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The phase diagrams of the spin-$3/2$ Blume-Capel model with competing shortand long-range interactions were studied through the free energy densityobtained by analytical methods. The competition emerges when positiveshort-range interactions of strength $K$ arranged in a linear chain tend toestablish 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, thesecond-order frontier separating the ferromagnetic and the paramagnetic phasesis transformed gradually into a first-order line, when $K/J$ is greater than acertain critical value. Accordingly, there is a subinterval of $K$, for whichtwo tricritical points appear restricting the length of the second-orderfrontier. Nevertheless, for greater values of $K/J$, theferromagnetic-paramagnetic frontier becomes wholly of first order. Also, thetipical coexistence line, which divides two different ferromagnetic phases ofmagnetization $m=3/2$ and $m=1/2$, becomes more complex by giving rise toanother line of coexistence with a reentrant behavior that encloses a thirdordered phase. In this case, the competition is such that there is a region inthe 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 absolutevalue of the magnetization per spin is one.

The spin-3/2 Blume-Capel model with competing short- and long-range interactions