  Employing the relativistic coupled-cluster method, comparative studies of the
parity non-conserving electric dipole amplitudes for the $7s \ ^2S_{1/2}
\rightarrow 6d \ ^2D_{5/2}$ transitions in $^{210}$Fr and $^{211}$Fr isotopes
have been carried out. It is found that these transition amplitudes, sensitive
only to the nuclear spin dependent effects, are enhanced by more than 3 orders
compared to the low-lying $S-D_{5/2}$ transitions in Ba$^+$ and Ra$^+$ owing to
the very large contributions from the electron core-polarization effects in Fr.
This translates to a relatively large and, in principle, measurable induced
light shift, which would be a signature of nuclear spin dependent parity
nonconservation that is dominated by the nuclear anapole moment in a heavy atom
like Fr. A plausible scheme to measure this quantity using the Cyclotron and
Radioisotope Center (CYRIC) facility at Tohoku University has been outlined.
