Experiments with Supersonic Beams as a Source of Cold Atoms

is the velocity, and ω is the angular velocity, dictates the design of the rotor, which
must be strong enough at every point to support the force exerted by the section of
the rotor at larger radius. The force at each position can be approximated (assuming
no transverse forces in the rotor) as
F =
rmax
ρA(r)ω 2 rdr, (3.5)
where rmax
rc
is the maximum radius of the rotor, rc is position the force is being
measured, ρ is the density of the rotor material, and A(r) is the cross sectional area
of the rotor. The force at each position must be compared to the yield tensile strength
of the rotor material times the cross sectional area. Equation 3.5 shows that a light
and strong material is ideal for the rotor. While composites would be ideal, their
lack of vacuum compatibility poses a serious problem. The chosen rotor material is
grade 5 titanium, a high strength alloy which is also vacuum compatible. Another
constraint on the rotor design is the vacuum itself, since the rotor must fit into a
practical vacuum chamber. This led to a compromise design with a rotor radius of
50.4 cm at the center of the crystal.
A CAD image of the rotor can be seen in figure 3.7. The overall length of the
rotor is 101.5 cm. Since the inside of the rotor needs to be stronger than the tips,
as seen in equation 3.5, the rotor is tapered linearly from a minimum thickness of
0.3175cm where the rotor joins the mirror holder, to a maximum thickness of 2.54cm,
with the rotor a constant thickness over its center 5.08 cm. This taper reduces the
mass at large radius, reducing the force exerted, while placing material where it is
needed near the center of the rotor. The crystal mirrors, which fit directly into the
rotor are 5 mm in diameter and are held in place by an aluminum set screw, allowing
for an exposed crystal diameter of 4 mm.
While an analytical calculation of the forces placed on a the rotor is pro-
hibitively complex, finite element analysis permits a numerical calculation of the
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