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:
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
- (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.
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.
Reciprocal LatticesThe 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.
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
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.
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
- 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]=
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.
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.