Abstract

The averaging effect of velocity-changing collisions reduces the Doppler broadening of isolated spectral lines and leads to one type of collisional narrowing. We present four collisionally narrowed profiles in standardized form using dimensionless parameters and estimate the quantitative effect of narrowing on the spectral line shape in terms of the magnitudes of these parameters. We show that a collisionally narrowed profile fitted by a Voigt function exhibits a characteristic signature on a plot of the residual errors in the fit. This provides a simple test for detectable narrowing effects. One of the simpler and better known models which includes collisional narrowing is the Galatry profile. We present sample plots of the residual errors resulting when theoretical profiles computed from other more elaborate models are fitted by a Galatry function.

© 1984 Optical Society of America

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References

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  1. R. H. Dicke, “The Effect of Collisions upon the Doppler Width of Spectral Lines,” Phys. Rev. 89, 472 (1953).
    [Crossref]
  2. J. R. Murray, A. Javan, “Effects of Collisions on Raman Line Profiles of Hydrogen and Deuterium Gas,” J. Mol. Spectrosc. 42, 1 (1972).
    [Crossref]
  3. R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
    [Crossref]
  4. A. S. Pine, “Collisional Narrowing of HF Fundamental Band Spectral Lines by Neon and Argon,” J. Mol. Spectrosc. 82, 435 (1980).
    [Crossref]
  5. P. L. Varghese, R. K. Hanson, “Tunable Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” submitted to J. Quant. Spectrosc. Radiat. Transfer.
  6. D-W. Chen, E. R. Niple, S. K. Poultney, “Determining Tunable Diode Laser Spectrometer Performance Through Measurements of N2O Line Intensities and Widths at 7.8 μm,” Appl. Opt. 21, 2906 (1982).
    [Crossref] [PubMed]
  7. S. G. Rautian, I. I. Sobelman, “Effect of Collisions on the Doppler Broadening of Spectral Lines,” Sov. Phys. Usp. 9, 701 (1967)
    [Crossref]
  8. F. Herbert, “Spectral Line Profile: A Generalized Voigt Function Including Collisional Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1974).
    [Crossref]
  9. E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).
  10. I. I. Sobelman, L. A. Vainshtein, E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, 1981).
    [Crossref]
  11. G. J. Nienhuis, “Effects of Radiator Motion in the Classical and Quantum Mechanical Theories of Collisional Spectral-Line Broadening,” J. Quant. Spectrosc. Radiat. Transfer 20, 275 (1978).
    [Crossref]
  12. P. L. Varghese, Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide, Report 6-83-T, HTGL, Stanford U., Stanford, Calif. (1983).
  13. R. G. Gordon, “Theory of the Width and Shift of Molecular Spectral Lines in Gases,” J. Chem. Phys. 44, 3083 (1966).
    [Crossref]
  14. L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
    [Crossref]
  15. P. R. Berman, “Quantum-Mechanical Transport Equation for Atomic Systems,” Phys. Rev. A 5, 927 (1972).
    [Crossref]
  16. H. M. Pickett, “Effects of Velocity Averaging on the Shapes of Absorption Lines,” J. Chem. Phys. 73, 6090 (1980).
    [Crossref]
  17. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1972).
  18. J. Humlicek, “An Efficient Method for Evaluation of the Complex Probability Function: the Voigt Function and its Derivatives,” J. Quant. Spectrosc. Radiat. Transfer 21, 309 (1979).
    [Crossref]
  19. A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
    [Crossref]
  20. C. D. Rodgers, “Collisional Narrowing: Its Effect on the Equivalent Widths of Spectral Lines,” Appl. Opt. 15, 714 (1976).
    [Crossref] [PubMed]
  21. R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

1982 (1)

1980 (2)

A. S. Pine, “Collisional Narrowing of HF Fundamental Band Spectral Lines by Neon and Argon,” J. Mol. Spectrosc. 82, 435 (1980).
[Crossref]

H. M. Pickett, “Effects of Velocity Averaging on the Shapes of Absorption Lines,” J. Chem. Phys. 73, 6090 (1980).
[Crossref]

1979 (1)

J. Humlicek, “An Efficient Method for Evaluation of the Complex Probability Function: the Voigt Function and its Derivatives,” J. Quant. Spectrosc. Radiat. Transfer 21, 309 (1979).
[Crossref]

1978 (2)

A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
[Crossref]

G. J. Nienhuis, “Effects of Radiator Motion in the Classical and Quantum Mechanical Theories of Collisional Spectral-Line Broadening,” J. Quant. Spectrosc. Radiat. Transfer 20, 275 (1978).
[Crossref]

1976 (1)

1974 (1)

F. Herbert, “Spectral Line Profile: A Generalized Voigt Function Including Collisional Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1974).
[Crossref]

1972 (3)

P. R. Berman, “Quantum-Mechanical Transport Equation for Atomic Systems,” Phys. Rev. A 5, 927 (1972).
[Crossref]

J. R. Murray, A. Javan, “Effects of Collisions on Raman Line Profiles of Hydrogen and Deuterium Gas,” J. Mol. Spectrosc. 42, 1 (1972).
[Crossref]

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

1967 (1)

S. G. Rautian, I. I. Sobelman, “Effect of Collisions on the Doppler Broadening of Spectral Lines,” Sov. Phys. Usp. 9, 701 (1967)
[Crossref]

1966 (1)

R. G. Gordon, “Theory of the Width and Shift of Molecular Spectral Lines in Gases,” J. Chem. Phys. 44, 3083 (1966).
[Crossref]

1961 (1)

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[Crossref]

1953 (1)

R. H. Dicke, “The Effect of Collisions upon the Doppler Width of Spectral Lines,” Phys. Rev. 89, 472 (1953).
[Crossref]

Armstrong, B. H.

A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
[Crossref]

Benedict, W. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Berman, P. R.

P. R. Berman, “Quantum-Mechanical Transport Equation for Atomic Systems,” Phys. Rev. A 5, 927 (1972).
[Crossref]

Burch, D. E.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Calawa, A. R.

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

Calfee, R. F.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Chapell, W. R.

E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).

Chen, D-W.

Clough, S. A.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Cooper, J.

E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).

Dicke, R. H.

R. H. Dicke, “The Effect of Collisions upon the Doppler Width of Spectral Lines,” Phys. Rev. 89, 472 (1953).
[Crossref]

Dillon, T.

E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).

Eng, R. S.

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

Fox, K.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Galatry, L.

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[Crossref]

Garing, J. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Gordon, R. G.

R. G. Gordon, “Theory of the Width and Shift of Molecular Spectral Lines in Gases,” J. Chem. Phys. 44, 3083 (1966).
[Crossref]

Hanson, R. K.

P. L. Varghese, R. K. Hanson, “Tunable Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” submitted to J. Quant. Spectrosc. Radiat. Transfer.

Harman, T. C.

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

Herbert, F.

F. Herbert, “Spectral Line Profile: A Generalized Voigt Function Including Collisional Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1974).
[Crossref]

Hui, A. K.

A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
[Crossref]

Humlicek, J.

J. Humlicek, “An Efficient Method for Evaluation of the Complex Probability Function: the Voigt Function and its Derivatives,” J. Quant. Spectrosc. Radiat. Transfer 21, 309 (1979).
[Crossref]

Javan, A.

J. R. Murray, A. Javan, “Effects of Collisions on Raman Line Profiles of Hydrogen and Deuterium Gas,” J. Mol. Spectrosc. 42, 1 (1972).
[Crossref]

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

Kelley, P. L.

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

McClatchey, R. A.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Murray, J. R.

J. R. Murray, A. Javan, “Effects of Collisions on Raman Line Profiles of Hydrogen and Deuterium Gas,” J. Mol. Spectrosc. 42, 1 (1972).
[Crossref]

Nienhuis, G. J.

G. J. Nienhuis, “Effects of Radiator Motion in the Classical and Quantum Mechanical Theories of Collisional Spectral-Line Broadening,” J. Quant. Spectrosc. Radiat. Transfer 20, 275 (1978).
[Crossref]

Niple, E. R.

Pickett, H. M.

H. M. Pickett, “Effects of Velocity Averaging on the Shapes of Absorption Lines,” J. Chem. Phys. 73, 6090 (1980).
[Crossref]

Pine, A. S.

A. S. Pine, “Collisional Narrowing of HF Fundamental Band Spectral Lines by Neon and Argon,” J. Mol. Spectrosc. 82, 435 (1980).
[Crossref]

Poultney, S. K.

Rautian, S. G.

S. G. Rautian, I. I. Sobelman, “Effect of Collisions on the Doppler Broadening of Spectral Lines,” Sov. Phys. Usp. 9, 701 (1967)
[Crossref]

Rodgers, C. D.

Rothman, L. S.

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

Smith, E. W.

E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).

Sobelman, I. I.

S. G. Rautian, I. I. Sobelman, “Effect of Collisions on the Doppler Broadening of Spectral Lines,” Sov. Phys. Usp. 9, 701 (1967)
[Crossref]

I. I. Sobelman, L. A. Vainshtein, E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, 1981).
[Crossref]

Vainshtein, L. A.

I. I. Sobelman, L. A. Vainshtein, E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, 1981).
[Crossref]

Varghese, P. L.

P. L. Varghese, Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide, Report 6-83-T, HTGL, Stanford U., Stanford, Calif. (1983).

P. L. Varghese, R. K. Hanson, “Tunable Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” submitted to J. Quant. Spectrosc. Radiat. Transfer.

Wray, A. A.

A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
[Crossref]

Yukov, E. A.

I. I. Sobelman, L. A. Vainshtein, E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, 1981).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

R. S. Eng, A. R. Calawa, T. C. Harman, P. L. Kelley, A. Javan, “Collisional Narrowing of Infrared Water Vapor Transitions,” Appl. Phys. Lett. 21, 303 (1972).
[Crossref]

J. Chem. Phys. (2)

R. G. Gordon, “Theory of the Width and Shift of Molecular Spectral Lines in Gases,” J. Chem. Phys. 44, 3083 (1966).
[Crossref]

H. M. Pickett, “Effects of Velocity Averaging on the Shapes of Absorption Lines,” J. Chem. Phys. 73, 6090 (1980).
[Crossref]

J. Mol. Spectrosc. (2)

A. S. Pine, “Collisional Narrowing of HF Fundamental Band Spectral Lines by Neon and Argon,” J. Mol. Spectrosc. 82, 435 (1980).
[Crossref]

J. R. Murray, A. Javan, “Effects of Collisions on Raman Line Profiles of Hydrogen and Deuterium Gas,” J. Mol. Spectrosc. 42, 1 (1972).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (5)

F. Herbert, “Spectral Line Profile: A Generalized Voigt Function Including Collisional Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1974).
[Crossref]

E. W. Smith, J. Cooper, W. R. Chapell, T. Dillon, “An Impact Theory for Doppler and Pressure Broadening—I. General Theory,” J. Quant. Spectrosc. Radiat. Transfer 11, 1547; “An Impact Theory for Doppler and Pressure Broadening—II. Atomic and Molecular Systems,” 11, 1567 (1971).

J. Humlicek, “An Efficient Method for Evaluation of the Complex Probability Function: the Voigt Function and its Derivatives,” J. Quant. Spectrosc. Radiat. Transfer 21, 309 (1979).
[Crossref]

A. K. Hui, B. H. Armstrong, A. A. Wray, “Rapid Computation of the Voigt and Complex Error Functions,” J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978).
[Crossref]

G. J. Nienhuis, “Effects of Radiator Motion in the Classical and Quantum Mechanical Theories of Collisional Spectral-Line Broadening,” J. Quant. Spectrosc. Radiat. Transfer 20, 275 (1978).
[Crossref]

Phys. Rev. (2)

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[Crossref]

R. H. Dicke, “The Effect of Collisions upon the Doppler Width of Spectral Lines,” Phys. Rev. 89, 472 (1953).
[Crossref]

Phys. Rev. A (1)

P. R. Berman, “Quantum-Mechanical Transport Equation for Atomic Systems,” Phys. Rev. A 5, 927 (1972).
[Crossref]

Sov. Phys. Usp. (1)

S. G. Rautian, I. I. Sobelman, “Effect of Collisions on the Doppler Broadening of Spectral Lines,” Sov. Phys. Usp. 9, 701 (1967)
[Crossref]

Other (5)

I. I. Sobelman, L. A. Vainshtein, E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin, 1981).
[Crossref]

P. L. Varghese, R. K. Hanson, “Tunable Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” submitted to J. Quant. Spectrosc. Radiat. Transfer.

P. L. Varghese, Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide, Report 6-83-T, HTGL, Stanford U., Stanford, Calif. (1983).

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1972).

R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, J. S. Garing, “AFCRL Atmospheric Absorption Line Parameters Compliation,” AFCRL-TR-73-0096 (1973).

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Figures (9)

Fig. 1
Fig. 1

Comparison of the standardized Voigt [V(x,y)] and Galatry [G(x,y,z)] functions for y = z = 1. The functions are normalized to area π. The lower curve is a plot of the difference between the two functions expressed as a percentage of the maximum value of G.

Fig. 2
Fig. 2

Percentage difference between the Galatry and Voigt functions at line center (x = 0) as a function of y and r(≡y/z). The difference is expressed as a percentage of G(x = 0).

Fig. 3
Fig. 3

Comparison of the generalized Galatry function (H) with the Galatry (G) and Voigt (V) functions when line shifts are negligible. H(x,y,z,ζ,s) exhibits less collisional narrowing than G(xs,y,z) but is narrower than V(xs,y). The profiles are calculated for y = 1, z = 1, ζ = 2, s = 0.

Fig. 4
Fig. 4

Comparison of the generalized Galatry function (H) with the Galatry (G) and Voigt (V) functions when line shifts are significant. H(x,y,z,ζ,s) is asymmetric while G(xs,y,z) and V(xs,y) are symmetric about the shifted center line. The profiles are calculated for y = 1, z = 1, ζ = 2, s = −0.05.

Fig. 5
Fig. 5

Least-squares Voigt profile fit (solid line) to the experimental data (circles) for the P(4) transition (1000 ← 0000 band) of HCN at 3299.53 cm−1 recorded at high resolution using a tunable diode laser.5 The abscissa is the dimensionless frequency separation from line center x, and the ordinate is the logarithm of the transmissivity which is proportional to the line shape function for the experimental conditions. The standardized broadening parameter obtained from a least-squares fit was 1.05.

Fig. 6
Fig. 6

Residual errors from the least-squares Voigt fit shown in Fig. 5 expressed as a percentage of the maximum value of the fitted function (solid line). This is compared to a least-squares Voigt fit of synthetic data computed from a Galatry function with y = 1.13 and z = 0.9 (dashed line). The choice of parameters is explained in the text.

Fig. 7
Fig. 7

Residual error when data computed from a hard collision model profile P(x,y,ζ) is fitted by a Galatry profile G(x,y,z). The input parameters for P were y = 1, ζ = 1; the least-squares fit gave y = 1.01, z = 1.18 for G. Note that the residual errors are about an order of magnitude smaller than those displayed in Fig. 6.

Fig. 8
Fig. 8

Residual error when the generalized Galatry profile H(x,y,ζ,s) displayed in Fig. 6 is fitted to a simple Galatry profile G(x,y,z). The input parameters for H were y = 1, z = 1, ζ = 2, s = −0.05; the least-squares fit gave y = 1.00, z = 0.48 for G. The asymmetry of the profile being fitted is clearly shown in the plot of residual error.

Fig. 9
Fig. 9

Division of the y-z plane for computation of the standardized Galatry function.

Tables (4)

Tables Icon

Table I Dimensionless Parameters for Standardized Line Profiles

Tables Icon

Table II Standardized Line Profiles

Tables Icon

Table III Limiting Behavior of the Generalized Galatry Function

Tables Icon

Table IV Coefficients of Asymptotic Expansion [Eq.(A3)]

Equations (28)

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- K ( x , y , ) d x = π .
Δ ν D = 2 ln 2 Δ ν D = 2 ln 2 α D / 2 π c = 7.162 × 10 - 7 T / m ν 0 .
x = ( ω - ω 0 ) / α D = ( ν - ν o ) / Δ ν D ,
y = Γ / α D = γ p / Δ ν D ,
s Δ / α D = δ p / Δ ν D .
z β / α D = β / 2 π c Δ ν D ;             β = k T / m D .
ζ = Ω / α d .
r y / z .
Z ( s = 0 ) = z eff = z ( 1 - y / ζ ) .
Z = z ( 1 - y / ζ ) - i s z / ζ ,
Q ( x , y , ζ , s = 0 ) P ( x , y , ζ - y ) ,
D ( x ) = exp ( - x 2 ) .
L ( x , y ) = 1 π y y 2 + x 2 ; x x - s .
V ( x , y ) y π - d ξ exp ( - ξ 2 ) y 2 + ( x - ξ ) 2 = Re [ w ( x , y ) ] .
G ( x ' , y , z ) = 1 π Re ( 0 d τ exp { - i x τ - y τ + 1 2 z 2 [ 1 - z τ - exp ( - z τ ] } ) = 1 π Re [ 1 1 2 z + y - i x M ( 1 ; 1 + 1 2 z 2 + y - i x z ; 1 2 z 2 ) ] .
H ( x , y , z , ζ , s ) = G ( x , y , Z ) ; Z z ( 1 - y / ζ ) - i s z / ζ
P ( x , y , ζ ) = R e [ w ( x , y + ζ ) 1 - π ζ w ( x , y + ζ ) ] .
Q ( x , y , ζ , s ) = Re [ w ( x , ζ ) 1 - π ( ζ - y - i s ) w ( x , ζ ) ]
G ( x , y , z ) = 2 z π Re { n = 0 1 [ 1 + 2 z ( y - i x ) ] [ 1 + 2 z ( y - i x + z ) ] [ 1 + 2 z ( y - i x + n z ) ] } .
G ( x , y , z ) = Re [ w ( q ) + n = 3 c n i n d n d q n w ( q ) ] ,
n = 0 c n τ n ~ exp [ - 1 2 z 2 n = 3 ( - z τ ) n n ! ] .
G ( x , y , z ) = Re [ w ( q ) + n = 3 n 1 c n ( z ) i n d n w ( q ) d q n ] ;             n 1 8 ,
d n w d q n = - 2 [ q d n - 1 w d q n - 1 + ( n - 1 ) d n - 2 w d q n - 2 ] ,             n 2 ,
d w d q = 2 i π - 2 q w .
G ( x , y , z ) = 1 π z Re [ n = 0 n 2 δ n θ ( θ + 1 ) ( θ + n ) ] , δ 1 2 z 2 , θ 1 2 z 2 + y - i x z .
n 2 = 4 + z - 1.05 [ 1 + 3 exp ( - 1.1 y ) ] .
G ( x , y , z ) = 1 π Re [ 1 y - i x + 1 · ½ z + y - i x + 2 · ½ 2 z + y - i x + ] ,
n 3 = 2 + 37 exp ( - 0.6 y ) .

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