Reciprocal space and the Ewald sphere

Drenth, J.

doi:10.1107/97809553602060000658

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 57-58 | 1 | 2 |

Section 2.1.5. Reciprocal space and the Ewald sphere

J. Drenth^a ^*

^a Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: j.drenth@chem.rug.nl

2.1.5. Reciprocal space and the Ewald sphere

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A most convenient tool in X-ray crystallography is the reciprocal lattice. Unlike real or direct space, reciprocal space is imaginary. The reciprocal lattice is a superior instrument for constructing the X-ray diffraction pattern, and it will be introduced in the following way. Remember that vector S(hkl) is perpendicular to a reflecting plane and has a length $[|{\bf S}(hkl)| = 2 \sin \theta/\lambda = 1/d (hkl)]$ (Section 2.1.4.5). This will now be applied to the boundary planes of the unit cell: the bc plane or (100), the ac plane or (010) and the ab plane or (001).

For the bc plane or (100): indices , and ; S(100) is normal to this plane and has a length . Vector S(100) will be called $[{\bf a}^{*}]$ .
For the ac plane or (010): indices , and ; S(010) is normal to this plane and has a length . Vector S(010) will be called $[{\bf b}^{*}]$ .
For the ab plane or (001): indices , and ; S(001) is normal to this plane and has a length . Vector S(001) will be called $[{\bf c}^{*}]$ .

From the definition of $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ and $[{\bf c}^{*}]$ and the Laue conditions [equation (2.1.4.7)], the following properties of the vectors $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ and $[{\bf c}^{*}]$ can be derived: $[{\bf a}^{*} \cdot {\bf a} = {\bf a} \cdot {\bf a}^{*} = {\bf a} \cdot {\bf S}(100) = h = 1.]$

Similarly $[{\bf b}^{*} \cdot {\bf b} = {\bf b} \cdot {\bf S}(010) = k = 1,]$ and $[{\bf c}^{*} \cdot {\bf c} = {\bf c} \cdot {\bf S}(001) = l = 1.]$

However, $[{\bf a}^{*} \cdot {\bf b} = 0]$ and $[{\bf a}^{*} \cdot {\bf c} = 0]$ because $[{\bf a}^{*}]$ is perpendicular to the (100) plane, which contains the b and c axes. Correspondingly, $[{\bf b}^{*} \cdot {\bf a} = {\bf b}^{*} \cdot {\bf c} = 0]$ and $[{\bf c}^{*} \cdot {\bf a} = {\bf c}^{*} \cdot {\bf b} = 0]$ .

Proposition. The endpoints of the vectors S(hkl) form the points of a lattice constructed with the unit vectors $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ and $[{\bf c}^{*}]$ .

Proof. Vector S can be split into its coordinates along the three directions $[a^{*}]$ , $[b^{*}]$ and $[c^{*}]$ : $[{\bf S} = X \cdot {\bf a}^{*} + Y \cdot {\bf b}^{*} + Z \cdot {\bf c}^{*}. \eqno(2.1.5.1)]$

Our proposition is true if X, Y and Z are whole numbers and indeed they are. Multiply equation (2.1.5.1) on the left and right side by a. $[\matrix{{\bf a} \cdot {\bf S} = &X \cdot {\bf a} \cdot {\bf a}^{*}\;\; + &Y \cdot {\bf a} \cdot {\bf b}^{*}\;\; + &Z \cdot {\bf a} \cdot {\bf c}^{*}\cr\vdots &\vdots &\vdots &\vdots\cr= h &= X \cdot 1 &= 0 &= 0.\cr}]$

The conclusion is that [X = h] , [Y = k] and [Z = l] , and, therefore, $[{\bf S} = h \cdot {\bf a}^{*} + k \cdot {\bf b}^{*} + l \cdot {\bf c}^{*}.]$

The diffraction by a crystal [equation (2.1.4.6)] is only different from zero if the Laue conditions [equation (2.1.4.7)] are satisfied. All vectors S(hkl) are vectors in reciprocal space ending in reciprocal-lattice points and not in between. Each vector S(hkl) is normal to the set of planes ( [hkl] ) in real space and has a length [1/d(hkl)] (Fig. 2.1.5.1).

Figure 2.1.5.1 | top | pdf |

A two-dimensional real unit cell is drawn together with its reciprocal unit cell. The reciprocal-lattice points are the endpoints of the vectors S(hk) [in three dimensions S(hkl)]; for instance, vector S(11) starts at O and ends at reciprocal-lattice point (11). Reproduced with permission from Drenth (1999). Copyright (1999) Springer-Verlag.

The reciprocal-lattice concept is most useful in constructing the directions of diffraction. The procedure is as follows:

Step 1: Draw the vector $[{\bf s}_{o}]$ indicating the direction of the incident beam from a point M to the origin, O, of the reciprocal lattice. As in Section 2.1.4.2, the length of $[{\bf s}_{o}]$ and thus the distance MO is $[1/\lambda]$ (Fig. 2.1.5.2).

Figure 2.1.5.2 | top | pdf |

The circle is, in fact, a sphere with radius $[1/\lambda]$ . $[{\bf s}_{O}]$ indicates the direction of the incident beam and has a length $[1/\lambda]$ . The diffracted beam is indicated by vector s, which also has a length $[1/\lambda]$ . Only reciprocal-lattice points on the surface of the sphere are in a reflecting position. Reproduced with permission from Drenth (1999). Copyright (1999) Springer-Verlag.

Step 2: Construct a sphere with radius $[1/\lambda]$ and centre M. The sphere is called the Ewald sphere. The scattering object is thought to be placed at M.
Step 3: Move a reciprocal-lattice point P to the surface of the sphere. Reflection occurs with $[{\bf s} = {\bf MP}]$ as the reflected beam, but only if the reciprocal-lattice point P is on the surface of the sphere, because only then does $[{\bf S}(hkl) = {\bf s} - {\bf s}_{o}]$ (Section 2.1.4.2). Noncrystalline objects scatter differently. Their scattered waves are not restricted to reciprocal-lattice points passing through the Ewald sphere. They scatter in all directions.

References

International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 57-58