Experiments with Supersonic Beams as a Source of Cold Atoms

y
x
V i
Figure A.1: The geometry used for calculating the reflection of an atom from the tip
of the rotor.
Since the situation is now one of an atom reflecting from an apparently sta-
tionary object, the initial and final speeds of the atom will be equal in this frame, and
the trajectory of the atom will be reflected about the normal vector of the crystal ˆn.
This calculation can be done quite simply in vector form and the final velocity is
V f
Vout = Vin − 2(Vin · ˆn)ˆn. (A.4)
In the chosen geometry, the normal vector of the crystal is
ˆn =[sin(θ + φ), cos (θ + φ)] . (A.5)
Inserting equations A.3 and A.5 into equation A.4 and simplifying gives the reflected
velocity in the frame of the crystal,
Vf−c = [Vxi cos (2(θ + φ)) − Vyi sin (2(θ + φ)) + r ˙ θ sin (θ +2φ), (A.6)
−Vyi cos (2(θ + φ)) − Vxi sin (2(θ + φ)) + r ˙ θ cos (θ +2φ)].
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φ
θ
r