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Inverse bremsstrahlung in the field of the lithium atom
J. Quonr.
Specwosc.
Rodiat.
Transfer.
Vol.
IO. pp. 6143.
Pergamon Press 1970. Printed in Great Britain
NOTE INVERSE
BREMSSTRAHLUNG LITHIUM B. Department
of Physics,
IN THE ATOM
FIELD
OF THE
YA’AKOBI
The Hebrew
University,
Jerusalem,
Israel
(Received 2 June 1969) Abstract-The for different
absorption temperatures
coefficient due to inverse bremsstrahlung in the field of the lithium atom is calculated and wavelengths employing s-waves phase shifts, including exchange and correlation.
ions of the alkali atoms are of great interest for atomic-theory calculations and for discharges in low-temperature alkali vapours. In particular, the electron affinity of the lithium atom (or the binding energy of the negative lithium ion) has been investigated experimentally”) and theoretically. w The cross-section for the photo-detachment of the negative ion of lithium has also been calculated. (3S4)It now remains to determine the cross-section for transitions in the continuum of the negative ion, or the cross-section for inverse bremsstrahlung of free electrons in the field of the neutral lithium atom. A comparison with the negative ion of hydrogen (5) shows that the cross-section under consideration can be expected to overwhelm the photo-detachment cross-section at sufficiently long wavelengths. The cross-section to be calculated can be related to the phase-shift of s-waves of electrons elastically scattered by the lithium atom. (WThis phase shift was calculated by GARRETT(~) taking into account exchange and correlation (or core polarization). The cross-section for elastic scattering of electrons by lithium atoms as calculated with these phase shifts yielded good agreement with the experiment. (*) These cross sections were here used to obtain interpolation expressions for the singlet and triplet s-phase shifts 6+ and 6- as a function of E, the energy of the incoming electron in Rydberg units : NEGATIVE
106(6+)-* = 2.1756E3 + 1*5559E*+ 2.0745E+ 1.7939 106(6 -)- * = 6.3516E3 + 2.5256E2 + 1.0022E + 1.9474.
(1)
The cross-section for the free-free transition of an electron from a state with wavenumber k. to a state with wave-number k (k, and k are in atomic units) is o(k;, Ak*) = ~ora~k,2k;1(Ak2)-3[/M(ko,
k,)[*+(M(kl,
k,)j*] cm5,
(2)
where k* = kf - ki and, therefore, the wavelength of the absorbed photon is L(A) = 911.31 k*; CIis the fine-structure constant and a, the Bohr radius. As in hydrogen, transitions 61
B.
62
YA’AKOBI
involving angular momentum quantum numbers 1 larger than 1 are neglected!” The matrix element M is derived in a way similar to that of OHMURA and OHMURA’@except that a 2s wave function rather than a 1s wave function was employed for the bound electron. We find that (M(kO, kJ2 = k:[3 sin’ 6-(k,)+sin2 d’(k,)]. (3) The cross-section was averaged over a Maxwellian distribution of electron velocities to obtain the absorption coefficient, by integrating numerically up to large enough values of kg to incorporate most electrons at any of the temperatures chosen. The results of these calculations are given in Table 1. For illustrative purposes, we show in Fig. 1 the absorption I
1
I
T=5000°K
B-
0
5000
10000 WAVELENGTH
15000
20000
6,
1. The absorption coefficients due to photo-detachment of the negative lithium ion (marked : bout&free) and due to inverse bremsstrahlung in the field of the lithium neutral atom (marked : freefree). These are computed per unit concentration of atoms and unit concentration of free electrons at the temperatures 5000°K (a) and 9000°K (b). Induced emission was accounted for. FIG.
Inverse bremsstrahlung in the field of the lithium atom
63
TABLE~.COMPUTEDABSORPTIONCOEFFICIENTPRODUCEDBYINVERSEBREMSSTRAHLUNGINTHEFIELDOFTHENEUTRAL LITHIUM ATOM, PER UNIT CONCENTRATION OF ATOMS AND UNIT CONCENTRATION OF FREE ELECTRONS, IN UNITS OF 10-40cMs(~D~cEDEMIssIo~Is~cc0uN~~~F0~) 44 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000
T("K) = 5000
6000
7000
8000
9000
10000
15ooo
20000
13.0 20.7 29.9 40.6 52.7 66.2 81.0 97.0 114.4 133.1 153.0 174.2 196.6 220.4 245.5 271.8 299.4 328.4
11.4 18.3 264 35.8 46.3 57.9 70.5 84.2 98.9 114.7 131.5 149.2 168.1 188.0 2090 231.0 254.1 278.3
10.2 16.3 23.5 31.7 40.9 50.9 61.7 73.4 85.9 99.3 113.4 128.5 144.4 161.2 178.9 197.5 217.0 237.4
9.1 14.6 21.0 28.3 36.2 44.9 54.2 64.3 75.0 86.4 98.5 111.4 125.0 139.3 154.4 1702 186.8 204.2
8.3 13.2 18.9 25.3 32.3 399 48.0 56.6 65.9 75.8 86.3 97.4 109.1 121.4 134.5 148.1 162.5 177.5
7.5 12.0 17.1 22.8 29.0 35.6 42.7 50.3 58.4 67.1 76.2 85.9 96.1 107.0 118.3 130.3 142.8 156.0
5.1 8.0 11.3 14.8 18.6 22.6 27.0 31.6 36.6 41.8 47.5 53.5 598 66.5 73.5 81.0 88.8 97.0
3.9 6.0 8.5 11.1 14.0 17.1 20.5 24.2 28.1 32.3 36.9 41.7 46.9 52.3 58.1 64.2 70.6 77.3
coefficients due to photo-detachment and inverse bremsstrahlung in the field of the lithium atom at two different temperatures. The former was obtained by averaging the photodetachment cross-section’4’ over a Maxwellian distribution of electron velocities. As the temperature rises, the absorption cross-section calculated here becomes predominant. Both coefficients decrease with a rise of temperature but, together, they may still dominate the infra-red absorption of ionized lithium gas (if the temperature is not too high). Thus, infra-red absorption due to recombination and bremsstrahlung in the field of singlyionized lithium is small.(g’ The close similarity of the present results with those obtained for the negative ion of hydrogen”’ is notable. REFERENCES 1. B. YA'AKOBI,Phys. Letters 23, 655 (1966). 2. A. W. WEISS,Phys. Rev. 122, 1826 (1961); ibid. 166,70 (1968). Additional references are given by B. L. MorSEIWITSCH,Advances in Atomic and Molecular Physics (Edited D. R. BATE and 1.ESTERMANN) p. 61.Academic Press,New York (1965). 3. S. GELTMAN, Phys. Rev. 104,346 (1956). 4. B. YA'AKOBI, Phys. Rev. 176,227 (1968). 5. S. CHANDRASEKHAR and F. H. BREEN,J. Astrophys. 104,430(1946). 6. T. OHMURA and H. OIIMURA, J.Astrophys. 131,8 (1960); Phys. Rev. 121, 513 (1961). 7. W. R. GARRETT, Phys. Rev. 140, A705 (1965). 8. J. PEREL,P.ENGLANDER and B.BEDERSON, Phys.Rev. 128,1148(1962). 9. B. YA'AKOBI, Proc. Phys. Sot. 92, 100 (1967).
Specwosc.
Rodiat.
Transfer.
Vol.
IO. pp. 6143.
Pergamon Press 1970. Printed in Great Britain
NOTE INVERSE
BREMSSTRAHLUNG LITHIUM B. Department
of Physics,
IN THE ATOM
FIELD
OF THE
YA’AKOBI
The Hebrew
University,
Jerusalem,
Israel
(Received 2 June 1969) Abstract-The for different
absorption temperatures
coefficient due to inverse bremsstrahlung in the field of the lithium atom is calculated and wavelengths employing s-waves phase shifts, including exchange and correlation.
ions of the alkali atoms are of great interest for atomic-theory calculations and for discharges in low-temperature alkali vapours. In particular, the electron affinity of the lithium atom (or the binding energy of the negative lithium ion) has been investigated experimentally”) and theoretically. w The cross-section for the photo-detachment of the negative ion of lithium has also been calculated. (3S4)It now remains to determine the cross-section for transitions in the continuum of the negative ion, or the cross-section for inverse bremsstrahlung of free electrons in the field of the neutral lithium atom. A comparison with the negative ion of hydrogen (5) shows that the cross-section under consideration can be expected to overwhelm the photo-detachment cross-section at sufficiently long wavelengths. The cross-section to be calculated can be related to the phase-shift of s-waves of electrons elastically scattered by the lithium atom. (WThis phase shift was calculated by GARRETT(~) taking into account exchange and correlation (or core polarization). The cross-section for elastic scattering of electrons by lithium atoms as calculated with these phase shifts yielded good agreement with the experiment. (*) These cross sections were here used to obtain interpolation expressions for the singlet and triplet s-phase shifts 6+ and 6- as a function of E, the energy of the incoming electron in Rydberg units : NEGATIVE
106(6+)-* = 2.1756E3 + 1*5559E*+ 2.0745E+ 1.7939 106(6 -)- * = 6.3516E3 + 2.5256E2 + 1.0022E + 1.9474.
(1)
The cross-section for the free-free transition of an electron from a state with wavenumber k. to a state with wave-number k (k, and k are in atomic units) is o(k;, Ak*) = ~ora~k,2k;1(Ak2)-3[/M(ko,
k,)[*+(M(kl,
k,)j*] cm5,
(2)
where k* = kf - ki and, therefore, the wavelength of the absorbed photon is L(A) = 911.31 k*; CIis the fine-structure constant and a, the Bohr radius. As in hydrogen, transitions 61
B.
62
YA’AKOBI
involving angular momentum quantum numbers 1 larger than 1 are neglected!” The matrix element M is derived in a way similar to that of OHMURA and OHMURA’@except that a 2s wave function rather than a 1s wave function was employed for the bound electron. We find that (M(kO, kJ2 = k:[3 sin’ 6-(k,)+sin2 d’(k,)]. (3) The cross-section was averaged over a Maxwellian distribution of electron velocities to obtain the absorption coefficient, by integrating numerically up to large enough values of kg to incorporate most electrons at any of the temperatures chosen. The results of these calculations are given in Table 1. For illustrative purposes, we show in Fig. 1 the absorption I
1
I
T=5000°K
B-
0
5000
10000 WAVELENGTH
15000
20000
6,
1. The absorption coefficients due to photo-detachment of the negative lithium ion (marked : bout&free) and due to inverse bremsstrahlung in the field of the lithium neutral atom (marked : freefree). These are computed per unit concentration of atoms and unit concentration of free electrons at the temperatures 5000°K (a) and 9000°K (b). Induced emission was accounted for. FIG.
Inverse bremsstrahlung in the field of the lithium atom
63
TABLE~.COMPUTEDABSORPTIONCOEFFICIENTPRODUCEDBYINVERSEBREMSSTRAHLUNGINTHEFIELDOFTHENEUTRAL LITHIUM ATOM, PER UNIT CONCENTRATION OF ATOMS AND UNIT CONCENTRATION OF FREE ELECTRONS, IN UNITS OF 10-40cMs(~D~cEDEMIssIo~Is~cc0uN~~~F0~) 44 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000
T("K) = 5000
6000
7000
8000
9000
10000
15ooo
20000
13.0 20.7 29.9 40.6 52.7 66.2 81.0 97.0 114.4 133.1 153.0 174.2 196.6 220.4 245.5 271.8 299.4 328.4
11.4 18.3 264 35.8 46.3 57.9 70.5 84.2 98.9 114.7 131.5 149.2 168.1 188.0 2090 231.0 254.1 278.3
10.2 16.3 23.5 31.7 40.9 50.9 61.7 73.4 85.9 99.3 113.4 128.5 144.4 161.2 178.9 197.5 217.0 237.4
9.1 14.6 21.0 28.3 36.2 44.9 54.2 64.3 75.0 86.4 98.5 111.4 125.0 139.3 154.4 1702 186.8 204.2
8.3 13.2 18.9 25.3 32.3 399 48.0 56.6 65.9 75.8 86.3 97.4 109.1 121.4 134.5 148.1 162.5 177.5
7.5 12.0 17.1 22.8 29.0 35.6 42.7 50.3 58.4 67.1 76.2 85.9 96.1 107.0 118.3 130.3 142.8 156.0
5.1 8.0 11.3 14.8 18.6 22.6 27.0 31.6 36.6 41.8 47.5 53.5 598 66.5 73.5 81.0 88.8 97.0
3.9 6.0 8.5 11.1 14.0 17.1 20.5 24.2 28.1 32.3 36.9 41.7 46.9 52.3 58.1 64.2 70.6 77.3
coefficients due to photo-detachment and inverse bremsstrahlung in the field of the lithium atom at two different temperatures. The former was obtained by averaging the photodetachment cross-section’4’ over a Maxwellian distribution of electron velocities. As the temperature rises, the absorption cross-section calculated here becomes predominant. Both coefficients decrease with a rise of temperature but, together, they may still dominate the infra-red absorption of ionized lithium gas (if the temperature is not too high). Thus, infra-red absorption due to recombination and bremsstrahlung in the field of singlyionized lithium is small.(g’ The close similarity of the present results with those obtained for the negative ion of hydrogen”’ is notable. REFERENCES 1. B. YA'AKOBI,Phys. Letters 23, 655 (1966). 2. A. W. WEISS,Phys. Rev. 122, 1826 (1961); ibid. 166,70 (1968). Additional references are given by B. L. MorSEIWITSCH,Advances in Atomic and Molecular Physics (Edited D. R. BATE and 1.ESTERMANN) p. 61.Academic Press,New York (1965). 3. S. GELTMAN, Phys. Rev. 104,346 (1956). 4. B. YA'AKOBI, Phys. Rev. 176,227 (1968). 5. S. CHANDRASEKHAR and F. H. BREEN,J. Astrophys. 104,430(1946). 6. T. OHMURA and H. OIIMURA, J.Astrophys. 131,8 (1960); Phys. Rev. 121, 513 (1961). 7. W. R. GARRETT, Phys. Rev. 140, A705 (1965). 8. J. PEREL,P.ENGLANDER and B.BEDERSON, Phys.Rev. 128,1148(1962). 9. B. YA'AKOBI, Proc. Phys. Sot. 92, 100 (1967).