Magnetic and magnetocaloric effects in the systems characterized by the first-order reversible phase transitions

Within the framework of the model of interacting parameters of magnetic and structural orders, a theoretical analysis of magnetostructural reversible first-order phase transitions is carried out. Reversible phase transitions are characterized by a jump-like appearance of magnetic order with decreasing temperature (as in a first-order phase transition), and with a reverse increase in temperature, the magnetic order gradually disappears (as in a second-order phase transition). Such transitions are observed in some alloys of the Mn$_{1-x}$Cr$_{x}$NiGe magnetocaloric system under pressure ($x$ = 0.11) and without ($x$ = 0.18) and are accompanied by specific magnetic and magnetocaloric features. A phenomenological description of these features is carried out within the concept of a soft mode for the structural subsystem undergoing first-order structural phase transition $(P6_{3}/mmc-P_{nma})$ and the Heisenberg model for the spin subsystem. For systems with magnetostructural instability within the molecular field approximation for the spin subsystem and the shifted harmonic oscillator approximation for the lattice subsystem, it is shown that the reversible phase transitions arise when the temperature of magnetic disordering is in the temperature hysteresis region of the 1st order structural phase transition $P6_{3}/mmc-P_{nma}$. It is also shown that the two-peak form of the isothermal entropy, which is characteristic of reversible transitions, is due to the separation of the structural and magnetic entropy contributions.