34.1: Introduction
- Page ID
- 34933
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Electrons can act as waves as well as particles; this is a consequence of quantum mechanics. A series of electrons hitting an object is exactly equivalent to a beam of electron waves hitting the object and it produces a diffraction pattern in the same way as a beam of X-rays does.
The two important differences between electron and X-ray diffraction are that (1) electrons have a much smaller wavelength than X-rays, and (2) the sample is very thin in the direction of the electron beam (of the order of 100 nm or less) - it has to be thin so that enough electrons can get through to form a diffraction pattern without being absorbed. These factors conspire to have a fortunate effect on the Ewald sphere construction (see The Ewald sphere in the Reciprocal Space TLP) and diffraction pattern:
- The thin sample makes the reciprocal lattice points longer in the reciprocal direction corresponding to the real-space dimension in which the sample is thin:
It should be noted that there is not necessarily always a particular plane oriented like this. However, it is usual for identification of crystalline phases in a sample to orient the sample so that the electron beam is parallel to a low index lattice direction, as this makes the electron diffraction pattern easier to interpret.
- The small electron wavelength makes the radius of the Ewald sphere very large (recall its radius is 1/λ). The small electron wavelength also makes the diffraction angles θ small (1-2°); this can be seen by substituting a wavelength of 2.51 x 10-12 m into the Bragg equation (see Bragg's law in the X-ray Diffraction TLP).
These make the Ewald sphere diagram look like this so that whole layers of the reciprocal lattice end up projected onto the film or screen:
Note that the large (strong) spot in the middle is the straight-through beam (the beam which has passed through the sample without diffracting). This always has the index 000.
Caution 1: systematic (kinetic) absences appear in electron diffraction patterns just as in X-ray diffraction patterns, for the same reason: the various features of the lattice or motif diffract electrons in the same direction but the phase factors from the various features cancel, leaving an absence.
Caution 2: sometimes where there should be a systematic absence, the spot appears to be still there. This is because of the strong interaction between electrons and atoms: there is a small but significant probability that an electron will be diffracted twice, from two planes one after another - i.e. in two different reciprocal lattice directions one after another. These two directions can add up so that the twice-diffracted electron may arrive at a position in reciprocal space where there is a systematic absence. As an example, the diagram below is a schematic of the [011] electron diffraction pattern of silicon: the 200 type reflections are systematically absent. The intensity at the 200 reflections is caused by double diffraction (arising from the addition of the two reciprocal lattice vectors shown). Thus, in words, intensity can occur in the 200 reflection from, firstly, diffraction from the 11 planes, followed by, secondly, diffraction by the 11 planes as the electron wave passes through the specimen.
