Two Types of the Effective Mass Divergence and the Gr\"uneisen Ratio in Heavy Fermion Metals

The behavior of the specific heat $c_p$, effective mass $M^*$, and the thermal expansion coefficient $\alpha$ of a Fermi system located near the fermion condensation quantum phase transition (FCQPT) is considered. We observe the first type behavior if the system is close to FCQPT: the specific heat $c_p\propto \sqrt{T}$, $M^*\propto1/\sqrt{T}$, while the thermal expansion coefficient $\alpha\propto \sqrt{T}$. Thus, the Gr\"uneisen ratio ${\rm \Gamma}(T)=\alpha/c_p$ does not diverges. At the transition region, where the system passes over from the non-Fermi liquid to the Landau Fermi liquid, the ratio diverges as ${\rm \Gamma}(T)\propto 1/\sqrt{T}$. When the system becomes the Landau Fermi liquid, ${\rm \Gamma}(T,r)\propto 1/r$, with $r$ being a distance from the quantum critical point. Provided the system has undergone FCQPT, the second type takes place: the specific heat behaves as $c_p\propto \sqrt{T}$, $M^*\propto 1/T$, and $\alpha=a+bT$ with $a,b$ being constants. Again, the Gr\"uneisen ratio diverges as ${\rm \Gamma}(T)\propto 1/\sqrt{T}$.