  We report a fourfold improvement in the determination of nuclear magnetic
moments for neutron-deficient isotopes of francium-207--213, reducing the
uncertainties from 2% for most isotopes to 0.5%. These are found by comparing
our high-precision calculations of hyperfine structure constants for the ground
states with experimental values. In particular, we show the importance of a
careful modeling of the Bohr-Weisskopf effect, which arises due to the finite
nuclear magnetization distribution. This effect is particularly large in Fr and
until now has not been modeled with sufficiently high accuracy. An improved
understanding of the nuclear magnetic moments and Bohr-Weisskopf effect are
crucial for benchmarking the atomic theory required in precision tests of the
standard model, in particular atomic parity violation studies, that are
underway in francium.
