Presentation on theme: "Reciprocal Space Learning outcomes"— Presentation transcript:
1 Reciprocal Space Learning outcomes By the end of this section you should:understand the derivation of the reciprocal latticebe able to derive the Laue condition for the reciprocal latticeunderstand how reciprocal space relates to the diffraction experimentbe able to use reciprocal space to make calculations
2 The reciprocal lattice A diffraction pattern is not a direct representation of the crystal latticeThe diffraction pattern is a representation of the reciprocal latticeWe 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-spacingThe 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 lengthG1G2These are called reciprocal lattice vectors G1 and G2Dimensions = 1/length
5 Reciprocal Lattice/Unit Cells We will use a monoclinic unit cell to avoid orthogonal axesWe 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 cellG102 G002G101 G001c*So a* = G and c* = G001and |a*| = 1/d100 and |c*| = 1/d001*G100 Oa*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 cThe cross product (b x c) defines a vector parallel to a* with modulus of the area defined by b and cThe volume of the unit cell is thus given by a.(bxc)We can define the reciprocal lattice, thus, as follows:
8 Reciprocal vs realReciprocal 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 + wcwhere uvw represents a complete unit cell translation and uvw are necessarily integersv, w = 0xu=1u=2
9 The K vectorWe define incident and reflected X-rays as ko and k respectively, with moduli 1/Then we define vector K = k - ko
10 K vectorAs k and ko are of equal length, 1/, the triangle O, O’, O’’ is isosceles.The angle between k and -ko is 2hkland the hkl plane bisects it.The length of K is given by:
11 The Laue conditionK is perpendicular to the (hkl) plane, so can be defined as:G is also perpendicular to (hkl) soBut 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 directionsConstructive interference only occurs when the scattering vector, K, coincides with a reciprocal lattice vector, GThis naturally leads to the Ewald Sphere construction
13 Ewald SphereWe superimpose the imaginary “sphere” of radiated radiation upon the reciprocal latticeFor 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 koReflection is only observed if sphere intersects a pointi.e. where K=G
14 What does this actually mean?! Relate to a real diffraction experiment with crystal at OK=G so scattered beam at angle 2Geometry:
15 PracticalitiesThis 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 sphereIn powder methods, we assume that the random orientation of the crystallites means that G takes up all orientations at once.
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 absencesRemember, h+k+l=2nl =0l =1
19 Indexing - Face Centred Again we have absencesRemember, h,k,l all odd or all evenl =0l =1
20 Notes on indexingAbsences mean that it’s not so straightforward - need to take many imagesThe real, body-centred lattice gives a face-centred reciprocal latticeThe real, face-centred lattice gives a body-centred recoprocal lattice
21 SummaryThe observed diffraction pattern is a view of the reciprocal latticeThe 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 conditionThis leads naturally to the imaginary Ewald sphere, which allows us to make calculations from a measured diffraction pattern.