Table 2.5.2.1

Cowley, J. M.

doi:10.1107/97809553602060000558

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 280 | 1 | 2 |

Table 2.5.2.1

J. M. Cowley^a ^‡

Table 2.5.2.1 | top | pdf |
Standard crystallographic and alternative crystallographic sign conventions for electron diffraction

	Standard	Alternative
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 $[\chi (U) = \pi \lambda \Delta fU^{2} + \textstyle{1 \over 2} \pi C_{s} \lambda^{3} U^{4}]$ , $[U^{2} = u^{2} + v^{2}]$ 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); $[C_{s} =]$ 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, $[f_{e}]$ , related to the atomic scattering factor for X-rays, $[f_{X}]$ , by the Mott formula $[f_{e} = (e/\pi U^{2}) (Z - f_{X})]$ . $[Q({\bf h})=]$ Fourier transform of periodic slice transmission function.