Читать книгу Solid State Physics - Philip Hofmann - Страница 26

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

(1.24)

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

(1.25)

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

(1.26)

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,

(1.27)

where is the number of unit cells in the crystal and we have used the lattice periodicity of and Eq. (1.14) in the last step. We now assume that the electron density in the unit cell is given by the sum of atomic electron densities that can be calculated from the atomic wave functions. In doing so, we neglect the fact that some of the electrons form bonds between the atoms and are no longer part of the spherical electron cloud around the atom. If the atoms are not too light, however, the number of these valence electrons is small compared to the total number of electrons and the approximation is appropriate. We can then write

(1.28)

where we sum over the different atoms in the unit cell (i.e. the basis) at positions . This permits us to rewrite the integral in Eq. (1.27) as a sum of integrals over the individual atoms in the unit cell

(1.29)

where . The two exponential functions give rise to two types of interference. The first describes the interference between the X‐rays scattered by the different atoms in the unit cell, and the second the interference between the X‐rays scattered by the electrons within one atom. The last integral is called the atomic form factor and can be calculated from the atomic properties alone. We therefore see how the diffracted intensity for an assumed structure can be calculated from the atomic form factors and the arrangement of the atoms.