Difference between revisions of "Electron-density distribution"
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| − | The '''electron-density distribution''' is the three-dimensional [[realspace]] arrangement of the electrons in the material. Because electrons are [[quantum mechanics|quantum mechanically]] [[wave packet|delocalized]], each electron occupies a 'fuzzy' region of space. The total electron density--the summation of the electron distribution for every electron in every atom--is thus inherently spread spatially; e.g a map of electron distribution will have diffuse boundaries. Nevertheless, when the electron distribution is visualized, it is often shown as a surface, representing an isosurface of constant electron-density. | + | The '''electron-density distribution''' is the three-dimensional [[realspace]] arrangement of the electrons in the material. Because electrons are [[quantum mechanics|quantum mechanically]] [[wave packet|delocalized]], each electron occupies a 'fuzzy' region of space ('''electron cloud'''). The total electron density--the summation of the electron distribution for every electron in every atom--is thus inherently spread spatially; e.g a map of electron distribution will have diffuse boundaries. Nevertheless, when the electron distribution is visualized, it is often shown as a surface, representing an isosurface of constant electron-density. |
The exact electron-density distribution within a [[unit cell]] of a crystal can be reconstructed by carefully fitting the peak heights measured in an [[x-ray]] [[diffraction]] experiment ([[crystallography|crystallographic]] 'structure solution'). | The exact electron-density distribution within a [[unit cell]] of a crystal can be reconstructed by carefully fitting the peak heights measured in an [[x-ray]] [[diffraction]] experiment ([[crystallography|crystallographic]] 'structure solution'). | ||
| − | The electron distribution at a larger scale can be similarly reconstructed by fitting small-angle [[scattering]] data ([[SAXS]] or [[GISAXS]]). In this case, one is probing the average electron-density distribution at the nanoscale, without resolving the exact arrangement of electron-distribution (atoms) within this nanostructure. | + | The electron distribution at a larger scale can be similarly reconstructed by fitting small-angle [[scattering]] data ([[SAXS]] or [[GISAXS]]). In this case, one is probing the average electron-density distribution at the nanoscale, without resolving the exact arrangement of electron-distribution (atoms) within this nanostructure. The specific electron-distribution within a measurement volume (as opposed to the average structure defined by a unit-cell) can be reconstructed using coherent methods such as [[CDI]] or [[ptychography]]. |
| + | |||
| + | ==Mathematical form== | ||
| + | The electron-density (number of electrons per unit volume) can be computed from the [[atomic scattering factor]], <math>f_1</math>; or from the [[refractive index]]: | ||
| + | :<math> | ||
| + | \begin{alignat}{2} | ||
| + | \rho_e & = \frac{\rho N_a f_1}{M_a} \\ | ||
| + | & = \frac{2 \pi}{\lambda^2 r_e} \delta | ||
| + | \end{alignat} | ||
| + | </math> | ||
==References== | ==References== | ||
* Philip Coppens, Bo Iversen, Finn Krebs Larsen [http://www.sciencedirect.com/science/article/pii/S0010854504000402 The use of synchrotron radiation in X-ray charge density analysis of coordination complexes] ''Coordination Chemistry Reviews'' '''2005''', 249 (1-2), 179-195 [http://dx.doi.org/10.1016/j.ccr.2004.02.019 doi: 10.1016/j.ccr.2004.02.019] | * Philip Coppens, Bo Iversen, Finn Krebs Larsen [http://www.sciencedirect.com/science/article/pii/S0010854504000402 The use of synchrotron radiation in X-ray charge density analysis of coordination complexes] ''Coordination Chemistry Reviews'' '''2005''', 249 (1-2), 179-195 [http://dx.doi.org/10.1016/j.ccr.2004.02.019 doi: 10.1016/j.ccr.2004.02.019] | ||
* Mads R. V. Jørgensen, Venkatesha R. Hathwar, Niels Bindzus, Nanna Wahlberg, Yu-Sheng Chen, Jacob Overgaard and Bo B. Iversen [http://journals.iucr.org/m/issues/2014/05/00/lc5060/index.html Contemporary X-ray electron-density studies using synchrotron radiation] ''IUCrJ'' '''2014''' 1 (5), 267-280 [http://dx.doi.org/10.1107/S2052252514018570 doi: 10.1107/S2052252514018570] | * Mads R. V. Jørgensen, Venkatesha R. Hathwar, Niels Bindzus, Nanna Wahlberg, Yu-Sheng Chen, Jacob Overgaard and Bo B. Iversen [http://journals.iucr.org/m/issues/2014/05/00/lc5060/index.html Contemporary X-ray electron-density studies using synchrotron radiation] ''IUCrJ'' '''2014''' 1 (5), 267-280 [http://dx.doi.org/10.1107/S2052252514018570 doi: 10.1107/S2052252514018570] | ||
Latest revision as of 14:36, 28 January 2015
The electron-density distribution is the three-dimensional realspace arrangement of the electrons in the material. Because electrons are quantum mechanically delocalized, each electron occupies a 'fuzzy' region of space (electron cloud). The total electron density--the summation of the electron distribution for every electron in every atom--is thus inherently spread spatially; e.g a map of electron distribution will have diffuse boundaries. Nevertheless, when the electron distribution is visualized, it is often shown as a surface, representing an isosurface of constant electron-density.
The exact electron-density distribution within a unit cell of a crystal can be reconstructed by carefully fitting the peak heights measured in an x-ray diffraction experiment (crystallographic 'structure solution').
The electron distribution at a larger scale can be similarly reconstructed by fitting small-angle scattering data (SAXS or GISAXS). In this case, one is probing the average electron-density distribution at the nanoscale, without resolving the exact arrangement of electron-distribution (atoms) within this nanostructure. The specific electron-distribution within a measurement volume (as opposed to the average structure defined by a unit-cell) can be reconstructed using coherent methods such as CDI or ptychography.
Mathematical form
The electron-density (number of electrons per unit volume) can be computed from the atomic scattering factor, ; or from the refractive index:
References
- Philip Coppens, Bo Iversen, Finn Krebs Larsen The use of synchrotron radiation in X-ray charge density analysis of coordination complexes Coordination Chemistry Reviews 2005, 249 (1-2), 179-195 doi: 10.1016/j.ccr.2004.02.019
- Mads R. V. Jørgensen, Venkatesha R. Hathwar, Niels Bindzus, Nanna Wahlberg, Yu-Sheng Chen, Jacob Overgaard and Bo B. Iversen Contemporary X-ray electron-density studies using synchrotron radiation IUCrJ 2014 1 (5), 267-280 doi: 10.1107/S2052252514018570
