The role of ionic sizes in inducing the cubic to tetragonal distortion   in AV$_{2}$O$_{4}$ and ACr$_{2}$O$_{4}$ (A=Zn, Mg and Cd) compounds
  Cubic to tetragonal distortion in spinel compounds have been a contentious issue for last two decades. Different groups have proposed different mechanisms to understand such a distortion in these spinels, which are: (i) spin lattice coupling mechanism known as the spin driven Jahn-Teller (JT) effect, (ii) the strong relativistic spin-orbit coupling, a moderate JT distortion and weak V-V interactions and (iii) the JT effect. Now, in order to know the possible cause for such a distortion, we have avoided these complexities (various interactions among spin, electronic, orbital and lattice degrees of freedom) by carrying out spin unpolarized calculations. The calculated values of bulk moduli for ZnV$_{2}$O$_{4}$ (ZnCr$_{2}$O$_{4}$), MgV$_{2}$O$_{4}$ (MgCr$_{2}$O$_{4}$) and CdV$_{2}$O$_{4}$ (CdCr$_{2}$O$_{4}$) are found to be $\sim$289 ($\sim$254), $\sim$244 ($\sim$243) and $\sim$230 ($\sim$233) GPa, respectively. For vanadates and chromates, the order of calculated values of lattice parameter $a$ are found to CdV$_{2}$O$_{4}$$>$MgV$_{2}$O$_{4}$$>$ZnV$_{2}$O$_{4}$ and CdCr$_{2}$O$_{4}$$>$MgCr$_{2}$O$_{4}$$>$ZnCr$_{2}$O$_{4}$, respectively and are consistent with the experimental results. The calculated values of cubic to tetragonal distortion (c/a), with c/a$<$1 for ZnV$_{2}$O$_{4}$ (ZnCr$_{2}$O$_{4}$), MgV$_{2}$O$_{4}$ (MgCr$_{2}$O$_{4}$) and CdV$_{2}$O$_{4}$ (CdCr$_{2}$O$_{4}$) are $\sim$0.996 ($\sim$0.997), $\sim$0.995 ($\sim$0.994) and $\sim$0.997 ($\sim$0.998), respectively. These values are in good agreement with the experimental data for ZnV$_{2}$O$_{4}$, MgV$_{2}$O$_{4}$, ZnCr$_{2}$O$_{4}$ and MgCr$_{2}$O$_{4}$ compounds. The present study clearly shows the role of ionic sizes in inducing the cubic to tetragonal distortion in these spinels. These mechanisms also appear to be responsible for deciding the other physical properties of these compounds. 
Sohan Lal and Sudhir K. Pandey
cond-mat.str-el