This is The Mote in God's Eye by Larry Niven and Jerry Pournelle, part of the latter's CoDominium series, and is actually a novel published in 1974. The aliens are called "Moties" because their home star appears to a nearby human colony as a speck in front of a red giant star, called Murcheson's Eye.
In the series, FTL travel happens by traveling to a particular point between two stars, then instantaneously jumping to the equivalent point near the other star. The Moties are trapped on their home planet because the only jump point available to them ends up inside the nearby red giant, where their ships get destroyed. The light-sail-powered craft is an attempt to escape the home planet by travelling to a different star nearby, which (fortunately) is already inhabited by humans.
The Moties are divided into several subspecies: we learn about Masters, Mediators, Engineers, Watchmakers and some others. The Moties' civilization is doomed to repeat the cycle of expansion and collapse because:
biologically, they have to reproduce or they will die, and competition between the different Masters ensures that the same mistakes (such as breeding Warrior Moties) will be reproduced on every cycle.
There is also a sequel, The Gripping Hand (published as The Moat around Murcheson's Eye in the UK), which deals with the Moties...
... escaping their home planet via a different route.