Scattering from a monolayer


Now we will apply the Fourier formalism to determine basic features of the scattering from a monolayer. The electron density of the atoms within a unit cell be again given by

ruc(r) = Si rat,i(r-ri)
The two-dimensional lattice can be described by
latt2D(r) = Smnd(R-Rmn) d(z)
where we choose our coordinate system such that R, Rmn are vectors in the surface plane, and z is the component of r in the direction of the surface normal n, so that we can write
r = R + z n
with n R = 0 and || n || = 1. The 2D lattice has the basis vectors a and b and thus
Rmn = ma + nb
The electron density of a monolayer is then given by
r(r) = ruc(r) * latt2D(r)
As before in the case of 3D, the Fourier transform of ruc(r) yields the structure factor of the unit cell Suc(q). The Fourier transform of latt2D(r) can also be readily obtained by exchanging summation and integration:
Latt2D(q) = FT{latt2D} = Smn exp(i q Rmn)
This sum vanishes for all q, except when q Rmn = 2p, where the sum diverges. If we introduce the reciprocal lattice vectors of the 2D lattice as
a* = 2p (b x n) / (a x b) n     b* = 2p (n x a) / (a x b) n
we can rewrite Latt2D(q) as
Latt2D(q) = Shkd(q-Ghk)
and
Ghk = h a* + k b*
Note that the Fourier transform of d(z) yields simply 1, i.e. there is a constant diffuse scattering intensity in qz-direction wherever the in-plane component of q coincides with a Ghk. These are the so-called scattering rods for a point lattice. If we now add in the atomic density distribution, we get
S(q) = Suc(q) Latt2D(q)
If the monolayer consists of a single atom, the scattering rods will fall off according to the atomic form factor. Neutrons in comparison are scattered by the point-like nuclei and the scattering rods are constant (except for the Debye-Waller factor, that we have left out of this consideration).

As an aside we mention, that in case Suc(q) has a complex structure, such as the one for a molecule, the scattering rod can be used to measure the structure factor of the unit cell Suc(q) along qz .