Correction of Some Erroneous Calculations of the Einstein A Coefficient for the 18 cm Transition of OH

Abstract

IN a recent note¹ Turner pointed out the confusion over the value of the Einstein A coefficient for the ²II_3/2, J = 3/2, Λ-doublet transition of OH. This coefficient is required to derive the interstellar abundance of OH from the intensity of the OH absorption lines observed in radio astronomy. Turner points out an error in previous calculations which may be traced to an incorrect matrix element in equation (2–16) of Townes and Schawlow². His own calculation, however, is also in error in some respects. The calculation of the transition probability can most conveniently be done in two steps. First the probability can be calculated for the pure Λ-doublet transition, without regard to hyperfine structure. Second, the effect of the hyperfine structure can be treated by well-known methods. The Einstein A coefficient for the Λ-doublet transition is related to the dipole matrix element μ_ij by where |μ_ij| is given by³ The summation is carried out only over M′, since A_Λ represents the probability that a molecule in one particular state M of the upper level will make a transition to any state M′ of the lower level. The dipole matrix element may be related to the line strength S_ij in the usual manner where μ is the permanent electric moment. S_ij is identical with the familiar symmetric rotor line strength⁴ if K (the component of angular momentum along the molecular axis) is replaced by <Ω> ; we thus have so that With the use of the most recent molecular constants of Radford⁵, which lead to Ω = 1.470 and the accurate dipole moment (1.660±0.010 D) of Powell and Lide⁶, we finally obtain for the ²II_3/2, J = 3/2 transition