Section 7.4.2.2.2. Calculation of α

Inserting (7.4.2.8) into (7.4.2.6) gives the TDS correction factor as where T_mn, an element of a 3 × 3 symmetric matrix T, is defined by Equation (7.4.2.9) can also be written in the matrix form with representing the transpose of H.

The components of H relate to orthonormal axes, whereas it is more convenient to express them in terms of Miller indices hkl and the axes of the reciprocal lattice. If S is the 3 × 3 matrix that transforms the scattering vector H from orthonormal axes to reciprocal-lattice axes, then where h^T = (h, k, l) . The final expression for α, from (7.4.2.11) and (7.4.2.12), is This is the basic formula for the TDS correction factor.

We have assumed that the entire one-phonon TDS under the Bragg peak contributes to the measured integrated intensity, whereas some of it is removed in the background subtraction. This portion can be calculated by taking the range of integration in (7.4.2.10) as that corresponding to the region of reciprocal space covered in the background measurement.

To evaluate T requires the integration of the function A⁻¹ over the scanned region in reciprocal space (see Fig. 7.4.2.2). Both the function itself and the scanned region are anisotropic about the reciprocal-lattice point, and so the TDS correction is anisotropic too, i.e. it depends on the direction of the diffraction vector as well as on .

Figure 7.4.2.2| top | pdf | Diagrams in reciprocal space illustrating the volume abcd swept out for (a) an ω scan, and (b) a θ/2θ, or ω/2θ, scan. The dimension of ab is determined by the aperture of the detector and of bc by the rocking angle of the crystal.

Computer programs for calculating the anisotropic TDS correction for crystals of any symmetry have been written by Rouse & Cooper (1969), Stevens (1974), Merisalo & Kurittu (1978), Helmholdt, Braam & Vos (1983), and Sakata, Stevenson & Harada (1983). To simplify the calculation, further approxi­mations can be made, either by removing the anisotropy associated with A⁻¹ or that associated with the scanned region. In the first case, the element T_mn is expressed as where the angle brackets indicate the average value over all directions. In the second case, where is the radius of the sphere that replaces the anisotropic region (Fig. 7.4.2.2) actually scanned in the experiment, and dS is a surface element of this sphere. q_m can be estimated by equating the volume of the sphere to the volume swept out in the scan.

If both approximations are employed, the correction factor is isotropic and reduces to with v_L representing the mean velocity of the elastic waves, averaged over all directions of propagation and of polarization.

Experimental values of α have been measured for several crystals by γ-ray diffraction of Mössbauer radiation (Krec & Steiner, 1984). In general, there is good agreement between these values and those calculated by the numerical methods, which take into account anisotropy of the TDS. The correction factors calculated analytically from (7.4.2.14) are less satisfactory.

The principal effect of not correcting for TDS is to underestimate the values of the atomic displacement parameters. Writing , we see from (7.4.2.14) that the overall displacement factor is increased from B to B + ΔB when the correction is made. ΔB is given by Typically, ΔB/B is 10–20%. Smaller errors occur in other parameters, but, for accurate studies of charge densities or bonding effects, a TDS correction of all integrated intensities is advisable (Helmholdt & Vos, 1977; Stevenson & Harada, 1983).