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How would I get a beam of silver atoms? I imagine this to be problematic, since the silver atoms are neutral. How would I accelerate them?

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    $\begingroup$ Didn't the original experiment use a thermal source (furnace) and a collimator? Indeed, the Wikipedia article shows exactly that in the image at the top of the page. $\endgroup$ – dmckee Aug 3 '15 at 21:43
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A common method is to do the same thing for when a beam of light is needed and no laser is to be found -- collimate by elimination. That is, have an omnidirectional source and just filter out the particles moving in the wrong direction.

This is illustrated in the below Wikimedia image:

Stern-Gerlach setup

A source is heated so that atoms fly off in all directions. Small apertures are then used to collimate the beam. Smaller apertures result in better collimation but with fewer atoms making the cut. Note that the cyan in the image should more accurately show a broad spread of particles coming out of the furnace with many being blocked by the first aperture.

This is exactly the setup I used in an undergraduate physics lab.

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