9.1 The scattering force 179

fall neatly into either category of the radiation force, i.e. (a) the scatter-
ing force that arises from absorption of light and spontaneous emission,
and (b) the dipole force, described in Section 9.6. The forces on micro-
scopic particles have properties similar to the forces on individual atoms,
and this analogy is used to introduce the dipole force in Section 9.5.
Atoms that have been pre-cooled by the other laser cooling techniques
described in this chapter can be cooled even further using Raman tran-
sitions (Section 9.8). Finally, Section 9.9 describes an atomic fountain
which requires laser-cooled atoms.

9.1 The scattering force

The idea that radiation has momentum (and energy) goes back to James
Clerk Maxwell in the nineteenth century. It follows from the conserva-
tion of momentum that when an object absorbs radiation its momentum
changes. The force on the object equals the rate of change of momen-
tum. Therefore the force equals the rate at which the light delivers
momentum—this is the same as the rate at which the light delivers en-
ergy divided by the speed of light.^4 Therefore radiation of intensityI^4 The energy of radiation divided by its
momentum equalsc. For a photon
energy
momentum

ω

k

ω
k
=c.
Of course, Maxwell showed this by clas-
sical electromagnetism, not in terms of
photons. (The ratio does not depend on

.) This and other aspects of the pres-
sure due to electromagnetic radiation
are discussed in Bleaney and Bleaney
(1976, Section 8.8).

exerts a force on areaAgiven by

Frad=

IA

c

. (9.1)

Equivalently, the radiation pressure isFrad/A = I/c.Thequantity
IAequals the power absorbed, e.g. forIA=1WtheforceisF =
3. 3 × 10 −^9 N. At a surface that reflects the radiation back on itself the
momentum change is twice as large and gives twice the force in eqn 9.1.
Although small, the radiation force has observable effects in astrophysics,
e.g. the outward radiation pressure balances gravity in stars, and the
tails of comets point away from the sun (rather than trailing behind
as for shooting stars in the atmosphere). Note, however, that although
radiation pressure does have some effect on the dust and ice particles that
form the tail of a comet, the solar wind is also important—the stream of
particles emanating from the sun hit the particles in the comet tail and
even the relatively low pressure in space leads to a force comparable to
that from radiation pressure.^5 Radiation forces have a dramatic effect^5 The radiation from the sun has an
intensity of 1.4kWm−^2 at the Earth.
Thus the radiation pressure at the
Earth’s orbit is 5× 10 −^6 Nm−^2 ,or
slightly less than∼ 10 −^10 times atmo-
spheric pressure.

on atoms because the peak absorption cross-sectionσabs(ω 0 )ismuch
greater than the physical size of the atom (see eqn 7.81).^6

(^6) Alkali atoms in a vapour have a large
fraction of their absorption strength
concentrated in a narrow range centred
at the frequency of the resonance line.
Lasers produce well-collimated monochromatic beams of light that
can slow atoms in an atomic beam, as illustrated in Fig. 9.1. A counter-
propagating laser beam exerts a force ofF =−σabsI/con an atom,
where the minus sign indicates a force in the opposite direction to the
motion. This expression in terms of the absorption cross-section shows
that the light does not have to be considered as quantised in order to
calculate the force, but it is convenient to describe the processes in
terms of photons. Each absorbed photon gives the atom a kick in the
direction opposite to its motion and spontaneously-emitted photons go
in all directions, so that the scattering of many photons gives an average