  In this paper, assuming the validity of distance duality relation,
$\eta=D_L(z)(1+z)^{-2}/D_A(z)=1$, where $D_A(z)$ and $D_L(z)$ are the angular
and the luminosity distance respectively, we explore two kinds of gas mass
density profiles of clusters: the isothermal $\beta$ model and the
non-isothermal double-$\beta$ model. In our analysis, performed on 38 massive
galaxy clusters observed by \textit{Chandra} (within the redshift range of
$0.14<z<0.89$), we use two types of cluster gas mass fraction data
corresponding to different mass density profiles fitted to the X-ray data.
Using two general parameterizations of $\eta(z)$ (phenomenologically allowing
for distance duality violation), we find that the non-isothermal double-$\beta$
model agrees better with the distance duality relation, while the isothermal
$\beta$ model tends to be marginally incompatible with the Etherington theorem
at 68.3% CL. However, current accuracy of the data does not allow to
distinguish between the two models for the gas-density distribution at a
significant level.
