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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 C6H6, CCl4, CO2, CS2
Polycrystalline ZnO, TlCl

Polycrystalline standards

Wave reflection on crystal lattice.
Scheme of diffraction experiment on a polycrystalline target.

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, dhkl 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 dhkl 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} . $