Occupancy factor, refined

Furthermore, the analysis of the values of the occupation factors refined for V2a, V2b, and 06 point to the following relationship ... 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Calculated atomic positions lattice parameters taken from reference 25. Atom displaced from la sites to 41 sites with quarter occupancy. SOccupancy factor refined to 0.94(1) atom/site. hSite composition fixed at 0.83 Ca + 0.17 Tl. 

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

One additional set of refinements was performed to test the sensitivity of the neutron profile technique to the ordering of Si and Al, which have scattering amplitudes of 0.415 and 0.345x10 12 cm respectively.(19) When the occupation factor f (defined as the fraction of "right atoms on Si and Al sites) was allowed to vary,... 

The oxygen atom was found to be disordered at the crystallographic inversion center and therefore refined with half occupation factors on each side of the B-B bond (model A, Space Group C2/c, No. 15). This disorder is connected with the centrosymmetry in the crystal lattice, which is also strictly valid for the C(SiMe3)3 substituents. In spite of the disorder of the oxygen atom in the crystal, the bond length of the B-B bond could be determined with satisfactory precision [1.601 (7)... 

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. 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

Refining site population factors this is similar to the previous approach but is a more appropriate way of testing for the scattering power of an atom because the multiplication of the atomic number of the element, currently present on a certain site, by its fractional occupation factor results in the approximate number of electrons in the element that should occupy the given site. 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by... 

For the determination of the meso- and hetero-octahedral MDO polytypes, a complete structure refinement is necessary, because the occupancy factors of the three... 

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. 

One constraint foimd in practically every refinement is toe site occupancy factor. In toe absence of disorder it is fixed to unity, which means that the atom site is fuUy occupied (in other words toe atom is present at that site in every unit cell). For atoms disordered over two sites in toe unit cell, toe ratio of toe two site occupancy factors can be refined, but generally their sum is still constrained to unity. 

As their name suggests, free variables can be used to refine a multitude of different parameters and facilitate the formulation of constraints and restraints. The first free variable is always the overall scale factor (osf), which is used to bring the reflections in the dataset to an absolute scale. The example in Section 4.4.3 shows the effects of incorrect scaling on the refinement. Additional free variables can be linked to the site occupancy factors (sof) of groups of disordered atoms (for details see Chapter 5), but can also be related to other atomic parameters (x, y, z, sof, U, etc.) and even interatomic distances, chiral volumes, and other parameters. 

The occupancy is refined with the help of a free variable, given in the. ins file. The line which directly precedes the first atom starts with fvar and contains the overall scale factor (osf), also known as first free variable. For the refinement of a disorder, the o /should be followed by a second free variable whose value is between 0 and 1, describing the fraction of unit cells in the crystal showing the conformation described under PART 1. This means the second free variable is equivalent to the occupancy of the atoms in component one. For example a value of 0.6 for the second free variable corresponds to a ratio of 0.6 0.4, describing a 60- 0% disorder. The values of the free variables are refined, but one must guess the initial value or estimate it from the peak height in the difference Fourier map. When in doubt, 0.6 is almost always a reasonable starting value. 

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