0198506961.pdf

6.2 Isotope shift 107

Reduced mass shift

(visible line)

Volume shift

Doppler width

Energy (eV)

1 10 100

1

10

102

10 −^1

10 −^2

10 −^3

10 −^4

10 −^5

10 −^6

10 −^7

10 −^8

10 −^9

10 −^10

Atomic number,

Fine structure

Gross structure

Residual electrostatic energy

Hyperfine structure (ground state)

Fig. 6.7A plot of the energy of var-

ious structures against atomic number

Z. This includes the data shown on

Fig. 6.5, but the scale has been changed

to show the effects of lower energy.

The contributions to the isotope shift

from the mass and volume effects are

plotted using eqns 6.21 and 6.26, re-

spectively. Although these effects arise

from completely independent physical

effects, they have a similar magnitude

for medium-heavy elements (and can

be comparable to the magnetic dipole

hyperfine structure of an excited state,

which is considerably smaller than the

ground-state splitting). The Doppler

width for a visible transition with en-

ergy 2 eV is plotted to indicate the

typical experimental limit in spectro-

scopic measurements using light; the

line has a slope of− 1 /2 because ∆fD∝

1 /

√

Mand the atomic massAis ap-

proximatelyA∼ 2 Z,sothat∆fD∝

Z−^1 /^2 (see eqn 8.7). For hydrogen

the Doppler width is almost equal to

the fine-structure splitting (and much

larger than hyperfine structure). The

Doppler broadening of visible lines in

sodium is comparable with its hyper-

fine structure; for heavier elements the

ground-state hyperfine structure can be

resolved in a vapour at room tempera-

ture. This gives an indication of the

importance of Doppler-free techniques

described in Chapter 8. The Zeeman

effect of a magnetic flux density ofB=

1 T is equivalent to a frequency shift of

14 GHz.

has zero potential energy, and for a distribution of nuclear chargeρN(r)
the potential energy is

EVol=

∫∫∫

ρNφed^3 r=

e

6  0

|ψ(0)|^2

∫∫∫

ρNr^2 d^3 r

=

Ze^2

6  0

|ψ(0)|^2

〈

rN^2

〉

. (6.24)

The integral gives the mean-square charge radius of the nucleus

〈

rN^2

〉

times the chargeZe. This volume effect only applies to configurations
with s-electrons. The liquid drop model gives a formula for the radius