Overall scale factor

The thermodynamic state is therefore considered equivalent to specification of the complete set of independent intensive properties 7 1 R2, Rn. The fact that state can be specified without reference to extensive properties is a direct consequence of the macroscopic character of the thermodynamic system, for once this character is established, we can safely assume that system size does not matter except as a trivial overall scale factor. For example, it is of no thermodynamic consequence whether we choose a cup-full or a bucket-full as sample size for a thermodynamic investigation of the normal boiling-point state of water, because thermodynamic properties of the two systems are trivially related. 

Although this equation is rather forbidding, it is actually a familiar equation (5.15) with the new parameters included. Equation (7.8) says that structure factor Fhk[ can be calculated (Fc) as a Fourier series containing one term for each atom j in the current model. G is an overall scale factor to put all Fcs on a convenient numerical scale. In the /th term, which describes the diffractive contribution of atom j to this particular structure factor, n- is the occupancy of atom j f- is its scattering factor, just as in Eq. (5.16) Xj,yjt and zf are its coordinates and Bj is its temperature factor. The first exponential term is the familiar Fourier description of a simple three-dimensional wave with frequencies h, k, and / in the directions x, y, and 7. The second exponential shows that the effect of Bj on the structure factor depends on the angle of the reflection [(sin 0)/X]. 

Quadrupole deformations (axial as well as non-axial) are introduced by employing the Rainwater method, where the average single-particle field oscillates at different frequencies in different directions. The three oscillator frequencies are related to (an overall scaling factor for the... 

Structure amplitudes are put on an absolute scale by a Wilson plot, which gives an overall scale factor and temperature factor. 

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. 

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. 

This is the purpose of the first free variable in SHELXL, which is also known as the overall scale factor. 

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. 

To adjust the overall scale factor, the solvent model and the position of the initial model, it may be advisable to begin a SHELXL refinement with one round of rigid body refinement. As the number of parameters is small (6 per rigid body), full matrix refinement (ke5nvord L. S.) can be used. The corresponding instmctions are ... 

The addition of the C-S-H phase to the refinement led to a considerable improvement of the fit, as represented in Figure 4.18c. The C-S-H phase was added as a so-called peaks phase. The contribution of the peaks phase is not calculated from an underlying crystal structure but is merely an ensemble of peaks with fixed intensities and peak widths. The overall scale factor of the peaks phase was refined to come to a best fit with the observed XRD scan. In a sense, the combination of peaks phases and Rietveld analysis engenders a hybrid quantification approach that joins elements from pattern decomposition and Rietveld analysis. 

Here a is the scaling factor and N the number of electrons in the system. The overall scaling factor has been incorporated so as to preserve normalization of the scaled wave function (see Exercise 4.3) ... 

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