Atomic displacement parameter values

NdOs4Sbi2 may undergo a displacive-type phase transition at -86 °C in which the Nd atoms freeze at off center positions (Evers et al., 1995). This transition was proposed on the basis of scanning calorimetry measurements and the huge room temperature value for the Nd atomic displacement parameter (Beq = 4 A2). 

Values of the e.s.d.s of parameters can be obtained, as shown in Figure 10.13, in the least-squares refinement from values of the diagonals of the inverse matrix. Similarly, any correlations between parameters, such as is often found to occur between occupancy and atomic displacement parameters, can be identified and taken into account in the description of the resulting molecular structure. The e.s.d.s for the refined parameters can then be used to calculate e.s.d.s of derived parameters, such as distances, angles, and torsion angles. ... 

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). 

The refinement of the overall B results in a significant reduction of all residuals (fourth line in Table 7.3), which is also expected since now we have a realistic global estimate of thermal motions of atoms in the crystal lattice of LaNi4,85Sno.i5. The refined value of the overall atomic displacement parameter is 5 = 1.32(2) A, where the number in parenthesis indicates a standard deviation in the last significant digit. 

At this point, we may begin to refine atomic displacement parameters of the individual atoms. This is done by substituting individual B s (which were kept at 0) by the refined overall atomic displacement parameter. The overall B after this substitution should be set to 0. The distribution of atoms in the model of the crystal structure of LaNi4,85Sno.i5 is such that two of the sites, 2(c) and 3(g), see Table 7.2, are occupied by different types of atoms simultaneously. Generally, atomic displacement parameters of different atoms occupying the same crystallographic site should be constrained at identical values. ... 

Whenever the external vibrations produee large anisotropic displacement parameter values for the scattering atoms it will exaggerate the impact of any given value of Q. The phonon wing envelope will move to even higher frequencies and the response will broaden. Only two characteristics of a sample bear on its anisotropic displacement parameter (with samples at low temperatures), the effective molecular mass, Hef[, and the Einstein frequency, see ( 2.6.2.1). The lighter the... 

Close inspection of the crystal structure (see Fig. 24) reveals much larger atomic displacement parameter (ADP) values for 02 and 03 in comparison with the other donor atoms. In addition, there is a systematic... 

The structures of the phases below 150 and 120 K were refined based on three possible symmetries—orthorhombic, monodinic, and tetragonal. The structure of the LT-V phase below 110 K may successfully be refined both in the tetragonal and monoclinic lattices with practically the same residuals (see Table 5). In spite of more adjustable parameters in the monodinic model, no improvement of the residuals was achieved. For the Fr and 03 atoms in the monodinic model, no deviation from the tetragonal s)mimetry was found within the imcertainties. Besides, physically meaningless values of atomic displacement parameters (adp)... 

The atomic scattering factors are the Fourier transforms of the spherical atomic electron distributions. They are considered as known from quantum-chemical calculations. The site occupation parameters may assume values different from unity if the structure is disordered. The Debye-Waller factors allow for the atomic thermal motions. They are functions of the atomic displacement parameters W. Omitting the atom index n and representing the Miller indices and lengths of the reciprocal lattice vectors by and a, respectively ... 

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... 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

Changes in parameters are found that increase the agreement between measured data and those calculated from the revised model that results from these changes. The positions x, y, and z and the atomic displacement (thermal) parameters B derived for each atom in a preliminary crystal structure are adjusted so as to improve the agreement between the observed structure amplitudes, F hkl) o, and those calculated from the determined model F hkl) c. The progress in improving this agreement is usually monitored by a residual index known as the R value, defined as ... 

Select the parameters xi,X2, - , Xm ore to he refined. These may be positional parameters, displacement parameters, occupancy factors. In this simplified example we consider only the positional parameters x , for the nth atom. Determine an initial value for each parameter Xj from the atomic arrangement at this stage and calculate structure factors. 

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