Standard
Contents
Standards in GED
The generalized scattering coordinate s in gas electron diffraction is the function of electron wavelength λ and geometric scattering angle φ:
$ s = \frac{4π}{λ}\sin (\frac{φ}{2}) $
Expressing angle φ through the distance from nozzle tip to detector L and radius on the detector r gives
$ s = \frac{4π}{λ}\sin (\frac{\arctan (\frac{r}{L})}{2}) $
Standards are used in GED for calibration of electron wavelength or nozzle position. There are two kinds of them, gas and polycrystalline.
Type of standard | Examples |
---|---|
Gas | C_{6}H_{6}, CCl_{4}, CO_{2}, CS_{2} |
Polycrystalline | ZnO, TlCl |
Polycrystalline standards
The most widely used in GED polycrystalline standard is the zinc oxide ZnO. Several reasons are behind its' prevalence.
- It is inexpensive and simple to prepare. The metallic Zn burned to yield ZnO in form of smoke which is condensed on the wire net substrate.
- It has only one stable crystal modification at conditions of GED experiments.
Theoretical considerations
Bragg-Wulff equation
The scattering on the polycrystalline sample can be described with the Bragg-Wulff equation:
$ 2d_{hkl} \sin \theta = n \lambda , $
where hkl — are Miller indices, d_{hkl} is the distance between hkl crystallographic planes, θ — is the reflection angle from the corresponding plane, n — is the positive integer. The GED usually operates with the scattering angle φ. The relationship between θ and φ is $ \varphi=2\theta $ (see Figures). As it can be easily seen from the illustration $ \varphi=\arctan( \frac{R}{L} ) $, where R is the ring radius, L is the nozzle-to-detector distance. Therefore the substitution of the scattering angle φ into the Bragg-Wulff formula results in the equation for the wavelength determination:
$ \lambda = \frac{2d_{hkl}}{n}\sin (\frac{1}{2} \arctan ( \frac{R}{L} ) ) . $
In real electron diffraction experiments only rings for n = 1 are observed.
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:
$ \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) ], $
where $ s= 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos\alpha \cos\beta \cos\gamma $. 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 $ a=b\neq c $, $ \alpha=\beta=90^{\circ} $, $ \gamma = 120^{\circ} $ the equation for d_{hkl} would be:
$ \frac{1}{d_{hkl}^2} = \frac{4}{3}\frac{h^2+hk+k^2}{a^2} + \frac{l^2}{c^2} . $
For cubic crystal system it is as simple as
$ \frac{1}{d_{hkl}^2} = \frac{h^2+k^2+l^2}{a^2} . $