0198506961.pdf

1.8 The Zeeman effect 13

Einstein devised a clever argument to find the relationship between the
A 21 -andB-coefficients and this allows a complete treatment of atoms in-
teracting with radiation. Einstein imagined what would happen to such
an atom in a region of black-body radiation, e.g. inside a box whose sur-
face acts as a black body. The energy density of the radiationρ(ω)dω
between angular frequencyωandω+dωdepends only on the tempera-
tureTof the emitting (and absorbing) surfaces of the box; this function
is given by the Planck distribution law:^2626 Planck was the first to consider radi-
ation quantised into photons of energy

ω. See Pais (1986).
ρ(ω)=

ω^3

π^2 c^3

1

exp(
ω/kBT)− 1

. (1.29)

Now we consider the level populations of an atom in this black-body
radiation. At equilibrium the rates of change ofN 1 andN 2 (in eqn 1.26)
are both zero and from eqn 1.25 we find that

ρ(ω 12 )=

A 21

B 21

1

(N 1 /N 2 )(B 12 /B 21 )− 1

. (1.30)

At thermal equilibrium the population in each of the states within the
levels are given by the Boltzmann factor (the population in each state
equals that of the energy level divided by its degeneracy):

N 2

g 2

=

N 1

g 1

exp

(

−

ω

kBT

)

. (1.31)

Combining the last three equations (1.29, 1.30 and 1.31) we find^27

(^27) These equations hold for allT,so
we can equate the parts that contain
exp(
ω/kBT) and the temperature-
independent factors separately to ob-
tain the two equations.

A 21 =

ω^3

π^2 c^3

B 21 (1.32)

and
B 12 =

g 2

g 1

B 21. (1.33)

The Einstein coefficients are properties of the atom.^28 Therefore these

(^28) This is shown explicitly in Chapter 7
by a time-dependent perturbation the-
ory calculation ofB 12.
relationships between them hold for any type of radiation, from narrow-
bandwidth radiation from a laser to broadband light. Importantly,
eqn 1.32 shows that strong absorptionis associated with strong emission.
Like many of the topics covered in this chapter, Einstein’s treatment cap-
tured the essential features of the physics long before all the details of
the quantum mechanics were fully understood.^29
(^29) To excite a significant fraction of the
population into the upper level of a visi-
ble transition would require black-body
radiation with a temperature compara-
ble to that of the sun, and this method
is not generally used in practice—such
transitions are easily excited in an elec-
trical discharge where the electrons im-
part energy to the outermost electrons
in an atom. (The voltage required to
excite weakly-bound outer electrons is
much less than for X-ray production.)

1.8 The Zeeman effect

This introductory survey of early atomic physics must include Zeeman’s
important work on the effect of a magnetic field on atoms. The obser-
vation of what we now call the Zeeman effect and three other crucial
experiments were carried out just at the end of the nineteenth century,
and together these discoveries mark the watershed between classical and
quantum physics.^30 Before describing Zeeman’s work in detail, I shall

(^30) Pais (1986) and Segr`e (1980) give his-
torical accounts.