9.1 The scattering force 181whereais positive. For constant deceleration from an initial velocityv 0
atz= 0, integration gives the velocity as a function of distance:
v 02 −v^2 =2az. (9.7)Typically, the deceleration is half the maximum valuea=amax/2to
ensure that atoms are not left behind.^9 Hence the stopping distance is^9 Fluctuations of the force about its av-
erage value arise from the randomness
in the number of photons scattered per
L 0 = unit time, as described in Section 9.3.1.
v 02
amax. (9.8)
A typical apparatus for slowing a sodium atom (M23 a.m.u.)usesthe
parameters in the following table, where the initial velocityv 0 is taken
to be the most probable velocity in a beam (given in Table 8.1).
Most probable velocity in beam (T= 900 K) v 0 1000 m s−^1
Resonance wavelength λ 589 nm
Lifetime of the excited state τ 16 ns
Recoil velocity vr=h/(λM)3cms−^1
Stopping distance (at half of 2 v 2
the maximum deceleration)^0 τ/vr^1 .1mA distance of 1 m is a convenient length for an experiment, and all
alkali metals require surprisingly similar stopping distances. Although
the heavier elements have a lower deceleration, they also have lower
initial velocities because of their higher mass and the lower temperature
required to give sufficient vapour pressure in an oven for a high-flux
atomic beam, e.g. eqn 9.8 givesL 0 =1.2 m for rubidium and a most
probable velocity ofv 0 =vbeam= 360 m s−^1 atT = 450 K (from the
data in Table 9.1).
The majority of laser cooling experiments have been carried out with
sodium or rubidium.^10
(^10) Other elements such as magnesium
(an alkaline earth metal) have ultravi-
olet transitions with shorter lifetimes
that scatter photons of higher momen-
tum. Thus the atoms stop in a shorter
distance; however, it is technically more
difficult to obtain continuous-wave ul-
traviolet radiation.
The calculation of the stopping distance assumes a constant decelera-
tion, but for a given laser frequency atoms only experience a strong force
over a narrow range of velocities, ∆v∼Γ/k, for which the atoms have a
Table 9.1Properties of some elements used in laser cooling experiments.
Element Atomic mass Wavelength of Lifetime of the
number resonance (nm) excited state (ns)
H 1 121. 61. 6
Li 7 671 27
Na 23 589 16
K 39 767 26
Rb 85, 87 780 27
Cs 133 852 31