alt="Schematic illustration of top: A chain with a lattice constant a as well as its reciprocal lattice, a chain with a spacing of 2Π/a. Middle and bottom: Two lattice-periodic functions ρ(x) in real space as well as their Fourier coefficients. The magnitude of the Fourier coefficients |ρn| is plotted on the reciprocal lattice vectors they belong to."/>
as well as its reciprocal lattice, a chain with a spacing of . Middle and bottom: Two lattice‐periodic functions in real space as well as their Fourier coefficients. The magnitude of the Fourier coefficients is plotted on the reciprocal lattice vectors they belong to.
The same ideas also work in three dimensions. In fact, one can use a Fourier sum for lattice‐periodic properties that corresponds to Eq. (1.20). For the lattice‐periodic electron concentration , we get
Thus we have seen that the reciprocal lattice is very useful for describing lattice‐periodic functions. But this is not the whole story: The reciprocal lattice can also simplify the treatment of waves in crystals in a very general sense. Such waves can be X‐rays, elastic lattice distortions, or even electronic wave functions. We will come back to this point at a later stage.
1.3.1.6 X‐Ray Diffraction from Periodic Structures
Turning back to the specific problem of X‐ray diffraction, we can now exploit the fact that the electron concentration is lattice‐periodic by inserting Eq. (1.23) in our expression from Eq. (1.11) for the diffracted intensity. This yields
Let us inspect the integrand. The exponential represents a plane wave with a wave vector . If the crystal is very big, the integration will average over the crests and troughs of this wave and the result of the integration will be very small (or zero for an infinitely large crystal). The only exception to this is the case where
that is, when the difference between incoming and scattered wave vector equals a reciprocal lattice vector. In this case, the exponential in the integral is 1, and the value of the integral is equal to the volume of the crystal. Equation 1.25 is often called the Laue condition. It is central to the description of X‐ray diffraction from crystals in that it describes the condition for the observation of constructive interference.
Looking back at Eq. (1.24), the observation of constructive interference for a chosen scattering geometry (or scattering vector ) clearly corresponds to a particular reciprocal lattice vector . The intensity measured at the detector is proportional to the square of the Fourier coefficient of the electron concentration . We could therefore think of measuring the intensity of the diffraction spots appearing for all possible reciprocal lattice vectors, obtaining the Fourier coefficients of the electron concentration and reconstructing this concentration from the coefficients. This would give us all the information needed and thus conclude the process of structure determination. Unfortunately, this straightforward approach does not work because the Fourier coefficients are complex numbers. Taking the square root of the intensity at the diffraction spot therefore gives the magnitude, but not the phase of . The phase is lost in the measurement; this is known as the phase problem in X‐ray diffraction. To solve an unknown structure, one has to find a way to work around it. One simple approach for this is to calculate the electron concentration for a structural model, obtain the magnitude of the values and thus also the expected diffracted intensity, and compare this to the experimental result. Based on the outcome, the model can be refined until the agreement is satisfactory.
In the following, we will describe in more detail how this can be achieved. We start with Eq. (1.11), the expression for the diffracted intensity that we had obtained before introducing the reciprocal lattice. But now we know that constructive interference is only observed in an arrangement that corresponds to meeting the Laue condition and we can therefore write the intensity for a particular diffraction spot as
We also know that the crystal consists of many identical unit cells at the positions of the Bravais lattice. Therefore, we can split the integral and write it as a sum of integrals over the individual unit cells,
