Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 301   | 1 | 2 |

Section Recommended sign conventions

J. M. Cowleya Recommended sign conventions

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There are two alternative sets of signs for the functions describing wave optics. Both sets have been widely used in the literature. There is, however, a requirement for internal consistency within a particular analysis, independently of which set is adopted. Unfortunately, this requirement has not always been met and, in fact, it is only too easy at the outset of an analysis to make errors in this way. This problem might have come into prominence somewhat earlier were it not for the fact that, for centrosymmetric crystals (or indeed for centrosymmetric projections in the case of planar diffraction), only the signs used in the transmission and propagation functions can affect the results. It is not until the origin is set away from a centre of symmetry that there is a need to be consistent in every sign used.

Signs for electron diffraction have been chosen from two points of view: (1) defining as positive the sign of the exponent in the structure-factor expression and (2) defining the forward propagating free-space wavefunction with a positive exponent.

The second of these alternatives is the one which has been adopted in most solid-state and quantum-mechanical texts.

The first, or standard crystallographic convention, is the one which could most easily be adopted by crystallographers accustomed to retaining a positive exponent in the structure-factor equation. This also represents a consistent International Tables usage. It is, however, realized that both conventions will continue to be used in crystallographic computations, and that there are by now a large number of operational programs in use.

It is therefore recommended (a) that a particular sign usage be indicated as either standard crystallographic or alternative crystallographic to accord with Table[link], whenever there is a need for this to be explicit in publication, and (b) that either one or other of these systems be adhered to throughout an analysis in a self-consistent way, even in those cases where, as indicated above, some of the signs appear to have no effect on one particular conclusion.

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Standard crystallographic and alternative crystallographic sign conventions for electron diffraction

Free-space wave [\exp [- i({\bf k} \cdot {\bf r} - \omega t)]] [\exp [+ i({\bf k} \cdot {\bf r} - \omega t)]]
Fourier transforming from real space to reciprocal space [\textstyle\int \psi ({\bf r}) \exp [+ 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf r}] [\textstyle\int \psi ({\bf r}) \exp [- 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf r}]
Fourier transforming from reciprocal space to real space [\psi ({\bf r}) = \textstyle\int \Psi ({\bf u}) \exp [- 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf u}] [\textstyle\int \Psi ({\bf u}) \exp [+ 2\pi i({\bf u} \cdot {\bf r})]\,\hbox{d}{\bf u}]
Structure factors [V({\bf h}) = (1/\Omega) \textstyle\sum_{j} f_{j} ({\bf h}) \exp (+ 2\pi i{\bf h} \cdot {\bf r}_{j})] [(1/\Omega) \textstyle\sum_{j} f_{j} ({\bf h}) \exp (- 2\pi i{\bf h} \cdot {\bf r}_{j})]
Transmission function (real space) [\exp [- i\sigma \varphi (x, y) \Delta z]] [\exp [+ i\sigma \varphi (x, y) \Delta z]]
Phenomenological absorption [\sigma \varphi ({\bf r}) - i\mu ({\bf r})] [\sigma \varphi ({\bf r}) + i\mu ({\bf r})]
Propagation function P(h) (reciprocal space) within the crystal [\exp (- 2\pi i\zeta_{\bf h} \Delta z)] [\exp (+ 2\pi i\zeta_{\bf h} \Delta z)]
Iteration (reciprocal space) [\Psi_{n + 1} ({\bf h}) = [\Psi_{n} ({\bf h}) \cdot P({\bf h})] \ast Q({\bf h})]  
Unitarity test (for no absorption) [T({\bf h}) = Q({\bf h}) \ast Q^{*} (- {\bf h}) = \delta ({\bf h})]  
Propagation to the image plane-wave aberration function, where [\chi (U) = \pi \lambda \Delta fU^{2} + \textstyle{1 \over 2} \pi C_{\rm s} \lambda^{3} U^{4}], [U^{2} = u^{2} + v^{2}] and [\Delta f] is positive for overfocus [\exp [i\chi (U)]] [\exp [- i\chi (U)]]

[\sigma =] electron interaction constant [= 2\pi me\lambda/h^{2}]; [m =] (relativistic) electron mass; [\lambda =] electron wavelength; [e =] (magnitude of) electron charge; [h =] Planck's constant; [k = 2\pi/\lambda]; [\Omega =] volume of the unit cell; [{\bf u} =] continuous reciprocal-space vector, components u, v; [{\bf h} =] discrete reciprocal-space coordinate; [\varphi (x, y) =] crystal potential averaged along beam direction (positive); [\Delta z =] slice thickness; [\mu ({\bf r}) =] absorption potential [positive; typically [\leq 0.1 \sigma \varphi ({\bf r})]]; [\Delta f =] defocus (defined as negative for underfocus); [C_{\rm s} =] spherical aberration coefficient; [\zeta_{\bf h} =] excitation error relative to the incident-beam direction and defined as negative when the point h lies outside the Ewald sphere; [f_{j} ({\bf h}) =] atomic scattering factor for electrons, [f_{\rm e}], related to the atomic scattering factor for X-rays, [f_{\rm X}], by the Mott formula [f_{\rm e} = (e/\pi U^{2}) (Z - f_{\rm X})]. [Q({\bf h})=] Fourier transform of periodic slice transmission function.

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