On low-dimensional models at NMR line shape analysis in nanomaterial systems

Recently, considerable progress has been made in studies on many-spin mechanisms responsible for the wings of NMR absorption line [1]. Now we present a model of localized spin dynamics for the low-dimensional solid-state spin... [ view full abstract ]

Recently, considerable progress has been made in studies on many-spin mechanisms responsible for the wings of NMR absorption line [1]. Now we present a model of localized spin dynamics for the low-dimensional solid-state spin system, which contains small ensembles of magnetic nuclei (N ~ 20). The standard spin Hamiltonian is the sum of the Zeeman term and the magnetic dipole interaction term. It is possible to truncate the interaction among the spins in a strong external magnetic field, keeping only the secular terms. The ¹⁹F spins in a single crystal of fluorapatite (Ca₅F(PO₄)₃) at room temperature have often been used to approximate a one-dimensional spin system. The ¹⁹F NMR line in fluorapatite is split up into three lines if the constant external field is parallel to the c axis. In this crystal orientation, the 3D ¹⁹F system may be treated as a collection of many identical spin chains. Following direct-product formalism theory [2] we make the next approximation, namely, considering the longitudinal part of the secular term, we suggest that transverse component of a spin in a certain site rotates in a constant local magnetic field. This field changes if the spin jumps to another site. On return, this spin continues to rotate in the former field. Such a simplified description is necessary to obtain a computationally tractable model. We expand the density matrix in a set of eigenoperators of the Zeeman Hamiltonian. Then we obtain a system of coupled differential equations for the expansion coefficients. The system of equations is then solved by straightforward numerical methods, and the fluorine NMR line shapes of fluorapatite for different chain lengths are calculated. The structure in the resonances of one-dimensional (Ca₅F(PO₄)₃) spin system is explained. The separation D between the satellites in fluorapatite was found 3.50 Oe in a good agreement with experiment [3] (the experimental value is 3.64 ± 0.20 Oe). References: [1] V.E. Zobov, M.M. Kucherov, JETP 124, 151 (2017). [2] A. A. Nevzorov, J. H. Freed, J. Chem. Phys. 115, 2401 (2001). [3] W. Van der Lugt, W.J. Caspers, Physica (Amsterdam) 30, 1658 (1964)