Inverse scattering problem

• allen1998 : Allen, L. J., Josefsson, T. W., & Leeb, H. (1998). Obtaining the Crystal Potential by Inversion from Electron Scattering Intensities. Acta Crystallographica Section A: Foundations of Crystallography, 54(4), 388–398. doi
• spence2020 : Donatelli, J. J., & Spence, J. C. H. (2020). Inversion of Many-Beam Bragg Intensities for Phasing by Iterated Projections: Removal of Multiple Scattering Artifacts from Diffraction Data. Physical Review Letters, 125(6), 65502. doi
• shen1998 : Shen, Q. (1998). Solving the phase problem using reference-Beam X-ray diffraction. Physical Review Letters, 80(15), 3268–3271. doi

Allen1998

Allen, L. J., Josefsson, T. W., & Leeb, H. (1998). Obtaining the Crystal Potential by Inversion from Electron Scattering Intensities. Acta Crystallographica Section A: Foundations of Crystallography, 54(4), 388–398. doi

Non linear system to solve from imposing the unitarity of $\bb S$ and imposing diagonal relationship of $\bb S=e^{2j\pi\bb A t}$ : $$\begin{eqnarray} \sum_f |S_{gf}||S_{hf}|\cos(\theta_{gf}-\theta_{hf}) &=& \delta_{g,h} ~\mbox{,}~ g \neq h\\ \sum_f |S_{gf}||S_{hf}|\sin(\theta_{gf}-\theta_{hf}) &=& 0 ~\mbox{,}~ g \neq h\\ \big(2K/jt\big)\big(ln S\big)_{gg} &=& -\big( \bb k_t+\bb g \big)^2 \\ \gamma^{(i)} t &=& \big(\theta^{(i)}+2n^{(i)}\pi\big) \end{eqnarray}$$

Where $S_{gf}=|S_{gf}|e^{j\theta_{gf}}$ and $ln S = \bb C \Gamma \big[j\gamma^{(i)}t\big] \bb C^{\dagger}$ and $t$ the thickness (assuming it is known). There are $2N^2+2N$ equations, $N(N-1)+2N$ independent equations and $N^2+N$ unknowns $\theta_{gf}$, $\theta_{gg}$ and $n^{(i)}$.

The measured intensity matrix $\bb M$ = $|\bb S|^2$ is simulated. The inverse scattering problem is then solved through the non linear equations using global Newton's method for :

• systematic row ZnS[111] at 100keV@100A (and 400A) for 5, 7 and 13 beams.
• zone axis case GaAs[110] at 400keV@112A.

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Spence2020

Inversion of Many-Beam Bragg Intensities for Phasing by Iterated Projections

The flowcharts for recovering the electrostatic potential from magnitude data $\bb M$. The methods differ from how the amplitude matrix $\bb S$ is computed from the lambda matrix $\bb A$ and from how the structure matrix $\tilde{\bb A}$ is updated from the corrected matrix $\tilde{\bb S}$.

Tests were conducted on $GaAs$ and spinel showing Pearson correlation coefficients with reference structure factors above 0.85 for 100keV and 100nm thickness.

Shen1998

Solves the phase problem for three-beam interaction in a 2-beam rotation experiment. The main beam is $\bb D_0$ the bragg excited beam is $\bb D_G$ and the measured diffracted beams are $\bb D_H$. $F_G, F_H, F_{H-G}$ are the structure factors and $R(\theta)$, $\nu(\theta)$ are measured amplitude and phase of $D_G/D_0$ with $\theta$ the rocking curve angle about $\bb G$. The anomalous integrated diffracted intensities I_H can be used to work out the phase $\delta_{HG}$.

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