96 TheLS-coupling scheme
scheme. Give a physical justification for three of
these rules.
Which of the following are allowed for electric
dipole transitions in theLS-coupling scheme:(a) 1s2s^3 S 1 –1s3d^3 D 1 ,
(b) 1s2p^3 P 1 –1s3d^3 D 3 ,
(c) 2s2p^3 P 1 –2p^23 P 1 ,
(d) 3p^23 P 1 –3p^23 P 2 ,
(e) 3p^61 S 0 –3p^5 3d^1 D 2?
The transition 4d^9 5s^22 D 5 / 2 –4d^10 5p^2 P 3 / 2 satis-
fies the selection rules forL, SandJbut it ap-
pearstoinvolvetwoelectronsjumpingatthesame
time. This arises from configuration mixing—the
residual electrostatic interaction may mix configu-
rations.^21 The commutation relations in eqns 5.2
and 5.3 imply thatHreonly mixes terms of the
sameL, SandJ. Suggest a suitable configuration
that gives rise to a^2 P 3 / 2 level that could mix with
the 4d^10 5p configuration to cause this transition.(5.10)The anomalous Zeeman effect
What selection rule governs ∆MJin electric dipole
transitions? Verify that the^3 S 1 –^3 P 2 transition
leads to the pattern of nine equally-spaced lines
shown in Fig. 5.13 when viewed perpendicular to
a weak magnetic field. Find the spacing for a mag-
netic flux density of 1 T.
(5.11)The anomalous Zeeman effect
Draw an energy-level diagram for the states of^3 S 1
and^3 P 1 levels in a weak magnetic field. Indi-
cate the allowed electric dipole transitions between
the Zeeman states. Draw the pattern of lines ob-
served perpendicular to the field on a frequency
scale (marked in units ofμBB/h).
(5.12)The anomalous Zeeman effect
22 3 22 3 22FrequencyThe above Zeeman pattern is observed for a spec-
tral line that originates from one of the levels of a(^3) P term in the spectrum of a two-electron system;
the numbers indicate the relative separations of
the lines, observed perpendicular to the direction
of the applied magnetic field. IdentifyL,SandJ
for the two levels in the transition.^22
(5.13)The anomalous Zeeman effect in alkalis
Note that atoms with one valence electron are not
discussed explicitly in the text.
(a) Give the value ofgJfor the one-electron levels
(^2) S 1 / 2 , (^2) P 1 / 2 and (^2) P 3 / 2.
(b) Show that the Zeeman pattern for the
3s^2 S 1 / 2 –3p^2 P 3 / 2 transition in sodium has
six equally-spaced lines when viewed perpen-
dicular to a weak magnetic field. Find the
spacing (in GHz) for a magnetic flux density
of 1 T. Sketch the Zeeman pattern observed
along the magnetic field.
(c) Sketch the Zeeman pattern observed per-
pendicular to a weak magnetic field for the
3s^2 S 1 / 2 –3p^2 P 1 / 2 transition in sodium.
(d) The two fine-structure components of the 3s–
3p transition in sodium in parts (b) and (c)
have wavelengths of 589.6 nm and 589.0 nm,
respectively. What magnetic flux density pro-
duces a Zeeman splitting comparable with the
fine structure?^23
(5.14)The Paschen–Back effect
In a strong magnetic field LandSprecess in-
dependently about the field direction (as shown
in Fig. 5.14), so thatJ andMJ are not good
quantum numbers and appropriate eigenstates are
|LMLSMS〉. This is called the Paschen–Back ef-
fect. In this regime theLS-coupling selection rules
are ∆ML=0,±1and∆MS= 0 (because the elec-
tric dipole operator does not act on the spin).^24
Show that the Paschen–Back effect leads to a pat-
tern of three lines with the same spacing as in the
normal Zeeman effect (i.e. the same as if we com-
pletely ignore spin).^25
(^21) In the discussion of theLS-couplingschemewetreatedHreas a perturbation on a configuration and assumed thatEre
is small compared to the energy separation between the configurations in the central field. This is rarely true for high-lying
configurations of complex atoms.
(^22) The relative intensities of the components have not been indicated.
(^23) This value is greater than 1 T so the assumption of a weak field in part (b) is valid.
(^24) The rules forJandMJare not relevant in this regime.
(^25) The Paschen–Back effect occurs when the valence electrons interact more strongly with the external magnetic field than
with the orbital field inHs−o.TheLS-coupling scheme still describes this system, i.e.LandSare good quantum numbers.