Scanning tunneling images of carbon nanotubes frequently show electron distributions which break the local six-fold symmetry of the graphene sheet. We present a theory of these images which relates the anisotropies to off diagonal correlations in the single particle density matrix induced by elastic scattering from tube defects. The theory reveals that there are three general effects which one can associate with elastic scattering from defects. Any defect can be characterized by a matrix of reflection coefficients which define its signature in the tunneling image. We provide a theory for tunneling images with broken translational symmetry, "primitive" broken symmetry images with broken rotational symmetry but no broken translational symmetry, and a novel "semiconductor effect" in which the symmetry of the tunneling image switches with the sign of the tunnel bias.
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