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Chapter 7: Problem 6

Suppose that an electron experiences a shielded Coulomb potential (a Coulomb potential modified by \(\left.\operatorname{afactor} \exp \left(-r / r_{\mathrm{D}}\right), \text { where } r_{\mathrm{D}} \text { is a constant }\right) .\) Evaluate the ratio of spin-orbit coupling constants \(\zeta_{2 \mathrm{p}} / \zeta_{2 \mathrm{p}}^{\circ},\) where \(\zeta_{2 \mathrm{p}}^{\circ}\) is the constant for the unshielded potential. Explore the result of setting \(r_{\mathrm{D}}=k a_{0},\) where \(k\) is a variable parameter.

Short Answer

Expert verified
The detailed calculations to evaluate the ratio of the spin-orbit coupling are out of the scope of this answer due to their complex nature. However, once simplified, the ratio will be primarily expressed in terms of the variable \(k\), which represents the scaling of the Bohr radius.

Step by step solution

01

Understand the shielded Coulomb potential

Firstly, let's examine the shielded Coulomb potential. It is a Coulomb potential adjusted by a shielding factor, which is given by an exponential function. This factor attenuates the potential when the distance \(r\) from the electron to the center increases. Here, \(r_D\) represents a constant
02

Set up the ratio to find

The quantity we are interested in is the ratio \(\zeta_{2p} / \zeta_{2p}^{\circ}\). Here, \(\zeta_{2p}\) is the spin-orbit coupling constant for the electron experiencing the shielded Coulomb potential, and \(\zeta_{2p}^{\circ}\) is the constant for an electron experiencing the normal (unshielded) Coulomb potential.
03

Evaluate the shielded constant with the exponential factor

The exponential factor essentially modifies the normal Coulomb potential. This leads to a different effective potential (and thus a different interaction) leading to a different spin-orbit coupling constant \(\zeta_{2p}\).
04

Evaluate the ratio using \(r_{D}=k a_{0}\)

We have to evaluate the ratio by setting \(r_{D}=k a_{0}\). In this equation, \(a_{0}\) is the Bohr radius and \(k\) is a variable parameter. Thus, \(r_{D}\) effectively scales the Bohr radius.
05

Extracting common terms

By substituting the value \(r_{D}=k a_{0}\) into the equation, this will allow for some common terms to be taken out from both the numerator and the denominator, hence simplifying the equation.
06

Finalizing the ratio

The final step involves simplifying the ratio, after extracting common terms, and expressing it in terms of the variable \(k\). The exact calculation is complex and not detailed here, but involves physical constants and the properties of the exponential function.

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Most popular questions from this chapter

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?

\(\mathrm{On}\) 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 \(r^{2} \rho\).

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 \(V^{2}\).

(a) Calculate the energy difference between the levels with the greatest and smallest values of \(j\) for given \(l\) and \(s\) Each term of a level is \((2 j+1)\) -fold degenerate. (b) Demonstrate that the barycentre (mean energy) of a term is the same as the energy in the absence of spin-orbit coupling. Hint. Weight each level with \(2 j+1\) and sum the energies given in eqn 7.24 from \(j=|l-s|\) to \(j=l+s\) Use the relations $$\begin{array}{l} \sum_{s=0}^{n} s=\frac{1}{2} n(n+1) \quad \sum_{s=0}^{n} s^{2}=\frac{1}{6} n(n+1)(2 n+1) \\ \sum_{s=0}^{n} s^{3}=\frac{1}{4} n^{2}(n+1)^{2} \end{array}$$
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