What does Basil mean by hesychia?

In reading through the letters and the Ascetical Writings of St. Basil the Great, there are numerous mentions of silence, solitude, tranquility, and stillness, but he uses various words in different ways and for different contexts. It would be helpful, for those of us who want to learn what we are able from the experience of our Father Basil, to dig down to that root meaning that ties them all together to really understand how we also may attend this "school of the soul" of which he speaks when he says, "More than anything, 'school' to the soul is the gracious granting of nightly hesychia (silence)." If we are going to go to school, we will have to know where it is. As a beginning to our search for Basil’s “Soul School of Silence”, we must first explore what he does not mean by hesychia (‘silence’). It is not just ‘staying quiet’, and indeed, Basil does not use the same word when he means merely staying quiet: “It was my purpose to maintain silence (ἀποσιωπᾷν) towards you”. And similarly, using a variation of that same word, “I have kept silence”, this time using σιωπήσομαι and τῆς σιωπῆς, again meaning “to keep quiet”. And in another place, he says, “silent (σιωπᾷ) is his tongue.” Keeping our mouths shut is not the definition of hesychastic silence. Interestingly, however, our guide Basil, in speaking of our overly talkative tendencies, does say that not remaining quiet can upset the hesychia of those around us: “[the monastic] must not talk idly, prattling of things which neither conduce to the benefit of his listeners nor to the activities that are indispensable and permitted us by God; so that both the workers may as far as possible have silence (ἠσυχίας, hesychia) in which to apply themselves zealously into their work…” Rather than this idle talk, in emphasizing the essential characteristics of the monastic life, he places hesychastic silence near the top of the list, admonishing monastics to be “quiet of demeanour, not hasty in speech, nor contentious, quarrelsome, vainglorious, nor given to interpreting texts; but be a man of trust, of few words, and always more ready to learn than to teach.” By far my personal favorite of the these uses of hesychia is in a letter to St. Athanasius of Alexandria. St. Athanasius was the hero of the Nicene Council as a young man, and now, though Athanasius is quite aged at this point, Basil is trying to coax him out of silence to speak out on the issues which Basil was facing, somewhat like trying to bring the sports star out of retirement to come back and help win the final game. Basil says, τοὺς δὲ καθησυχάσαι, or (paraphrased) “you need to hesychize them”, translated in the Loeb Classical Library as “tranquillize”. All I can picture is St. Athanasius in (true-to-his-Alexandrian roots) African-safari bishop’s garb, face steely-eyed and set on the heretics, with a tranquilizer rifle over his shoulder. Cartoonish images aside, assuming Basil is speaking of hesychastic silence as more than just an outward quietness, but rather an inward quietness, he uses the word “hesychia” with κατα pinned on the front to make the word “give an inner silence/stillness to them”, to suggest that the authority of right belief, distilled in the experience and person of such a trusted source, could potentially bring forth hesychastic silence in others. The bickering school children are immediately silenced when the venerable teacher enters the room.