Experiments with Supersonic Beams as a Source of Cold Atoms

Figure 4.4: This figure illustrates the geometry of the coil which produces the magnetic
field calculated in equations 4.19 and 4.20. The B field is expressed in its
components along the ρ and z directions (there is no component in the φ direction).
Bρ = μ0I z
2πρ (r + ρ) 2 +(z) 2
−K(k 2 )+ r2 + ρ2 − z2 (r − ρ) 2 + z2 E(k2
)
(4.20)
where
k 2 4rρ
=
(R + ρ) 2 + z2 (4.21)
is the argument of the complete elliptic integrals K and E of the first and second
kind respectively. The situation described by the equations above is illustrated in
figure 4.4. Finding exact analytical solutions to these equations is often difficult, or
even impossible depending on the argument of the elliptic integrals, and solutions are
typically found numerically.
Another reason the magnetic fields of the coils used in the experiments de-
scribed here are typically found numerically is that the above equations are for coils
and fields in vacuum. While the particles in the beam are traveling through vacuum
and are of a low enough density not to appreciably change the character of the field,
the materials near and around the coils will have a significant effect on the fields
produced. To account for this, Maxwell’s equations can be modified, resulting in the
64