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