When referencing a sprocket, there are two dimensions that are pertinant. 1) Bottom diameter 2) Pitch diameter
The Bottom Diameter is the circle around the sprocket that coincides with where the rollers sit in the sprocket.
The Pitch Diameter is the circle around the sprocket where the pin centers of the chain are when the chain is properly tensioned and nested on the sprocket.
The pitch of a #520 chain is 5/8", or .625".
For the sprocket to properly handle a chain, the teeth must be cut so that they coincide with that .625" pitch.
If you have a 52 tooth sprocket using a #520 chain, the circumference of the pitch diameter will be 10.351". The circumference of the 10.351" dia. is 32.512".
You'll find that if you divide 32.512" by 52 (teeth) you have .625", which is the correct pitch for #520 chain.
When machining the sprocket, after the first gear valley is cut, the holder needs to advance 6.923 degrees to cut the next valley. (360 deg. / 52 = 6.923 deg.)
When the sprocket rotates on the bike, the working face of the tooth on the driver sprocket pushes on the chain roller causing it to advance. Inversely, the chain roller on the driven sprocket pushes against the working face of the tooth, causing the sprocket to advance.
In a perfect world when no additional forces are at work, this perfectly matched pitch between chain and sprocket tooth will never become deformed and can continue to produce work forever.
However, there is friction that occurs on the chain parts that causes wear, and as a result, the pitch of the chain starts to change. When the chain pitch starts to change, the chain can no longer fit perfectly in the valley between the teeth.
You now have a miss-match of pitch between chain and sprocket. When this happens, the bottom of the chain roller will no longer sit in the bottom diameter of the sprocket.
It actually starts to ride up on the working face of the tooth.
The further towards the outside diameter of the sprocket you go, the less metal there is on the tooth, and less meat there is on the working face. On an aluminum sprocket, the energy that the chain places on the tooth can actually start to cause the tooth to deform. The extent of this deformation is in relationship to how much the pitch has elongated from it's original pitch.
When a hard sprocket is used, the elongated chain tends not to be able to deform the sprocket tooth. However, due to miss-match of pitches between chain and sprocket, the chain still rides up on the working face. Also, there is still the same amount of force placed on the hard tooth as there would be on the aluminum.
Since there is always an equal and opposite reaction to every action (who ever thunk that up had it together, dun you thin?) the energy that would normally deform the tooth is placed back on the chain parts themselves.
This actually accelerates the wear on the chain. Even moreso than what you would have seen with the aluminum.
From all this you get a couple of things...for one, and probably the most important, no matter what type of sprocket you run, the longer amount of time you can keep the chain from elongating from wear, the longer all parts of your chain drive will last.
What happens lots of times is that a rider has no idea of what length his chain is. He only knows that with an aluminum sprocket he gets a visual from the teeth wearing and knows from that, that it's time to change out both chain and sprocket.
With a hard sprocket, the rider doesn't have a visual. So he continues to run the chain, even though it's worn and elongated past where it should be. Even running it to a point where it's in danger of actually breaking. It will eventually start to deform the hard sprocket too. But, you will probably have a dangerously worn chain by that time.
Bottom line....no matter what sprocket you run, if you measure your chain and never run it past ~1.5% of it's original pitch, you will never wear out the sprocket....aluminum, carbon steel, or titanium.
If you are happy with paying outrageous prices for chains and sprockets that last one year, then so be it.
I'll be happy running mine 2-3 tmes longer and paying half as much as you.
We will all be happy as clams, si?