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Ptychography data analysis with SQUARREL
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Multimodal x-ray and electron microscopy of the Allende meteorite

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To derive quantitative elemental information from the complex (amplitude and phase) ptychography images, here, we introduced a new semiquantitative analysis method in x-ray ptychography, named Scattering QUotient Analysis to REtrieve the Ratio of ELements (SQUARREL). In this method, the complex ptychographic images are used to calculate the scattering quotient map (fq) (4648), which is defined as $fq≡ln(∣T(x,y;E)∣)ϕ(x,y;E)=∑iNi(x,y;E)βi(E)∑iNi(x,y;E)δi(E)=∑iNi(x,y;E)fIi(E)∑iNi(x,y;E)fRi(E)$. ∣T(x, y; E)∣ and ϕ(x, y; E) are the transmission magnitude image (0 ≤ ∣T∣ ≤ 1) and the phase retardation image (ϕ ≤ 0) of the complex transmission function, or T(x, y; E) = ∣ T(x, y; E) ∣ exp [iϕ(x, y; E)], that was acquired by ptychography at an x-ray photon energy (E). Ni is the atomic number density of the ith element in the sample. β and δ are the imaginary and real part of the complex refractive index (n) decrement, respectively. fIi and fRi are the imaginary and real part of complex atomic scattering factors, respectively, where n = 1 − δ + iβ ∝ (1 − fRi + ifIi). Note that the complex atomic scattering factors are averaged values along the projection through the sample and fq ≥ 0.

The scattering quotient is, in principle, independent of sample thickness variation since the thickness has been canceled out in the quotient (48, 49). As a result, the scattering quotient is especially suitable for studying inhomogeneous specimens where conventional analysis methods cannot distinguish thickness variations from changes in refractive indices of different compositions, since both thickness and composition variations contribute to changes in light absorption and phase retardation. The scattering quotient map images have been used as a previously unidentified contrast mechanism in materials (5052) and as an image segmentation and classification method for biological samples (48, 49). The key idea to SQUARREL is to convert a scattering quotient map to a two-element ratio map, Ra = Ra(fq; E), given a fixed amount of a third element as a priori knowledge. The two-element ratio conversion function (Ra) is derived by a direct comparison to a theoretical calculation, which is based on the tabulated complex refractive indices from The Center for X-Ray Optics (CXRO) or National Institute of Standards and Technology (NIST) Standard Reference Database (53) of the mixture in atomic %.

For example, given a priori knowledge of a possible compound or a mixture Z (Z = XpYqVr) with X, Y, V elements and with p, q, r atoms, the conversion function can be calculated to provide a two-element ratio, e.g., a q-to-p ratio at a given r. The mixture’s atomic number density reads NZ = NAV(pρX + qρY + rρV)/(pMX + qMY + rMV), where NAV is the Avogadro constant, ρX, Y, V are the individual constituent’s mass number density, and MX, Y, V are the individual constituent’s atomic mass number. We can then get the individual constituent’s atomic number density, NX = pNZ, NY = qNZ, NV = rNZ, to finish the theoretical scattering quotient calculation of the mixture Z. After choosing an appropriate x-ray photon energy (E), i.e., a horizontal line out from fig. S15, the two-element conversion function, Ra(E), behaves as a smooth and monotonic function, which correlates fq and Ra. Therefore, we can convert scattering quotient maps into two-element ratio maps.

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