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Resonances in neutral bremsstrahlung
NOTES
AND DISCUSSIONS
The Editors of the Journal of Quantitative Spectroscopy and Radiative Transfer welcome the submittal of brief, original contributions on timely topics. These manuscripts will generally be published promptly and without Editorial review if they are submitted through one of the Associate Editors of JQSRT
.I. Quanr. Spectrosc. Rodiat.
Transfer. Vol. 6, pp. 369-370.
Pergamon
Press
Ltd.,
1966. Printed
in Great
Britain
NOTE RESONANCES
IN NEUTRAL
BREMSSTRAHLUNG*
B. KIVEL Avco-Everett
Research
Laboratory.
Everett.
Massachusetts
(Received 16 December 1965)
THE purpose of this note is to point out that the possibility of resonances should be considered in estimating bremsstrahlung. In particular, there is a well-known resonance in the elastic scattering cross section of low energy electrons by molecular nitrogenoV2’ which may be significant in estimating radiation from air. For air the bremsstrahlung from electrons scattered by neutral species accounts for about 5 per cent of the total radiation intensity of an optically thin gas c3y4)at a temperature of 8000°K and a density of one atmosphere. Molecular nitrogen is an important source of the neutral bremsstrahlung according to intensity measurements in the infrared.“’ The molecular nitrogen resonance intensity may be estimated in a manner analogous to a transition between two bound states. To do this one needs the density of radiating states and thef-number for the transition. We estimate the density of electrons in resonant states using NNP -= N&N,
g, exp[
-JVW
(2rcmkTJh2)3/2
(1)
where E, is the resonance energy, g, is spatial degeneracy of the resonance state N;*, T the temperature, Ni the ith particle density and we have assumed that the core partition functions for vibration and rotation cancel ; that the N; state is a doublet so that its spin degeneracy cancels that of the free electron; and that the electronic state of N, is ‘C which has a degeneracy of 1. Using this expression we can relate the spectral intensity for a resonance to the Kramers’ continuum by I, (resonance) 1, (Kramers) where g is the spatial degeneracy of the lower state of the transition, AB is the spectral * This work has been supported by Headquarters, Air Force Special Weapons Center, Air Force Systems Command,
United
States Air Force,
under
Contract
AF29(601)-6488. 369
370
B. K~VEL
width of the resonance radiation, I is the spectral wavelength, f is the f-number for the resonance transition, T is the gas temperature, 2 is the effective nuclear charge for the Kramers’ prediction (TAYLOR")gives 2’ = O-022for Nz and 2’ = OX@9for N) and E is the final energy of the free electron. We see that when all of the terms in parentheses take on the value 1, that the resonance intensity is approximately equal to the Kramers’ continuum value. At shorter wavelengths and lower values of E this ratio will be larger. We also note both intensities are proportional to the product of electron and molecule densities so that these quantities do not appear in Eq. (2). As usual, the f-number is a key unknown. Thus, a strong resonance transition in molecular nitrogen may enhance the spectral radiation moderately in the neighborhood of 1~. At smaller wavelengths our simple expression, equation (2), allows for a greater increase, however the fact that the elastic scattering cross section goes to a small value in the neighborhood of zero energy’@ (an anti-resonance) probably implies a small f-number. The effect on the total radiation is smaller since the resonance covers a smaller spectral range. We estimate the ratio of the emissivity attributed to the resonance to that of the integrated Kramers’ continuum by E/L (resonance) E/L (Kramers) In this case when the parenthetical terms have the value 1, then the contribution of the resonance to the emissivity is 30 per cent of that from the Kramers’ continuum. CO* may have a large resonance radiation (near 0.3~) since it has resonances in elastic scattering at low energy as well as in the neighborhood of 4 eV.(” REFERENCES 1. V. C. RAMSAUER and R. KOLLATH,Annln. Phys. 4, 91 (1930). 2. G. J. SCHULTZ,Phys. Rev. 135, A998 (1964). 3. B. Kwe~, H. MAYER and H. BETHE, Ann. Phys. 2, 57 (1957). J. KECK, J. CAMM,B. KIVEL and T. WENTINKJR., Ann. Phys. 7, 1 (1959). 4. B. KIVEL and K. BAILEY, “Tables of Radiation from High Temperature Air”, Avco-Everett Laboratory, RR 21, December 1957. 5. R. L. TAYLOR, J. chew Phys. 39, 2354 (1963). 6. L. S. FRUIT and A. V. PHELPS, Phys. Rev. 127, 1621 (1962).
Research
AND DISCUSSIONS
The Editors of the Journal of Quantitative Spectroscopy and Radiative Transfer welcome the submittal of brief, original contributions on timely topics. These manuscripts will generally be published promptly and without Editorial review if they are submitted through one of the Associate Editors of JQSRT
.I. Quanr. Spectrosc. Rodiat.
Transfer. Vol. 6, pp. 369-370.
Pergamon
Press
Ltd.,
1966. Printed
in Great
Britain
NOTE RESONANCES
IN NEUTRAL
BREMSSTRAHLUNG*
B. KIVEL Avco-Everett
Research
Laboratory.
Everett.
Massachusetts
(Received 16 December 1965)
THE purpose of this note is to point out that the possibility of resonances should be considered in estimating bremsstrahlung. In particular, there is a well-known resonance in the elastic scattering cross section of low energy electrons by molecular nitrogenoV2’ which may be significant in estimating radiation from air. For air the bremsstrahlung from electrons scattered by neutral species accounts for about 5 per cent of the total radiation intensity of an optically thin gas c3y4)at a temperature of 8000°K and a density of one atmosphere. Molecular nitrogen is an important source of the neutral bremsstrahlung according to intensity measurements in the infrared.“’ The molecular nitrogen resonance intensity may be estimated in a manner analogous to a transition between two bound states. To do this one needs the density of radiating states and thef-number for the transition. We estimate the density of electrons in resonant states using NNP -= N&N,
g, exp[
-JVW
(2rcmkTJh2)3/2
(1)
where E, is the resonance energy, g, is spatial degeneracy of the resonance state N;*, T the temperature, Ni the ith particle density and we have assumed that the core partition functions for vibration and rotation cancel ; that the N; state is a doublet so that its spin degeneracy cancels that of the free electron; and that the electronic state of N, is ‘C which has a degeneracy of 1. Using this expression we can relate the spectral intensity for a resonance to the Kramers’ continuum by I, (resonance) 1, (Kramers) where g is the spatial degeneracy of the lower state of the transition, AB is the spectral * This work has been supported by Headquarters, Air Force Special Weapons Center, Air Force Systems Command,
United
States Air Force,
under
Contract
AF29(601)-6488. 369
370
B. K~VEL
width of the resonance radiation, I is the spectral wavelength, f is the f-number for the resonance transition, T is the gas temperature, 2 is the effective nuclear charge for the Kramers’ prediction (TAYLOR")gives 2’ = O-022for Nz and 2’ = OX@9for N) and E is the final energy of the free electron. We see that when all of the terms in parentheses take on the value 1, that the resonance intensity is approximately equal to the Kramers’ continuum value. At shorter wavelengths and lower values of E this ratio will be larger. We also note both intensities are proportional to the product of electron and molecule densities so that these quantities do not appear in Eq. (2). As usual, the f-number is a key unknown. Thus, a strong resonance transition in molecular nitrogen may enhance the spectral radiation moderately in the neighborhood of 1~. At smaller wavelengths our simple expression, equation (2), allows for a greater increase, however the fact that the elastic scattering cross section goes to a small value in the neighborhood of zero energy’@ (an anti-resonance) probably implies a small f-number. The effect on the total radiation is smaller since the resonance covers a smaller spectral range. We estimate the ratio of the emissivity attributed to the resonance to that of the integrated Kramers’ continuum by E/L (resonance) E/L (Kramers) In this case when the parenthetical terms have the value 1, then the contribution of the resonance to the emissivity is 30 per cent of that from the Kramers’ continuum. CO* may have a large resonance radiation (near 0.3~) since it has resonances in elastic scattering at low energy as well as in the neighborhood of 4 eV.(” REFERENCES 1. V. C. RAMSAUER and R. KOLLATH,Annln. Phys. 4, 91 (1930). 2. G. J. SCHULTZ,Phys. Rev. 135, A998 (1964). 3. B. Kwe~, H. MAYER and H. BETHE, Ann. Phys. 2, 57 (1957). J. KECK, J. CAMM,B. KIVEL and T. WENTINKJR., Ann. Phys. 7, 1 (1959). 4. B. KIVEL and K. BAILEY, “Tables of Radiation from High Temperature Air”, Avco-Everett Laboratory, RR 21, December 1957. 5. R. L. TAYLOR, J. chew Phys. 39, 2354 (1963). 6. L. S. FRUIT and A. V. PHELPS, Phys. Rev. 127, 1621 (1962).
Research