90% of the time everyone assumes the alternator when in fact it's actually the charging circuit that is the problem.

Is the battery light on the dash working? If not then this is your problem, this light is part of the charging circuit, it is common for the wire for this bulb to break at the alt end (Blue wire).

What happens if the charge excites the field windings which starts the charge process obviously if the bulb is not working this can't happen, if you rev to 4000 RPM this also will start the charge process.

11.7V is not charging, to charge the charge needs to be higher than the open circuit voltage of the battery so 13V +

Martin