5.2.3: Transition Intensities
- Page ID
- 419789
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Transition Intensities
The relative intensity of the P- and R-branch lines depends on the thermal distribution of rotational states; more specifically, they depend on the population of the \(J"\) states. The intensity of a spectral line depends not only on the transition probability (selection rules) and the frequency \(\nu\) (energy of radiation), but also on the number of molecules in any initial state. Since most infrared (IR) spectra are observed under conditions of thermal equilibrium, we need only consider the distribution of the molecules over the different quantum states in thermal equilibrium.
According to the Boltzmann equation, the number of molecules \(dN_E\) that have a classical vibrational energy between \(E\) and \(E + dE\) is proportional to \( \large e^{-\frac{E}{k_B T}}\normalsize dE\), where \(k_B\) is Boltzmann's constant and \(T\) is the absolute temperature. In quantum theory, only discrete values, \(E_{(\nu)}\), are possible for the vibrational energy. The number of molecules in each of the vibrational states is then proportional to the Boltzmann factor \(\large e^{-\frac{E_{(\nu)}}{k_b T}}\). The zero-point energy can be left out, since to add this to the exponent would only result in a multiplication factor that would be the same for all the vibrational levels (including the zero level).
The thermal distribution of the rotational levels (unlike that of the vibrational levels) is not simply given by the Boltzmann factor. We must account for the fact that, according to quantum theory, each state of an atomic system with total angular momentum J consists of 2J+1 energy levels which are degenerate in the absence of an external field; that is, the state has a (2J+1)-fold degeneracy. The number of molecules NJ in the rotational level J of the lowest vibrational state at the temperature T is thus proportional to
\[ \large N_J \propto \left ( 2J+1 \right )e^{-\huge \frac{F_{(J)}hc}{k_B T}} \]
where F(J) is the rotational term value (equal to \(\large \frac{E_{(J)}}{hc}\)). For sufficiently large T or small rotational constant B, the population of the Jth rotational level, for N molecules, is given by the following equation [see p. 125 in reference (3)]
\[ \large N_J =N\frac{hcB}{kT}\left ( 2J+1 \right )e^{-\huge \frac{BJ \left ( J+1 \right )hc}{kT}} \]
Since the factor 2J+1 increases linearly with J, the number of molecules in the different rotational levels goes through a maximum before decreasing with the rotational quantum number. It can be shown3 that this maximum lies at
\[ \large J_{max}=\sqrt{\frac{kT}{2Bhc}}-\frac{1}{2}=0.5896\sqrt{\frac{T}{B}}-\frac{1}{2} \]
This expression clearly shows that Jmax increases with decreasing B and increasing T. It should be noted that the number of molecules in the lowest rotational level, J = 0, is not zero. With increasing temperature, the band extends over a wider frequency range and the intensity maxima of the two rotational branches move outward and at the same time become flatter.
This is the reason that rovibrational spectral lines increase in energy to a maximum as J increases, then decrease to zero as J continues to increase, as seen in Figure 5.2.3.2 .
From this relationship, we can also deduce that in heavier molecules, \(B\) will decrease because the moment of inertia will increase, and the decrease in the exponential factor is less pronounced. This results in the population distribution shifting to higher values of J. Similarly, as temperature increases, the population distribution will shift towards higher values of \(J\).