Abstract

Pure rotational coherent anti-Stokes Raman scattering measurements of pure CO2 have been performed in a temperature range from 300 to 773 K and for pressure from 0.1 to 5 MPa for the purpose of time-resolved CO2 thermometry. Particular emphasis was put on the comparison of several linewidth approximations to model the experimental spectra. Generally good agreement of the temperature mean values with the thermocouple reference has been found for all models over almost the whole pressure and temperature range investigated. The standard deviations, which increased with temperature, were comparable with or better than the results gained for single-shot measurements of pure N2 or O2–N2 mixtures. Yet for high particle densities close to the critical point of CO2 the limitation of the models became obvious, owing to the strongly increased influence of motional narrowing effects. The characteristics of these effects have been demonstrated by measurements even closer to the critical conditions.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. R. Regnier, J.-P. E. Taran, “On the possibilities of measuring gas concentration by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240–242 (1973).
    [CrossRef]
  2. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon & Breach, 1996), Vol. 3.
  3. T. Seeger, A. Leipertz, “Experimental comparison of single-shot broadband vibrational and dual-broadband pure rotational coherent anti-Stokes-Raman scattering in hot air,” Appl. Opt. 35, 2665–2671 (1996).
    [CrossRef] [PubMed]
  4. A. Thumann, M. Schenk, J. Jonuscheit, T. Seeger, A. Leipertz, “Simultaneous temperature and relative nitrogen–oxygen concentration measurements in air with pure rotational coherent anti-Stokes-Raman scattering for temperatures to as high as 2050 K,” Appl. Opt. 36, 3500–3505 (1997).
    [CrossRef] [PubMed]
  5. L. Martinsson, P.-E. Bengtsson, M. Alden, “Oxygen concentration and temperature measurements in N2–O2 mixtures using rotational coherent anti-Stokes-Raman spectroscopy,” Appl. Phys. B 62, 29–37 (1996).
    [CrossRef]
  6. M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
    [CrossRef]
  7. J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
    [CrossRef]
  8. M. Afzelius, C. Brackmann, F. Vestin, P.-E. Bengtsson, “Pure rotational coherent anti-Stokes Raman spectroscopy in mixtures of CO and N2,” Appl. Opt. 43, 6664–6665 (2004).
    [CrossRef]
  9. M. Schenk, “Simultane Temperatur- und Konzentrationsmessung in binären und ternären Gemischen mittels Rotations-CARS-Spektroskopie,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, A. Liepertz, ed. (ESYTEC Energie und Systemtechnik, 2000), Vol. 2000.2, pp. 98–118.
  10. M. Schenk, T. Seeger, A. Leipertz, “CO2-thermometry and simultaneous temperature and relative CO2/N2-concentration measurements using single-shot dual broadband pure rotational CARS,” in Proceedings of the XVIth International Conference on Raman Spectroscopy, A. M. Heynes, ed. (Wiley, 1998), pp. 160–161.
  11. M. Schenk, T. Seeger, A. Leipertz, “Simultaneous and time resolved temperature and relative CO2-N2and O2-CO2-N2 concentration measurements using pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 44, 5582–5593 (2005).
    [CrossRef] [PubMed]
  12. A. Weber, “High resolution Raman studies of gases,” in The Raman Effect, A. Anderson, ed. (Marcel Dekker, 1973), Vol. 2, pp. 543–757.
  13. H. H. Nielsen, “The quantum mechanical Hamiltonian for the linear polyatomic molecule treated as a limiting case of the non-linear polyatomic molecule,” Phys. Rev. 66, 282–287 (1944).
    [CrossRef]
  14. K. Altmann, W. Klöckner, G. Strey, “Der Intensitätsverlauf im reinen Rotations-Raman-Spektrum von CO2 und N2O unter Berücksichtigung des 0110-Niveaus,” Z. Naturforsch.31a, 1311–1317 (1976).
  15. J. J. Barrett, A. Weber, “Pure-rotational Raman scattering in a CO2 electric discharge,” J. Opt. Soc. Am. 60, 70–77 (1970).
    [CrossRef]
  16. H. P. Godfried, I. F. Silvera, “Rotational R-branch spectroscopy in CO2,” J. Chem. Phys. 78, 121–123 (1983).
    [CrossRef]
  17. L. A. Rahn, R. E. Palmer, “Studies of nitrogen self-broadening at high temperature with inverse Raman spectroscopy,” J. Opt. Soc. Am. B 3, 1164–1169 (1986).
    [CrossRef]
  18. R. L. Farrow, R. Trebino, R. E. Palmer, “High-resolution CARS measurements of temperature profiles and pressure in a tungsten lamp,” Appl. Opt. 26, 331–335 (1987).
    [CrossRef] [PubMed]
  19. B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
    [CrossRef]
  20. T. A. Brunner, D. Pritchard, “Fitting laws for rotationally inelastic collisions,” in Dynamics of the Excited State, K. P. Lawley, ed. (Wiley, 1982), Vol. L, pp. 589–641.
  21. A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
    [CrossRef]
  22. L. Rosenmann, J. M. Hartmann, M. Y. Perrin, J. Taine, “Accurate calculated tabulations of IR and Raman CO2 line broadening by CO2, H2O, O2 in the 300–2400-K temperature range,” Appl. Opt. 27, 3902–3907 (1988).
    [CrossRef] [PubMed]
  23. R. Span, W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 8000 MPa,” J. Phys. Chem. Ref. Data 25, 1509–1596 (1996).
    [CrossRef]
  24. R. Span, Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (personal communication to K. Kraft, 1994).
  25. VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen, VDI-Wärmeatlas (Springer, Berlin, 1988).
  26. L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
    [CrossRef]
  27. L. S. Rothman, L. D. G. Young, “Infrared energy levels and intensities of carbon dioxide-II,” J. Quant. Spectrosc. Radiat. Transfer 25, 505–524 (1981).
    [CrossRef]
  28. C. M. Penney, R. L. St. Peters, M. Lapp, “Absolute rotational Raman cross sections for N2, O2, and CO2,” J. Opt. Soc. Am. 64, 712–716 (1974).
    [CrossRef]
  29. M. C. Drake, G. M. Rosenblatt, “Rotational Raman scattering from premixed and diffusion flames,” Combust. Flame 33, 179–196 (1978).
    [CrossRef]
  30. J. D. Drake, “Rotational Raman intensity-correction factors due to vibrational anharmonicity: their effect on temperature measurements,” Opt. Lett. 7, 440–441 (1982).
    [CrossRef] [PubMed]
  31. E. Magens, “Nutzung von Rotations-CARS zur Temperatur-und Konzentrationsmessung in Flammen,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, (ESYTEC Energie und Systemtechnik, Germany, 1993), Vol. 93.2.
  32. D. A. Greenhalgh, “Quantitative CARS spectroscopy,” in Advances in Nonlinear Spectroscopy, R. J. H. Clark, R. E. Hester, eds. (Wiley, 1988), Vol. 15, pp. 193–251.
  33. T. Lasser, “An alternative method for CARS-spectra calculation,” Opt. Commun. 35, 447–450 (1980).
    [CrossRef]
  34. L. Martinsson, “Theoretical development of rotational CARS for combustion diagnostics,” Ph.D. dissertation (Lund Institute of Technology, 1994).
  35. M. Alden, P.-E. Bengtsson, D. Nilsson, H. Edner, S. Kröll, “Rotational CARS—a comparison of different techniques with emphasis on accuracy in temperature determination,” Appl. Opt. 28, 3206–3219 (1989).
    [CrossRef]
  36. M. Schenk, A. Thumann, T. Seeger, A. Leipertz, “Pure rotational coherent anti-Stokes Raman scattering: comparison of evaluation techniques for determining single-shot temperature and relative N2–O2 concentration,” Appl. Opt. 37, 5659–5671 (1998).
    [CrossRef]
  37. R. C. H. Tam, A. D. May, “Motional narrowing of the rotational Raman band of compressed CO, N2, and CO2,” Can. J. Phys. 61, 1558–1566 (1983).
    [CrossRef]
  38. R. L. Armstrong, “Line Mixing in the ν2 band of CO2,” Appl. Opt. 21, 2141–2145 (1982).
    [CrossRef] [PubMed]
  39. I. N. Levine, Molecular Spectroscopy (Wiley, 1975).
  40. G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra of Polyatomic Molecules (Krieger, 1991).
  41. H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
    [CrossRef]
  42. E. Fermi, “Über den Ramaneffekt des Kohlendioxyds,” Z. Phys. 71, 250–259 (1931).
    [CrossRef]
  43. D. A. Long, Raman Spectroscopy (McGraw-Hill International, 1977).

2005

2004

2000

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
[CrossRef]

J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
[CrossRef]

1998

1997

1996

T. Seeger, A. Leipertz, “Experimental comparison of single-shot broadband vibrational and dual-broadband pure rotational coherent anti-Stokes-Raman scattering in hot air,” Appl. Opt. 35, 2665–2671 (1996).
[CrossRef] [PubMed]

L. Martinsson, P.-E. Bengtsson, M. Alden, “Oxygen concentration and temperature measurements in N2–O2 mixtures using rotational coherent anti-Stokes-Raman spectroscopy,” Appl. Phys. B 62, 29–37 (1996).
[CrossRef]

R. Span, W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 8000 MPa,” J. Phys. Chem. Ref. Data 25, 1509–1596 (1996).
[CrossRef]

1992

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

1990

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

1989

1988

1987

1986

1983

R. C. H. Tam, A. D. May, “Motional narrowing of the rotational Raman band of compressed CO, N2, and CO2,” Can. J. Phys. 61, 1558–1566 (1983).
[CrossRef]

H. P. Godfried, I. F. Silvera, “Rotational R-branch spectroscopy in CO2,” J. Chem. Phys. 78, 121–123 (1983).
[CrossRef]

1982

1981

L. S. Rothman, L. D. G. Young, “Infrared energy levels and intensities of carbon dioxide-II,” J. Quant. Spectrosc. Radiat. Transfer 25, 505–524 (1981).
[CrossRef]

1980

T. Lasser, “An alternative method for CARS-spectra calculation,” Opt. Commun. 35, 447–450 (1980).
[CrossRef]

1979

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

1978

M. C. Drake, G. M. Rosenblatt, “Rotational Raman scattering from premixed and diffusion flames,” Combust. Flame 33, 179–196 (1978).
[CrossRef]

1977

H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
[CrossRef]

1974

1973

P. R. Regnier, J.-P. E. Taran, “On the possibilities of measuring gas concentration by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240–242 (1973).
[CrossRef]

1970

1944

H. H. Nielsen, “The quantum mechanical Hamiltonian for the linear polyatomic molecule treated as a limiting case of the non-linear polyatomic molecule,” Phys. Rev. 66, 282–287 (1944).
[CrossRef]

1931

E. Fermi, “Über den Ramaneffekt des Kohlendioxyds,” Z. Phys. 71, 250–259 (1931).
[CrossRef]

Afzelius, M.

Alden, M.

J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
[CrossRef]

L. Martinsson, P.-E. Bengtsson, M. Alden, “Oxygen concentration and temperature measurements in N2–O2 mixtures using rotational coherent anti-Stokes-Raman spectroscopy,” Appl. Phys. B 62, 29–37 (1996).
[CrossRef]

M. Alden, P.-E. Bengtsson, D. Nilsson, H. Edner, S. Kröll, “Rotational CARS—a comparison of different techniques with emphasis on accuracy in temperature determination,” Appl. Opt. 28, 3206–3219 (1989).
[CrossRef]

Altmann, K.

K. Altmann, W. Klöckner, G. Strey, “Der Intensitätsverlauf im reinen Rotations-Raman-Spektrum von CO2 und N2O unter Berücksichtigung des 0110-Niveaus,” Z. Naturforsch.31a, 1311–1317 (1976).

Armstrong, R. L.

Augustin, S. D.

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

Barrett, J. J.

Bengtsson, P. E.

J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
[CrossRef]

Bengtsson, P.-E.

Berger, H.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Bonamy, J.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Bonamy, L.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Bood, J.

J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
[CrossRef]

Brackmann, C.

Brunner, T. A.

T. A. Brunner, D. Pritchard, “Fitting laws for rotationally inelastic collisions,” in Dynamics of the Excited State, K. P. Lawley, ed. (Wiley, 1982), Vol. L, pp. 589–641.

DePristo, A. E.

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

Drake, J. D.

Drake, M. C.

M. C. Drake, G. M. Rosenblatt, “Rotational Raman scattering from premixed and diffusion flames,” Combust. Flame 33, 179–196 (1978).
[CrossRef]

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon & Breach, 1996), Vol. 3.

Edner, H.

Farrow, R. L.

Fermi, E.

E. Fermi, “Über den Ramaneffekt des Kohlendioxyds,” Z. Phys. 71, 250–259 (1931).
[CrossRef]

Finsterhölzl, H.

H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
[CrossRef]

Gamache, R. R.

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

Godfried, H. P.

H. P. Godfried, I. F. Silvera, “Rotational R-branch spectroscopy in CO2,” J. Chem. Phys. 78, 121–123 (1983).
[CrossRef]

Greenhalgh, D. A.

D. A. Greenhalgh, “Quantitative CARS spectroscopy,” in Advances in Nonlinear Spectroscopy, R. J. H. Clark, R. E. Hester, eds. (Wiley, 1988), Vol. 15, pp. 193–251.

Hartmann, J. M.

Hawkins, R. L.

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

Herzberg, G.

G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra of Polyatomic Molecules (Krieger, 1991).

Hohenbleicher, J. G.

H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
[CrossRef]

Jonuscheit, J.

Klöckner, W.

K. Altmann, W. Klöckner, G. Strey, “Der Intensitätsverlauf im reinen Rotations-Raman-Spektrum von CO2 und N2O unter Berücksichtigung des 0110-Niveaus,” Z. Naturforsch.31a, 1311–1317 (1976).

Kraft, K.

R. Span, Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (personal communication to K. Kraft, 1994).

Kröll, S.

Lapp, M.

Lasser, T.

T. Lasser, “An alternative method for CARS-spectra calculation,” Opt. Commun. 35, 447–450 (1980).
[CrossRef]

Lavorel, B.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Leipertz, A.

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous and time resolved temperature and relative CO2-N2and O2-CO2-N2 concentration measurements using pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 44, 5582–5593 (2005).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
[CrossRef]

M. Schenk, A. Thumann, T. Seeger, A. Leipertz, “Pure rotational coherent anti-Stokes Raman scattering: comparison of evaluation techniques for determining single-shot temperature and relative N2–O2 concentration,” Appl. Opt. 37, 5659–5671 (1998).
[CrossRef]

A. Thumann, M. Schenk, J. Jonuscheit, T. Seeger, A. Leipertz, “Simultaneous temperature and relative nitrogen–oxygen concentration measurements in air with pure rotational coherent anti-Stokes-Raman scattering for temperatures to as high as 2050 K,” Appl. Opt. 36, 3500–3505 (1997).
[CrossRef] [PubMed]

T. Seeger, A. Leipertz, “Experimental comparison of single-shot broadband vibrational and dual-broadband pure rotational coherent anti-Stokes-Raman scattering in hot air,” Appl. Opt. 35, 2665–2671 (1996).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “CO2-thermometry and simultaneous temperature and relative CO2/N2-concentration measurements using single-shot dual broadband pure rotational CARS,” in Proceedings of the XVIth International Conference on Raman Spectroscopy, A. M. Heynes, ed. (Wiley, 1998), pp. 160–161.

Levine, I. N.

I. N. Levine, Molecular Spectroscopy (Wiley, 1975).

Long, D. A.

D. A. Long, Raman Spectroscopy (McGraw-Hill International, 1977).

Magens, E.

E. Magens, “Nutzung von Rotations-CARS zur Temperatur-und Konzentrationsmessung in Flammen,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, (ESYTEC Energie und Systemtechnik, Germany, 1993), Vol. 93.2.

Martinsson, L.

L. Martinsson, P.-E. Bengtsson, M. Alden, “Oxygen concentration and temperature measurements in N2–O2 mixtures using rotational coherent anti-Stokes-Raman spectroscopy,” Appl. Phys. B 62, 29–37 (1996).
[CrossRef]

L. Martinsson, “Theoretical development of rotational CARS for combustion diagnostics,” Ph.D. dissertation (Lund Institute of Technology, 1994).

May, A. D.

R. C. H. Tam, A. D. May, “Motional narrowing of the rotational Raman band of compressed CO, N2, and CO2,” Can. J. Phys. 61, 1558–1566 (1983).
[CrossRef]

Millot, G.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Nielsen, H. H.

H. H. Nielsen, “The quantum mechanical Hamiltonian for the linear polyatomic molecule treated as a limiting case of the non-linear polyatomic molecule,” Phys. Rev. 66, 282–287 (1944).
[CrossRef]

Nilsson, D.

Palmer, R. E.

Penney, C. M.

Perrin, M. Y.

Pritchard, D.

T. A. Brunner, D. Pritchard, “Fitting laws for rotationally inelastic collisions,” in Dynamics of the Excited State, K. P. Lawley, ed. (Wiley, 1982), Vol. L, pp. 589–641.

Rabitz, H.

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

Rahn, L. A.

Ramaswany, R.

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

Regnier, P. R.

P. R. Regnier, J.-P. E. Taran, “On the possibilities of measuring gas concentration by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240–242 (1973).
[CrossRef]

Robert, D.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Rosenblatt, G. M.

M. C. Drake, G. M. Rosenblatt, “Rotational Raman scattering from premixed and diffusion flames,” Combust. Flame 33, 179–196 (1978).
[CrossRef]

Rosenmann, L.

Rothman, L. S.

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

L. S. Rothman, L. D. G. Young, “Infrared energy levels and intensities of carbon dioxide-II,” J. Quant. Spectrosc. Radiat. Transfer 25, 505–524 (1981).
[CrossRef]

Saint-Loup, R.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

Schenk, M.

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous and time resolved temperature and relative CO2-N2and O2-CO2-N2 concentration measurements using pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 44, 5582–5593 (2005).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
[CrossRef]

M. Schenk, A. Thumann, T. Seeger, A. Leipertz, “Pure rotational coherent anti-Stokes Raman scattering: comparison of evaluation techniques for determining single-shot temperature and relative N2–O2 concentration,” Appl. Opt. 37, 5659–5671 (1998).
[CrossRef]

A. Thumann, M. Schenk, J. Jonuscheit, T. Seeger, A. Leipertz, “Simultaneous temperature and relative nitrogen–oxygen concentration measurements in air with pure rotational coherent anti-Stokes-Raman scattering for temperatures to as high as 2050 K,” Appl. Opt. 36, 3500–3505 (1997).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “CO2-thermometry and simultaneous temperature and relative CO2/N2-concentration measurements using single-shot dual broadband pure rotational CARS,” in Proceedings of the XVIth International Conference on Raman Spectroscopy, A. M. Heynes, ed. (Wiley, 1998), pp. 160–161.

M. Schenk, “Simultane Temperatur- und Konzentrationsmessung in binären und ternären Gemischen mittels Rotations-CARS-Spektroskopie,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, A. Liepertz, ed. (ESYTEC Energie und Systemtechnik, 2000), Vol. 2000.2, pp. 98–118.

Seeger, T.

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous and time resolved temperature and relative CO2-N2and O2-CO2-N2 concentration measurements using pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 44, 5582–5593 (2005).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
[CrossRef]

M. Schenk, A. Thumann, T. Seeger, A. Leipertz, “Pure rotational coherent anti-Stokes Raman scattering: comparison of evaluation techniques for determining single-shot temperature and relative N2–O2 concentration,” Appl. Opt. 37, 5659–5671 (1998).
[CrossRef]

A. Thumann, M. Schenk, J. Jonuscheit, T. Seeger, A. Leipertz, “Simultaneous temperature and relative nitrogen–oxygen concentration measurements in air with pure rotational coherent anti-Stokes-Raman scattering for temperatures to as high as 2050 K,” Appl. Opt. 36, 3500–3505 (1997).
[CrossRef] [PubMed]

T. Seeger, A. Leipertz, “Experimental comparison of single-shot broadband vibrational and dual-broadband pure rotational coherent anti-Stokes-Raman scattering in hot air,” Appl. Opt. 35, 2665–2671 (1996).
[CrossRef] [PubMed]

M. Schenk, T. Seeger, A. Leipertz, “CO2-thermometry and simultaneous temperature and relative CO2/N2-concentration measurements using single-shot dual broadband pure rotational CARS,” in Proceedings of the XVIth International Conference on Raman Spectroscopy, A. M. Heynes, ed. (Wiley, 1998), pp. 160–161.

Silvera, I. F.

H. P. Godfried, I. F. Silvera, “Rotational R-branch spectroscopy in CO2,” J. Chem. Phys. 78, 121–123 (1983).
[CrossRef]

Span, R.

R. Span, W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 8000 MPa,” J. Phys. Chem. Ref. Data 25, 1509–1596 (1996).
[CrossRef]

R. Span, Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (personal communication to K. Kraft, 1994).

St. Peters, R. L.

Strey, G.

H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
[CrossRef]

K. Altmann, W. Klöckner, G. Strey, “Der Intensitätsverlauf im reinen Rotations-Raman-Spektrum von CO2 und N2O unter Berücksichtigung des 0110-Niveaus,” Z. Naturforsch.31a, 1311–1317 (1976).

Taine, J.

Tam, R. C. H.

R. C. H. Tam, A. D. May, “Motional narrowing of the rotational Raman band of compressed CO, N2, and CO2,” Can. J. Phys. 61, 1558–1566 (1983).
[CrossRef]

Taran, J.-P. E.

P. R. Regnier, J.-P. E. Taran, “On the possibilities of measuring gas concentration by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240–242 (1973).
[CrossRef]

Thumann, A.

Trebino, R.

Vestin, F.

Wagner, W.

R. Span, W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 8000 MPa,” J. Phys. Chem. Ref. Data 25, 1509–1596 (1996).
[CrossRef]

Wattson, R. B.

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

Weber, A.

J. J. Barrett, A. Weber, “Pure-rotational Raman scattering in a CO2 electric discharge,” J. Opt. Soc. Am. 60, 70–77 (1970).
[CrossRef]

A. Weber, “High resolution Raman studies of gases,” in The Raman Effect, A. Anderson, ed. (Marcel Dekker, 1973), Vol. 2, pp. 543–757.

Young, L. D. G.

L. S. Rothman, L. D. G. Young, “Infrared energy levels and intensities of carbon dioxide-II,” J. Quant. Spectrosc. Radiat. Transfer 25, 505–524 (1981).
[CrossRef]

Appl. Opt.

R. L. Armstrong, “Line Mixing in the ν2 band of CO2,” Appl. Opt. 21, 2141–2145 (1982).
[CrossRef] [PubMed]

R. L. Farrow, R. Trebino, R. E. Palmer, “High-resolution CARS measurements of temperature profiles and pressure in a tungsten lamp,” Appl. Opt. 26, 331–335 (1987).
[CrossRef] [PubMed]

L. Rosenmann, J. M. Hartmann, M. Y. Perrin, J. Taine, “Accurate calculated tabulations of IR and Raman CO2 line broadening by CO2, H2O, O2 in the 300–2400-K temperature range,” Appl. Opt. 27, 3902–3907 (1988).
[CrossRef] [PubMed]

A. Thumann, M. Schenk, J. Jonuscheit, T. Seeger, A. Leipertz, “Simultaneous temperature and relative nitrogen–oxygen concentration measurements in air with pure rotational coherent anti-Stokes-Raman scattering for temperatures to as high as 2050 K,” Appl. Opt. 36, 3500–3505 (1997).
[CrossRef] [PubMed]

M. Schenk, A. Thumann, T. Seeger, A. Leipertz, “Pure rotational coherent anti-Stokes Raman scattering: comparison of evaluation techniques for determining single-shot temperature and relative N2–O2 concentration,” Appl. Opt. 37, 5659–5671 (1998).
[CrossRef]

T. Seeger, A. Leipertz, “Experimental comparison of single-shot broadband vibrational and dual-broadband pure rotational coherent anti-Stokes-Raman scattering in hot air,” Appl. Opt. 35, 2665–2671 (1996).
[CrossRef] [PubMed]

M. Alden, P.-E. Bengtsson, D. Nilsson, H. Edner, S. Kröll, “Rotational CARS—a comparison of different techniques with emphasis on accuracy in temperature determination,” Appl. Opt. 28, 3206–3219 (1989).
[CrossRef]

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous temperature and relative O2–N2 concentration measurements by pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 39, 6918–6925 (2000).
[CrossRef]

M. Afzelius, C. Brackmann, F. Vestin, P.-E. Bengtsson, “Pure rotational coherent anti-Stokes Raman spectroscopy in mixtures of CO and N2,” Appl. Opt. 43, 6664–6665 (2004).
[CrossRef]

M. Schenk, T. Seeger, A. Leipertz, “Simultaneous and time resolved temperature and relative CO2-N2and O2-CO2-N2 concentration measurements using pure rotational coherent anti-Stokes Raman scattering for pressures as great as 5 MPa,” Appl. Opt. 44, 5582–5593 (2005).
[CrossRef] [PubMed]

Appl. Phys. B

L. Martinsson, P.-E. Bengtsson, M. Alden, “Oxygen concentration and temperature measurements in N2–O2 mixtures using rotational coherent anti-Stokes-Raman spectroscopy,” Appl. Phys. B 62, 29–37 (1996).
[CrossRef]

J. Bood, P. E. Bengtsson, M. Alden, “Temperature and concentration measurements in acetylene–nitrogen mixtures in the range 300–600 K using dual-broadband rotational CARS,” Appl. Phys. B 70, 607–620 (2000).
[CrossRef]

Appl. Phys. Lett.

P. R. Regnier, J.-P. E. Taran, “On the possibilities of measuring gas concentration by stimulated anti-Stokes scattering,” Appl. Phys. Lett. 23, 240–242 (1973).
[CrossRef]

Can. J. Phys.

R. C. H. Tam, A. D. May, “Motional narrowing of the rotational Raman band of compressed CO, N2, and CO2,” Can. J. Phys. 61, 1558–1566 (1983).
[CrossRef]

Combust. Flame

M. C. Drake, G. M. Rosenblatt, “Rotational Raman scattering from premixed and diffusion flames,” Combust. Flame 33, 179–196 (1978).
[CrossRef]

J. Chem. Phys.

A. E. DePristo, S. D. Augustin, R. Ramaswany, H. Rabitz, “Quantum number and energy scaling for nonreactive collisions,” J. Chem. Phys. 71, 850–865 (1979).
[CrossRef]

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, D. Robert, “Study of collision effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. I. Rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2176–2184 (1990).
[CrossRef]

H. P. Godfried, I. F. Silvera, “Rotational R-branch spectroscopy in CO2,” J. Chem. Phys. 78, 121–123 (1983).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. Chem. Ref. Data

R. Span, W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 8000 MPa,” J. Phys. Chem. Ref. Data 25, 1509–1596 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

L. S. Rothman, R. L. Hawkins, R. B. Wattson, R. R. Gamache, “Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands,” J. Quant. Spectrosc. Radiat. Transfer 48, 537–566 (1992).
[CrossRef]

L. S. Rothman, L. D. G. Young, “Infrared energy levels and intensities of carbon dioxide-II,” J. Quant. Spectrosc. Radiat. Transfer 25, 505–524 (1981).
[CrossRef]

J. Raman Spectrosc.

H. Finsterhölzl, J. G. Hohenbleicher, G. Strey, “Intensity distribution in pure rotational Raman spectra of linear molecules in the ground and vibrational Π states: application to acetylene,” J. Raman Spectrosc. 6, 13–19 (1977).
[CrossRef]

Opt. Commun.

T. Lasser, “An alternative method for CARS-spectra calculation,” Opt. Commun. 35, 447–450 (1980).
[CrossRef]

Opt. Lett.

Phys. Rev.

H. H. Nielsen, “The quantum mechanical Hamiltonian for the linear polyatomic molecule treated as a limiting case of the non-linear polyatomic molecule,” Phys. Rev. 66, 282–287 (1944).
[CrossRef]

Z. Phys.

E. Fermi, “Über den Ramaneffekt des Kohlendioxyds,” Z. Phys. 71, 250–259 (1931).
[CrossRef]

Other

D. A. Long, Raman Spectroscopy (McGraw-Hill International, 1977).

I. N. Levine, Molecular Spectroscopy (Wiley, 1975).

G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra of Polyatomic Molecules (Krieger, 1991).

L. Martinsson, “Theoretical development of rotational CARS for combustion diagnostics,” Ph.D. dissertation (Lund Institute of Technology, 1994).

E. Magens, “Nutzung von Rotations-CARS zur Temperatur-und Konzentrationsmessung in Flammen,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, (ESYTEC Energie und Systemtechnik, Germany, 1993), Vol. 93.2.

D. A. Greenhalgh, “Quantitative CARS spectroscopy,” in Advances in Nonlinear Spectroscopy, R. J. H. Clark, R. E. Hester, eds. (Wiley, 1988), Vol. 15, pp. 193–251.

K. Altmann, W. Klöckner, G. Strey, “Der Intensitätsverlauf im reinen Rotations-Raman-Spektrum von CO2 und N2O unter Berücksichtigung des 0110-Niveaus,” Z. Naturforsch.31a, 1311–1317 (1976).

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon & Breach, 1996), Vol. 3.

M. Schenk, “Simultane Temperatur- und Konzentrationsmessung in binären und ternären Gemischen mittels Rotations-CARS-Spektroskopie,” in Berichte zur Energie- und Verfahrenstechnik -BEV-, A. Liepertz, ed. (ESYTEC Energie und Systemtechnik, 2000), Vol. 2000.2, pp. 98–118.

M. Schenk, T. Seeger, A. Leipertz, “CO2-thermometry and simultaneous temperature and relative CO2/N2-concentration measurements using single-shot dual broadband pure rotational CARS,” in Proceedings of the XVIth International Conference on Raman Spectroscopy, A. M. Heynes, ed. (Wiley, 1998), pp. 160–161.

A. Weber, “High resolution Raman studies of gases,” in The Raman Effect, A. Anderson, ed. (Marcel Dekker, 1973), Vol. 2, pp. 543–757.

T. A. Brunner, D. Pritchard, “Fitting laws for rotationally inelastic collisions,” in Dynamics of the Excited State, K. P. Lawley, ed. (Wiley, 1982), Vol. L, pp. 589–641.

R. Span, Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (personal communication to K. Kraft, 1994).

VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen, VDI-Wärmeatlas (Springer, Berlin, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Relative Raman line strength of the rotational R and S branch of the (01101e)–(01101f) levels of CO2 as a function of Raman shift (anti-Stokes side).

Fig. 2
Fig. 2

Theoretical CARS spectra of N2 and O2 at a temperature of 2000 K and of CO2 at 300 and 1000 K and a pressure of 0.1 MPa, respectively. The signatures of the particular hot bands are clearly visible for the high-temperature spectra.

Fig. 3
Fig. 3

Linewidth data for CO2–CO2 collisional broadening according to Ref. 22 in comparison with the values modeled by the ECS-P and the MEG law and a polynomial approximation of fourth order.

Fig. 4
Fig. 4

Comparison of experimental spectra of CO2 (accumulation of 200 single-shot spectra, solid curves) with the particular theoretical spectra (dashed curves, obscured) at 300 K and 0.1 and 1.0 MPa, respectively. The differences of the particular theoretical and experimental spectra are likewise depicted. Compared from top to bottom are the ECS-P and the MEG law (Gordon matrix model) and the fourth-order polynomial approximation (isolated line model).

Fig. 5
Fig. 5

Pressure dependence of the resulting temperature mean values for the ECS-P (right) and the MEG law (left). Top, results of the Gordon matrix model; bottom, the isolated line model. The error bars represent twice the statistical error of the single shot results, i.e. ±σ. The particular reference temperatures are displayed as solid lines.

Fig. 6
Fig. 6

Pressure dependence of the resulting temperature mean values for the fourth-order polynomial approximation (isolated line model). The error bars represent twice the statistical error of the single-shot results, i.e. ±σ. The particular reference temperatures are displayed as solid lines.

Fig. 7
Fig. 7

Pressure dependence of the resulting single-shot temperature standard deviation for the ECS-P law. Top, results of the Gordon matrix model; bottom, the isolated line model. Left, obtained by means of a constantly weighted least square fit; right, by a mainly inversely weighted LSF.

Fig. 8
Fig. 8

Enlarged view of the pressure dependence of the resulting single-shot temperature standard deviation for the ECS-P and the MEG law at 300 K. Open symbols, results of the isolated line model; filled symbols, results of the Gordon matrix model, obtained by means of a constantly weighted LSF. The results of the polynomial approach are depicted as well. The pressures on the x axis represent the manometric values; the pressures labels in the plot panel represent the pressures that an ideal gas of the same particle density would have at 300 K.

Fig. 9
Fig. 9

Accumulated spectra (50 shots) of the CARS and coherent Stokes Raman scattering (CSRS) signal of CO2 at 300 K and manometric pressures of 6.5 MPa (left) and 0.1 MPa (right). The corresponding ideal-gas pressure is also stated in the legend. The spectra are detected for the polarization of the pump, with the pump, Stokes, probe, and signal beams being parallel polarized. In the right-hand plot the frequency of the probe beam was covered by a thin rod within the spectrometer for camera protection. The spectra are corrected with respect to the background but not to the excitation profile, which is merely flat in this Raman shift region. The minor irregularities in the left-hand spectrum result from small local irregularities in the photocathode of the intensified diode-array detector.

Tables (1)

Tables Icon

Table 1 Parameters γ|m| (300 K) and N|m| of the Polynomial Correlations [Eqs. (2) and (3)]

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

γ m ( T ) = γ m ( T 0 ) ( T 0 / T ) N m
γ m ( T 0 ) = a 4 m 4 + a 3 m 3 + a 2 m 2 + a 1 m + a 0 ,
N m = b 4 m 4 + b 3 m 3 + b 2 m 2 + b 1 m + b 0 .
k g k i e k - i t k ( T , c ) 2 = min ( T , c i ) .
g k = 1 c 0 + c 1 i e k + c 2 ( i e k ) 2 .
h ν 2 [ ( v 2 a + 1 2 ) + ( v 2 b + 1 2 ) ] = h ν 2 ( v 2 + 1 ) .
l = m N .
l = v 2 , v 2 - 2 , v 2 - 4 , , 1 or 0.
E vib = i 3 h ν i , e ( v i + d i 2 ) + i 3 k i h x i k ( v i + d i 2 ) ( v k + d k 2 ) + h g 22 l 2 + ,
F ( v , J ) = B [ v ] [ J ( J + 1 ) - l 2 ] + A [ v ] l 2 - D [ v ] [ J ( J + 1 ) - l 2 ] 2 + ,
Ψ ( + ) = 1 2 [ Ψ ( + l ) + Ψ ( - l ) ] , Ψ ( - ) = 1 2 [ Ψ ( + l ) - Ψ ( - l ) ] .
Δ l - doubling = ± q t J ( J + 1 ) 2 .
T ( v , J ) + G ( v ) + B [ v ] J ( J + 1 ) - D [ v ] [ J ( J + 1 ) ] 2 + H [ v ] [ J ( J + 1 ) ] 3 + ,
α b a 2 ~ b J J + Δ J γ e 2 .
b J J + 1 = 3 l 2 [ ( J + 1 ) 2 - l 2 ] J ( J + 1 ) ( J + 2 ) ( 2 J + 1 ) ,
b J J + 2 = 3 [ ( J + 1 ) 2 - l 2 ] [ ( J + 2 ) 2 - l 2 ] 2 ( J + 1 ) ( J + 2 ) ( 2 J + 1 ) ( 2 J + 3 ) .
b J J + 2 = 3 ( J + 1 ) ( J + 2 ) 2 ( 2 J + 1 ) ( 2 J + 3 ) ,
b J J + 1 = 3 ( J + 1 ) ( 2 J + 1 )             ( R branch ) ,
b J J + 2 = 3 J ( J + 3 ) 2 ( 2 J + 1 ) ( 2 J + 3 )             ( S branch ) .

Metrics