### Hydrogenic Species

*Precise quantum-mechanical calculations*exist only for hydrogenic species. The total power ε_{cont}radiated (per unit source volume and per unit solid angle, and expressed in SI units) in the wavelength interval Δλ is the sum of radiation due to the recombination of a free electron with a bare ion (free-bound transitions) and bremsstrahlung (free-free transitions):

(44) where

*N*_{e}is the electron density,*N*the number density of hydrogenic (bare) ions of nuclear charge_{Z}*Z*,*I*_{H}the ionization energy of hydrogen,*n*′ the principal quantum number of the lowest level for which adjacent levels are so close that they approach a continuum and summation over*n*may be replaced by an integral. (The choice of*n*′ is rather arbitrary;*n*′ as low as 6 is found in the literature.) γ_{f b}and γ_{f f}are the Gaunt factors, which are generally close to unity. (For the higher free-bound continua, starting with*n*′ + 1, an average Gaunt factor_{f b}is used.) For neutral hydrogen, the recombination continuum forming*H*becomes important, too [35].^{-}In the equation above, the value of the constant factor is

6.065 × 10 [Numerical example: For atomic hydrogen (^{-55}W m^{4}J^{1/2}sr^{-1}.*Z*= 1), the quantity ε_{cont}has the value 2.9 W m^{-3}sr^{-1}under the following conditions: λ = 3 × 10^{-7}m; Δλ = 1 × 10^{-10}m;*N*_{e}(=*N*_{Z=1}) = 1 × 10^{21}m^{-3};*T*= 12 000 K. The lower limit of the summation index*n*is 2; the upper limit*n*′ has been taken to be 10. All Gaunt factors γ_{f b},_{f b}, γ_{f f}have been assumed to be unity.]

### Many-Electron Systems

For many-electron systems, only approximate theoretical treatments exist, based on the quantum-defect method (for results of calculations for noble gases, see, e.g., Ref. [36]). Experimental work is centered on the noble gases [37]. Modifications of the continuum by autoionization processes must also be considered.Near the ionization limit, the

*f*values for bound-bound transitions of a spectral series (*n*′ → ∞) make a smooth connection to the differential oscillator strength distribution*df*/*d*ε in the continuum [38].