Particle statistics

For quantitative phase analysis it is generally accepted that the peak intensities need to be measured to an accuracy of about 1 — 2% relative. The ability to achieve this is strongly influenced by the size of the crystallites in the sample reproducible diffraction intensities require a small crystallite size to ensure that there is uniform intensity around the Debye-Scherrer cone. 

Elton and Salt have used both theoretical and experimental methods to estimate the number of crystallites diffracting (Adiff) in a sample. Fluctuations in line intensity between replicate samples arise largely from statistical variation in the number of particles contributing to the diffraction process. It has been shown that small changes to the instrumental and sample configurations can 

For a given sample, several methods can be used to increase the number of crystallites contributing to the diffraction pattern, including  

The basis for calculation of powder diffraction intensities relies on the sample being a randomly orientated powder that is, all reflections have an equal 

Notably, the correction algorithms are only approximations and may not adequately correct for extreme preferred orientation. In this case it may be better to eliminate, or at least minimize, preferred orientation before data collection begins through appropriate selection of sample packing technique or instrument geometry.  

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For non-mteracting particles in a box, the result depends on the particle statistics Fenni, Bose of Boltzmamr. The state of a quanPim system can be specified by the wavefrmction for that state, Tv(Qi> Q2 . qyy). is the vth eigensolution to the Scln-ddinger equation for an A -particle system. If the particles are noninteracting, then the wavefrmction can be expressed in temis of the single-particle wavefrinctions given... 

G. Herdan, Small Particle Statistics , Butterworths, London (1960). 

Second Derivation of the Boltzmann Equation.—The derivation of the Boltzmann equation given in the first sections of this chapter suffers from the obvious defect that it is in no way connected with the fundamental law of statistical mechanics, i.e., LiouviUe s equation. As discussed in Section 12.6of The Mathematics of Physics and Chemistry, 2nd Ed.,22 the behavior of all systems of particles should be compatible with this equation, and, thus, one should be able to derive the Boltzmann equation from it. This has been avoided in the previous derivation by implicitly making statistical assumptions about the behavior of colliding particles that the number of collisions between particles of velocities v1 and v2 is taken proportional to /(v.i)/(v2) implies that there has been no previous relation between the particles (statistical independence) before collision. As noted previously, in a... 

As X 00, Batchelor (1950) has shown that the variance of the instantaneous relative concentration distribution must become independent of two-particle statistics. As a consequence, and o-,. Consequently, for all X, the power law... 

Any of the four conditions has an infinity of solutions. Actually, the energy is stationary for any eigenstate of the Hamiltonian, so one has to specify in which state one is interested. This will usually be done at the iteration start. Moreover, the stationarity conditions do not discriminate between pure states and ensemble states. The stationarity conditions are even independent of the particle statistics. One must hence explicitly take care that one describes an n-fermion state. The hope that by means of the CSE or one of the other sets of conditions the n-representability problem is automatically circumvented has, unfortunately, been premature. 

Herdan, G., Small Particle Statistics, pp. 84-85, Academic Press, New... 

Another image-processing method that is especially applicable to many pharmaceutical sample images is particle statistics analysis. This technique works well when the individual... 

Figure 2.9 The figure is a binary image in which the caffeine domains created using particle statistics are visualized. Other sample components are set to zero, so only the caffeine domains are apparent.

Once collected, particles can be sized by a variety of means. Optical and electron microscopy are probably best known and are quite reliable. Yet they involve tedious scanning of many samples to obtain sufficient counts to provide meaningful particle statistics. Microscopic techniques are suitable for solid particles and for nonvolatile liquids. Volatility creates a significant uncertainty unless the particles are trapped in a substrate that reveals a signal of the impacted particle. 

We first review the fundamentals of small particle statistics as these apply to synthetic polymers. This is mainly concerned with the use of statistical moments to characterize molecular weight distributions. One of the characteristics of such a distribution is its central tendency, or average, and the following main topic shows how it is possible to determine various of these averages from measure-... 

Various molecular weight averages are current in polymer science. We show here that these are simply arithmetic means of molecular weight distributions. It Tiiay be mentioned in passing that the concepts of small particle statistics that are discussed here apply also to other systems, such as soils, emulsions, and carbon black, in which any sample contains a distribution of elements with different sizes. 

Herden, G. (I960), Small particle statistics, Butterworths, 38 Deleuil, M. (1994), Handbook of powder technology, No 9, Powder Technology and Pharmaceutical Processes, Ch. 1, ed. D. Chulia, M. 

Herein, G. Small Particle Statistics, London Academic Press 1960... 

Up to now we have tacitly ignored the particle statistics and derived the formalism as if the particles were distinguishable. As far as the Hamiltonian... 

The fundamental relationships between diffracted peak intensity in a powder diffraction pattern and the quantity of phase in a mixture producing that peak are well established. However, many factors impinge upon these relationships. These are generally experimental and arise from sample and instrument related effects. Such factors include counting errors, particle statistics, preferred orientation, microabsorption and, the most hazardous of all, operator error. 

Table 11.1 summarizes the work of Smith into the effect of particle statistics in a static sample. This work showed that a powder of particle size 40 pm may have as few as 12 crystallites diffracting in a sample volume of 20mm. This results in a value of aps of about 0.3, which is insufficient for reproducible pattern statistics. Reduction of the particle size to 1 pm reduces Gps to a more acceptable value of 0.005. Notably, the number of diffracting... 

Reduction of particle size. Further discussion can be found Section 11.4.1 (Particle Statistics) and in Buhrke et al ... 

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