Magnetic momentum density, Fermi surface and directional magnetic Compton profiles in LaSr$_{2}$Mn$_{2}$O$_{7}$ and La$_{1.2}$Sr$_{1.8}$Mn$_2$O$_7$

We have carried out first principles, all-electron computations of the magnetic momentum density $\rho_{mag}({\mathbf p})$ and magnetic Compton profiles (MCPs) for momentum transfer along the [100], [001], and [110] directions in LaSr$_2$Mn$_2$O$_7$ and La$_{1.2}$Sr$_{1.8}$Mn$_2$O$_7$ within the local spin density approximation (LSDA) based band theory framework. Parallel measurements of these three MCPs from a single crystal of La$_{1.2}$Sr$_{1.8}$Mn$_2$O$_7$ at 5 K in a magnetic field of 7 T are also reported. $\rho_{mag}({\mathbf p})$ is shown to contain distinct peaks arising from the occupied majority-spin $t_{2g}$ electrons and to display images of the Fermi surface (FS) in the first and higher Brillouin zones (BZs). The overall shape of the MCPs, $J_{mag}(p_z)$, obtained by integrating $\rho_{mag}({\mathbf p})$ over $p_x$ and $p_y$, is found to be dominated by the majority-spin $t_{2g}$ states. The FS-related fine structure in the MCPs is however substantial only in the [100] MCP, which contains features arising from the large majority-spin hole sheets. The overall shapes and widths of the experimental MCPs along all three directions investigated are in reasonably good accord with theoretical predictions, although some discrepancies indicating inadequacy of the LSDA in treating the magnetic states can be identified. We discuss details of the FS-related signatures in the first and higher BZs in the [100] MCP and show that high resolution magnetic Compton scattering experiments with a momentum resolution of 0.1 a.u. FWHM (full-width-at-half-maximum) or better will be necessary to observe this fine structure. We comment also on the feasibility of using positron annihilation spectroscopy in this connection.