  We investigate the asymmetry of the scattering cross section of radiation
around Ly$\alpha$ by atomic hydrogen, which may be applied to analyses of
scattering media with high column neutral hydrogen densities including damped
Ly$\alpha$ absorption systems of quasars. The exact scattering cross section is
given by the Kramers-Heisenberg formula obtained from the fully quantum
mechanical second-order time dependent theory, where, in the case of hydrogen,
each matrix element is given in a closed analytical form. The asymmetric
deviation of the scattering cross section from the Lorentzian near the line
center is computed by expanding the Kramers-Heisenberg formula in terms of
$\Delta\omega /\omega_{Ly\alpha}$,where $\omega_{Ly\alpha}$ is the angular
frequency of the Ly$\alpha$ transition and $\Delta\omega$ is the deviation of
incident radiation from $\omega_{Ly\alpha}$. To the first order of
$\Delta\omega/\omega_{Ly\alpha}$, we obtain $\sigma(\omega) = \sigma_T (0.5
f_{12} \omega_{Ly\alpha}/\Delta\omega)^2 (1-1.79\Delta\omega /
\omega_{Ly\alpha})$, where $\sigma_T$ is the Thomson scattering cross section
and $f_{12}=0.4162$ is the oscillator strength for the Ly$\alpha$ transition.
With this deviation, the line center of the damped wing profile apparently
shifts blueward of the true Ly$\alpha$ line center. In the case of a
dampedLy$\alpha$ system with a H I column density $5\times 10^{21}{\rm
cm^{-2}}$, the apparent line center shift relative to the true center amounts
to $0.2{\rm \AA}$ resulting in an underestimation of redshift by $\Delta z\sim
10^{-4}$. A measurable underestimation by an amount of $\Delta z\sim 10^{-3}$
is expected for absorbing systems with $N_{HI}\ge 4\times 10^{22}{\rm
cm^{-2}}$.
