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a consequence of the refractive index of materials in the x-ray regime being less than unity. The atomic scattering factor (also called the atomic form factor) f describes the total scattering amplitude of an atom as a function of sin θ/λ (which is proportional to the scattering vector Q) and is expressed in units of the scattering amplitude produced by one electron. It falls off quasimonotonically with increasing Q . As already stated, in the forward scattering direction, f = f (0) is nothing more than the electron density, integrated over the atom’s electron cloud, and is therefore equal to Z , the atomic number. Note, however, that in the case of an ionic species, f (0) will differ from Z by an amount equal to the amount of electron transfer to or from neighbouring ions. Hence cations have f (0)-values less than Z and anions values greater than Z . It is conventionally assumed that the electron density of an atom is roughly spherically symmetric and hence f depends only on the magnitude of Q and not on its orientation relative to the scattering atom. Values for f (sin θ/λ) can be either directly found in the International Tables for Crystallography , or can be calculated using nine tabulated coefficients and the expression f (sin θ/λ) = 4∑ i=1 ai exp(−bi sin2 θ/λ2) + c. (2.8) Examples of the dependence of f on sin θ/λ are given in Figure 2.8. Note that, for both La and B, f (0) does indeed equal Z . 2.5.2 Correction Terms for the Atomic Scattering Factor The above description assumes that the electrons are essentially free, that is, their move- ment is not damped by the fact that they are bound in an atom. In reality, the response of bound electrons in an atom, molecule or bulk material, to the electric field of the incoming x-rays is such that additional energy-dependent terms must be added to more completely describe the structure factor. We consider these now. Electrons bound in material have discrete energy levels. If the incident x-ray photon has an energy much less than the binding energy of the tightly-bound core electrons, the response (i.e. oscillation amplitude) of these electrons to the driving EM-field will be damped because they are bound. The real part of the elastic scattering factor, which we The Interaction of X-rays with Matter 23 0 0.2 0.4 0.6 0.8 sinθ/λ 20 30 40 50 60 f 0 0.2 0.4 0.6 0.8 sinθ/λ 1 2 3 4 5 La B Figure 2.8 The real parts of the atomic scattering factors for La and B as a function of sin θ/λ. e− scatteredincoming Figure 2.9 Elastic scattering of an x-ray by a bound electron. The amplitude of the scattered wave decreases as one approaches a discrete energy level of an electron bound in an atom. Far from resonant edges, the electrons are essentially free and the damping term f ′ is insignificant. henceforth refer to as f1 (in order to distinguish it from our ‘start’ definition f for a free electron) will therefore be reduced by an additional term f ′, that is f1 = f − f ′. (2.9) In contrast, at photon energies far above absorption resonances, the electrons are essen- tially free and f ′ = 0 (see Figure 2.9). Close to an absorption edge, the x-rays will be partially absorbed. After a delay, some x-radiation is re-emitted, which interferes with the directly (but also elastically) scattered part, thereby altering both the phase and the amplitude. This is expressed in the description of the atomic form factor by introducing a second additional term accounting for the increasing phase lag relative to the driving electromagnetic field of the x-rays as the photon energy approaches an absorption edge. Note that the interference between 24 An Introduction to Synchrotron Radiation 100 1000 10000 photon energy [eV] −10 0 10 20 f 1 K L 0.01 0.1 1 10 f 2 Figure 2.10 The real (f1 = f − f ′) and imaginary (f2 = f ′′) components of the atomic form factor for Si as a function of photon energy. Note that f1 converges to a constant value at photon energies far above absorption energies equal to the number of electrons in the atom, Z = 14. f2, on the other hand, falls away as E−2. the incoming and emitted radiation causes energy loss (i.e. absorption), and the new second additional term if ′′ is imaginary. The mathematical justification for this is given in Section 2.6.3. This term shows sharp increases at absorption edges, but otherwise falls off as E−2 (note the gradient of −2 in the double logarithmic plot of Figure 2.10). More precisely, f ′′ is related to the photoabsorption cross-section σa at that photon energy by f ′′(0) = σa 2r0λ . (2.10) From Figure 2.4, we can see that, for example, at 12.4 keV photon energy (1 Å wave- length), the photoabsorption cross-section of Ba is of the order of 10−24 m2 and hence f ′′ ∼ 1. We refer to the imaginary component of the total (i.e. complex) atomic scattering factor as f2 = f ′′, and hence ftot = f1 + if2. (2.11) The atomic form factors for forward scattering for Si are shown in Figure 2.10. We can qualitatively understand the curves as follows. At the binding energies of the K - and L-shells, there are sharp decreases in f1 and corresponding sharp increases in f2. Mid between these resonances, coupling of the x-rays to the material is inefficient, and f1 increases again. At high photon energies, there is only a weak interaction of the x-ray photons with the atoms, and f1 ≈ Z = 14 and f2 (which, remember, represents absorption), falls off to very small values. The Interaction of X-rays with Matter 25 2.6 The Refractive Index, Reflection and Absorption 2.6.1 The Refractive Index The index of refraction n at a wavelength λ describes the response of electrons in matter to electromagnetic radiation, and is a complex quantity which we express as n = nR + inI , (2.12) In the x-ray regime, n is related to the atomic scattering factors of the individual atoms in a material by n = 1 − r0 2π λ2 ∑ i Ni fi (0), (2.13) where Ni is the number of atoms of type i per unit volume and fi (0) is the complex atomic scattering factor in the forward direction of the i th atom. Because the phenomena of refraction and absorption both depend on the response of the electrons in a material to the oscillating electric field of an electromagnetic wave prop- agating through that material, the real and imaginary components of f (0) are coupled,3 as can be clearly seen in Figure 2.10. 2.6.2 Refraction and Reflection We have argued that when electromagnetic radiation passes through a solid, it interacts with the electrons within the material. For visible light, this normally means a reduction of the group velocity by a factor equal to the real part of the refractive index n of that material. This can, for example, be used to bend or focus light with optical lenses. But n is a function of the wavelength of the radiation; in other words, the material disperses the light. This is why rainbows occur – they are manifestations of refraction and reflection (and are not , as commonly misconceived, a diffraction phenomenon). Normally, as an absorption maximum of a material is approached from lower frequencies, the refractive index increases. Therefore blue light is refracted more by water or quartz than is red light. Past the absorption maximum, however, the refractive index drops to values less than unity. In such cases, it is the phase velocity that has the value c/n . This is greater than the velocity of light, but it is the group velocity that carries the energy, and this remains below c – the laws of relativity remain inviolate! This is the region of so-called anomalous dispersion . This means that (a) as radiation enters a material with a refractive index less than unity, it is refracted to shallower angles to the interface, and (b) the scattered wave is phase-shifted by π radians with respect to the incident wave. We express the complex refractive index as n = 1 − δ + iβ, (2.14) 3 The precise relationship between these properties is described by the Kramers–Kronig relation, though it lies