Lattices interplanar spacings

Now envision this lattice of imaginary points in the same space occupied by the crystal. For a small real unit cell, interplanar spacings dhkl are small, and hence the lines from the origin to the reciprocal lattice points are long. Therefore, the reciprocal unit cell is large, and lattice points are widely spaced. On the other hand, if the real unit cell is large, the reciprocal unit cell is small, and reciprocal space is densely populated with reciprocal lattice points. 

The reciprocal lattice is spatially linked to the crystal because of the way the lattice points are defined, so if we rotate the crystal, the reciprocal lattice rotates with it. So now when you think of a crystal, and imagine the many identical unit cells stretching out in all directions (real space), imagine also a lattice of points in reciprocal space, points whose lattice spacing is inversely proportional to the interplanar spacings within the crystal. 

Note that any mixing can hardly take place at either of the interfaces between reacting solid phases. Therefore, the basic assumptions of R. Pretorius et al.261,262 about a limiting element and its concentration at an interface seem to be somewhat artificial. In fact, in the reactions under consideration, there is no reaction volume, with all the interactions proceeding onto the phase surfaces exposed to each other. The distances between those surfaces scarcely exceed considerably the usual values of interplanar spacings in the crystal lattices of chemical compounds. Hence, the concept concentration of a limiting element at an interface does not appear to have any real physical meaning. 

Relations between Interplanar Spacing, Miller Indexes, and Lattice Parameters... 

In the Bragg model, it is assumed the electron density is in the lattice planes, that is, the atoms coincide with the lattice points. This may or may not be the case. Nonetheless, a simple formula may be deduced relating the angle that the incident and reflected rays make with a given family of parallel lattice planes of interplanar spacing, dhu-... 

In calculating the interplanar spacing, or perpendicular distance between adjacent planes of given indices, dku, in the direct lattice (whether or not these planes coincide with lattice points), it is helpful to consider the reciprocal lattice, which defines a crystal in terms of the vectors that are the normals to sets of planes in the direct lattice and whose lengths are the inverse of dku- The relationship between the interplanar spacing and the magnitude of the reciprocal lattice vectors, a, b, c, is given by ... 

It may be recalled that an alternative description for a crystal stmcture can be made in terms of sets of lattice planes, which intersect the unit cell axes at ua, VU2, and was-The reciprocals of the coefficients are transformed to the smallest three integers having the same ratios, h, k, and I, which are used to denote the plane (hkl). Of course, the lattice planes may or may not coincide with the layers of atoms. Any such set of planes is completely specihed by the interplanar spacing, dhU7 and the unit vector normal to the set, since the former is given by the projection of, for example, u ui onto n kh that is dhki = u ui- n ki- The reciprocal lattice vector is defined as ... 

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