Evidence for Two-Dimensional Thermal Fluctuations of the Vortex Structure in ${\mathrm{Bi}}_{2.15}{\mathrm{Sr}}_{1.85}{\mathrm{CaCu}}_2{O}_{8+\delta }$ from Muon Spin Rotation Experiments

In the mixed state of ${\mathrm{Bi}}_{2.15}{\mathrm{Sr}}_{1.85}{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ the temperature dependence of the muon spin rotation ( $\mu \mathrm{SR}$) linewidth is found to have a strong field dependence, which is attributed to the effect on the line shapes resulting from thermal fluctuations of the vortex positions. This provides important information concerning the form of the fluctuations and moreover gives a measure of the anisotropy parameter $\gamma$. By combining this result with a $\mu \mathrm{SR}$ determination of the penetration depth $\lambda \left(T\right)$ and the melting line $B_m\left(T\right)$, we are further able to obtain the Lindemann number $c_L$.