Vectors in reciprocal space

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. 

Here /i g(u) is the Fourier components of the image intensity function. u is the lattice vector in reciprocal space. F (u) is the Fourier spectmm of the projected potential y>,(x, y). In crystallography, F (u) is the structure factor of the crystal. 

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

This is now in the form of a vector in reciprocal space but, at this point, no particular significance is attached to the parameters h, k, and /. They are continuously variable and may assume any values, integral or nonintegral. Equation (6) now becomes... 

The solid-state equivalent of the divide-and-conquer method has been known for a long time. The natural subunits are the unit cells, which are repeated many times to form the macroscopic crystal. The wave equations need to be solved only for a single cell. The connection between the cells is made by multiplying by the factor exp (ik-R) when one goes from any point in one unit cell to the corresponding point in another unit cell. R is a vector marking the distance of the second cell from the first and k is a wave vector in reciprocal space. This... 

The Bloch vector k which may be used to label the one-electron states is conveniently viewed as a vector in reciprocal space. A lattice vector G in... 

Here U(q/c) is the amplitude which is independent of the unit cell index /, Cl) is the frequency, and q is the wave vector which is a vector in reciprocal space and designates the modes of the system. Its components q, q,) have values... 

Section 15-15 Two-Dimensional Periodicity and Vectors in Reciprocal Space... 

For a three-dimensional crystal, we label the translation vectors for a unit cell in real space ai, a2, and as. Any point r in the unit cell can then be expressed in terms of vectors a. For the vectors in reciprocal space, we use the symbols bi, b2, and bs. The position k in reciprocal space can then be expressed in terms of vectors b. Vectors a have dimensions of length, and b of reciprocal length. 

Next we consider the effects of stacking two-dimensional sheets to form the three-dimensional crystal. As Fig. 15-30a indicates, the stacking pattern repeats the orientation of a sheet after one intervening layer in an ABABAB stacking pattern. This means that the three-dimensional unit cell must contain carbon atoms from two layers. The unit cell now has three associated translation vectors. The two intrasheet translations ai and a2 are as before, and the intersheet translation as is perpendicular to the planes of the sheets and 7.0 A long. Because as is the longest vector in real space, bs is the shortest vector in reciprocal space, leading to a reciprocal lattice where sheets... 

Introducing the reciprocal lattice vectors as bi, bs, bs in place of a, b, c and a vector in reciprocal space as Ilhki = hbi + fcbs + /bs, it can be shown that the diffraction vector (S — So) in reciprocal space can be related to the reciprocal lattice vector Hhki as... 

Here the parameter rj can be chosen at will, jy is a vector in reciprocal space... 

Thus the primitive translation vector in reciprocal space, G = h A + fc B + C for a bcc direct lattice (fee reciprocal lattice) is... 

In plane waves (PW) ealeulations, the function w (r) is expanded in a set of plane waves with the same periodicity of the system, as indieated in eq. (12), where Cc are eoefficients and G are vectors in reciprocal space. 

---