Factor multiplicity

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26], 

The multiplicity factor, m, specifies the number of equivalent lattice planes that may all cause reflection at the same Bragg angle position, that is, the number of equally spaced planes cutting a unit cell in a particular, Qikl), crystalline plane family. In the case of low symmetry systems, the multiplicity factor will be low every time. On the other hand, for high symmetry systems, a single family of 

In this regard, for the (100) reflection of a powdered cubic material, some crystallites will be oriented in such a way that the (100) reflection will occur, and for others in such a position the (010) or (001) reflection will occur, but since dm = dmo = dm, these reflections contribute to the same peak [21], 

As we established earlier, a powder diffraction pattern is one-dimensional but the associated reciprocal lattice is three-dimensional. This translates into scattering from multiple reciprocal lattice vectors at identical Bragg angles. Consider two points in a reciprocal lattice, 00/ and 00/. By examining Eqs. 2.29 to 2.34 it is easy to see that in any crystal system l/c/ (00/) = l/c/ (00/). Thus, Bragg reflections from these two reciprocal lattice points will be observed at exactly the same Bragg angle. 

Now consider the orthorhombic crystal system. Simple analysis of Eq. 2.32 indicates that the following groups of reciprocal lattice points will have identical reciprocal lattice vector lengths and thus, are equivalent in terms of the corresponding Bragg angle  

Nature (London) 150, 151 (1942)], consideration of which is beyond the scope of this book. 

The correctness of the scale factor is dependent on many parameters. The most critical are the photon flux in the incident beam remains identical during measurements at any Bragg angle the volume of the material producing scattered intensity is constant the number of crystallites approaches infinity and their orientations are completely random the background is accounted precisely the absorption of x-rays (when relevant) is accounted. 

Reciprocal lattices and therefore, diffraction patterns are generally centrosymmetric regardless of whether the corresponding direct lattices are centrosymmetric or not Thus, pairs of reflections with the opposite signs of indices, hkl) and hkl) -ihe so-called Friedel pairs - usually have equal intensity. Yet, they may be different in the presence of atoms that scatter anomalously (see section 2.11.4) and this phenomenon should be taken into account when multiplicity factors are evaluated comprehensively. Relevant details associated with the effects of anomalous scattering on the multiplicity factor will be considered below in section 2.12.2. 

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The nuclear chain reaction can be modeled mathematically by considering the probable fates of a typical fast neutron released in the system. This neutron may make one or more coUisions, which result in scattering or absorption, either in fuel or nonfuel materials. If the neutron is absorbed in fuel and fission occurs, new neutrons are produced. A neutron may also escape from the core in free flight, a process called leakage. The state of the reactor can be defined by the multiplication factor, k, the net number of neutrons produced in one cycle. If k is exactly 1, the reactor is said to be critical if / < 1, it is subcritical if / > 1, it is supercritical. The neutron population and the reactor power depend on the difference between k and 1, ie, bk = k — K closely related quantity is the reactivity, p = bk jk. i the reactivity is negative, the number of neutrons declines with time if p = 0, the number remains constant if p is positive, there is a growth in population. 

The practice of estabHshing empirical equations has provided useflil information, but also exhibits some deficiencies. Eor example, a single spray parameter, such as may not be the only parameter that characterizes the performance of a spray system. The effect of cross-correlations or interactions between variables has received scant attention. Using the approach of varying one parameter at a time to develop correlations cannot completely reveal the tme physics of compHcated spray phenomena. Hence, methods employing the statistical design of experiments must be utilized to investigate multiple factors simultaneously. 

Multiple-factor methods include the cost contributions for each given activity, which can be added together to give an overall factor. This factor can be used to multiply the total cost of dehvered equipment X (Ce(j)del lo produce an estimate of the total fixed-capital investment either for grass-roots or for battery-hmit plants. The costs may be divided into four groups ... 

A multiple-factor method for predesign cost estimating has been put forward by D. H. Allen and R. C. Paffe [Chem. Eng., 82, 142-150 (Mar. 3, 1975)] for fluid-type plants (F) that include some vapor processing. The method requires the following input information ... 

Multiple-Factor Methods That Separate Materials and Labor... 

A particular strength of Equation (7) is that the intensity ratio is formed between mea-surements of the same X-ray energy in both the unknown and standard. This procedure has significant advant es First, there is no need to know the spectrometer s efficiency, a value that is very difficult to calibrate absolutely, since it appears as a multiplicative factor in both terms and therefore cancels. Second, an exact knowledge of the inner shell ionization cross section or fluorescence yields is not needed, since they also cancel in the ratio. 

In a boundary layer equation the mass center is considered with the help of the velocity (u, Uy, u ) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity v, giving a formula for Vax /x, but not for /ax - x useful approach is offered by Eq. (4.268c), using the artificial multiplication factor (v - ax /... 

Number of Slacks of a Given Height Multiplication Factor... 

We must thus multiply the Gaussian concentrations by approximately three to obtain the short-term peaks. Other sources have suggested that the multiplication factor may be higher. 

These simple values for the first few coefficients are the result of our choice of number density and temperature as multiplicative factors in the zero-order distribution function, and of defining the expansion variable in terms of the mean velocity and temperature. 

The departure of these segments from spherical is given by the terms set out in the Tab. 1 in Ch. 8. Table 1 simply gives a similar Table of the peak-to-valley departure for the inner and outer rings of segments for the aberrations using those equations (and multiplying by 2 to get P-V except for spherical aberration where the multiplication factor is 1.5). 

Multiple factors interact in complex ways to result in opioid dependence. It is difficult to delineate, even for a specific individual, the precise etiology of dependence. In addition, each of the etiologic factors discussed in this section may play variable roles in initiation of use, maintenance of use, relapse, and recovery. Keeping in mind all of these potential factors is essential when formulating a treatment plan for each individual. 

The number of active sites is a multiplicative factor in the rate of the main reaction. See for example Equations (10.11) and (10.16). Thus, the decline in reaction rate can be modeled using a time-dependent effectiveness. A reasonable functional form for the time-dependent effectiveness factor is... 

Although the redox potential of Rieske-type clusters is approximately 400 mV lower than that of Rieske clusters, it is 300 mV more positive than the redox potential of plant-type ferredoxins (approximately -400 mV). Multiple factors have been considered to be essential for the redox potential of iron sulfur proteins ... 

This large value may be experimental error. The dpm s above background for the adrenals of the three rats were 17, 29 and 90. Since the total amount of tissue was 1 ess than 20 mg, the multiplication factor may have magnified the error many fold. 

On the basis of hydrodynamic arguments first suggested by Brady, Johnson et al. [186] proposed that the hydrodynamic and steric effects that influence the diffusivity of a macromolecule in a fibrous medium can be separated into two multiplicative factors. The overall functional dependence is of the form... 

MULTIPLE FACTORS STABILIZE TERTIARY QUATERNARY STRUCTURE... 

MULTIPLE FACTORS AFFECT THE RATES OF ENZYME-CATALYZED REACTIONS... 

At first these effects of activators may appear complicated, requiring multiple factors to explain the total response, but reference to the kinetic model for channel gating (Figure 2) clarifies the complexity. Activators may produce spontaneous opening... 

Chin-A-Woeng TFC, D van den Broek, G de Voer, KMGK van der Drift, S Tuinman, JE Thomas-Oates, BJJ Lugtenberg, GV Bloemberg (2001) Phenazine-l-carboxamide production in the biocontrol strain Pseudomonas chlororaphis PC L 1391 is regulated by multiple factors secreted into the growth medium. Mol Plant-Microbe Interact 14 869-879. 

The factor /(0)//iiJm(0) ensures that the solution of the eq.(15) is independent of a multiplicative factor (if / is a solution, then A/ is also a solution for any number A) and that / is proportional to Rum when e=Ci. It turns out that no useful comparison with the molecular case can be made in the absence of this factor. 

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