Bye the way, on several occasions I have personally downed buck deer with a .22 rimfire. The trick is to use a very good scope and hit the deer on it's spine as close to the head as possible. It doesn't kill the animal at this point but does paralyze it so you can move in close and kill it with a couple shots to the head (shoot for the lower part of the brain from the back just a couple inches above where the spinal cord enters the skull. This is the most vulnerable part of the brain and will cause the animal to stop breathing among other deadly functions.
I have also downed a deer by first hitting a rear knee cap, then the opposite front knee cap. The animal has to be standing still to get off a good shot, and one must place the second shot very quickly before the deer realizes what's going on. With both knee caps shattered it can't run and again you can then move in for the kill. With both methods hunt as if you were using a bow and arrow - use camouflage and shoot from no more than 40 yards. If you only get the rear knee cap the deer can still run, but not nearly as fast and as far. It will hole up and hide.
Tracking the deer is much easier than with only a bow because you not only have blood sign, but also the tracks are unmistakable. Look for brush quite close to where the deer was originally shot. Try to get in close enough for either the correct spine or head shot or even the opposite knee before spooking the deer again.
Offered by Ron.