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Chapter 7: Problem 18

An excited state of atomic calcium has the electron configuration \(1 \mathrm{s}^{2} 2 \mathrm{s}^{2} 2 \mathrm{p}^{6} 3 \mathrm{s}^{2} 3 \mathrm{p}^{6} 3 \mathrm{d}^{1} 4 \mathrm{f}^{1}\) (a) Derive all the term symbols (with the appropriate specifications of \(S, L \text { and } J)\) for the electron configuration. (b) Which term symbol corresponds to the lowest energy of this electron configuration? (c) Consider a \(^{3} \mathrm{F}_{2}\) level of calcium derived from a different electron configuration than that shown above. Which of the term symbols determined in part (a) can participate in spectroscopic transitions to this \(^{3} \mathrm{F}_{2}\) level?

Short Answer

Expert verified
The derived term symbols are \(^{2}D\), \(^{2}F\), \(^{4}F\), \(^{2}G\), \(^{4}G\), \(^{2}H\), \(^{4}H\). The term symbol with the lowest energy for this configuration is \(^{4}H\). The only terms that can participate in spectroscopic transitions to \(^{3}\mathrm{F}_{2}\) are \(^{4}F_{1}\) and \(^{4}F_{2}\).

Step by step solution

01

Derive All Term Symbols

For two unpaired electrons, determine combinations of different values of their spin and orbital quantum numbers. Spin quantum number \(s = ±1/2\) and orbital quantum number \(l\) can range from 0 to 3 (or s, p, d, f). Calculate the term symbol for each combination using the formula: \(^{2S+1}L_{J}\), where \(S\) is the total spin quantum number, \(L\) is the total orbital quantum number, and \(J\) is the total angular momentum quantum number. For this electron configuration, the possible term symbols are \(^{2}D\), \(^{2}F\), \(^{4}F\), \(^{2}G\), \(^{4}G\), \(^{2}H\), and \(^{4}H\).
02

Determine the Lowest Energy Term Symbol

According to Hund's rules, the term symbol with the largest spin multiplicity \(2S+1\) has the lowest energy. If there is a tie, the one with the larger \(L\) value is lower in energy, and if there’s still a tie, the option with lower \(J\) value (for less than half-filled shells) or higher \(J\) value (for more than half-filled shells) is lower in energy. Here, the term withlargest spin multiplicity and highest \(L\) would have the lowest energy. Therefore, the term symbol with the lowest energy for this configuration is \(^{4}H\).
03

Find Possible Spectroscopic Transitions

Spectroscopic transitions can occur if the change in \(J\) is 0, ±1 (except \(J=0\) to \(J=0\)) and the spin multiplicity does not change. From the derived term symbols specified in step 1, the only term that can participate in spectroscopic transitions to \(^{3}\mathrm{F}_{2}\) are \(^{4}F_{1}\) and \(^{4}F_{2}\).

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Most popular questions from this chapter

Take a trial function for the helium atom as \(\psi=\) \(\psi(1) \psi(2),\) with \(\psi(1)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta r_{1}}\) and \(\psi(2)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta_{2}}, \zeta\) being a parameter, and find the best ground-state energy for a function of this form, and the corresponding value of \(\zeta\). Calculate the first and second ionization energies. Hint. Use the variation theorem. All the integrals are standard; the electron repulsion term is calculated in Example 7.2 Interpret \(Z\) in terms of a shielding constant. The experimental ionization energies are \(24.58 \mathrm{eV}\) and \(54.40 \mathrm{eV}\).

Find the first-order corrections to the energies of the hydrogen atom that result from the relativistic mass increase of the electron. Hint. The energy is related to the momentum by \(E=\left(p^{2} c^{2}+m^{2} c^{4}\right)^{1 / 2}+V .\) When \(p^{2} c^{2} \ll m^{2} c^{4}\), \(E \approx \mu c^{2}+p^{2} / 2 \mu+V-p^{4} / 8 \mu^{3} c^{2},\) where the reduced mass \(\mu\) has replaced \(m\). Ignore the rest energy \(\mu c^{2},\) which simply fixes the zero. The term \(-p^{4} / 8 \mu^{3} c^{2}\) is a perturbation; hence calculate \(\left\langle n l m_{l}\left|H^{(1)}\right| n l m_{l}\right\rangle=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(p^{2} / 2 \mu\right)^{2}\right| n l m_{l}\right\rangle\) \(=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(E_{n l m_{l}}-V\right)^{2}\right| n l m_{l}\right\rangle .\) We know \(E_{n l m} ;\) therefore calculate the matrix elements of \(V=-e^{2} / 4 \pi \varepsilon_{0} r\) and \(V^{2}\).

Demonstrate that for one-electron atoms the selection rules are \(\Delta l=\pm 1, \Delta m_{l}=0,\pm 1,\) and \(\Delta n\) unlimited. Hint. Evaluate the electric-dipole transition moment \(\left\langle n^{\prime}\left|m_{l}^{\prime}\right| \mu | n l m_{l}\right\rangle\) with \(\mu_{x}=-e r \sin \theta \cos \varphi, \mu_{y}=-e r \sin \theta \sin \varphi,\) and \(\mu_{z}=-e r \cos \theta\). The easiest way of evaluating the angular integrals is to recognize that the components just listed are proportional to \(Y_{l m,}\) with \(l=1,\) and to analyse the resulting integral group theoretically.

Consider a one-dimensional square well containing two electrons. One electron has \(n=1\) and the other has \(n=2 .\) Plot a two-dimensional contour diagram of the probability distribution of the electrons when their spins are (a) parallel, (b) antiparallel. Devise a measure of the radius of the Fermi hole. Hint. Recall the discussion in Section \(7.11 .\) When the spins are parallel (for example, \(\alpha \alpha\) ) the antisymmetric combination \(\psi_{1}(1) \psi_{2}(2)-\psi_{2}(1) \psi_{1}(2)\) must be used, and when the spins are antiparallel, the symmetric combination must be used. In each case plot \(\psi^{2}\) against axes labelled \(x_{1}\) and \(x_{2}\). Computer graphics may be used to obtain striking diagrams, but a sketch is sufficient.

\(\mathrm{On}\) the basis of the Thomas-Fermi model of an atom, evaluate the radius within which there is a 50 per cent probability of finding the electron density and evaluate it for the Period 2 elements. Hint. Use the radial distribution function \(r^{2} \rho\).
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