Plane-spacing equations, table

The powder pattern of zinc made with Cu A a radiation (Fig. 3-13) will serve to illustrate how the pattern of a hexagonal substance is indexed. Thirteen lines were observed on this pattern their sin 0 values and relative intensities are listed in Table 10-2. A fit was obtained on a Hull-Davey chart for hexagonal close-packed lattices at an approximate cja ratio of 1.87. The chart lines disclosed the indices listed in the fourth column of the table. In the case of line 5, two chart lines (10- 3 and 11-0) almost intersect at cja = 1.87, so the observed line is evidently the sum of two lines, almost overlapping, one from the (10 3) planes and the other from (11 -0) planes. The same is true of line 11. Four lines on the chart, namely, 20 0, 10 4, 21 -0, and 20 4, do not appear on the pattern, and it must be inferred that these are too weak to be observed. On the other hand, all the observed lines are accounted for, so we may conclude that the lattice of zinc is actually hexagonal close-packed. The next step is to calculate the lattice parameters. Combination of the Bragg law and the plane-spacing equation gives... 

The particle diameter D is related to the full width at half maximum A by the Debye-Scherrer equation D = 0.9 XIA cos0, where 20 is the diffraction angle and X is the X-ray wavelength. Table 27.1 lists the particle size and lattice plane spacing calculated using the strongest (h,k,l) peak for the Fe, W, Mo carbides, nitrides, oxynitrides and oxycarbides. It is important to note that the calculated particle size using the Debye-... 

For emissions into half space, the concentrahon given by Equation 4.4 is again doubled (see Item 6 of Table 4.1). Soluhons to continuous plane sources are, as before, similar in form (Items 7 and 8), with the emission rate Mca riow expressed in units of kilogram per square meter second (kg/m s). Table 4.1 also shows that concentrahons due to instantaneous sources all contain negative exponentials (the distribution curve), while those due to continuous sources all contain error functions (area under distribution curve). 

This equation, due to Higbie, was originally derived to describe mass transfer between rising gas bubbles and a surrounding liquid Tran. AIChE, 31,368 [1935]). It applies quite generally to situations where the contact time between the phases is short and the penetration (or depletion) depth is so small that transfer may be viewed as taking place from a plant to a semiinfinite domain. In Section 4.1.2.3 we will provide a quantitative criterion for this approach, which is also referred to as the Penetration Theory. It also describes both the short- and long-term behavior in diffusion between a plane and a semi-infinite space, and we used this property in Chapter 1, Table 1.4, to help us set upper and lower bounds to mass transfer coefficients and "film" thickness Zp j. 

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