Experiments with Supersonic Beams as a Source of Cold Atoms

magnetic field will adiabatically follow the field so long as the Larmor frequency is
much larger than the rate of change of the direction of the magnetic field. If this
condition, expressed as
ωL ≫ d
B
,
dt |B|
(4.16)
is met, then the equations for the Hamiltonian and for the energy shift do not depend
on B, but instead will depend on | B|, which is hereafter simply referred to as B.
This condition is easily met for most magnetic fields described in this dissertaion,
and special note is made of situation where it may not be met.
4.2 Magnetic Fields from Electromagnetic Coils
Pulsed electromagnetic coils are used to produce the high magnetic fields nec-
essary to effectively control the velocity of a supersonic beam. From the differential
form of Ampère’s law
∇× B = μ0 J, (4.17)
where J is current density and μ0 is the permeability of free space, it is possible to
derive the Biot-Savart law
B = μ0
I d
4π C
ℓ × x
|x| 3 , (4.18)
where I is the current and d ℓ is the infinitesimal path of the current [74] (this also
assumes a wire with zero cross-sectional area). From this, the magnetic field from
a particular current path can be found. Of particular interest is the magnetic field
from a circular loop of wire of radius r. The cylindrically symmetric field generated
by this loop can be found exactly as [75]
Bz = μ0I
2π
1
K(k
(r + ρ) 2 +(z) 2
2 )+ r2 − ρ2 − z2 (r − ρ) 2 + z2 E(k2
)
63
(4.19)