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280 8 ATOMIC STRUCTURE AND SPECTRA
�e Pfund series corresponds to n1 = 5. �e lowest energy transition, which
would involve a photon with the longest wavelength, is to the next highest
energy level which has n2 = 6 . Transitions to higher energy levels involvemore
anmore energy, and the limit of this is the transition to n2 =∞ which involves
the greatest possible energy change and hence the shortest wavelength.
E8C.2(b) he energy levels of a hydrogenic atom are En = −hcZ2R̃Nn−2, where Z is the
atomic number; for all but the most precise work it is su�cient to approximate
R̃N by R̃∞. �e wavenumber of the transition between states with quantum
numbers n1 and n2 in the Li2+ ion is given by amodi�ed version of [8A.1–304],
ν̃ = Z2R̃∞(n−21 − n−22 ). For the 5→ 4 transition and with Z = 3
ν̃ = 32 × (1.0974 × 105 cm−1) × (4−2 − 5−2) = 2.22 × 105 cm−1
λ = ν̃−1 = 1/[32 × (1.0974 × 105 cm−1) × (4−2 − 5−2)]
= 4.49... × 10−5 cm = 450 nm
ν = c/λ = (2.9979 × 108ms−1)/(4.49... × 10−7 m) = 666 THz
E8C.3(b) �e selection rules for a many-electron atom are given in [8C.8–335]. For a
single electron these reduce to ∆l = ±1; there is no restriction on changes in n.
(i) 5d (n = 5, l = 2)→ 2s (n = 2, l = 0) has ∆l = −2, and so is forbidden .
(ii) 5p (n = 5, l = 1)→ 3s (n = 1, l = 0) has ∆l = −1, and so is allowed .
(iii) 6p (n = 3, l = 1)→ 4f (n = 2, l = 3) has ∆l = +2, and so is forbidden .
E8C.4(b) �e single electron in a f orbital has l = 3 and hence L = 3, and s = 1
2 hence
S = 1
2 . �e spin multiplicity is 2S + 1 = 2. Using the Clebsh–Gordon series,
[8C.5–332], the possible values of J are J = L + S , L + S − 1, . . . ∣L − S∣ = 7
2 ,
5
2 .
Hence, the term symbols for the levels are 2F7/2, 2F5/2 .
E8C.5(b) For a p electron l = 1 and s = 1
2 . Using the Clebsh–Gordon series, [8C.5–332],
the possible values of j are l + s, l + s − 1, . . . ∣l − s∣, which in this case are
j = 3
2 ,
1
2 .
For an h electron l = 5 and s = 1
2 hence j =
11
2 ,
9
2 .
E8C.6(b) �e Clebsch–Gordan series [8C.5–332] gives the possible values of J as J =
j1 + j2 , j1 + j2 − 1, . . . ∣ j1 − j2∣. With j1 = 5, j2 = 3, the possible values of J are
J = 8, 7, 6, 5, 4, 3, 2 .
E8C.7(b) �e symbol F implies that the total orbital angular momentum L = 3 , the su-
perscript 3 implies that the multiplicity 2S+1 = 3, so that the total spin angular
momentum S = 1 . �e subscript 4 implies that the total angular momentum
J = 4 .