Population factor

Considering a randomly chosen unit cell, each available position with a specific coordinate triplet (x , y, z ) may be only occupied by one atom (fractional population parameter g = 1) or it may be left unoccupied (g = 0). On the other hand, even very small crystals contain a nearly infinite number of unit cells (e.g. a crystal in the form of a cube with 1 pm side will contain 

10 cubic unit cells with a= 10 A) and diffraction is observed from all of the unit cells simultaneously. Hence, the resulting structure amplitude is normalized to a certain mean unit cell, which represents the distribution of atoms averaged over the entire volume of the studied sample. In the majority of compounds, each crystallite is fully ordered and the content of every unit cell may be assumed identical throughout the whole crystal, so the population factor remains imity for every atom. Occasionally, population factor(s) may be lower than one but greater than zero, and some of the common reasons of why this occurs are briefly discussed below. 

It is possible that in some of the unit cells atoms are missing. Thus, instead of a complete occupancy, the corresponding site in an average unit cell will contain only a fraction (0 g 1) of the f atom. In cases like that, it is said that the lattice has defects and g smaller than 1 reflects a fraction of the unit cells where a specific position is occupied. Obviously, a fraction of the unit cells where the same position is empty complements g to unity and it is equal to 1 -g.  

This usually means that in some of the unit cells atom j is located on one side of the mirror plane, while in the others it is positioned on the opposite side of the same plane. If this is the case, then in the absence of conventional defects, the population factors g and g are related as g = 1 -g , and g = g because of mirror symmetry. 

middle and right). When defects are present in addition to the overlap, g is no longer equal to g and two or more population factors may become independent. 

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At higher densities, the population factor in Eq. (3.65) ceases to be proportional to the collisional excitation rate, but is rather given (in a two-level approximation) by... 

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Population factors of pairs. The probability of finding a pair of molecules in the state E ) of relative motion is given by a Boltzmann factor like... 

In this equation, the energies , and / of the initial and final states, i) and I/), and the dipole moment all refer to a pair of diatomic molecules hcvij = Ef — Ei is Bohr s frequency condition. With isotropic interaction, rotation and translation may be assumed to be independent so that the rotational and translational wavefuntions, population factors, etc., factorize. Furthermore, we express the position coordinates of the pair in terms of center-of-mass and relative coordinates as this was done in Chapter 5. 

Equation 10.141 introduces K(n, T) as a Boltzmann population factor, the fraction of the population at energy level n this factor will be useful in later analysis. 

The quantum mechanical result predicts a complex temperature dependence for kobs arising from the statistical population factors in equation (28). However, in the limit that the classical approximation works reasonably well, and assuming that the temperature dependence of vet is negligible, kobs is predicted to vary with T as shown in equation (43). 

Pgl and the 584-A/N2 photoionization electron spectra.81 For the three different electronic final states, energetically accessible in the Pgl transition from the single entry channel potential curve, rather narrow unshifted individual distributions and vibrational populations very similar to those for 584-A photoionization are observed. In Fig. 31 the population factors —differently normalized bt v)—for Pgl and photoionization are compared for some systems with well-resolved vibrational lines.48,74... 

Nj (Strong Line, Including Fractional Population Factor,... 

Table 11.6. Calculated photodissociation rates R of the hypetfine sublevels of the three lowest rotational levels of the H molecular ion, for the electric light vector aligned parallel to the static magnetic field. Relative residual population factors are listed after irradiation for four dissociation time constants x...

In the formulation above, each atom of a structure model has typically nine parameters (three coordinates, one to six thermal motion parameters, plus optionally one population factor) which must be refined by means of a minimization technique since the measured intensities are affected by errors. A rule of thumb is that 5 to 10 structure factors must be observed per parameter to be refined. As we shall see in Section VI.C, this ratio... 

In all cases, considered in this section, population factor(s) could be refined even though some of them may be constrained by symmetry or other relationships. For example when g° -0 and w = 3 in Eq. 2.90, the following constraint should be in effect = 1 - g - Given the sensitivity of the... 

Fractional population factors in GSAS are treated as g s (see Eq. 7.8), while site multiplicities are automatically accounted for and cannot be changed. Site populations, however, can be refined when needed. For example, a population factor g = 0.75 for an A atom in a site with multiplicity 4 means that 75% of the site is occupied and that there are 3 A atoms in the unit cell. Obviously, the fractional population factor cannot be greater than unity or less than zero. When the refined value is out of the range 0 < g < 1, this usually points to the incorrect assignments of atom types or incorrectly located atom(s). 

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. 

The first term, Gexp is a constant characteristic of the experimental set-up used, while the remaining terms are predicted theoretically. B o), T) is a temperature function related to the Boltzman population factor n co)... 

The temperature dependences of the integrated intensities of the 6.4 meV and 1.6 meV peaks are well described in terms of the Boltzmann thermal population factors of the split ground-state levels, both for the neutron-energy gain and neutron-energy loss. On the other hand, the observed temperature dependences of the peak intensities differ qualitatively from those expected for phonons or harmonic oscillators [124]. 

From Equation 13.5 it is clear that the resonant CARS intensity varies with temperature through its dependence on the number density N as well as population factor and Raman line width Tj, which are functions of temperature. The total CARS intensity also depends on the non-resonant contribution, which in turn varies with species concentration and temperature. These features are the basis of temperafure and species concentration measurements using the CARS technique. 

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