Water H2O

The water molecule has a relatively large dipole moment, almost 20 times that of CO. Here, the associated vector f is directed along the symmetry axis through the oxygen atom (Figure 5.15). Within the ground vibrational state are a large number of allowed rotational transitions at far-infrared and millimeter wavelengths. Excitation of these levels followed by prompt radiative decay provides an important cooling mechanism in shock-heated clouds, where accelerated chemical reactions produce a relatively high abundance of the molecule. Absorption by atmospheric water is an impediment that has prevented H2O from becoming a primary diagnostic of cloud conditions. Spaceborne observations, first by the SWAS satellite, have yielded an upper bound to the H2O abundance in quiescent clouds (Table 5.1). A number of higher transitions had previously been detected, starting with the important 22.2 GHz (1.35 cm) line discovered in 1969. This line, along with others subsequently found, are actually maser transitions, in which an enhanced population of the upper state creates extraordinarily strong emission. We will discuss water masers and their application in Chapter 14.

5.5.1 The Asymmetric Top

The rotational emission spectrum of H2O is more complex than that of NH3 because the molecule is an asymmetric top, with three unequal moments of inertia along its principal axes (Figure 5.15). Classically, the quantities conserved during rotation are the total vector angu-

Water Molecule Asymmetrical
Figure 5.13 left panel: The inversion of NH3, as the nitrogen atom tunnels through the plane of hydrogens. right panel: The potential energy of the molecule, shown as a function of the nitrogen atom's distance from the hydrogen plane. Note the central energy barrier.

NH3 (1,1) Hyperfine Splitting

Inversion Electric Magnetic Interaction

Quadrupole

Figure 5.14 Splitting of the inversion line in the NH3 (1,1) state. The various frequency differences are indicated, along with allowed transitions.

Axis B

Axis B

Axis A

Figure 5.15 Molecular structure of H2O, showing the three principal axes and the electric dipole moment f.

Axis C

Axis C

Axis A

Figure 5.15 Molecular structure of H2O, showing the three principal axes and the electric dipole moment f.

lar momentum J and its projection along an axis fixed in inertial space, but not the projection along any axis tied to the molecule itself. Thus there is no second quantum number beyond J to parametrize the rotational energy. This energy may be expressed as a complicated function of J and the rotational constants A, B, and C, which correspond to the three principal axes in Figure 5.15. It is conventional to order the constants so that A > B > C. Since B is numerically closer in value to C than to A, the molecule is more prolate than oblate.

Generalizing from the JK notation for symmetric tops, the rotational states are labeled with three numbers in the form JK-1kKi . The first subscript is the K-value of that prolate symmetric top state obtained if the rotational constant B were changed to C. Similarly, K1 is the subscript for the oblate configuration created by letting B tend toward A.2 The dipole selection rules allow J to change by 0 or ±1, while K_i or K1 can each change by ±1 or ±3. The so-called eo states, i. e., those with even K_1 and odd K1, can only change into oe states and vice versa, while ee and oo states are similarly linked. Physically, these two separate classes are distinguished by the sign change of the molecular wave function under a 180° rotation about the symmetry axis.

Any given rotational state has a number of equal-energy sublevels corresponding to different orientations of the spins of the two hydrogen nuclei. This degeneracy is three times greater for the eo and oe ("ortho") class of states, which are correspondingly more populated than the ee and oo ( "para") class. The energy-level diagram of Figure 5.16 displays separately the lower ortho- and para-rotational states. These are arranged so that states with a common J-value occupy the same column.

5.5.2 Observed Rotational Lines

The water molecule's large dipole moment implies that many downward transitions have relatively high A-values, particularly those between levels with the same J and neighboring K-values. As a consequence, it is rather difficult to excite the higher levels collisionally. For example, the 110 ^ 101 transition has A = 3.5 x 10_3 s-1 and a collisional deexcitation rate of 2.0 x 10_10 cm_3 s-1 at Tkin = 20 K. The corresponding value of ncr;t is 2x 107 cm_3. This is far greater than densities in quiescent clouds, but attainable in shocked regions near massive

2 The rather cumbersome notation for the subscripts reflects the fact that a standard parameter measuring molecular asymmetry tends toward — 1 for prolate configurations and toward +1 for oblate structures.

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