Abstract

Laser-induced incandescence from soot was analyzed with a time-dependent, numerical model of particle heating and cooling processes that includes spatial and temporal intensity profiles associated with laser sheet illumination. For volume fraction measurements, substantial errors result primarily from changes in gas temperature and primary soot particle size. The errors can be reduced with the proper choice of detection wavelength, prompt gating, and high laser intensities. Two techniques for primary particle size measurements, based on ratios of laser-induced incandescence signals from a single laser pulse, were also examined. Compared with the ratio of two integration times, the newly proposed ratio of two detection wavelengths is better suited for simultaneous volume fraction and size measurements, because it is less temperature sensitive and produces stronger signals with, however, a lower sensitivity to size changes.

© 1997 Optical Society of America

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References

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  1. A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. 48, 4473–4479 (1977).
    [CrossRef]
  2. L. A. Melton, “Soot diagnostics based on laser heating,” Appl. Opt. 23, 2201–2208 (1984).
    [CrossRef] [PubMed]
  3. B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
    [CrossRef]
  4. R. L. Vander Wal, K. J. Weiland, “Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction,” Appl. Phys. B 59, 445–452 (1994).
    [CrossRef]
  5. J. E. Dec, A. O. zur Loye, D. L. Siebers, “Soot distribution in a D.I. diesel engine using 2-D laser-induced incandescence imaging,” SAE paper 910224 (Society of Automotive Engineers, Warrendale, Pa., 1991).
  6. J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).
  7. F. Cignoli, S. Benecchi, G. Zizak, “Time-delayed detection of laser-induced incandescence for the two-dimensional visualization of soot in flames,” Appl. Opt. 33, 5778–5782 (1994).
    [CrossRef] [PubMed]
  8. P.-E. Bengtsson, M. Aldén, “Soot-visualization strategies using laser techniques,” Appl. Phys. B 60, 51–59 (1995).
    [CrossRef]
  9. T. Ni, J. A. Pinson, S. Gupta, R. J. Santoro, “Two-dimensional imaging of soot volume fraction by the use of laser-induced incandescence,” Appl. Opt. 34, 7083–7091 (1995).
    [CrossRef] [PubMed]
  10. C. M. Sorensen, J. Cai, N. Lu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
    [CrossRef] [PubMed]
  11. R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.
  12. J. Lahaye, G. Prado, “Morphology and internal structure of soot and carbon blacks,” in Particulate Carbon: Formation During Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), pp. 33–55.
    [CrossRef]
  13. C. J. Dasch, “Continuous-wave probe laser investigation of laser vaporization of small soot particles in a flame,” Appl. Opt. 23, 2209–2215 (1984).
    [CrossRef] [PubMed]
  14. D. L. Hofeldt, “Real-time soot concentration measurement technique for engine exhaust streams,” SAE Paper 930079 (Society of Automotive Engineers, Warrendale, Pa., 1993).
  15. H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
    [CrossRef]
  16. R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
    [CrossRef]
  17. R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
    [CrossRef]
  18. R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
    [CrossRef]
  19. B. Mewes, J. M. Seitzman, “Analysis of laser-induced incandescence and novel soot measurement approaches,” presented at the 34th Aerospace Sciences Meeting, Reno, Nev., 15–18 January 1996, paper AIAA-96-0538.
  20. S. Will, S. Schraml, A. Leipertz, “Two-dimensional soot-particle sizing by time-resolved laser induced incandescence,” Opt. Lett. 20, 2342–2344 (1995).
    [CrossRef]
  21. F. M. White, Viscous Fluid Flow, 2nd ed. (McGraw-Hill, New York, 1991), pp. 28–32.

1995

R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
[CrossRef]

P.-E. Bengtsson, M. Aldén, “Soot-visualization strategies using laser techniques,” Appl. Phys. B 60, 51–59 (1995).
[CrossRef]

S. Will, S. Schraml, A. Leipertz, “Two-dimensional soot-particle sizing by time-resolved laser induced incandescence,” Opt. Lett. 20, 2342–2344 (1995).
[CrossRef]

T. Ni, J. A. Pinson, S. Gupta, R. J. Santoro, “Two-dimensional imaging of soot volume fraction by the use of laser-induced incandescence,” Appl. Opt. 34, 7083–7091 (1995).
[CrossRef] [PubMed]

1994

F. Cignoli, S. Benecchi, G. Zizak, “Time-delayed detection of laser-induced incandescence for the two-dimensional visualization of soot in flames,” Appl. Opt. 33, 5778–5782 (1994).
[CrossRef] [PubMed]

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

R. L. Vander Wal, K. J. Weiland, “Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction,” Appl. Phys. B 59, 445–452 (1994).
[CrossRef]

1993

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

1992

1987

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

1984

1977

A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. 48, 4473–4479 (1977).
[CrossRef]

1973

H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
[CrossRef]

Aldén, M.

P.-E. Bengtsson, M. Aldén, “Soot-visualization strategies using laser techniques,” Appl. Phys. B 60, 51–59 (1995).
[CrossRef]

Benecchi, S.

Bengtsson, P.-E.

P.-E. Bengtsson, M. Aldén, “Soot-visualization strategies using laser techniques,” Appl. Phys. B 60, 51–59 (1995).
[CrossRef]

Cai, J.

Choi, M. Y.

R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
[CrossRef]

Cignoli, F.

Dasch, C. J.

Dec, J. E.

J. E. Dec, A. O. zur Loye, D. L. Siebers, “Soot distribution in a D.I. diesel engine using 2-D laser-induced incandescence imaging,” SAE paper 910224 (Society of Automotive Engineers, Warrendale, Pa., 1991).

Dobbins, R. A.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Eckbreth, A. C.

A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. 48, 4473–4479 (1977).
[CrossRef]

Gupta, S.

Hofeldt, D. L.

D. L. Hofeldt, “Real-time soot concentration measurement technique for engine exhaust streams,” SAE Paper 930079 (Society of Automotive Engineers, Warrendale, Pa., 1993).

Horvath, J. J.

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

Krikorian, O. H.

H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
[CrossRef]

Lahaye, J.

J. Lahaye, G. Prado, “Morphology and internal structure of soot and carbon blacks,” in Particulate Carbon: Formation During Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), pp. 33–55.
[CrossRef]

Lee, K.-O.

R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
[CrossRef]

Lee, T. W.

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

Leider, H. R.

H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
[CrossRef]

Leipertz, A.

Litzinger, T. A.

J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).

Lu, N.

Melton, L. A.

Mewes, B.

B. Mewes, J. M. Seitzman, “Analysis of laser-induced incandescence and novel soot measurement approaches,” presented at the 34th Aerospace Sciences Meeting, Reno, Nev., 15–18 January 1996, paper AIAA-96-0538.

Mitchell, D. L.

J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).

Ni, T.

T. Ni, J. A. Pinson, S. Gupta, R. J. Santoro, “Two-dimensional imaging of soot volume fraction by the use of laser-induced incandescence,” Appl. Opt. 34, 7083–7091 (1995).
[CrossRef] [PubMed]

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

Pinson, J. A.

T. Ni, J. A. Pinson, S. Gupta, R. J. Santoro, “Two-dimensional imaging of soot volume fraction by the use of laser-induced incandescence,” Appl. Opt. 34, 7083–7091 (1995).
[CrossRef] [PubMed]

J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).

Prado, G.

J. Lahaye, G. Prado, “Morphology and internal structure of soot and carbon blacks,” in Particulate Carbon: Formation During Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), pp. 33–55.
[CrossRef]

Puri, R.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

Quay, B.

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

Richardson, T. F.

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

Santoro, R. J.

T. Ni, J. A. Pinson, S. Gupta, R. J. Santoro, “Two-dimensional imaging of soot volume fraction by the use of laser-induced incandescence,” Appl. Opt. 34, 7083–7091 (1995).
[CrossRef] [PubMed]

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Schraml, S.

Seitzman, J. M.

B. Mewes, J. M. Seitzman, “Analysis of laser-induced incandescence and novel soot measurement approaches,” presented at the 34th Aerospace Sciences Meeting, Reno, Nev., 15–18 January 1996, paper AIAA-96-0538.

Semerjian, H. G.

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Siebers, D. L.

J. E. Dec, A. O. zur Loye, D. L. Siebers, “Soot distribution in a D.I. diesel engine using 2-D laser-induced incandescence imaging,” SAE paper 910224 (Society of Automotive Engineers, Warrendale, Pa., 1991).

Sorensen, C. M.

Vander Wal, R. L.

R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
[CrossRef]

R. L. Vander Wal, K. J. Weiland, “Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction,” Appl. Phys. B 59, 445–452 (1994).
[CrossRef]

Weiland, K. J.

R. L. Vander Wal, K. J. Weiland, “Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction,” Appl. Phys. B 59, 445–452 (1994).
[CrossRef]

White, F. M.

F. M. White, Viscous Fluid Flow, 2nd ed. (McGraw-Hill, New York, 1991), pp. 28–32.

Will, S.

Yeh, T. T.

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

Young, D. A.

H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
[CrossRef]

Zizak, G.

zur Loye, A. O.

J. E. Dec, A. O. zur Loye, D. L. Siebers, “Soot distribution in a D.I. diesel engine using 2-D laser-induced incandescence imaging,” SAE paper 910224 (Society of Automotive Engineers, Warrendale, Pa., 1991).

Appl. Opt.

Appl. Phys. B

P.-E. Bengtsson, M. Aldén, “Soot-visualization strategies using laser techniques,” Appl. Phys. B 60, 51–59 (1995).
[CrossRef]

R. L. Vander Wal, K. J. Weiland, “Laser-induced incandescence: development and characterization towards a measurement of soot-volume fraction,” Appl. Phys. B 59, 445–452 (1994).
[CrossRef]

Carbon

H. R. Leider, O. H. Krikorian, D. A. Young, “Thermodynamic properties of carbon up to the critical point,” Carbon 11, 555–563 (1973).
[CrossRef]

Combust. Flame

R. L. Vander Wal, M. Y. Choi, K.-O. Lee, “The effects of rapid heating of soot: implications when using laser-induced incandescence for soot diagnostics,” Combust. Flame 102, 200–204 (1995).
[CrossRef]

R. Puri, T. F. Richardson, R. J. Santoro, R. A. Dobbins, “Aerosol dynamic processes of soot aggregates in a laminar ethylene diffusion flame,” Combust. Flame 92, 320–333 (1993).
[CrossRef]

B. Quay, T. W. Lee, T. Ni, R. J. Santoro, “Spatially resolved measurements of soot volume fraction using laser-induced incandescence,” Combust. Flame 97, 384–392 (1994).
[CrossRef]

Combustion Sci. Technol.

R. J. Santoro, T. T. Yeh, J. J. Horvath, H. G. Semerjian, “The transport and growth of soot particles in laminar diffusion flames,” Combustion Sci. Technol. 53, 89–115 (1987).
[CrossRef]

J. Appl. Phys.

A. C. Eckbreth, “Effects of laser-modulated particulate incandescence on Raman scattering diagnostics,” J. Appl. Phys. 48, 4473–4479 (1977).
[CrossRef]

Opt. Lett.

Other

D. L. Hofeldt, “Real-time soot concentration measurement technique for engine exhaust streams,” SAE Paper 930079 (Society of Automotive Engineers, Warrendale, Pa., 1993).

J. E. Dec, A. O. zur Loye, D. L. Siebers, “Soot distribution in a D.I. diesel engine using 2-D laser-induced incandescence imaging,” SAE paper 910224 (Society of Automotive Engineers, Warrendale, Pa., 1991).

J. A. Pinson, D. L. Mitchell, R. J. Santoro, T. A. Litzinger, “Quantitative, planar soot measurements in a D.I. diesel engine using laser-induced incandescence and light scattering,” SAE paper 932650 (Society of Automotive Engineers, Warrendale, Pa., 1993).

B. Mewes, J. M. Seitzman, “Analysis of laser-induced incandescence and novel soot measurement approaches,” presented at the 34th Aerospace Sciences Meeting, Reno, Nev., 15–18 January 1996, paper AIAA-96-0538.

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

J. Lahaye, G. Prado, “Morphology and internal structure of soot and carbon blacks,” in Particulate Carbon: Formation During Combustion, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), pp. 33–55.
[CrossRef]

F. M. White, Viscous Fluid Flow, 2nd ed. (McGraw-Hill, New York, 1991), pp. 28–32.

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Figures (7)

Fig. 1
Fig. 1

Comparison of experimental4 and predicted time histories of the LII signal from a Nd:YAG laser sheet with Gaussian temporal (5-ns FWHM) and spatial (500-µm FWHM) laser intensity distributions, and an equivalent laser intensity of 36 MW/cm2.

Fig. 2
Fig. 2

Soot volume fraction in a laminar ethylene–air diffusion flame (3.85 cm3/s of ethylene and 713.3 cm3/s of air) measured by extinction,17 compared with predicted LII signals [Eq. (3)] for frequency-doubled (2×) and fundamental outputs of a Nd:YAG laser with I eq = 100 MW/cm2 and detection at 400 nm. The simulated LII signal is converted to volume fraction by a calibration to the extinction data at a height of 30 mm; ppm, parts in 106.

Fig. 3
Fig. 3

Error in soot volume fraction measurements for frequency-doubled Nd:YAG illumination with an equivalent laser intensity of 100 MW/cm2 for three different time gates (in nanoseconds). The top figure is for a 50% increase in primary particle diameter (from 20 to 30 nm); the bottom is for a 33% increase in local gas temperature (from 1500 K to 2000 K).

Fig. 4
Fig. 4

Instantaneous error ε′ for soot volume fraction measurement with a frequency-doubled Nd:YAG laser sheet having a Gaussian intensity profile, 100 MW/cm2 equivalent laser intensity, and with a detection wavelength at 400 nm, for a 50% increase in primary particle diameter (20 to 30 nm) and a 33% increase in local gas temperature (1500 to 2000 K). The LII signal is shown for a particle at the calibration condition (1500 K, 20-nm diameter).

Fig. 5
Fig. 5

Error in soot volume fraction measurements with different excitation sources and detection at 700 nm, as a function of equivalent laser intensity: a, for a 50% increase in primary particle diameter; b, for a 33% increase in local gas temperature.

Fig. 6
Fig. 6

Ratio of LII signal detected at the 700-nm to 400-nm signal, as a function of primary particle diameter for three different local gas temperatures. The excitation source is a frequency-doubled Nd:YAG with an equivalent laser intensity of 50 MW/cm2; a, prompt gate from 0 to 50 ns; b, delayed gate from 50 to 450 ns.

Fig. 7
Fig. 7

Sensitivity of time gated signal ratio to primary particle size and gas temperature. Results are shown for the ratio of a 50-ns to a 400-ns delayed gate, both with a 50-ns duration. Excitation is from a 100 MW/cm2 frequency-doubled Nd:YAG laser and detection is at 400 nm (note the exponential scaling for R t ).

Equations (7)

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

fV=π6 Nnpdp3.
Spt, T, Tp, 0, dp, 0, I1, ΔtL, λdet=πdp2t×Ω4π-Gλλdetελ,pdpteb,λTptdλ,
St1, t2, T, dp, 0, Ieq, ΔtL, ΔxL, λdet=AmvNnp-t1t2×Spt, T, dp, 0, I1xmv, ΔtL, λdetdtdxmv,
C=SfV=AmvΩ4πNnp-t1t2Spt, T, dp, 0, I1xmv, ΔtL, λdetdtdxmvπ6Nnpdp,03.
ε  fV,meas-fV,realfV,real=Creal-CcalCcal,
Rλλa, λb=-t1t2Spt, T, dp, 0, I1xmv, ΔtL, λbdtdxmv-t1t2Spt, T, dp, 0, I1xmv, ΔtL, λadtdxmv.
Rtt1,a, t2,a, t1,b, t2,b=-t1bt2bSpt, T, dp, 0, I1xmv, ΔtL, λdetdtdxmv-t1at2a Spt, T, dp, 0, I1xmv, ΔtL, λdetdtdxmv,

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