<p>$$\textit{In this exercise we will study the rotational and vibrational modes of diatomic molecules. We will see how key quantities such as interatomic distance and atomic mass can be extracted from the emission spectrum.}$$</p>
<p>where $\psi(R)$ is the radial part of the wavefunction, $\mu$ is the reduced mass and $V(R)$ is the energy of the system with fixed nuclei position. We have seen two approximations of $V(R)$ - the harmonic potential</p>
<p>a.) Show that $V_{mor}(R)$ is minimal at $R = R_e$. Expand the potential up to second order at the minimum and express $k$ as a function of $D_e$ and $a$. Show that this approximation recovers exactly the harmonic term of the Morse potential spectrum (i.e. the anharmonicity term with $x_e$ is excluded).</p>
<p>b.) Taking $D_e \approx 1$ eV and $a \approx 1/$Ang and $\mu = 1u$ as typical orders of magnitudes, estimate the energy scale of the vibrational spectrum. Express the characteristic energy in the units of temperature, photon wavelength and photon frequency.</p>
<p>c.) Using that $R_e \approx 1$ Ang and $\mu \approx 1u$, estimate the energy scale of the rotational spectrum, as well as the corresponding temperature, photon wavelength and frequency.</p>
<p>Since the rotational energy scale is much smaller than the vibrational energy scale one might add the rotational energy as a first order perturbation in the full Hamiltonian:</p>
where $E_{vib, n}$ is the pure vibrational energy of the $n$-th state, and express $B_n$ in terms of $\langle \psi_n|\frac{1}{R^2}|\psi_n \rangle$. Further express the emission spectrum of the $n = 1 \to 0$ transition in terms of $\Delta E_{vib} = E_{vib, 1}
-E_{vib,0}$ and $B_1$ and $B_0$.</p>
<p>$\textit {Remark: Recall that only the $\Delta J = \pm 1$ transitions are allowed. This means that every transition either belongs to the P-branch (where the angular momentum of the $n = 1$ vibrational}$
$\textit{mode is lower than the $n = 0$ vibrational mode) or the R-branch (where the angular momentum of the $n = 1$ vibrational mode is higher than the $n = 0$ vibrational mode).}$</p>
<p>We will now turn our attention to a qualitative analysis of $B_e$, $B_0$ and $B_1$.</p>
<p>e.) Which approximation of $\langle \psi_n|\frac{1}{R^2}|\psi_n \rangle$ was used in the rigid rotor assumption to get $B_e = B_0 = B_1$?</p>
<p>f.) Assume we go beyond the rigid rotor assumption, such that $B_n$ is not only defined by the position of the potential minimum $R_e$ but requires information of the entire potential $V(R)$. Order the energies $B_e$, $B_0$ and $B_1$ for a parabolic potential $V_{par}(R)$.</p>
<p>g.) Sketch a potential characterized by $B_e > B_0 > B_1$.</p>
<p>$\textit{Hint: The Morse potentialfulfills this condition. Explain which characteristics of the potential are essential for this ordering.}$</p>
<p>In the last part we address the information that can be obtained from the emission spectrum. This is often used in astrophysics e.g. to study the composition and temperature of interstallar medium (see Figure 1). The following spectrum (Figure 2) is obtained from a diatomic molecule in this cloud, and the transition in question is $n =1 \to 0$. The rigid rotor approximation is used.</p>
<p>$\textit{Technical Remark on the numerical Fourier Transform}$:</p>
<p>Imagine we want to calculate $ g(k) := \hat{f}(2 \pi k) = \int e^{2\pi i kx} f(x) dx$, where $\hat{f}$ is the usual Fourier Transform of $f$.
Numerically this is obtained as</p>
<p>$$g[m] = \sum_{n = 0}^{N-1}e^{-\frac{2 \pi i mn}{N}} f[n]$$</p>
<p>where we have dropped the normalization for the sake of simplicity. Writing $x[n] = n dv$ we see that defining $k[m] = \frac{m}{dv N}$ ensures consistency between the numerical and analytical result. Here $N$ (the number of data points) was the length of the arrays $f$ and $g$ , and $dv$ is the spacing between the $x$-values.</p>
<p>With regards to the following code we have:</p>
<p>$f$ is written using the variable fs (the photon count)</p>
<p>$x$ is written using the variable vs (the photon frequency in GHz)</p>
<p>$g$ is written using the variable F (the numerical Fourier Transform of fs)</p>
<p>i) A close up of the Fourier Transform shows 4 fundamental frequencies (all around $0.01$ GHz$^{-1}$) corresponding to 4 dominant rotational modes. Determine their relative strength and the corresponding photon frequency.</p>
<p>$\textit{Remark: We recommend analyzing the higher harmonics, where the signal is more spread out and the 4 rotational modes can be better distinguished. The code snipplet takes e.g. the 7-th harmonics (from $0.06-0.66$ GHz$^{-1}$).}$</p>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Close up of the Fourier Transform |F(ν)|'</span><span class="p">)</span>
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"/>
<p>j) Assume that the interstellar medium only contains two of the four elements in the table. Deduce from the ratio of the photon frequencies in the emission spectrum what these two elements are. What are the relative abundances of the isotopes in the interstellar medium.</p>
<p>$\textit{Hint: Use that in the rigid rotor approximation the reduced mass is proportional to the photon wavelength. Furthermore assume that the emission spectrum has $\delta$-peak like signals, such that, ignoring temperature dependences (which are rather small), the magnitudes}$
$\textit{of the (fundamental or) higher harmonics directly reflects the isotope abundance.}$</p>
<span class="c1">#Compare np.sort(1/rot_fs) and np.sort(redms)/ratio with a method of your liking</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The abundance ratio of the first element is'</span><span class="p">)</span> <span class="c1">#Deduce using h1/h2, h1/h3, h1/h4</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The abundance ratio of the second element is'</span><span class="p">)</span> <span class="c1">#Deduce using h1/h2, h1/h3, h1/h4</span>
<p>l) At last we are intersted in the temperature of the ISM. To deduce it we will concentrate solely on the R-branch of the diatomic molecule using the most abundant isotope combination.</p>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The temperature in the ISM is'</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s1">'K'</span><span class="p">)</span>
<p>$\textit{The equilibrium distance $R_e$ and the spring constant $k$ used in this exercise are both real experimental values.} $
$\textit{Furthermore, while the temperature of the ISM is realistic, the isotope ratios are not, and were solely chosen to simplify the numerical analysis.}$</p>
<p>a.) Assume that the polarizability $\alpha(x)$ is a function of the interatomic distance. Use a first order expansion and show that the dipole moment oscillates with 3 different
frequencies, $\nu$, $\nu + \nu_{vib}$ and $\nu- \nu_{vib}$.</p>
<p>$\textit{Remark: Picking up these frequency using a detector allows the measurement of $\nu_{vib}$. This is the underlying principle of Raman spectroscopy from a classical viewpoint.}$</p>
<p>We will now follow the same logic for the rotational spectrum.</p>
<p>b.) We write the polarizability matrix in the principle coordinates as</p>
<p>$$ \alpha = \begin{pmatrix} \alpha_{\perp} & 0 & 0 \\ 0 & \alpha_{\perp} & 0 \\ 0 & 0& \alpha_{\parallel} \end{pmatrix}$$</p>
<p>where the $\alpha_{\parallel}$ point along the axis connecting the two atoms. Express the induced dipole moment $\mu_{E}$ collinear to the incident electrical field. What is the corresponding $\alpha_E$.</p>
<p>$\textit{Use spherical coordinates and express everything in terms of $\theta$, defined as the angle between the incident light and the interatomic axis.}$</p>