Berrington, Cooling rate of thermal electrons by electron impact excitation of ®ne structure levels of atomic oxygen

The atomic oxygen fine structure cooling rate of thermal electrons based on new effective collision strengths for electron impact excitation of the ground-state 3P fine-structure levels in atomic oxygen have been fitted to an analytical expression which is available to the researcher for quick reference and accurate computer modeling with a minimum of calculations. We found that at the F region altitudes of the ionosphere the new cooling rate is much less than the currently used fine structure cooling rates (up to a factor of 2–4), and this cooling rate is not the dominant electron cooling process in the F region of the ionosphere at middle latitudes.


Introduction
The electron temperature in the ionosphere is of great signi®cance in that it usually controls the rates of many physical and chemical ionospheric processes. The theoretical computation of electron temperature distribution in the ionosphere requires the knowledge of various heating and cooling rates, and heat transport through conduction. Schunk and Nagy (1978) have reviewed the theory of these processes and presented the generally accepted electron cooling rates. Pavlov (1998a, c) has revised and evaluated the electron cooling rates by vibrational and rotational excitation of N 2 and O 2 and concluded that the generally accepted electron cooling rates of Prasad and Furman (1973) due to the excitation of O 2 1 D g and O 2 1 R g are negligible in comparison with those for vibrational excitation of O 2 .
The thermal electron impact excitation of the ®ne structure levels of the 3 ground state of atomic oxygen is presently believed to be one of the dominant electron cooling processes in the F region of the ionosphere (Dalgarno and Degges, 1968;Hoegy, 1976;Schunk and Nagy, 1978;Carlson and Mantas, 1982;Richards et al., 1986;Richards and Khazanov, 1997). To evaluate the energy loss rate for this process, Dalgarno and Degges (1968) have employed the theoretical O 3 excitation cross sections given by Breig and Lin (1966). The electron cooling rates of Hoegy (1976) and Carlson and Mantas (1982) which are currently used in models of the ionosphere are based on the excitation cross sections calculated by Tambe andHenry (1974, 1976) and Le Dourneuf and Nesbet (1976). The shortcomings of the theoretical approach of Tambe andHenry (1974, 1976) and Le Dourneuf and Nesbet (1976) were summarised by Berrington (1988) and Bell et al. (1998) Bell et al. (1998 improved the work of Berrington (1988) and presented the numerical calculations of the rate coecient of this electron cooling rate for the electron temperature, e 200, 500, 1000, 2000, and 3000 K and the neutral temperature, n 100, 300, 1000, and 2000 K. The primary object of this study is to use the theoretical O 3 excitation cross sections of Bell et al. (1998) to calculate and to ®t to a new analytical expression for atomic oxygen ®ne structure cooling rate of thermal electrons.
2 The electron cooling rate by electron impact excitation of ®ne-structure levels of atomic oxygen The O 3 ground state is split into three ®ne structure levels 3 i i 2Y 1Y 0 with the level energies given by Radzig and Smirnov (1980) as i 2 0, i 1 227X7 K (or 0.01962 eV), and i 0 326X6 K (or 0.02814 eV). Collisions of thermal electrons with the ground state of atomic oxygen produce transitions among the O 3 i ®ne structure levels and the electron cooling. Sharma et al. (1994) found that within an accuracy of 1±2% the Correspondence to: A. V. Pavlov ®ne structure levels are in local thermodynamic equilibrium at the local neutral atom translation temperature, n , for altitudes up to 400 km: where g i 2i 1 is the statistical weight of the i-th level, is the full number density of atomic oxygen.
This study has been also conducted assuming that the velocity distribution of electrons is described by a Maxwellian distribution with a thermal electron gas of temperature, e . In this approximation the oxygen ®ne structure cooling rate is given by the expression (1) of Stubbe and Varnum (1972) as k is Boltzmann's coecient, the i 2 ground level with i 2 0 and the i 1 level with i 1 0X01962 eV of O 3 i are excited by thermal electrons, the O 3 j deexcitation levels are the j 0 upper level with i 0 0X02814 eV and the j 1 level, i ij i j À i i b 0, x ik e À1 , i is the energy of electrons, and m e denotes the mass of electrons, r ij is the cross section for excitation by electrons of O 3 from i-th to j-th state. It should be noted that deexcitation j 3 i cross sections of O 3 are related with excitation i 3 j cross sections of O 3 through the principle of detailed balancing. As a result, the excitation term of the electron cooling rate is de®ned as N e Oh À1 2 i1 j`i ij , and the deexcitation term of the electron cooling rate is N e Oh À1 2 i1 j`i ij exp i ij À1 e À À1 n À Á k À1 Â Ã . It follows from this de®nition of the cooling rate that in the energy balance equation for electrons, the cooling rate is subtracted from the electron heating rate received by thermal electrons from photoelectrons. The value of v is positive when e b n and negative when e`n .
The cross section, r ij i, for the transition i 3 j is obtained from the collision strength, X ij i, by (see Eq. 1 of Hoegy, 1976) where 0 is the Bohr radius, and y is the Rydberg constant. Using Eq. (4), we conclude that where the eective collision strength, ij e , is determined as ij e I 0 X ij x expÀxdx 6 Carlson and Mantas (1982) found that the collision strengths calculated by Tambe andHenry (1974, 1976) and Le Dourneuf and Nesbet (1976) can be approximated by the empirical formula where the constants e ij , f ij , g ij , and q ij are given in Table 1 of Carlson and Mantas (1982).
In the approximation of Eq. (7) the eective collision strength is calculated as ij e g ij k e f ij e ij 1 k e q ij À1 8 Figure 1 shows the eective collision strengths for the ®ne structure transitions 2 3 1 (left panel), 2 3 0  and Nesbet (1976) and Tambe andHenry (1974, 1976) calculated in the approximation of Eq. (8) with the constants e ji , f ji , g ji and q ji given in Table 1 of Carlson and Mantas (1982)