Experiments with Supersonic Beams as a Source of Cold Atoms

Appendix A
Reflection From a Spinning Rotor
The equations of motion for an atom reflecting from the rotor are more com-
plicated than the 1D simplification given in chapter 3. In the experiment, the atoms
can hit the crystal at an angle, and the velocity of the crystal may not be square to
the normal of the crystal face.
The key to finding the equations that define the reflection of an atom from a
moving crystal is to move into the frame of the crystal. In this reference frame, the
speed of the atom is unchanged by the reflection, and the angle of incidence is equal
to the angle of reflection. The solution to the problem can thus be found finding
the velocity of the atom in the frame of the crystal, reflecting the atom from the
apparently stationary crystal, and transforming the velocity of the atom back into
the laboratory frame.
The geometry used in deriving these equation is shown in figure A.1. The
atom starts with an initial velocity
and in the geometry chosen, the crystal has velocity
Vc =
Vi =[Vxi ,Vxi ] (A.1)
r ˙ θ sin θ, r ˙
θ cos θ , (A.2)
where r is the radius of the rotor. Thus in the frame of the rotor, the atom has a
velocity of
Vi−c = Vxi − r ˙ θ sin θ, Vxi − r ˙
θ cos θ . (A.3)
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