  The $g$ factors of $11/2^-$ isomers in semimagic $^{109}$Sn and $^{111}$Sn
(isomeric lifetimes $\tau = 2.9(3)$ ns and $\tau = 14.4(7)$ ns, respectively)
were measured by an extension of the Time Differential Perturbed Angular
Distribution technique, which uses \LaBr detectors and the hyperfine fields of
a gadolinium host to achieve precise measurements in a new regime of
short-lived isomers. The results, $g(11/2^-; {^{109}\textrm{Sn}}) = -0.186(8)$
and $g(11/2^-; {^{111}\textrm{Sn}}) = -0.214(4)$, are significantly lower in
magnitude than those of the $11/2^-$ isomers in the heavier isotopes and depart
from the value expected for a near pure neutron $h_{11/2}$ configuration.
Broken-symmetry density functional theory calculations applied to the sequence
of $11/2^-$ states reproduce the magnitude and location of this deviation. The
$g(11/2^-)$ values are affected by shape core polarization; the odd $0h_{11/2}$
neutron couples to $J^{\pi}=2^+,4^+,6^+...$ configurations in the
weakly-deformed effective core, causing a decrease in the $g$-factor
magnitudes.
