Penetration Depth Anisotropy in MgB$_{2}$ measured by Small-Angle Neutron Scattering

Traditionally the anisotropy of a type-II superconductor is described either by $\gamma _{\lambda }=\lambda _{c}$/$\lambda _{ab}$ or $\gamma _{H}$~=~$H_{ab}$/$H_{c}$~=~$\xi _{ab}$/$\xi _{c}$. with the two considered to be identical. However, in materials with anisotropic gaps this is generally not the case. MgB$_{2}$ represents an extreme case in which $\gamma _{\lambda }$~$\ne $~$\gamma _{H}$. While there is consensus on the value of $\gamma _{H}(T)$, measurements of $\gamma _{\lambda }$ are still contradictory. Here we demonstrate a novel use of small-angle neutron scattering to determine $\gamma _{\lambda }$ in MgB$_{2}$, by measuring the misalignment between the applied field and the direction of the flux-line lattice as the field is rotated between the $c$ axis and the basal plane. Using a two-band/two-gap model we can fit the angular dependence of the misalignment, yielding $\gamma _{\lambda }$~=~1.1 $\pm $ 0.2 at 4.9 K and 0.4 T.