Heat capacity of URu$_{2-x}$Os$_x$Si$_2$ at low temperatures

We perform measurements of the heat capacity as a function of temperature in URu$_{2-x}$Os$_x$Si$_2$ alloys. Our experimental results show that the critical temperature of the second-order phase transition increases while the value of the Sommerfeld coefficient in the ordered state decreases with an increase in osmium concentration. We also observe the increase in the values of the heat capacity at the critical temperature as well as a broadening of the critical fluctuations region with an increase in $x$. We analyze the experimental data using the Haule-Kotliar model which, in particular, identifies the 'hidden order' transition in the parent material URu$_2$Si$_2$ as a transition to a state with nonzero hexadecapole moment. We demonstrate that our experimental results are consistent with the predictions of that model.