  At very low temperatures, the tunnelling theory for amorphous solids predicts
a thermal conductivity $\kappa\propto T^p$, with $p = 2$. We have studied the
effect of the Nuclear Quadrupole moment on the thermal conductivity of glasses
at very low temperatures. We developed a theory that couples the tunnelling
motion to the nuclear quadrupoles moment in order to evaluate the thermal
conductivity. Our result suggests a cross over between two different regimes at
the temperature close to the nuclear quadrupoles energy. Below this temperature
we have shown that the thermal conductivity is larger than the standard
tunneling result and therefore we have $p < 2$. However, for temperatures
higher than the nuclear quadrupoles energy, the result of standard tunnelling
model has been found.
