4. Quantitative X-ray dark-field imaging
4.2. Implications of SAXS theory for the dark-field signal
unique for each individual experimental setup. The purple curve in gure 4.5 represents such a reference pattern. If a sample is placed between G1 and G2, a fraction of the beam is scattered by a scattering vector Q~ according to the dierential scattering cross-sectiondσ/dΩ(Q)~ (see equation 4.2). The interferometer is not sensitive to scattering in y-direction because the grating lamellae run along the y-axis. We will therefore neglect the y-component Qy in the following considerations. Furthermore, the z-component Qz vanishes due to the assumption of small scattering angles [62, 66].
Figure 4.5.: This gure depicts the general layout of a Talbot-Lau interferometer and the observed reference pattern and a scattered pattern. The interferometer consists of a source of intensity I0, three gratings and a detector pixel. The purple solid curve represents the unscattered reference pattern. A sample positioned at a distance dS,G2 to grating G2 interacts with the X-rays. The orange dashed curve illustrates the pattern originating from X-rays scattered in x-direction at an angle 2θ. Both curves are shifted along the x-axis by ∆x.
The scattered X-ray fraction gives rise to a periodic pattern similar to the reference pattern of the unscattered beam given by equation (4.19). In contrast to the reference pattern, this scattered pattern has less intensity, and it is slightly shifted by∆x along the x-axis due to scattering in this direction. An exemplary pattern of scattered X-rays is illustrated by the orange dashed curve in gure 4.5. It depends on Qx and is formulated as follows:
Is(Qx, x) I0
=a0tdσ
dΩ(Qx) +a1tdσ
dΩ(Qx) cos 2π
p2
(x+ ∆x)
. (4.20)
4. Quantitative X-ray dark-field imaging
The dierential scattering cross-section accounts for the fraction of intensity that is scat-tered byQx, while ∆xaccounts the scattered pattern's shift in x-direction. In the follow-ing, we refer to Is as the scattered pattern denoted by the index s. Here, the thickness of the sample t is introduced in order to account for extended objects. In equation 4.2, dσ/dΩof a single object is normalized to its volume. Hence, the scattered fraction of the whole sample is given by a multiplication of the dierential cross-section with the sample thickness. X-ray attenuation is here neglected for reasons of simplicity. However, it is easily included by multiplication of equation 4.20 with an exponential term according to the Beer-Lambert law as stated in equation 2.14.
Besides being scattered, a certain fraction of the beam is transmitted, and superimposes the scattered pattern at the site of the detector. The pattern Ius of this unscattered fraction is obtained by multiplying the reference pattern stated in equation 4.19 with a factor of (1−σt). We obtain the superposition Is+us by adding the unscattered pattern Ius to the scattered patternIs stated in equation 4.20. This procedure results in following representation for the superposition Is+us:
Is+us(Qx, x)
The superposition of scattered and unscattered pattern depends on Qx. Thus, we inte-grate Is+us(Qx, x) over the full Qx-range because all scattering vectors contribute to the total observed intensity patternItot:
48
4.2. Implications of SAXS theory for the dark-field signal
Under the assumption that the dierential scattering cross-section is an even function, i.e.
dσ
dΩ(Qx) = dΩdσ(−Qx), the cosine term cos [x+ ∆x] can be split up according to following identity:
cos [x+ ∆x] + cos [x−∆x]
2 = cos [x] cos [∆x] . (4.23)
Using equation 4.23 and the denition of the total scattering cross-section σ given by equation 4.13, equation 4.22 is rewritten as follows:
Itot(x)
A detailed derivation of equation 4.24 is given in appendix A. Here, we also used the following relation between ∆x and the scattering vector Qx under the small-angle ap-proximation for the scattering angle2θ:
4. Quantitative X-ray dark-field imaging
∆x= sin (2θ)dS,G2 ≈ λQxdS,G2
2π . (4.25)
This relation is derived based on simple geometric considerations shown in gures 4.5 and 4.2. Equation 4.24 can be further simplied, and using the correlation functionG(x)(see equation 4.15) nally yields the following expression for the Itot:
Itot(x)
Remarkably, this resembles the reference pattern given by equation 4.19 with just a slight modication. An additional term including the correlation functionG(x)is multiplied to the cosine term. This term is equal or less than unity, due to the denition of G(x), and it consequently reduces the amplitude, or visibility, of the observed pattern Itot.
Here, we emphasize that the argument λdS,G2/p2 of the correlation function in equa-tion 4.26 corresponds to the beam separaequa-tion of the rst diracequa-tion order as reported in references [71, 72]. From this point of view, the sample is scanned by a split beam and spatial correlations within the sample are probed at the length scale of this beam separa-tion. We refer to the beam separation asξcorr in the remainder of this work. If the sample is placed in front of the grating G1, the beam separation equals to ξcorr = λdG0,S/p0. Here, p0 is the period of G0, anddG0,S represents the distance between the source grating and sample. This corresponds to ndings of previous publications [26, 27, 73]. Scanning the sample with a split beam is another characteristic that grating-based X-ray dark-eld imaging shares with SESANS. Here, the neutron's spin wave is split into diraction orders by the magnetic elds and the sample is scanned with this split beam [72]. Figure 4.6 illustrates how a sample is scanned by two beams spatially separated by ξcorr.
The last line of equation 4.24 implies that a grating interferometer transforms the scatter-ing functiondσ/dΩback into the system's correlation space. According to equation 4.15, this results in a direct relation between the dark-eld signal and the correlation function G(ξcorr).
The dark-eld signal is dened as visibility (see equation 2.29) obtained with the sample normalized to the reference pattern's visibility. Dividing equation 4.26 by equation 4.19 results in the following expression for the dark-eld signal DF:
50
4.2. Implications of SAXS theory for the dark-field signal
x z
ξcorr ξcorr
dS,G2 dS,G0
Figure 4.6.: This gure illustrates how a sample is scanned by a split beam in grating-based X-ray imaging. The gratings split the beam into the rst diraction order. The parameter ξcorr represents the spatial separation of these diraction orders. Correlations within the sample are probed on the length scale of the beam separation.
DF(ξcorr) = 1−σt+σtG(ξcorr)
= exp [σt(G(ξcorr)−1)]
= exp
Z
VSample
σ(z0) (G(ξcorr, z0)−1)dz0
. (4.27)
The second line of equation 4.27 represents the case of scattering by many particles [74, 75], while the third line represents the case of inhomogeneous samples[27]. Here, the sample thickness t is replaced by a line integral through the sample equivalent to the line integral given in equation 2.24 in chapter 2.28. A similar result has been obtained in reference [76] under the assumption of a Gaussian dierential scattering cross-section and the limitation to single scattering. Thus, the presented mathematical formalism provides a more general understanding of the dark-eld signal.
Equation 4.27 directly relates the dark-eld signal to the real-space correlation function G(ξcorr) of the sample's microstructure. The correlation lengthξcorr determines at which specic length correlations within the sample are probed by dark-eld measurements. As already outlined in the previous text, ξcorr is calculated as follows:
4. Quantitative X-ray dark-field imaging
Figure 4.7.: This gure illustrates the experimental procedure to measure the correlation functionG(x)with a laboratory-based X-ray grating interferometer. The sample is placed either betweenG0 andG1, or betweenG1 andG2, and the correlation lengthξcorr is tuned by varying the sample's position. For each sample position, a dark-eld image is acquired.
ξcorr =
λdS,G2
p2 , if the sample is placed between G1 and G2 ,
λdG0,S
p0 , if the sample is placed between G0 and G1 . (4.28) According to equation 4.28, the correlation function G(x) can be evaluated for a certain range of the correlation length by tuning the parameter ξcorr within the experimentally feasible limitations.