Graphene's structure bears on both the material's electronic properties and
fundamental questions about long range order in two-dimensional crystals. We
present an analytic calculation of selected area electron diffraction from
multi-layer graphene and compare it with data from samples prepared by chemical
vapor deposition and mechanical exfoliation. A single layer scatters only 0.5%
of the incident electrons, so this kinematical calculation can be considered
reliable for five or fewer layers. Dark-field transmission electron micrographs
of multi-layer graphene illustrate how knowledge of the diffraction peak
intensities can be applied for rapid mapping of thickness, stacking, and grain
boundaries. The diffraction peak intensities also depend on the mean-square
displacement of atoms from their ideal lattice locations, which is
parameterized by a Debye-Waller factor. We measure the Debye-Waller factor of a
suspended monolayer of exfoliated graphene and find a result consistent with an
estimate based on the Debye model. For laboratory-scale graphene samples,
finite size effects are sufficient to stabilize the graphene lattice against
melting, indicating that ripples in the third dimension are not necessary.Comment: 10 pages, 4 figure
