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[[File:Dhkl vs elbeam.png|200px|thumb|Wave reflection on crystal lattice.]] | [[File:Dhkl vs elbeam.png|200px|thumb|Wave reflection on crystal lattice.]] | ||
[[File:Elbeam and ring for crystals.png |200px|thumb|Scheme of diffraction experiment on a polycrystalline target.]] | [[File:Elbeam and ring for crystals.png |200px|thumb|Scheme of diffraction experiment on a polycrystalline target.]] | ||
==== Theoretical considerations ==== | |||
==== Bragg-Wolf equation ==== | |||
The scattering on the polycrystalline sample can be described with the Bragg-Wolf equation: | |||
<math> | |||
2d_{hkl} \sin \theta = n \lambda , | |||
</math> | |||
where ''hkl'' - are Miller indexes, ''d''<sub>''hkl''</sub> is a distance between ''hkl'' crystallographic planes, ''θ'' - is a reflection angle from the corresponding plane, ''n'' - is positive integer. The GED usually operates with the scattering angle ''φ''. The relationship between ''θ'' and ''φ'' is <math>\varphi=2\theta</math> (see pictures). The scattering angle ''φ'' is usually calculated in GED from the scattered ring radius ''R'' and distance between scattering center and the registration device ''L''. As it is easily seen from the illustration the formula for this angle is: <math>\varphi=\arctan( \frac{R}{L} )</math>. Therefore the substitution of the scattering angle ''φ'' into the Bragg-Wolf formula results the equation for the wavelength determination: | |||
<math> | |||
\lambda = \frac{2d_{hkl}}{n}\sin (\frac{1}{2} \arctan ( \frac{R}{L} ) ) . | |||
</math> | |||
==== Calculation of the interplane distance ==== | |||
The distances between the cystallographic planes can be calculated from the lattice parameters: | |||
* unit cell edges lengths '''''a''''', '''''b''''', '''''c''''', | |||
* angles between edges ''α'', ''β'', ''γ''. | |||
The general formula is: | |||
<math> | |||
\frac{1}{d_{hkl}^2} = \frac{1}{s} [ (\frac{h\sin\alpha}{a})^2 + (\frac{k\sin\beta}{b})^2 +\\ | |||
+ (\frac{l\sin\gamma}{c})^2 + 2 \frac{hk}{ab}(\cos\alpha \cdot \cos\beta - \cos\gamma) + 2 \frac{hl}{ac}(\cos\alpha \cdot \cos\gamma - \cos\beta) + | |||
2 \frac{kl}{bc}(\cos\beta \cdot \cos\gamma - \cos\alpha) ], | |||
</math> | |||
where <math>s= 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos\alpha \cos\beta \cos\gamma </math>. | |||
This formula is easily simplified to smaller equations for the cases of lattices more symmetrical then the triclinic one. | |||
For example, in the case of hexagonal lattice, where <math>a=b\neq c</math>, <math>\alpha=\beta=90^{\circ}</math>, <math>\gamma = 120^{\circ}</math> the equation for ''d''<sub>''hkl''</sub> would be: | |||
<math> | |||
\frac{1}{d_{hkl}^2} = \frac{4}{3}\frac{h^2+hk+k^2}{a^2} + \frac{l^2}{c^2} . | |||
</math> | |||
==== The common polycrystall standards ==== | |||
The most widely used polycrystalline standard is the zinc oxide ZnO. Several reasons are behind its' prevalence. | |||
* It is simple to prepare. The metal Zn is burned to yielding a ZnO in a form of smoke that is condensed on the wire net substrate. | |||
* It has only one stable crystal modification in the usual GED experimental conditions. | |||
* It is cheap. | |||
* ??? | |||
* PROFIT!!! |
Revision as of 16:44, 8 October 2015
Standards in GED
Standards are used in GED for calibration of electron wavelength or nozzle position. There are two kinds of them:
Type of standard | Examples |
---|---|
Gas | C6H6, CCl4, CO2, CS2 |
Polycrystalline | ZnO, TlCl |
Polycrystalline standards
Theoretical considerations
Bragg-Wolf equation
The scattering on the polycrystalline sample can be described with the Bragg-Wolf equation:
<math> 2d_{hkl} \sin \theta = n \lambda , </math>
where hkl - are Miller indexes, dhkl is a distance between hkl crystallographic planes, θ - is a reflection angle from the corresponding plane, n - is positive integer. The GED usually operates with the scattering angle φ. The relationship between θ and φ is <math>\varphi=2\theta</math> (see pictures). The scattering angle φ is usually calculated in GED from the scattered ring radius R and distance between scattering center and the registration device L. As it is easily seen from the illustration the formula for this angle is: <math>\varphi=\arctan( \frac{R}{L} )</math>. Therefore the substitution of the scattering angle φ into the Bragg-Wolf formula results the equation for the wavelength determination:
<math> \lambda = \frac{2d_{hkl}}{n}\sin (\frac{1}{2} \arctan ( \frac{R}{L} ) ) . </math>
Calculation of the interplane distance
The distances between the cystallographic planes can be calculated from the lattice parameters:
- unit cell edges lengths a, b, c,
- angles between edges α, β, γ.
The general formula is:
<math> \frac{1}{d_{hkl}^2} = \frac{1}{s} [ (\frac{h\sin\alpha}{a})^2 + (\frac{k\sin\beta}{b})^2 +\\ + (\frac{l\sin\gamma}{c})^2 + 2 \frac{hk}{ab}(\cos\alpha \cdot \cos\beta - \cos\gamma) + 2 \frac{hl}{ac}(\cos\alpha \cdot \cos\gamma - \cos\beta) + 2 \frac{kl}{bc}(\cos\beta \cdot \cos\gamma - \cos\alpha) ], </math> where <math>s= 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos\alpha \cos\beta \cos\gamma </math>. This formula is easily simplified to smaller equations for the cases of lattices more symmetrical then the triclinic one. For example, in the case of hexagonal lattice, where <math>a=b\neq c</math>, <math>\alpha=\beta=90^{\circ}</math>, <math>\gamma = 120^{\circ}</math> the equation for dhkl would be:
<math> \frac{1}{d_{hkl}^2} = \frac{4}{3}\frac{h^2+hk+k^2}{a^2} + \frac{l^2}{c^2} . </math>
The common polycrystall standards
The most widely used polycrystalline standard is the zinc oxide ZnO. Several reasons are behind its' prevalence.
- It is simple to prepare. The metal Zn is burned to yielding a ZnO in a form of smoke that is condensed on the wire net substrate.
- It has only one stable crystal modification in the usual GED experimental conditions.
- It is cheap.
- ???
- PROFIT!!!