Gas electron diffraction

Diffraction occurs because the wavelength of electrons accelerated by a potential of a few thousand volts is of the same order of magnitude as internuclear distances in molecules. The principle is the same as that of other electron diffraction methods such as LEED and RHEED, but the obtainable diffraction pattern is considerably weaker than those of LEED and RHEED because the density of the target is about one thousand times smaller. Since the orientation of the target molecules relative to the electron beams is random, the internuclear distance information obtained is one-dimensional. Thus only relatively simple molecules can be completely structurally characterized by electron diffraction in the gas phase. It is possible to combine information obtained from other sources, such as rotational spectra, NMR spectroscopy or high-quality quantum-mechanical calculations with electron diffraction data, if the latter are not sufficient to determine the molecule's structure completely.

The total scattering intensity in GED is given as a function of the momentum transfer, which is defined as the difference between the wave vector of the incident electron beam and that of the scattered electron beam and has the reciprocal dimension of length.[4] The total scattering intensity is composed of two parts: the atomic scattering intensity and the molecular scattering intensity. The former decreases monotonically and contains no information about the molecular structure. The latter has sinusoidal modulations as a result of the interference of the scattering spherical waves generated by the scattering from the atoms included in the target molecule. The interferences reflect the distributions of the atoms composing the molecules, so the molecular structure is determined from this part.

GED can be described by scattering theory. The outcome if applied to gases with randomly oriented molecules is provided here in short:

Scattering occurs at each individual atom (${\displaystyle I_{a}(s)}$), but also at pairs (also called molecular scattering) (${\displaystyle I_{m}(s)}$), or triples (${\displaystyle I_{t}(s)}$), of atoms.

${\displaystyle s}$ is the scattering variable or change of electron momentum and its absolute value defined as

${\displaystyle \mid s\mid ={\frac {4\pi }{\lambda }}\sin(\theta /2)}$, with ${\displaystyle \lambda }$ being the electron wavelength defined above and ${\displaystyle \theta }$ being the scattering angle

The above mentionend contributions of scattering add up to the total scattering (${\displaystyle I_{tot}(s)}$):

${\displaystyle I_{tot}(s)=I_{a}(s)+I_{m}(s)+I_{t}(s)+I_{b}(s)}$, whereby (${\displaystyle I_{b}(s)}$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.

${\displaystyle I_{a}(s)={\frac {K^{2}}{R^{2}}}I_{0}\sum _{i=1}^{N}\mid f_{i}(s)\mid ^{2}}$ , with ${\displaystyle K={\frac {8\pi ^{2}me^{2}}{h^{2}}}}$, ${\displaystyle R}$ being the distance between the point of scattering and the detector, ${\displaystyle I_{0}}$ being the intensity of the primary electron beam and ${\displaystyle f_{i}(s)}$ 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. ${\displaystyle I_{a}(s)}$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)

${\displaystyle I_{m}(s)={\frac {K^{2}}{R^{2}}}I_{0}\sum _{i=1}^{N}\sum _{j=1,i\neq j}^{N}\mid f_{i}(s)\mid \mid f_{j}(s)\mid {\frac {\sin[s(r_{ij}-\kappa s^{2})]}{sr_{ij}}}e^{-(1/2l_{ij}s^{2})}\cos[\eta _{i}(s)-\eta _{i}(s)]}$ with ${\displaystyle r_{ij}}$ being the parameter of main interest: the atomic distance between two atoms, ${\displaystyle l_{ij}}$ being the mean square amplitude of vibration between the two atoms, ${\displaystyle \kappa }$ the anharmonicity constant (correcting the vibration description for deviations from a purely harmonic model), and ${\displaystyle \eta }$ 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.

${\displaystyle I_{t}(s)}$ is negligible in most cases and not described here in more detail and ${\displaystyle I_{b}(s)}$ 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.

Some selected examples of important contributions to the structural chemistry of molecules are provided here:

• Structure of diborane B₂H₆[6]
• Structure of the planar trisilylamine[7]
• Determinations of the structures of gaseous elemental phosphorus P₄ and of the binary P₃As[8]
• Determination of the structure of C₆₀[9] and C₇₀[10]
• Structure of tetranitromethane[11]
• Absence of local C₃ symmetry in the simplest phosphonium ylide H₂C=PMe₃[12] and in amino-phosphanes like P(NMe₂₎₃ and ylides H₂C=P(NMe₂)₃[13]
• Determination of intramolecular London dispersion interaction effects on gas-phase and solid-state structures of diamondoid dimers[14]