Experiments with Supersonic Beams as a Source of Cold Atoms

Using equation 2.17 to modify equation 2.10 gives dP
ρ = v2 dρ
s ρ
substituted into the momentum conservation law stated in equation 2.14 gives
dρ
ρ
which when
w
= −
v2 dw. (2.19)
s
This relationship can be substituted into the mass conservation law of equation 2.12
along with equation 2.17 giving
dA
A
+ dw
w
1 − M 2 =0. (2.20)
Much of the behavior of a 1D flow can be determined from this equation. A cold
beam, or one with a high degree of directed motion as opposed to internal thermal
motion, will have a high Mach number. Starting in the reservoir, the Mach number
is clearly much less than one, and so (1 − M 2 ) > 0. This means that if the flow
is subsonic, the flow velocity increases with decreasing cross-sectional area, and vice
versa. Alternatively, if the flow is supersonic, then (1 − M 2 ) < 0 and the flow will
increase in velocity with increasing area. Finally, if there is a sufficient pressure
gradient to accelerate the flow to supersonic velocities, at the location where M =1,
(1 − M 2 )=0andsodA= 0, implying that the cross sectional area is at a minimum.
A
This means that to create a supersonic beam, a nozzle must converge to a minimum
before diverging again.
2.1.3 Supersonic Beam Velocities
Having determined the qualitative character of the supersonic flow, it is helpful
to turn to the specifics of beam velocity and temperature. The energy conservation
law, equation 2.16, can be integrated to find the flow velocity at various locations,
w 2 1 − w 2 0
2
= h1 − h0 =
T0
T1
cP dT = cP T0
1 − T1
, (2.21)
assuming that cP is a constant. Selecting the initial location to be well inside the
gas reservoir, where the flow velocity is negligible and the gas is at the reservoir
12
T0