There is a useful parallel between a lithium atom and a hydrogen atom in its first excited state: the lithium nucleus, shielded by the two n = 1 electrons, appears to have the same net charge as the proton in hydrogen, so the n = 2 electron moves in a similar electric field to that experienced by an electron with n = 2 in hydrogen. We can test this parallel quantitatively by comparing the ionisation energy of lithium (5 . 39 eV) with the energy of H with n = 2 (3 . 40 eV). This agreement is not terribly good because the n = 2 ,l = 0 wavefunction that forms the ground state of lithium overlaps
238 Chapter 10: Helium and the periodic table
10.3 Periodic table 239 significantly with the n = 1 wavefunction, and therefore has exposure to the full nuclear charge. There is a more satisfying parallel between the first excited state of lithium, in which the n = 2 electron has l = 1 and the corresponding state of hydrogen: in this state lithium has ionisation energy 3 . 54 eV. Consider now the effect of transmuting lithium into beryllium by simul- taneously increasing the nuclear charge by one unit and adding a second electron to the n = 2, l = 0 state. The parallel that we have just described suggests that this operation will be analogous to moving up from hydrogen to helium, and will significantly increase the ionisation energy of the atom. Experiment bears out this expectation, for the ionisation energy of beryl- lium is 9 . 32 eV, 1 . 7 times that of lithium. As in helium, the ground state of beryllium has total spin zero, while the first excited states have spin one. However, whereas in the excited states of helium the two electrons have dif- ferent values of n , in beryllium they both have n = 2, and they differ in their values of l . Consequently, the overlap between the single-electron states that form the beryllium triplet is significantly larger than the corresponding over- lap in helium. This fact makes the exchange integral in equations (10.32) large and causes the singlet excited state to lie 2 . 5 eV above the triplet of excited states. If we add a unit of charge to the nucleus of a beryllium atom, we create an atom of singly ionised boron. The four electrons with l = 0 that envelop the ion’s nucleus screen the nuclear charge to a considerable extent from the perspective of the lowest-energy unfilled single-particle state, which is a 2 p state ( n = 2, l = 1). The screening is far from complete, however, so the nuclear charge Z that the outermost electron perceives is greater than unity and the dynamics of the outermost electron of boron is similar to that of the electron in a hydrogenic atom with Z > 1. The ionisation energy from the n = 2 level of hydrogen is 1 4 Z 2 R = 3 . 40 Z 2 eV, while that of boron is 8 . 30 eV, so Z ∼ 1 . 6. Spin-orbit coupling causes the ground state of boron to form the 2 P 1 / 2 term in which the electron’s spin and orbital angular momenta are antipar- allel. At this early stage in the periodic table, spin-orbit coupling is weak, so the excited states of the 2 P 3 / 2 term lie only 0 . 0019 eV above the ground state. C +
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