http://gisaxs.com/api.php?action=feedcontributions&user=68.194.136.6&feedformat=atomGISAXS - User contributions [en]2024-03-29T09:12:05ZUser contributionsMediaWiki 1.31.7http://gisaxs.com/index.php?title=Example:P3HT_orientation_analysis&diff=877Example:P3HT orientation analysis2014-06-23T00:20:29Z<p>68.194.136.6: /* P3HT orientation */</p>
<hr />
<div>This tutorial describes how to quantify the orientation distribution of the semiconducting polymer [[P3HT]] using [[GIWAXS]]. This orientation analysis is meant to determine the relative amounts of the material oriented in different ways. The same kind of analysis can be applied to other semiconducting polymers or small-molecules. In fact, the same conceptual steps can be applied more broadly to determining any kind of orientation distribution (though one must be careful in interpreting the relationship between [[reciprocal-space]] peaks, which will be different for each material's specific [[unit cell]]).<br />
<br />
==P3HT orientation==<br />
P3HT crystallizes into the [[unit cell]] shown below. P3HT crystals are anisotropic, often appearing as long needle-like structures. Note that this unit cell is highly idealized: the alkyl side-chains have considerable freedom and thus disorder (they may even be liquid-like). Moreover, real samples of P3HT organize in a more disordered state than shown here. Although the lamellar stacking is well-defined, and the aromatic (''π''-''π'') stacking is also well-defined, one typically does not observe any scattering peak along the backbone direction: i.e. there is usually no well-defined persistence of order in this direction. It is also likely that subsequent chains do not organize with the well-defined registry shown in the cartoon. Overall, P3HT is thus better thought of as being semi-crystalline or liquid-crystalline: it exhibits considerable molecular order (giving rise to scattering peaks), but is not an extended well-ordered crystal.<br />
[[Image:Cell3-main2.png|thumb|center|300px|Cartoon of an idealized P3HT [[unit cell]].]]<br />
* The lamellar stacking has a spacing of ~1.6 nm, [[Q value|giving rise]] to a peak (100) at ''q'' ~0.4 Å<sup>−1</sup>.<br />
* The aromatic (''π''-''π'') stacking has a spacing of ~0.39 nm, [[Q value|giving rise]] to a peak (010) at ''q'' ~1.6 Å<sup>−1</sup>.<br />
* The spacing along the ring direction is too disordered to give a well-defined 001 peak.<br />
<br />
When P3HT is cast as a thin-film, this unit cell can adopt a variety of orientations. Although most materials form 3D powders when cast from solution, P3HT adopts a relatively well-defined orientation with respect to the substrate. It seems that the alkyl side-chains preferentially segregate to the film-air and film-substrate interfaces (presumably in order to lower the interfacial energy), which drives the material overall into the ''edge-on'' orientation shown below:<br />
<br />
{|<br />
| [[Image:Cell3-edge on2.png|200px|thumb|edge-on]]<br />
| [[Image:Cell3-face on2.png|200px|thumb|face-on]]<br />
| [[Image:Cell3-end on2.png|200px|thumb|end-on]]<br />
|}<br />
<br />
<br />
Absent any other driving force, the above configurations are assumed to be in-plane isotropic (2D powders). In such a case, one would expect to see [[reciprocal-space]]s corresponding to these states of:<br />
<br />
{|<br />
| [[Image:P3ht_rs-edge-on.png|200px|thumb|edge-on]]<br />
| [[Image:P3ht_rs-face-on.png|200px|thumb|face-on]]<br />
| [[Image:P3ht_rs-end-on.png|200px|thumb|end-on]]<br />
| [[Image:P3ht_rs-powder.png|200px|thumb|isotropic]]<br />
|}<br />
<br />
Of course, what one observes on the detector is only where the [[Ewald sphere]] intersects [[reciprocal-space]]:<br />
<br />
{|<br />
| [[Image:P3ht_det-edge-on.png|200px|thumb|edge-on]]<br />
| [[Image:P3ht_det-face-on.png|200px|thumb|face-on]]<br />
| [[Image:P3ht_det-end-on.png|200px|thumb|end-on]]<br />
| [[Image:P3ht_det-powder.png|200px|thumb|isotropic]]<br />
|}<br />
<br />
Summarizing:<br />
* '''Edge-on''' involves the lamellar side-chains wetting the interfaces, which means the lamellar stacking is vertical (100 peak along ''q<sub>z</sub>''), and the ''π''-''π'' stacking is then in-plane (010 peak along ''q<sub>r</sub>'').<br />
* '''Face-on''' involves the aromatic rings facing the substrate, which means the lamellar stacking is in-plane (100 peak along ''q<sub>r</sub>''), and the ''π''-''π'' stacking is in the film normal direction (010 peak along ''q<sub>z</sub>'').<br />
* '''End-on''' involves the chain ends pointing towards the substrate, which means the lamellar stacking is in-plane (100 peak along ''q<sub>r</sub>''), and the ''π''-''π'' stacking is also in-plane (010 peak along ''q<sub>r</sub>''). Since the chain axis doesn't give a well-defined peak, there is no scattering along ''q<sub>z</sub>''.<br />
* '''Isotropic''' involves a complete orientational averaging of the unit cell orientation. The lamellar (100) peak and the ''π''-''π'' (010) peak will both appear to be rings of uniform intensity.<br />
<br />
==Example GIWAXS==<br />
Shown below is a typical [[GIWAXS]] image of P3HT cast on a [[Material:Silicon|silicon]] substrate:<br />
<br />
[[Image:P3ht-generic giwaxs02.png|450px|center]]<br />
<br />
Because of the ''edge-on'' orientation, the lamellar peaks appear along the ''q<sub>z</sub>'' axis. Below is a comparison of P3HT with different orientation distributions.<br />
<br />
{|<br />
| [[Image:P3ht-generic giwaxs.png|300px|center|thumb|[[Material:P3HT|P3HT]] with predominantly '''edge-on''' orientation. The 100 peak is along ''q<sub>z</sub>'', while the 010 peak is along ''q<sub>x</sub>''.]]<br />
| [[Image:P3ht-face-on giwaxs.png|300px|center|thumb|P3HT with predominantly '''face-on''' orientation. Notice the 100 peak is along ''q<sub>x</sub>'', while a weak 010 peak can be seen near the ''q<sub>z</sub>'' axis.]]<br />
| [[Image:P3ht-isotropic giwaxs.png|300px|center|thumb|P3HT with nearly '''isotropic''' orientation. Notice the extremely broad (nearly uniform) distribution of intensity for all the peaks.]]<br />
|}<br />
<br />
==Analysis: In-plane powder==<br />
In order to quantify the orientation distribution of P3HT, we need to measure the intensity of the scattering as a function of angle. We define the azimuthal angle to be ''χ'': this is the angle with respect to the ''q<sub>z</sub>'' axis. I.e. ''χ'' = 0° is the out-of-plane (''q<sub>z</sub>'') axis, while ''χ'' = 90° is the in-plane (''q<sub>r</sub>'') axis.<br />
===Step 1: Conversion to ''q''-space===<br />
The 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:<br />
# The [[X-ray energy|x-ray wavelength]]. This is known since it is set by the x-ray source (and/or monochromator).<br />
# 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 reasonable accurate), or by noting the pixel position of a ring in calibration standard.<br />
# Direct beam position (i.e. the pixel position of the direct beam). Even with the beam block 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.<br />
# Detector tilt. This can be assessed by noting the curvature of scattering rings from a standard sample.<br />
With the above information, one can convert from the raw image into data in [[reciprocal-space]] (various pieces of [[software]] will do this for you).<br />
<br />
Notice that one can either display the data as <math>(q_x,q_z)</math> or as <math>(q_r,q_z)</math>, where:<br />
:<math><br />
q_r = \sqrt{ q_x^2 + q_y^2 }<br />
</math><br />
The <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 '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).<br />
<br />
{|<br />
| [[Image:P3ht calibration01.png|thumb|250px|Raw detector image.]]<br />
| [[Image:P3ht calibration02.png|thumb|300px|Data converted to ''q''-space.]]<br />
| [[Image:P3ht calibration03.png|thumb|300px|Data converted to ''q''-space, taking into account the [[Ewald sphere]].]]<br />
|}<br />
<br />
The conversion to ''q''-space is necessary for all subsequent analysis steps. Moreover, we typically are interested in ''χ'' defined in [[reciprocal-space]] (i.e. with respect to the sample coordinates, not the instrument reference frame), so we should compute ''χ'' in the <math>(q_r,q_z)</math> representation shown above.<br />
<br />
===Step 2: Linecut===<br />
From the converted data, we can take a linecut along a ''χ''-arc. In the present analysis, the most useful place to take such a cut is along the 100 lamellar peak (and then along the 010 aromatic stacking peak). When taking the linecut, one should consider:<br />
# When extracting the intensities along the arc, one should try to integrate the full peak width. The integration must be wide enough to take the full peak into account for all ''χ'' (otherwise the intensity will change along ''χ'' simply due to the fraction of the peak one is probing.<br />
# The intensity data should be [[background]]-subtracted. In this case, subtracting the [[Background#Local_background|local background]] makes the most sense: i.e. subtract the intensity just outside the arc of interest. (This must be done at each corresponding ''χ'', since the background may not be isotropic.)<br />
# The ''χ''-scale should be shifted in order to take into account the above-noted curvature of the Ewald sphere. Thus, there will be a small gap in the data near ''χ'' = 0°.<br />
# One should ignore the [[Yoneda]] streak, since it exhibits an intensity enhancement. Any data at ''χ'' beyond the Yoneda should also be discarded since this data comes from below the [[horizon]], and is thus attenuated. For the P3HT 100 peak, this usually means eliminating ''χ'' > ~82°.<br />
# In principle, one should re-adjust the ''χ''-scale to take into account [[refraction distortion]]s: the grazing-incidence geometry induces substantial beam refractions, which causes [[reciprocal-space]] to look warped on the detector image. This distortion only affects the ''q<sub>z</sub>'' component, and is greatest near the Yoneda. This distortion is substantial in [[GISAXS]], but is usually negligible in [[GIWAXS]].<br />
<br />
[[Image:P3ht chi example.png|400px|thumb|center|Example of integrated peak intensity along ''χ''-arcs for an edge-on P3HT sample. The dark line is the lamellar 100 peak, the grey line is the aromatic 010 peak.]]<br />
<br />
===Step 3: Account for sample symmetry===<br />
The ''I'' vs. ''χ'' curve shown above can be thought of as the orientation distribution of the P3HT. However, one must be careful in directly interpreting the intensity as representing the ''amount of material''. The reason for this (as usual) is that the [[Ewald sphere]] is probing only a single slice through the 3D reciprocal-space. So we are not probing the full scattering intensity for any given population.<br />
<br />
How to properly account for the full scattering intensity depends on sample symmetry. In the most general sense, we can rotate the sample, accumulate multiple images, and reconstruct the full 3D reciprocal-space. However, it is far easier to account for this mathematically, if we know the sample symmetry. It turns out, that for an in-plane powder, we must simply multiply the above curve by <math>\sin(\chi)</math>. For a detailed explanation of why this is the case, refer to [[Integrated intensity|integrated intensity]]. Roughly, this factor arises from the spherical coordinate system, aligned with the symmetry axis of the 2D powder. More heuristically, one can imagine that observed scattering intensity near ''q<sub>r</sub>'' actually represents a whole ring of scattering in the 3D reciprocal-space. This 'ring' becomes smaller at larger ''χ''; the <math>\sin(\chi)</math> factor exactly accounts for this variation.<br />
<br />
{|<br />
| [[Image:p3ht_chi_example02.png|350px]]<br />
| [[Image:p3ht_chi_example03.png|350px]]<br />
|}<br />
<br />
The corrected data (right panel above) now has the intensity being proportional to the ''amount of material'' oriented at that ''χ'' angle. This curve can be reported as an orientation distribution for the material (closely related to [[pole figures]]).<br />
<br />
===Step 4: Integrate populations===<br />
One can now integrate the intensity in the corrected curve, to come up with estimates of the relative amount of material. There are many ways one could choose to do this. In this example, we divide the scattering intensity into three populations:<br />
* '''Isotropic''': The scattering intensity arising from the subset of grains that form an isotropic distribution.<br />
* '''Edge-on''': The scattering intensity arising from the edge-on grains. We include all grains with an orientation χ<45°. <br />
* '''Face-on''': The scattering intensity arising from the face-on grains. We include all grains with an orientation χ>45°.<br />
<br />
Note that we are here assuming that all the intensity from the 100 scattering along ''q<sub>r</sub>'' arises from entirely from ''face-on'' material. However, both ''face-on'' and ''end-on'' material would give rise to scattering at that position in [[reciprocal-space]]. To differentiate between those two possibilities, one must invoke additional data. Specifically, one can look at the 010 (''π''-''π'') peak: for ''end-on'', the ''π''-''π'' peak will also be in-plane (along ''q<sub>r</sub>''), whereas for ''face-on'', one should observe substantial intensity of the ''π''-''π'' peak in the out-of-plane direction (near ''q<sub>z</sub>'' axis).<br />
<br />
For the present example, the ''end-on'' orientation was excluded as a possibility, and therefore we ascribe all of the ''q<sub>r</sub>'' scattering to the ''face-on'' orientation. We compute integrated quantities by summing up the area under the curve, and distributing it into the three categories noted above. Importantly, the isotropic population appears as a uniform baseline (isotropic ring) in the original data, and thus appears as a <math>\sin(\chi)</math> baseline in the corrected data.<br />
<br />
{|<br />
| [[Image:Pop schematict03.png|300px|]]<br />
| [[Image:Pop schematict04b.png|300px|]]<br />
|}<br />
<br />
Note again that only the intensities in the corrected curve (right panel above) are linearly representative of the ''amount of material''. We can compare the three integrated values to the total, and thereby compute a percentage for edge-on, face-on, and isotropic.<br />
<br />
==Analysis: In-plane aligned==<br />
Note that for a sample that ''isn't'' an in-plane powder (e.g. P3HT aligned in a grating), then the above symmetry arguments would change. In that case, one would need to instead account for the orientation distribution in the in-plane direction (''ϕ''). Note that if the material were highly aligned with the grating axis ((''ϕ'' = 0°), then a single measurement in this aligned geometry would be sufficient to assess the full orientation distribution. In other words, the original uncorrected data would correctly represent the relative amounts of material:<br />
<br />
[[Image:Pop schematict03.png|300px|]]<br />
<br />
See also [[grating alignment]] for caveats related to the in-plane angle (''ϕ'').<br />
<br />
==Literature==<br />
<br />
===Development/description of analysis method===<br />
* [http://pubs.acs.org/doi/abs/10.1021/nn202515z Nanoimprint-Induced Molecular Orientation in Semiconducting Polymer Nanostructures] Htay Hlaing, Xinhui Lu, Tommy Hofmann, [[Kevin Yager|Kevin G. Yager]], Charles T. Black, and Benjamin M. Ocko ''ACS Nano'' 2011, 5 (9), 7532-7538 [http://dx.doi.org/10.1021/nn202515z doi: 10.1021/nn202515z]<br />
* [http://scitation.aip.org/content/aip/journal/apl/99/16/10.1063/1.3651509 Enhanced charge collection in confined bulk heterojunction organic solar cells] Jonathan E. Allen, [[Kevin G. Yager]], Htay Hlaing, Chang-Yong Nam, Benjamin M. Ocko and Charles T. Black ''Applied Physics Letters'' 2011, 99, 163301. [http://dx.doi.org/10.1063/1.3651509 doi: 10.1063/1.3651509] '''c.f. [ftp://ftp.aip.org/epaps/appl_phys_lett/E-APPLAB-99-015142/080111_TemplatedOrganicSolarCells_SupplementaryInformation_APL_final.pdf Supplementary Information]'''<br />
* [http://pubs.acs.org/doi/abs/10.1021/nl301759j One-Volt Operation of High-Current Vertical Channel Polymer Semiconductor Field-Effect Transistors] Johnston, D.E.; [[Kevin Yager|Yager, K.G.]]; Nam, C.-Y.; Ocko, B.M.; Black, C.T. ''Nano Letters'' 2012, 8, 4181–4186 [http://dx.doi.org/10.1021/nl301759j doi: 10.1021/nl301759j]<br />
* '''[http://pubs.acs.org/doi/abs/10.1021/nn4060539 Nanostructured Surfaces Frustrate Polymer Semiconductor Molecular Orientation]''' Johnston, D.E.; [[Kevin Yager|Yager, K.G.]]; Hlaing, H.; Lu, X.; Ocko, B.M.; Black, C.T. ''ACS Nano'' 2014 [http://dx.doi.org/10.1021/nn4060539 doi: 10.1021/nn4060539]<br />
<br />
===Application of method===<br />
* [http://pubs.acs.org/doi/abs/10.1021/cm501251n Stable and Controllable Polymer/Fullerene Composite Nanofibers through Cooperative Noncovalent Interactions for Organic Photovoltaics] Fei Li, [[Kevin G. Yager]], Noel M. Dawson, Ying-Bing Jiang, Kevin J. Malloy, and Yang Qin ''Chemistry of Materials'' 2014 [http://dx.doi.org/10.1021/cm501251n doi: 10.1021/cm501251n]<br />
===Related papers===<br />
====Angular correction (curvature of [[Ewald sphere]])====<br />
* '''[http://pubs.acs.org/doi/abs/10.1021/la904840q Quantification of Thin Film Crystallographic Orientation Using X-ray Diffraction with an Area Detector]''' Jessy L. Baker, Leslie H. Jimison, Stefan Mannsfeld, Steven Volkman, Shong Yin, Vivek Subramanian, Alberto Salleo, A. Paul Alivisatos and Michael F. Toney ''Langmuir'' 2010, 26 (11), 9146-9151. [http://dx.doi.org/10.1021/la904840q doi: 10.1021/la904840q]<br />
* [http://scripts.iucr.org/cgi-bin/paper?S0021889808001064 Simulating X-ray diffraction of textured films] D. W. Breiby, O. Bunk, J. W. Andreasen, H. T. Lemke and M. M. Nielsen ''J. Appl. Cryst.'' 2008, 41, 262-271. [http://dx.doi.org/10.1107/S0021889808001064 doi: 10.1107/S0021889808001064]<br />
====sin(angle) correction====<br />
* [http://pubs.acs.org/doi/abs/10.1021/nl501233g Confinement-Driven Increase in Ionomer Thin-Film Modulus] Kirt A. Page, Ahmet Kusoglu, Christopher M. Stafford, Sangcheol Kim, R. Joseph Kline, and Adam Z. Weber ''Nano Letters'' 2014, 14 (5), 2299-2304. [http://dx.doi.org/10.1021/nl501233g doi: 10.1021/nl501233g]<br />
====Other orientation analyses====<br />
* [http://scripts.iucr.org/cgi-bin/paper?S0021889806038957 Evaluation of equatorial orientation distributions] C. Burger and W. Ruland ''J. Appl. Cryst.'' 2006, 39, 889-891. [http://dx.doi.org/10.1107/S0021889806038957 doi: 10.1107/S0021889806038957]<br />
* [http://scripts.iucr.org/cgi-bin/paper?S0021889807010503 Two-dimensional small-angle X-ray scattering of self-assembled nanocomposite films with oriented arrays of spheres: determination of lattice type, preferred orientation, deformation and imperfection] W. Ruland and B. M. Smarsly ''J. Appl. Cryst.'' 2007, 40, 409-417. [http://dx.doi.org/10.1107/S0021889807010503 doi: 10.1107/S0021889807010503]<br />
* [http://pubs.acs.org/doi/abs/10.1021/nl903187v Device-Scale Perpendicular Alignment of Colloidal Nanorods] Jessy L. Baker, Asaph Widmer-Cooper, Michael F. Toney, Phillip L. Geissler and A. Paul Alivisatos ''Nano Letters'' 2010, 10 (1), 195-201. [http://dx.doi.org/10.1021/nl903187v doi: 10.1021/nl903187v]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=332Technical articles2014-06-05T02:19:15Z<p>68.194.136.6: /* Scattering concepts */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]/[[SANS]]<br />
** [[CD-SAXS]] ([[RSANS]])<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
* [[Scattering intensity]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Fourier transform]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
* [[Refractive index]]<br />
* [[Absorption length]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=X9&diff=331X92014-06-05T02:17:04Z<p>68.194.136.6: /* Flux */</p>
<hr />
<div>'''X9''' is an [[x-ray]] [[synchrotron]] [[beamline]] at the National Synchrotron Light Source ([[NSLS]]), Brookhaven National Lab (BNL). It is a joint beamline between the NSLS and the [http://www.bnl.gov/cfn/ Center for Functional Nanomaterials] (which is a partner user). The instrument is optimized for doing Small-Angle X-ray Scattering ([[SAXS]]) and Grazing-Incidence Small-Angle X-ray Scattering ([[GISAXS]]), as well as wide-angle Grazing-Incidence X-ray Diffraction (GID/GIXD/[[GIWAXS]]).<br />
<br />
[[Image:X9_NSLS_01.jpg|400px]][[Image:X9_NSLS_02.jpg|400px]]<br />
<br />
==Capabilities==<br />
[[Image:X9-scale.png|center|thumb|600px|X9 hutch layout]]<br />
X9 can perform a variety of x-ray scattering experiments:<br />
* [[SAXS]] and [[WAXS]] on powders or solutions in capillaries. Holder can heat from RT to 80°C.<br />
* [[SAXS/WAXS]] (simultaneous) on proteins or other macromolecules in solution, using a full-vacuum path and a flow-through cell (for robust background subtraction).<br />
* [[GISAXS]] and [[GIWAXS]] on thin films. Sample holder can accommodate multiple samples (for automated measurements), and can heat from RT to 220°C.<br />
<br />
==Performance==<br />
===Energy range===<br />
Can perform experiments from 6 keV to 20 keV.<br />
<br />
===q-range===<br />
The instrument can capture data on both a SAXS and WAXS detector (either simultaneously or sequentially). The '''q'''-range depends on x-ray energy and the sample-detector distance. For SAXS, one can typically obtain (e.g. for 13.5 keV):<br />
* '''5.4 m''': 0.002 Å<sup>−1</sup> to 0.1 Å<sup>−1</sup> (314 nm to 6.3 nm)<br />
* '''4.1 m''': 0.005 Å<sup>−1</sup> to 0.2 Å<sup>−1</sup> (125 nm to 3.5 nm)<br />
* '''3.0 m''': 0.004 Å<sup>−1</sup> to 0.2 Å<sup>−1</sup> (157 nm to 3.5 nm)<br />
* '''2.0 m''': 0.010 Å<sup>−1</sup> to 0.4 Å<sup>−1</sup> (63 nm to 1.6 nm)<br />
* '''1.5 m''': 0.011 Å<sup>−1</sup> to 0.45 Å<sup>−1</sup> (63 nm to 1.4 nm)<br />
<br />
For WAXS, standard configurations yields approximately (for 13.5 keV):<br />
* '''0.235 m''' (sample inside chamber): 0.13 Å<sup>−1</sup> to 4.1 Å<sup>−1</sup> (4.8 nm to 0.15 nm)<br />
* '''~0.4 m''' (sample upstream of chamber): 0.15 Å<sup>−1</sup> to 2.1 Å<sup>−1</sup> (4.2 nm to 0.3 nm)<br />
<br />
===Resolution===<br />
* SAXS: Pixel-limited resolution at 5 m (sample-detector) is ~0.0017°, or 0.0002 Å<sup>−1</sup> (at ~13.5 keV).<br />
* WAXS: Pixel-limited resolution is ~0.04°, or 0.0046 Å<sup>−1</sup> (at ~13.5 keV). Actual resolution depends on experimental setup and sample size.<br />
<br />
===Flux===<br />
* At 10 keV, flux capture by mirrors is ~2×10<sup>12</sup> ph/s/0.01%bw<br />
* At 10 keV, flux at sample is ~5×10<sup>11</sup> ph/s/0.01%bw<br />
* At 10 keV, microfocused (10 µm), flux at sample is ~2×10<sup>11</sup> ph/s/0.01%bw<br />
<br />
===Beam Size===<br />
* TSAXS: Typical beam size is ~150 µm horizontal and ~100 µm vertical.<br />
* GISAXS: Typical beam size is ~120 µm horizontal and ~50 µm vertical.<br />
* Microbeam: Beam can be focused to ~15 µm (horizontal and vertical).<br />
<br />
==Detectors==<br />
====Dectris Pilatus 1M (SAXS)====<br />
* Hybrid pixel detector.<br />
* [https://www.dectris.com/pilatus3_specifications.html#main_head_navigation Specs]<br />
* Format: 981×1043 = 1,023,183 pixels.<br />
* Pixel size: 172×172 µm<sup>2</sup>.<br />
* Area: 168.7×179.4 mm<sup>2</sup>.<br />
<br />
====Mar CCD (SAXS)====<br />
* Fiber-coupled CCD detector.<br />
* [http://www.mar-usa.com/support/downloads/sx_series.pdf Specs]<br />
* Outputs 16-bit Grayscale TIFF files with MSB endianess.<br />
* Pixel size (using 1024×1024 binning) is 161 μm (software says 158 um).<br />
* Active area is 165 mm diameter.<br />
<br />
====Dectris Pilatus 300k (SAXS)====<br />
* Hybrid pixel detector.<br />
* [http://www.dectris.com/sites/pilatus300k.html Specs]<br />
* Format: 487×619 = 301,453 pixels.<br />
* Pixel size: 172×172 µm<sup>2</sup>.<br />
* Area: 83.8×106.5 mm<sup>2</sup>.<br />
<br />
====Photonic Science (WAXS)====<br />
* Fiber-coupled CCD detector.<br />
* Outputs 16-bit Grayscale TIFF files with MSB endianess.<br />
* Pixel size (1042×1042): 101.7 μm.<br />
* Detector image is 106 mm in diameter.<br />
<br />
==Access==<br />
X9 is available through [http://www.bnl.gov/ps/nsls/users/access/beamtime-new_users.asp NSLS] or [http://www.bnl.gov/cfn/user/ CFN] user programs. However, with [[NSLS]] shutting down permanently in September 2014, new proposals are no longer being accepted.<br />
<br />
==Safety==<br />
Users must complete [http://www.bnl.gov/ps/nsls/users/access/training.asp NSLS safety training], as well as receive beamline-specific training:<br />
* [http://beamlines.ps.bnl.gov/forms/BLOSA/X9.pdf X9 BLOSA form]<br />
<br />
==See Also==<br />
* [http://www.bnl.gov/ps/x9/ X9 NSLS Official Site]</div>68.194.136.6http://gisaxs.com/index.php?title=Diffuse_scattering&diff=330Diffuse scattering2014-06-05T02:14:16Z<p>68.194.136.6: /* Analysis: low-q */</p>
<hr />
<div>'''Diffuse scattering''' is the scattering that arises from any departure of the material structure from that of a perfectly regular [[lattice]]. One can think of it as the signal that arises from disordered structures, and it appears in experimental data as scattering spread over a wide ''q''-range (diffuse). Diffuse scattering is generally difficult to quantify, because of the wide variety of effects that contribute to it.<br />
<br />
Bragg diffraction occurs when scattering amplitudes add ''constructively''. If there is a defect in a crystal lattice (e.g. atom missing or in a slightly 'wrong' position), then the amplitude of the Bragg peak decreases. This 'lost' scattering intensity is redistributed into diffuse scattering. The diffuse scattering thus arises from the local (short range) configuration of the material (not the long-range structural order).<br />
<br />
In the limit of disorder, one entirely lacks a realspace lattice and thus scattering does not generate any Bragg peaks. However, a disordered structure will still give rise to diffuse scattering. The [[Fourier transform]] of a disordered structure will not give any well-defined peaks, but will give a distribution of scattering intensity over a wide range of ''q''-values. Thus samples with an inherently disordered structure (polymer blends, randomly packed nanoparticles, etc.) will only generate diffuse scattering.<br />
<br />
==Causes==<br />
This is only a partial list of sources of diffuse scattering:<br />
* '''Thermal motion''' causes atoms to jitter about their ideal unit cell positions, which decorelates them. This suppresses the intensity of the Bragg peaks, especially the higher-order peaks (see [[Debye-Waller factor]]), and instead generates high-''q'' diffuse scattering. (One can also think of this in terms of phonons: in ordered systems the diffuse scattering is probing phonon modes.)<br />
* '''Static disorder''' in crystals (vacancy defects, substitutional defects, stacking faults, etc.) similarly creates diffuse scattering.<br />
* '''Grain structure''' in otherwise ordered materials will also contribute. The grains themselves can count as 'scattering objects', but since their size is ill-defined, the grain boundaries give rise to diffuse scattering.<br />
* '''Nanoscale disorder''' gives rise to low-''q'' diffuse scattering. For instance, a disordered polymer blend (or a bulk heterojunction) or a random packing of nanoparticles, will generate substantial low-''q'' diffuse scattering.<br />
* '''Surface roughness''' in thin films measured by [[GISAXS]] gives rise to low-''q'' diffuse scattering in GISAXS. Roughness will tend to broaden (and increase the intensity of) the specular rod, and will also generate intense low-''q'' scattering.<br />
* '''Particle size/shape polydispersity''' introduces a diffuse background.<br />
* '''Polymer chains in solution''' generate scattering without a well-defined size-scale. This is normally interpreted in terms of the [[form factor]] of the polymer chain. However one can also think of it as the polymer chains having disordered arrangements and thus giving rise to diffuse scattering (c.f. [[definitional boundaries]]).<br />
<br />
==Analysis: low-''q''==<br />
Diffuse scattering can be difficult to quantify, since so many different effects contribute to it. Nevertheless, if one has a good understanding of the expected kind of disorder, one can fit the diffuse scattering with a model.<br />
<br />
===Ornstein-Zernike model===<br />
Yields correlation length (''ξ''):<br />
::<math><br />
I(q) \propto \frac{1}{ 1 + q^2 \xi^2 }<br />
</math><br />
* References:<br />
** M. Kahlweit , R. Strey , P. Firman "[http://pubs.acs.org/doi/abs/10.1021/j100276a038 Search for tricritical points in ternary systems: water-oil-nonionic amphiphile]" ''J. Phys. Chem.'' 1986, 90, 4, 674. [http://dx.doi.org/10.1021/j100276a038 doi: 10.1021/j100276a038]<br />
** M. Teubner and R. Strey "[http://scitation.aip.org/content/aip/journal/jcp/87/5/10.1063/1.453006 Origin of the scattering peak in microemulsions]" '' J. Chem. Phys.'' 1987, 87, 3195. [http://dx.doi.org/10.1063/1.453006 doi: 10.1063/1.453006]<br />
** U. Nellen, J. Dietrich, L. Helden, S. Chodankar, K. Nygård, J. Friso van der Veenbc, C. Bechingerad "[http://pubs.rsc.org/en/Content/ArticleLanding/2011/SM/c1sm05103b#!divAbstract Salt-induced changes of colloidal interactions in critical mixtures]" ''Soft Matter'' 2011, 7, 5360. [http://dx.doi.org/10.1039/c1sm05103b doi: 10.1039/c1sm05103b]<br />
<br />
===Debye-Bueche random two-phase model===<br />
Yields correlation length (''a''):<br />
::<math><br />
\frac{\mathrm{d}S}{\mathrm{d}W} = \frac{A}{ (1 + (qa)^2 )^2 }<br />
</math><br />
<br />
* References:<br />
** P. Debye, A.M. Bueche "[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. [http://link.aip.org/link/doi/10.1063/1.1698419 doi: 10.1063/1.1698419]<br />
** P. Debye, H.R. Anderson, H. Brumberger "[http://scitation.aip.org/content/aip/journal/jap/28/6/10.1063/1.1722830 Scattering by an Inhomogeneous Solid. II. The Correlation Function and Its Application]" ''J. Appl. Phys.'' 1957, 28, 679. [http://link.aip.org/link/doi/10.1063/1.1722830 doi: 10.1063/1.1722830]<br />
** [http://www.ncnr.nist.gov/resources/sansmodels/DAB.html NCNR page]<br />
<br />
===Guinier model===<br />
Yields average radius of gyration (''R<sub>g</sub>''):<br />
::<math><br />
I(q) \propto e^{ R_g^2 q^2 / 3 }<br />
</math><br />
<br />
==Analysis: high-''q''==<br />
===Porod law===<br />
For high-''q'', gives specific surface area (''S''):<br />
::<math><br />
I(q) \propto S q^{-4}<br />
</math><br />
* References:<br />
** W. Ruland "[http://scripts.iucr.org/cgi-bin/paper?S0021889871006265 Small-angle scattering of two-phase systems: determination and significance of systematic deviations from Porod's law]" ''J. Appl. Cryst.'' 1971, 4, 70. [http://dx.doi.org/10.1107/S0021889871006265 doi: 10.1107/S0021889871006265]<br />
** J. T. Koberstein, B. Morra and R. S. Stein "[http://scripts.iucr.org/cgi-bin/paper?S0021889880011478 The determination of diffuse-boundary thicknesses of polymers by small-angle X-ray scattering]" ''J. Appl. Cryst.'' 1980, 13, 34. [http://dx.doi.org/10.1107/S0021889880011478 doi: 10.1107/S0021889880011478]<br />
** [http://en.wikipedia.org/wiki/Porod%27s_law Wikipedia: Porod law]<br />
<br />
===Porod fractal law===<br />
For high-''q'', gives specific surface area:<br />
:: <math>\lim_{q \rightarrow \infty} I(q) \propto S' q^{-(6-d)}</math><br />
* References:<br />
** [http://en.wikipedia.org/wiki/Porod%27s_law Wikipedia: Porod law]<br />
<br />
==See Also==<br />
* [http://neutrons.ornl.gov/conf/nxs2011/pdf/lectures/Diffuse11-GeneIce.pdf Diffuse Scattering] from ORNL<br />
* [http://www.neutron.ethz.ch/education/Lectures/neutronfall/Lecture_4-2 Diffuse scattering in crystals] from ETH Zurich</div>68.194.136.6http://gisaxs.com/index.php?title=Diffuse_scattering&diff=329Diffuse scattering2014-06-05T02:13:20Z<p>68.194.136.6: /* Guinier model */</p>
<hr />
<div>'''Diffuse scattering''' is the scattering that arises from any departure of the material structure from that of a perfectly regular [[lattice]]. One can think of it as the signal that arises from disordered structures, and it appears in experimental data as scattering spread over a wide ''q''-range (diffuse). Diffuse scattering is generally difficult to quantify, because of the wide variety of effects that contribute to it.<br />
<br />
Bragg diffraction occurs when scattering amplitudes add ''constructively''. If there is a defect in a crystal lattice (e.g. atom missing or in a slightly 'wrong' position), then the amplitude of the Bragg peak decreases. This 'lost' scattering intensity is redistributed into diffuse scattering. The diffuse scattering thus arises from the local (short range) configuration of the material (not the long-range structural order).<br />
<br />
In the limit of disorder, one entirely lacks a realspace lattice and thus scattering does not generate any Bragg peaks. However, a disordered structure will still give rise to diffuse scattering. The [[Fourier transform]] of a disordered structure will not give any well-defined peaks, but will give a distribution of scattering intensity over a wide range of ''q''-values. Thus samples with an inherently disordered structure (polymer blends, randomly packed nanoparticles, etc.) will only generate diffuse scattering.<br />
<br />
==Causes==<br />
This is only a partial list of sources of diffuse scattering:<br />
* '''Thermal motion''' causes atoms to jitter about their ideal unit cell positions, which decorelates them. This suppresses the intensity of the Bragg peaks, especially the higher-order peaks (see [[Debye-Waller factor]]), and instead generates high-''q'' diffuse scattering. (One can also think of this in terms of phonons: in ordered systems the diffuse scattering is probing phonon modes.)<br />
* '''Static disorder''' in crystals (vacancy defects, substitutional defects, stacking faults, etc.) similarly creates diffuse scattering.<br />
* '''Grain structure''' in otherwise ordered materials will also contribute. The grains themselves can count as 'scattering objects', but since their size is ill-defined, the grain boundaries give rise to diffuse scattering.<br />
* '''Nanoscale disorder''' gives rise to low-''q'' diffuse scattering. For instance, a disordered polymer blend (or a bulk heterojunction) or a random packing of nanoparticles, will generate substantial low-''q'' diffuse scattering.<br />
* '''Surface roughness''' in thin films measured by [[GISAXS]] gives rise to low-''q'' diffuse scattering in GISAXS. Roughness will tend to broaden (and increase the intensity of) the specular rod, and will also generate intense low-''q'' scattering.<br />
* '''Particle size/shape polydispersity''' introduces a diffuse background.<br />
* '''Polymer chains in solution''' generate scattering without a well-defined size-scale. This is normally interpreted in terms of the [[form factor]] of the polymer chain. However one can also think of it as the polymer chains having disordered arrangements and thus giving rise to diffuse scattering (c.f. [[definitional boundaries]]).<br />
<br />
==Analysis: low-''q''==<br />
Diffuse scattering can be difficult to quantify, since so many different effects contribute to it. Nevertheless, if one has a good understanding of the expected kind of disorder, one can fit the diffuse scattering with a model.<br />
<br />
===Ornstein-Zernike model===<br />
Yields correlation length (''ξ''):<br />
::<math><br />
I(q) \propto \frac{1}{ 1 + q^2 \xi^2 }<br />
</math><br />
* References:<br />
** M. Kahlweit , R. Strey , P. Firman "[http://pubs.acs.org/doi/abs/10.1021/j100276a038 Search for tricritical points in ternary systems: water-oil-nonionic amphiphile]" ''J. Phys. Chem.'' 1986, 90, 4, 674. [http://dx.doi.org/10.1021/j100276a038 doi: 10.1021/j100276a038]<br />
** M. Teubner and R. Strey "[http://scitation.aip.org/content/aip/journal/jcp/87/5/10.1063/1.453006 Origin of the scattering peak in microemulsions]" '' J. Chem. Phys.'' 1987, 87, 3195. [http://dx.doi.org/10.1063/1.453006 doi: 10.1063/1.453006]<br />
** U. Nellen, J. Dietrich, L. Helden, S. Chodankar, K. Nygård, J. Friso van der Veenbc, C. Bechingerad "[http://pubs.rsc.org/en/Content/ArticleLanding/2011/SM/c1sm05103b#!divAbstract Salt-induced changes of colloidal interactions in critical mixtures]" ''Soft Matter'' 2011, 7, 5360. [http://dx.doi.org/10.1039/c1sm05103b doi: 10.1039/c1sm05103b]<br />
<br />
===Debye-Bueche random two-phase model===<br />
Yields correlation length (''a''):<br />
::<math><br />
\frac{\mathrm{d}S}{\mathrm{d}W} = \frac{A}{ (1 + (qa)^2 )^2 }<br />
</math><br />
<br />
* References:<br />
** P. Debye, A.M. Bueche "[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. [http://link.aip.org/link/doi/10.1063/1.1698419 doi: 10.1063/1.1698419]<br />
** P. Debye, H.R. Anderson, H. Brumberger "[http://scitation.aip.org/content/aip/journal/jap/28/6/10.1063/1.1722830 Scattering by an Inhomogeneous Solid. II. The Correlation Function and Its Application]" ''J. Appl. Phys.'' 1957, 28, 679. [http://link.aip.org/link/doi/10.1063/1.1722830 doi: 10.1063/1.1722830]<br />
** [http://www.ncnr.nist.gov/resources/sansmodels/DAB.html NCNR page]<br />
<br />
===Guinier model===<br />
Yields average radius of gyration<br />
<br />
==Analysis: high-''q''==<br />
===Porod law===<br />
For high-''q'', gives specific surface area (''S''):<br />
::<math><br />
I(q) \propto S q^{-4}<br />
</math><br />
* References:<br />
** W. Ruland "[http://scripts.iucr.org/cgi-bin/paper?S0021889871006265 Small-angle scattering of two-phase systems: determination and significance of systematic deviations from Porod's law]" ''J. Appl. Cryst.'' 1971, 4, 70. [http://dx.doi.org/10.1107/S0021889871006265 doi: 10.1107/S0021889871006265]<br />
** J. T. Koberstein, B. Morra and R. S. Stein "[http://scripts.iucr.org/cgi-bin/paper?S0021889880011478 The determination of diffuse-boundary thicknesses of polymers by small-angle X-ray scattering]" ''J. Appl. Cryst.'' 1980, 13, 34. [http://dx.doi.org/10.1107/S0021889880011478 doi: 10.1107/S0021889880011478]<br />
** [http://en.wikipedia.org/wiki/Porod%27s_law Wikipedia: Porod law]<br />
<br />
===Porod fractal law===<br />
For high-''q'', gives specific surface area:<br />
:: <math>\lim_{q \rightarrow \infty} I(q) \propto S' q^{-(6-d)}</math><br />
* References:<br />
** [http://en.wikipedia.org/wiki/Porod%27s_law Wikipedia: Porod law]<br />
<br />
==See Also==<br />
* [http://neutrons.ornl.gov/conf/nxs2011/pdf/lectures/Diffuse11-GeneIce.pdf Diffuse Scattering] from ORNL<br />
* [http://www.neutron.ethz.ch/education/Lectures/neutronfall/Lecture_4-2 Diffuse scattering in crystals] from ETH Zurich</div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor&diff=328Form Factor2014-06-05T02:05:28Z<p>68.194.136.6: /* Equations */</p>
<hr />
<div>The '''Form Factor''' is the scattering which results from the ''shape'' of a particle. When particles are distributed without any particle-particle correlations (e.g. dilute solution of non-interacting particles, freely floating), then the scattering one observes is entirely the form factor. By comparison, when particles are in a well-defined structure, the scattering is dominated by the [[structure factor]]; though even in these cases the form factor continues to contribute, since it modulates both the structure factor and the [[diffuse scattering]].<br />
<br />
When reading discussions of scattering modeling, one must be careful about the usage of the term 'form factor'. This same term is often used to describe three different (though related) quantities:<br />
* <math>F(\mathbf{q})</math>, the '''form factor amplitude''' arising from a single particle; this can be thought of as the 3D [[reciprocal-space]] of the particle, and is thus in general anisotropic.<br />
* <math>|F(\mathbf{q})|^2</math>, the '''form factor intensity'''; whereas the amplitude cannot be measured experimentally, the form factor intensity in principle can be.<br />
* <math>P(q) = \left\langle |F(\mathbf{q})|^2 \right\rangle </math>, the '''isotropic form factor intensity'''; i.e. the orientational averaged of the form factor square. This is the 1D scattering that is measured for, e.g., particles freely distributed distributed in solution (since they tumble randomly and thus average over all possible orientations).<br />
<br />
==Equations==<br />
In the most general case of an arbitrary distribution of [[Scattering Length Density|scattering density]], <math>\rho(\mathbf{r})</math>, the form factor is computed by integrating over all space:<br />
<br />
:<math><br />
F_{j}(\mathbf{q}) = \int \rho_j(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V <br />
</math><br />
<br />
The subscript denotes that the form factor is for particle ''j''; in multi-component systems, each particle has its own form factor. For a particle of uniform density and volume ''V'', we denote the scattering contrast with respect to the ambient as <math>\Delta \rho</math>, and the form factor is simply:<br />
<br />
:<math><br />
F_{j}(\mathbf{q}) = \Delta \rho \int\limits_{V} e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V <br />
</math><br />
<br />
For monodisperse particles, the average (isotropic) form factor intensity is an average over all possible particle orientations:<br />
:<math><br />
\begin{alignat}{2}<br />
P_j(q) & = \left\langle |F_j(\mathbf{q})|^2 \right\rangle \\<br />
& = \int\limits_{\phi=0}^{2\pi}\int\limits_{\theta=0}^{\pi} | F_j(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi<br />
\end{alignat}<br />
</math><br />
<br />
Note that in cases where particles are not monodisperse, then the above average would also include averages over the distritubions in particle size and/or shape. Note that for <math>q=0</math>, we expect:<br />
:<math><br />
\begin{alignat}{2}<br />
F(0) & = \int\limits_{\mathrm{all\,\,space}} \rho(\mathbf{r}) e^{0} \mathrm{d}\mathbf{r} = \rho_{\mathrm{total}} \\<br />
& = \Delta \rho \int\limits_{V} e^{0} \mathrm{d}\mathbf{r} = \Delta \rho V<br />
\end{alignat}<br />
</math><br />
And so:<br />
:<math><br />
\begin{alignat}{2}<br />
P(0) & = \left\langle \left| F(0) \right|^2 \right\rangle \\<br />
& = 4 \pi \Delta \rho^2 V^2<br />
\end{alignat}<br />
</math><br />
As expected, scattering intensity scales with the square of the scattering contrast and the particle volume. For multi-component systems, this has the effect of greatly emphasizing larger particles. For instance, a 2-fold increase in particle diameter results in a <math>(2^3)^2 = 64</math>-fold increase in scattering intensity.<br />
<br />
==Form Factor Equations==<br />
* [[Form Factor:Sphere|Sphere]]<br />
* [[Form Factor:Ellipsoid of revolution|Ellipsoid of revolution]]<br />
* [[Form Factor:Cube|Cube]]<br />
<br />
==Form Factor Equations in the Literature==<br />
The following is a partial list of form factors that have been published in the literature:<br />
* [http://www.ncnr.nist.gov/resources/old_applets/index.html NCNR SANS solution form factors]:<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Sphere.html Sphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyHardSphere.html PolyHardSphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyRectSphere.html PolyRectSphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/CoreShell.html CoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyCoreShell.html PolyCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyCoreShellRatio.html PolyCoreShellRatio]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Cylinder.html Cylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/HollowCylinder.html HollowCylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/CoreShellCylinder.html CoreShellCylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Ellipsoid.html Ellipsoid]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/OblateCoreShell.html OblateCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/ProlateCoreShell.html ProlateCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/TwoHomopolymerRPA.html TwoHomopolymerRPA]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/DAB.html DAB]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/TeubnerStrey.html TeubnerStrey]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Lorentz.html Lorentz]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PeakLorentz.html PeakLorentz]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PeakGauss.html PeakGauss]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PowerLaw.html PowerLaw]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/BE_RPA.html BE_RPA]<br />
<br />
* [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors] (see also Gilles Renaud, Rémi Lazzari,Frédéric Leroy "[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVY-4X36TK4-1&_user=2422869&_coverDate=08%2F31%2F2009&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1702575836&_rerunOrigin=google&_acct=C000057228&_version=1&_urlVersion=0&_userid=2422869&md5=d5f357bfcbf9a39bb8e42cfeac555359&searchtype=a Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering]" Surface Science Reports, 64 (8), 31 August 2009, 255-380 [http://dx.doi.org/10.1016/j.surfrep.2009.07.002 doi:10.1016/j.surfrep.2009.07.002]):<br />
** Parallelepiped<br />
** Pyramid<br />
** Cylinder<br />
** Cone<br />
** Prism 3<br />
** Tetrahedron<br />
** Prism 6<br />
** cone 6<br />
** Sphere<br />
** Cubooctahedron<br />
** Facetted sphere<br />
** Full sphere<br />
** Full spheroid<br />
** Box<br />
** Anisotropic pyramid<br />
** Hemi-ellipsoid<br />
<br />
* [http://scripts.iucr.org/cgi-bin/paper?S0021889811011691 Scattering functions of Platonic solids] Xin Li, Roger Pynn, Wei-Ren Chen, et al. Journal of Applied Crystallography 2011, 44, p.1 [http://dx.doi.org/10.1107/S0021889811011691 doi:10.1107/S0021889811011691]<br />
*# Tetrahedron<br />
*# Hexahedron (cube, parallelepiped, etc.)<br />
*# Octahedron<br />
*# Dodecahedron<br />
*# Icosahedron<br />
<br />
* Pedersen Review: [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
*# Homogeneous sphere<br />
*# Spherical shell<br />
*# Spherical concentric shells<br />
*# Particles consisting of spherical subunits<br />
*# Ellipsoid of revolution<br />
*# Tri-axial ellipsoid<br />
*# Cube and rectangular parallelepipedons<br />
*# Truncated octahedra<br />
*# Faceted sphere<br />
*# Cube with terraces<br />
*# Cylinder<br />
*# Cylinder with elliptical cross section<br />
*# Cylinder with spherical end-caps<br />
*# Infinitely thin rod<br />
*# Infinitely thin circular disk<br />
*# Fractal aggregates<br />
*# Flexible polymers with Gaussian statistics<br />
*# Flexible self-avoiding polymers<br />
*# Semi-flexible polymers without self-avoidance<br />
*# Semi-flexible polymers with self-avoidance<br />
*# Star polymer with Gaussian statistics<br />
*# Star-burst polymer with Gaussian statistics<br />
*# Regular comb polymer with Gaussian statistics<br />
*# Arbitrarily branched polymers with Gaussian statistics<br />
*# Sphere with Gaussian chains attached<br />
*# Ellipsoid with Gaussian chains attached<br />
*# Cylinder with Gaussian chains attached<br />
<br />
* [http://www.nature.com/nmat/journal/v9/n11/extref/nmat2870-s1.pdf Supplementary Information] of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin "[http://www.nature.com/nmat/journal/v9/n11/full/nmat2870.html DNA-nanoparticle superlattices formed from anisotropic building blocks]" Nature Materials '''9''', 913-917, '''2010'''. [http://dx.doi.org/10.1038/nmat2870 doi: 10.1038/nmat2870]<br />
*# Pyramid<br />
*# Cube<br />
*# Cylinder<br />
*# Octahedron<br />
*# Rhombic dodecahedron (RD)<br />
*# Triangular prism<br />
<br />
* Other:<br />
** [http://www.eng.uc.edu/~gbeaucag/Classes/Analysis/Chapter8.html This tutorial] lists sphere, rod, disk, and Gaussian polymer coil.<br />
** '''Block-Copolymer Micelles''' [http://pubs.acs.org/doi/abs/10.1021/ma9512115 Scattering Form Factor of Block Copolymer Micelles] Jan Skov Pedersen* and Michael C. Gerstenberg, Macromolecules, 1996, 29 (4), pp 1363–1365 [http://dx.doi.org/10.1021/ma9512115 DOI: 10.1021/ma9512115]<br />
** '''Capped cylinder''' [http://scripts.iucr.org/cgi-bin/paper?S0021889804000020 Scattering from cylinders with globular end-caps]. H. Kaya. J. Appl. Cryst. (2004). 37, 223-230 [http://dx.doi.org/10.1107/S0021889804000020 doi: 10.1107/S0021889804000020]<br />
** '''Lens-shaped disc''' [http://scripts.iucr.org/cgi-bin/paper?aj5016 Scattering from capped cylinders. Addendum.] H. Kaya and N.-R. de Souza. J. Appl. Cryst. (2004). 37, 508-509 [http://dx.doi.org/10.1107/S0021889804005709 doi: 10.1107/S0021889804005709 ]</div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor:Cube&diff=327Form Factor:Cube2014-06-05T02:00:11Z<p>68.194.136.6: Created page with "==Equations== For cubes of edge-length 2''R'' (volume <math>V_{cube}=(2R)^3</math>): ===Form Factor Amplitude=== ::<math> F_{cube}(\mathbf{q}) = \left\{ \begin{array..."</p>
<hr />
<div>==Equations==<br />
For cubes of edge-length 2''R'' (volume <math>V_{cube}=(2R)^3</math>):<br />
===Form Factor Amplitude===<br />
::<math><br />
F_{cube}(\mathbf{q}) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
\Delta\rho V_{cube} \mathrm{sinc}(q_x R) \mathrm{sinc}(q_y R) \mathrm{sinc}(q_z R)<br />
& \mathrm{when} \,\, \mathbf{q}\neq(0,0,0)\\<br />
\Delta\rho V_{cube}<br />
& \mathrm{when} \,\, \mathbf{q}=(0,0,0) \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
===Isotropic Form Factor Intensity===<br />
<br />
::<math><br />
P_{cube}(q) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
\frac{16 \Delta\rho^2 V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{1}{\sin\theta}\left( \frac{ \sin(q_zR) }{ \sin(2\theta) } \right)^2<br />
\int_{0}^{2\pi} <br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
\mathrm{d}\phi \mathrm{d}\theta<br />
<br />
& \mathrm{when} \,\, q\neq0\\<br />
4\pi \Delta\rho^2 V_{cube}^2<br />
& \mathrm{when} \,\, q=0 \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
<br />
==Sources==<br />
====Byeongdu Lee (APS)====<br />
From [http://www.nature.com/nmat/journal/v9/n11/extref/nmat2870-s1.pdf Supplementary Information] of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin "[http://www.nature.com/nmat/journal/v9/n11/full/nmat2870.html DNA-nanoparticle superlattices formed from anisotropic building blocks]" Nature Materials '''9''', 913-917, '''2010'''. [http://dx.doi.org/10.1038/nmat2870 doi: 10.1038/nmat2870]<br />
:<math><br />
F_{cube}(\mathbf{q}) = V_{cube} \mathrm{sinc}(q_xR) \mathrm{sinc}(q_yR) \mathrm{sinc}(q_zR)<br />
</math><br />
Where ''2R'' is the edge length of the cube, such that the volume is:<br />
::<math> V_{cube} = \left(2R \right)^3 </math><br />
and sinc is the [http://en.wikipedia.org/wiki/Sinc_function unnormalized sinc function]:<br />
::<math><br />
\mathrm{sinc}(x) = \left\{<br />
\begin{array}{c l}<br />
1 & \mathrm{when} \,\, x=0 \\<br />
\frac{\sin x}{x} & \mathrm{when} \,\, x\neq0<br />
\end{array}<br />
\right.<br />
</math><br />
====Pedersen====<br />
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
For a rectangular [http://en.wikipedia.org/wiki/Parallelepiped parallelepipedon] with edges ''a'', ''b'', and ''c'':<br />
:<math><br />
P(q, a,b,c) = \frac{2}{\pi}\int_{0}^{\pi/2}\int_{0}^{\pi/2} <br />
\frac { \sin(q a \sin\alpha\cos\beta) }{ q a \sin\alpha\cos\beta }<br />
\frac { \sin(q b \sin\alpha\cos\beta) }{ q b \sin\alpha\sin\beta }<br />
\frac { \sin(q c \cos\alpha) }{ q c \cos\alpha }<br />
\sin\alpha \mathrm{d}\alpha \mathrm{d}\beta<br />
</math><br />
For a cube of edge length ''a'' this would be:<br />
:<math><br />
P_{cube}(q,a) = \frac{2}{\pi q^3 a^3}\int_{0}^{\pi/2}\int_{0}^{\pi/2} <br />
\frac { \sin(q a \sin\alpha\cos\beta) }{ \sin\alpha\cos\beta }<br />
\frac { \sin(q a \sin\alpha\cos\beta) }{ \sin\alpha\sin\beta }<br />
\frac { \sin(q a \cos\alpha) }{ \cos\alpha }<br />
\sin\alpha \mathrm{d}\alpha \mathrm{d}\beta<br />
</math><br />
<br />
==Derivations==<br />
===Form Factor===<br />
For a cube of edge-length 2''R'', the volume is:<br />
::<math> V_{cube} = \left(2R \right)^3 </math><br />
We integrate over the interior of the cube, using [http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian coordinates]:<br />
::<math>\mathbf{q} = (q_x, q_y, q_z)</math><br />
::<math>\mathbf{r} = (x, y, z)</math><br />
::<math>\mathbf{q}\cdot\mathbf{r} = q_x x + q_y y + q_z z</math><br />
Such that:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{cube}(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\<br />
<br />
& = \int_{z=-R}^{R}\int_{y=-R}^{R}\int_{x=-R}^{R} e^{i (q_x x + q_y y + q_z z) } \mathrm{d}x \mathrm{d}y \mathrm{d}z \\<br />
& = \int_{-R}^{R} e^{i q_x x} \mathrm{d}x \int_{-R}^{R} e^{i q_y y} \mathrm{d}y \int_{-R}^{R} e^{i q_z z} \mathrm{d}z <br />
\end{alignat}<br />
</math><br />
Each integral is of the same form:<br />
::<math><br />
<br />
\begin{alignat}{2}<br />
<br />
f_{cube,x}(q_x) & = \int_{-R}^{R} e^{i q_x x} \mathrm{d}x \\<br />
& = \int_{-R}^{R} \left[\cos(q_x x) + i \sin(q_x x)\right] \mathrm{d}x \\<br />
& = \left[\frac{-1}{q_x}\sin(q_x x) + \frac{i}{q_x} \cos(q_x x)\right]_{x=-R}^{R} \\<br />
& = \left[ \frac{-1}{q_x}\sin(q_x R) + \frac{i}{q_x} \cos(q_x R) - \frac{-1}{q_x}\sin(-q_x R) - \frac{i}{q_x} \cos(-q_x R) \right] \\<br />
& = \left[ -\frac{1}{q_x}\sin(q_x R) - \frac{1}{q_x}\sin(q_x R) \right] \\<br />
& = -\frac{2}{q_x}\sin(q_x R) \\<br />
\end{alignat}<br />
</math><br />
Which gives:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{cube}(\mathbf{q}) & = \frac{2}{q_x}\sin(q_x R) \frac{2}{q_y}\sin(q_y R) \frac{2}{q_z}\sin(q_z R) \\<br />
& = 2^3R^3 \frac{\sin(q_x R)}{q_x R} \frac{\sin(q_y R)}{q_y R} \frac{\sin(q_z R)}{q_z R} \\<br />
& = V_{cube} \mathrm{sinc}(q_x R) \mathrm{sinc}(q_y R) \mathrm{sinc}(q_z R)<br />
\end{alignat}<br />
</math><br />
===Form Factor at ''q''=0===<br />
At small ''q'':<br />
:<math><br />
F_{cube}\left(0\right) = V_{cube}<br />
</math><br />
<br />
===Isotropic Form Factor===<br />
To average over all possible orientations, we note:<br />
::<math>\mathbf{q}=(q_x,q_y,q_z)=(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)</math><br />
and use:<br />
:<math><br />
\begin{alignat}{2}<br />
\left\langle F_{cube}(\mathbf{q}) \right\rangle_{\mathrm{iso}} & = \int\limits_{S} F_{cube}(\mathbf{q}) \mathrm{d}\mathbf{s} \\<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} F_{cube}(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta) \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = V_{cube} \int_{0}^{2\pi}\int_{0}^{\pi} \mathrm{sinc}(q_x R) \mathrm{sinc}(q_y R) \mathrm{sinc}(q_z R) \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
<br />
& = V_{cube} \int_{0}^{\pi} \sin\theta \left( \frac{\sin(q_z R)}{q_z R} \right) <br />
\int_{0}^{2\pi} <br />
\left( \frac{\sin(q_x R)}{q_x R} \right)<br />
\left( \frac{\sin(q_y R)}{q_y R} \right)<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
& = \frac{V_{cube}}{ (qR)^3 } \int_{0}^{\pi} \frac{\sin\theta \sin(q_z R)}{\cos \theta} <br />
\int_{0}^{2\pi} <br />
\frac{\sin(q_x R) \sin(q_y R) }{- \sin \theta \cos \phi \sin \theta \sin \phi}<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
& = - \frac{V_{cube}}{ (qR)^3 } \int_{0}^{\pi} \frac{\sin(q_z R)}{\sin\theta \cos\theta} <br />
\int_{0}^{2\pi} <br />
\frac{\sin(q_x R) \sin(q_y R) }{\sin\phi \cos\phi }<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
& = - \frac{4 V_{cube}}{ (qR)^3 } \int_{0}^{\pi} \frac{\sin(q_z R)}{\sin(2\theta)} <br />
\int_{0}^{2\pi} <br />
\frac{\sin(q_x R) \sin(q_y R) }{\sin(2\phi)}<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
\end{alignat}<br />
</math><br />
From symmetry, it is sufficient to integrate over only one of the eight octants: <br />
:<math><br />
\begin{alignat}{2}<br />
\left\langle F_{cube}(\mathbf{q}) \right\rangle_{\mathrm{iso}}<br />
<br />
& = - \frac{32 V_{cube}}{ (qR)^3 } \int_{0}^{\pi/2} \frac{\sin(q_z R)}{\sin(2\theta)} <br />
\int_{0}^{\pi/2} <br />
\frac{\sin(q_x R) \sin(q_y R) }{\sin(2\phi)}<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity===<br />
To average over all possible orientations, we note:<br />
::<math>\mathbf{q}=(q_x,q_y,q_z)=(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)</math><br />
and use:<br />
:<math><br />
\begin{alignat}{2}<br />
P_{cube}(q) & = \int\limits_{S} | F_{cube}(\mathbf{q}) |^2 \mathrm{d}\mathbf{s} \\<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{cube}(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = V_{cube}^2 \int_{0}^{2\pi}\int_{0}^{\pi} | \mathrm{sinc}(q_x R) \mathrm{sinc}(q_y R) \mathrm{sinc}(q_z R) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
<br />
& = V_{cube}^2 \int_{0}^{\pi} \sin\theta \left( \frac{\sin(q\cos(\theta)R)}{q \cos(\theta)R} \right)^2 <br />
\int_{0}^{2\pi} <br />
\left( \frac{\sin(-q\sin(\theta)\cos(\phi)R)}{-q \sin(\theta)\cos(\phi)R} \right)^2<br />
\left( \frac{\sin(q\sin(\theta)\sin(\phi)R)}{q \sin(\theta)\sin(\phi)R} \right)^2<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
& = \frac{V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{\sin\theta \sin^2(q\cos(\theta)R)}{\cos^2(\theta)\sin^2(\theta)\sin^2(\theta)}<br />
\int_{0}^{2\pi} <br />
\frac{\sin^2(-q\sin(\theta)\cos(\phi)R)\sin^2(q\sin(\theta)\sin(\phi)R)}<br />
{\cos^2(\phi)\sin^2(\phi)}<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
& = \frac{V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{ \sin^2(q\cos(\theta)R) }{ \sin^3(\theta)\cos^2(\theta) }<br />
\int_{0}^{2\pi} <br />
\frac{\sin^2(-q\sin(\theta)\cos(\phi)R)\sin^2(q\sin(\theta)\sin(\phi)R)}<br />
{ ( \frac{1}{2} \sin(2\phi) )^2 }<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
\end{alignat}<br />
</math><br />
Solving integrals that involve nested trigonometric functions is not generally possible. However we can simplify in preparation for performing the integrals numerically:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
P_{cube}(q)<br />
& = \frac{V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{ \sin^2(q_zR) }{ \sin^3(\theta)\cos^2(\theta) }<br />
\int_{0}^{2\pi} <br />
\frac{\sin^2(q_xR)\sin^2(q_yR)}<br />
{ ( \frac{1}{2} \sin(2\phi) )^2 }<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
& = \frac{2^2 V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{ \sin^2(q_zR) }{ \sin(\theta)(\frac{1}{2}\sin(2\theta) )^2 }<br />
\int_{0}^{2\pi} <br />
\frac{\sin^2(q_xR)\sin^2(q_yR)}<br />
{ ( \sin(2\phi) )^2 }<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
& = \frac{16 V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi}<br />
\frac{1}{\sin\theta}\left( \frac{ \sin(q_zR) }{ \sin(2\theta) } \right)^2<br />
\int_{0}^{2\pi} <br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
<br />
<br />
<br />
\end{alignat}<br />
</math><br />
<br />
From symmetry, it is sufficient to integrate over only one of the eight octants:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
P_{cube}(q)<br />
& = \frac{128 V_{cube}^2}{ (qR)^6 } \int_{0}^{\pi/2}<br />
\frac{1}{\sin\theta}\left( \frac{ \sin(q_zR) }{ \sin(2\theta) } \right)^2<br />
\int_{0}^{\pi/2} <br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
\mathrm{d}\phi \mathrm{d}\theta \\<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity contribution when <math>\phi</math>=0===<br />
The integrand of the <math>\phi</math>-integral becomes:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
& = <br />
\left(<br />
\frac{\sin(-q \sin(\theta)\cos(\phi)R)\sin(q \sin(\theta) \sin(\phi) R)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
\\<br />
& = <br />
\left(<br />
\frac{\sin(-q \sin(\theta)\cos(\phi)R)\sin(q \sin(\theta) \sin(\phi) R)}<br />
{ 2 \sin(\phi) \cos(\phi) }<br />
\right)^2<br />
\\<br />
\end{alignat}<br />
</math><br />
For small <math>\phi</math>, the various <math>\sin(\phi)</math> can be replaced by <math>\phi</math>, and the various <math>\cos(\phi)</math> can be replaced by <math>1</math>:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
\lim_{\phi\to0}<br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
& = <br />
\left(<br />
\frac{\sin(-q \sin(\theta)R)\sin(q \sin(\theta) \phi R)}<br />
{ 2 \phi }<br />
\right)^2<br />
\\<br />
& = <br />
\left(<br />
\frac{\sin(-q \sin(\theta)R) q \sin(\theta) \phi R}<br />
{ 2 \phi }<br />
\right)^2<br />
\\<br />
& = <br />
\left(<br />
\frac{\sin(-q \sin(\theta)R) q \sin(\theta) R}<br />
{ 2 }<br />
\right)^2<br />
\\<br />
\end{alignat}<br />
</math><br />
Which is a constant (with respect to <math>\phi</math>). The part of the <math>\phi</math>-integral near <math>\phi=0</math> has the contribution:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
\int_{\phi=0}^{\phi=0+\delta} <br />
\left(<br />
\frac{\sin(q_xR)\sin(q_yR)}<br />
{ \sin(2\phi) }<br />
\right)^2<br />
\mathrm{d}\phi<br />
& =<br />
\left(<br />
\frac{\sin(-q \sin(\theta)R) q \sin(\theta) R}<br />
{ 2 }<br />
\right)^2<br />
\int_{\phi=0}^{\phi=0+\delta} <br />
\mathrm{d}\phi \\<br />
& =<br />
\left(<br />
\frac{\sin(-q \sin(\theta)R) q \sin(\theta) R}<br />
{ 2 }<br />
\right)^2<br />
\delta \\<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity at ''q''=0===<br />
At very small ''q'':<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
P_{cube}(0) & = V_{cube}^2 \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = 4\pi V_{cube}^2 \\<br />
<br />
\end{alignat}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor:Ellipsoid_of_revolution&diff=326Form Factor:Ellipsoid of revolution2014-06-05T01:57:26Z<p>68.194.136.6: </p>
<hr />
<div>An '''ellipsoid of revolution''' is a 'squashed' or 'stretched' [[Form Factor:Sphere|sphere]]; technically an oblate or prolate spheroid, respectively.<br />
<br />
==Equations==<br />
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta} & = \sqrt{ R_z^2 \cos^2 \theta + R_r^2(1- \cos^2\theta) } \\<br />
& = R_r \sqrt{ 1 + (\epsilon^2-1) \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta }<br />
\end{alignat}<br />
</math><br />
The ellipsoid is also characterized by:<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
===Form Factor Amplitude===<br />
::<math><br />
F_{ell}(\mathbf{q}) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
3 \Delta\rho V_{ell} \frac{ \sin(q R_{\theta})-q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 }<br />
& \mathrm{when} \,\, q\neq0\\<br />
\Delta\rho V_{ell}<br />
& \mathrm{when} \,\, q=0 \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
===Isotropic Form Factor Intensity===<br />
<br />
<br />
::<math><br />
P_{ell}(q) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
18 \pi \Delta\rho^2 V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
& \mathrm{when} \,\, q\neq0\\<br />
4\pi \Delta\rho^2 V_{ell}^2<br />
& \mathrm{when} \,\, q=0\\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
<br />
==Sources==<br />
====NCNR====<br />
From [http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html NCNR SANS Models documentation]:<br />
:<math><br />
\begin{alignat}{2}<br />
P(q) & =\frac{ \rm{scale} }{ V_{ell} }(\rho_{ell}-\rho_{solv})^2 \int_0^1 f^2 [ qr_b(1+x^2(v^2-1))^{1/2} ] dx + bkg \\<br />
<br />
f(z) & = 3 V_{ell} \frac{(\sin z - z \cos z)}{z^3} \\<br />
<br />
V_{ell} & = \frac{4 \pi}{3} r_a r_b^2 \\<br />
v & = \frac{r_a}{r_b} \\<br />
<br />
\end{alignat}<br />
</math><br />
* ''Parameters:''<br />
*# <math>\rm{scale}</math> : Intensity scaling<br />
*# <math>r_a</math> : rotation axis (Å)<br />
*# <math>r_b</math> : orthogonal axis (Å)<br />
*# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)<br />
*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)<br />
<br />
====Pedersen====<br />
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
:<math><br />
\begin{alignat}{2}<br />
& P(q, R, \epsilon)= \int_{0}^{\pi / 2} F_{sphere}^2[q,r(R,\epsilon,\alpha)] \sin \alpha d\alpha \\<br />
& r(R,\epsilon,\alpha) = R \left( \sin^2\alpha + \epsilon^2 \cos^2 \alpha \right)^{1/2}<br />
\end{alignat}<br />
</math><br />
Where:<br />
:<math>F_{sphere} = \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 } <br />
</math><br />
* ''Parameters:''<br />
*# <math>R</math> : radius (Å)<br />
*# <math>\epsilon R</math> : orthogonal size (Å)<br />
<br />
====IsGISAXS====<br />
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:<br />
:<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math><br />
:<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math><br />
:<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math><br />
Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:<br />
::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau<br />
</math><br />
<br />
====Sjoberg Monte Carlo Study====<br />
From [http://scripts.iucr.org/cgi-bin/paper?S0021889899006640 Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics], Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. [http://dx.doi.org/10.1107/S0021889899006640 doi 10.1107/S0021889899006640]<br />
<br />
:<math>F(\mathbf{q}) = 3 \frac{ \sin(qs) - qs \cos(qs) }{(qs)^3}<br />
</math><br />
where:<br />
::<math>s=\left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2}<br />
</math><br />
where <math>\gamma</math> is the angle between <math>\mathbf{q}</math> and the ''a''-axis vector of the ellipsoid of revolution (which also has axes ''b'' = ''c''); <math>\cos\gamma</math> is the inner product of unit vectors parallel to <math>\mathbf{q}</math> and the ''a''-axis. In some sense, ''s'' is the 'equivalent size' of a sphere that would lead to the scattering for a particular <math>\mathbf{q}</math>: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the <math>\mathbf{q}</math>-vector.<br />
<br />
Note that for <math> a = \epsilon b</math>:<br />
<br />
::<math><br />
\begin{alignat}{2}<br />
s & = \left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\<br />
& = \left[ b^2\epsilon^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\ <br />
& = b \left[ \epsilon^2\cos^2\gamma + (1-\cos^2\gamma) \right]^{1/2} \\<br />
& = b \left[ \epsilon^2\cos^2\gamma + \sin^2\gamma \right]^{1/2} \\<br />
& = b \left[ 1 + (\epsilon^2-1)\cos^2\gamma \right]^{1/2}<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Derivations==<br />
===Form Factor===<br />
For an ellipsoid oriented along the ''z''-axis, we denote the size in-plane (in ''x'' and ''y'') as <math>R_r</math> and the size along ''z'' as <math>R_z=\epsilon R_r</math>. The parameter <math>\epsilon</math> denotes the shape of the ellipsoid: <math>\epsilon=1</math> for a sphere, <math>\epsilon<1</math> for an oblate spheroid and <math>\epsilon>1</math> for a prolate spheroid. The volume is thus:<br />
<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates <math>(r_{xy},z)</math> (where <math>r_{xy}</math> is a distance in the ''xy''-plane):<br />
::<math><br />
\begin{alignat}{2}<br />
r_{xy} & = R_r \sin\theta \\<br />
z & = R_z \cos\theta = \epsilon R_r \cos\theta<br />
\end{alignat}<br />
</math><br />
Where <math>\theta</math> is the angle with the ''z''-axis. This lets us define a useful quantity, <math>R_{\theta}</math>, which is the distance to the point from the origin:<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta}<br />
& = \sqrt{ (R_r \sin\theta)^2 + (R_z \cos \theta)^2 } \\<br />
& = \sqrt{ R_r^2 \sin^2\theta + \epsilon^2 R_r^2 \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta } \\<br />
\end{alignat}<br />
</math><br />
<br />
The form factor is:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{ell}(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\<br />
<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 2 \pi \int_{0}^{\pi} \left [ \int_{0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \right ] \sin\theta \mathrm{d}\theta \\<br />
<br />
\end{alignat}</math><br />
<br />
Imagine instead that we compress/stretch the ''z'' dimension so that the ellipsoid becomes a sphere:<br />
::<math><br />
\begin{alignat}{2}<br />
x^{\prime} & = x \\<br />
y^{\prime} & = y \\<br />
z^{\prime} & = z R_r/R_z=z/\epsilon \\<br />
r^{\prime} & = \left| \mathbf{r}^{\prime} \right| = r \frac{R_r}{R_{\gamma}} \\<br />
\mathrm{d}V & = \mathrm{d}x\mathrm{d}y\mathrm{d}z = \mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\epsilon\mathrm{d}z^{\prime} = \epsilon \mathrm{d}V^{\prime}<br />
\end{alignat}<br />
</math><br />
<br />
This implies a coordinate transformation for the <math>\mathbf{q}</math>-vector of:<br />
::<math><br />
\begin{alignat}{2}<br />
q_x^{\prime} & = q_x \\<br />
q_y^{\prime} & = q_y \\<br />
q_z^{\prime} & = q_z R_z/R_r = q_z \epsilon \\<br />
q^{\prime} & = \left| \mathbf{q}^{\prime} \right| = q \frac{R_{\gamma}}{R_r}<br />
\end{alignat}<br />
</math><br />
Where <math>R_{\gamma}</math> is the <math>R_{\theta}</math> relation for a ''q''-vector tilted at angle <math>\gamma</math> with respect to the ''z'' axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular <math>\mathbf{q}</math> vector sees a sphere-like scatterer with size (length-scale) given by <math>R_{\gamma}</math>.<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) <br />
& = \epsilon \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r^{\prime}=0}^{R_r} <br />
e^{i \mathbf{q}^{\prime} \cdot \mathbf{r}^{\prime} } r^{\prime 2} \mathrm{d}r^{\prime}<br />
\sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 3 \left( \frac{4 \pi}{3} \epsilon R_r^3 \right) \frac{ \sin(q^{\prime} R_r) - q^{\prime} R_r \cos(q^{\prime} R_r) }{ (q^{\prime} R_r)^3 }<br />
<br />
\end{alignat}</math><br />
<br />
We can then convert back:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 }<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity===<br />
To average over all possible orientations, we use:<br />
:<math><br />
\begin{alignat}{2}<br />
P_{ell}(q)<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{ell}(\mathbf{q}) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{ell} \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\<br />
& = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Approximating by a Sphere==<br />
One can approximate a spheroid using an isovolumic [[Form Factor:Sphere|sphere]] of radius ''R''<sub>effective</sub>:<br />
:<math>V_{ell}<br />
= \frac{ 4\pi }{ 3 } R_z R_r^2 </math><br />
:<math><br />
\begin{alignat}{2}<br />
R_{\mathrm{effective}}<br />
& = \left( \frac{ 3 V_{ell} }{ 4 \pi } \right)^{1/3} \\<br />
& = ( R_z R_r^2 )^{1/3} \\<br />
\end{alignat}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor:Ellipsoid_of_revolution&diff=325Form Factor:Ellipsoid of revolution2014-06-05T01:57:11Z<p>68.194.136.6: /* Approximating by a Sphere */</p>
<hr />
<div>An '''ellipsoid of revolution''' is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.<br />
<br />
==Equations==<br />
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta} & = \sqrt{ R_z^2 \cos^2 \theta + R_r^2(1- \cos^2\theta) } \\<br />
& = R_r \sqrt{ 1 + (\epsilon^2-1) \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta }<br />
\end{alignat}<br />
</math><br />
The ellipsoid is also characterized by:<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
===Form Factor Amplitude===<br />
::<math><br />
F_{ell}(\mathbf{q}) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
3 \Delta\rho V_{ell} \frac{ \sin(q R_{\theta})-q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 }<br />
& \mathrm{when} \,\, q\neq0\\<br />
\Delta\rho V_{ell}<br />
& \mathrm{when} \,\, q=0 \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
===Isotropic Form Factor Intensity===<br />
<br />
<br />
::<math><br />
P_{ell}(q) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
18 \pi \Delta\rho^2 V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
& \mathrm{when} \,\, q\neq0\\<br />
4\pi \Delta\rho^2 V_{ell}^2<br />
& \mathrm{when} \,\, q=0\\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
<br />
==Sources==<br />
====NCNR====<br />
From [http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html NCNR SANS Models documentation]:<br />
:<math><br />
\begin{alignat}{2}<br />
P(q) & =\frac{ \rm{scale} }{ V_{ell} }(\rho_{ell}-\rho_{solv})^2 \int_0^1 f^2 [ qr_b(1+x^2(v^2-1))^{1/2} ] dx + bkg \\<br />
<br />
f(z) & = 3 V_{ell} \frac{(\sin z - z \cos z)}{z^3} \\<br />
<br />
V_{ell} & = \frac{4 \pi}{3} r_a r_b^2 \\<br />
v & = \frac{r_a}{r_b} \\<br />
<br />
\end{alignat}<br />
</math><br />
* ''Parameters:''<br />
*# <math>\rm{scale}</math> : Intensity scaling<br />
*# <math>r_a</math> : rotation axis (Å)<br />
*# <math>r_b</math> : orthogonal axis (Å)<br />
*# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)<br />
*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)<br />
<br />
====Pedersen====<br />
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
:<math><br />
\begin{alignat}{2}<br />
& P(q, R, \epsilon)= \int_{0}^{\pi / 2} F_{sphere}^2[q,r(R,\epsilon,\alpha)] \sin \alpha d\alpha \\<br />
& r(R,\epsilon,\alpha) = R \left( \sin^2\alpha + \epsilon^2 \cos^2 \alpha \right)^{1/2}<br />
\end{alignat}<br />
</math><br />
Where:<br />
:<math>F_{sphere} = \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 } <br />
</math><br />
* ''Parameters:''<br />
*# <math>R</math> : radius (Å)<br />
*# <math>\epsilon R</math> : orthogonal size (Å)<br />
<br />
====IsGISAXS====<br />
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:<br />
:<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math><br />
:<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math><br />
:<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math><br />
Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:<br />
::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau<br />
</math><br />
<br />
====Sjoberg Monte Carlo Study====<br />
From [http://scripts.iucr.org/cgi-bin/paper?S0021889899006640 Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics], Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. [http://dx.doi.org/10.1107/S0021889899006640 doi 10.1107/S0021889899006640]<br />
<br />
:<math>F(\mathbf{q}) = 3 \frac{ \sin(qs) - qs \cos(qs) }{(qs)^3}<br />
</math><br />
where:<br />
::<math>s=\left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2}<br />
</math><br />
where <math>\gamma</math> is the angle between <math>\mathbf{q}</math> and the ''a''-axis vector of the ellipsoid of revolution (which also has axes ''b'' = ''c''); <math>\cos\gamma</math> is the inner product of unit vectors parallel to <math>\mathbf{q}</math> and the ''a''-axis. In some sense, ''s'' is the 'equivalent size' of a sphere that would lead to the scattering for a particular <math>\mathbf{q}</math>: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the <math>\mathbf{q}</math>-vector.<br />
<br />
Note that for <math> a = \epsilon b</math>:<br />
<br />
::<math><br />
\begin{alignat}{2}<br />
s & = \left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\<br />
& = \left[ b^2\epsilon^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\ <br />
& = b \left[ \epsilon^2\cos^2\gamma + (1-\cos^2\gamma) \right]^{1/2} \\<br />
& = b \left[ \epsilon^2\cos^2\gamma + \sin^2\gamma \right]^{1/2} \\<br />
& = b \left[ 1 + (\epsilon^2-1)\cos^2\gamma \right]^{1/2}<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Derivations==<br />
===Form Factor===<br />
For an ellipsoid oriented along the ''z''-axis, we denote the size in-plane (in ''x'' and ''y'') as <math>R_r</math> and the size along ''z'' as <math>R_z=\epsilon R_r</math>. The parameter <math>\epsilon</math> denotes the shape of the ellipsoid: <math>\epsilon=1</math> for a sphere, <math>\epsilon<1</math> for an oblate spheroid and <math>\epsilon>1</math> for a prolate spheroid. The volume is thus:<br />
<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates <math>(r_{xy},z)</math> (where <math>r_{xy}</math> is a distance in the ''xy''-plane):<br />
::<math><br />
\begin{alignat}{2}<br />
r_{xy} & = R_r \sin\theta \\<br />
z & = R_z \cos\theta = \epsilon R_r \cos\theta<br />
\end{alignat}<br />
</math><br />
Where <math>\theta</math> is the angle with the ''z''-axis. This lets us define a useful quantity, <math>R_{\theta}</math>, which is the distance to the point from the origin:<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta}<br />
& = \sqrt{ (R_r \sin\theta)^2 + (R_z \cos \theta)^2 } \\<br />
& = \sqrt{ R_r^2 \sin^2\theta + \epsilon^2 R_r^2 \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta } \\<br />
\end{alignat}<br />
</math><br />
<br />
The form factor is:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{ell}(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\<br />
<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 2 \pi \int_{0}^{\pi} \left [ \int_{0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \right ] \sin\theta \mathrm{d}\theta \\<br />
<br />
\end{alignat}</math><br />
<br />
Imagine instead that we compress/stretch the ''z'' dimension so that the ellipsoid becomes a sphere:<br />
::<math><br />
\begin{alignat}{2}<br />
x^{\prime} & = x \\<br />
y^{\prime} & = y \\<br />
z^{\prime} & = z R_r/R_z=z/\epsilon \\<br />
r^{\prime} & = \left| \mathbf{r}^{\prime} \right| = r \frac{R_r}{R_{\gamma}} \\<br />
\mathrm{d}V & = \mathrm{d}x\mathrm{d}y\mathrm{d}z = \mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\epsilon\mathrm{d}z^{\prime} = \epsilon \mathrm{d}V^{\prime}<br />
\end{alignat}<br />
</math><br />
<br />
This implies a coordinate transformation for the <math>\mathbf{q}</math>-vector of:<br />
::<math><br />
\begin{alignat}{2}<br />
q_x^{\prime} & = q_x \\<br />
q_y^{\prime} & = q_y \\<br />
q_z^{\prime} & = q_z R_z/R_r = q_z \epsilon \\<br />
q^{\prime} & = \left| \mathbf{q}^{\prime} \right| = q \frac{R_{\gamma}}{R_r}<br />
\end{alignat}<br />
</math><br />
Where <math>R_{\gamma}</math> is the <math>R_{\theta}</math> relation for a ''q''-vector tilted at angle <math>\gamma</math> with respect to the ''z'' axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular <math>\mathbf{q}</math> vector sees a sphere-like scatterer with size (length-scale) given by <math>R_{\gamma}</math>.<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) <br />
& = \epsilon \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r^{\prime}=0}^{R_r} <br />
e^{i \mathbf{q}^{\prime} \cdot \mathbf{r}^{\prime} } r^{\prime 2} \mathrm{d}r^{\prime}<br />
\sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 3 \left( \frac{4 \pi}{3} \epsilon R_r^3 \right) \frac{ \sin(q^{\prime} R_r) - q^{\prime} R_r \cos(q^{\prime} R_r) }{ (q^{\prime} R_r)^3 }<br />
<br />
\end{alignat}</math><br />
<br />
We can then convert back:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 }<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity===<br />
To average over all possible orientations, we use:<br />
:<math><br />
\begin{alignat}{2}<br />
P_{ell}(q)<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{ell}(\mathbf{q}) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{ell} \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\<br />
& = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Approximating by a Sphere==<br />
One can approximate a spheroid using an isovolumic [[Form Factor:Sphere|sphere]] of radius ''R''<sub>effective</sub>:<br />
:<math>V_{ell}<br />
= \frac{ 4\pi }{ 3 } R_z R_r^2 </math><br />
:<math><br />
\begin{alignat}{2}<br />
R_{\mathrm{effective}}<br />
& = \left( \frac{ 3 V_{ell} }{ 4 \pi } \right)^{1/3} \\<br />
& = ( R_z R_r^2 )^{1/3} \\<br />
\end{alignat}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor:Ellipsoid_of_revolution&diff=324Form Factor:Ellipsoid of revolution2014-06-05T01:56:31Z<p>68.194.136.6: </p>
<hr />
<div>An '''ellipsoid of revolution''' is a 'squashed' or 'stretched' sphere; technically an oblate or prolate spheroid, respectively.<br />
<br />
==Equations==<br />
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta} & = \sqrt{ R_z^2 \cos^2 \theta + R_r^2(1- \cos^2\theta) } \\<br />
& = R_r \sqrt{ 1 + (\epsilon^2-1) \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta }<br />
\end{alignat}<br />
</math><br />
The ellipsoid is also characterized by:<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
===Form Factor Amplitude===<br />
::<math><br />
F_{ell}(\mathbf{q}) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
3 \Delta\rho V_{ell} \frac{ \sin(q R_{\theta})-q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 }<br />
& \mathrm{when} \,\, q\neq0\\<br />
\Delta\rho V_{ell}<br />
& \mathrm{when} \,\, q=0 \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
===Isotropic Form Factor Intensity===<br />
<br />
<br />
::<math><br />
P_{ell}(q) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
18 \pi \Delta\rho^2 V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
& \mathrm{when} \,\, q\neq0\\<br />
4\pi \Delta\rho^2 V_{ell}^2<br />
& \mathrm{when} \,\, q=0\\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
<br />
==Sources==<br />
====NCNR====<br />
From [http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html NCNR SANS Models documentation]:<br />
:<math><br />
\begin{alignat}{2}<br />
P(q) & =\frac{ \rm{scale} }{ V_{ell} }(\rho_{ell}-\rho_{solv})^2 \int_0^1 f^2 [ qr_b(1+x^2(v^2-1))^{1/2} ] dx + bkg \\<br />
<br />
f(z) & = 3 V_{ell} \frac{(\sin z - z \cos z)}{z^3} \\<br />
<br />
V_{ell} & = \frac{4 \pi}{3} r_a r_b^2 \\<br />
v & = \frac{r_a}{r_b} \\<br />
<br />
\end{alignat}<br />
</math><br />
* ''Parameters:''<br />
*# <math>\rm{scale}</math> : Intensity scaling<br />
*# <math>r_a</math> : rotation axis (Å)<br />
*# <math>r_b</math> : orthogonal axis (Å)<br />
*# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)<br />
*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)<br />
<br />
====Pedersen====<br />
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
:<math><br />
\begin{alignat}{2}<br />
& P(q, R, \epsilon)= \int_{0}^{\pi / 2} F_{sphere}^2[q,r(R,\epsilon,\alpha)] \sin \alpha d\alpha \\<br />
& r(R,\epsilon,\alpha) = R \left( \sin^2\alpha + \epsilon^2 \cos^2 \alpha \right)^{1/2}<br />
\end{alignat}<br />
</math><br />
Where:<br />
:<math>F_{sphere} = \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 } <br />
</math><br />
* ''Parameters:''<br />
*# <math>R</math> : radius (Å)<br />
*# <math>\epsilon R</math> : orthogonal size (Å)<br />
<br />
====IsGISAXS====<br />
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:<br />
:<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math><br />
:<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math><br />
:<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math><br />
Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:<br />
::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau<br />
</math><br />
<br />
====Sjoberg Monte Carlo Study====<br />
From [http://scripts.iucr.org/cgi-bin/paper?S0021889899006640 Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics], Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. [http://dx.doi.org/10.1107/S0021889899006640 doi 10.1107/S0021889899006640]<br />
<br />
:<math>F(\mathbf{q}) = 3 \frac{ \sin(qs) - qs \cos(qs) }{(qs)^3}<br />
</math><br />
where:<br />
::<math>s=\left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2}<br />
</math><br />
where <math>\gamma</math> is the angle between <math>\mathbf{q}</math> and the ''a''-axis vector of the ellipsoid of revolution (which also has axes ''b'' = ''c''); <math>\cos\gamma</math> is the inner product of unit vectors parallel to <math>\mathbf{q}</math> and the ''a''-axis. In some sense, ''s'' is the 'equivalent size' of a sphere that would lead to the scattering for a particular <math>\mathbf{q}</math>: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the <math>\mathbf{q}</math>-vector.<br />
<br />
Note that for <math> a = \epsilon b</math>:<br />
<br />
::<math><br />
\begin{alignat}{2}<br />
s & = \left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\<br />
& = \left[ b^2\epsilon^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\ <br />
& = b \left[ \epsilon^2\cos^2\gamma + (1-\cos^2\gamma) \right]^{1/2} \\<br />
& = b \left[ \epsilon^2\cos^2\gamma + \sin^2\gamma \right]^{1/2} \\<br />
& = b \left[ 1 + (\epsilon^2-1)\cos^2\gamma \right]^{1/2}<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Derivations==<br />
===Form Factor===<br />
For an ellipsoid oriented along the ''z''-axis, we denote the size in-plane (in ''x'' and ''y'') as <math>R_r</math> and the size along ''z'' as <math>R_z=\epsilon R_r</math>. The parameter <math>\epsilon</math> denotes the shape of the ellipsoid: <math>\epsilon=1</math> for a sphere, <math>\epsilon<1</math> for an oblate spheroid and <math>\epsilon>1</math> for a prolate spheroid. The volume is thus:<br />
<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates <math>(r_{xy},z)</math> (where <math>r_{xy}</math> is a distance in the ''xy''-plane):<br />
::<math><br />
\begin{alignat}{2}<br />
r_{xy} & = R_r \sin\theta \\<br />
z & = R_z \cos\theta = \epsilon R_r \cos\theta<br />
\end{alignat}<br />
</math><br />
Where <math>\theta</math> is the angle with the ''z''-axis. This lets us define a useful quantity, <math>R_{\theta}</math>, which is the distance to the point from the origin:<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta}<br />
& = \sqrt{ (R_r \sin\theta)^2 + (R_z \cos \theta)^2 } \\<br />
& = \sqrt{ R_r^2 \sin^2\theta + \epsilon^2 R_r^2 \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta } \\<br />
\end{alignat}<br />
</math><br />
<br />
The form factor is:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{ell}(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\<br />
<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 2 \pi \int_{0}^{\pi} \left [ \int_{0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \right ] \sin\theta \mathrm{d}\theta \\<br />
<br />
\end{alignat}</math><br />
<br />
Imagine instead that we compress/stretch the ''z'' dimension so that the ellipsoid becomes a sphere:<br />
::<math><br />
\begin{alignat}{2}<br />
x^{\prime} & = x \\<br />
y^{\prime} & = y \\<br />
z^{\prime} & = z R_r/R_z=z/\epsilon \\<br />
r^{\prime} & = \left| \mathbf{r}^{\prime} \right| = r \frac{R_r}{R_{\gamma}} \\<br />
\mathrm{d}V & = \mathrm{d}x\mathrm{d}y\mathrm{d}z = \mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\epsilon\mathrm{d}z^{\prime} = \epsilon \mathrm{d}V^{\prime}<br />
\end{alignat}<br />
</math><br />
<br />
This implies a coordinate transformation for the <math>\mathbf{q}</math>-vector of:<br />
::<math><br />
\begin{alignat}{2}<br />
q_x^{\prime} & = q_x \\<br />
q_y^{\prime} & = q_y \\<br />
q_z^{\prime} & = q_z R_z/R_r = q_z \epsilon \\<br />
q^{\prime} & = \left| \mathbf{q}^{\prime} \right| = q \frac{R_{\gamma}}{R_r}<br />
\end{alignat}<br />
</math><br />
Where <math>R_{\gamma}</math> is the <math>R_{\theta}</math> relation for a ''q''-vector tilted at angle <math>\gamma</math> with respect to the ''z'' axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular <math>\mathbf{q}</math> vector sees a sphere-like scatterer with size (length-scale) given by <math>R_{\gamma}</math>.<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) <br />
& = \epsilon \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r^{\prime}=0}^{R_r} <br />
e^{i \mathbf{q}^{\prime} \cdot \mathbf{r}^{\prime} } r^{\prime 2} \mathrm{d}r^{\prime}<br />
\sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 3 \left( \frac{4 \pi}{3} \epsilon R_r^3 \right) \frac{ \sin(q^{\prime} R_r) - q^{\prime} R_r \cos(q^{\prime} R_r) }{ (q^{\prime} R_r)^3 }<br />
<br />
\end{alignat}</math><br />
<br />
We can then convert back:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 }<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity===<br />
To average over all possible orientations, we use:<br />
:<math><br />
\begin{alignat}{2}<br />
P_{ell}(q)<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{ell}(\mathbf{q}) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{ell} \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\<br />
& = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Approximating by a Sphere==<br />
One can approximate a spheroid using an isovolumic sphere of radius ''R''<sub>effective</sub>:<br />
:<math>V_{ell}<br />
= \frac{ 4\pi }{ 3 } R_z R_r^2 </math><br />
:<math><br />
\begin{alignat}{2}<br />
R_{\mathrm{effective}}<br />
& = \left( \frac{ 3 V_{ell} }{ 4 \pi } \right)^{1/3} \\<br />
& = ( R_z R_r^2 )^{1/3} \\<br />
\end{alignat}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor:Ellipsoid_of_revolution&diff=323Form Factor:Ellipsoid of revolution2014-06-05T01:53:49Z<p>68.194.136.6: Created page with "==Equations== For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid..."</p>
<hr />
<div>==Equations==<br />
For an ellipsoid of revolution, the size ('radius') along one direction will be distinct, whereas the other two directions will be identical. Assume an ellipsoid aligned along the ''z''-direction (rotation about ''z''-axis, i.e. sweeping the <math>\phi</math> angle in spherical coordinates), such that the size in the ''xy''-plane is <math>R_r</math> and along ''z'' is <math>R_z = \epsilon R_r</math>. A useful quantity is <math>R_{\theta}</math>, which is the distance from the origin to the surface of the ellipsoid for a line titled at angle <math>\theta</math> with respect to the ''z''-axis. This is thus the half-distance between tangent planes orthogonal to a vector pointing at the given <math>\theta</math> angle, and provides the 'effective size' of the scattering object as seen by a ''q''-vector pointing in that direction.<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta} & = \sqrt{ R_z^2 \cos^2 \theta + R_r^2(1- \cos^2\theta) } \\<br />
& = R_r \sqrt{ 1 + (\epsilon^2-1) \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta }<br />
\end{alignat}<br />
</math><br />
The ellipsoid is also characterized by:<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
===Form Factor Amplitude===<br />
::<math><br />
F_{ell}(\mathbf{q}) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
3 \Delta\rho V_{ell} \frac{ \sin(q R_{\theta})-q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 }<br />
& \mathrm{when} \,\, q\neq0\\<br />
\Delta\rho V_{ell}<br />
& \mathrm{when} \,\, q=0 \\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
===Isotropic Form Factor Intensity===<br />
<br />
<br />
::<math><br />
P_{ell}(q) = \left\{<br />
<br />
\begin{array}{c l}<br />
<br />
18 \pi \Delta\rho^2 V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
& \mathrm{when} \,\, q\neq0\\<br />
4\pi \Delta\rho^2 V_{ell}^2<br />
& \mathrm{when} \,\, q=0\\<br />
\end{array}<br />
<br />
\right.<br />
</math><br />
<br />
==Sources==<br />
====NCNR====<br />
From [http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html NCNR SANS Models documentation]:<br />
:<math><br />
\begin{alignat}{2}<br />
P(q) & =\frac{ \rm{scale} }{ V_{ell} }(\rho_{ell}-\rho_{solv})^2 \int_0^1 f^2 [ qr_b(1+x^2(v^2-1))^{1/2} ] dx + bkg \\<br />
<br />
f(z) & = 3 V_{ell} \frac{(\sin z - z \cos z)}{z^3} \\<br />
<br />
V_{ell} & = \frac{4 \pi}{3} r_a r_b^2 \\<br />
v & = \frac{r_a}{r_b} \\<br />
<br />
\end{alignat}<br />
</math><br />
* ''Parameters:''<br />
*# <math>\rm{scale}</math> : Intensity scaling<br />
*# <math>r_a</math> : rotation axis (Å)<br />
*# <math>r_b</math> : orthogonal axis (Å)<br />
*# <math>\rho_{ell}-\rho_{solv}</math> : scattering contrast (Å<sup>−2</sup>)<br />
*# <math>\rm{background}</math> : incoherent background (cm<sup>−1</sup>)<br />
<br />
====Pedersen====<br />
From Pedersen review, [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
:<math><br />
\begin{alignat}{2}<br />
& P(q, R, \epsilon)= \int_{0}^{\pi / 2} F_{sphere}^2[q,r(R,\epsilon,\alpha)] \sin \alpha d\alpha \\<br />
& r(R,\epsilon,\alpha) = R \left( \sin^2\alpha + \epsilon^2 \cos^2 \alpha \right)^{1/2}<br />
\end{alignat}<br />
</math><br />
Where:<br />
:<math>F_{sphere} = \frac{ 3 \left[ \sin(qr)-qr \cos(qr) \right ] }{ (qr)^3 } <br />
</math><br />
* ''Parameters:''<br />
*# <math>R</math> : radius (Å)<br />
*# <math>\epsilon R</math> : orthogonal size (Å)<br />
<br />
====IsGISAXS====<br />
From [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors]:<br />
:<math> F_{ell}(\mathbf{q}, R, W, H, \alpha) = 2 \pi RWH \frac{ J_1 (\gamma) }{ \gamma } \sin_c(q_z H/2) \exp( i q_z H/2 )</math><br />
:<math>\gamma = \sqrt{ (q_x R)^2 + (q_y W)^2 } </math><br />
:<math> V_{ell} = \pi RWH, \, S_{anpy} = \pi R W , \, R_{anpy} = Max(R,W) </math><br />
Where (presumably) ''J'' is a [http://en.wikipedia.org/wiki/Bessel_function Bessel function]:<br />
::<math> J_1(\gamma) = \frac{1}{\pi} \int_0^\pi \cos (\tau - x \sin \tau) \,\mathrm{d}\tau<br />
</math><br />
<br />
====Sjoberg Monte Carlo Study====<br />
From [http://scripts.iucr.org/cgi-bin/paper?S0021889899006640 Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics], Bo Sjöberg, J.Appl. Cryst. (1999), 32, 917-923. [http://dx.doi.org/10.1107/S0021889899006640 doi 10.1107/S0021889899006640]<br />
<br />
:<math>F(\mathbf{q}) = 3 \frac{ \sin(qs) - qs \cos(qs) }{(qs)^3}<br />
</math><br />
where:<br />
::<math>s=\left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2}<br />
</math><br />
where <math>\gamma</math> is the angle between <math>\mathbf{q}</math> and the ''a''-axis vector of the ellipsoid of revolution (which also has axes ''b'' = ''c''); <math>\cos\gamma</math> is the inner product of unit vectors parallel to <math>\mathbf{q}</math> and the ''a''-axis. In some sense, ''s'' is the 'equivalent size' of a sphere that would lead to the scattering for a particular <math>\mathbf{q}</math>: it is half the distance between the tangential planes that bound the ellipsoid, perpendicular to the <math>\mathbf{q}</math>-vector.<br />
<br />
Note that for <math> a = \epsilon b</math>:<br />
<br />
::<math><br />
\begin{alignat}{2}<br />
s & = \left[ a^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\<br />
& = \left[ b^2\epsilon^2\cos^2\gamma + b^2(1-\cos^2\gamma) \right]^{1/2} \\ <br />
& = b \left[ \epsilon^2\cos^2\gamma + (1-\cos^2\gamma) \right]^{1/2} \\<br />
& = b \left[ \epsilon^2\cos^2\gamma + \sin^2\gamma \right]^{1/2} \\<br />
& = b \left[ 1 + (\epsilon^2-1)\cos^2\gamma \right]^{1/2}<br />
\end{alignat}<br />
</math><br />
<br />
<br />
==Derivations==<br />
===Form Factor===<br />
For an ellipsoid oriented along the ''z''-axis, we denote the size in-plane (in ''x'' and ''y'') as <math>R_r</math> and the size along ''z'' as <math>R_z=\epsilon R_r</math>. The parameter <math>\epsilon</math> denotes the shape of the ellipsoid: <math>\epsilon=1</math> for a sphere, <math>\epsilon<1</math> for an oblate spheroid and <math>\epsilon>1</math> for a prolate spheroid. The volume is thus:<br />
<br />
::<math>V_{ell} = \frac{ 4\pi }{ 3 } R_z R_r^2 = \frac{ 4\pi }{ 3 } \epsilon R_r^3 <br />
</math><br />
We also note that the cross-section of the ellipsoid (an ellipse) will have coordinates <math>(r_{xy},z)</math> (where <math>r_{xy}</math> is a distance in the ''xy''-plane):<br />
::<math><br />
\begin{alignat}{2}<br />
r_{xy} & = R_r \sin\theta \\<br />
z & = R_z \cos\theta = \epsilon R_r \cos\theta<br />
\end{alignat}<br />
</math><br />
Where <math>\theta</math> is the angle with the ''z''-axis. This lets us define a useful quantity, <math>R_{\theta}</math>, which is the distance to the point from the origin:<br />
::<math><br />
\begin{alignat}{2}<br />
R_{\theta}<br />
& = \sqrt{ (R_r \sin\theta)^2 + (R_z \cos \theta)^2 } \\<br />
& = \sqrt{ R_r^2 \sin^2\theta + \epsilon^2 R_r^2 \cos^2 \theta } \\<br />
& = R_r \sqrt{ \sin^2\theta + \epsilon^2 \cos^2 \theta } \\<br />
\end{alignat}<br />
</math><br />
<br />
The form factor is:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F_{ell}(\mathbf{q}) & = \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\<br />
<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 2 \pi \int_{0}^{\pi} \left [ \int_{0}^{R_{\theta}} e^{i \mathbf{q} \cdot \mathbf{r} } r^2 \mathrm{d}r \right ] \sin\theta \mathrm{d}\theta \\<br />
<br />
\end{alignat}</math><br />
<br />
Imagine instead that we compress/stretch the ''z'' dimension so that the ellipsoid becomes a sphere:<br />
::<math><br />
\begin{alignat}{2}<br />
x^{\prime} & = x \\<br />
y^{\prime} & = y \\<br />
z^{\prime} & = z R_r/R_z=z/\epsilon \\<br />
r^{\prime} & = \left| \mathbf{r}^{\prime} \right| = r \frac{R_r}{R_{\gamma}} \\<br />
\mathrm{d}V & = \mathrm{d}x\mathrm{d}y\mathrm{d}z = \mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\epsilon\mathrm{d}z^{\prime} = \epsilon \mathrm{d}V^{\prime}<br />
\end{alignat}<br />
</math><br />
<br />
This implies a coordinate transformation for the <math>\mathbf{q}</math>-vector of:<br />
::<math><br />
\begin{alignat}{2}<br />
q_x^{\prime} & = q_x \\<br />
q_y^{\prime} & = q_y \\<br />
q_z^{\prime} & = q_z R_z/R_r = q_z \epsilon \\<br />
q^{\prime} & = \left| \mathbf{q}^{\prime} \right| = q \frac{R_{\gamma}}{R_r}<br />
\end{alignat}<br />
</math><br />
Where <math>R_{\gamma}</math> is the <math>R_{\theta}</math> relation for a ''q''-vector tilted at angle <math>\gamma</math> with respect to the ''z'' axis. Considered in this way, the integral reduces to the form factor for a sphere. In effect, a particular <math>\mathbf{q}</math> vector sees a sphere-like scatterer with size (length-scale) given by <math>R_{\gamma}</math>.<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) <br />
& = \epsilon \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r^{\prime}=0}^{R_r} <br />
e^{i \mathbf{q}^{\prime} \cdot \mathbf{r}^{\prime} } r^{\prime 2} \mathrm{d}r^{\prime}<br />
\sin\theta \mathrm{d}\theta \mathrm{d}\phi \\<br />
<br />
& = 3 \left( \frac{4 \pi}{3} \epsilon R_r^3 \right) \frac{ \sin(q^{\prime} R_r) - q^{\prime} R_r \cos(q^{\prime} R_r) }{ (q^{\prime} R_r)^3 }<br />
<br />
\end{alignat}</math><br />
<br />
We can then convert back:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
F_{ell}(\mathbf{q}) & = 3 V_{ell} \frac{ \sin(q R_{\gamma}) - q R_{\gamma} \cos(q R_{\gamma}) }{ (q R_{\gamma})^3 }<br />
\end{alignat}<br />
</math><br />
<br />
===Isotropic Form Factor Intensity===<br />
To average over all possible orientations, we use:<br />
:<math><br />
\begin{alignat}{2}<br />
P_{ell}(q)<br />
& = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F_{ell}(\mathbf{q}) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{ell} \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\<br />
& = 9 V_{ell}^2 \int_{0}^{2\pi}\mathrm{d}\phi \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta \\<br />
& = 18 \pi V_{ell}^2 \int_{0}^{\pi} \left( \frac{ \sin(q R_{\theta}) - q R_{\theta} \cos(q R_{\theta}) }{ (q R_{\theta})^3 } \right)^2 \sin\theta\mathrm{d}\theta<br />
\end{alignat}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Form_Factor&diff=322Form Factor2014-06-05T01:53:35Z<p>68.194.136.6: /* Form Factor Equations */</p>
<hr />
<div>The '''Form Factor''' is the scattering which results from the ''shape'' of a particle. When particles are distributed without any particle-particle correlations (e.g. dilute solution of non-interacting particles, freely floating), then the scattering one observes is entirely the form factor. By comparison, when particles are in a well-defined structure, the scattering is dominated by the [[structure factor]]; though even in these cases the form factor continues to contribute, since it modulates both the structure factor and the [[diffuse scattering]].<br />
<br />
When reading discussions of scattering modeling, one must be careful about the usage of the term 'form factor'. This same term is often used to describe three different (though related) quantities:<br />
* <math>F(\mathbf{q})</math>, the '''form factor amplitude''' arising from a single particle; this can be thought of as the 3D [[reciprocal-space]] of the particle, and is thus in general anisotropic.<br />
* <math>|F(\mathbf{q})|^2</math>, the '''form factor intensity'''; whereas the amplitude cannot be measured experimentally, the form factor intensity in principle can be.<br />
* <math>P(q) = \left\langle |F(\mathbf{q})|^2 \right\rangle </math>, the '''isotropic form factor intensity'''; i.e. the orientational averaged of the form factor square. This is the 1D scattering that is measured for, e.g., particles freely distributed distributed in solution (since they tumble randomly and thus average over all possible orientations).<br />
<br />
==Equations==<br />
In the most general case of an arbitrary distribution of [[Scattering Length Density|scattering density]], <math>\rho(\mathbf{r})</math>, the form factor is computed by integrating over all space:<br />
<br />
:<math><br />
F_{j}(\mathbf{q}) = \int \rho_j(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V <br />
</math><br />
<br />
The subscript denotes that the form factor is for particle ''j''; in multi-component systems, each particle has its own form factor. For a particle of uniform density and volume ''V'', we denote the scattering contrast with respect to the ambient as <math>\Delta \rho</math>, and the form factor is simply:<br />
<br />
:<math><br />
F_{j}(\mathbf{q}) = \Delta \rho \int\limits_{V} e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V <br />
</math><br />
<br />
For monodisperse particles, the average (isotropic) form factor intensity is an average over all possible particle orientations:<br />
:<math><br />
\begin{alignat}{2}<br />
P_j(q) & = \left\langle |F_j(\mathbf{q})|^2 \right\rangle \\<br />
& = \int\limits_{\phi=0}^{2\pi}\int\limits_{\theta=0}^{\pi} | F_j(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi<br />
\end{alignat}<br />
</math><br />
<br />
Note that in cases where particles are not monodisperse, then the above average would also include averages over the distritubions in particle size and/or shape. Note that for <math>q=0</math>, we expect:<br />
:<math><br />
\begin{alignat}{2}<br />
F(0) & = \int\limits_{\mathrm{all\,\,space}} \rho(\mathbf{r}) e^{0} \mathrm{d}\mathbf{r} = \rho_{\mathrm{total}} \\<br />
& = \Delta \rho \int\limits_{V} e^{0} \mathrm{d}\mathbf{r} = \Delta \rho V<br />
\end{alignat}<br />
</math><br />
And so:<br />
:<math><br />
\begin{alignat}{2}<br />
P(0) & = \left\langle \left| F(0) \right|^2 \right\rangle<br />
& = \Delta \rho^2 V^2<br />
\end{alignat}<br />
</math><br />
As expected, scattering intensity scales with the square of the scattering contrast and the particle volume. For multi-component systems, this has the effect of greatly emphasizing larger particles. For instance, a 2-fold increase in particle diameter results in a <math>(2^3)^2 = 64</math>-fold increase in scattering intensity.<br />
<br />
==Form Factor Equations==<br />
* [[Form Factor:Sphere|Sphere]]<br />
* [[Form Factor:Ellipsoid of revolution|Ellipsoid of revolution]]<br />
* [[Form Factor:Cube|Cube]]<br />
<br />
==Form Factor Equations in the Literature==<br />
The following is a partial list of form factors that have been published in the literature:<br />
* [http://www.ncnr.nist.gov/resources/old_applets/index.html NCNR SANS solution form factors]:<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Sphere.html Sphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyHardSphere.html PolyHardSphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyRectSphere.html PolyRectSphere]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/CoreShell.html CoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyCoreShell.html PolyCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PolyCoreShellRatio.html PolyCoreShellRatio]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Cylinder.html Cylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/HollowCylinder.html HollowCylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/CoreShellCylinder.html CoreShellCylinder]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Ellipsoid.html Ellipsoid]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/OblateCoreShell.html OblateCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/ProlateCoreShell.html ProlateCoreShell]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/TwoHomopolymerRPA.html TwoHomopolymerRPA]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/DAB.html DAB]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/TeubnerStrey.html TeubnerStrey]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/Lorentz.html Lorentz]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PeakLorentz.html PeakLorentz]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PeakGauss.html PeakGauss]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/PowerLaw.html PowerLaw]<br />
** [http://www.ncnr.nist.gov/resources/old_applets/sansmodels/BE_RPA.html BE_RPA]<br />
<br />
* [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS, Born form factors] (see also Gilles Renaud, Rémi Lazzari,Frédéric Leroy "[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVY-4X36TK4-1&_user=2422869&_coverDate=08%2F31%2F2009&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1702575836&_rerunOrigin=google&_acct=C000057228&_version=1&_urlVersion=0&_userid=2422869&md5=d5f357bfcbf9a39bb8e42cfeac555359&searchtype=a Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering]" Surface Science Reports, 64 (8), 31 August 2009, 255-380 [http://dx.doi.org/10.1016/j.surfrep.2009.07.002 doi:10.1016/j.surfrep.2009.07.002]):<br />
** Parallelepiped<br />
** Pyramid<br />
** Cylinder<br />
** Cone<br />
** Prism 3<br />
** Tetrahedron<br />
** Prism 6<br />
** cone 6<br />
** Sphere<br />
** Cubooctahedron<br />
** Facetted sphere<br />
** Full sphere<br />
** Full spheroid<br />
** Box<br />
** Anisotropic pyramid<br />
** Hemi-ellipsoid<br />
<br />
* [http://scripts.iucr.org/cgi-bin/paper?S0021889811011691 Scattering functions of Platonic solids] Xin Li, Roger Pynn, Wei-Ren Chen, et al. Journal of Applied Crystallography 2011, 44, p.1 [http://dx.doi.org/10.1107/S0021889811011691 doi:10.1107/S0021889811011691]<br />
*# Tetrahedron<br />
*# Hexahedron (cube, parallelepiped, etc.)<br />
*# Octahedron<br />
*# Dodecahedron<br />
*# Icosahedron<br />
<br />
* Pedersen Review: [http://linkinghub.elsevier.com/retrieve/pii/S0001868697003126 Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting] Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. [http://dx.doi.org/10.1016/S0001-8686(97)00312-6 doi: 10.1016/S0001-8686(97)00312-6]<br />
*# Homogeneous sphere<br />
*# Spherical shell<br />
*# Spherical concentric shells<br />
*# Particles consisting of spherical subunits<br />
*# Ellipsoid of revolution<br />
*# Tri-axial ellipsoid<br />
*# Cube and rectangular parallelepipedons<br />
*# Truncated octahedra<br />
*# Faceted sphere<br />
*# Cube with terraces<br />
*# Cylinder<br />
*# Cylinder with elliptical cross section<br />
*# Cylinder with spherical end-caps<br />
*# Infinitely thin rod<br />
*# Infinitely thin circular disk<br />
*# Fractal aggregates<br />
*# Flexible polymers with Gaussian statistics<br />
*# Flexible self-avoiding polymers<br />
*# Semi-flexible polymers without self-avoidance<br />
*# Semi-flexible polymers with self-avoidance<br />
*# Star polymer with Gaussian statistics<br />
*# Star-burst polymer with Gaussian statistics<br />
*# Regular comb polymer with Gaussian statistics<br />
*# Arbitrarily branched polymers with Gaussian statistics<br />
*# Sphere with Gaussian chains attached<br />
*# Ellipsoid with Gaussian chains attached<br />
*# Cylinder with Gaussian chains attached<br />
<br />
* [http://www.nature.com/nmat/journal/v9/n11/extref/nmat2870-s1.pdf Supplementary Information] of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin "[http://www.nature.com/nmat/journal/v9/n11/full/nmat2870.html DNA-nanoparticle superlattices formed from anisotropic building blocks]" Nature Materials '''9''', 913-917, '''2010'''. [http://dx.doi.org/10.1038/nmat2870 doi: 10.1038/nmat2870]<br />
*# Pyramid<br />
*# Cube<br />
*# Cylinder<br />
*# Octahedron<br />
*# Rhombic dodecahedron (RD)<br />
*# Triangular prism<br />
<br />
* Other:<br />
** [http://www.eng.uc.edu/~gbeaucag/Classes/Analysis/Chapter8.html This tutorial] lists sphere, rod, disk, and Gaussian polymer coil.<br />
** '''Block-Copolymer Micelles''' [http://pubs.acs.org/doi/abs/10.1021/ma9512115 Scattering Form Factor of Block Copolymer Micelles] Jan Skov Pedersen* and Michael C. Gerstenberg, Macromolecules, 1996, 29 (4), pp 1363–1365 [http://dx.doi.org/10.1021/ma9512115 DOI: 10.1021/ma9512115]<br />
** '''Capped cylinder''' [http://scripts.iucr.org/cgi-bin/paper?S0021889804000020 Scattering from cylinders with globular end-caps]. H. Kaya. J. Appl. Cryst. (2004). 37, 223-230 [http://dx.doi.org/10.1107/S0021889804000020 doi: 10.1107/S0021889804000020]<br />
** '''Lens-shaped disc''' [http://scripts.iucr.org/cgi-bin/paper?aj5016 Scattering from capped cylinders. Addendum.] H. Kaya and N.-R. de Souza. J. Appl. Cryst. (2004). 37, 508-509 [http://dx.doi.org/10.1107/S0021889804005709 doi: 10.1107/S0021889804005709 ]</div>68.194.136.6http://gisaxs.com/index.php?title=Fourier_transform&diff=321Fourier transform2014-06-05T01:49:52Z<p>68.194.136.6: /* Scattering */</p>
<hr />
<div>The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.<br />
<br />
The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.<br />
<br />
==Mathematical form==<br />
The Fourier transform is typically given by:<br />
:<math>\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx</math><br />
The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:<br />
:<math>f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi</math><br />
<br />
==Scattering==<br />
The fundamental equation in scattering is:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\<br />
\end{alignat}<br />
</math><br />
Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average over the intensities of all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\<br />
\end{alignat}<br />
</math><br />
The inner component can be thought of as the [[reciprocal-space]]:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\<br />
<br />
\end{alignat}<br />
</math><br />
As as be seen, this is mathematically identical to the Fourier transform operation previously described.<br />
<br />
==Phase problem==<br />
Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.<br />
<br />
There are many interesting research thrusts aimed at 'solving' the phase problem. For instance, one can conduct repeated measurements on the same (or highly similar) sample using different contrast conditions. For neutrons, this can be accomplished by isotopic substitution or by using polarized neutrons interacting with magnetic layers. For x-rays, one can vary the x-ray energy to modify the relative scattering contrasts. In any case, one then attempts to simultaneously fit the combined dataset in a self-consistent way; this narrows the range of possible solutions and thus recovers some phase information.<br />
<br />
Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. [[Coherent Diffraction Imaging]]).</div>68.194.136.6http://gisaxs.com/index.php?title=Fourier_transform&diff=320Fourier transform2014-06-05T01:35:12Z<p>68.194.136.6: /* Phase problem */</p>
<hr />
<div>The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.<br />
<br />
The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.<br />
<br />
==Mathematical form==<br />
The Fourier transform is typically given by:<br />
:<math>\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx</math><br />
The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:<br />
:<math>f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi</math><br />
<br />
==Scattering==<br />
The fundamental equation in scattering is:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\<br />
\end{alignat}<br />
</math><br />
Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average over the intensities of all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous density of the scattering <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\<br />
\end{alignat}<br />
</math><br />
The inner component can be thought of as the [[reciprocal-space]]:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\<br />
<br />
\end{alignat}<br />
</math><br />
As as be seen, this is mathematically identical to the Fourier transform operation previously described.<br />
<br />
==Phase problem==<br />
Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.<br />
<br />
There are many interesting research thrusts aimed at 'solving' the phase problem. For instance, one can conduct repeated measurements on the same (or highly similar) sample using different contrast conditions. For neutrons, this can be accomplished by isotopic substitution or by using polarized neutrons interacting with magnetic layers. For x-rays, one can vary the x-ray energy to modify the relative scattering contrasts. In any case, one then attempts to simultaneously fit the combined dataset in a self-consistent way; this narrows the range of possible solutions and thus recovers some phase information.<br />
<br />
Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. [[Coherent Diffraction Imaging]]).</div>68.194.136.6http://gisaxs.com/index.php?title=Fourier_transform&diff=319Fourier transform2014-06-05T01:34:48Z<p>68.194.136.6: /* Phase problem */</p>
<hr />
<div>The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.<br />
<br />
The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.<br />
<br />
==Mathematical form==<br />
The Fourier transform is typically given by:<br />
:<math>\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx</math><br />
The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:<br />
:<math>f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi</math><br />
<br />
==Scattering==<br />
The fundamental equation in scattering is:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\<br />
\end{alignat}<br />
</math><br />
Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average over the intensities of all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous density of the scattering <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\<br />
\end{alignat}<br />
</math><br />
The inner component can be thought of as the [[reciprocal-space]]:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\<br />
<br />
\end{alignat}<br />
</math><br />
As as be seen, this is mathematically identical to the Fourier transform operation previously described.<br />
<br />
==Phase problem==<br />
Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.<br />
<br />
There are many interesting research thrusts aimed at 'solving' the phase problem. For instance, one can conduct repeated measurements on the same (or highly similar) sample using different contrast conditions. For neutrons, this can be accomplished by isotopic substitution or by using polarized neutrons interacting with magnetic layers. For x-rays, one can vary the x-ray energy to modify the relative scattering contrasts. In any case, one then attempts to simultaneously fit the combined dataset in a self-consistent way; this narrows the range of possible solutions and thus recovers some phase information.<br />
<br />
Alternatively, some methods exploit a high degree of coherence in the incident radiation in order to reconstruct the realspace structure (e.g. [[Coherent Diffractive Imaging]]).</div>68.194.136.6http://gisaxs.com/index.php?title=Fourier_transform&diff=318Fourier transform2014-06-05T01:30:50Z<p>68.194.136.6: </p>
<hr />
<div>The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.<br />
<br />
The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.<br />
<br />
==Mathematical form==<br />
The Fourier transform is typically given by:<br />
:<math>\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx</math><br />
The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:<br />
:<math>f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi</math><br />
<br />
==Scattering==<br />
The fundamental equation in scattering is:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\<br />
\end{alignat}<br />
</math><br />
Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average over the intensities of all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous density of the scattering <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\<br />
\end{alignat}<br />
</math><br />
The inner component can be thought of as the [[reciprocal-space]]:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\<br />
<br />
\end{alignat}<br />
</math><br />
As as be seen, this is mathematically identical to the Fourier transform operation previously described.<br />
<br />
==Phase problem==<br />
Note that experimentally, we can only measure the squared amplitude, ''I''('''q'''); we can never directly measure the ''true'' reciprocal-space, given by ''F''('''q'''). This is known as the ''phase-problem'': our detector only measures the intensity of the scattered radiation, and not the phase of the scattered waves. As such, the scattering dataset is incomplete. If we had both the amplitude and phase of reciprocal-space, we could simply invert the data (using an inverse Fourier transform), and thereby recover the realspace structure of the sample. Lacking the phase information, we cannot do this. Instead, scattering data must be fit to candidate models, with concordant models being deemed 'more likely' to represent the sample. However, interpreting scattering data is a formally ill-prosed problem: there are multiple possible realspace models that will fit a particular experimental dataset. Abstractly, it is impossible to know which model is the right one. In practice, one can use auxilliary knowledge, such as other measurements (AFM, electron microscopy) or physical constraints, in order to exclude some candidate models, and thereby identify the correct sample structure.</div>68.194.136.6http://gisaxs.com/index.php?title=Fourier_transform&diff=317Fourier transform2014-06-05T01:25:25Z<p>68.194.136.6: Created page with "The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stre..."</p>
<hr />
<div>The '''Fourier transform''' is a mathematical transformation that appears in many branches of physics. One feature of this transform is that periodic signals in the input stream are converted into well-defined peaks in the output Fourier space.<br />
<br />
The Fourier transform is central in scattering. In scattering, an incident wave travels through a sample. All of the entities in the sample act as scattering sources, giving rise to secondary waves that interfere with one another. This wave-interference phenomenon is essentially physically performing the Fourier transform operation. Thus, the observed scattering pattern (patterns of constructive and destructive interference) is the Fourier transform of the realspace density profile probed by the wave. The full 3D [[reciprocal-space]] is the Fourier transform of the sample's structure.<br />
<br />
==Mathematical form==<br />
The Fourier transform is typically given by:<br />
:<math>\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx</math><br />
The transform inverts the units of the input variable. For instance, when the input stream represents time, the Fourier space will represent frequency (1/time). When the input stream represents space, the Fourier space will represent inverse-space (1/distance). The Fourier transform can also be inverted:<br />
:<math>f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi</math><br />
<br />
==Scattering==<br />
The fundamental equation in scattering is:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\<br />
\end{alignat}<br />
</math><br />
Where the observed scattering intensity (''I'') in the 3D reciprocal space ('''q''') is given by an ensemble average over the intensities of all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the scattering contribution (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous density of the scattering <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space:<br />
:<math><br />
\begin{alignat}{2}<br />
I(\mathbf{q}) <br />
& = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\<br />
\end{alignat}<br />
</math><br />
The inner component can be thought of as the [[reciprocal-space]]:<br />
:<math><br />
\begin{alignat}{2}<br />
<br />
F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\<br />
<br />
\end{alignat}<br />
</math><br />
As as be seen, this is mathematically identical to the Fourier transform operation previously described.</div>68.194.136.6http://gisaxs.com/index.php?title=Lattices&diff=316Lattices2014-06-05T01:06:17Z<p>68.194.136.6: </p>
<hr />
<div>In x-ray scattering, we frequently study materials which have constituents arranged on a well-defined '''lattice'''. For instance, an atomic crystal has atoms which occupy well-defined sites within a representative [[unit cell]], which then repeats in all three directions throughout space. Nanoparticle superlattices are a nanoscale analogue, where each lattice site is occupied by a nanoparticle. Other kinds of nanostructures systems can be considered similarly. Block-copolymers mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice).<br />
<br />
Well-define realspace lattices (repeating structures) [[Fourier transform|give rise]] to well-defined peaks in [[reciprocal-space]], which makes it possible to determine the realspace lattice by considering the arrangement (symmetry) of the scattering peaks. <br />
<br />
==Notation==<br />
* '''Real space''':<br />
** Crystal ''planes'':<br />
*** (hkl) denotes a plane of the crystal structure (and repetitions of that plane, with the given spacing). In cubic systems (but not others), the normal to the plane is [hkl]<br />
*** {hkl} denotes the set of all planes that are equivalent to (hkl) by the symmetry of the lattice<br />
** Crystal ''directions'':<br />
*** [hkl] denotes a direction of a vector (in the basis of the direct lattice vectors)<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all directions that are equivalent to [hkl] by symmetry (e.g. in cubic system &#12296;100&#12297; means [100], [010], [001], [-100], [0-10], [00-1])<br />
** hkl denotes a diffracting plane<br />
<br />
* '''[[Reciprocal space]]''':<br />
** Reciprocal ''planes'':<br />
*** [hkl] denotes a plane<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all planes that are equivalent to [hkl]<br />
** Reciprocal ''directions'':<br />
*** (hkl) denotes a particular direction (normal to plane (hkl) in real space)<br />
*** {hkl} denotes the set of all directions that are equivalent to (hkl)<br />
** hkl denotes an indexed reflection<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Crystal_structure Wikipedia: Crystal Structure]<br />
* [[Lattice:Packing fraction]]</div>68.194.136.6http://gisaxs.com/index.php?title=Lattices&diff=315Lattices2014-06-05T01:05:35Z<p>68.194.136.6: </p>
<hr />
<div>In x-ray scattering, we frequently study materials which have constituents arranged on a well-defined '''lattice'''. For instance, an atomic crystal has atoms which occupy well-defined sites within a representative [[unit cell]], which then repeats in all three directions throughout space. Nanoparticle superlattices are a nanoscale analogue, where each lattice site is occupied by a nanoparticle. Other kinds of nanostructures systems can be considered similarly. Block-copolymers mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice).<br />
<br />
Well-define realspace lattices (repeating structures) give rise to well-defined peaks in [[reciprocal-space]], which makes it possible to determine the realspace lattice by considering the arrangement (symmetry) of the scattering peaks. <br />
<br />
==Notation==<br />
* '''Real space''':<br />
** Crystal ''planes'':<br />
*** (hkl) denotes a plane of the crystal structure (and repetitions of that plane, with the given spacing). In cubic systems (but not others), the normal to the plane is [hkl]<br />
*** {hkl} denotes the set of all planes that are equivalent to (hkl) by the symmetry of the lattice<br />
** Crystal ''directions'':<br />
*** [hkl] denotes a direction of a vector (in the basis of the direct lattice vectors)<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all directions that are equivalent to [hkl] by symmetry (e.g. in cubic system &#12296;100&#12297; means [100], [010], [001], [-100], [0-10], [00-1])<br />
** hkl denotes a diffracting plane<br />
<br />
* '''[[Reciprocal space]]''':<br />
** Reciprocal ''planes'':<br />
*** [hkl] denotes a plane<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all planes that are equivalent to [hkl]<br />
** Reciprocal ''directions'':<br />
*** (hkl) denotes a particular direction (normal to plane (hkl) in real space)<br />
*** {hkl} denotes the set of all directions that are equivalent to (hkl)<br />
** hkl denotes an indexed reflection<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Crystal_structure Wikipedia: Crystal Structure]<br />
* [[Lattice:Packing fraction]]</div>68.194.136.6http://gisaxs.com/index.php?title=Lattices&diff=314Lattices2014-06-05T01:04:05Z<p>68.194.136.6: /* See Also */</p>
<hr />
<div>In x-ray scattering, we frequently study materials which have constituents arranged on a well-defined '''lattice'''. For instance, an atomic crystal has atoms which occupy well-defined sites within a representative [[unit cell]], which then repeats in all three directions throughout space. Nanoparticle superlattices are a nanoscale analogue, where each lattice site is occupied by a nanoparticle. Other kinds of nanostructures systems can be considered similarly. Block-copolymers mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice).<br />
<br />
Every lattice has a particular symmetry, which defines the [[reciprocal-space]] peaks which will appear.<br />
<br />
==Notation==<br />
* '''Real space''':<br />
** Crystal ''planes'':<br />
*** (hkl) denotes a plane of the crystal structure (and repetitions of that plane, with the given spacing). In cubic systems (but not others), the normal to the plane is [hkl]<br />
*** {hkl} denotes the set of all planes that are equivalent to (hkl) by the symmetry of the lattice<br />
** Crystal ''directions'':<br />
*** [hkl] denotes a direction of a vector (in the basis of the direct lattice vectors)<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all directions that are equivalent to [hkl] by symmetry (e.g. in cubic system &#12296;100&#12297; means [100], [010], [001], [-100], [0-10], [00-1])<br />
** hkl denotes a diffracting plane<br />
<br />
* '''[[Reciprocal space]]''':<br />
** Reciprocal ''planes'':<br />
*** [hkl] denotes a plane<br />
*** <math>\left\langle hkl\right\rangle</math> denotes the set of all planes that are equivalent to [hkl]<br />
** Reciprocal ''directions'':<br />
*** (hkl) denotes a particular direction (normal to plane (hkl) in real space)<br />
*** {hkl} denotes the set of all directions that are equivalent to (hkl)<br />
** hkl denotes an indexed reflection<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Crystal_structure Wikipedia: Crystal Structure]<br />
* [[Lattice:Packing fraction]]</div>68.194.136.6http://gisaxs.com/index.php?title=Lattice:Packing_fraction&diff=313Lattice:Packing fraction2014-06-05T00:56:27Z<p>68.194.136.6: </p>
<hr />
<div>The '''packing fraction''' (or particle volume fraction) for a [[Lattices|lattice]] is given by:<br />
:<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math><br />
Where ''N'' is the number of particles per unit cell (which has volume <math>v_{\mathrm{cell}}</math>). For a sphere, the volume is <math>V=4\pi R^3/3</math> so:<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }</math><br />
For a [[Lattice:Cubic|cubic]] [[unit cell]] of edge-length ''a'':<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math><br />
===Examples===<br />
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=16 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{2}/6\approx0.740</math> when <math>R=a/(2\sqrt{2})</math><br />
For a [[Lattice:BCC#Symmetry|BCC lattice]], the packing fraction is 0.680:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=8 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/8\approx0.680</math> when <math>R=a\sqrt{3}/4</math><br />
For a [[Lattice:Diamond#Symmetry|diamond lattice]], the packing fraction is 0.340:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=32 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/16\approx0.340</math> when <math>R=a\sqrt{3}/8</math></div>68.194.136.6http://gisaxs.com/index.php?title=Lattice:Packing_fraction&diff=312Lattice:Packing fraction2014-06-05T00:55:56Z<p>68.194.136.6: /* Examples */</p>
<hr />
<div>The '''packing fraction''' (or particle volume fraction) for a [[Lattices|lattice]] is given by:<br />
:<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math><br />
Where ''N'' is the number of particles per unit cell (which has volume <math>v_{\mathrm{cell}}</math>). For a sphere, the volume is <math>V=4\pi R^3/3</math> so:<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }</math><br />
For a [[Lattice:Cubic|cubic]] cell of edge-length ''a'':<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math><br />
===Examples===<br />
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=16 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{2}/6\approx0.740</math> when <math>R=a/(2\sqrt{2})</math><br />
For a [[Lattice:BCC#Symmetry|BCC lattice]], the packing fraction is 0.680:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=8 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/8\approx0.680</math> when <math>R=a\sqrt{3}/4</math><br />
For a [[Lattice:Diamond#Symmetry|diamond lattice]], the packing fraction is 0.340:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=32 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/16\approx0.340</math> when <math>R=a\sqrt{3}/8</math></div>68.194.136.6http://gisaxs.com/index.php?title=Lattice:Packing_fraction&diff=311Lattice:Packing fraction2014-06-05T00:55:01Z<p>68.194.136.6: </p>
<hr />
<div>The '''packing fraction''' (or particle volume fraction) for a [[Lattices|lattice]] is given by:<br />
:<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math><br />
Where ''N'' is the number of particles per unit cell (which has volume <math>v_{\mathrm{cell}}</math>). For a sphere, the volume is <math>V=4\pi R^3/3</math> so:<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }</math><br />
For a [[Lattice:Cubic|cubic]] cell of edge-length ''a'':<br />
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math><br />
===Examples===<br />
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=16 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{2}/6\approx0.740</math> when <math>R=a/(2\sqrt{2})</math><br />
For a [[Lattice:BCC#Symmetry|BCC lattice]], the packing fraction is 0.680:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle volume fraction: <math>\phi=8 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/8\approx0.680</math> when <math>R=a\sqrt{3}/4</math><br />
For a [[Lattice:Diamond#Symmetry|diamond lattice]], the packing fraction is 0.340:<br />
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math><br />
* Assuming spherical particles of radius ''R'':<br />
** Particle [[Lattice:Packing fraction|volume fraction]]: <math>\phi=32 \pi R^3/\left(3a^3\right)</math><br />
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/16\approx0.340</math> when <math>R=a\sqrt{3}/8</math></div>68.194.136.6http://gisaxs.com/index.php?title=Refractive_index&diff=310Refractive index2014-06-05T00:50:21Z<p>68.194.136.6: /* Refraction */</p>
<hr />
<div>The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index. It describes how strongly wave propagation is altered within the given material. The modification of the wave's phase velocity in turn causes the propagation direction to be altered (i.e. the wave is refracted).<br />
<br />
==Total external reflection==<br />
X-rays interact rather weakly with matter (e.g. [[absorption lengths]] are on the order of millimeters); x-ray refractive indices are thus extremely close to 1. In fact, x-ray refractive indices tend to be slightly ''smaller'' than 1, giving rise to total ''external'' reflection at sufficiently small angle. This can be compared to total ''internal'' reflection typically observed for visible light.<br />
<br />
Visible light can be totally reflected when travelling from a denser medium into a less denser medium (e.g. light traveling in glass towards a glass-air interface will be totally reflected when the incident angle (measured from the interface) is less than ~50°). Whereas for x-rays, the beam can be totally reflected when travelling from a less dense medium into a denser medium (e.g. x-rays traveling in vacuum towards a vaucuum-[[Material:Silicon|silicon]] interface will be totally reflected if the grazing-incidence angle is less than ~0.2°).<br />
<br />
Notice that because the refractive index is very close to 1, the [[critical angle]]s are very small (as measured from the interface). The critical angle depends on the x-ray energy and the material, but is typically on the order of 0.1° to 0.5°.<br />
<br />
==Mathematical form==<br />
The complex refractive index is customarily denoted as:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
The real and imaginary components, ''𝛿'' and ''β'', describe the dispersive and absorptive aspects of the wave-matter interaction. These components can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
==Refraction==<br />
The refraction of x-rays follows the same laws as for refraction of visible light. I.e. [http://en.wikipedia.org/wiki/Snell%27s_law snell's law]. However, in [[GISAXS]] and [[reflectivity]], we conventionally describe the incident angle with respect to the plane of the interface (rather than being measured with respect to the film normal, as done in visible light optics). As such, the sine version of snell's law is converted into a cosine version:<br />
:<math><br />
\frac{\cos \theta_1}{\cos \theta_2} = \frac{n_2}{n_1}<br />
</math><br />
<br />
==See Also==<br />
* [[Critical angle]]</div>68.194.136.6http://gisaxs.com/index.php?title=Refractive_index&diff=309Refractive index2014-06-05T00:49:59Z<p>68.194.136.6: </p>
<hr />
<div>The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index. It describes how strongly wave propagation is altered within the given material. The modification of the wave's phase velocity in turn causes the propagation direction to be altered (i.e. the wave is refracted).<br />
<br />
==Total external reflection==<br />
X-rays interact rather weakly with matter (e.g. [[absorption lengths]] are on the order of millimeters); x-ray refractive indices are thus extremely close to 1. In fact, x-ray refractive indices tend to be slightly ''smaller'' than 1, giving rise to total ''external'' reflection at sufficiently small angle. This can be compared to total ''internal'' reflection typically observed for visible light.<br />
<br />
Visible light can be totally reflected when travelling from a denser medium into a less denser medium (e.g. light traveling in glass towards a glass-air interface will be totally reflected when the incident angle (measured from the interface) is less than ~50°). Whereas for x-rays, the beam can be totally reflected when travelling from a less dense medium into a denser medium (e.g. x-rays traveling in vacuum towards a vaucuum-[[Material:Silicon|silicon]] interface will be totally reflected if the grazing-incidence angle is less than ~0.2°).<br />
<br />
Notice that because the refractive index is very close to 1, the [[critical angle]]s are very small (as measured from the interface). The critical angle depends on the x-ray energy and the material, but is typically on the order of 0.1° to 0.5°.<br />
<br />
==Mathematical form==<br />
The complex refractive index is customarily denoted as:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
The real and imaginary components, ''𝛿'' and ''β'', describe the dispersive and absorptive aspects of the wave-matter interaction. These components can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
==Refraction==<br />
The refraction of x-rays follows the same laws as for refraction of visible light. I.e. [http://en.wikipedia.org/wiki/Snell%27s_law snell's law]. However, in [[GISAXS]] and [[reflectivity]], we conventionally describe the incident angle with respect to the plane of the interface (rather than being measured with respect to the film normal, as done in visible light optics). As such, the sine version of snell's law is converted into a cosine version:<br />
:<math><br />
\frac{\theta_1}{\theta_2} = \frac{n_2}{n_1}<br />
</math><br />
<br />
==See Also==<br />
* [[Critical angle]]</div>68.194.136.6http://gisaxs.com/index.php?title=Critical_angle&diff=308Critical angle2014-06-05T00:49:37Z<p>68.194.136.6: </p>
<hr />
<div>In [[GISAXS]], the '''critical angle''' for a thin film is the incident angle below which one gets [[Refractive_index#Total_external_reflection|total external reflection]] of the x-ray beam.<br />
<br />
Below the critical angle, the beam is fully reflected from the film. The x-ray field probes a short distance into the film surface (due to the evanescent wave); on the order of a few nanometers. Thus, a GISAXS measurement below the critical angle is inherently probing only the film surface. A measurement well above the critical angle, by comparison, penetrates through the film and thus measures the average of the structure through the whole film. Close to the critical angle, the refracted beam is nearly parallel to the film interface; in other words the beam is coupled into [[X-ray waveguide|waveguide]] modes. This increases the effective path-length of the beam through the sample, which thereby increases the intensity of the scattering.<br />
<br />
Normally in GISAXS experiments, it is useful to do measurements both below and above the critical angle; by comparing the two patterns, you can ascertain whether the structures observed at the surface (e.g. as seen by AFM or SEM) are representative of the entire film. Measurements close to the critical angle are useful because of the intensity enhancement.<br />
<br />
==Examples==<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon|Si]]<br />
| 2.3290<br />
| 2.0<br />
| 6.20<br />
| 0.824<br />
| 0.0291<br />
| 16.89<br />
|-<br />
| <br />
| <br />
| 4.0<br />
| 3.10<br />
| 0.451<br />
| 0.0319<br />
| 20.28<br />
|-<br />
| <br />
| <br />
| 8.0<br />
| 1.55<br />
| 0.224<br />
| 0.0317<br />
| 20.07<br />
|-<br />
| <br />
| <br />
| 12.0<br />
| 1.03<br />
| 0.149<br />
| 0.0317<br />
| 19.92<br />
|-<br />
<br />
| <br />
| <br />
| 16.0<br />
| 0.77<br />
| 0.112<br />
| 0.0316<br />
| 19.84<br />
|-<br />
| <br />
| <br />
| 24.0<br />
| 0.52<br />
| 0.07426<br />
| 0.0315<br />
| 19.77<br />
|-<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon dioxide|SiO2]]<br />
| 2.648<br />
| 2.0<br />
| 6.20<br />
| 0.927<br />
| 0.0328<br />
| 21.42<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 0.480<br />
| 0.0340<br />
| 22.96<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.239<br />
| 0.0338<br />
| 22.71<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.159<br />
| 0.0337<br />
| 22.58<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.119<br />
| 0.0337<br />
| 22.53<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.079<br />
| 0.0336<br />
| 22.48<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Gold|Au]]<br />
| 19.32<br />
| 2.0<br />
| 6.20<br />
| 1.830<br />
| 0.0647<br />
| 83.44<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 1.097<br />
| 0.0776<br />
| 119.89<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.560<br />
| 0.0792<br />
| 124.86<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.348<br />
| 0.0738<br />
| 108.29<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.282<br />
| 0.0797<br />
| 126.51<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.191<br />
| 0.0811<br />
| 130.80<br />
|}<br />
<br />
==Calculating==<br />
The critical angle for a material can be calculated in a variety of ways. For an elemental substance, [http://henke.lbl.gov/optical_constants/pert_form.html this online tool] will calculate it for you.<br />
===From SLD===<br />
The [[Scattering Length Density]] (SLD) for any given material can be computed using tabulated values. From this, the critical angle can be computed. The critical scattering vector is:<br />
:<math>q_c = \sqrt{ 16 \pi \mathrm{SLD} }</math><br />
In reflection-mode, the scattering vector is:<br />
::<math> \begin{alignat}{2}<br />
\mathbf{q} & = \mathbf{k}_o-\mathbf{k}_i \\<br />
|\mathbf{q}| & = q = 2 |k| \sin\theta_i = \frac{4 \pi}{\lambda}\sin\theta_i<br />
\end{alignat}<br />
</math><br />
Where ''λ'' is the wavelength of the x-rays. So:<br />
:<math><br />
\begin{alignat}{2}<br />
\theta_c & = \arcsin\left( \frac{ q_c \lambda }{4 \pi} \right) \\<br />
& = \arcsin\left( \frac{ \lambda \sqrt{16 \pi \mathrm{SLD} } }{4 \pi} \right) \\<br />
& \approx \sqrt{\frac{\lambda^2 \mathrm{SLD} }{\pi}}<br />
\end{alignat}<br />
</math><br />
<br />
===From refractive index===<br />
The critical angle is simply a result of the refractive index contrast between the film and the ambient. Of course, in this case we are talking about the '''x-ray [[refractive index]]''' (not the usual refractive index for visible light). In the case of neutrons, the film similarly exhibits a neutron refractive index. The refractive index is complex:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
For x-rays, the values of ''𝛿'' and ''β'' can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
Once the real part of the refractive index (''𝛿'') is known, conversion to critical angle is straightforward:<br />
:<math><br />
\theta_c = \sqrt{ 2 \delta }<br />
</math><br />
Note that the resultant critical angle has units of ''radians''. For a multi-component system, one simply sums the contributions from each element:<br />
:<math><br />
\theta_c = \sqrt{ \frac{\rho N_a r_e \lambda^2 \sum_{i=1}^{N} f_{1i} }{ \pi \sum_{i=1}^{N} M_i } }<br />
</math><br />
<br />
==See Also==<br />
* [[Scattering Length Density]]<br />
* [[Refractive index]]</div>68.194.136.6http://gisaxs.com/index.php?title=Refractive_index&diff=307Refractive index2014-06-05T00:48:48Z<p>68.194.136.6: </p>
<hr />
<div>The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index: it describes how strongly wave propagation is altered within the given material. The modification of the wave's phase velocity in turn causes the propagation direction to be altered (i.e. the wave is refracted).<br />
<br />
==Total external reflection==<br />
X-rays interact rather weakly with matter (e.g. [[absorption lengths]] are on the order of millimeters); x-ray refractive indices are thus extremely close to 1. In fact, x-ray refractive indices tend to be slightly ''smaller'' than 1, giving rise to total ''external'' reflection at sufficiently small angle. This can be compared to total ''internal'' reflection typically observed for visible light.<br />
<br />
Visible light can be totally reflected when travelling from a denser medium into a less denser medium (e.g. light traveling in glass towards a glass-air interface will be totally reflected when the incident angle (measured from the interface) is less than ~50°). Whereas for x-rays, the beam can be totally reflected when travelling from a less dense medium into a denser medium (e.g. x-rays traveling in vacuum towards a vaucuum-[[Material:Silicon|silicon]] interface will be totally reflected if the grazing-incidence angle is less than ~0.2°).<br />
<br />
Notice that because the refractive index is very close to 1, the [[critical angle]]s are very small (as measured from the interface). The critical angle depends on the x-ray energy and the material, but is typically on the order of 0.1° to 0.5°.<br />
<br />
==Mathematical form==<br />
The complex refractive index is customarily denoted as:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
The real and imaginary components, ''𝛿'' and ''β'', describe the dispersive and absorptive aspects of the wave-matter interaction. These components can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
==Refraction==<br />
The refraction of x-rays follows the same laws as for refraction of visible light. I.e. [http://en.wikipedia.org/wiki/Snell%27s_law snell's law]. However, in [[GISAXS]] and [[reflectivity]], we conventionally describe the incident angle with respect to the plane of the interface (rather than being measured with respect to the film normal, as done in visible light optics). As such, the sine version of snell's law is converted into a cosine version:<br />
:<math><br />
\frac{\theta_1}{\theta_2} = \frac{n_2}{n_1}<br />
</math><br />
<br />
==See Also==<br />
* [[Critical angle]]</div>68.194.136.6http://gisaxs.com/index.php?title=Refractive_index&diff=306Refractive index2014-06-05T00:48:32Z<p>68.194.136.6: /* Total external reflection */</p>
<hr />
<div>The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index: it describes how strongly wave propagation is altered within the given material. The modification of the wave's phase velocity in turn causes the propagation direction to be altered (i.e. the wave is refracted).<br />
<br />
==Total external reflection==<br />
X-rays interact rather weakly with matter (e.g. [[absorption lengths]] are on the order of millimeters); x-ray refractive indices are thus extremely close to 1. In fact, x-ray refractive indices tend to be slightly ''smaller'' than 1, giving rise to total ''external'' reflection at sufficiently small angle. This can be compared to total ''internal'' reflection typically observed for visible light.<br />
<br />
Visible light can be totally reflected when travelling from a denser medium into a less denser medium (e.g. light traveling in glass towards a glass-air interface will be totally reflected when the incident angle (measured from the interface) is less than ~50°). Whereas for x-rays, the beam can be totally reflected when travelling from a less dense medium into a denser medium (e.g. x-rays traveling in vacuum towards a vaucuum-[[Material:Silicon|silicon]] interface will be totally reflected if the grazing-incidence angle is less than ~0.2°).<br />
<br />
Notice that because the refractive index is very close to 1, the [[critical angle]]s are very small (as measured from the interface). The critical angle depends on the x-ray energy and the material, but is typically on the order of 0.1° to 0.5°.<br />
<br />
==Mathematical form==<br />
The complex refractive index is customarily denoted as:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
The real and imaginary components, ''𝛿'' and ''β'', describe the dispersive and absorptive aspects of the wave-matter interaction. These components can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
==Refraction==<br />
The refraction of x-rays follows the same laws as for refraction of visible light. I.e. [http://en.wikipedia.org/wiki/Snell%27s_law snell's law]. However, in [[GISAXS]] and [[reflectivity]], we conventionally describe the incident angle with respect to the plane of the interface (rather than being measured with respect to the film normal, as done in visible light optics). As such, the sine version of snell's law is converted into a cosine version:<br />
:<math><br />
\frac{\theta_1}{\theta_2} = \frac{n_2}{n_1}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Refractive_index&diff=305Refractive index2014-06-05T00:47:31Z<p>68.194.136.6: Created page with "The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index: it describes how strongly wave propagation is altered within ..."</p>
<hr />
<div>The '''refractive index''' for x-rays is strictly analogous to the conventional (visible light) refractive index: it describes how strongly wave propagation is altered within the given material. The modification of the wave's phase velocity in turn causes the propagation direction to be altered (i.e. the wave is refracted).<br />
<br />
==Total external reflection==<br />
X-rays interact rather weakly with matter (e.g. [[absorption lengths]] are on the order of millimeters); x-ray refractive indices are thus extremely close to 1. In fact, x-ray refractive indices tend to be slightly ''smaller'' than 1, giving rise to total ''external'' reflection at sufficiently small angle. This can be compared to total ''internal'' reflection typically observed for visible light.<br />
<br />
Visible light can be totally reflected when travelling from a denser medium into a less denser medium (e.g. light traveling in glass towards a glass-air interface will be totally reflected when the incident angle (measured from the interface) is less than ~50°). Whereas for x-rays, the beam can be totally reflected when travelling from a less dense medium into a denser medium (e.g. x-rays traveling in vacuum towards a vaucuum-[[Material:Silicon|silicon]] interface will be totally reflected if the grazing-incidence angle is less than ~0.2°).<br />
<br />
Notice that because the refractive index is very close to 1, the critical angles are very small (as measured from the interface). The critical angle depends on the x-ray energy and the material, but is typically on the order of 0.1° to 0.5°.<br />
<br />
==Mathematical form==<br />
The complex refractive index is customarily denoted as:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
The real and imaginary components, ''𝛿'' and ''β'', describe the dispersive and absorptive aspects of the wave-matter interaction. These components can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
==Refraction==<br />
The refraction of x-rays follows the same laws as for refraction of visible light. I.e. [http://en.wikipedia.org/wiki/Snell%27s_law snell's law]. However, in [[GISAXS]] and [[reflectivity]], we conventionally describe the incident angle with respect to the plane of the interface (rather than being measured with respect to the film normal, as done in visible light optics). As such, the sine version of snell's law is converted into a cosine version:<br />
:<math><br />
\frac{\theta_1}{\theta_2} = \frac{n_2}{n_1}<br />
</math></div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=304Technical articles2014-06-05T00:33:28Z<p>68.194.136.6: /* Important quantities */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]/[[SANS]]<br />
** [[CD-SAXS]] ([[RSANS]])<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
* [[Scattering intensity]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
* [[Refractive index]]<br />
* [[Absorption length]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Critical_angle&diff=303Critical angle2014-06-05T00:29:27Z<p>68.194.136.6: /* From refractive index */</p>
<hr />
<div>In [[GISAXS]], the '''critical angle''' for a thin film is the incident angle below which one gets total external reflection of the x-ray beam.<br />
<br />
Below the critical angle, the beam is fully reflected from the film. The x-ray field probes a short distance into the film surface (due to the evanescent wave); on the order of a few nanometers. Thus, a GISAXS measurement below the critical angle is inherently probing only the film surface. A measurement well above the critical angle, by comparison, penetrates through the film and thus measures the average of the structure through the whole film. Close to the critical angle, the refracted beam is nearly parallel to the film interface; in other words the beam is coupled into [[X-ray waveguide|waveguide]] modes. This increases the effective path-length of the beam through the sample, which thereby increases the intensity of the scattering.<br />
<br />
Normally in GISAXS experiments, it is useful to do measurements both below and above the critical angle; by comparing the two patterns, you can ascertain whether the structures observed at the surface (e.g. as seen by AFM or SEM) are representative of the entire film. Measurements close to the critical angle are useful because of the intensity enhancement.<br />
<br />
==Examples==<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon|Si]]<br />
| 2.3290<br />
| 2.0<br />
| 6.20<br />
| 0.824<br />
| 0.0291<br />
| 16.89<br />
|-<br />
| <br />
| <br />
| 4.0<br />
| 3.10<br />
| 0.451<br />
| 0.0319<br />
| 20.28<br />
|-<br />
| <br />
| <br />
| 8.0<br />
| 1.55<br />
| 0.224<br />
| 0.0317<br />
| 20.07<br />
|-<br />
| <br />
| <br />
| 12.0<br />
| 1.03<br />
| 0.149<br />
| 0.0317<br />
| 19.92<br />
|-<br />
<br />
| <br />
| <br />
| 16.0<br />
| 0.77<br />
| 0.112<br />
| 0.0316<br />
| 19.84<br />
|-<br />
| <br />
| <br />
| 24.0<br />
| 0.52<br />
| 0.07426<br />
| 0.0315<br />
| 19.77<br />
|-<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon dioxide|SiO2]]<br />
| 2.648<br />
| 2.0<br />
| 6.20<br />
| 0.927<br />
| 0.0328<br />
| 21.42<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 0.480<br />
| 0.0340<br />
| 22.96<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.239<br />
| 0.0338<br />
| 22.71<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.159<br />
| 0.0337<br />
| 22.58<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.119<br />
| 0.0337<br />
| 22.53<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.079<br />
| 0.0336<br />
| 22.48<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Gold|Au]]<br />
| 19.32<br />
| 2.0<br />
| 6.20<br />
| 1.830<br />
| 0.0647<br />
| 83.44<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 1.097<br />
| 0.0776<br />
| 119.89<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.560<br />
| 0.0792<br />
| 124.86<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.348<br />
| 0.0738<br />
| 108.29<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.282<br />
| 0.0797<br />
| 126.51<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.191<br />
| 0.0811<br />
| 130.80<br />
|}<br />
<br />
==Calculating==<br />
The critical angle for a material can be calculated in a variety of ways. For an elemental substance, [http://henke.lbl.gov/optical_constants/pert_form.html this online tool] will calculate it for you.<br />
===From SLD===<br />
The [[Scattering Length Density]] (SLD) for any given material can be computed using tabulated values. From this, the critical angle can be computed. The critical scattering vector is:<br />
:<math>q_c = \sqrt{ 16 \pi \mathrm{SLD} }</math><br />
In reflection-mode, the scattering vector is:<br />
::<math> \begin{alignat}{2}<br />
\mathbf{q} & = \mathbf{k}_o-\mathbf{k}_i \\<br />
|\mathbf{q}| & = q = 2 |k| \sin\theta_i = \frac{4 \pi}{\lambda}\sin\theta_i<br />
\end{alignat}<br />
</math><br />
Where ''λ'' is the wavelength of the x-rays. So:<br />
:<math><br />
\begin{alignat}{2}<br />
\theta_c & = \arcsin\left( \frac{ q_c \lambda }{4 \pi} \right) \\<br />
& = \arcsin\left( \frac{ \lambda \sqrt{16 \pi \mathrm{SLD} } }{4 \pi} \right) \\<br />
& \approx \sqrt{\frac{\lambda^2 \mathrm{SLD} }{\pi}}<br />
\end{alignat}<br />
</math><br />
<br />
===From refractive index===<br />
The critical angle is simply a result of the refractive index contrast between the film and the ambient. Of course, in this case we are talking about the '''x-ray [[refractive index]]''' (not the usual refractive index for visible light). In the case of neutrons, the film similarly exhibits a neutron refractive index. The refractive index is complex:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
For x-rays, the values of ''𝛿'' and ''β'' can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
Once the real part of the refractive index (''𝛿'') is known, conversion to critical angle is straightforward:<br />
:<math><br />
\theta_c = \sqrt{ 2 \delta }<br />
</math><br />
Note that the resultant critical angle has units of ''radians''. For a multi-component system, one simply sums the contributions from each element:<br />
:<math><br />
\theta_c = \sqrt{ \frac{\rho N_a r_e \lambda^2 \sum_{i=1}^{N} f_{1i} }{ \pi \sum_{i=1}^{N} M_i } }<br />
</math><br />
<br />
==See Also==<br />
* [[Scattering Length Density]]<br />
* [[Refractive index]]</div>68.194.136.6http://gisaxs.com/index.php?title=Scattering_Length_Density&diff=302Scattering Length Density2014-06-05T00:27:10Z<p>68.194.136.6: </p>
<hr />
<div>The '''Scattering Length Density''' ('''SLD''', sometimes denoted ''N<sub>b</sub>'') is a measure of the scattering power of a material. It increases with the physical density (how tightly packed the scattering entities are), as well as the intrinsic scattering power of the 'scattering entities'. For x-rays, the scattering arises from the electron density, whereas for neutrons, the scattering arises from the nuclear scattering lengths.<br />
<br />
==Calculating==<br />
SLD can be computed from the scattering lengths and material densities. Specifically:<br />
:<math>\mathrm{SLD} = \frac{\sum_{i=1}^{N}b_i}{V_m}</math><br />
where we sum the scattering length contributions (''b<sub>i</sub>'') from the ''N'' atoms within the [[unit cell]], and divide by the volume, ''V<sub>m</sub>'' of this 'unit cell'. Note that the 'unit cell' need not be a crystallographic unit cell: it is simply a representative volume of the material in question: the atoms within it may be arranged randomly (e.g. amorphous material) or in a crystalline way. For instance for a single molecule, ''V<sub>m</sub>'' is simply the molecular volume, and the sum includes all the atoms in the molecule. In general we don't know what the appropriate 'molecular volume' should be, but we can compute it from the known bulk density of the material (<math>\rho</math>) and the molecular weight (''MW''):<br />
::<math>V_m = \frac{ \mathrm{MW} }{\rho N_a}</math><br />
Where ''N<sub>a</sub>'' is the Avogadro constant. Thus:<br />
:<math>\mathrm{SLD} = \frac{\sum_{i=1}^{N}b_i}{V_m} = \frac{\rho N_a \sum_{i=1}^{N}b_i}{\sum_{i=1}^{N}\mathrm{M}_i} </math><br />
where M<sub>''i''</sub> is the atomic molar mass for each element.<br />
<br />
===Neutron scattering lengths===<br />
For neutrons, the ''b<sub>i</sub>'' in the above equation are the neutron scattering lengths for the atoms in question. This value depends on the strength of the interaction between a neutron and a given nucleus. It is in general not easy to predict this quantity theoretically; instead experiments are performed to tabulate the scattering lengths. Refer to the [http://www.ill.eu/quick-links/publications/ neutron data booklet] (starting on page 1.1-8) for tables of ''b<sub>i</sub>'' for known isotopes.<br />
<br />
===X-ray scattering lengths===<br />
For x-rays, the scattering arises from the interaction between the incident wave and the electron clouds of the atoms in the material. Thus, the scattering power scales with electron density in the material (with corrections for additional effects, such as [[resonant scattering]], a.k.a. absorption lines). In practice, one can look up tabulated values of the [[atomic scattering factor]] for the atom in question. From this, one can compute the scattering length using:<br />
:<math><br />
\begin{alignat}{2}<br />
b_i & = \frac{ e^2 }{ 4 \pi \varepsilon_0 m_e c^2}f_1 \\<br />
& = r_e f_1<br />
\end{alignat}<br />
</math><br />
Where ''f<sub>''1</sub> is the real part of the atomic scattering factor, ''e'' is the [http://en.wikipedia.org/wiki/Elementary_charge charge of the electron] (1.602176565×10<sup>−19</sup> coulombs), ''ε''<sub>0</sub> is the [http://en.wikipedia.org/wiki/Vacuum_permittivity permittivity of free space] (8.854187817x10<sup>−12</sup> F/m), ''m<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Electron_rest_mass mass of the electron] (9.10938215×10<sup>−31</sup> kg), and ''c'' is the [http://en.wikipedia.org/wiki/Speed_of_light speed of light] (299,792,458 m/s).<br />
<br />
The prefactor to ''f<sub>''1</sub> is simply the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (''r<sub>e</sub>''), and has the value 2.8179403x10<sup>−15</sup> m.<br />
<br />
==Converting==<br />
SLD is typically used in the neutron scattering community, but is less common in x-ray scattering. The SLD can be converted to the x-ray [[refractive index]] using:<br />
:<math><br />
\delta = \frac{\lambda^2}{2 \pi} \mathrm{Re}(\mathrm{SLD})<br />
</math><br />
:<math><br />
\beta = \frac{\lambda^2}{2 \pi} \mathrm{Im}(\mathrm{SLD})<br />
</math><br />
<br />
==See Also==<br />
* [[Critical angle]]<br />
* [http://www.ncnr.nist.gov/resources/activation/ SLD calculator]: NIST Center for Neutron Research calculator for predicting SLD.</div>68.194.136.6http://gisaxs.com/index.php?title=Critical_angle&diff=301Critical angle2014-06-05T00:24:40Z<p>68.194.136.6: /* Calculating */</p>
<hr />
<div>In [[GISAXS]], the '''critical angle''' for a thin film is the incident angle below which one gets total external reflection of the x-ray beam.<br />
<br />
Below the critical angle, the beam is fully reflected from the film. The x-ray field probes a short distance into the film surface (due to the evanescent wave); on the order of a few nanometers. Thus, a GISAXS measurement below the critical angle is inherently probing only the film surface. A measurement well above the critical angle, by comparison, penetrates through the film and thus measures the average of the structure through the whole film. Close to the critical angle, the refracted beam is nearly parallel to the film interface; in other words the beam is coupled into [[X-ray waveguide|waveguide]] modes. This increases the effective path-length of the beam through the sample, which thereby increases the intensity of the scattering.<br />
<br />
Normally in GISAXS experiments, it is useful to do measurements both below and above the critical angle; by comparing the two patterns, you can ascertain whether the structures observed at the surface (e.g. as seen by AFM or SEM) are representative of the entire film. Measurements close to the critical angle are useful because of the intensity enhancement.<br />
<br />
==Examples==<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon|Si]]<br />
| 2.3290<br />
| 2.0<br />
| 6.20<br />
| 0.824<br />
| 0.0291<br />
| 16.89<br />
|-<br />
| <br />
| <br />
| 4.0<br />
| 3.10<br />
| 0.451<br />
| 0.0319<br />
| 20.28<br />
|-<br />
| <br />
| <br />
| 8.0<br />
| 1.55<br />
| 0.224<br />
| 0.0317<br />
| 20.07<br />
|-<br />
| <br />
| <br />
| 12.0<br />
| 1.03<br />
| 0.149<br />
| 0.0317<br />
| 19.92<br />
|-<br />
<br />
| <br />
| <br />
| 16.0<br />
| 0.77<br />
| 0.112<br />
| 0.0316<br />
| 19.84<br />
|-<br />
| <br />
| <br />
| 24.0<br />
| 0.52<br />
| 0.07426<br />
| 0.0315<br />
| 19.77<br />
|-<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon dioxide|SiO2]]<br />
| 2.648<br />
| 2.0<br />
| 6.20<br />
| 0.927<br />
| 0.0328<br />
| 21.42<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 0.480<br />
| 0.0340<br />
| 22.96<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.239<br />
| 0.0338<br />
| 22.71<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.159<br />
| 0.0337<br />
| 22.58<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.119<br />
| 0.0337<br />
| 22.53<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.079<br />
| 0.0336<br />
| 22.48<br />
|}<br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Gold|Au]]<br />
| 19.32<br />
| 2.0<br />
| 6.20<br />
| 1.830<br />
| 0.0647<br />
| 83.44<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 1.097<br />
| 0.0776<br />
| 119.89<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.560<br />
| 0.0792<br />
| 124.86<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.348<br />
| 0.0738<br />
| 108.29<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.282<br />
| 0.0797<br />
| 126.51<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.191<br />
| 0.0811<br />
| 130.80<br />
|}<br />
<br />
==Calculating==<br />
The critical angle for a material can be calculated in a variety of ways. For an elemental substance, [http://henke.lbl.gov/optical_constants/pert_form.html this online tool] will calculate it for you.<br />
===From SLD===<br />
The [[Scattering Length Density]] (SLD) for any given material can be computed using tabulated values. From this, the critical angle can be computed. The critical scattering vector is:<br />
:<math>q_c = \sqrt{ 16 \pi \mathrm{SLD} }</math><br />
In reflection-mode, the scattering vector is:<br />
::<math> \begin{alignat}{2}<br />
\mathbf{q} & = \mathbf{k}_o-\mathbf{k}_i \\<br />
|\mathbf{q}| & = q = 2 |k| \sin\theta_i = \frac{4 \pi}{\lambda}\sin\theta_i<br />
\end{alignat}<br />
</math><br />
Where ''λ'' is the wavelength of the x-rays. So:<br />
:<math><br />
\begin{alignat}{2}<br />
\theta_c & = \arcsin\left( \frac{ q_c \lambda }{4 \pi} \right) \\<br />
& = \arcsin\left( \frac{ \lambda \sqrt{16 \pi \mathrm{SLD} } }{4 \pi} \right) \\<br />
& \approx \sqrt{\frac{\lambda^2 \mathrm{SLD} }{\pi}}<br />
\end{alignat}<br />
</math><br />
<br />
===From refractive index===<br />
The critical angle is simply a result of the refractive index contrast between the film and the ambient. Of course, in this case we are talking about the '''x-ray refractive index''' (not the usual refractive index for visible light). In the case of neutrons, the film similarly exhibits a neutron refractive index. The refractive index is complex:<br />
:<math><br />
n = 1 - \delta + i \beta<br />
</math><br />
For x-rays, the values of ''𝛿'' and ''β'' can be calculated from the [[atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>) using:<br />
:<math><br />
\delta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_1<br />
</math><br />
:<math><br />
\beta = \frac{ n_a r_e \lambda^2 }{2 \pi} f_2<br />
</math><br />
Where ''r<sub>e</sub>'' is the [http://en.wikipedia.org/wiki/Classical_electron_radius classical electron radius] (2.8179403x10<sup>−15</sup> m), ''λ'' is the wavelength of the probing x-rays, and ''n<sub>a</sub>'' is the number density:<br />
:<math><br />
n_a = \frac{\rho N_a}{M_a}<br />
</math><br />
where ''ρ'' is the physical density (e.g. in g/cm<sup>3</sup>), ''N<sub>a</sub>'' is the [http://en.wikipedia.org/wiki/Avogadro_constant Avogadro constant] (6.02214129x10<sup>23</sup> mol<sup>−1</sup>), and ''M<sub>a</sub>'' is molar mass (e.g. in g/mol). There are also [http://henke.lbl.gov/optical_constants/getdb2.html online tools] that will compute the refractive index for you.<br />
<br />
Once the real part of the refractive index (''𝛿'') is known, conversion to critical angle is straightforward:<br />
:<math><br />
\theta_c = \sqrt{ 2 \delta }<br />
</math><br />
Note that the resultant critical angle has units of ''radians''. For a multi-component system, one simply sums the contributions from each element:<br />
:<math><br />
\theta_c = \sqrt{ \frac{\rho N_a r_e \lambda^2 \sum_{i=1}^{N} f_{1i} }{ \pi \sum_{i=1}^{N} M_i } }<br />
</math><br />
<br />
==See Also==<br />
* [[Scattering Length Density]]<br />
* [[Refractive index]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=300Technical articles2014-06-05T00:24:12Z<p>68.194.136.6: /* Important quantities */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]/[[SANS]]<br />
** [[CD-SAXS]] ([[RSANS]])<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
* [[Scattering intensity]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
* [[Refractive index]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Material:Silicon&diff=299Material:Silicon2014-06-04T23:49:44Z<p>68.194.136.6: /* Properties */</p>
<hr />
<div>'''Silicon''' is crystalline solid with a diamond cubic crystal structure. Silicon wafers are frequently used as substrates for samples used in [[GISAXS]]. Si wafers make ideal substrates because they are very smooth at the atomic/nano scale, and are also very flat across larger (macroscale) distances. Note, however, that sample preparation (e.g. spin coating) may stress the wafer and 'kink' it, with effects that can be visible in GISAXS and especially [[reflectivity]] experiments.<br />
<br />
==Scattering==<br />
Because silicon is normally a single-crystal, it leads to no discernible peaks on the detector unless the crystal lattice is aligned to satisfy the Bragg condition (i.e. the [[Ewald sphere]] must intercept a peak in the [[reciprocal-space|reciprocal lattice]]).<br />
<br />
==Properties==<br />
* Density: 2.3290 g/cm<sup>3</sup><br />
* Neutron [[SLD]]: 2.074×10<sup>−6</sup> Å<sup>−2</sup><br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon|Si]]<br />
| 2.3290<br />
| 2.0<br />
| 6.20<br />
| 0.824<br />
| 0.0291<br />
| 16.89<br />
|-<br />
| <br />
| <br />
| 4.0<br />
| 3.10<br />
| 0.451<br />
| 0.0319<br />
| 20.28<br />
|-<br />
| <br />
| <br />
| 8.0<br />
| 1.55<br />
| 0.224<br />
| 0.0317<br />
| 20.07<br />
|-<br />
| <br />
| <br />
| 12.0<br />
| 1.03<br />
| 0.149<br />
| 0.0317<br />
| 19.92<br />
|-<br />
<br />
| <br />
| <br />
| 16.0<br />
| 0.77<br />
| 0.112<br />
| 0.0316<br />
| 19.84<br />
|-<br />
| <br />
| <br />
| 24.0<br />
| 0.52<br />
| 0.07426<br />
| 0.0315<br />
| 19.77<br />
|-<br />
|}<br />
<br />
<br />
[[Image:Silicon-atomic scatt factor.png|400px|[[Atomic scattering factors]] (''f''<sub>1</sub> and ''f''<sub>2</sub>).]][[Image:Silicon-n.png|400px]]<br />
<br />
[[Image:Silicon-crit.png|400px]][[Image:Silicon-crit_zoom.png|400px]]<br />
<br />
[[Image:Silicon-critq.png|400px]][[Image:Silicon-SLD.png|400px]]<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Silicon Wikipedia: Silicon]</div>68.194.136.6http://gisaxs.com/index.php?title=SLD&diff=298SLD2014-06-04T23:49:20Z<p>68.194.136.6: Redirected page to Scattering Length Density</p>
<hr />
<div>#REDIRECT [[Scattering Length Density]]</div>68.194.136.6http://gisaxs.com/index.php?title=Material:Silicon_dioxide&diff=297Material:Silicon dioxide2014-06-04T23:49:03Z<p>68.194.136.6: /* Properties */</p>
<hr />
<div>'''Silicon dioxide''' (SiO<sub>2</sub>), also known as silica, spontaneously forms on the surface of a [[Material:Silicon|silicon]] wafer exposed to air. This 'native oxide' must be included in calculations of (for example) the x-ray [[reflectivity]] curve for a Si surface. In terms of x-ray properties, SiO<sub>2</sub> can be used to approximate a variety of related materials (fused quartz, glass, etc.).<br />
<br />
==Properties==<br />
* Density: 2.648 g/cm<sup>3</sup><br />
* Neutron [[SLD]]: 4.183×10<sup>−6</sup> Å<sup>−2</sup><br />
<br />
{| class="wikitable"<br />
|-<br />
! Material<br />
! density (g/cm<sup>3</sup>)<br />
! X-ray energy (keV)<br />
! X-ray wavelength (Å)<br />
! critical angle (°)<br />
! ''q<sub>c</sub>'' (Å<sup>−1</sup>)<br />
! SLD (10<sup>−6</sup>Å<sup>−2</sup>)<br />
|-<br />
| [[Material:Silicon dioxide|SiO2]]<br />
| 2.648<br />
| 2.0<br />
| 6.20<br />
| 0.927<br />
| 0.0328<br />
| 21.42<br />
|-<br />
|<br />
|<br />
| 4.0<br />
| 3.10<br />
| 0.480<br />
| 0.0340<br />
| 22.96<br />
|-<br />
|<br />
|<br />
| 8.0<br />
| 1.55<br />
| 0.239<br />
| 0.0338<br />
| 22.71<br />
|-<br />
|<br />
|<br />
| 12.0<br />
| 1.03<br />
| 0.159<br />
| 0.0337<br />
| 22.58<br />
|-<br />
|<br />
|<br />
| 16.0<br />
| 0.77<br />
| 0.119<br />
| 0.0337<br />
| 22.53<br />
|-<br />
|<br />
|<br />
| 24.0<br />
| 0.52<br />
| 0.079<br />
| 0.0336<br />
| 22.48<br />
|}<br />
<br />
[[Image:SiO2-n.png|400px]]<br />
<br />
[[Image:SiO2-crit.png|400px]][[Image:SiO2-crit_zoom.png|400px]]<br />
<br />
[[Image:SiO2-critq.png|400px]][[Image:SiO2-SLD.png|400px]]<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Silicon_dioxide Wikipedia: Silicon dioxide]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=296Technical articles2014-06-04T23:07:52Z<p>68.194.136.6: /* Scattering techniques */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]/[[SANS]]<br />
** [[CD-SAXS]] ([[RSANS]])<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
* [[Scattering intensity]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=295Technical articles2014-06-04T22:58:39Z<p>68.194.136.6: /* Kinds of scattering */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
* [[Scattering intensity]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=294Technical articles2014-06-04T22:58:12Z<p>68.194.136.6: /* Scattering techniques */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[X-ray waveguiding]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=293Technical articles2014-06-04T22:57:25Z<p>68.194.136.6: /* Scattering techniques */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[Reflectivity]]<br />
* [[Resonant scattering]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=292Technical articles2014-06-04T22:57:07Z<p>68.194.136.6: /* Scattering concepts */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[Reflectivity]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
* [[Ewald sphere]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=291Technical articles2014-06-04T22:56:45Z<p>68.194.136.6: /* Scattering techniques */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
* [[Reflectivity]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=290Technical articles2014-06-04T22:56:35Z<p>68.194.136.6: /* Important quantities */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
<br />
==Important quantities==<br />
* [[Atomic scattering factor]]<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Technical_articles&diff=289Technical articles2014-06-04T22:34:03Z<p>68.194.136.6: /* Scattering concepts */</p>
<hr />
<div>This page lists the various technical topics that are described on this wiki. Note that many of the topics are currently empty (red links); if you feel qualified, please jump in and contribute!<br />
<br />
==Scattering techniques==<br />
* [[SAXS]]<br />
* [[WAXS]]<br />
* [[GISAXS]]<br />
* [[GIWAXS]]<br />
<br />
==Kinds of scattering==<br />
* [[Form Factor]]<br />
* [[Structure Factor]]<br />
* [[Diffuse scattering]]<br />
* [[Debye-Waller factor]]<br />
<br />
==Scattering concepts==<br />
* [[Momentum transfer]]<br />
* [[Lattices|Lattice]]<br />
* [[Unit cell]]<br />
* [[Reciprocal-space]]<br />
<br />
==Important quantities==<br />
* [[Scattering Length Density]] (SLD)<br />
* [[Critical angle]]<br />
<br />
==Analysis==<br />
* [[Scherrer grain size analysis]]<br />
<br />
==Theory==<br />
* [[DWBA]]</div>68.194.136.6http://gisaxs.com/index.php?title=Beamlines&diff=288Beamlines2014-06-04T22:29:04Z<p>68.194.136.6: /* Available Labscale Instruments */</p>
<hr />
<div>The following is a list of x-ray scattering beamlines available to the user community:<br />
<br />
==Available Synchrotron Beamlines==<br />
<br />
{| class="wikitable"<br />
|-<br />
! Beamline<br />
! Facility<br />
! Status<br />
! Description<br />
! Techniques<br />
! Energy range<br />
! ''q''-range<br />
! Access<br />
|-<br />
| '''[[X9]]'''<br />
| [[NSLS]] at [[BNL]]<br />
| Running until Sept. 2014<br />
| [[X9]] is an undulator-based (high-flux) and high-resolution beamline that can perform transmission and grazing-incidence scattering across a wide ''q''-range (simultaneous SAXS/WAXS possible). Typical beam size is 150 µm for TSAXS and 50 µm for GISAXS (focus to ~15 µm is possible). Measurements in air or vaccum are possible. Sample environments for performing in-situ sample heating (RT-220°C for GISAXS, RT-80°C for TSAXS) are available. Accomodations for in-situ experiments (solvent annealing, electrochemical cells) are also possible. Contact [http://staff.ps.bnl.gov/staff.aspx?id=21212 Lin Yang], [http://staff.ps.bnl.gov/staff.aspx?id=10076 Masa Fukuto], or [http://www.bnl.gov/cfn/people/staff.php?q=150 Kevin Yager] for details.<br />
| [[SAXS]], [[WAXS]], [[GISAXS]], [[GIWAXS]]<br />
| 6 keV to 20 keV<br />
| 0.002 Å<sup>−1</sup> to 4.2 Å<sup>−1</sup><br />
| Through [http://www.bnl.gov/ps/nsls/users/access/beamtime-new_users.asp NSLS] or [http://www.bnl.gov/cfn/user/ CFN] user programs. (No longer accepting proposals.)<br />
|-<br />
|-<br />
| '''[[7.3.3]]'''<br />
| [http://www-als.lbl.gov/index.php/beamlines/beamlines-directory.html ALS] at [http://www.lbl.gov/ LBNL]<br />
| Running <br />
| [[7.3.3]] is a SAXS/WAXS/GISAXS/GIWAXS beamline, covering a wide ''q'' range (0.004 - 2.5 Å<sup>−1</sup>) and length scale (2.5 - 1500 angstrom). See more details at [http://www-als.lbl.gov/index.php/beamlines/beamlines-directory.html ALS website]. New users, please contact Alex Hexemer, Eric Schaible, or Chenhui Zhu for details. <br />
| [[SAXS]], [[WAXS]], [[GISAXS]], [[GIWAXS]]<br />
| 10 keV<br />
| 0.004 Å<sup>−1</sup> to 2 Å<sup>−1</sup><br />
| Through ALS general user program. Rapid access is also available. Please contact beamline staff for details. <br />
|-<br />
|}<br />
<br />
==Available Labscale Instruments==<br />
{| class="wikitable"<br />
|-<br />
! Instrument<br />
! Facility<br />
! Status<br />
! Description<br />
! Techniques<br />
! Energy range<br />
! ''q''-range<br />
! Access<br />
|-<br />
| '''[[CFN Bruker Nanostar|Bruker Nanostar]]'''<br />
| [[CFN]] at [[BNL]]<br />
| Running<br />
| This labscale rotating-anode instrument is available to users. Of course a labscale instrument will not have the flux or resolution of a synchrotron beamline; nevertheless this instrument is optimized for resolution, while providing sufficient flux for measuring strongly scattering systems (e.g. nanoparticles). The sample environment is under vacuum, and offers both transmission-mode and grazing-incidence stages. Contact [http://www.bnl.gov/cfn/people/staff.php?q=124 Dmytro Nykypanchuk] or [http://www.bnl.gov/cfn/people/staff.php?q=150 Kevin Yager] for details.<br />
| [[SAXS]], [[WAXS]], [[GISAXS]], [[GIWAXS]]<br />
| [[Cu K-alpha|8.04 keV]]<br />
| 0.005 Å<sup>−1</sup> to 0.3 Å<sup>−1</sup> (up to ~2 Å<sup>−1</sup> using image plate).<br />
| Available through [http://www.bnl.gov/cfn/user/ CFN user program].<br />
|-<br />
|}<br />
<br />
==Future Synchrotron Beamlines==<br />
{| class="wikitable"<br />
|-<br />
! Beamline<br />
! Facility<br />
! Status<br />
! Description<br />
! Techniques<br />
! Energy range<br />
! ''q''-range<br />
! Access<br />
|-<br />
| '''[[CMS]]'''<br />
| [[NSLS-II]] at [[BNL]]<br />
| Under construction.<br />
| [[CMS]] is an bending-magnet beamline being constructed at [[NSLS-II]], based in part on the design and hardware at [[X9]] ([[NSLS]]). The beamline will be able to perform transmission and grazing-incidence scattering across a wide ''q''-range (simultaneous SAXS/WAXS possible), and will emphasize automation, high-throughput, and in-situ measurements. Contact [http://staff.ps.bnl.gov/staff.aspx?id=10076 Masa Fukuto], or [http://www.bnl.gov/cfn/people/staff.php?q=150 Kevin Yager] for details.<br />
| [[SAXS]], [[WAXS]], [[GISAXS]], [[GIWAXS]]<br />
| 10 keV to 17 keV<br />
| 0.002 Å<sup>−1</sup> to 4.2 Å<sup>−1</sup><br />
| Will be available through NSLS-II user program.<br />
|-<br />
| '''[[SMI]]'''<br />
| [[NSLS-II]] at [[BNL]]<br />
| Under construction.<br />
| [[SMI]] is a undulator-based beamline being constructed at [[NSLS-II]]. SMI will be a high-flux and high-resolution instrument optimized for grazing-incidence measurements on interfaces, though also capable of performing world-class transmission scattering experiments. [http://staff.ps.bnl.gov/staff.aspx?id=20298 Elaine DiMasi], [http://staff.ps.bnl.gov/staff.aspx?id=83687 Mikhail Zhernenkov], or [http://www.bnl.gov/cfn/people/staff.php?q=150 Kevin Yager] for details.<br />
| [[SAXS]], [[WAXS]], [[GISAXS]], [[GIWAXS]], [[XR]]<br />
| ~2 keV to 24 keV<br />
| Very wide ''q''-range (SAXS and WAXS).<br />
| Will be available through NSLS-II user program.<br />
|-<br />
|}</div>68.194.136.6http://gisaxs.com/index.php?title=Debye-Waller_factor&diff=185Debye-Waller factor2014-06-04T02:01:59Z<p>68.194.136.6: /* Mathematical form */</p>
<hr />
<div>The '''Debye-Waller factor''' is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-''q'' peaks). This scattering intensity then appears as [[diffuse scattering]]. Conceptually, thermal fluctuations create disorder, because the atoms/particles oscillate about their equilibrium positions and thus the lattice is never (instantaneously) perfect.<br />
<br />
==Mathematical form==<br />
For a lattice-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
G(q) & = e^{-\langle u^2 \rangle q^2} \\<br />
& = e^{-\sigma_{\mathrm{rms}}^2q^2} \\<br />
& = e^{-\sigma_a^2a^2q^2}<br />
\end{alignat}<br />
</math><br />
Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement.<br />
<br />
Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a [[diffuse scattering]] term, which appears in the [[structure factor]] (<math>S(q)</math>) as:<br />
:<math><br />
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]<br />
</math><br />
And thus appears in the overall intensity as:<br />
:<math><br />
I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]<br />
</math><br />
where <math>P(q)</math> is the [[form factor]].<br />
<br />
In the high-''q'' limit, form factors frequently exhibit a <math>q^{-4}</math> scaling (c.f. [[Form_Factor:Sphere#Isotropic_Form_Factor_Intensity_at_large_q|sphere form factor]]), in which case one expects (since <math>G(q \rightarrow \infty) = 1</math>):<br />
:<math><br />
I_{\mathrm{diffuse}}(q \rightarrow \infty) \propto q^{-4}<br />
</math><br />
(Which reproduces the scaling of the [[Diffuse_scattering#Porod_law|Porod law]].)<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]</div>68.194.136.6http://gisaxs.com/index.php?title=Debye-Waller_factor&diff=184Debye-Waller factor2014-06-04T01:58:42Z<p>68.194.136.6: /* Mathematical form */</p>
<hr />
<div>The '''Debye-Waller factor''' is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-''q'' peaks). This scattering intensity then appears as [[diffuse scattering]]. Conceptually, thermal fluctuations create disorder, because the atoms/particles oscillate about their equilibrium positions and thus the lattice is never (instantaneously) perfect.<br />
<br />
==Mathematical form==<br />
For a lattice-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
G(q) & = e^{-\langle u^2 \rangle q^2} \\<br />
& = e^{-\sigma_{\mathrm{rms}}^2q^2} \\<br />
& = e^{-\sigma_a^2a^2q^2}<br />
\end{alignat}<br />
</math><br />
Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement.<br />
<br />
Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a [[diffuse scattering]] term, which appears in the [[structure factor]] (<math>S(q)</math>) as:<br />
:<math><br />
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]<br />
</math><br />
And thus appears in the overall intensity as:<br />
:<math><br />
I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]<br />
</math><br />
where <math>P(q)</math> is the [[form factor]].<br />
<br />
In the high-''q'' limit, form factors frequently exhibit a <math>q^{-4}</math> scaling (c.f. [[Form_Factor:Sphere#Isotropic_Form_Factor_Intensity_at_large_q|sphere form factor]]), in which case one expects (since <math>G(q \rightarrow \infty) = 1</math>):<br />
:<math><br />
I_{\mathrm{diffuse}}(q \rightarrow \infty) \propto q^{-4}<br />
</math><br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]</div>68.194.136.6http://gisaxs.com/index.php?title=Debye-Waller_factor&diff=183Debye-Waller factor2014-06-04T01:52:19Z<p>68.194.136.6: </p>
<hr />
<div>The '''Debye-Waller factor''' is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-''q'' peaks). This scattering intensity then appears as [[diffuse scattering]]. Conceptually, thermal fluctuations create disorder, because the atoms/particles oscillate about their equilibrium positions and thus the lattice is never (instantaneously) perfect.<br />
<br />
==Mathematical form==<br />
For a lattice-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
G(q) & = e^{-\langle u^2 \rangle q^2} \\<br />
& = e^{-\sigma_{\mathrm{rms}}^2q^2} \\<br />
& = e^{-\sigma_a^2a^2q^2}<br />
\end{alignat}<br />
</math><br />
Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement.<br />
<br />
Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a [[diffuse scattering]] term, which appears in the [[structure factor]] (<math>S(q)</math>) as:<br />
:<math><br />
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]<br />
</math><br />
And thus appears in the overall intensity as:<br />
:<math><br />
I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]<br />
</math><br />
where <math>P(q)</math> is the [[form factor]].<br />
<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]</div>68.194.136.6http://gisaxs.com/index.php?title=Debye-Waller_factor&diff=182Debye-Waller factor2014-06-04T01:38:11Z<p>68.194.136.6: </p>
<hr />
<div>The '''Debye-Waller factor''' is a term (in scattering equations) which accounts for how thermal fluctuations extinguish scattering intensity (especially high-''q'' peaks). This scattering intensity then appears as [[diffuse scattering]]. Conceptually, thermal fluctuations create disorder, because the atoms/particles oscillate about their equilibrium positions and thus the lattice is never (instantaneously) perfect.<br />
<br />
==Mathematical form==<br />
For a lattice-size ''a'', the constituent entities (atoms, particles, etc.) will oscillate about their equilibrium positions with an rms width <math>\sigma_a</math>, attenuating structural peaks like:<br />
<br />
:<math><br />
\begin{alignat}{2}<br />
G(q) & = e^{-\langle u^2 \rangle q^2} \\<br />
& = e^{-\sigma_{\mathrm{rms}}^2q^2} \\<br />
& = e^{-\sigma_a^2a^2q^2}<br />
\end{alignat}<br />
</math><br />
Where <math>\sigma_{\mathrm{rms}} \equiv \sqrt{ \langle u^2 \rangle }</math> is the [http://en.wikipedia.org/wiki/Root_mean_square root-mean-square] displacement of the lattice-spacing ''a'' (such that the spacing at time ''t'' is <math>a+u(t)</math>), and <math>\sigma_a \equiv \sigma_{\mathrm{rms}}/a</math> is the relative displacement.<br />
<br />
Thus, the intensity of the structural peaks is multiplied by <math>G(q)</math>, which attenuates the higher-order (high-''q'') peaks, and redistributes this intensity into a diffuse scattering term, which appears in the [[structure factor]] (<math>S(q)</math>) as:<br />
:<math><br />
S_{\mathrm{diffuse}}(q) = \left[ 1- G(q) \right]<br />
</math><br />
And thus appears in the overall intensity as:<br />
:<math><br />
I_{\mathrm{diffuse}}(q) = P(q) \left[ 1- G(q) \right]<br />
</math><br />
where <math>P(q)</math> is the [[form factor]].<br />
<br />
<br />
==See Also==<br />
* [http://en.wikipedia.org/wiki/Debye%E2%80%93Waller_factor Wikipedia: Debye-Waller factor]</div>68.194.136.6