Lecture 7: Systematic Absences

1 Crystallography Supplementary SubjectLecture 7: Systematic Absences1 Lecture 7: Systematic AbsencesThe reason that we have focussed on crystal symmetry for the last four lectures is thatsymmetry is going to help us to simplify the interpretation of crystallographic crux of this Lecture is to understand how the various symmetry elements Bravaislattice, translational symmetry and point symmetry affect diffraction patterns. What weare leading towards is being able to use the symmetry of an observed diffraction pattern inreciprocal space to deduce the symmetry about the crystal in real begin by making two simple but important observations.

2 (i) The structure factors at reciprocal space vectorsQand Qare in fact complex con-jugates; (hkl) =F ( h k l). This result, known as Friedel s law , means thatthe corresponding diffraction intensities are equal:I(hkl) =I( h k l). Importantly, thediffraction pattern of a crystal is necessarily centrosymmetric irrespective of whetheror not the crystal itself has a centre of symmetry.(ii) The point symmetry of an object is preserved in its diffraction pattern. As a simpleexample, let us imagine that we have a system that possesses a 2-fold axis paralleltoz, running through the origin.

3 Then we can divide all the atoms in our systeminto three groups. The first group contains all those atoms that lie on the axis itself,and we know that these will have coordinates(0,0,z). We then divide the remainingatoms into two equal groups such that each atom in one group will be mapped ontoan atom in the second group by the 2-fold axis. That is, for each atom at(x,y,z), weknow that there is an equivalent atom at( x, y,z), and we place one atom in oursecond group, and one atom in our third group. Taking an arbitrary point in reciprocalspace(h,k,l), we will show that the Fourier component at this point is the same asthat at( h, k,l), and hence the diffraction pattern will also contain a 2-fold axis, thistime lying perpendicular toc.

4 Crystallography Supplementary SubjectLecture 7: Systematic Absences2F(hkl) = jfjexp[2 i(hxj+kyj+lzj)](1)= j group1fjexp[2 i(lzj)] + j group2fjexp[2 i(hxj+kyj+lzj)]+ j group3fjexp[2 i( hxj kyj+lzj)](2)= j group1fjexp[2 i(lzj)] + j group2fjexp[2 i(( h)xj+ ( k)yj+lzj)]+ j group3fjexp[2 i( ( h)xj ( k)yj+lzj)](3)=F( h kl)(4)(here, the notation( h kl)is shorthand for( h, k,l)).Taken together, these two results mean thatthe point group of a diffraction patternis the centrosymmetric parent of the point group of the point group of adiffraction pattern is usually called the Laue class of the diffraction pattern.

5 The relation-ship between Laue classes and point groups is shown in the table system Laue class Point groups of the Laue groupTriclinic 11 1 Monoclinic2/m2m2/mOrthorhombicmmm222mm2m mmTrigonal 33 3 3m323m 3mTetragonal4/m4 44/m4/mmm422 4mm 42m4/mmmHexagonal6/m6 66/m6/mmm622 6mm 6m26/mmmCubicm323m3m 3m432 43m m 3mYou should be able to convince yourself that the diffraction pattern for a crystal inI41cdwill have4/mmmpoint symmetry; likewise, that of a crystal inP62will have6 established the effect of point symmetry on the diffraction pattern, we now proceedto show that the translational symmetry elements do not affect the symmetry of the diffrac-tion patternper se, but do result in what we call Systematic Absences the absence ofany diffraction intensity at specific sets of reciprocal lattice Supplementary SubjectLecture 7: Systematic Absences3 The first type of Systematic Absences we will address are those that arise due to latticecentering.

6 Let us consider a face-centred lattice (presumably of orthorhombic or cubiclattice symmetry, but this doesn t matter). What we know is that for each atomjin theunit cell at(xj,yj,zj)there are corresponding atomsj ,j andj at(xj,yj+12,zj+12),(xj+12,yj,zj+12)and(x j+12,yj+12,zj), respectively. Consequently, we can split ourscattering equations into four parts:F(hkl) = jfj{exp[2 i(hxj+kyj+lzj)]+ exp[2 i(hxj+k{yj+12}+l{zj+12})]+ exp[2 i(h{xj+12}+kyj+l{zj+12})]+ exp[2 i(h{xj+12}+k{yj+12}+lzj)]}(5)On factorising we obtainF(hkl) ={1 + exp[ i(k+l)] + exp[ i(h+l)] + exp[ i(h+k)]} jfjexp[2 i(hxj+kyj+lzj)](6)It is not difficult (and a good exercise) to show that the prefactor is zero for allh,k,lexceptwhenever the three indices are all even or are all odd (when it equals four).

7 This meansthat for a face centred crystal we do not expect to observe any intensity (100),(321),..reflections. Let us use this result to visualise reciprocal space for a face-centredlattice:What we find is that the reciprocal lattice of a face-centred cubic lattice is itself a body-centred cubic lattice in reciprocal space, a result that we met in Lecture 1. It is a goodCrystallography Supplementary SubjectLecture 7: Systematic Absences4exercise to check that the reverse also holds true; that is, to confirm that a body-centredlattice is face-centred in reciprocal space.

8 Consequently, the centering of a diffractionpattern we observe experimentally will tell us what particular type of centering exists inreal space. This enables us to start determining the space group for our other two translational symmetry operations, namely screw axes and glide planes,also give rise to Systematic Absences . The mathematics involved is very similar, if a littletedious. We cover some representative derivations here only really to explain from wherethese results arise; what is important is only that the Absences occur, and that we knowhow to recognise and interpret these in a diffraction us address quickly the mathematics for screw axes, and we will use as our examplea crystal that contains a21screw axis parallel tob.

9 This will have the effect of replicatingeach atomj, originally at(xj,yj,zj), at( xj,12+yj, zj). The structure factor is then givenasF(hkl) = jfj{exp[2 i(hxj+kyj+lzj)] + exp[2 i( hxj+k{12+yj} lzj)]}.(7)What we do is to consider the intensity at reciprocal lattice points of the type(0k0):F(0k0)= jfj{exp(2 ikyj) + exp[2 ik(12+yj)]}(8)=[1 + ( 1)k] jfjexp(2 ikyj)(9){=6=0 ifk= 2n+ 1 (odd)0 ifk= 2n(even),(10)wherenis an integer. What this tells us is that(0k0)reflections with odd values ofkwillnot be observed: a new set of Systematic Absences that we should be able to observe ina diffraction pattern.}

10 Similar Systematic Absences would arise from screw axes along completeness we will cover a similar calculation for glide planes. This time our examplewill be ac-glide perpendicular tob. This will replicate each atomj, originally at(xj,yj,zj),at(xj, yj,12+zj). Writing out the structure factor explicitly we obtainF(hkl) = jfj{exp[2 i(hxj+kyj+lzj)] + exp[2 i(hxj kyj+l{12+zj})]}.(11)Now, for(h0l)reflections we haveCrystallography Supplementary SubjectLecture 7: Systematic Absences5F(h0l)= jfj{exp[2 i(hxj+lzj)] + exp[2 i(hxj+l{12+zj})]}(12)=[1 + ( 1)l] jfjexp[2 i(hxj+lzj)](13){=6=0 ifl= 2n+ 1 (odd)0 ifl= 2n(even),(14)where againnis an integer.}