Difference between revisions of "Realspace"

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[[Image:BCP etch.jpg|thumb|300px|Example realspace image (SEM) of a [[block-copolymer]] pattern etched into silicon.]]
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[[Image:BCP etch.jpg|thumb|300px|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 distribution|electron-density spatial distribution]] (as imaged in SEM, TEM, AFM, STM, etc.).
 
'''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 distribution|electron-density spatial distribution]] (as imaged in SEM, TEM, AFM, STM, etc.).
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Samples of scientific interest often exhibit well-defined packing in realspace, and are thus described by their [[lattice]] and/or [[unit cell]]. This gives rise to well-defined [[scattering features|features]] in reciprocal-space (peaks, rings, etc.).
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==Realspace functions==
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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
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===Indirect Fourier Transform (IFT)===
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* Glatter, O. ''Acta Phys. Austriaca,'' '''1977''' 47, 83–102
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* Glatter, O. [http://scripts.iucr.org/cgi-bin/paper?S0021889877013879 A new method for the evaluation of small-angle scattering data] ''J. Appl. Cryst.'' '''1977''', 10, 415–421 [http://dx.doi.org/10.1107/S0021889877013879 doi: 10.1107/S0021889877013879]
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* Moore, P. B. [http://scripts.iucr.org/cgi-bin/paper?S002188988001179X Small-angle scattering. Information content and error analysis] ''J. Appl. Cryst.'' '''1980''', 13, 168–175 [http://dx.doi.org/10.1107/S002188988001179X doi: 10.1107/S002188988001179X]
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* Svergun, D. I., Semenyuk, A. V. & Feigin, L. A.  [http://scripts.iucr.org/cgi-bin/paper?S0108767387011255 Small-angle-scattering-data treatment by the regularization method] ''Acta Cryst. A'' '''1988''' A44, 244–250 [http://dx.doi.org/10.1107/S0108767387011255 doi: 10.1107/S0108767387011255]
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===Pair Distribution Function ([[PDF]])===
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* Porod, G. ''Acta Phys. Austriaca'' '''1948''', 2, 255–292.
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* 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.
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* 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]
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*  Kirsten M. Ø. Jensen, Anders B. Blichfeld, Sage R. Bauers, Suzannah R. Wood, Eric Dooryhée, David C. Johnson, Bo B. Iversen and Simon J. L. Billinge [http://journals.iucr.org/m/issues/2015/05/00/yu5008/index.html Demonstration of thin film pair distribution function analysis (tfPDF) for the study of local structure in amorphous and crystalline thin films] ''IUCrJ'' '''2015''', 2 (5). [http://dx.doi.org/10.1107/S2052252515012221 doi: 10.1107/S2052252515012221]
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===Autocorrelation Function===
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* 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]
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===Other===
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* 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]
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** C. J. Gommes, Y. Jiao, A. P. Roberts and D. Jeulin [http://scripts.iucr.org/cgi-bin/paper?in5019 Chord-length distributions cannot generally be obtained from small-angle scattering] ''J. Appl. Cryst.'' '''2020''', 53. [https://doi.org/10.1107/S1600576719016133 doi: 10.1107/S1600576719016133]
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* 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]
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===Two-dimensional===
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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.
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* 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]
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==See Also==
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* [[Q value]]: Conversion from [[reciprocal-space]] values to realspace distances can be accomplished using <math>\scriptstyle d = 2 \pi / q</math>

Latest revision as of 11:38, 8 January 2020

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.).

Samples of scientific interest often exhibit well-defined packing in realspace, and are thus described by their lattice and/or unit cell. This gives rise to well-defined features in reciprocal-space (peaks, rings, etc.).

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)

Pair Distribution Function (PDF)

Autocorrelation Function

Other

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.

See Also

  • Q value: Conversion from reciprocal-space values to realspace distances can be accomplished using