  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.
