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Even the NIST website for the standard F-1 clock and the new F-2 do not explain, exactly, why it helps (or is necessary) to make the cesium atoms appear, at least temporarily, weightless....

Wouldn't their absorption and emissions be redshifted and blueshifted by the motion?

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According to the description on the nist.gov website of the NIST-F1 Cesium Fountain Atomic Clock the atomic fountain setup was chosen to obtain a longer observation window. According to the article the atoms have a round trip time of a second.

Quote:

Traditional cesium clocks measure room-temperature atoms moving at several hundred meters per second. Since the atoms are moving so fast, the observation time is limited to a few milliseconds.

Summerizing:
As I understand that description: the atomic fountain setup is used for two reasons:
-the velocities of the observed cesium atoms are much slower than is the case with what is in the article referred to as traditional cesium clock setup.
-the observation time is much longer

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