Page 79 - High Power Laser Handbook

Chemical Lasers 49

Figure 3.3
Diatomic molecule
motions that
determine energy
levels.

x y

The partition function Z is approximately proportional to the
temperature T and has a weak dependence on the vibrational
level v, because B is also weakly dependent on v. When las-
J
ing, although deviations from this simple distribution can be
taken into account in advanced models, the general trend
remains.
3. Vibrational motion: Classically, vibrational motion can be
viewed as a simple harmonic motion springlike mode. Quan-
tum mechanically, it is characterized as a simple harmonic
oscillator that sometimes includes anharmonic terms. The
associated energy levels are simply:

ω(v + 1/2) (3.4)

where ω is vibrational quantum energy and v is vibrational
quantum number.

In contrast to translational and rotational degrees of freedom, vir-
tually all molecules are in the vibrational ground state (v = 0) in the
absence of chemical pumping. When pumping or lasing is occurring,
the assumption of a simple thermal behavior correlated with the
translational temperature is invalid.
The rotational and vibrational energy level expressions were sim-
plified and do not include the higher-order terms necessary to accu-
rately determine energy levels. Equation (3.5) is a more accurate
expression:

E(v, J) = ω(v + 1/2) + X(v + 1/2) + B × J × ( J + 1)
2
J
+ B (v + 1/2) × J × ( J + 1) + higher-order terms (3.5)
1J
Typical values for HF and DF in the energy unit wave number are
shown in Table 3.2.

48