Experiments with Supersonic Beams as a Source of Cold Atoms

Since there will be no heat exchange between the gas and the environment during the
expansion, equations 2.5 and 2.7 become
and
0=deint + P d
1
, (2.8)
ρ
0=dh− 1
dP. (2.9)
ρ
The specific heats at constant pressure, cP , and at constant volume, cV ,are
given by cP = dh
dT
P and cV = deint
dT
V
, and their ratio, γ = cP
cV ,isaconstantforan
ideal gas that depends only on the number of degrees of freedom of a gas molecule.
Using these relations, equations 2.8 and 2.9, can be re-expressed and combined as [8]
γ dρ
ρ
− dP
P
This last relation is of particular use in examining the gas flow.
=0. (2.10)
There are several conservation laws that can be used to simplify the problem
of gas flow through a nozzle. Clearly, conservation of mass applies in the flow, gas
particles are not being spontaneously created or annihilated. This is expressed as
or simplified for 1D flow as
dw
w
∇ (ρw) = 0 (2.11)
+ dρ
ρ
+ dA
A
= 0 (2.12)
where w is the velocity of the flow, ρ is the density of the gas, and A is the cross-
sectional area of the flow. Momentum is also conserved in the flow, which is essentially
an expression of Newton’s second law in the gas. Assuming no external forces on the
gas
Dw
Dt
= −∇P
ρ
10
(2.13)