6.1. Summary In this Lecture we cover the theory of x-ray diffraction, which gives direct information about the atomic structure of crystals. In these experiments, the wavelength of the incident beam must be comparable to, or smaller than, interatomic distances. In a crystal, atoms are typically spaced at distances of the order 10 10 m (the Angstrom, Å) and X-rays have a wavelength in the range 0.1 to 6 Å. We will learn how x-ray diffraction can be employed to determine a crystal s lattice parameters, space group and atomic fractional coordinates. Furthermore, the energy levels of bound electrons are such that the elements have characteristic absorption lines in the x-ray region of the electromagnetic spectrum, which can also be exploited. 6.2. Elastic X-ray scattering The elementary scatterer of x-rays is the electron. Here we consider an elastic scattering process in which the energy of the incident x-ray is the same as that scattered, and momentum is conserved through momentum transfer to the scattering medium. It is sufficient to consider a classical model in which the electric field of the incident radiation exerts a force on the electron, causing it to accelerate, oscillate, and hence emit the scattered x-ray. In a typical experiment an x-ray beam quantified by its flux, Φ 0, (rate of photons passing through a unit area) is incident on the scatterer. The intensity of the scattered x-rays, I, (rate of scattered photons) is measured by a detector that subtends a solid angle, Ω, positioned some distance, R, away from the scatterer. The efficiency of the scattering process is quantified by the differential scattering cross-section defined as: = I dω Φ 0 Ω The differential scattering cross-section for one electron can be shown to be ( ) ( dσ e 2 ) 2 = dω 4πɛ 0 mc 2 P = r0p 2 where r 0 is the Thomson scattering length, or the classical radius of the electron (r 0 = 2.82x10 15 m). Accordingly, the scattering process takes the name Thomson scattering. P = ˆɛ ˆɛ 2 is a polarisation factor, where ˆɛ and ˆɛ are the polarisation of the incident and scattered x- rays, respectively. We define a scattering amplitude, A, such that = AA dω and A electron = r 0 P 1 2. Page 1 of 6
Figure 1: The atomic form factors for hydrogen and neon. The scattering of x-rays by an atom is calculated by considering scattering from the electron charge density of the atom, ρ(r), in units of the electron charge. We assume that the incident and scattered x-ray beams are plane waves with wave vectors k and k, respectively. X-rays scattered from a volume element of the charge density at some vector r from the origin will acquire a phase difference of φ(r) = (k k ) r = Q r, where Q is the scattering vector. Integrating over the full charge density we have f 0 (Q) = ρ(r)e iq r dr where f 0 (Q) is the atomic form factor. At Q = 0, all electrons scatter in-phase, and f 0 (Q) = Z, the total number of electrons. As Q tends to infinity, f 0 (Q) tends to zero. The hydrogen and neon form factors are plotted in Figure 1 Itinerant electrons contribute weakly to the scattering of x-rays, as the atomic form factor decays rapidly with increasing Q. The majority of x-ray scattering occurs through interaction with tightly bound core electrons, and it is important to consider the quantisation of energy levels below the itinerant continuum. When the incident x-ray energy is near to an atomic resonance (excitation of a core electron into the continuum followed by instantaneous decay back to the core state) we see a departure from energy-independent Thomson scattering. Such resonant effects on the scattering process can be accounted for by including energy dependent, complex terms in the atomic form factor, known as anomalous terms: f(q) = f 0 (Q) + f ( hω) + if ( hω) The anomalous terms are independent of Q, but strongly dependent on the x-ray energy hω, and become large near atomic resonances. Below an atomic resonance the core electrons are tightly bound, whereas above the resonance energy they become essentially free. Their ability to respond to the driving force of Page 2 of 6
Figure 2: Anomalous terms in the neon atomic form factor. the incident x-ray s electric field is therefore reduced below the resonance, and increased above. This is captured by f ( hω). Atomic resonances coincide with x-ray absorption events, and the respective dispersion is contained in the complex term if ( hω). Both terms are plotted as a function of energy for neon in Figure 2. Finally, the scattering amplitude for an atom is the atomic form factor times the electron scattering amplitude. Hence, the differential scattering cross-section for the atom is = A atom (Q)A dω atom(q) = f(q) 2 r0p 2 6.3. X-ray scattering from a perfect crystal The scattering amplitude of an atom is equal to the Fourier transform of the charge density times the electron scattering length times a polarisation factor. The same is true for a crystal, but now the charge density is periodic by the translational symmetry of the lattice, and we might have multiple atoms in the basis. We saw in Lecture 4 that the Fourier transform of a real space lattice is the reciprocal space lattice. Therefore, the scattering amplitude of a crystal is proportional to the reciprocal lattice of the crystal weighted by the structure factor: A crystal = r 0 P 1 2 F (Q) where the summation is over all reciprocal lattice points, R, and F (Q) = j f j(q)e Q r j is the structure factor summed over all j atoms in a single unit cell. The differential cross-section is then: = A crystal (Q)A crystal(q) = r 20P N (2π)3 δ(q τ ) F (Q) 2 dω V In summary, we find that the differential cross-section for a crystal is proportional to the squared modulus of the structure factor and to the number of unit cells in the sample. It is only non-zero if Q is equal to a reciprocal lattice vector, τ. R e iq R The delta function δ(q τ ) is traditionally expressed as the Laue equation, Q = τ which describes the condition for diffraction, where Q = ha + kb + lc. τ Page 3 of 6
Figure 3: The scattering triangle Figure 4: The diffraction condition (expanded scattering triangle). By geometry (see Figure 3) we find that the length of the scattering vector, Q = k k, is related to the total scattering angle, 2θ, between k and k by Q = 4π λ sin(θ) There exists a family of crystal planes perpendicular to Q that we also label with the Miller indices (h, k, l). The spacing between the planes is d = 2π Q Combining the above two equations we obtain Bragg s Law, λ = 2d sin(θ) which is a real space analogy to the Laue equation. The Laue equation is more attractive as it allows us to use integer values of h, k, and l. However, the Bragg equation is perhaps more intuitive, and helps us understand the experimental condition for diffraction. We must orient our crystal planes of spacing d with respect to the incident x-ray beam at the scattering angle, θ, such that the diffraction condition is met. We must then position our detector to measure the peak diffraction intensity at an angle 2θ with respect to the incident beam. We can see this geometry by expanding the scattering triangle, shown in Figure 4. Page 4 of 6
6.4. Determining the lattice parameters By measuring the scattering angle of diffraction peaks with known h, k, and l, and with non-parallel scattering vectors, we can determine the lattice parameters of the crystal (using monochromatic x-rays): sin hkl (θ) = λ h [h, k, l] 2 G k l where the reciprocal metric tensor, G is a symmetric 3x3 tensor, written in terms of the lattice parameters with 6 unknowns. Aside: It is often the case that one cannot identify the Miller indices of a reflection without prior knowledge of the lattice parameters. In a typical experiment on an unknown crystal one has to measure a very large number of diffraction peaks, and use computer algorithms to fit the lattice parameters. 6.5. Determining the space group The space group is determined as follows: a) Having determined the lattice parameters, label every measured diffraction peak with Miller indices (hkl). b) Identify if any peaks are systematically absent, and determine whether or not the absences are consistent with a lattice centring extinction condition or roto-translation extinction conditions, or both. c) Determine the relative value of F (hkl) 2 for each reflection. The measured diffraction intensity for a given peak is related to F (hkl) 2 by I = Ψ 0 L(θ)P A(θ, λ)r 2 0N F (hkl) 2 Here, L(θ) is a correction, known as the Lorentz correction, which is applied in order to transform intensities measured in terms of angles of the experiment, into peaks in reciprocal space (it is a Jacobian), and A(θ, λ) is an absorption correction. The correction terms L(θ), P, and A(θ, λ) are included here for completeness, but they are non-examinable. You would be given F (Q) 2 directly. d) Identify any symmetry in the values of F (hkl) 2 in reciprocal space in order to determine the Laue Class. By combining the Laue Class with the observed extinction conditions we can determine the space group up to the presence of inversion symmetry. 6.6. Determining the atomic fractional coordinates We have shown that the structure factor is proportional to the Fourier transform of the charge density, ρ(r), integrated over the unit cell. Therefore, by taking the inverse Fourier transform of the structure factor we can find an expression for the charge density, which is dependent upon the element types and their fractional coordinates. ρ(r) = 1 F (Q)e iq r V Q Page 5 of 6
To determine the charge density exactly one needs in principle to know the structure factor at every point of the reciprocal lattice. However, for a given wavelength we can only measure up to a value of Q max = 2 k i, giving our structure solution a real space resolution of 2π/ Q max. A direct reconstruction of the charge density from diffraction data is impossible. In a diffraction experiment we measure intensities, which are proportional to the modulus square of the structure factor. Therefore, all phase information in the scattering amplitude, Q r, is lost. This is known as the phase problem. Mathematical tools have been developed, known as direct methods, that attempt to reconstruct the charge density from diffraction data based upon physical constraints on the inverse Fourier transform. For example, the charge density must be positive (for x-rays), and it must resemble a periodic array of atoms. In many cases one has a good guess at the crystal structure, owing largely to years of work by crystallographers who have assembled an extensive database. In these cases it is sufficient to refine a good guess directly against F (Q) 2 values using non-linear least squares methods, so long as the crystal structure can be quantified by a modest number of free parameters. For example, consider a 1D diatomic chain: We want to measure the position of atom B with respect to atom A, which we define with the fractional coordinate, x. F (h) = f A (h) + f B (h)e 2πihx I(h) F (h) 2 = f 2 A(h) + f 2 B(h) + 2f A (h)f B (h)cos(2πhx) So if we measure the series of diffraction peaks h = 1, 2, 3, 4..., we will find that their relative intensity is modulated with a periodicity dependent upon the fractional coordinate, x. Page 6 of 6