Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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


J. M. Cowleya

Table| top | pdf |
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_{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.