Small Angle Neutron Scattering Studies of the Vortex Lattice in CeCoIn$_5$ with $H \perp c$

We report on small-angle neutron scattering measurements on the vortex lattice (VL) in the mixed state of CeCoIn$_5$ with the magnetic field ($H$) along $[100]$ and $[110]$. For both field orientations a distorted hexagonal VL is observed, reflecting the penetration depth anisotropy of the screening current plane. With $H \parallel [100]$ the VL is oriented with Bragg reflections along the [001]-axis at all fields. For $H \parallel [110]$ the same VL orientation is observed at low fields, followed by a $90^{\circ}$ first-order reorientation transition as $H$ is increased. We attribute this behavior to Fermi surface anisotropy coupled with non-local effects. For $H \parallel [100]$ we obtain the field dependence of the form factor ($|F|^2$) both within (50~mK) and outside (350~mK) the magnetic $Q$-phase. At both temperatures $|F|^2$ varies with $H$ in a manner similar to $H \parallel [001]$ [J.S. White {\em et al.}. New J. Phys. {\bf 12}, 023026 (2010)], due to the competition between Pauli paramagnetism and the antiparallel spin alignment of $d$-wave pairing giving rise to ``magnetized" VL cores.