In this article, we present the novel application of the nuclear spin bath model and the cluster correlation expansion method on studying the matrix material structure via embedded electron spin decoherence. Profiles of embedded electron spin decoherence under the Carr-Purcell-Meiboom-Gill dynamical decoupling pulse series in a model system for organic solids (malonic acid) are calculated for different structures. Resulting decay profiles exhibit a strong correlation to the variations of an adjacent proton environment among them. In addition, the decoherence behavior of embedded spin in proton spin bath(s) of organic solids is found to be significantly different from bath models with other nuclei through the violation of the even-odd pulse parity, which characterizes the influence of large dipolar coupling between protons at the quantum level. Theoretical predictions of decoherence profiles in polycrystalline, the relative distribution of Hahn echo signal decay time scales among single crystal orientations, and the reduction in Hahn echo signal decay time scale by disorder are positively verified by experiments.

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This setting would also comply with the early concept of the diffusion barrier,12 which states that per distance the very close protons do not contribute significantly to decoherence through the bath dynamics channel.

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Notice that by replacing the spin operator Ii outside the cluster with its expectation value Ii=I|Ii|I, one still retains part of their influence through Iiz. In LCE, this corresponds to the renormalization of the spin propagator.16 

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Note that these statistics are calculated from the fitting results for the profile of each orientation, while the inset in Fig. 3(d) is calculated from the averaged profile. Certain deviation does exist between these two.

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