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lecture notes6

The rotational spectral lines are often labeled with

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Unformatted text preview: n from Diatomic Molecules We are now ready to consider the selection rules for radiation from diatomic molecules. We may consider three types of transitions: rotational, vibrational, and electronic, in order of increasing energy. In the latter cases, the quantum numbers associated with lower ­energy phenomena also may change, e.g. a vibrational transition is often accompanied by a change in angular momentum. In all cases, the exact selection rules apply. For example, in electric dipole transitions (the most common type), we must have |ΔJ|≤1 and a change in parity. A. ROTATION SPECTRA We focus on the rotation spectra of closed ­shell molecules in their ground electronic and vibrational state. The typical transitions are electric dipole decays with ΔJ=−1. The frequency of the emitted photon can be obtained by subtracting the energies of the levels: 1 Ⱥ 2 J ( J + 1) 2 ( J u − 1) J u Ⱥ h ν = Ⱥ u u − J u . Ⱥ = h Ⱥ 2I 2I Ⱥ 4 πI The quantity h/8πI is often denoted by B and is called the rotational constant. The spectral lines of the molecule are at frequencies 2B, 4B, 6B, etc.: they are evenly € spaced. A slight deviation from this pattern occurs due to rotational deformation of the molecule (the bond lengthens as the molecule rotates faster), so that at high quantum numbers the frequencies are packaged more closely together. The decay rate is obtained from the usual formula, 2 4ω 3 AJ →J −1 = 3 ∑ J − 1, M f µ JM i . 3c M f Here μ is the dipole moment operator. The matrix element is an integral over the spherical harmonics, which evaluates to € AJ →J −1 = € 4ω 3 ∑ J − 1, M f µ JM i 3c 3 M f 2 = 4ω 3µ 2 ∑ 3c 3 M f 2 ∫ nYJ*−1,M f (n)YJM i (n)d 2n = 4ω 3µ 2 J . 3c 3...
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