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Pure Rotation

Pure rotational transitions are between states where only rotational quantum numbers are changed,
$\displaystyle \vert Km_K, v, \alpha\rangle \to \vert K'm_K', v, \alpha\rangle$     (2.2)

leaving the vibrational quantum number(s) \bgroup\color{col1}$ v$\egroup and the electronic quantum number(s) \bgroup\color{col1}$ \alpha$\egroup unchanged. Such transitions then depend on matrix elements of the dipole operator,
$\displaystyle \langle Km_K\vert {\bf d} \vert K'm_K'\rangle.$     (2.3)

The calculation of this matrix element, using spherical harmonics, yields the purely rotational selection rules
$\displaystyle \Delta K = \pm 1,\quad \Delta m_K = 0,\pm 1.$     (2.4)

Writing the rotational part of the energy as
    $\displaystyle \varepsilon_{\rm rot}(K) = BK(K+1)$ (2.5)
  $\displaystyle \leadsto$ $\displaystyle \Delta\varepsilon_{\rm rot}(K)\equiv
B(K+1)(K+2) - BK(K+1) = 2B (K+1).$  

The distance between the corresponding spectral lines is constant, \bgroup\color{col1}$ \Delta\varepsilon_{\rm vib}(K+1)-\Delta\varepsilon_{\rm vib}(K)=2B$\egroup.


next up previous contents index
Next: Pure Vibration Up: Selection Rules Previous: Dipole Approximation   Contents   Index
Tobias Brandes 2005-04-26