  We study muon-spin rotation (muSR) spectra in the mixed phase of highly
anisotropic layered superconductors, specifically Bi_2+xSr_2-xCaCu_2O_8+delta
(BSCCO), by modeling the fluid and solid phases of pancake vortices using
liquid-state and density functional methods. The role of thermal fluctuations
in causing motional narrowing of muSR lineshapes is quantified in terms of a
first-principles theory of the flux-lattice melting transition. The effects of
random point pinning are investigated using a replica treatment of liquid state
correlations and a replicated density functional theory. Our results indicate
that motional narrowing in the pure system, although substantial, cannot
account for the remarkably small linewidths obtained experimentally at
relatively high fields and low temperatures. We find that satisfactory
agreement with the muSR data for BSCCO in this regime can be obtained through
the ansatz that this ``phase'' is characterized by frozen short-range
positional correlations reflecting the structure of the liquid just above the
melting transition. This proposal is consistent with recent suggestions of a
``pinned liquid'' or ``glassy'' state of pancake vortices in the presence of
pinning disorder. Our results for the high-temperature liquid phase indicate
that measurable linewidths may be obtained in this phase as a consequence of
density inhomogeneities induced by the pinning disorder. The results presented
here comprise a unified, first-principles theoretical treatment of muSR spectra
in highly anisotropic layered superconductors in terms of a controlled set of
approximations.
