Lattice intercepts

Plotting r versus 1/n gives kTJqJ as the intercept and (kTJqJ)( -y) as the slope from which and y can be determined. Figure A2.3.29 illustrates the method for lattices in one, two and tliree dimensions and compares it with mean-field theory which is independent of the dimensionality. 

Vector notation is being used here because this is the easiest way to define the unit-cell. The reason for using both unit lattice vectors and translation vectors lies in the fact that we can now specify unit-cell parameters in terms of a, b, and c (which are the intercepts of the translation vectors on the lattice). These cell parameters are very useful since they specify the actual length eind size of the unit cell, usually in A., as we shall see. 

Thus, these intercepts are given in terms of the actual unit-cell length found for the specific structure, and not the lattice itself. The Miller Indices are thus the indices of a stack of planes within the lattice. Planes are important in solids because, as we will see, they are used to locate atom positions within the lattice structure. 

The factors given in both 2.2.4. and Table 2-1 arise due to the unit-cell axes, intercepts and angles involved for a given crystal lattice structure. Also given are the lattice symbols which are generally used. The axes and angles given for each system are the restrictions on the unit cell to make... 

Step 4 Separation of Distortions of 1st and 2nd Kind. From Eq. (8.27) the graphical method for the separation of small lattice distortions of the first and the second kind is obvious. It is sketched in Fig. 8.5. In a plot of fo (r/L) vs. r/L the amount of lattice distortions of the second kind is determined from the intercept. Lattice distortions of the first kind are computed from the slope of the observed... 

The direction of planes in a lattice is described in a manner that, at first sight, seems rather strange but which is, in fact, derived directly from standard techniques in 3-D geometry. In essence, the unit cell is drawn and the plane of interest translated until it intercepts all three axes within the unit cell but as far away from the origin as possible. The point of intersection of the plane with the axes then determines the label given to the plane if the intersection takes place at a fraction (1 jh) of the a-axis, jk) of the fc-axis and (1/0 of the c-axis, then the plane is referred to as the hkl) plane . As indicated, this apparently rather strange method arises because if the axes a, A, c of the unit cell are mutually perpendicular, and of equal length, then the equation of any point x, y, z in the (hkl) plane can always be written ... 

In general, the lattice points forming a three-dimensional space lattice should be visualized as occupying various sets of parallel planes. With reference to the axes of the unit cell (Fig. 16.2), each set of planes has a particular orientation. To specify the orientation, it is customary to use the Miller indices. Those are defined in the following manner Assume that a particular plane of a given set has intercepts p, q, and r... 

Plotting Oq xct vs then leads to a straight line with an intercept equal to the Gibbs energy difference between the f.c.c. and b.c.c. forms of Cr, at the temperature where measurements were made (Fig. 6.5), while the slope of the line yields the associated regular solution interaction parameter. The lattice stability and the interaction parameter are conjugate quantities and, therefore, if a different magnitude... 

To decide which of these many vectors to use, it is usual to specify the points at which the plane intersects the three axes of the material s primitive cell or the conventional cell (either may be used). The reciprocals of these intercepts are then multiplied by a scaling factor that makes each reciprocal an integer and also makes each integer as small as possible. The resulting set of numbers is called the Miller index of the surface. For the example in Fig. 4.4, the plane intersects the z axis of the conventional cell at 1 (in units of the lattice constant) and does not intersect the x and y axes at all. The reciprocals of these intercepts are(l/oo,l/oo,l/l), and thus the surface is denoted (001). No scaling is needed for this set of indices, so the surface shown in the figure is called the (001) surface. 

In addition to cell coordinates and directions, crystal planes are very important for the determination and analysis of structure. We begin with the cell s coordinate system, with axes x, y, and z. Recall that the axes are not necessarily orthogonal and that a, b, and c are the lattice parameters. Look at Figure 1.24. The equation of an arbitrary plane with intercepts A, B, and C, relative to the lattice parameters is given by... 

Consider now any set of real lattice planes having indices hid. If one plane passes through the origin ()i the next plane JRST makes an intercept of cjl on the OZ axis. Draw a perpendicular to this plane, meeting it at N (ON = d, the spacing of the planes), and produce ON to P, where OP = Ajd. P is the reciprocal lattice point corresponding to the set of real lattice planes hid. 

N is Avogadro s number and is the Debye temperature) and whose intercept at T = 0 is y. In the Debye model, the slope has a dependency (actually, r" for an n-dimen-sional solid) owing to the lattice or phonon contribution to the heat capacity. Of course, the heat capacity normally measured is Cp, the heat capacity at constant pressure. However, for solids the difference between Cp and Cy is typically less than 1 percent at low temperatures and thus can be neglected. 

In principle, rotations around three axis (one of them 6) would be sufficient to bring any lattice plane into an orientation satislying the Bragg condition (see above) and to intercept the resulting diffracted beam with the detector. However, the fourth degree of freedom makes possible an azimuthal scan of a reftection, during which the plane remains in a reftecting position but the crystal rotates around the normal to this plane (so-called ir axis). 

For simplicity, fractional coordinates are used to describe the lattice positions in terms of crystallographic axes, o, b, and c. For instance, the fractional coordinates are (1/2,1/2,1/2) for an object perfectly in the middle of a rmit cell, midway between all three crystallographic axes. To characterize crystallographic planes, integers known as Miller indices are used. These numbers are in the format (hkl), and correspond to the interception of unit cell vectors at (alh,blk,cll). Figure 2.8 illustrates... 

FIGURE 7.9. Use of the Ewald sphere to predict which Bragg reflections will bf-observed in a particular X-ray diffraction experiment (see Figure 3.17, Chapter 3). (ai Variation in the orientation of the crystal (by oscillation or rotation about (f>). The two limits of oscillation are shown, (b) Variation in the wavelength of the radiation used, cis in a Laue photograph, with a stationary crystal. The limits for two wavelengths are shown by the two circles. In both cases all Bragg reflections in the shaded area will be observed (where the surface of the Ewald sphere intercepts reciprocal lattice points). 

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