The spin-3/2 Blume-Capel model with competing short- and long-range   interactions
  The phase diagrams of the spin-$3/2$ Blume-Capel model with competing short and long-range interactions were studied through the free energy density obtained by analytical methods. The competition emerges when positive short-range interactions of strength $K$ arranged in a linear chain tend to establish an anti-parallel spin order, whereas negative long-range interactions $-J$ tend to align them in parallel. Thus, no ferromagnetic order exists for $K/J > 0.25$. So, the phase-diagrams were scanned by varying the values of $K$ in this interval. As in other similar study done for the spin-1 case, the second-order frontier separating the ferromagnetic and the paramagnetic phases is transformed gradually into a first-order line, when $K/J$ is greater than a certain critical value. Accordingly, there is a subinterval of $K$, for which two tricritical points appear restricting the length of the second-order frontier. Nevertheless, for greater values of $K/J$, the ferromagnetic-paramagnetic frontier becomes wholly of first order. Also, the tipical coexistence line, which divides two different ferromagnetic phases of magnetization $m=3/2$ and $m=1/2$, becomes more complex by giving rise to another line of coexistence with a reentrant behavior that encloses a third ordered phase. In this case, the competition is such that there is a region in the phase diagram, where for each spin $i$ with $S_{i}=3/2$ ($S_{i}=-3/2$), there is another one spin $j$ with $S_{j}=-1/2$ ($S_{j}=1/2$), so the absolute value of the magnetization per spin is one. 
Octavio D. Rodriguez Salmon, Jose Ricardo de Sousa, Minos A. Neto,   Igor T. Padilha, J. Roberto Viana Azevedo and Francisco...
cond-mat.stat-mech