Reciprocal cells

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

Figure 3 (a) The reciprocating cylinder apparatus (Bio-Dis) and (b) reciprocating cell. 

Fig. 107. The systematic absences in this reciprocal lattice indicate that a larger reciprocal cell (that is, a smaller real cell) can be chosen. The new reciprocal cell is heavily outlined.

Expressions for calculating the dimensions of the real cell from those of the reciprocal cell are given in International Tables (1952), and by Buerger (1942). 

Comparison of equations 3 and 4 shows immediately how to calculate the reciprocal cell parameters from the coefficients in 4. From these, the unit cell parameters may be calculated using standard expressions (17.). The weight of each observation is calculated by... 

Because there is one lattice point per reciprocal unit cell (one-eighth of each lattice point lies within each of the eight unit-cell vertices), the number of reflections within the limiting sphere is approxiniately the number of reciprocal unit cells within this sphere. So the number N of possible reflections equals the volume of the limiting sphere divided by the volume Vrecip of one reciprocal cell. The volume of a sphere of radius r is (4ir/3)r3, and r for the limiting sphere is 2/X, so... 

The d-values of the electron diffractogram allowed us to fully index the X-ray fiber diagram using a triclinic unit cell. Without the electron diffraction reciprocal cell parameters (a, b, Y ) the task of indexing the fiber diagram would have been more difficult because of the paucity of hkO reflections on the equator. 

Figure 6. Methane reaction rate as a function of reciprocal cell temperature [4]...

Show also that, if an atom is at fractional coordinates ra, rb, rc along the direct unit cell axes ea, eb, ec and is also at fractional coordinates ra., rb., rc, along the reciprocal cell axes ea,r eb>, ec>, then the following holds ... 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

Intensity and phase of the diffraction spot at the reciprocal lattice point with indices hkl Distance 1 /d Angle 180 — a Reciprocal cell dimensions... 

Once the reciprocal cell parameters have been calculated, the direct cell is easily derived. The goodness of indexing results depends on the quality of [Qhki], and therefore on the accuracy of the peak positions. The importance of data accuracy has been emphasized by de Wolff ... 

Figure 1.3. The reciprocal cell of the bcc unit cell is actually an fee (face centered cubic) unit cell, but with smaller dimensions. In the same way is the fee reciprocal cell a bcc unit cell.

The distance between spots A is related to the reciprocal cell parameter a by the formula... 

The reciprocal cell is defined by three vectors bi, hi and bs derived from the Ui, 82 and as vectors of the direct cell, and obeying the orthonormality condition a,by = 2jt8 y. 

The reciprocal cell of a cubic cell with side length L is also a cube, with the side length 2k/L. The equivalent of a unit cell in reciprocal space is called the (first) Brillouin zone. Just as a point in real space may be described by a vector r, a point in reciprocal space may be described by a vector k. Since k has units of inverse length, it is often called a wave vector. It is also closely related to the momentum and energy, e.g. the momentum and kinetic energy of a (free) particle described by a plane wave of the form e is k and respectively. 

The reciprocal space is a very useful concept that simplifies calculations in crystallography and diffraction. Let a, b and c be the translation vectors of the unit cell. A set of vectors a, b and c, the vectors of the reciprocal cell, exists that fulfils the conditions ... 

---