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            "489": {
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                "title": "Realspace",
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                        "*": "[[Image:BCP etch.jpg|thumb|300px|Example realspace image (SEM) of a [[block-copolymer]] pattern etched into [[silicon]].]]\n'''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.).\n\nSamples 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.).\n\n==Realspace functions==\nScattering 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\n\n===Indirect Fourier Transform (IFT)===\n* Glatter, O. ''Acta Phys. Austriaca,'' '''1977''' 47, 83\u2013102\n* 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\u2013421 [http://dx.doi.org/10.1107/S0021889877013879 doi: 10.1107/S0021889877013879]\n* 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\u2013175 [http://dx.doi.org/10.1107/S002188988001179X doi: 10.1107/S002188988001179X]\n* 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\u2013250 [http://dx.doi.org/10.1107/S0108767387011255 doi: 10.1107/S0108767387011255]\n\n===Pair Distribution Function ([[PDF]])===\n* Porod, G. ''Acta Phys. Austriaca'' '''1948''', 2, 255\u2013292. \n* 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.\n* 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]\n*  Kirsten M. \u00d8. Jensen, Anders B. Blichfeld, Sage R. Bauers, Suzannah R. Wood, Eric Dooryh\u00e9e, 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]\n\n===Autocorrelation Function===\n* 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\u2013525 [http://dx.doi.org/10.1063/1.1698419 doi: 10.1063/1.1698419]\n\n===Other===\n* 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\u20131196 [http://dx.doi.org/10.1107/S0021889803014262 doi: 10.1107/S0021889803014262]\n** 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]\n\n* 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]\n\n===Two-dimensional===\nTwo-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.\n* 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]\n\n==See Also==\n* [[Q value]]: Conversion from [[reciprocal-space]] values to realspace distances can be accomplished using <math>\\scriptstyle d = 2 \\pi / q</math>"
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            "39": {
                "pageid": 39,
                "ns": 0,
                "title": "Reciprocal-space",
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                        "*": "[[Image:P3ht_rs-face-on.png|200px|thumb|Example reciprocal-space for a material (in this case, face-on [[P3HT]]), shown in blue. The [[Ewald sphere]] is shown as a purple sheet.]]\n\n'''Reciprocal-space''' is a conceptual three-dimensional space which contains the full 3D scattering pattern of a given sample. It is the 3D [[Fourier transform]] of the sample's [[realspace]] electron-density distribution. It may also be called '''inverse-space''', ''q''-'''space''', or '''Fourier space'''.\n\nThe distribution of intensity in reciprocal-space can be arbitrarily complex. For a sample with a distinct and non-trivial internal structure, the Fourier transform will be highly complex and difficult to interpret. However, samples of scientific interest generally have certain regularities, which give rise to a reciprocal-space with recognizable [[scattering features|features]], making the data easier to understand and analyze. In particular, a well-defined repeating structure in realspace gives rise to a sharp peak in reciprocal-space. The position of the peak [[Q value|encodes]] the repeat-spacing (the peak width [[Scherrer grain size analysis|encodes]] the correlation length, etc.).\n\n==Detector Image==\nThe concept of reciprocal-space is extremely useful, because it allows one to connect between experiment geometry and the scattering equations. In particular, the scattering observed on a 2D [[detector]] plane during an experiment is actually a particular 'slice' through the 3D reciprocal-space. Thus, one can probe different regions of reciprocal-space by [[sample orientation|reorienting]] the sample with respect to the beam.\n\nThe 'slice' through reciprocal-space is not strictly a plane: it is actually the surface of a sphere, known as the [[Ewald sphere]].\n\n[[Image:R000.png|400px|thumb|center|Example of the reciprocal-space of a crystal (blue), which contains an array of peaks. The only peaks that are observed on the detector are those that intersect the Ewald sphere.]]\n\n===Conversion to ''q''-space===\nThe raw detector image needs to be converted into [[reciprocal-space]]. This is typically done by using a [[Materials#Calibration_standards|calibration standard]], which has rings at known ''q''-position. The purpose of calibration is to convert from pixel position to <math>(q_x,q_z)</math> value. One needs to know:\n# The [[X-ray energy|x-ray wavelength]]. This is known since it is set by the x-ray source (and/or monochromator).\n# The unitless detector distance D/d, where D is the detector distance and d is the detector width. Since d is typically known, one can either measure D directly (even using a tape measure is reasonably accurate), or by noting the pixel position of a ring in calibration standard.\n# Direct beam position (i.e. the pixel position of the direct beam). Even with the beam blocked by a [[beamstop]], low-''q'' beam spillover is usually sufficient to note the beam position. One can also determine the center of scattering rings to define the beam position.\n# Detector tilt. This can be assessed by noting the curvature of scattering rings from a standard sample.\nWith the above information, one can convert from the raw image into data in [[reciprocal-space]] (various pieces of [[software]] will do this for you).\n\nNotice that one can either display the data as <math>(q_x,q_z)</math> or as <math>(q_r,q_z)</math>, where:\n:<math>\nq_r = \\sqrt{ q_x^2 + q_y^2 }\n</math>\nThe <math>(q_x,q_z)</math> representation ignores the ''q<sub>y</sub>'' component of the [[momentum transfer]]. For small-angle measurements ([[GISAXS]]), this is a reasonable approximation. However for wide-angle ([[GIWAXS]]) measurements, this is a poor approximation: the curvature of the [[Ewald sphere]] means that the part of reciprocal-space probed by the detector is curving away from the <math>(q_x,q_z)</math> plane. As such, in the <math>(q_r,q_z)</math> representation, we note a '[[GI missing wedge|missing wedge]]' of data near the ''q<sub>z</sub>'' axis. In fact, in grazing-incidence geometry, we do not probe the ''true'' ''q<sub>z</sub>'' axis, except at two points (the direct beam position, and the specularly-reflected beam position).\n\n{|\n| [[Image:P3ht calibration01.png|thumb|250px|Raw detector image.]]\n| [[Image:P3ht calibration02.png|thumb|300px|Data converted to ''q''-space.]]\n| [[Image:P3ht calibration03.png|thumb|300px|Data converted to ''q''-space, taking into account the [[Ewald sphere]].]]\n|}\n\nThe conversion to ''q''-space is necessary for all subsequent analysis steps. Moreover, we typically are interested in ''\u03c7'' defined in [[reciprocal-space]] (i.e. with respect to the sample coordinates, not the instrument reference frame), so we should compute ''\u03c7'' in the <math>(q_r,q_z)</math> representation shown above.\n\n===Caveats===\nAlthough it is extremely useful to imagine that the detector image is a slice through reciprocal-space, one must keep in mind other effects can influence what one observes on the detector. For instance, anything which affects the scattered beams as they travel to the detector will influence the image one sees. For example, ambient air will scatter the beam, giving rise to a [[background]] signal (which will look like [[diffuse scattering]]). As another example, portions of the instrument (e.g. sample windows) can attenuate or obstruct the x-ray beam.\n\nThe sample itself can also affect the scattered rays, again making the detector image not strictly a map of reciprocal-space. The scattering may be attenuated as it transits through the sample; [[Attenuation correction for sample shape|this attenuation may be non-uniform]]. In [[GISAXS]], the observed image is distorted due to [[Refractive_index|refraction]] of both the incident beam and the scattered rays. This [[refraction distortion]] is non-trivial. [[Dynamic scattering]] also gives rise to multiple scattering peaks (some due to the direct beam, and some due to the reflected beam), as well as more complex effects such as the [[Yoneda]] streak and [[waveguide]] streaks.\n\nThus, especially in GISAXS, one must be careful about naively interpreting the detector image as being a direct replica of reciprocal-space. On the other hand, all of the complications noted above can be accounted for; thus GISAXS data can be quantitatively mapped into the 'true' reciprocal-space.\n\n[[Image:Detector example02.png|thumb|center|300px|An example of a detector image affected by the experimental setup: the dark regions on the right side of this [[GIWAXS]] image are actually the shadow of a screw (sitting on top of washers) that was at the corner of the sample-holding bar. It blocked the scattering from the sample. Although silly, this image is a good reminder that the experimental setup (sample chamber windows, air scattering, etc.) affects the x-ray scattering image.]]\n\n==Accessible Regions==\nTBD"
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