Abstract

Methods for the calculation of spectral absorption coefficients for combined Doppler and Lorentz broadening are summarized. The “curves of growth” have been extended to cover the ranges of parameters which arise in spectroscopic studies on flames.

© 1953 Optical Society of America

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References

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  1. See, for example, A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, Cambridge, 1934), Appendix I.
  2. For details see, for example, D. G. Kendall, Z. Astrophys. 16, 308 (1938).
  3. R. Ladenburg, Z. Physik 65, 200 (1930);R. D. Cowan and G. H. Dieke, Revs. Modern Phys. 20, 418 (1948).
    [Crossref]
  4. This expansion was obtained by H. S. Tsien.
  5. Kavanagh, Björnerud, and Penner, J. Opt. Soc. Am. 43, 380 (1953)
    [Crossref]
  6. R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
    [Crossref]
  7. W. M. Elsasser, Harvard Meteorological Studies No. 6, Milton (Massachusetts), 1942.
  8. R. W. Kavanagh and S. S. Penner, J. Opt. Soc. Am. 43, 483 (1953).
  9. F. Reiche, Verhandl. deut. physik. Ges. 15, 3 (1913).
  10. See also, A. Unsöld, Physik der Sternatmosphären (Edwards Brothers, Inc., Ann Arbor, 1948), pp. 159–169.
  11. M. Born, Oplik (J. Springer, Berlin, 1933), pp. 482–486.
  12. D. L. Harris, Astrophys. J. 108, 112 (1948).
    [Crossref]
  13. W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 28771931).
  14. M. W. Zemansky, Phys. Rev. 36, 219 (1930).
    [Crossref]
  15. H. E. Salzer, “Mathematical Tables and Other Aids to Computation,” National Research Council, April, 1951.
  16. See, for example, E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves (Dover Publications, New York, 1943), pp. 34–35.
  17. Reference 1, p. 132; reference 10, p. 168.
  18. E. M. F.v. d. Held, Z. Physik 70, 508 (1931).
    [Crossref]

1953 (2)

R. W. Kavanagh and S. S. Penner, J. Opt. Soc. Am. 43, 483 (1953).

Kavanagh, Björnerud, and Penner, J. Opt. Soc. Am. 43, 380 (1953)
[Crossref]

1948 (1)

D. L. Harris, Astrophys. J. 108, 112 (1948).
[Crossref]

1938 (1)

For details see, for example, D. G. Kendall, Z. Astrophys. 16, 308 (1938).

1931 (2)

E. M. F.v. d. Held, Z. Physik 70, 508 (1931).
[Crossref]

W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 28771931).

1930 (2)

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[Crossref]

R. Ladenburg, Z. Physik 65, 200 (1930);R. D. Cowan and G. H. Dieke, Revs. Modern Phys. 20, 418 (1948).
[Crossref]

1913 (2)

R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

F. Reiche, Verhandl. deut. physik. Ges. 15, 3 (1913).

Björnerud,

Born, M.

M. Born, Oplik (J. Springer, Berlin, 1933), pp. 482–486.

Elsasser, W. M.

W. M. Elsasser, Harvard Meteorological Studies No. 6, Milton (Massachusetts), 1942.

Emde, F.

See, for example, E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves (Dover Publications, New York, 1943), pp. 34–35.

Gordon, A. R.

W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 28771931).

Harris, D. L.

D. L. Harris, Astrophys. J. 108, 112 (1948).
[Crossref]

Held, E. M. F.v. d.

E. M. F.v. d. Held, Z. Physik 70, 508 (1931).
[Crossref]

Jahnke, E.

See, for example, E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves (Dover Publications, New York, 1943), pp. 34–35.

Kavanagh,

Kavanagh, R. W.

R. W. Kavanagh and S. S. Penner, J. Opt. Soc. Am. 43, 483 (1953).

Kendall, D. G.

For details see, for example, D. G. Kendall, Z. Astrophys. 16, 308 (1938).

Ladenburg, R.

R. Ladenburg, Z. Physik 65, 200 (1930);R. D. Cowan and G. H. Dieke, Revs. Modern Phys. 20, 418 (1948).
[Crossref]

R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Miller, W. L.

W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 28771931).

Mitchell, A. C. G.

See, for example, A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, Cambridge, 1934), Appendix I.

Penner,

Penner, S. S.

R. W. Kavanagh and S. S. Penner, J. Opt. Soc. Am. 43, 483 (1953).

Reiche, F.

F. Reiche, Verhandl. deut. physik. Ges. 15, 3 (1913).

R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Salzer, H. E.

H. E. Salzer, “Mathematical Tables and Other Aids to Computation,” National Research Council, April, 1951.

Unsöld, A.

See also, A. Unsöld, Physik der Sternatmosphären (Edwards Brothers, Inc., Ann Arbor, 1948), pp. 159–169.

Zemansky, M. W.

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[Crossref]

See, for example, A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, Cambridge, 1934), Appendix I.

Ann. Physik (1)

R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Astrophys. J. (1)

D. L. Harris, Astrophys. J. 108, 112 (1948).
[Crossref]

J. Opt. Soc. Am. (2)

R. W. Kavanagh and S. S. Penner, J. Opt. Soc. Am. 43, 483 (1953).

Kavanagh, Björnerud, and Penner, J. Opt. Soc. Am. 43, 380 (1953)
[Crossref]

J. Phys. Chem. (1)

W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 28771931).

Phys. Rev. (1)

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[Crossref]

Verhandl. deut. physik. Ges. (1)

F. Reiche, Verhandl. deut. physik. Ges. 15, 3 (1913).

Z. Astrophys. (1)

For details see, for example, D. G. Kendall, Z. Astrophys. 16, 308 (1938).

Z. Physik (2)

R. Ladenburg, Z. Physik 65, 200 (1930);R. D. Cowan and G. H. Dieke, Revs. Modern Phys. 20, 418 (1948).
[Crossref]

E. M. F.v. d. Held, Z. Physik 70, 508 (1931).
[Crossref]

Other (8)

See, for example, A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, Cambridge, 1934), Appendix I.

W. M. Elsasser, Harvard Meteorological Studies No. 6, Milton (Massachusetts), 1942.

This expansion was obtained by H. S. Tsien.

See also, A. Unsöld, Physik der Sternatmosphären (Edwards Brothers, Inc., Ann Arbor, 1948), pp. 159–169.

M. Born, Oplik (J. Springer, Berlin, 1933), pp. 482–486.

H. E. Salzer, “Mathematical Tables and Other Aids to Computation,” National Research Council, April, 1951.

See, for example, E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves (Dover Publications, New York, 1943), pp. 34–35.

Reference 1, p. 132; reference 10, p. 168.

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Figures (1)

F. 1
F. 1

Extended curves of growth.

Tables (1)

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Table I Comparison of notation used by different authors for equivalent physical quantities.

Equations (41)

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P ( ω ) = P ( a / π ) + [ a 2 + ( ξ y ) 2 ] 1 [ exp ( y 2 ) ] d y ,
P = ( S / ω 0 ) ( m c 2 / 2 π k T ) 1 2 , a = [ ( b N + b C ) / ω 0 ] ( m c 2 / 2 k T ) 1 2 = ( b N + b C ) ( ln 2 ) 1 2 / b D ,
ξ = [ ( ω ω 0 ) / ω 0 ] ( m c 2 / 2 k T ) 1 2 = [ ( ω ω 0 ) / b D ] ( ln 2 ) 1 2 .
A = Δ ω R 0 ( ω ) { 1 exp [ P ( ω ) X ] } d ω ,
A / R 0 = + { 1 exp [ P ( ω ) X ] } d ω ,
+ P ( ω ) d ω = S .
P ( ω ) = P exp ( ξ 2 ) .
A / R 0 = ( S X ) n = 0 [ ( n + 1 ) 1 2 ( n + 1 ) ! ] 1 ( P X ) n .
A / R 0 = S X exp [ 1 2 ( P X ) 1 2 ] .
A / R 0 = 2 π 1 2 ( S / P ) z 1 2 × { 1 [ 2 Γ ( 1 2 ) ] 1 n = 1 [ Γ ( n ) ( 1 ) Γ ( n 1 2 ) / ( n ! z n ) ] } ,
Γ ( m ) = 0 e t t m 1 d t .
S = 2.3789 × 10 7 ( 273.1 / T ) f .
P ( ω ) = ( b / π ) [ ( ω ω 0 ) 2 + b 2 ] 1 S
A / R 0 = 2 π b f ( x ) .
f ( x ) = x [ exp ( x ) ] [ J 0 ( i x ) i J 1 ( i x ) ]
x = S X / 2 π b .
A / R 0 ~ S X for small values of x
A / R 0 ~ 2 ( S b X ) 1 2 for large values of x ,
P ( ω ) / P = π 1 2 0 { exp [ a x ( x 2 / 4 ) ] } cos ξ x d x .
P ( ω 0 ) = P [ exp ( a 2 ) ] [ erfc ( a ) ] ,
erfc ( a ) = ( 2 / π 1 2 ) a [ exp ( x 2 ) ] d x = 1 erf ( a ) .
P ( ω ) / P ~ [ a / π 1 2 ( a 2 + ξ 2 ) ] [ 1 + ( 3 / 2 ξ 2 ) ] .
P ( ω ) / P = [ exp ( a 2 ) ] [ cos ( 2 a ξ ) ] [ exp ( ξ 2 ) ] + 2 π 1 2 [ exp ( a 2 ) ] [ sin ( 2 a ξ ) ] F ( ξ ) 2 a π 1 2 [ exp ( a 2 ) ] [ cos ( 2 a ξ ) ] × m = 0 [ a 2 m H 2 m ( i ξ ) / ( 2 m + 1 ) ! ] + 2 a π 1 2 [ exp ( a 2 ) ] [ i sin ( 2 a ξ ) ] × m = 0 [ a 2 m 1 H 2 m 1 ( i ξ ) / ( 2 m ) ! ] ,
P ( ω ) / P = exp ( ξ 2 ) 2 a π 1 2 [ 1 2 ξ F ( ξ ) ] + a 2 ( 1 2 ξ 2 ) exp ( ξ 2 ) 2 a 3 π 1 2 { ( 2 / 3 ) ( 1 ξ 2 ) 2 ξ [ 1 ( 2 / 3 ) ξ 2 ] F ( ξ ) } + a 4 [ 1 2 2 ξ 2 + ( 2 / 3 ) ξ 4 ] exp ( ξ 2 ) + ,
F ( ξ ) = exp ( ξ 2 ) 0 ξ [ exp ( x 2 ) ] d x .
P ( ω ) / P = ( 1 a π 1 1 + ( ξ / a ) 2 ) × { 1 1 2 ( 1 a ) 2 1 ( 3 2 ) ( ξ / a ) 2 [ 1 + ( ξ / a ) 2 ] 2 + 1 × 3 2 2 ( 1 a ) 4 × 1 ( 5 2 ) ( ξ / a ) 2 + ( 5 4 ) ( ξ / a ) 4 [ 1 + ( ξ / a ) 2 ] 4 + } .
P ( ω ) / P = 2 π exp ( a 2 ξ 2 ) × { sin 2 a ξ { ξ a 3 1 ! 3 [ ( 3 1 ) ξ a ( 3 3 ) ( ξ a ) 3 ] + a 5 2 ! 5 [ ( 5 1 ) ξ a ( 5 3 ) ( ξ a ) 3 + ( 5 3 ) ( ξ a ) 5 ] + } + cos 2 a ξ { π 2 a + a 3 1 ! 3 [ 1 ( 3 2 ) ( ξ a ) 2 ] a 5 2 ! 5 [ 1 ( 5 2 ) ( ξ a ) 2 + ( 5 4 ) ( ξ a ) 4 ] + } } .
P ( ω ) / P = ( 2 / π 1 2 ) [ exp ( a 2 ξ 2 ) ] × a { exp [ t 2 + ( a 2 ξ 2 / t 2 ) ] } d t .
P ( ω ) / P = n = 0 I n ( a ) ( n ! ) 1 ( ξ ) 2 n [ exp ( ξ 2 ) ] ,
I n ( a ) = [ a / ( 2 n 1 ) ] [ ( 2 / π 1 2 ) 2 a I n 1 ( a ) ] ,
I 0 ( a ) = [ exp ( a 2 ) ] [ erfc ( a ) ] .
P ( ω ) / P = R { [ exp ( z 2 ) ] [ erfc ( z ) ] } ,
P ( ω ) / P = [ exp ( a 2 ξ 2 ) ] { [ cos ( 2 a ξ ) ] [ erfc ( a ) ] + ( 2 π a ) 1 [ exp ( a 2 ) ] [ 1 cos ( 2 a ξ ) ] ( 2 / π ) [ exp ( a 2 ) ] n = 1 [ exp ( n 2 / 4 ) ] ( n 2 + 4 a 2 ) 1 × { 2 a [ cos ( 2 a ξ ) cosh ( n ξ ) ] } .
P ( ω ) / P = [ exp ( a 2 ξ 2 ) ] × { [ 1 C ( 2 π 1 2 a ) S ( 2 π 1 2 a ) ] cos ( 2 a 2 ) + [ C ( 2 π 1 2 a ) S ( 2 π 1 2 a ) ] sin ( 2 a 2 ) } ,
P ( ω ) / P ( π 1 2 a ) 1 n = 0 ( 1 / 2 ) ( 3 / 2 ) [ ( 2 n 1 ) / 2 ] a 2 n × ( sin 2 n + 1 θ ) [ sin ( 2 n + 1 ) θ ] , π / 4 < θ < π / 2 ,
ν = ν / c
1 2 ( Δ ν N / c )
( δ / 4 π c ) ( ln 2 ) 1 2
( ln 2 ) 1 2 Δ ω D / 2 π c
1 2 ( Δ ν D / c )
( ω ω 0 ) / 1 2 δ