Gas electron diffraction: Difference between revisions

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== Theory ==

GED can be described by scattering theory. The outcome if applied to gases with randomly oriented molecules is provided here in short:<ref>{{Cite book |title=Stereochemical Applications of Gas‐Phase Electron Diffraction, Part A: The Electron Diffraction Technique |last=Hargittai |first=I. |publisher=VCH Verlagsgesellschaft |year=1988 |___location=Weinheim}}. {{|isbn|3-527-26691-7|=0-89573-337-4}}</ref><ref name=":1" />

Scattering occurs at each individual atom (<math>I_\text{a}(s)</math>), but also at pairs I_a(also called molecular scattering) (<math>I_\text{m}(s)</math>), or triples (<math>I_\text{t}(s)</math>), of atoms.

</math>), but also at pairs (also called molecular scattering) (<math> I_m(s)

</math>), or triples (<math> I_t(s)

</math>), of atoms.

<math>s</math> is the scattering variable or change of electron momentum, and its absolute value is defined as

<math> s

: <math> \mid |s\mid| = \frac{4\pi}{\lambda} \sin (\theta / 2) ,</math>

</math> is the scattering variable or change of electron momentum and its absolute value defined as

with <math>\lambda</math> being the electron wavelength defined above, and <math> \theta</math> being the scattering angle.

The above mentioned contributions of scattering add up to the total scattering

<math> \mid s\mid = \frac{4\pi}{\lambda}\sin (\theta / 2)

: <math>I_\text{tot}(s) = I_\text{a}(s) + I_\text{m}(s) + I_\text{t}(s) + I_\text{b}(s),</math>

</math>, with <math> \lambda

where <math>I_\text{b}(s)</math> is the experimental background intensity, which is needed to describe the experiment completely.

</math> being the electron wavelength defined above and <math> \theta

</math> being the scattering angle

The abovecontribution mentionendof contributionsindividual ofatom scattering addis upcalled to the totalatomic scattering and (<math> I_{tot}(s)easy to calculate:

: <math> I_aI_\text{a}(s) = \frac{K^2}{R^2} I_0 \sum_{i=1}^N \mid |f_i(s)\mid|^2 ,</math>

</math>):

with <math>K = \frac{8 \pi^2 me^2}{h^2}</math>, <math>R</math> being the distance between the point of scattering and the detector, <math>I_0</math> being the intensity of the primary electron beam, and <math>f_i(s)</math> being the scattering amplitude of the ''i''-th atom. In essence, this is a summation over the scattering contributions of all atoms independent of the molecular structure. <math>I_\text{a}(s)</math> is the main contribution and easily obtained if the atomic composition of the gas (sum formula) is known.

The most interesting contribution is the molecular scattering, because it contains information about the distance between all pairs of atoms in a molecule (bonded or non-bonded):

<math> I_{tot}(s) = I_a(s)+I_m(s)+I_t(s)+I_b(s)

: <math> I_mI_\text{m}(s) = \frac{K^2}{R^2} I_0 \sum_{i=1}^N \sum_{j=1,i\neq j}^N \mid |f_i(s)|\mid\mid ,|f_j(s)\mid| \frac{\sin [s(r_{ij} - \kappa s^2)]}{sr_{ij}}e^{-(\frac{1/}{2} l_{ij} s^2)} \cos [\eta _i eta_i(s) -\eta _i \eta_i(s)] ,</math>

</math>, whereby (<math> I_b(s)

with <math>r_{ij}</math> being the parameter of main interest: the atomic distance between two atoms, <math>l_{ij}</math> being the mean square amplitude of vibration between the two atoms, <math>\kappa</math> the anharmonicity constant (correcting the vibration description for deviations from a purely harmonic model), and <math>\eta</math> is a phase factor, which becomes important if a pair of atoms with very different nuclear charge is involved.

</math>is the experimental background intensity, which is needed to describe the experiment completely

The contribution of individual atom scattering is called atomic scattering and easy to calculate.

<math> I_a(s) = \frac{K^2}{R^2}I_0\sum_{i=1}^N \mid f_i(s)\mid^2

</math> , with <math> K = \frac{8 \pi ^2me^2}{h^2}

</math>, <math> R

</math> being the distance between the point of scattering and the detector, <math> I_0

</math> being the intensity of the primary electron beam and <math> f_i(s)

</math> being the scattering amplitude of the i-th atom. In essence theis is a summation over the scattering contributions of all atoms independent of the molecular structure. <math> I_a(s)

</math>is the main contribution and easily obtained if the atomic composition of the gas (sum formula) is known.

The most interesting contribution is the molecular scattering, because it contains information about the distance between all pairs of atoms in a molecule (bonded or non-bonded)

<math> I_m(s) = \frac{K^2}{R^2}I_0\sum_{i=1}^N \sum_{j=1,i\neq j}^N \mid f_i(s)\mid\mid f_j(s)\mid \frac{\sin [s(r_{ij}-\kappa s^2)]}{sr_{ij}}e^{-(1/2 l_{ij} s^2)} \cos [\eta _i (s) -\eta _i (s)]

</math> with <math> r_{ij}

</math> being the parameter of main interest: the atomic distance between two atoms, <math> l_{ij}

</math> being the mean square amplitude of vibration between the two atoms, <math> \kappa

</math> the anharmonicity constant (correcting the vibration description for deviations from a purely harmonic model), and <math> \eta

</math> is a phase factor which becomes important if a pair of atoms with very different nuclear charge is involved.

The first part is similar to the atomic scattering, but contains two scattering factors of the involved atoms. Summation is performed over all atom pairs.

<math>I_\text{t}(s)</math> is negligible in most cases and not described here in more detail. <math>I_\text{b}(s)</math> is mostly determined by fitting and subtracting smooth functions to account for the background contribution.

<math> I_t(s)

</math> is negligible in most cases and not described here in more detail and <math> I_b(s)

</math> is mostly determined by fitting and subtracting smooth functions to account for the background contribution.

So it is the molecular scattering intensity that is of interest, and this is obtained by calculation all other contributions and subtracting them from the experimentally measured total scattering function.