Difference between revisions of "Realspace"
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Scattering data can be converted into a corresponding realspace representation. Normally, scattering data cannot be simply inverted (via [[Fourier transform]]) to recover the exact realspace structure. This is known as the [[Fourier_transform#Phase_problem|phase problem]]: experiments typically record the intensity (but not phase) of scattered radiation, making unambiguous inversion impossible (note that coherent techniques, such as [[CDI]] or [[ptychography]] attempt to work around this). However, the inverse-space data of a scattering experiment can at least be converted into a statistical (or average) realspace representation. Total one-dimensional scattering can be converted into a realspace function that describes the amount of correlation across various distances | Scattering data can be converted into a corresponding realspace representation. Normally, scattering data cannot be simply inverted (via [[Fourier transform]]) to recover the exact realspace structure. This is known as the [[Fourier_transform#Phase_problem|phase problem]]: experiments typically record the intensity (but not phase) of scattered radiation, making unambiguous inversion impossible (note that coherent techniques, such as [[CDI]] or [[ptychography]] attempt to work around this). However, the inverse-space data of a scattering experiment can at least be converted into a statistical (or average) realspace representation. Total one-dimensional scattering can be converted into a realspace function that describes the amount of correlation across various distances | ||
| − | ===Pair Distribution Function | + | ===Indirect Fourier Transform (IFT)=== |
| + | * Glatter, O. ''Acta Phys. Austriaca,'' '''1977''' 47, 83–102 | ||
| + | * Glatter, O. ''J. Appl. Cryst.'' '''1977''', 10, 415–421 | ||
| + | * Moore, P. B. ''J. Appl. Cryst.'' '''1980''', 13, 168–175 | ||
| + | * Svergun, D. I., Semenyuk, A. V. & Feigin, L. A. ''Acta Cryst. A'' '''1988''' A44, 244–250 | ||
| + | |||
| + | ===Pair Distribution Function ([[PDF]])=== | ||
* Porod, G. ''Acta Phys. Austriaca'' '''1948''', 2, 255–292. | * Porod, G. ''Acta Phys. Austriaca'' '''1948''', 2, 255–292. | ||
* Billinge, S. J. L. & Thorpe, M. F. [http://link.springer.com/book/10.1007%2Fb119172 Local Structure from Diffraction], edited by S. J. L. Billinge. '''1998''' New York: Plenum Press. | * Billinge, S. J. L. & Thorpe, M. F. [http://link.springer.com/book/10.1007%2Fb119172 Local Structure from Diffraction], edited by S. J. L. Billinge. '''1998''' New York: Plenum Press. | ||
* P. F. Peterson, M. Gutmann, Th. Proffen and S. J. L. Billinge [http://scripts.iucr.org/cgi-bin/paper?ks0034 PDFgetN: a user-friendly program to extract the total scattering structure factor and the pair distribution function from neutron powder diffraction data] ''J. Appl. Cryst.'' '''2000''', 33, 1192 [http://dx.doi.org/10.1107/S0021889800007123 doi: 10.1107/S0021889800007123] | * P. F. Peterson, M. Gutmann, Th. Proffen and S. J. L. Billinge [http://scripts.iucr.org/cgi-bin/paper?ks0034 PDFgetN: a user-friendly program to extract the total scattering structure factor and the pair distribution function from neutron powder diffraction data] ''J. Appl. Cryst.'' '''2000''', 33, 1192 [http://dx.doi.org/10.1107/S0021889800007123 doi: 10.1107/S0021889800007123] | ||
| + | |||
| + | ===Autocorrelation Function=== | ||
| + | * Debye, P. & Bueche, A. M. [http://scitation.aip.org/content/aip/journal/jap/20/6/10.1063/1.1698419 Scattering by an inhomogeneous solid] ''J. Appl. Phys.'' '''1949''', 20, 518–525 [http://dx.doi.org/10.1063/1.1698419 doi: 10.1063/1.1698419] | ||
| + | |||
| + | ===Other=== | ||
| + | * Hansen, S. [http://journals.iucr.org/j/issues/2003/05/00/issconts.html Estimation of chord length distributions from small-angle scattering using indirect Fourier transformation] ''J. Appl. Cryst.'' '''2003''', 36, 1190–1196 [http://dx.doi.org/10.1107/S0021889803014262 doi: 10.1107/S0021889803014262] | ||
| + | * Fritz, G. [http://scitation.aip.org/content/aip/journal/jcp/124/21/10.1063/1.2202325 Determination of pair correlation functions of dense colloidal systems by means of indirect Fourier transformation] ''J. Chem. Phys.'' '''2006''' 124, 214707 [http://dx.doi.org/10.1063/1.2202325 doi: 10.1063/1.2202325] | ||
===Two-dimensional=== | ===Two-dimensional=== | ||
Two-dimensional scattering data can in principle also be converted into a two-dimensional realspace function. This realspace representation contains the same information, but emphasizes spatial distance correlations. | Two-dimensional scattering data can in principle also be converted into a two-dimensional realspace function. This realspace representation contains the same information, but emphasizes spatial distance correlations. | ||
* G. Fritz-Popovski [http://scripts.iucr.org/cgi-bin/paper?ks5437 Interpretation of two-dimensional real-space functions obtained from small-angle scattering data of oriented microstructures] ''J. Appl. Cryst.'' '''2015''', 48 [http://dx.doi.org/10.1107/S1600576714024972 doi: 10.1107/S1600576714024972] | * G. Fritz-Popovski [http://scripts.iucr.org/cgi-bin/paper?ks5437 Interpretation of two-dimensional real-space functions obtained from small-angle scattering data of oriented microstructures] ''J. Appl. Cryst.'' '''2015''', 48 [http://dx.doi.org/10.1107/S1600576714024972 doi: 10.1107/S1600576714024972] | ||
Revision as of 15:31, 19 December 2014
Example realspace image (SEM) of a block-copolymer pattern etched into silicon.
Realspace or direct-space is simply the regular 3D space that we inhabit. In scattering, it is introduced as a term to differentiate from reciprocal-space (a.k.a. inverse-space). Whereas reciprocal-space refers to the Fourier transform of the sample's structure, realspace refers to the actual structure: the electron-density spatial distribution (as imaged in SEM, TEM, AFM, STM, etc.).
Contents
Realspace functions
Scattering data can be converted into a corresponding realspace representation. Normally, scattering data cannot be simply inverted (via Fourier transform) to recover the exact realspace structure. This is known as the phase problem: experiments typically record the intensity (but not phase) of scattered radiation, making unambiguous inversion impossible (note that coherent techniques, such as CDI or ptychography attempt to work around this). However, the inverse-space data of a scattering experiment can at least be converted into a statistical (or average) realspace representation. Total one-dimensional scattering can be converted into a realspace function that describes the amount of correlation across various distances
Indirect Fourier Transform (IFT)
- Glatter, O. Acta Phys. Austriaca, 1977 47, 83–102
- Glatter, O. J. Appl. Cryst. 1977, 10, 415–421
- Moore, P. B. J. Appl. Cryst. 1980, 13, 168–175
- Svergun, D. I., Semenyuk, A. V. & Feigin, L. A. Acta Cryst. A 1988 A44, 244–250
Pair Distribution Function (PDF)
- Porod, G. Acta Phys. Austriaca 1948, 2, 255–292.
- Billinge, S. J. L. & Thorpe, M. F. Local Structure from Diffraction, edited by S. J. L. Billinge. 1998 New York: Plenum Press.
- P. F. Peterson, M. Gutmann, Th. Proffen and S. J. L. Billinge PDFgetN: a user-friendly program to extract the total scattering structure factor and the pair distribution function from neutron powder diffraction data J. Appl. Cryst. 2000, 33, 1192 doi: 10.1107/S0021889800007123
Autocorrelation Function
- Debye, P. & Bueche, A. M. Scattering by an inhomogeneous solid J. Appl. Phys. 1949, 20, 518–525 doi: 10.1063/1.1698419
Other
- Hansen, S. Estimation of chord length distributions from small-angle scattering using indirect Fourier transformation J. Appl. Cryst. 2003, 36, 1190–1196 doi: 10.1107/S0021889803014262
- Fritz, G. Determination of pair correlation functions of dense colloidal systems by means of indirect Fourier transformation J. Chem. Phys. 2006 124, 214707 doi: 10.1063/1.2202325
Two-dimensional
Two-dimensional scattering data can in principle also be converted into a two-dimensional realspace function. This realspace representation contains the same information, but emphasizes spatial distance correlations.
- G. Fritz-Popovski Interpretation of two-dimensional real-space functions obtained from small-angle scattering data of oriented microstructures J. Appl. Cryst. 2015, 48 doi: 10.1107/S1600576714024972
