Reciprocal Space

1. Reciprocal Space Learning outcomes By the end of this section you should: • understand the derivation of the reciprocal lattice • be able to derive the Laue condition for the reciprocal lattice • understand how reciprocal space relates to the diffraction experiment • be able to use reciprocal space to make calculations

2. The reciprocal lattice • A diffraction pattern is not a direct representation of the crystal lattice • The diffraction pattern is a representation of the reciprocal lattice We have already considered some reciprocal features - Miller indices were derived as the reciprocal (or inverse) of unit cell intercepts.

3. Reciprocal Lattice vectors Any set of planes can be defined by: (1) their orientation in the crystal (hkl) (2) their d-spacing The orientation of a plane is defined by the direction of a normal (vector product)

4. Defining the reciprocal Take two sets of planes: Draw directions normal: These lines define the orientation but not the length G1 G2 These are called reciprocal lattice vectors G1 and G2 Dimensions = 1/length

5. Reciprocal Lattice/Unit Cells We will use a monoclinic unit cell to avoid orthogonal axes We define a plane and consider some lattice planes (001) (100) (002) (101) (101) (102)

6. Reciprocal lattice vectors Look at the reciprocal lattice vectors as defined above: These vectors give the outline of the reciprocal unit cell  G002 G102  G001 G101 c* So a* = G100 and c* = G001 and |a*| = 1/d100 and |c*| = 1/d001 * G100  O a* a* and c* are not parallel to a and c - this only happens in orthogonal systems * is the complement of 

7. Vectors and the reciprocal unit cell From the definitions, it should be obvious that: a.a* = 1 a*.b = 0 a*.c = 0 etc. i.e. a* is perpendicular to both b and c The cross product (b x c) defines a vector parallel to a* with modulus of the area defined by b and c The volume of the unit cell is thus given by a.(bxc) We can define the reciprocal lattice, thus, as follows:

8. Reciprocal vs real Reciprocal lattice vectors can be expressed in terms of the reciprocal unit cell a* b* c* For hkl planes: Ghkl = ha* + kb* + lc* compared with real lattice: uvw = ua + vb + wc where uvw represents a complete unit cell translation and uvw are necessarily integers v, w = 0 x u=2 u=1

9. The K vector We define incident and reflected X-rays as ko and k respectively, with moduli 1/ Then we define vector K = k - ko

10. K vector As k and koare of equal length, 1/, the triangle O, O’, O’’ is isosceles. The angle between k and -kois 2hkland the hkl plane bisects it. The length of K is given by:

11. The Laue condition K is perpendicular to the (hkl) plane, so can be defined as: G is also perpendicular to (hkl) so But Bragg: 2dsin =  So K = Ghkl the Laue condition

12. What does this mean?! Laue assumed that each set of atoms could radiate the incident radiation in all directions Constructive interference only occurs when the scattering vector, K, coincides with a reciprocal lattice vector, G This naturally leads to the Ewald Sphere construction

13. Ewald Sphere We superimpose the imaginary “sphere” of radiated radiation upon the reciprocal lattice For a fixed direction and wavelength of incident radiation, we draw -ko(=1/) e.g. along a* Draw sphere of radius 1/ centred on end of ko Reflection is only observed if sphere intersects a point i.e. where K=G

14. What does this actually mean?! Relate to a real diffraction experiment with crystal at O K=G so scattered beam at angle 2 Geometry:

15. Practicalities This allows us to convert distances on the film to lengths of reciprocal lattice vectors. Indexing the pattern (I.e. assigning (hkl) values to each spot) allows us to deduce the dimensions of the reciprocal lattice (and hence real lattice) In single crystal methods, the crystal is rotated or moved so that each G is brought to the surface of the Ewald sphere In powder methods, we assume that the random orientation of the crystallites means that G takes up all orientations at once.

16. Examples Q1 In handout:

17. Indexing - Primitive Each “spot” represents a set of planes. A primitive lattice with no absences is straightforward to index - merely count out from the origin

18. Indexing - Body Centred In this case we have absences Remember, h+k+l=2n l =0 l =1

19. Indexing - Face Centred Again we have absences Remember, h,k,l all odd or all even l =0 l =1

20. Notes on indexing Absences mean that it’s not so straightforward - need to take many images The real, body-centred lattice gives a face-centred reciprocal lattice The real, face-centred lattice gives a body-centred recoprocal lattice

21. Summary • The observed diffraction pattern is a view of the reciprocal lattice • The reciprocal lattice is related to the real lattice and a*=1/a, a*.b=0, a*.c=0 etc. • By considering the Bragg construction in terms of the reciprocal lattice, we can show that K = G for constructive interference - the Laue condition • This leads naturally to the imaginary Ewald sphere, which allows us to make calculations from a measured diffraction pattern.