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

We present a model of localized spin dynamics at room temperature for the low-dimensional solid-state spin system, which contains small ensembles of magnetic nuclei (N ~ 40). The standard spin Hamiltonian (XXZ model) is the sum of the Zeeman term in a strong external magnetic field and the magnetic dipole interaction secular term. The ¹⁹F spins in a single crystal of fluorapatite [Ca₅(PO₄)₃F] have often been used to approximate a one-dimensional spin system. If the constant external field is parallel to the c axis, the 3D ¹⁹F system may be treated as a collection of many identical spin chains. When 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. Then we expand the density matrix in a set of eigenoperators of the Zeeman Hamiltonian. A system of coupled differential equations for the expansion coefficients then solved by straightforward numerical methods, and the fluorine NMR line shapes of fluorapatite for different chain lengths are calculated.