  Using first-principles density functional theory (DFT), we studied the issue
of water incorporation in iron bearing wadsleyite (Mg$_2$SiO$_4$)mineral at
transition zone pressures and temperature under varying conditions of vacancy
concentration and oxygen fugacity. We have considered the effect of varying
vacancy concentration by considering single and double vacancy. The affect of
varying oxygen fugacity has been modelled through varying ratio of ferric and
ferrous ions in the cell. Our first-principles calculations have confirmed that
about 30% of the Fe$^{3+}$ can be found at the Si tetrahedral site, which is in
accordance with the previous experimental observation by Bolfan-Casanova et al.
[2012]. Using different structures of Fe bearing wadsleyite, representing
vacancy concentration and oxidizing condition, we calculated elastic
properties, such as bulk and shear moduli, phase and shear wave velocity, as a
function of pressure and temperature. By comparison of our theoretical
prediction with the Preliminary Reference Earth Model (PREM) data, we conclude
that the water content of the transition zone decreases with increasing depth,
corroborating the previously proposed theories on mantle convection [Bercovici
and Karato, 2003].
