Surface Diffraction and the Reciprocal Lattice

The diffraction of either photons or electrons (sometimes neutrons) is one of the most powerful techniques for surface structure determination. Unfortunately, the diffraction pattern is not a direct representation of the real-space arrangement of the atoms in a solid or on a surface. The most convenient way to link the real structure of the material to it's diffraction pattern is through the reciprocal lattice.

In order for measureable diffraction to occur, the wavelength of the interrogating wave-particle should be on the same order as the periodicity of the features. For atoms or molecules in a crystalline solid, this periodicity is a few Angstroms. This means that if we are using photons to examine the lattice spacing of a solid, their wavelength should be a few Angstroms (x-rays). Consider a 5 keV photon:

Now consider a similar calculation for the de Broglie wavelength of a 20 eV electron:

So electrons of a few tens of eV and x-rays of a few keV are suitable for diffraction from an atomic lattice. These energy particles are more strongly backscattered from a solid towards the source than transmitted. In the case of x-rays the technique is often called Laue or von Laue Backscatter Diffraction and in the case of electrons the technique is called low energy electron diffraction (LEED).
The angle of diffraction of the wave-particle is governed by the Bragg equation. Constructive interference between two out-going waves only occurs if the pathlength difference between them is equal to an intergral number of wavelengths. For this to occur, the Bragg equation

must be satisfied.

There are three important features of the Bragg equation worth noting

(A) Sin[theta] is proportional to 1/d, which implies a large atomic spacing will produce a small diffraction angle and a small atomic spacing will produce a large diffraction angle. This already gives us a hint about a reciprocal relationship between the real arrangement and the diffraction pattern.
(B) Sin[theta] is proportional to 1/(eV)^1/2, which implies the size of the diffraction pattern will vary with incident wave-particle energy. The diffraction angle becomes smaller with increasing incident energy.
(C) Diffraction has same probability for n=1 and n=-1, which implies the diffraction pattern should possess some sort of symmetry.

In fact, for two isolated atoms, the diffraction intensity varies smoothly between zero when pathlength=(n+1/2)[lambda] and a maximum when pathlength=n[lambda]. For many atoms, all with the same separation, the diffraction intensity is almost zero at all angles except when very close to the pathlength being equal to an integral number of wavelengths.
It is possible to think of surface diffraction in terms of rows of atoms and draw a qualitative picture of the diffraction pattern merely by considering the spacing of these rows. We can pick any pair of rows at any angle and measure the distance between them. Remebering note (A) above, we can deduce that the row spacing in the diffraction pattern will be 1/(real space distance) as shown below. But this method is cumbersome and it is easy to get confused when drawing the pattern.

An alternate (and much used) idea is to make use of a reciprocal lattice which is directly related to the diffraction pattern.

Reciprocal Lattices

The real space surface lattice (net) can be defined by two lattice vectors a₁ and a₂. These vectors may be chosen in any arbitrary direction but it is usual to pick conventional lattice vectors which are often obvious from the symmetry of the lattice. These conventional lattice vectors are chosen so that a₁ and a₂ are arranged in an anti-clockwise fashion (see below) and if they are not the same magnitude, a₂ labels the largest vector. This should create a unique set of vectors.

In the same way, we can define reciprocal lattice vectors which we will use to generate the reciprocal lattice. These are given the symbols a'₁ and a'₂ for the surface and b'₁ and b'₂ for any adsorbate structure.
But which way do the reciprocal vectors point and how long are they? The rules for determining the magnitude and direction of the reciprocal lattice vectors are

and

Remember that in vector algebra, dot product means "the magnitude of x multiplied by the magnitude of y multiplied by the cosine of the angle between them." In effect the two rules can be stated

For example, when a=0' (ie when a₁ and a'₁ are parallel)

and the reciprocal relationship between the real and reciprocal lattice vectors becomes obvious.
Note: Physicists often use a value of 2[pi] instead of 1 in the relationships above but we won't.
These ideas can be applied to create the reciprocal lattices of the square real space lattices below. If we knew the length of a₁ in term of Angstroms, we could calculate the length of a'₁in terms of reciprocal Angstroms.
FCC(100):

In this case, the real and reciprocal lattices look very similar.
FCC(110):

Now, the reciprocal lattice looks like the real one but rotated by 90 degrees!
It is important to note that

a₁ and a₂ perpendicular
a₁ and a'₂ perpendicular
a₁ and a'₁ parallel
Since [alpha]=0', Cos[alpha]=1 and a'₁=1/a₁

Things become a little more complicated when dealing with a non-square real lattice but the same rules can be applied to create the reciprcal lattice as shown below.
FCC(111):

and once more, the real and reciprocal lattices have the same symmetry.
This time

a₁ and a₂ not perpendicular
but a₁ and a'₂ perpendicular
and a₂ and a'₁ perpendicular
but a₁ and a'₁ not parallel
Since [alpha]=30', Cos[alpha]=
and

The diffraction pattern is just a scaled version of the reciprocal lattice!
The same rules can be applied when dealing with an adsorbate on a surface. The diffraction pattern will now be composed of two parts: one due to adsorbate lattice diffraction and one due to substrate lattice diffraction.

It is important to note that this method tells us nothing about the intensity of the diffraction spots although we might reasonably assume that the adsorbate spots will be weaker than the substrate spots. Much more sophisticated theories (multiple scattering theories) are needed to account for the diffraction spot intensities.
The reciprocal lattice approach is a very useful way to determine the diffraction pattern. If the dimensions of the diffractometer and the wavelength or energy of the incident wave-particles are known, the diffraction pattern can be "backconverted" into the real space arrangement. Historically, LEED has been the primary method for determining adsorbate and surface structures.