Ewald sphere - Revision history

KevinYager: /* Equations of GISAXS Geometry */

2015-12-21T14:03:34Z

‎Equations of GISAXS Geometry

KevinYager: /* TSAXS */

2014-12-16T23:05:55Z

‎TSAXS

KevinYager: /* Mathematics */

2014-10-07T20:40:33Z

‎Mathematics

KevinYager at 14:26, 18 September 2014

2014-09-18T14:26:03Z

KevinYager at 14:15, 18 September 2014

2014-09-18T14:15:33Z

KevinYager: /* Literature */

2014-06-24T14:34:28Z

‎Literature

KevinYager: /* Conceptual Understanding of Ewald sphere */

2014-06-24T14:32:34Z

‎Conceptual Understanding of Ewald sphere

KevinYager: /* Out-of-plane scattering only */

2014-06-24T14:32:10Z

‎Out-of-plane scattering only

KevinYager: /* Reflectivity */

2014-06-24T14:31:51Z

‎Reflectivity

KevinYager: /* GISAXS */

2014-06-24T14:17:26Z

‎GISAXS

@@ Line 129: / Line 129: @@
 ===Equations of GISAXS Geometry===
 * [http://gisaxs.de/index.html gisaxs.de]: Brief introduction to the theory of GISAXS.
-* [http://ln-www.insp.upmc.fr/axe4/Oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS Manual]: Provides a complete description (including equations) of the GISAXS theory required to model data.
+* [http://www.insp.jussieu.fr/oxydes/IsGISAXS/figures/doc/manual.html IsGISAXS Manual]: Provides a complete description (including equations) of the GISAXS theory required to model data.
 * [http://www.sciencedirect.com/science/article/pii/S0167572909000399 Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering] Gilles Renaud, Rémi Lazzari, Frédéric Leroy, ''Surface Science Reports'' 2009, 64 (8), 255-380. [http://dx.doi.org/10.1016/j.surfrep.2009.07.002 doi: 10.1016/j.surfrep.2009.07.002]
 * [http://www.springerlink.com/content/883152564460n600/ A Basic Introduction to Grazing Incidence Small-Angle X-Ray Scattering] P. Müller-Buschbaum [http://dx.doi.org/10.1007/978-3-540-95968-7 DOI 10.1007/978-3-540-95968-7] (Chapter 3 in [http://link.springer.com/book/10.1007/978-3-540-95968-7 Applications of Synchrotron Light to Scattering and Diffraction in Materials and Life Sciences])

@@ Line 81: / Line 81: @@
 q  = \frac{4 \pi}{\lambda} \sin(\theta_s)
 </math>
-In the more general case of probing an anisotropic material (e.g. [[CD-SAXS]]), one must take into account the full ''q''-vector, and in particular the relative orientation of the incident beam and the sample: i.e. the relative orientation of the Ewald sphere and the reciprocal-space.
+In the more general case of probing an anisotropic material (e.g. [[CD-SAXS]]), one must take into account the full ''q''-vector, and in particular the relative [[sample orientation|orientation]] of the incident beam and the sample: i.e. the relative orientation of the Ewald sphere and the reciprocal-space.
 ===[[GISAXS]]===

@@ Line 16: / Line 16: @@
 \begin{alignat}{2}
 q_x & = \frac{2 \pi}{\lambda} \cos(\alpha_f)\sin(2\theta_f) \\
-q_y & = \frac{2 \pi}{\lambda} \left[ \cos(\alpha_f)\cos(2\theta_f) - \cos(\alpha_f) \right ]\\
+q_y & = \frac{2 \pi}{\lambda} \left[ \cos(\alpha_f)\cos(2\theta_f) - \cos(\alpha_i) \right ]\\
-q_x & = \frac{2 \pi}{\lambda} \left[ \sin(\alpha_f) + \sin(\alpha_f) \right ]
+q_z & = \frac{2 \pi}{\lambda} \left[ \sin(\alpha_f) + \sin(\alpha_i) \right ]
 \end{alignat}
 </math>
 ==Derivation==

← Older revision		Revision as of 14:26, 18 September 2014
Line 1:		Line 1:
	The '''Ewald sphere''' is the surface, in [[reciprocal-space]], that all experimentally-observed scattering arises from. (Strictly, only the ''elastic'' scattering comes from the Ewald sphere; inelastic scattering is so-called 'off-shell'.) A peak observed on the detector indicates that a reciprocal-space peak is intersecting with the Ewald sphere.		The '''Ewald sphere''' is the surface, in [[reciprocal-space]], that all experimentally-observed scattering arises from. (Strictly, only the ''elastic'' scattering comes from the Ewald sphere; inelastic scattering is so-called 'off-shell'.) A peak observed on the detector indicates that a reciprocal-space peak is intersecting with the Ewald sphere.
		+
		+	Although the Ewald sphere may seem like an abstract concept, it is simply the geometric representation of the scattering equation (i.e. it is the surface that satisfies the scattering condition).

	[[Image:3D ewald example.png\|center\|450px\|thumb\|Example of a reciprocal-lattice (blue dots) being probed in an x-ray scattering experiment. The signal observed on the detector comes from the intersection of the Ewald sphere with the reciprocal-space peaks.]]		[[Image:3D ewald example.png\|center\|450px\|thumb\|Example of a reciprocal-lattice (blue dots) being probed in an x-ray scattering experiment. The signal observed on the detector comes from the intersection of the Ewald sphere with the reciprocal-space peaks.]]

← Older revision		Revision as of 14:15, 18 September 2014
Line 1:		Line 1:
	The '''Ewald sphere''' is the surface, in [[reciprocal-space]], that all experimentally-observed scattering arises from. (Strictly, only the ''elastic'' scattering comes from the Ewald sphere; inelastic scattering is so-called 'off-shell'.) A peak observed on the detector indicates that a reciprocal-space peak is intersecting with the Ewald sphere.		The '''Ewald sphere''' is the surface, in [[reciprocal-space]], that all experimentally-observed scattering arises from. (Strictly, only the ''elastic'' scattering comes from the Ewald sphere; inelastic scattering is so-called 'off-shell'.) A peak observed on the detector indicates that a reciprocal-space peak is intersecting with the Ewald sphere.
		+
		+	[[Image:3D ewald example.png\|center\|450px\|thumb\|Example of a reciprocal-lattice (blue dots) being probed in an x-ray scattering experiment. The signal observed on the detector comes from the intersection of the Ewald sphere with the reciprocal-space peaks.]]
		+
		+	[[Image:R000.png\|450px\|thumb\|center\|Simplified 2D representation of a scattering experiment. The reciprocal-space of a crystal (blue), contains an array of peaks. The only peaks that are observed on the detector are those that intersect the Ewald sphere.]]

	==Mathematics==		==Mathematics==

← Older revision		Revision as of 14:34, 24 June 2014
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	* [http://www.sciencedirect.com/science/article/pii/S0167572909000399 Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering] Gilles Renaud, Rémi Lazzari, Frédéric Leroy, ''Surface Science Reports'' 2009, 64 (8), 255-380. [http://dx.doi.org/10.1016/j.surfrep.2009.07.002 doi: 10.1016/j.surfrep.2009.07.002]		* [http://www.sciencedirect.com/science/article/pii/S0167572909000399 Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering] Gilles Renaud, Rémi Lazzari, Frédéric Leroy, ''Surface Science Reports'' 2009, 64 (8), 255-380. [http://dx.doi.org/10.1016/j.surfrep.2009.07.002 doi: 10.1016/j.surfrep.2009.07.002]
	* [http://www.springerlink.com/content/883152564460n600/ A Basic Introduction to Grazing Incidence Small-Angle X-Ray Scattering] P. Müller-Buschbaum [http://dx.doi.org/10.1007/978-3-540-95968-7 DOI 10.1007/978-3-540-95968-7] (Chapter 3 in [http://link.springer.com/book/10.1007/978-3-540-95968-7 Applications of Synchrotron Light to Scattering and Diffraction in Materials and Life Sciences])		* [http://www.springerlink.com/content/883152564460n600/ A Basic Introduction to Grazing Incidence Small-Angle X-Ray Scattering] P. Müller-Buschbaum [http://dx.doi.org/10.1007/978-3-540-95968-7 DOI 10.1007/978-3-540-95968-7] (Chapter 3 in [http://link.springer.com/book/10.1007/978-3-540-95968-7 Applications of Synchrotron Light to Scattering and Diffraction in Materials and Life Sciences])
		+
		+	==See Also==
		+	* [http://en.wikipedia.org/wiki/Ewald%27s_sphere Wikipedia: Ewald sphere]

@@ Line 118: / Line 118: @@
 ===Conceptual Understanding of Ewald sphere===
 * [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]
-* '''[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]
+* [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]
 ===Equations of GISAXS Geometry===

@@ Line 110: / Line 110: @@
 </math>
-===Out-of-plane scattering only===
+====Out-of-plane scattering only====
-Simplifying the above expression yields:
+TBD
-:<math>
+====General form====
-\begin{alignat}{2}
+TBD
 ==Literature==

@@ Line 84: / Line 84: @@
 </math>
-===Reflectivity===
+====Reflectivity====
 Consider for a moment that the ''q''-vector is confined to the <math>(q_y,q_z)</math> plane:
 :<math>

@@ Line 77: / Line 77: @@
 ===[[GISAXS]]===
-Now let us assume that the incident beam strikes a thin film mounted to a substrate. The incident beam is in the grazing-incidence geometry (e.g. [[GISAXS]], and we denote the angle between the incident beam and the film surface as <math>\theta_i</math>. The reciprocal-space of the sample is thus rotated by <math>\theta_i</math> with respect to the beam reciprocal-space coordinates. We denote the sample's reciprocal coordinate system by uppercase, <math>(Q_x,Q_y,Q_z)</math>, and note that the equation of the Ewald sphere becomes (the center of the sphere is at <math>(0,k \cos\theta_i, k \sin\theta_i)</math>):
+Now let us assume that the incident beam strikes a thin film mounted to a substrate. The incident beam is in the grazing-incidence geometry (e.g. [[GISAXS]], and we denote the angle between the incident beam and the film surface as <math>\theta_i</math>. The reciprocal-space of the sample is thus rotated by <math>\theta_i</math> with respect to the beam reciprocal-space coordinates. We convert to the sample's reciprocal coordinate space; still denoted by <math>(Q_x,Q_y,Q_z)</math>. The equation of the Ewald sphere becomes (the center of the sphere is at <math>\langle 0,k \cos\theta_i, k \sin\theta_i \rangle</math>):
 :<math>\begin{alignat}{2}
-& Q_x^2 + \left(Q_y-k \cos\theta_i \right)^2 + \left(Q_z-k \sin\theta_i\right)^2 - k^2 = 0 \\
+& q_x^2 + \left(q_y-k \cos\theta_i \right)^2 + \left(q_z-k \sin\theta_i\right)^2 - k^2 = 0 \\
-& Q_y = +k\cos\theta_i  + \sqrt{ k^2 - Q_x^2 - \left(Q_z-k\sin\theta_i\right)^2 }
+& q_y = +k\cos\theta_i  + \sqrt{ k^2 - q_x^2 - \left(q_z-k\sin\theta_i\right)^2 }
 \end{alignat}
 </math>