Ptychography data analysis with SQUARREL

YL Yuan Hung Lo CL Chen-Ting Liao JZ Jihan Zhou AR Arjun Rana CB Charles S. Bevis GG Guan Gui BE Bjoern Enders KC Kevin M. Cannon YY Young-Sang Yu RC Richard Celestre KN Kasra Nowrouzi DS David Shapiro HK Henry Kapteyn RF Roger Falcone CB Chris Bennett MM Margaret Murnane JM Jianwei Miao

This protocol is extracted from research article:

Multimodal x-ray and electron microscopy of the Allende meteorite

**
Sci Adv**,
Sep 20, 2019;
DOI:
10.1126/sciadv.aax3009

Multimodal x-ray and electron microscopy of the Allende meteorite

Procedure

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 (*f*_{q}) (*46*–*48*), which is defined as ${f}_{\mathrm{q}}\equiv \frac{\text{ln}(\mid T(x,y;E)\mid )}{\mathrm{\varphi}(x,y;E)}=\frac{{\sum}_{i}{N}_{i}(x,y;E){\mathrm{\beta}}_{i}(E)}{{\sum}_{i}{N}_{i}(x,y;E){\mathrm{\delta}}_{i}(E)}=\frac{{\sum}_{i}{N}_{i}(x,y;E){f}_{\mathrm{I}i}(E)}{{\sum}_{i}{N}_{i}(x,y;E){f}_{\mathrm{R}i(E)}}$. ∣*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 [*i*ϕ(*x*, *y*; *E*)], that was acquired by ptychography at an x-ray photon energy (*E*). *N _{i}* is the atomic number density of the

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 (*50*–*52*) 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, *R*_{a} = *R*_{a}(*f*_{q}; *E*), given a fixed amount of a third element as a priori knowledge. The two-element ratio conversion function (*R*_{a}) 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* = *X _{p}Y_{q}V_{r}*) with

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