Size of the crystals

In order to describe the inflnence of this defect on the diffraction signal, we will determine how size modifies the width of the intensity distribntion. This method, which will also be apphed to the description of the effects of microstrains, was developed by Stokes and Wilson [STO 42, STO 44] and reviewed by Warren [WAR 69]. 

4 In practice, the greater the size, the smaller the effeet on the intensity distribution. So, elearly, the limit to what size ean be measured depends on the instrument s angular resolution. With traditional laboratory diffraetometers, the limit size is commonly considered to be in the range of 100 nm. The most reeent instruments developed for synchrotron radiation source achieve angular resolutions of up to a few thousandths of a degree [MAS 03], in which case the limit size that ean be measured is in the range of a few micrometers. 

The width of a diffraction peak can be defined as the ratio between the integral of the peak and its maximum value, something called the integral breadth [LAU 26], 

We will determine the maximum value of the intensity distribution and, therefore, lay out first the relation that gives the expression of this intensity distribution. We showed in Chapter 1 that the integrated intensity of a diffraction peak for a polyciystalline sample, when taking into account all of the diffracting grains, is expressed as  

1(20) being the distribution of the intensity diffracted about the Bragg position. 

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To develop this model into a quantitative relationship between T j, and the thickness of the crystal, we begin by realizing that for the transition crystal liquid, AG is the sum of two contributions. One of these is AG , which applies to the case of a crystal of infinite (superscript °o) size the other AG arises specifically from surface (superscript s) effects which reflect the finite size of the crystal ... 

Sihca and aluminosihcate fibers that have been exposed to temperatures above 1100°C undergo partial conversion to mullite and cristobaUte (1). Cristobahte is a form of crystalline siUca that can cause siUcosis, a form of pneumoconiosis. lARC has deterrnined that cristobaUte should be classified as 2A, a probable carcinogen. The amount of cristobahte formed, the size of the crystals, and the nature of the vitreous matrix in which they are embedded are time- and temperature-dependent. Under normal use conditions, refractory ceramic fibers are exposed to a temperature gradient, thus only the hottest surfaces of the material may contain appreciable cristobahte. Manufacturers Material Safety Data Sheets (MSDS) should be consulted prior to handling RCF materials. 

Tailoring of the particle size of the crystals from industrial crystallizers is of significant importance for both product quality and downstream processing performance. The scientific design and operation of industrial crystallizers depends on a combination of thermodynamics - which determines whether crystals will form, particle formation kinetics - which determines how fast particle size distributions develop, and residence time distribution, which determines the capacity of the equipment used. Each of these aspects has been presented in Chapters 2, 3, 5 and 6. This chapter will show how they can be combined for application to the design and performance prediction of both batch and continuous crystallization. 

This is known as regime I growth. It is independent of the size of the crystal if Lp > L. When growth is argued to take place in regime /, then application of z < 1 must lead to a reasonable estimate for Lp. Notice that the introduction of regime I does not affect the concentration arguments of the previous section. 

First, on a purely physical basis, we determine when departures from a linear growth rate may be expected, that is when the crystal dimensions do not increase proportionally with time, but also depend on the size of the crystal (and maybe even other factors). Then we show how these limits relate to the possible values of i and g. 

The most characteristic and unique property of crystalline solids is however, neither the shape of their crystals nor the relative size of the crystal faces, but the angle between any pair of crystal facets. For any substance, the angle between the crystal facets is constant and invariable, regardless of the overall shape or size of the crystals. Under some circumstances a substance may form short, wide crystals, while under others, the... 

In the field of scattering a simplified version of the Fourier breadth corollary Eq. (2.44) is known as the Scherrer equation21. As a result, the inverse of the integral breadth of a peak or reflection is the size of the crystal in the direction perpendicular to the netplanes that are related to the reflection. 

In calculations the values of 8j = 82= 0.98 were used. The average size of the crystals ( rx 6.2pm) have been defined with the help of a photomicrography of the active mass. As concentration of electrolyte did not change during the reaction, then Ck= 1, C = 1. The change of solid reagents... 

Fig. 9.14 (A) Photographs of biodegradability of neat PLA and PLA-based nanocomposite recovered from compost with time. Initial size of the crystallized samples was 3 x 10 x 0.1 cm3.

The equilibrium shape of a crystal is, as described above, a polyhedron where the size of the crystal facets is inversely proportional to their surface energy, ysg. In the present section we will consider other types of interfaces as well and we will show that the interface energies determine the equilibrium morphology of interfaces in general. 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

Depending upon the size of the crystal field term //cf in comparison to these three free ion terms, different approaches can be considered to the solution of Equation (5.1) by perturbation methods ... 

In the production of materials for use in DPI products, however, the particle size of the crystallized product is normally too large. Subsequent size reduction is necessary which can significantly alter the physical nature of the material [16]. 

X-Ray Diffraction (XRD) is well known technique based on scattering of X-rays. It can be applied to semiciystalline or crystaline polymers and structural changes induced by modification. Occasionally observed shifts in the X-ray peak positions might indicate a distortion of the crystal structure due to increasing strain. The width of the X-ray reflection peaks yields information on the size of the crystals rmder investigation. 

Quantum Free-Electron Theory Constant-Potential Model, The simple quantum free-electron theory (1) is based on the electron-in-a-box model, where the box is the size of the crystal. This model assumes that (1) the positively charged ions and all other electrons (nonvalence electrons) are smeared out to give a constant background potential (a potential box having a constant interior potential), and (2) the electron cannot escape from the box boundary conditions are such that the wavefunction if/ is... 

As discussed in the previous section and summarized in Table II, a drawback of the MCFT method is that the mass accumulation in the fines removal system cannot be simulated, therefore we examined whether this mass accumulation has a notlcable effect on the process dynamics. In the simulation the fines removal is simulated with a cut sizes of 150 ]m. The fines flow rate and the recycle flow rate Q were 1.25 and. 75 liter per second. The results are sho%m in Flgi e 7 It is clear that the mass accumulation has Indeed an effect on the process dynamics. Even on the mean size of the crystals a clear shift in the response is seen. It appears that the effect is strongly dependent on the value of the recycle flow rate (not shown). The conclusion from these results is that the effects of mass accumulation in the fines system are present, and can only be neglected at low cutsizes and low fines recycle rates. 

Results for Commercial Operations The content of a-form was up to 99% and average size of the crystal was about 24-35 jum. The formation of 3-form crystal In commercial operation Induced considerable Increase of the viscosity of the suspension. The features of the semi-batch cooling crystallization process are as follows. Even if crystallization temperature is considerably lowered in order to avoid the formation of 3-form crystal, and also even if the feed solution is highly concentrated at high temperature above -SSSK, obtained crystal size is large enough to separate the solvent by centrifuge. 

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