Surface X-Ray Diffraction

For a bulk crystal, which is periodic in all three lattice vector directions, the diffraction pattern consists of discrete peaks whose widths depend on the quality and the size of the crystal. Also, due to the finite-sized electron density distribution within the atoms, the diffraction intensity decays as a function of the distance from the origin of reciprocal space (Figure 1, left column). A crystal with a well-defined surface, however, is not infinitely periodic in all directions, because the periodicity in the surface normal direction is broken at the surface, and a different diffraction pattern is expected for surface diffraction. A well-defined crystal surface can be constructed by multiplying the infinite crystal with a one-dimensional step function (Figure 1, upper row). For convenience, let the surface normal be along the z direction. The diffraction pattern from this crystal surface can now be obtained by taking the Fourier transform of this product. The convolution theorem states that the Fourier transform of the product of two functions is equal to the convolution of the Fourier transforms of each function. Therefore, the diffraction pattern from a crystal with a well-defined surface can be represented as the convolution of the bulk diffraction pattern with a 1/z shape function, which is the Fourier transform of the step function.

Figure 1. Qualitative schematics for how CTRs are formed due to the presence of a well-defined crystal surface.

As described in Figure 1, the final diffraction pattern has strong peaks at the reciprocal lattice positions, which are broadened along the surface normal direction and result in streaks of continuous intensity along the out-of-plane direction. Since these streaks of continuous diffraction intensities, which originate from truncation of the crystal, look like rods along the surface normal, they are called crystal truncation rods (CTRs). Again, without the presence of a crystal surface, the diffraction pattern is a set of discrete Bragg peaks in reciprocal space. The continuous CTR intensities between the Bragg peaks, on the other hand, originate purely from the existence of a surface. Therefore, the CTR intensities are very sensitive to the surface structure of a crystal, and the structure can be determined by analyzing the CTRs. This CTR-based surface characterization method is called surface x-ray diffraction (SXRD).