Abstract

We give a simple two-transition model of Faraday modulation spectrometry (FAMOS) addressing the electronic X2Π(ν=0)A2Σ+(ν=0) band in nitric oxide. The model is given in terms of the integrated line strength, S, and first Fourier coefficients for the magnetic-field-modulated dispersive line shape function. Although the two states addressed respond differently to the magnetic field (they adhere to the dissimilar Hund coupling cases), it is shown that the technique shares some properties with FAMOS when rotational-vibrational Q-transitions are targeted: the line shapes have a similar form and the signal strength has an analogous magnetic field and pressure dependence. The differences are that the maximum signal appears for larger magnetic field amplitudes and pressures, 1500G and 200Torr, respectively.

© 2010 Optical Society of America

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  1. W. T. Piver, “Global atmospheric changes,” Environ. Health Perspect. 96, 131–137 (1991).
    [CrossRef] [PubMed]
  2. J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley-Interscience, 1998), p. 1326.
  3. Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).
  4. A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
    [CrossRef] [PubMed]
  5. G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
    [CrossRef]
  6. W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
    [CrossRef]
  7. A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
    [CrossRef]
  8. T. A. Blake, C. Chackerian, and J. R. Podolske, “Prognosis for a mid-infrared magnetic rotation spectrometer for the in situ detection of atmospheric free radicals,” Appl. Opt. 35, 973–985 (1996).
    [CrossRef] [PubMed]
  9. H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
    [CrossRef]
  10. H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
    [CrossRef]
  11. R. Gäbler and J. Lehmann, “Sensitive and isotope selective (NO14/NO15) online detection of nitric oxide by Faraday-laser magnetic resonance spectroscopy,” Methods Enzymol. 396, 54–60 (2005).
    [CrossRef] [PubMed]
  12. T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
    [CrossRef]
  13. R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.
  14. T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
    [CrossRef]
  15. R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
    [CrossRef] [PubMed]
  16. H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
    [CrossRef]
  17. C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
    [CrossRef] [PubMed]
  18. W. Urban and W. Herrmann, “Zeeman modulation spectroscopy with spin-flip Raman laser,” J. Appl. Phys. 17, 325–330(1978).
    [CrossRef]
  19. J. Shao, L. Lathdavong, P. Thavixay, and O. Axner, “Detection of nitric oxide at low ppb·m concentrations by differential absorption spectrometry using a fully diode-laser-based ultraviolet laser system,” J. Opt. Soc. Am. B 24, 2294–2306 (2007).
    [CrossRef]
  20. S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
    [CrossRef]
  21. D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. I,” J. Chem. Phys. 46, 4525–4529 (1967).
    [CrossRef]
  22. D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. II,” J. Chem. Phys. 50, 5018–5026 (1969).
    [CrossRef]
  23. J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner, “Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+ band: II. experiment,” Appl. Opt. , 49, 5614–5625 (2010).
    [CrossRef] [PubMed]
  24. Although Robinson provided a theoretical description of the MRS from the eight lines originating from a particular rotational state in the ground configuration (the J′′=13/2 state) for a given set of conditions (a given pressure and magnetic field) , the description does not provide any general information useful for predicting or modeling FAMOS signals from other states or under other conditions in any systematic manner.
  25. Although “optimum conditions” often refer to the situations when the signal-to-noise ratio is maximized, we will not do so here. Because we do not consider the noise in the system in this work, “optimum conditions” will refer to the cases for which the signal is maximized.
  26. J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
    [CrossRef]
  27. We have here utilized the same nomenclature as in Ref. , i.e., a tilde sign indicates that the entity is given in units of inverse centimeters, whereas an overbar shows that the entity is dimensionless. The superscript D indicates that it is normalized with respect to δν˜D/ln⁡2.
  28. When the magnetic field changes direction, the propagation of the light alters between being parallel and antiparallel to the magnetic field. If the quantization axis is taken as the direction of the magnetic field, B, as is customary, LHCP light should alter between inducing ΔM=+1 and ΔM=−1 transitions. However, it is mathematically inconvenient to have a quantization axis whose direction is periodically reversed and thereby to periodically alter the transition rules for a given helicity of the light. We have here instead let the quantization axis be fixed along the direction of B0, even though the magnetic field changes direction.
  29. Using ordinary angular momentum coupling, the transition dipole moment squared of a transition between two states expressed in the same basis sets can be expressed in terms of the square of a Clebsch–Gordan coefficient, which in turn can be expressed in terms of the square of a 3-j symbol.
  30. J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
    [CrossRef]
  31. As has been discussed previously , this expression for the FAMOS signal differs from that of Ganser et al. as well as Herrmann et al. by (at least) a factor of 3/(2J+1) and that of Fritsch et al. by a factor of 3.
  32. There is an inherent property of a 3-j symbol of the type given in Eq. that its square summed over all possible M′′ values (and thereby all possible M′ values) yields a value of 1/3. This implies that the sum of all relative dipole moments squared becomes unity.
  33. B. W. Shore, The Theory of Coherent Atomic Excitation, Volume 2, Multilelevel Atoms and Incoherence (Wiley, 1990).
  34. G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed. (van Nostrand Reinhold, 1950).
  35. N is the quantum number that corresponds to the rotational energy of a state adhering to the Hund case (b) that lacks orbital angular momentum (Λ=0), given by BN(N+1), where B is the rotational constant and is associated with the operator (J−S)2, where J and S are the operators for the total angular momentum and the electronic spin, respectively.
  36. As a consequence of a weak coupling between the rotation of the nuclei and the orbital angular momentum of the electron, each lower state is additionally split into two states with opposite symmetry (+ and −, respectively) by a so-called Λ doubling. Because this splitting is smaller than the spin splitting as well as the separation between consecutive rotational levels and transitions only are allowed between states of dissimilar symmetry, this splitting does not give rise to any additional transitions; it can therefore be seen as a perturbation that only shifts the transitions slightly in frequency.
  37. K. Takazawa and H. Abe, “Electronic spectra of gaseous nitric oxide in magnetic fields up to 10T,” J. Chem. Phys. 110, 9492–9499 (1999).
    [CrossRef]
  38. K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
    [CrossRef]
  39. Takazawa et al. investigated the Q11(J′′) and P11(J′′) branches originating from the Π1/22(J′′) state and the Q12(J′′) and P12Q(J′′) branches from the Π3/22(J′′) state (referred to as Q21(J′′) and P21(J′′) by the authors) and their analysis dealt with light inducing ΔM=0 transitions .
  40. Because the total line strength of a transition is not altered by a splitting of the level, it is possible to conclude that S¯Π,MJ,Σ,1/2L+S¯Π,MJ,Σ,−1/2L=S¯Π,ΣL=1 and SΠ,MJ,Σ,1/2R+SΠ,MJ,Σ,−1/2R=S¯Π,ΣR=1, which, in turn, leads to S¯Π,MJ,Σ,1/2L−S¯Π,MJ,Σ,1/2R=S¯Π,MJ,Σ,−1/2R−S¯Π,MJ,Σ,−1/2L, which is Eq. .
  41. S. So, E. Jeng, and G. Wysocki, “VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection,” Appl. Phys. B DOI: 10.1007/s00340-010-4002-1 (2010).
    [CrossRef]
  42. SΠ,Σ and SΠ,Σ′ are related to each other through the relation SΠ,ΣNNO=SΠ,Σ′pNO.
  43. With a splitting of the transition of gSμBB0, and a value of the Bohr magneton of 4.6710−5cm−1/G, a splitting of 25cm−1 for a magnetic field of 25×104G provides a value of the gS factor of 2.1.
  44. C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).
  45. The model is appropriate for the cases when the magnetic splitting of the upper level significantly exceeds that of the lower state, which it does for all states except for those with the lowest rotational quantum number (it is valid primarily for states with J>6.5). However, because the states with lowest rotational quantum number in general also have lower line strengths than those with larger J, this is not a severe limitation.
  46. Because the final detectability of a technique is given by a number of entities, including the available laser power at the transition used, properties of the polarizers as well as the detectors, including noise and disturbances, this analysis cannot yet fully assess the detectability and thereby the true applicability of the FAMOS technique when electronic transitions are addressed. Despite this, the present work, with its characterization of the optimum conditions for FAMOS addressing electronic transitions has provided a first step toward such an assessment.

2010

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner, “Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+ band: II. experiment,” Appl. Opt. , 49, 5614–5625 (2010).
[CrossRef] [PubMed]

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

2009

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

2008

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

2007

2005

R. Gäbler and J. Lehmann, “Sensitive and isotope selective (NO14/NO15) online detection of nitric oxide by Faraday-laser magnetic resonance spectroscopy,” Methods Enzymol. 396, 54–60 (2005).
[CrossRef] [PubMed]

2004

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

2003

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
[CrossRef]

2002

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

2000

K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
[CrossRef]

1999

K. Takazawa and H. Abe, “Electronic spectra of gaseous nitric oxide in magnetic fields up to 10T,” J. Chem. Phys. 110, 9492–9499 (1999).
[CrossRef]

Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).

1996

1992

J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
[CrossRef]

1991

W. T. Piver, “Global atmospheric changes,” Environ. Health Perspect. 96, 131–137 (1991).
[CrossRef] [PubMed]

1982

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

1980

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
[CrossRef]

1978

W. Urban and W. Herrmann, “Zeeman modulation spectroscopy with spin-flip Raman laser,” J. Appl. Phys. 17, 325–330(1978).
[CrossRef]

1972

A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
[CrossRef] [PubMed]

1969

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. II,” J. Chem. Phys. 50, 5018–5026 (1969).
[CrossRef]

1967

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. I,” J. Chem. Phys. 46, 4525–4529 (1967).
[CrossRef]

Abe, H.

K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
[CrossRef]

K. Takazawa and H. Abe, “Electronic spectra of gaseous nitric oxide in magnetic fields up to 10T,” J. Chem. Phys. 110, 9492–9499 (1999).
[CrossRef]

Alkemade, C. T. J.

C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).

Anderson, T. N.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Aust, O.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Axner, O.

Barron-Jimenez, R.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Blake, T. A.

Bohle, W.

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

Bouba, O.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

Brouzos, P.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Brown, A. M.

H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
[CrossRef]

Carter, C. D.

J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
[CrossRef]

Caton, J. A.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Chackerian, C.

Curl, R. F.

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

Dion, C. M.

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

Doty, J.

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

Doty, J. H.

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

Fritsch, T.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

Gäbler, R.

R. Gäbler and J. Lehmann, “Sensitive and isotope selective (NO14/NO15) online detection of nitric oxide by Faraday-laser magnetic resonance spectroscopy,” Methods Enzymol. 396, 54–60 (2005).
[CrossRef] [PubMed]

Ganser, H.

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
[CrossRef]

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Halmer, D.

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

Hanna, S. F.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Heinrich, K.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Hering, P.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Herrmann, W.

W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
[CrossRef]

W. Urban and W. Herrmann, “Zeeman modulation spectroscopy with spin-flip Raman laser,” J. Appl. Phys. 17, 325–330(1978).
[CrossRef]

Herzberg, G.

G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed. (van Nostrand Reinhold, 1950).

Hinz, A.

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

Hollander, T.

C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).

Horstjann, M.

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Hou, Y. C.

Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).

Janczuk, A.

Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).

Jeng, E.

S. So, E. Jeng, and G. Wysocki, “VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection,” Appl. Phys. B DOI: 10.1007/s00340-010-4002-1 (2010).
[CrossRef]

Kaldor, A.

A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
[CrossRef] [PubMed]

Kelm, M.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Kleinbongard, P.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Kluczynski, P.

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner, “Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+ band: II. experiment,” Appl. Opt. , 49, 5614–5625 (2010).
[CrossRef] [PubMed]

Kolb-Bachofen, V.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Kroncke, K. D.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Lathdavong, L.

Laurendeau, N. M.

J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
[CrossRef]

Lehmann, J.

R. Gäbler and J. Lehmann, “Sensitive and isotope selective (NO14/NO15) online detection of nitric oxide by Faraday-laser magnetic resonance spectroscopy,” Methods Enzymol. 396, 54–60 (2005).
[CrossRef] [PubMed]

Lewicki, R.

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

Litfin, G.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Lucht, R. P.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Lundqvist, S.

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner, “Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+ band: II. experiment,” Appl. Opt. , 49, 5614–5625 (2010).
[CrossRef] [PubMed]

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

Mahotka, C.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Maki, A. G.

A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
[CrossRef] [PubMed]

Murtz, M.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Olson, W. B.

A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
[CrossRef] [PubMed]

Onana, M. B.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

Pandis, S. N.

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley-Interscience, 1998), p. 1326.

Pfeiffer, J.

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

Piver, W. T.

W. T. Piver, “Global atmospheric changes,” Environ. Health Perspect. 96, 131–137 (1991).
[CrossRef] [PubMed]

Podolske, J. R.

Pollock, C. R.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Rassaf, T.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Reisel, J. R.

J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. II,” J. Chem. Phys. 50, 5018–5026 (1969).
[CrossRef]

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. I,” J. Chem. Phys. 46, 4525–4529 (1967).
[CrossRef]

Rohrbeck, W.

W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
[CrossRef]

Sabana,

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

Sabana, H.

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

Schnorr, O.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Schroeder, P.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Seinfeld, J. H.

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley-Interscience, 1998), p. 1326.

Shao, J.

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation, Volume 2, Multilelevel Atoms and Incoherence (Wiley, 1990).

Sies, H.

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Snelleman, W.

C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).

So, S.

S. So, E. Jeng, and G. Wysocki, “VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection,” Appl. Phys. B DOI: 10.1007/s00340-010-4002-1 (2010).
[CrossRef]

Suschek, C. V.

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

Takazawa, K.

K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
[CrossRef]

K. Takazawa and H. Abe, “Electronic spectra of gaseous nitric oxide in magnetic fields up to 10T,” J. Chem. Phys. 110, 9492–9499 (1999).
[CrossRef]

Thavixay, P.

Tittel, F. K.

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

Urban, W.

H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
[CrossRef]

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
[CrossRef]

W. Urban and W. Herrmann, “Zeeman modulation spectroscopy with spin-flip Raman laser,” J. Appl. Phys. 17, 325–330(1978).
[CrossRef]

Wada, H.

K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
[CrossRef]

Walther, T.

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

Wang, P. G.

Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).

Westberg, J.

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner, “Faraday modulation spectrometry of nitric oxide addressing its electronic X2Π−A2Σ+ band: II. experiment,” Appl. Opt. , 49, 5614–5625 (2010).
[CrossRef] [PubMed]

Wysocki, G.

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

S. So, E. Jeng, and G. Wysocki, “VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection,” Appl. Phys. B DOI: 10.1007/s00340-010-4002-1 (2010).
[CrossRef]

Zeegers, P. J. T.

C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).

Appl. Opt.

Appl. Phys. B

S. F. Hanna, R. Barron-Jimenez, T. N. Anderson, R. P. Lucht, J. A. Caton, and T. Walther, “Diode-laser-based ultraviolet absorption sensor for nitric oxide,” Appl. Phys. B 75, 113–117 (2002).
[CrossRef]

H. Ganser, M. Horstjann, C. V. Suschek, P. Hering, and M. Murtz, “Online monitoring of biogenic nitric oxide with a QC laser-based Faraday modulation technique,” Appl. Phys. B 78, 513–517 (2004).
[CrossRef]

T. Fritsch, M. Horstjann, D. Halmer, Sabana, P. Hering, and M. Murtz, “Magnetic Faraday modulation spectroscopy of the 1-0 band of NO14 and NO15,” Appl. Phys. B 93, 713–723(2008).
[CrossRef]

H. Sabana, T. Fritsch, M. B. Onana, O. Bouba, P. Hering, and M. Murtz, “Simultaneous detection of NO14 and NO15 using Faraday modulation spectroscopy,” Appl. Phys. B 96, 535–544 (2009).
[CrossRef]

Chem. Phys. Lett.

K. Takazawa, H. Abe, and H. Wada, “Zeeman electronic spectra of gaseous NO in very high magnetic fields up to 25T,” Chem. Phys. Lett. 329, 405–411 (2000).
[CrossRef]

Curr. Pharm. Design

Y. C. Hou, A. Janczuk, and P. G. Wang, “Current trends in the development of nitric oxide donors,” Curr. Pharm. Design 5, 417–441 (1999).

Environ. Health Perspect.

W. T. Piver, “Global atmospheric changes,” Environ. Health Perspect. 96, 131–137 (1991).
[CrossRef] [PubMed]

FASEB Journal

C. V. Suschek, P. Schroeder, O. Aust, H. Sies, C. Mahotka, M. Horstjann, H. Ganser, M. Murtz, P. Hering, O. Schnorr, K. D. Kroncke, and V. Kolb-Bachofen, “The presence of nitrite during UVA irradiation protects from apoptosis,” FASEB Journal 17, 2342–2344 (2003).
[CrossRef] [PubMed]

J. Appl. Phys.

W. Urban and W. Herrmann, “Zeeman modulation spectroscopy with spin-flip Raman laser,” J. Appl. Phys. 17, 325–330(1978).
[CrossRef]

W. Herrmann, W. Rohrbeck, and W. Urban, “Line-shape analysis for Zeeman modulation spectroscopy,” J. Appl. Phys. 22, 71–75 (1980).
[CrossRef]

J. Chem. Phys.

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. I,” J. Chem. Phys. 46, 4525–4529 (1967).
[CrossRef]

D. W. Robinson, “Magnetic rotation spectrum of the A2Σ+−X2Πr transition in NO. II,” J. Chem. Phys. 50, 5018–5026 (1969).
[CrossRef]

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitivity enhancement of laser-absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

K. Takazawa and H. Abe, “Electronic spectra of gaseous nitric oxide in magnetic fields up to 10T,” J. Chem. Phys. 110, 9492–9499 (1999).
[CrossRef]

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

J. R. Reisel, C. D. Carter, and N. M. Laurendeau, “Einstein coefficients for rotational lines of the (0, 0) band of the NO A2Σ+−X2Π system,” J. Quant. Spectrosc. Radiat. Transfer 47, 43–54 (1992).
[CrossRef]

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvist, and O. Axner, “Quantitative description of Faraday modulation spectrometry in terms of the integrated line strength and 1st Fourier coefficients of the modulated line shape function,” J. Quant. Spectrosc. Radiat. Transfer 111, 2415–2433 (2010).
[CrossRef]

Methods Enzymol.

R. Gäbler and J. Lehmann, “Sensitive and isotope selective (NO14/NO15) online detection of nitric oxide by Faraday-laser magnetic resonance spectroscopy,” Methods Enzymol. 396, 54–60 (2005).
[CrossRef] [PubMed]

Mol. Phys.

A. Hinz, J. Pfeiffer, W. Bohle, and W. Urban, “Mid-infrared laser magnetic-resonance using the Faraday and Voigt effects for sensitive detection,” Mol. Phys. 45, 1131–1139 (1982).
[CrossRef]

H. Ganser, W. Urban, and A. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101, 545–550 (2003).
[CrossRef]

Nitric Oxide: Biol. Chem.

T. Fritsch, P. Brouzos, K. Heinrich, M. Kelm, T. Rassaf, P. Hering, P. Kleinbongard, and M. Murtz, “NO detection in biological samples: differentiation of NO14 and NO15 using infrared laser spectroscopy,” Nitric Oxide: Biol. Chem. 19, 50–56 (2008).
[CrossRef]

Proc. Natl. Acad. Sci. USA

R. Lewicki, J. H. Doty, R. F. Curl, F. K. Tittel, and G. Wysocki, “Ultrasensitive detection of nitric oxide at 5.33μm by using external cavity quantum cascade laser-based Faraday rotation spectroscopy,” Proc. Natl. Acad. Sci. USA 106, 12587–12592 (2009).
[CrossRef] [PubMed]

Science

A. Kaldor, A. G. Maki, and W. B. Olson, “Pollution monitor for nitric oxide: a laser device based on the Zeeman modulation of absorption,” Science 176, 508–510 (1972).
[CrossRef] [PubMed]

Other

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley-Interscience, 1998), p. 1326.

R. Lewicki, G. Wysocki, J. Doty, R. F. Curl, and F. K. Tittel, “Ultra-sensitive detection of nitric oxide at 5.33μm using an external cavity QCL based Faraday rotation spectroscopic sensor platform,” in 2008 Conference on Lasers and Electro-Optics & Quantum Electronics and Laser Science Conference (2008), Vols. 1–9, pp. 514–515.

We have here utilized the same nomenclature as in Ref. , i.e., a tilde sign indicates that the entity is given in units of inverse centimeters, whereas an overbar shows that the entity is dimensionless. The superscript D indicates that it is normalized with respect to δν˜D/ln⁡2.

When the magnetic field changes direction, the propagation of the light alters between being parallel and antiparallel to the magnetic field. If the quantization axis is taken as the direction of the magnetic field, B, as is customary, LHCP light should alter between inducing ΔM=+1 and ΔM=−1 transitions. However, it is mathematically inconvenient to have a quantization axis whose direction is periodically reversed and thereby to periodically alter the transition rules for a given helicity of the light. We have here instead let the quantization axis be fixed along the direction of B0, even though the magnetic field changes direction.

Using ordinary angular momentum coupling, the transition dipole moment squared of a transition between two states expressed in the same basis sets can be expressed in terms of the square of a Clebsch–Gordan coefficient, which in turn can be expressed in terms of the square of a 3-j symbol.

As has been discussed previously , this expression for the FAMOS signal differs from that of Ganser et al. as well as Herrmann et al. by (at least) a factor of 3/(2J+1) and that of Fritsch et al. by a factor of 3.

There is an inherent property of a 3-j symbol of the type given in Eq. that its square summed over all possible M′′ values (and thereby all possible M′ values) yields a value of 1/3. This implies that the sum of all relative dipole moments squared becomes unity.

B. W. Shore, The Theory of Coherent Atomic Excitation, Volume 2, Multilelevel Atoms and Incoherence (Wiley, 1990).

G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed. (van Nostrand Reinhold, 1950).

N is the quantum number that corresponds to the rotational energy of a state adhering to the Hund case (b) that lacks orbital angular momentum (Λ=0), given by BN(N+1), where B is the rotational constant and is associated with the operator (J−S)2, where J and S are the operators for the total angular momentum and the electronic spin, respectively.

As a consequence of a weak coupling between the rotation of the nuclei and the orbital angular momentum of the electron, each lower state is additionally split into two states with opposite symmetry (+ and −, respectively) by a so-called Λ doubling. Because this splitting is smaller than the spin splitting as well as the separation between consecutive rotational levels and transitions only are allowed between states of dissimilar symmetry, this splitting does not give rise to any additional transitions; it can therefore be seen as a perturbation that only shifts the transitions slightly in frequency.

Although Robinson provided a theoretical description of the MRS from the eight lines originating from a particular rotational state in the ground configuration (the J′′=13/2 state) for a given set of conditions (a given pressure and magnetic field) , the description does not provide any general information useful for predicting or modeling FAMOS signals from other states or under other conditions in any systematic manner.

Although “optimum conditions” often refer to the situations when the signal-to-noise ratio is maximized, we will not do so here. Because we do not consider the noise in the system in this work, “optimum conditions” will refer to the cases for which the signal is maximized.

Takazawa et al. investigated the Q11(J′′) and P11(J′′) branches originating from the Π1/22(J′′) state and the Q12(J′′) and P12Q(J′′) branches from the Π3/22(J′′) state (referred to as Q21(J′′) and P21(J′′) by the authors) and their analysis dealt with light inducing ΔM=0 transitions .

Because the total line strength of a transition is not altered by a splitting of the level, it is possible to conclude that S¯Π,MJ,Σ,1/2L+S¯Π,MJ,Σ,−1/2L=S¯Π,ΣL=1 and SΠ,MJ,Σ,1/2R+SΠ,MJ,Σ,−1/2R=S¯Π,ΣR=1, which, in turn, leads to S¯Π,MJ,Σ,1/2L−S¯Π,MJ,Σ,1/2R=S¯Π,MJ,Σ,−1/2R−S¯Π,MJ,Σ,−1/2L, which is Eq. .

S. So, E. Jeng, and G. Wysocki, “VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection,” Appl. Phys. B DOI: 10.1007/s00340-010-4002-1 (2010).
[CrossRef]

SΠ,Σ and SΠ,Σ′ are related to each other through the relation SΠ,ΣNNO=SΠ,Σ′pNO.

With a splitting of the transition of gSμBB0, and a value of the Bohr magneton of 4.6710−5cm−1/G, a splitting of 25cm−1 for a magnetic field of 25×104G provides a value of the gS factor of 2.1.

C. T. J. Alkemade, T. Hollander, W. Snelleman, and P. J. T. Zeegers, Metal Vapours in Flames (Pergamon, 1982).

The model is appropriate for the cases when the magnetic splitting of the upper level significantly exceeds that of the lower state, which it does for all states except for those with the lowest rotational quantum number (it is valid primarily for states with J>6.5). However, because the states with lowest rotational quantum number in general also have lower line strengths than those with larger J, this is not a severe limitation.

Because the final detectability of a technique is given by a number of entities, including the available laser power at the transition used, properties of the polarizers as well as the detectors, including noise and disturbances, this analysis cannot yet fully assess the detectability and thereby the true applicability of the FAMOS technique when electronic transitions are addressed. Despite this, the present work, with its characterization of the optimum conditions for FAMOS addressing electronic transitions has provided a first step toward such an assessment.

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration the energy level structure associated with a Q 22 ( J ) and a R 12 Q ( J ) transition in NO. The Π 2 ground state is first split into two states because of spin splitting (denoted by Π 1 / 2 2 and Π 3 / 2 2 ). Each such state is then split into various rotational states according to its total angular momentum, J. Each rotational state is then split into two with opposite parity because of Λ doubling. The upper Σ 2 state is first split in various rotational states according to the rotation of the nuclei, N. Each rotational state is then split into two (denoted by F 1 and F 2 ) because of a coupling between the electronic spin and the rotation of the nuclei, sometimes referred to as ρ-type doubling. The Q 22 ( J ) and the R 12 Q ( J ) lines correspond to transitions for which Δ N = Δ J = 0 (for the former) and Δ N = 0 and Δ J = + 1 (for the latter). The two types of transition originate from a given Π 3 / 2 2 ( J ) state (for which N = J + 1 / 2 ), but they couple to dissimilar excited states, Σ 2 ( F 2 , N = N ) (for which J = N 1 / 2 = N 1 / 2 = J ), and Σ 2 ( F 1 , N = N ) (for which J = N + 1 / 2 = N + 1 / 2 = J + 1 ), respectively. Because NO has a very small (or no) ρ-type doubling, these two transitions are strongly (often considered fully) overlapping.

Fig. 2
Fig. 2

Simplified model for FAMOS addressing an electronic transition in NO. The two panels represent the case (a) without and (b) with the magnetic field present. It is assumed that the upper state, Σ 2 ( F i , J , N ) , is split by the magnetic field into two states, Σ 2 ( F i , J , N , M S = 1 / 2 ) and Σ 2 ( F i , J , N , M S = 1 / 2 ) , denoted by Σ 2 ( 1 / 2 ) and Σ 2 ( 1 / 2 ) for short, that are shifted an amount Δ E M S = M S g S μ B B 0 . The lower state, Π i 2 ( J , N ) , is assumed not to be split. Both the LHCP (L) and RHCP (R) light can induce transitions to each of the two upper states, although to a dissimilar extent. S ¯ Π , Σ L and S ¯ Π , Σ R denote the relative line strengths for the transitions induced by LHCP and RHCP light from a lower Π i 2 ( J , N ) state to an upper Σ 2 ( F i , J , N ) state in the absence of a magnetic field (which both are equal to unity), whereas S ¯ Π , M J , Σ , 1 / 2 L / R and S ¯ Π , M J , Σ , 1 / 2 L / R are the corresponding relative line strengths for the transitions induced by LHCP/RHCP light to the M S = 1 / 2 and M S = 1 / 2 states, respectively, in the presence of a magnetic field.

Fig. 3
Fig. 3

Normalized FAMOS signal, S ¯ Π , Σ F ( Δ ν ˜ Π , Σ 0 ) , from a gaseous sample with a given concentration of NO in N 2 at four different total pressures as a function of frequency detuning, Δ ν ˜ Π , Σ 0 , expressed in units of inverse centimeters, for a set of magnetic field amplitudes. The 18 curves in each panel correspond to magnetic field amplitudes of 25, 50, 100, 150, 250, 400, 600, 750, 1000, 1250, 1500, 2000, 2500, 3000, 3500, 4000, 4500, and 5000 G , respectively. The four panels, (a)–(d), represent total pressures of 1, 100, 300, and 760 Torr , respectively. The shifts of the curves in the various panels are due to a pressure (collision) shift of the transition. The collision broadening and shift have been taken as 0.582 cm 1 / atm (FWHM) and 0.174 cm 1 / atm , respectively [19]. In this wavelength region, 1 cm 1 corresponds roughly to 5 pm .

Fig. 4
Fig. 4

Peak value of the normalized FAMOS signal, represented by the pressure-shifted on-resonance value of the signal, as a function of magnetic field amplitude for a variety of total pressures, ranging from 1 to 760 Torr . The ten curves in each panel represent, from the bottom to the top (at the highest magnetic field amplitudes), total pressures of 1, 10, 25, 50, 100, 200, 300, 400, 550, and 760 Torr , respectively.

Fig. 5
Fig. 5

Normalized FAMOS signal, S ¯ Π , Σ F ( Δ ν ˜ Π , Σ 0 ) , from a sample with a given concentration of NO in N 2 pumped down to various total pressures (ranging from 5 to 760 Torr ) as a function of frequency detuning, Δ ν ˜ Π , Σ 0 , for four different magnetic field amplitudes. The four panels, (a)–(d), represent magnetic field amplitudes of 100, 750, 1250, and 2500 G , respectively. The 12 curves in each panel correspond to pressures of 5, 10, 20, 35, 50, 75, 100, 150, 250, 350, 500, and 760 Torr , respectively. As for Fig. 3, the collision broadening and shift have been taken as 0.582 cm 1 / atm (FWHM) and 0.174 cm 1 / atm , respectively [19]. 1 cm 1 corresponds roughly to 5 pm .

Fig. 6
Fig. 6

Peak value of the normalized FAMOS signal, represented by the pressure-shifted on-resonance value of the signal, as a function of total pressure for a variety of magnetic field amplitudes, ranging from 50 to 5000 G . The 14 curves in each panel represent, from the bottom to the top (at the highest pressure), magnetic field amplitudes of 50, 100, 200, 300, 500, 750, 1000, 1500, 2000, 2500, 3000, 3500, 4000, and 5000 G , respectively.

Fig. 7
Fig. 7

(a) Magnetic field amplitude that provides the maximum FAMOS signal (pressure-shifted on-resonance peak-value value) from a gaseous sample with a given concentration of NO for a given total pressure, referred to as the “optimum magnetic field.” (b) Total pressure that provides the maximum FAMOS signal for a given magnetic field amplitude, denoted the “optimum total pressure.”

Fig. 8
Fig. 8

(a) Maximum FAMOS signal (pressure-shifted on- resonance peak-value value) as a function of pressure (thus for the optimum magnetic field). (b) Maximum FAMOS signal as a function of magnetic field amplitude (thus for the optimum pressure).

Equations (19)

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S i , j F ( ν ˜ ) = S i , j F , 0 · χ ¯ 1 F ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , δ ν ¯ L D ) .
S i , j F , 0 = η S i , j N NO L χ ^ 0 8 sin ( 2 φ ) I 0 ,
χ ¯ 1 F ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , δ ν ¯ L D ) = M , M [ S ¯ i , M , j , M L · χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , M , j , M a , D , δ ν ¯ L D ) S ¯ i , M , j , M R · χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , M , j , M a , D , δ ν ¯ L D ) ] ,
χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , M , j , M a , D , δ ν ¯ L D ) = 2 τ 0 τ χ ¯ disp [ Δ ν ¯ i , M , j , M D ( t ) , δ ν ¯ L D ] cos ( ω t ) d t ,
Δ ν ¯ i , M , j , M D ( t ) = Δ ν ¯ i , j D , 0 ν ¯ i , M , j , M a , D cos ( ω t ) ,
ν ¯ i , j a , D , L / R = ± ν ¯ i , j a , D = ± g J μ B B 0 ln 2 / δ ν ˜ D .
S ¯ i , M , j , M L / R = 3 | J M | μ ¯ ± | J M | 2 = 3 ( J 1 J M J ± 1 M J ) 2 .
χ ¯ 1 F , 2 ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , δ ν ¯ L D ) = χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , L , δ ν ¯ L D ) χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , R , δ ν ¯ L D ) ,
χ ¯ 1 F , 2 ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , δ ν ¯ L D ) = 2 χ ¯ 1 disp , even ( Δ ν ¯ i , j D , 0 , ν ¯ i , j a , D , L , δ ν ¯ L D ) .
Δ E M S = M S g S μ B B 0 ,
χ ¯ 1 F , E ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = S ¯ Π , M J , Σ , 1 / 2 L · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) + S ¯ Π , M J , Σ , 1 / 2 L · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) S ¯ Π , M J , Σ , 1 / 2 R · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) S ¯ Π , M J , Σ , 1 / 2 R · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) = Δ S ¯ Π , M J , Σ , 1 / 2 net · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) Δ S ¯ Π , M J , Σ , 1 / 2 net · χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) ,
ν ¯ Π , M J , Σ , M S a , D = M S ν ¯ Π , Σ a , D ,
Δ S ¯ Π , M J , Σ , 1 / 2 net = Δ S ¯ Π , M J , Σ , 1 / 2 net
χ ¯ 1 F , E ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = Δ S ¯ net [ χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) ] = 2 Δ S ¯ net χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) = Δ S ¯ net χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) ,
S Π , Σ F , E ( Δ ν ¯ Π , Σ D , 0 ) = S Π , Σ F , 0 χ ¯ 1 F , E ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = S Π , Σ F , 0 Δ S ¯ net χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = S Π , Σ F , E , 0 χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) .
S Π , Σ F , E ( Δ ν ¯ Π , Σ D , 0 ) = η S Π , Σ p NO L χ ^ 0 Δ S ¯ net 8 sin ( 2 φ ) I 0 χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = η S Π , Σ p 0 L χ ^ 0 Δ S ¯ net 8 sin ( 2 φ ) I 0 c NO p tot p 0 χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = S Π , Σ F , E , 0 , atm c NO p tot p 0 χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) ,
S ¯ Π , Σ F ( Δ ν ¯ Π , Σ D , 0 ) = p tot p 0 χ ¯ 1 F , 2 ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , Σ a , D , δ ν ¯ L D ) = 2 p tot p 0 χ ¯ 1 disp , even ( Δ ν ¯ Π , Σ D , 0 , ν ¯ Π , M J , Σ , 1 / 2 a , D , δ ν ¯ L D ) .
B E B Q = 2 g J g S δ ν ˜ D E δ ν ˜ D Q g J ν ˜ Π , Σ E ν ˜ i , j Q = 18 ,
max [ S Π , Σ F , E ( Δ ν ¯ Π , Σ D , 0 ) ] max [ S i , j F , Q ( Δ ν ¯ i , j D , 0 ) ] = S Π , Σ E S i , j Q δ ν ˜ D Q δ ν ˜ D E Δ S ¯ net δ ν ¯ L D , Q ( p 0 ) δ ν ¯ L D , E ( p 0 ) = S Π , Σ E S i , j Q γ Q γ E Δ S ¯ net ,

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