34.3: Laue Zones
- Page ID
- 34936
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)So far we have been looking at the central region of the diffraction pattern. This is only a part of the total diffraction pattern. If we look again at the Ewald sphere construction, we have:
We have been indexing the portion in the middle with the 000 spot in it. However, there are also areas of diffraction spots at the edges of the film, caused by the Ewald sphere intersecting points in an adjacent parallel plane containing reciprocal lattice points. (If the film was small or the camera length large it is possible that it did not catch these spots at the side, so that we sometimes only have the middle part.)
These outlying parts of the diffraction pattern are called Higher Order Laue Zones (HOLZs). Each of the HOLZs can be described by an equation of the general form
\[hu+kv+lw=N\]
where:
- N is always an integer, and is called the order of the Laue zone.
- [uvw] is the direction of the incident electron beam.
- hkl are the co-ordinates of an allowed reflection in the Nth order Laue zone.
The middle part of the diffraction pattern, with 000 in it, is the zero order Laue zone (ZOLZ), because it comes from the plane for which N = 0: an allowed reflection hkl in the ZOLZ is joined to the origin 000 by a reciprocal lattice vector that lies in the ZOLZ. For the ZOLZ the electron beam [uvw] and the allowed reflection hkl satisfy the Weiss zone law hu + kv + lw = 0. The next layer up has a value N = 1, then N = 2, and so on, as shown.
From the geometry of the way in which the Ewald sphere intersects the HOLZs, the radius of the Nth HOLZ ring, Rn, in reciprocal space, is given to a very good approximation by the formula
\[R_{n}=\sqrt{\left(\frac{2 N}{\lambda|u v w|}\right)}\]
assuming that the wavelength of the electrons is much less than the modulus |uvw| of the direction [uvw] in the crystal parallel to the electron beam direction.
Thus, HOLZs are seen more easily at lower voltages (e.g. 100 kV rather than 300 kV) and when the electron beam is parallel to a relatively high index direction in a crystal.
It is possible to index the reflections in the HOLZs on a diffraction pattern. Examples of such indexing are given in the book Transmission Electron Microscopy of Materials by D B Williams and C B Carter.