Scherrer grain size analysis - Revision history

KevinYager: /* K constant */

2016-01-05T17:33:15Z

‎K constant

KevinYager: /* K constant */

2016-01-05T17:06:54Z

‎K constant

KevinYager: /* K constant */

2016-01-05T17:06:21Z

‎K constant

KevinYager: /* K constant */

2016-01-05T17:03:22Z

‎K constant

KevinYager: /* References */

2015-07-19T16:16:24Z

‎References

KevinYager: /* Literature */

2015-07-19T15:30:33Z

‎Literature

KevinYager: /* References */

2015-07-19T15:30:20Z

‎References

KevinYager: /* Origin of Peak Broadening */

2015-01-13T15:13:16Z

‎Origin of Peak Broadening

KevinYager at 19:01, 11 October 2014

2014-10-11T19:01:50Z

KevinYager at 19:01, 11 October 2014

2014-10-11T19:01:39Z

@@ Line 18: / Line 18: @@
 :<math>
-K = 2[ln(2)/\pi]^{1/2} \simeq 0.9394
+K = 2 \sqrt{\ln(2)/\pi} \simeq 0.9394
 </math>
 This value is appropriate for spherical crystals with cubic symmetry, where the peak width is defined using the FWHM. If one instead uses the integral breadth, then the constant would be 0.89. In general, the constant ''K'' varies between 0.62 and 2.08.
@@ Line 25: / Line 25: @@
 :<math>
-K = \frac{2[ln(2)/\pi]^{1/2}}{2 \sqrt{2 \ln(2)}} = \frac{1}{\sqrt{2 \pi}} \simeq 0.3989
+K = \frac{2\sqrt{\ln(2)/\pi}}{2 \sqrt{2 \ln(2)}} = \frac{1}{\sqrt{2 \pi}} \simeq 0.3989
 </math>

@@ Line 25: / Line 25: @@
 :<math>
-K = \frac{2[ln(2)/\pi]^{1/2}}{2 \sqrt{2 \ln(2)}} ={1}{\sqrt{2 \pi}} \simeq 0.3989
+K = \frac{2[ln(2)/\pi]^{1/2}}{2 \sqrt{2 \ln(2)}} = \frac{1}{\sqrt{2 \pi}} \simeq 0.3989
 </math>

@@ Line 25: / Line 25: @@
 :<math>
-K = 2[ln(2)/\pi]^{1/2} \simeq 0.9394
+K = \frac{2[ln(2)/\pi]^{1/2}}{2 \sqrt{2 \ln(2)}} ={1}{\sqrt{2 \pi}} \simeq 0.3989
 </math>

@@ Line 21: / Line 21: @@
 </math>
 This value is appropriate for spherical crystals with cubic symmetry, where the peak width is defined using the FWHM. If one instead uses the integral breadth, then the constant would be 0.89. In general, the constant ''K'' varies between 0.62 and 2.08.
 ==Origin of Peak Broadening==

@@ Line 48: / Line 48: @@
 Importantly, if the peak are extremely sharp, and thus instrumental-limited, then the Scherrer analysis cannot be applied.
 ==References==
 ===Literature===

@@ Line 52: / Line 52: @@
 * '''Original publication''': P. Scherrer, "[https://eudml.org/doc/59018 Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen], ''Nachr. Ges. Wiss. Göttingen'' 26 (1918) pp 98-100.
 * '''K constant''': J.I. Langford and A.J.C. Wilson, "[http://scripts.iucr.org/cgi-bin/paper?a16745 Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size], ''J. Appl. Cryst.'' 11 (1978) pp 102-113. [http://dx.doi.org/10.1107/S0021889878012844 doi: 10.1107/S0021889878012844]
-* '''Review''': P. Scardi, M. Leoni and R. Delhez [Line broadening analysis using integral breadth methods: a critical review] ''J. Appl. Cryst.'' '''2004''', 37, 381-390. [http://dx.doi.org/10.1107/S0021889804004583 doi: 10.1107/S0021889804004583]
+* '''Review''': P. Scardi, M. Leoni and R. Delhez [http://scripts.iucr.org/cgi-bin/paper?S0021889804004583 Line broadening analysis using integral breadth methods: a critical review] ''J. Appl. Cryst.'' '''2004''', 37, 381-390. [http://dx.doi.org/10.1107/S0021889804004583 doi: 10.1107/S0021889804004583]
 * '''Correction for area detectors''': D.-M. Smilgies [http://scripts.iucr.org/cgi-bin/paper?S0021889809040126 Scherrer grain-size analysis adapted to grazing-incidence scattering with area detectors] J. Appl. Cryst. (2009). 42, 1030-1034    [http://dx.doi.org/10.1107/S0021889809040126 doi:10.1107/S0021889809040126]
 * '''Phase space notation''': D.-M. Smilgies [http://www.opticsinfobase.org/abstract.cfm?URI=ao-47-22-E106 Compact matrix formalism for phase space analysis of complex optical systems] Appl. Opt. (2008). 47 (22), E106-E115 [http://dx.doi.org/10.1364/AO.47.00E106 doi: 10.1364/AO.47.00E106]

@@ Line 39: / Line 39: @@
 * Temperature Factors
-Note that differentiating these effects from one another is in general difficult. The instrumental resolution can be independently calculated or measured. The other effects can sometimes be disentangled by gather appropriate data. For instance, detailed analysis of '''[[peak shape]]''' can be beneficial. Moreover, measuring the peak shape and peak width as a function of peak order, can be used to differentiate various effects.
+Note that differentiating these effects from one another is in general difficult. The instrumental resolution can be independently calculated or measured. The other effects can sometimes be disentangled by gathering appropriate data. For instance, detailed analysis of '''[[peak shape]]''' can be beneficial. Moreover, measuring the peak shape and peak width as a function of peak order, can be used to differentiate various effects.
 ==Resolution Limit==

← Older revision		Revision as of 19:01, 11 October 2014
Line 1:		Line 1:
−	The average '''grain size''' in a material may be estimated from the '''peak width'''. Sharp peaks indicate large grains, whereas broad peaks indicate small grains. This inverse relationship is, as always, due to the inverse nature of [[reciprocal-space]]. It can also be rationalized in terms of the [[Fourier transform]]: an infinitely large grain means that the [[lattice]] repeats forever without defect; this periodic oscillation can be described by a single peak in Fourier space. However, a finite grain requires additional Fourier information to encode this truncation to the repetition. A broad Fourier peak describes this 'localized periodicity' (c.f. [wave packet]s).	+	The average '''grain size''' in a material may be estimated from the '''peak width'''. Sharp peaks indicate large grains, whereas broad peaks indicate small grains. This inverse relationship is, as always, due to the inverse nature of [[reciprocal-space]]. It can also be rationalized in terms of the [[Fourier transform]]: an infinitely large grain means that the [[lattice]] repeats forever without defect; this periodic oscillation can be described by a single peak in Fourier space. However, a finite grain requires additional Fourier information to encode this truncation to the repetition. A broad Fourier peak describes this 'localized periodicity' (c.f. [[wave packet]]s).

	Note that in scattering analysis, we often convert peak width into a distance, and call this the 'grain size' or 'crystallite size'. This is conceptually meant to refer to the average size of the region over which the repeating lattice is similarly-aligned. This average size thus might describe the average distance between one grain boundary and another. However, in some types of systems, the material may not have well-defined grain boundaries. For instance, in [[block-copolymers]], the domains may 'meander', in which case the 'grain size' is instead a characteristic distance over which the lattice directionality is roughly preserved (an orientational correlation length). More generally still, some materials, the inverse peak width is perhaps better thought of as simply a '''correlation length''': a distance over which the repeating lattice is well-correlated.		Note that in scattering analysis, we often convert peak width into a distance, and call this the 'grain size' or 'crystallite size'. This is conceptually meant to refer to the average size of the region over which the repeating lattice is similarly-aligned. This average size thus might describe the average distance between one grain boundary and another. However, in some types of systems, the material may not have well-defined grain boundaries. For instance, in [[block-copolymers]], the domains may 'meander', in which case the 'grain size' is instead a characteristic distance over which the lattice directionality is roughly preserved (an orientational correlation length). More generally still, some materials, the inverse peak width is perhaps better thought of as simply a '''correlation length''': a distance over which the repeating lattice is well-correlated.