Reciprocal lattice space

FIGURE 2.7 The position of the three top layers of the (111) cut of a face-centered cubic lattice. Top-layer atoms are indicated by black dots, second-layer atoms by triangles, and third-layer atoms by squares. 

By using the eqs. (2.9) we can determine the two angles. For simple cubic lattices the components of the vector 1 are simply given by the Miller indices. This can be used to obtain a general expression for the angles a and /8 (see exercise 2.8). 

In a rectangular coordinate system the vectors a, have the three components aix, ttiy and Thus the vector (or lattice point) can be written as 

The basis vectors are given for the three cubic lattices and the closed-packed hexagonal (CPH) in Table 2.4 (see also Fig. 2.3). 

It will also be necessary to introduce the so-called reciprocal lattice space. Since the lattice is periodic we can represent any function as, for example, the potential, which depends upon the lattice positions, by a Fourier series or a Fourier integral. 

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Figure 4.26 Reflection spacings on the film are directly proportional to reciprocal-lattice spacings, and so they are inversely proportional to unit-cell dimensions.

The inverse relationship between the crystal s real space lattice and its reciprocal lattice defines the distances between adjacent reflections along reciprocal lattice rows and columns in the diffraction pattern. Conversely, measurement of the reciprocal lattice spacings yields the unit cell parameters. Angles between the axes of the reciprocal lattice can similarly be used to determine unit cell axial angles. 

This shows that diffraction occurs when the magnitude of the scattering vector h is an integral number of reciprocal lattice spacings jd. We define a vector d ... 

For some purposes, it is convenient to multiply the length of the reciprocal axes by a constant. Thus, physics texts frequently use a reciprocal lattice spacing 27t times that given above, that is ... 

Based on the components of the vector in the reciprocal lattice space the angle 9 is found as ... 

Dw = 35 pm] it is satisfied only for Qmi = 3.8 A-1, or 2 reciprocal lattice spacings for a calcite CTR. In other words, the range of reciprocal space that is inaccessible to quantitative measurements varies strongly with the experimental conditions. 

We suppose that jR is a symmetry operator that corresponds to some proper or improper rotation, and that r is a vector in the real lattice. The vector Rr is also a vector in the real lattice since K is a symmetry operator. There are as many points in reciprocal lattice space as in the direct lattice, and each direct lattice vector corresponds to a definite vector in the reciprocal lattice. It follows that Rr corresponds to a reciprocal lattice point if r is a reciprocal lattice vector. Thus the operators R, S,. . . , that form the rotational parts of a space group are also the rotational parts of the reciprocal lattice space group. It now follows that the direct and reciprocal lattices must belong to the same crystal class, although not necessarily to the same type of translational lattice (see Eqs. 10.28-10.31). 

The crystal planes hkl in the real crystal lattice define the coordinates of points of the reciprocal lattice space, also called fe-space. A plane in the real-space maps to a point in the reciprocal space and on the contrary, so there is one-to-one correspondence between planes in the real space and points in the reciprocal space. 

This concept of the diffraction pattern as a map in spatial frequency space (reciprocal lattice space and Fourier space are other names), is somewhat abstract and mathematically complex [3-7]. It is nevertheless extremely useful, as it gives a physical insight into many facets of microscope optics. 

It is easy to see by analogy that the integral over the electron density function times the phase shift is just a complex 3D Fourier transform that transforms a periodic function in direct lattice space to reciprocal lattice space. Therefore, one may think of the diffraction pattern as a Fourier transform of the direct lattice. In principal, one should then be able to measure the intensities and locations of the reflections in the diffraction pattern, take the inverse Fourier transform, and recover the electron density function. The inverse transform is obtained by... 

Diffracted beam directions are determined by intersecting the reciprocal lattice points with the sphere of reflection. All the reciprocal lattice points lying in any one layer of the reciprocal lattice layer perpendicular to the axis of rotation intersect the sphere of the reflection in a circle. The height of the circle above the equatorial plane is proportional to the vertical reciprocal lattice spacing. By remounting the crystal successively around different axes, we can determine the complete distribution of reciprocal lattice points. Of course, one mounting is sufficient if the crystal is cubic, but two or more may be needed if the crystal has lower symmetry. 

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