  Whereas disclination defects are energetically prohibitive in two-dimensional
flat crystals, their existence is necessary in crystals with spherical
topology, such as viral capsids, colloidosomes or fullerenes. Such a
geometrical frustration gives rise to large elastic stresses, which render the
crystal unstable when its size is significantly larger than the typical lattice
spacing. Depending on the compliance of the crystal with respect to stretching
and bending deformations, these stresses are alleviated by either a local
increase of the intrinsic curvature in proximity of the disclinations or by the
proliferation of excess dislocations, often organized in the form of
one-dimensional chains known as "scars". The associated strain field of the
scars is such to counterbalance the one resulting from the isolated
disclinations. Here, we develop a continuum theory of dislocation screening in
two-dimensional closed crystals with genus one. Upon modeling the flux of scars
emanating from a given disclination as an independent scalar field, we
demonstrate that the elastic energy of closed two-dimensional crystals with
various degrees of asphericity can be expressed as a simple quadratic function
of the screened topological charge of the disclinations, both at zero and
finite temperature. This allows us to predict the optimal density of the excess
dislocations as well as the minimal stretching energy attained by the crystal.
