[chris (OP)]

An electron microscope has a very fine tip; we're told it's a single atom at the end. How do they a) make that and b) know that they've achieved it?

[Bored chemist]

The tip is, pretty much by definition, the end of the needle. There has to be an atom there, so the tip is an atom.
I know they used to make them by electrolytically etching tungsten wire, but they may have other technique too.

[evan_au]

If the tip of your needle was a hemisphere of 0.5mm diameter, you would not be able to pick up the patterns of metal on a semiconductor chip, let alone the surface undulations of individual silicon or aluminium atoms on that chip.

If you make the needle from wire which is very fine, you can be sure the tip is not 0.5mm in diameter. So the construction method is important. But then you can test it.

[Bored chemist]

Here's a method
http://www.phy.ohiou.edu/~asmith/lab/stm-tip-prep.pdf

[chris (OP)]

Thanks all!