The rotational state populations start at . Their statistical weight factor initially favors higher , while the Boltzmann factor eventually suppresses them, producing one maximum followed by a tail tending to zero. The physically allowed values occur at integer ; a continuous curve only guides the eye.
Set , the rotational temperature. For a continuous nonnegative variable ,Equating this derivative to zero gives the most populated rotational levelThe second derivative is , so this is a maximum when . At lower temperatures the constrained continuous maximum is . For the physical integer maximum compare the nearby allowed levels explicitly.
The calculated rotational constant gives . At room temperature, taken as , the continuous maximum is ; comparing the adjacent integer populations gives . Taking instead gives and the same integer answer.
At , . In particular,and , so . The integer comparison avoids treating a noninteger angular momentum quantum number as a physical state.
Low rotational energy spacings can be excited in cold interstellar gas when electronic and most vibrational excitations are frozen out. A polar molecule such as carbon monosulfide has allowed electric dipole transition lines whose frequencies depend on its moment of inertia, providing a distinctive pattern for rotational spectral line assignment. Several matching spectral lines and their isotope shifts give a much stronger identification than one coincidence. The method is especially effective for polar species; a molecule without a permanent electric dipole moment lacks an ordinary pure electric-dipole rotational spectrum.
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