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114 Hyperfine structure and isotope shift

we consider the Doppler width ∆fDof a line of wavelengthλ:

∆fD

2 u

λ

, (6.38)

whereuis a typical atomic velocity. The factor of 2 appears because

atoms can move towards and away from the observer. For this estimate

we shall use the most probable velocityu=

√

(^34) This gives a value of ∆fDthat differs 2 kBT/m. (^34) For cadmium
by only
√
ln 2 = 0.8 from the exact re-
sult for the full width at half maximum
(FWHM) derived in Chapter 8.
atT= 300 K, we haveu200 m s−^1 and thus the Doppler width of
the lines with a wavelength ofλ= 468 nm is ∆fD=2u/λ=0.9GHz,
whereasA=7.9 GHz for the 5s6s^3 S 1 level. More generally, the Doppler
width for a visible transition is plotted in Fig. 6.7 as a function of the
atomic number. At room temperature the optical transitions of hydro-
gen, the lightest element, have a Doppler width slightly less than the
fine-structure splitting of the first excited state, so in this case the low-
field Zeeman effect cannot be observed even for fine structure (let alone
(^35) The Zeeman splitting of a spectral hyperfine structure). (^35) Figure 6.12 shows the results of an experimen-
line can only be resolved when the field
is strong enough to give the Paschen–
Back effect.
tal observation of the hyperfine structure and isotope shift of tin by a
technique of Doppler-free laser spectroscopy (that will be described in
Section 8.2).
Doppler broadening is much less of a problem in direct measurements
of the separation between hyperfine levels with microwave techniques
(at frequencies of gigahertz), or the even smaller splitting of the Zeeman
sub-levels that correspond to radio-frequency transitions. An example
of a radio-frequency and microwave spectroscopy technique is outlined
in the next section.

6.4.1 The atomic-beam technique

An atomic-beam apparatus can be understood as an extension of the

original Stern–Gerlach apparatus illustrated in Fig. 6.13. In the original

Stern–Gerlach experiment a beam of silver atoms was sent through a

region of strong gradient of the magnetic field and further downstream

the atoms were deposited on a glass plate. Upon inspecting the plate,

Stern and Gerlach found that the atoms appeared in two distinct places,

showing that the atomic beam was split into two directions. This told

them that angular momentum is quantised and that it can have values

Fig. 6.12Doppler-free laser spectro-
scopy of the 5p^23 P 0 –5p6s^3 P 1 line of
tin; the reduction in Doppler broad-
ening (cf. Fig. 6.11) reveals the iso-
tope shifts between the even isotopes,
in addition to the hyperfine splitting
of the odd isotopes. Each peak is
labelled by the relative atomic mass.
Each odd isotope gives rise to two peaks
because of hyperfine structure (as in
Fig. 6.11). For further details see Baird
et al. (1983).

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Frequency