. 

The mode specific vibrational root mean squared displacement vector is thus replaced by the mean square displacement tensor of that atom in that specific mode, it has units of A. The tensor is represented in Eq. (A2.85) by a matrix, in more compact notation it is written, for a specific vibrational mode and atom Bi. The colon-operator ( ) or tensor contraction operation is found by taking the trace (Tr) of the product of the two tensors, see Eq. (2.50). Further we write the total mean square displacement tensor of an atom as the sum over all the individual vibrational contributions. 

An exponential power series is recuperated by dividing through by the first term, Eq. (A2.91). Where the total mean square displacement tensor y4 is a spheroid that deviates little from the isotropic, i.e. one that is almost a sphere, then terms linear in A will be adequate [6]. This is the almost isotropic approximation and Eq. (A2.87) becomes Eq. (2,41) for = 1. 

A, 5.18 mean square displacement tensor summed over all modes, internal and external A ... 

This gives 29% population of the n (NO) orbitals, in good agreement with other estimates. Since the 3i/-electron population is effectively only 4-5, and there is a 45-population of about 0-5, the electron density at the nucleus is much higher than usual and confers on sodium nitroprusside its unusually low chemical isomer shift. The electric field gradient and mean square displacement tensors have been completely determined in sodium nitroprusside single-crystal absorbers from the polarisation dependence of the absorption cross-section [37, 38]. The principal axes of these tensors do not coincide. 

Expanding the energy of the R ion in a crystal field in a series in the components of the displacement tensor, one can present the operator of electron-deformation interaction in the following form (Al tshuler et al. 1985, Dohm and Fulde 1975) ... 

Since the spatial configuration of a lattice with excitation of long-wave acoustical vibrations can be described by the dynamical displacement tensor u p q and dynamical displacements of sublattices w k,qj (with the index of the acoustical branch of the vibrational spectrum), which are represented as... 

The free energy of a crystal in a magnetically ordered phase or in an applied magnetic field may be presented only as an expansion in powers of displacement tensor components Uafi (see eqs. 92 and 96) the contributions of finite deformations and electron-rotation interactions to isothermal elastic constants. ... 

U is one third of the trace of the orthogonalised anisotropic displacement tensor. 

There is a rigorous theory for the extraction of rigid-body displacement tensors from ADPs (Schomaker, V. Trueblood, K. N. Acta Cryst. 1968, B24,63-76, as well as for the calculation of ADPs from lattice dynamics (Gramaccioli, C. M. Filippini, G. Lattice-dynamical evaluation of temperature factors in non-rigid molecular crystals a first application to aromatic hydrocarbons, Acta Cryst. 1983, A39, 784-791). 

Most crystallographic refinements require parameter constraints due to site symmetries. For example, the position of an atom on a mirror plane is defined by two independent coordinates only. The displacement tensor of such an atom is defined by four, rather than six, values, since one of its eigenvectors must be perpendicular to the plane. Modern refinement programs find and apply such constraints automatically for all possible site symmetries. In some space groups the origin of the coordinate system... 

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