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Chapter 6: Problem 26

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).

Short Answer

Expert verified
The coefficients \(A\) and \(B\) are dependent on the atomic number in the order of \(Z^4\) and \(Z^2\) respectively, while the density of states scales with \(Z^3\). The stimulated emission rate in black-body radiation at \(1000 \mathrm{K}\) depicts a dependence of approximately \(\propto Z^5 \cdot T^4\).

Step by step solution

01

Relation between Atomic Number and Coefficients

Using the quantum mechanical model of hydrogenic atoms, the coefficients \(A\) for spontaneous emission and \(B\) for stimulated emission can be shown to depend on the atomic number as \(A \propto Z^4\) and \(B \propto Z^2\). This is due to the increased strength of the electrical field caused by the increased positive charge as the atomic number increases, leading to stronger interaction with the emission or absorption of photons.
02

Density of States Dependency

The density of states also shows a dependency on the atomic number \(Z\), scaling as \(Z^3\). This is a consequence of the increased energy levels available for transitions as the atomic number increases.
03

Stimulated Emission Rate

The stimulated emission rate depends on the density of the states and the absorption coefficient \(B\). Given that both show dependency on \(Z\) and using Einstein's coefficients relationship, the stimulated emission rate can be shown to also depend on \(Z\), being approximately \(\propto Z^5\).
04

Effect of Black-body Radiation

For an atom exposed to black-body radiation at \(1000 \mathrm{K}\), the stimulated emission rate will further depend on the intensity of the incident radiation, which is proportional to the fourth power of the temperature, \(\propto T^4\), according to Stefan–Boltzmann law. Thus, taking into account the already established \(Z^5\) dependence, and considering the temperature factor, the stimulated emission rate will show a final dependence of \(\propto Z^5 \cdot T^4\).

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Most popular questions from this chapter

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

\(\mathrm{A}\) biradical is prepared with its two electrons in a singlet state. A magnetic field is present, and because the two electrons are in different environments their interaction with the field is \(\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} .\) Evaluate the time-dependence of the probability that the electron spins will acquire a triplet configuration (that is, the probability that the \(S=1, M_{s}=0\) state will be populated). Examine the role of the energy separation \(h J\) of the singlet state and the \(M_{S}=0\) state of the triplet. Suppose \(g_{1}-g_{2}\) \(\approx 1 \times 10^{-3}\) and \(J \approx 0 ;\) how long does it take for the triplet state to emerge when \(\mathscr{B}=1.0 \mathrm{T}\) ? Hint. Use eqn 6.63 ; take \(|0,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta-\beta \alpha)\) and \(|1,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta+\beta \alpha) .\) See Problem 4.26 for the significance of \(\mu_{\mathrm{B}}\) and \(g\)

A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1 s-orbital (but with \(Z=2\) ). The electronelectron interaction is regarded as a perturbation, and calculation gives $$\int \psi_{1 s}^{2}\left(r_{1}\right)\left(\frac{e^{2}}{4 \pi \varepsilon_{0} r_{12}}\right) \psi_{1 s}^{2}\left(r_{2}\right) \mathrm{d} \tau=\frac{5}{4}\left(\frac{e^{2}}{4 \pi \varepsilon_{0} a_{0}}\right)$$ (see Example 7.2 ). Estimate (a) the binding energy of helium, (b) its first ionization energy. Hint. Use eqn 6.15 with \(E_{1}=E_{2}=E_{1 \mathrm{s}} .\) Be careful not to count the electronelectron interaction energy twice.

The symmetry of the ground electronic state of the water molecule is \(\mathrm{A}_{1}\). (a) An electric field, (b) a magnetic field is applied perpendicular to the molecular plane. What symmetry species of excited states may be mixed into the ground state by the perturbations? Hint. The electric interaction has the form \(H^{(1)}=a x ;\) the magnetic interaction has the form \(H^{(1)}=b l_{x^{*}}\)
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