Isotropic atomic displacement parameters

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. 

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

Table 1. Fractional coordinates, occupancies, g, and isotropic atomic displacement parameters, U, of Na CoOVvHjO (x 0.35, y 1.3).

The atomic environment within the crystal is usually far from isotropic, and the next simplest model of atomic motion (after the isotropic model just described) is one in which the atomic motion is represented by the axes of an ellipsoid this means that the displacements have to be described by six parameters (three to define the lengths of three mutually perpendicular axes describing the displacements in these directions, and three to define the orientation of these ellipsoidal axes relative to the crystal axes), rather than just one parameter, as in the isotropic case. Atomic displacement parameters, and their relationship to thermal vibrations and spatial disorder in crystals are covered in more detail in Chapter 13. 

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. 

Isotropic temperature parameter An atomic displacement parameter (q.v.) that represents an equal amplitude of vibration of an atom in all directions through a crystal. 

Individual atomic displacement parameters in isotropic approximation plus all peak shape parameters, background, zero shift, unit cell, scale 6.77 9.95 3.98 3.61... 

It is obvious that atoms in the 2(c) site have much larger atomic displacement parameters than identical atoms in the 3(g) site. This situation is quite unusual for a simple intermetallic compound and likely indicates that our assumption about a statistical distribution of Ni and Sn in both crystallographic sites was incorrect. The enhanced isotropic atomic displacement parameter in the 2(c) site points to a lower scattering ability, while the reduced atomic displacement parameter in the 3(g) site points to a higher scattering factor when compared to the current distribution of atoms. Indeed, we may speculate that only 3(g) sites contain Sn atoms, which have greater scattering ability than Ni atoms. Another possibility is that the 2(c)... 

Proceeding with the refinement of the individual atomic displacement parameters after removing Sn atoms from the 2(c) site and setting its occupancy to Nii = 0.08333 (the file Ch7Ex01d.inp is located on the CD), we find that the individual isotropic atomic displacement parameters of all atoms become much closer to one another. The resultant residuals are listed in row seven of Table 7.3. 

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. 

The overall isotropic atomic displacement parameter was assumed at 17iso= 0.015 A ... 

Refining coordinates of all atoms except Mn2 and HI and individual isotropic atomic displacement parameters results in little improvement of the... 

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. 

C(13) and C(15)) but rather two atoms before them (right before Ga(l) and Ga(2)). Also make sure that the atoms are all in the right order The similarity restraints SIMU and DELU (DELU is ignored by SHELXL if the named atoms are refined isotropically) make the atomic displacement parameters more reasonable. The critical portion of the new file, ga-03.ins, looks like this ... 

Scattering experiments using either neutron or X-ray sources can serve information on the atomic displacement parameters of atoms at specific crystallographic sites. The framework forming atoms can be considered as Debye sohd, wherein the isotropic ADPs (Ujso) are related to Gq via. 

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]. 

F ure 1.29 Relation between the isotropic atomic displacement parameter of O atoms U 0) and the diffusion coefficient of O atoms in solid solutions. ... 

The types of data included in ICSD entries contain the compound name and formula, as well as mineral names, bibliographic data, the complete structural description as published, i.e., unit cell parameters, space group symbol, atomic coordinates, and displacement parameters, isotropic or anisotropic. In addition there are remarks on experimental conditions such as single crystal or powder data, the diffracted radiation, as well as pressure and temperature, if other than standard conditions. 

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-... 

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... 

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. 

Table 2.2b Atomic Coordinates and Isotropic Displacement Parameters for si, si I, and sH Hydrates si Hydrate, CH4 5 75D2Oa ...

Table 2 Atomic coordinates and equivalent isotropic displacement parameters (Angstroms2) l for [ Cu(im idazole) ClJCl, Ueq is defined as one third of the trace of the orthogonalized lfJ tensor.

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