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An Introduction to Synchrotron Radiation Techniques and Applications by Philip Willmott (z-lib org)

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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
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