To describe scattering from matter, an enormous number of electrons and their scattered waves need to be considered. Given that the work presented in this thesis exclusively deals with scattering at small angles 2θ ! 1, several simplifications can be introduced here.
First of all, a consequence of only considering scattering in the near-forward direction is that inelastic Compton scattering is comparatively weak and can be neglected [AlsNielsen, 2011, pp. 121-122]. Secondly, the polarization factor is approximately constant for small scattering angles. Within the small-angle approximation, dσdΩe
therefore is independent of the scattering angle and proportional tor20 [Feigin, 1987, p. 15][Feldkamp, 2011, p. 30].
In order to simplify further considerations we set dσdΩe
1. If one is interested in quantitative results, the exact values of equation 2.4 can be reintroduced. Lastly, since the typical wavelength of X-rays used for scattering experiments is in the order of ˚A, small angle scattering is insensitive to variations on atomic length-scales. From this follows that the intensity scattered by a single atom to small-angles only depends on the number of electrons, which is given by the atomic numberZ [Feigin, 1987, p. 16]. The arrangement of individual electrons within an atom does not affect the scattering profile in the forward direction.
Our goal is to derive an expression for the scattered intensity as the sum of the waves originating from all electrons that make up matter. We will also make use of the single-scattering approximation, known as the first Born approximation, in which subsequent interactions of scattered waves are not considered [Feigin, 1987, p. 5]. This approximation is valid, owing to the weak interaction of X-rays with matter.
Given that atomic distances do not matter for small-angle scattering, it makes sense to employ the concept of electron density. In this picture, matter is seen as a continuous distribution of electrons that is described by an electron density function ρprq. The amplitude of the secondary wave emitted at a position r therefore is proportional to the local electron density ρprq. Following the previous section, the total scattered amplitude Apqqfor a certain scattering vectorqis given as the superposition of all secondary waves.
Hence we need to determine the relative phase for each secondary wave. Just like in the previous section, the phase for a secondary wave depends on the relative position inside the sample from which it originates and can be determined according to equation 2.6 as exppiqrq. We use the exponential function to account for phase differences greater than 2π. Integration of ρprq over the entire scattering volume V, weighted by the phase factor
2.2 Small-angle X-ray scattering
Figure 2.5:Small-angle approximation. a) For elastic scattering, only q-vectors that end on a circle with radius k can be seen for a given ki. b) The curvature of the circle is negligible for small scattering angles and the two-dimensional slice of the reciprocal space perpendicular to ki is recorded in SAXS.
exppiqrq, then yields Apqq[Glatter, 1982, pp. 19-20]:
Apqq
»
ρprqeiqrdV Fpρprqq. (2.9) Mathematically, this integral resembles the Fourier transform of the electron density, Fpρprqq. As only the scattered intensityISAXSpqq can be measured in an experiment, the result of equation2.9 has to be squared [Glatter, 1982, p. 19]:
ISAXSpqq |Apqq|2 |Fpρprqq|2. (2.10) Equation 2.10 reveals an important result: the intensity distribution measured in the far-field resembles the square of the Fourier transform of the electron density distribution inside an object. Just like for a single electron, a microscopic differential cross section
dσ dΩ
pqq can be used to describe the distribution of the scattered intensity:
ISAXSpqq9 dσ
dΩ
pqq. (2.11)
From the properties of Fourier transforms it follows that equal intensity is scattered to q-vectors of opposite sign. This is also known as Friedel’s law [AlsNielsen, 2011, p. 290]:
dσ
When recording a single diffraction pattern, the incident radiation is fully described by the wave-vectorki. Both the direction and wavelength of the incident radiation are fixed.
As there is no energy transfer in elastic scattering, only q-vectors that end on a sphere with radius |kf| give rise to elastic scattering. Figure 2.5 a) illustrates this fact. For the case of scattering to very small angles, the curvature of this sphere can be neglected.
With the definition of ki along the z, and therefore qz direction here, only q-vectors with qz ! qx, qy fall into the SAXS regime. From this follows that in the small-angle
LT
Σ
Aillum
2LT D
a) b)
R
λ
Figure 2.6: Spatial coherence. a) The spatial coherence length, LT, can be calculated only from geometric parameters. For a source of size D and X-rays of wave-length λ it only depends on the distance away from the source R. b) In a typical small-angle X-ray scattering experiment, a corresponding coherence area πL2T is much smaller than the area illuminated by the beam Aillum. The recorded SAXS signal therefore is the incoherent sum of the intensities of many diffraction patterns arising from difference coherence areas.
approximation qz 0, only q-vectors in the slice perpendicular to ki are probed by a single SAXS measurement [AlsNielsen, 2011, p. 136][Paganin, 2006, p. 90]. The intensity of the SAXS pattern measured for ki p0,0, kq therefore is a function of two variables:
ISAXSpqx, qy,0q.
2.2.1 Spatial coherence
So far, we have assumed a perfect illumination. The derivations in section 2.1.2 require that the phase of the incident X-rays does not vary in the plane perpendicular to their propagation direction. This generally is not the case and a constant phase-relation be-tween two points across the X-ray beam is only assured up to a certain distance bebe-tween them. An X-ray beam can be characterized by its spatial - or transverse - coherence length, LT. It is a measure of the distance between two points for which the phase of the X-ray beam varies by less than π. An illustration is given in figure 2.6 a). If the waves emitted by two distinct sources are exactly in phase at one point, LT is defined as half the distance to the next point at which the waves are in phase again. The spatial coherence length depends on the wavelengthλof the X-rays, the size of the source,D, and the distance from the source, R. It can be calculated from these geometric parameters [AlsNielsen, 2011, pp. 25-27] as:
LT λ 2
R
D. (2.13)
The spatial coherence length therefore does not only depend on the X-ray beam, but rather the entire geometry of an experimental setup [Feldkamp, 2011, p. 32]. For a symmetric source we can define a coherence area as Acoh πL2T. In the case of an asymmetric source, as is often the case for synchrotron sources, Acoh is asymmetric accordingly. The coherence area is the two dimensional analogue to the coherence length. For typical