Lecture 6 — Defects in Crystals.

1 Lecture 6 Defects in IntroductionIn the first five lectures, we have focussed our attentions on the methodologies used to describecrystal symmetry and on the scattering techniques employed to study the arrangement of atomsand spins in crystals (electronic, nuclear and spin densities). Throughout this discussion, wehave explicitly assumed that the crystalline order is perfect in other words, that translationalsymmetry is strictly valid. The only allowance we made for deviations from one unit cell to thenext is through the introduction of the Debye-Waller factors, the effect of which, as we have seen,is to spread out the scattering densities in a fashion similar to the atomic scattering factors. Inthis Lecture , we will explicitly consider crystal imperfections of different kinds, starting fromthe simplest form of translational symmetry breaking (the effects arising from the finite sizeof the crystals) and considering in turn other, more complex types of Defects : point Defects ,correlated Defects and line or plane Defects (dislocations or stacking faults).

2 As we shallsee, all these Defects give rise to particular kinds of scattering, most often away from the Braggpeaks. We will also introduce a new experimental technique transmission electron microscopy and describe its relevance in the study of crystal Space and time scales: coherenceWhen dealing with crystal imperfections, one of the first questions one should ask is whetherthey will contributecoherently( , amplitudes are summed) orincoherently( , intensities aresummed) to the diffraction us make this clear with a simple example: consider an inhomogeneous alloy, in which thecomposition and the lattice constants varies on the scale of millimetres. Clearly, the differentregions of the crystal will scatter independently, each region being in essence a perfect crystal .The result, therefore, with be theincoherentsuperposition of different patterns.

3 In this example,eachhklwill produce a set of independent Bragg peaks (corresponding to the different latticeconstants), which will typically appear as abroadeningof the scattered reflections. Let us nowimagine to shrink the lengthscale of the composition and lattice parameters fluctuations downto nanometre sizes. Here, the different compositions will scattercoherently, and the appropriatepicture is that of anaverage lattice. In the different regions, the atoms will be displaced awayfrom the average positions, and this, as we know by now, will produce areductionof the Braggintensity at highq, in complete analogy to the Debye-Waller factors. Furthermore, the displace-ments will becorrelated( , nearby atoms will tend to be displaced in the same direction). Later1in this Lecture , we will learn that this producesdiffuse scatteringaway from the Bragg peaks.

4 Atdifferent lengthscales, he diffraction patterns will be thereforequalitatively different. At whatpoint, between millimetres and nanometres, does this qualitative transition occur? We can intuitthat the lengthscale where the transition occurs might be set by theprobe, , that there mat beintrinsiccoherence lengthsfor X-rays, neutrons, electrons etc. A very similar argument maybe construed fortimescales. If the positions of the atoms in the crystal fluctuate , , dueto phonons, do the different configurations occurring at different times contribute coherently orincoherently to the diffraction pattern? If the timescale of the fluctuations if of the order of sec-onds, the latter will be true, but, as the timescale of the fluctuations is reduced, this will no longerbe the case. What is the typicalcoherence timeof the different probes?

5 A complete treatment of the correlation lengths and times is to complex to be presented here, butthe following points should provide a good idea of the issues involved. One distinguishes betweentransverseandlongitudinalcoherenc e lengths, perpendicular andparallel, respectively, to the direction of the beam. For a quasi-parallel beam geometry (as it is typical of a diffraction instrument), thetrans-verse coherence length tis proportional to the wavelength and inversely proportional tothe beam divergence ( t= / ). Here, the beam divergence is defined asthe anglesubtended by the source as seen from the sample. The derivation is analogous to that ofthe double-slit experiment, and corresponds to the distance between slits where the inter-ference pattern is lost. For for typical diffraction instruments, the beam divergence variesbetween a few mrad (lab diffractometers) down to a tenth of mrad (synchrotrons), so for = 1 A, tvaries between a few tens of nm up to about 1 m.

6 Highly coherent X-raybeams (several tens of m) are employed for special studies (lensless imaging, speckle patterns), revealing the shape and internal structure of large crystal domains. Neutronbeams have comparatively relaxed divergences, and typical coherence lengths are 1000-2000 A. For geometries employing focussing, ( , electron diffraction) it is possible to manipulate thecoherence domain by varying the focal plane where the diffraction pattern is formed (seebelow). If thediffraction patternis in focus, the coherence length is essentially limited bythe aberration of the lenses, and can be as large as for X-rays, in spite of the fact that thewavelength employed are 1-2 orders of magnitude smaller. If thesampleis in focus, thetransverse coherence length is much smaller (down to atomic sizes), so that each detectorpixel receives a coherent contribution of acolumnof atoms in the direction of the this case, the coherence domain coincides with theresolving powerof the instrument(see below).

7 2 Thelongitudinal coherence lengthis inversely proportional to the relative wavelength spreadof the beam: l=12 /( / ). High-resolution monochromators at synchrotron sourcestypically yield / 1 10 4, so at 1 A lis about 1 m. The wavelength spreadof neutron and electron beams is typically 10 times and 100 times larger, respectively. Thecoherence timecan be calculated using the uncertainty principle as =~/ Eor fromthe relation = l/v, yielding the same result apart for a factor of the order 1. Adoptingthe first approach, we obtain =1 2m 2~( ) 1for particles with mass =12 c( ) 1for photons(1)The prefactor 10 14[sec A 2]for neutrons, 10 17[sec A 2]for electrons and5 10 20[sec A 1]for photons. Using the wavelengths and monochromaticity values men-tioned above, we find that coherence times for neutrons, photons and 100 Kev electronsare in the picosecond, femtosecond and tens of attosecond ranges, Finite size effectsUp to this point, we have consider the crystal lattice to be of infinite extent.

8 As we have seen, thecross section becomes then a series of delta functions, centered on theRLnodes. We can relaxour approximation by starting form eq. 6 in Lecture 5 (the cross section) and solve explicitly forthe finite summations. in fact, we need not make any particular approximation other than the factthat the crystal should be composed ofN1,N2andN3unit cell in thea1,a2anda3directions,respectively. Remembering thatN n= Ne inx=sin((2N+ 1)x2)sinx2(2)after some straightforward math (in each direction,x=q ai) we obtaind d =[ isin2(Ni12q ai)sin2(12q ai)]|F(q)|2[ ]2(3)3 The diffraction pattern from a small single crystal taken withcoherentradiation, , radia-tion withcoherence length(=wavepacket length/width) larger than the crystal , will display3-dimensional fringes, as typical of the functionsin2(Nx)/sin2x. However, for typical experi-ments with incoherent radiation, the number of unit cells is set by the coherence length of eachphoton rather than by the crystal size, and the oscillations will be smeared out.

9 In this case, wecan replace the oscillatory functions with aGaussianfunction with the same maximum and samearea:sin2 Nxsin2x N2e (Nx)2/ (4)Observing that iNi=Nc, the total number of unit cell in the crystal , we obtaind d =N2c[ ie (Ni12q ai)2/ ]|F(q)|2[ ]2(5)where the Gaussian functions have variance and FWHM 2i=2 N2ia2iFWHM=4 ln 2 Niai(6)We can therefore conclude that: Thecross section at a given qis proportional toN2c. Thewidthinqisinversely proportional to the number of unit cells along that direction. Theintegrated cross sectionin three dimensions (remember the Gaussian integral 2 2)is therefore proportional toNc, which reproduces the result we obtained for the infinitecrystal (eq. 8, Lecture 5).3 Beyond the perfect- crystal approximation: diffuse scatter-ingFinite-size effects only generate scattering in the vicinity of the ideal Bragg positions, leav-ing the integrated intensity of the reflection unaltered.

10 In this section, we will demonstrate that4additionalscattering is generated whenever either the atomic scattering factors or the positionsof individual atoms deviate from the average values and/or the ideal lattice sites. In general,this additional scattering is present throughout reciprocal space, and is therefore known asdif-fuse scattering. In the remainder of this section, we will consider the deviations from perfectperiodicity asstatic. In other words, we will imagine that the scattering process occurs coher-ently from afrozen snapshotof the crystal . To estimate the full scattered intensity, at the end,we will undertake a time/thermal averaging process on the scattered intensities. This is a verygood approximation in the case of X-rays (and electrons), since the coherence times aremuchshorter than phonon frequencies. For neutrons, where this is clearly not the case, a more complextreatment is required, but qualitatively similar considerations (with some caveats) can be Classification of disorder in crystalsWe will here consider two types of disorder: Substitutional disorderis typical of alloys and, more generally, of solid solutions.