Exercises 265
7.7 Spin-Orbit Coupling- Why is the ground state of the hydrogen atom not split into
two sublevels by spin-orbit coupling? - The spin-orbit effect splits the 3P→ 3 Stransition in sodium
(which gives rise to the yellow light of sodium-vapor highway
lamps) into two lines, 589.0 nm corresponding to 3P 3 2 → 3 S 1 2
and 589.6 nm corresponding to 3P 1 2 → 3 S 1 2. Use these wave-
lengths to calculate the effective magnetic field experienced by the
outer electron in the sodium atom as a result of its orbital motion.
7.8 Total Angular Momentum- An atom has a single electron outside closed inner shells. What
total angular momentum Jcan the atom have if it is in a P
state? In a Dstate? - If j ^52 , what values of lare possible?
- (a) What are the possible values of Lfor a system of two elec-
trons whose orbital quantum numbers are l 1 1 and l 2 3?
(b) What are the possible values of S? (c) What are the possible
values of J? - What must be true of the subshells of an atom which has a^1 S 0
ground state? - Find the S, L,and Jvalues that correspond to each of the fol-
lowing states:^1 S 0 ,^3 P 2 ,^2 D 3 2 ,^5 F 5 ,^6 H 5 2. - The lithium atom has one 2selectron outside a filled inner
shell. Its ground state is^2 S 1 2. (a) What are the term symbols of
the other allowed states, if any? (b) Why would you think the
(^2) S 1 2 state is the ground state?
- The magnesium atom has two 3selectrons outside filled inner
shells. Find the term symbol of its ground state. - The aluminum atom has two 3selectrons and one 3pelectron
outside filled inner shells. Find the term symbol of its ground
state. - In a carbon atom, only the two 2pelectrons contribute to its
angular momentum. The ground state of this atom is^3 P 0 , and
the first four excited states, in order of increasing energy, are
(^3) P 1 , (^3) P 2 , (^1) D 2 , and (^1) S 0. (a) Give the L, S,and Jvalues for each
of these five states. (b) Why do you think the^3 P 0 state is the
ground state?
- Why is it impossible for a 2^2 D 3 2 state to exist?
- (a) What values can the quantum number jhave for a delec-
tron in an atom whose total angular momentum is provided by
this electron? (b) What are the magnitudes of the corresponding
angular momenta of the electron? (c) what are the angles
between the directions of Land Sin each case? (d) What are
the term symbols for this atom?
35. Answer the questions of Exercise 34 for an felectron in an
atom whose total angular momentum is provided by this
electron.
36. Show that if the angle between the directions of Land Sin
Fig. 7.15 is ,
cos- The magnetic moment Jof an atom in which LScoupling
holds has the magnitude
JJ(J1)gJB
where Be 2 mis the Bohr magneton andgJ 1 is the Landé g factor.(a) Derive this result with the help of the
law of cosines starting from the fact that averaged over time,
only the components of Land Sparallel to Jcontribute to
J. (b) Consider an atom that obeys LScoupling that is in a
weak magnetic field Bin which the coupling is preserved. How
many substates are there for a given value of J? What is the
energy difference between different substates?- The ground state of chlorine is^2 P 3 2. Find its magnetic moment
(see previous exercise). Into how many substates will the
ground state split in a weak magnetic field?
7.9 X-Ray Spectra- Explain why the x-ray spectra of elements of nearby atomic
numbers are qualitatively very similar, although the optical
spectra of these elements may differ considerably. - What element has a K x-ray line of wavelength 0.144 nm?
- Find the energy and the wavelength of the K x-rays of
aluminum. - The effective charge experienced by an M (n3) electron in an
atom of atomic number Zis about (Z7.4)e. Show that the
frequency of the L x-rays of such an element is given by
5 cR(Z7.4)^2 36.
Appendix: Atomic Spectra- Distinguish between singlet and triplet states in atoms with two
outer electrons. - Which of the following elements would you expect to have
energy levels divided into singlet and triplet states:
Ne, Mg, Cl, Ca, Cu, Ag, Ba?
J(J1)L(L1)S(S1)
2 J(J1)j(j1)l(l1)s(s1)
2 l(l 1 ) s(s1)bei48482_Ch07.qxd 1/23/02 9:03 AM Page 265