Anisotropic atomic displacement parameters

Note-. Cubic space group Pniim (No. 221). Number of formula units of Lao,6Sro.4Co03 in a unit cell Z= 1. Unit-cell parameters a = b = c = 3.9496 (3) A, a = P = y = 90° unit-cell volume 61.612(9) A g, occupancy x, y, z, fractional coordinates U, isotropic atomic displacement parameters equivalent isotropic atomic displacement parameters anisotropic atomic displa-... 

High resolution (between 1.4 and 2.0 A) Automated model building with ARP/wARP should work with most phase sets. RESOLVE, which uses a template-based rather than atom-based approach, should also perform well but may be computationally more consuming. Refinement can best be carried out with REEMAC or PHENIX using isotropic ADPs since the amount of data is no longer sufficient for an anisotropic description of atomic displacement parameters. The use of TLS (Winn et ah, 2003) is highly recommended. A use of NCS restraints should be critically evaluated and in most cases the refinement can proceed without them. Double conformations of side chains should be visible and modelled. Ordered solvent can be modelled automatically. 

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. 

The crystal structure found represents an average of these two possibilities. If the anisotropic displacement parameters do not correspond to ellipsoids but to other quadratic surfaces that are not everywhere positive, the atomic displacement parameters may lose their physical significance (they become nonpositive definite). 

More often than not, the anisotropic atomic displacement parameters determined from powder diffraction data affected by preferred orientation will be incorrect (unphysical). 

A brief description of the anharmonic approximation is included here for completeness since rarely, if ever, it is possible to obtain reasonable atomic displacement parameters of this complexity from powder diffraction data the total number of atomic displacement parameters of an atom in the fourth order anharmonic approximation may reach 31 (6 anisotropic + 10 third order +15 fourth order). The major culprits preventing their determination in powder diffraction are uncertainty of the description of Bragg peak shapes, non-ideal models to account for the presence of preferred orientation, and the inadequacy of accounting for porosity. 

Individual atomic displacement parameters in anisotropic approximation plus all peak shape parameters, background, zero shift, unit cell, scale (i.e. all free variables) 6.26 9.57 3.10 3.34... 

The only remaining degree of freedom in this crystal structure is to refine the displacement parameters of all atoms in the anisotropic approximation (the presence of preferred orientation is quite imlikely since the used powder was spherical and we leave it up to the reader to verify its absence by trying to refine the texture using available experimental data). As noted in Chapter 2, special positions usually mandate certain relationships between the anisotropic atomic displacement parameters of the corresponding atoms. In the space group P6/mmm, the relevant constraints are as follows ... 

After the individual isotropic atomic displacement parameters were replaced by the properly constrained individual anisotropic displacement parameters (LHPM-Rietica uses Pij, see Eq. 2.93), the refinement converges to the residuals listed in row 8 of Table 7.3. The parameters of the fully refined structure are found in the file Ch7Ex01e.inp on the CD. 

For every atom in the model that is located on a general position in the unit cell, there are three atomic coordinates and one or six atomic displacement parameters (one for isotropic, six for anisotropic models) to be refined. In addition there is one overall scale factor per structure (osf, or the first free variable in SHELXL see Section 2.7) and possibly several additional scale factors, like tbe batch scale factors in the refinement of twirmed structures, the Flack-x parameter for non-centrosymmetric structures, one parameter for extinction, etc. In addition to the overall scale factor, SHELXL allows for up to 98 additional free variables to be refined independently. These variables can be tied to site occupancy factors (see Chapter 5) and a variety of other parameters such as interatomic distances. 

Figure 14.12a demonstrates the Rietveld refinement pattern for time-of-flight (TOF) neutron diffraction data measured at room temperature for LiFeP04. Fitting was satisfactory (/ p = 2.66%, Rf - 0.46%, 5=1.34) with accurately refined atomic positions as well as anisotropic atomic displacement parameters for all atoms under the classical harmonic oscillation model. 

Important information is included in the anisotropic atomic displacement parameters for lithium, which determine the overall anisotropy of the thermal vibration by the shape of ellipsoid. Green ellipsoids shown in Figs. 14.11a, c and 13 represent the refined lithium vibration. The preferable direction of fhennal displacement is toward the face-shared vacant tetrahedra. The expected curved one-dimensional continuous chain of lithium atoms is drawn in Fig. 14.13 and is consistent with the computational prediction by Morgan et al. [22] and Islam et al. [23]. Such anisotropic thermal vibratiOTis of lithium were further supported by the Fourier synthesis of the model-independent nuclear distribution of lithium (see Fig. 14.14). 

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]. 

Rietveld analysis of LSCF6482 was performed using the neutron diffraction data taken at 667 K in the 26 range of 20°-153° by a trigonal R3c perovskite-type structure. La and Sr atoms were placed at the special position 6a 0,0,1 /4 of the R3c symmetry. Co and Fe atoms were put at the 6b 0,0,0 site. O atom was placed at the 18e x, 0, 1/4. In a preliminary analysis, the refined occupancy factor of O atoms at the 18e site g(0) was unity within the estimated standard deviation in the Rietveld analysis Thus, the g(0) was fixed to be unity in the final refinement. Isotropic and anisotropic atomic displacement parameters were used for the cations and anions, respectively. The calculated profile agreed well with the observed one [13]. The refined crystal parameters and reliability factors are shown in Table 6.4 [13]. The averaged valence of the Co and Fe cations was estimated to be 3.4 from the occupancy factor at 667 K, which is consistent with the calculated bond valence sum (BVS) value of 3.3. Here the average value of the bond valence parameter of 1.7118 was used for the... 

For cubic Lao.6Sro.4Co03 s at 1531 K and Lao.6Sro.4Coo.sFeo.2O3 g at 1533 K, the refined anisotropic atomic displacement parameters and the... 

An isotropic extinction parameter, of type I and Lorentzian distribution (in the formalism of Becker and Coppens [16]), was also refined. The motions of the non-H atoms were described by anisotropic parameters, while those of the H atoms by isotropic B s. All these displacement parameters were included among the refinable quantities of the model, for a total of 1161 variables in a single least-squares matrix. 

One of the most popular refinement programs is the state-of-the-art package Refmac (Murshudov et ah, 1997). Refmac uses atomic parameters (xyz, B, occ) but also offers optimization of TLS and anisotropic displacement parameters. The objective function is a maximum likelihood derived residual that is available for structure factor amplitudes but can also include experimental phase information. Refmac boasts a sparse-matrix approximation to the normal matrix and also full matrix calculation. The program is extremely fast, very robust, and is capable of delivering excellent results over a wide range of resolutions. 

Munshi P, Madsen A0, Spackman MA, Larsen S, Destro R (2008) Estimated H-atom anisotropic displacement parameters a comparison between different methods and with neutron diffraction results. Acta Crystallogr A 64 465 75... 

The C60 molecules were found to be executing large amplitude reorientations at room temperature, so that large anisotropic thermal displacement factors of the C60 carbon atoms were found. The thermal displacement parameters for some of the C60 carbon atoms at room temperature are, in fact, so large that the C60 atomic coordinates may well represent only an average over one or more disordered structures involving fractional atomic occupancy. On the other hand, the TDAE N and C atomic coordinates are well-defined already at room temperature. 

Insufficient data. A good LS refinement requires at least 5, and preferably 10 reflections per refined variable. Each refined atom contributes three coordinates and 1 or 6 displacement parameters (in isotropic or anisotropic refinement, respectively) ... 

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