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I learn that a band head is a region of maximum intensity due to many lines falling together. It can be determined by differentiating the expression for the R branch.

To no avail, I cannot find any explanation on why band heads arise after going through a few textbooks and Internet resources.

My guess: It might be due to the breakdown of Born-Oppenheimer Approximation which explain the unequal spacing in vib-rot spectra. The rotational constants decreases at higher vibrational states, to a limit?

Why does the absorption lines of R branch seem to 'U-turn' at higher J'' value?

Attached is the vib-rot spectra. This is one of the materials I searched from the Internet.(https://cefrc.princeton.edu/sites/g/files/toruqf1071/files/Files/2015%20Lecture%20Notes/Hanson/pLecture2.pdf , slide 28)enter image description here

FYI, I am at the entry-level of learning molecular spectroscopy. Any explanation is really appreciated.

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It's been a long time since I did this, but from my old notes... this is caused by the rotational constant $B$ being different in the two vibrational states $v = 0$ and $v = 1$. Generally we have that $B_1 < B_0$ (where $B_n$ denotes $B$ in the vibrational state $v = n$). This makes sense because $B = h/8\pi^2cI$ (expressed in wavenumber units) is inversely proportional to the moment of inertia $I$, and we expect $I$ to increase with vibrational excitation.

So, if $v_0$ is the vibrational frequency, then the R-branch transition frequencies (for the $(J, v) \to (J + 1, v+1)$ transition) are given by

$$\begin{align} R(J) &= v_0 + B_1(J+1)(J+2) - B_0J(J+1) \\ &= v_0 + J^2(B_1 - B_0) + J(3B_1 - B_0) + 2B_1 \end{align}$$

and the spacing between successive transitions is

$$\begin{align} R(J + 1) - R(J) &= [(J+1)^2 - J^2](B_1 - B_0) + [(J+1) - J](3B_1 - B_0) \\ &= (2J + 1)(B_1 - B_0) + 3B_1 - B_0 \\ &= 2J(B_1 - B_0) + 4B_1 - 2B_0 \end{align}$$

Since $B_1 - B_0 < 0$, this separation decreases with increasing $J$ as observed. (Notice that if we let $B_1 = B_0 = B$ then we recover the "simple" behaviour where the spacing between lines is $2B$.) Naturally, exactly the opposite scenario is seen in the P-branch; I'll leave the maths to the reader.

A very thorough book for spectroscopy is Hollas, Modern Spectroscopy (4th ed.), which explains all of this in detail.

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