Abstract

The question of the reasonable form of a line intensity distribution function for representation of molecular spectra is discussed. A distribution function which is proportional to 1/S in the low-intensity region is shown to be reasonable on a physical basis. A curve of growth is obtained for a continuous distribution of Lorentz lines which is proportional to 1/S, but cut off at a small value of S and decreases exponentially for large S to permit normalization. The limiting curve of growth for no cutoff (i.e., for a continuous exponential-tailed 1/S intensity distribution for 0<S<∞) is found to be a simple, conveniently handled expression. A comparison is made with the curve of growth presented by Godson for a distribution proportional to 1/S up to a maximum S and zero above. A comparison is also made with the curve of growth for a model with a discrete 1/S distribution (consisting of lines whose intensities are in geometric progression), which is calculated numerically from existing tables of the Ladenburg–Reiche function for several values of the intensity ratio. Criteria for the applicability of the curves of growth for the 1/S models are discussed; the curve developed here and the Ladenburg–Reiche curve form (approximate) lower and upper limits, respectively, to the curve of growth for any physically reasonable band model consisting of Lorentz lines.

© 1967 Optical Society of America

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References

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  1. W. M. Elsasser, Phys. Rev. 54, 126 (1938).
    [CrossRef]
  2. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [CrossRef]
  3. R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
    [CrossRef]
  4. W. L. Godson, J. Meteorol. 12, 272, 533 (1955).
    [CrossRef]
  5. W. L. Godson, Proceedings of the Toronto Meteorological Conference, 1953 (Royal Meteorological Society, London, 1954), p. 35.
  6. R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).
    [CrossRef]
  7. L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
    [CrossRef]
  8. A curve of growth equivalent to the r=0.1 curve is presented by G. N. Plass, Appl. Opt. 3, 859 (1964).
    [CrossRef]
  9. U. P. Oppenheim and Y. Ben-Aryeh, J. Opt. Soc. Am. 53, 344 (1963).
    [CrossRef]
  10. W. Malkmus, J. Opt. Soc. Am. 53, 951 (1963).
    [CrossRef]

1964 (1)

1963 (2)

1958 (1)

1956 (1)

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

1955 (1)

W. L. Godson, J. Meteorol. 12, 272, 533 (1955).
[CrossRef]

1952 (1)

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

1938 (1)

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[CrossRef]

1913 (1)

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

Ben-Aryeh, Y.

Eggers, D. F.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

Elsasser, W. M.

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[CrossRef]

Godson, W. L.

W. L. Godson, J. Meteorol. 12, 272, 533 (1955).
[CrossRef]

W. L. Godson, Proceedings of the Toronto Meteorological Conference, 1953 (Royal Meteorological Society, London, 1954), p. 35.

Goody, R. M.

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

Kaplan, L. D.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

Ladenburg, R.

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

Malkmus, W.

Oppenheim, U. P.

Plass, G. N.

Reiche, F.

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

Ann. Physik (1)

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

Appl. Opt. (1)

J. Chem. Phys. (1)

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

J. Meteorol. (1)

W. L. Godson, J. Meteorol. 12, 272, 533 (1955).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. (1)

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[CrossRef]

Quart. J. Roy. Meteorol. Soc. (1)

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

Other (1)

W. L. Godson, Proceedings of the Toronto Meteorological Conference, 1953 (Royal Meteorological Society, London, 1954), p. 35.

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Figures (5)

Fig. 1
Fig. 1

Line intensity probability distribution functions. Curve C is the exponential intensity distribution P ( S ) = S ¯ - 1 exp ( - S / S ¯ ). The dashed curves A and B are the truncated S−1 distributions P(S)=(S lnR)−1(SM/RSSM), P(S)=0 (otherwise). The solid curves A and B are the exponential-tailed S−1 distributions P(S)=(S lnR)−1 [exp(−S/SM)−exp(−RS/SM)]. For curves B, R is 103; for curves A, R is 106. In all cases, P(S) is normalized and S ¯ denotes the mean intensity: S ¯=∫SP(S)dS. For A and B, S ¯ is equal to (R−1) (R lnR)−1SM. The figure demonstrates the greater emphasis placed on weaker lines relative to stronger lines in the distributions proportional to S−1.

Fig. 2
Fig. 2

Curves of growth for random-band models composed of Lorentz lines for four different intensity distribution functions: (1) f(xE)=xE exp(−xE)[I0(xE)+I1(xE)] for P(S)=δ(SSE); (2) e ( x E ) = x E ( 1 + 1 2 π x E ) - 1 2 for P(S)=4π−1SE−1 exp(−4S/πSE); (3) g ( x E ) = 2 x E exp ( - 4 x E ) [ I 0 ( 4 x E ) + I 1 ( 4 x E ) ] + 1 4 exp ( - 4 x E ) I 0 ( 4 x E ) - 1 4 for P(S) ∝ S−1(S≤4SE), P(S)=0(S>4SE); (4) h ( x E ) = π - 1 [ ( 1 + 2 π x E ) 1 2 - 1 ] for P(S) ∝ S−1 exp(−S/πSE). The quantity SE is defined in the text for each case so that all curves have the same asymptotes.

Fig. 3
Fig. 3

Transition regions of curves of growth for random band models composed of Lorentz lines for various intensity distribution functions. [In (A) through (D) the function F(xE,r) is defined by Eq. (67), for P(S) defined by Eq. (55).] (A) r=0[F(xE,0)=f(xE)]; (B) r=0.01; (C) r=0.1; (D) r=0.5 or 1.0 (indistinguishable in this figure); [F(xE,1)=g(xE)]; (E) h(xE).

Fig. 4
Fig. 4

Experimentally measured values of −lnT/p for CO2 at ω=2273 cm−1 and T=1200°K with Ladenburg–Reiche curve f(x) (dashed) as fitted by Oppenheim and Ben-Aryeh.9 The solid curve is the exponential-tailed S−1 curve of growth h(x) [Eq. (44)] fitted to the same points. Both curves fit the points well, but the asymptotic regions (particularly the square-root region) are significantly different. The experimental points are labeled as follows: (B) R. H. Tourin and H. J. Babrov, J. Chem. Phys. 37, 581 (1962); (F) C. C. Ferriso, J. Chem. Phys. 37, 1955 (1962); (S) M. Steinberg and W. A. Davies, J. Chem. Phys. 34, 1373 (1961); (T) R. H. Tourin, J. Opt. Soc. Am. 51, 175 (1961); (+) see Ref. 9.

Fig. 5
Fig. 5

Comparison of theoretical values10 of 2 α 0 1 2 S 1 2 / d for CO2 at ω=2273 cm−1 and T=1200°K (solid curve) with experimental values9 (dashed curve) based on use of the Ladenburg–Reiche curve of growth f(x) [cf. Fig. 4]. The arrow and dot show the change of the experimentally determined value which results from use of the curve of growth h(x)[Eq. (44)].

Equations (71)

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P 0 ( S ) = { ( 1 / ln R ) S - 1 ( for S M / R S S M ) 0 ( for S < S M / R and S > S M ) ,
0 P 0 ( S ) d S = 1.
P ( S ) = 0 P 0 ( S ) P ( S , S ) d S ,
P ( S , S ) = ( 1 / S ) exp ( - S / S ) .
P ( S ) = S M / R S M 1 ln R ( S ) - 2 exp ( - S / S ) d S
P ( S ) = 1 ln R S - 1 [ exp ( - S / S M ) - exp ( - R S / S M ) ] .
P ( S ) ( 1 / ln R ) S - 1 .
P ( S ) ( 1 / ln R ) S - 1 exp ( - S / S M ) ,
P ( S ) ( 1 / ln R ) ( R - 1 ) / S M P 0 ( S M / R )
S ¯ = 0 S P ( S ) d S ,
S ¯ = [ ( R - 1 ) / R ln R ] S M .
P ( S ) = 1 ln R S - 1 [ exp ( - R - 1 R ln R S S ¯ ) - exp ( - R - 1 ln R S S ¯ ) ] .
W ( S , u ) = { 1 - exp [ - k ( ω ) u ] } d ω ,
= 1 - exp ( - W ¯ / d ) ,
W ¯ = W ( S , u ) P ( S ) d S .
k ( ω ) = S b / π ( ω - ω 0 ) 2 + b 2 ,
W = 2 π b f ( x ) ,
f ( x ) = x exp ( - x ) [ I 0 ( x ) + I 1 ( x ) ] ,
x = S u / 2 π b .
W ( S , u ) = S u exp ( - S u 2 π b ) [ I 0 ( S u 2 π b ) + I 1 ( S u 2 π b ) ] .
W ¯ exp = 0 W ( S , u ) 1 S 0 exp ( - S / S 0 ) d S ,
W ¯ exp = S 0 u ( 1 + S 0 u / π b ) - 1 2 .
W ¯ = S u exp ( - S u 2 π b ) [ I 0 ( S u 2 π b ) + I 1 ( S u 2 π b ) ] × 1 ln R S - 1 [ exp ( - S S M ) - exp ( - R S S M ) ] d S .
W ¯ = S = 0 S u exp ( - S u 2 π b ) [ I 0 ( S u 2 π b ) + I 1 ( S u 2 π b ) ] × 1 ln R S = S M / R S M ( S ) - 2 exp ( - S / S ) d S d S .
W ¯ = 1 ln R S = S M / R S M ( S ) - 2 S = 0 S u exp ( - S u / 2 π b ) × [ I 0 ( S u 2 π b ) + I 1 ( S u 2 π b ) ] exp ( - S / S ) d S d S .
W ¯ = 1 ln R S = S M / R S M u ( 1 + S u π b ) - 1 2 d S .
W ¯ = 2 π b ln R [ ( 1 + S M u π b ) 1 2 - ( 1 + S M u R π b ) 1 2 ] .
W ¯ d = 2 π b d ln R [ ( 1 + R ln R S ¯ u ( R - 1 ) π b ) 1 2 - ( 1 + ln R S ¯ u ( R - 1 ) π b ) 1 2 ] .
W ¯ / d S ¯ u / d
W ¯ / d 2 ( π b ) 1 2 ( S ¯ u ) 1 2 d 1 ( ln R ) 1 2 R 1 2 - 1 ( R - 1 ) 1 2 .
lim u 0 [ ( W ¯ / d ) / u ] = S E / d E
lim u [ ( W ¯ / d ) / u 1 2 ] = 2 ( S E b ) 1 2 / d E .
S E = S ¯ ( ln R / π ) ( R 1 2 + 1 ) / ( R 1 2 - 1 )
d E = d ( ln R / π ) ( R 1 2 + 1 ) / ( R 1 2 - 1 ) .
P ( S ) = 1 ln R S - 1 { exp [ - ( 1 + R - 1 2 ) 2 π S S E ] - exp [ - ( R 1 2 + 1 ) 2 π S S E ] }
W ¯ d = R 1 2 + 1 R 1 2 - 1 2 b d E [ ( 1 + S E u ( 1 + R - 1 2 ) 2 b ) 1 2 - ( 1 + S E u ( R 1 2 + 1 ) 2 b ) 1 2 ] .
W ¯ d = 2 b d E [ ( 1 + S E u / b ) 1 2 - 1 ] .
x E = S E u / 2 π b ,
β E = 2 π b / d E ,
W ¯ / d = ( β E / π ) [ ( 1 + 2 π x E ) 1 2 - 1 ] .
P ( S ) S - 1 exp [ - ( 1 / π ) S / S E ] ,
¯ = 1 - exp { - ( 2 b / d E ) [ ( 1 + S E u / b ) 1 2 - 1 ] } ,
¯ = 1 - exp { - ( β E / π ) [ ( 1 + 2 π x E ) 1 2 - 1 ] } .
( W ¯ / d ) / β E = h ( x E ) = ( 1 / π ) [ ( 1 + 2 π x E ) 1 2 - 1 ] .
W / 2 π b = f ( x ) = x e - x [ I 0 ( x ) + I 1 ( x ) ] .
W ¯ d = S E u d E ( 1 + S E u 4 b ) - 1 2 ,
S E = ( 4 / π ) S 0
d E = ( 4 / π ) d .
( W / d ) / β E = e ( x E ) = x E ( 1 + 1 2 π x E ) - 1 2 ,
P ( S ) { = ( 1 / ln R ) S - 1 for S M / R S S M = 0 for             S < S M / R and S > S M
( W ¯ d ) / β E = R 1 2 + 1 R 1 2 - 1 { 2 x E ( 1 + R - 1 2 ) 2 exp ( - 4 x E ( 1 + R - 1 2 ) 2 ) [ I 0 ( 4 x E ( 1 + R - 1 2 ) 2 ) + I 1 ( 4 x E ( 1 + R - 1 2 ) 2 ) ] - 2 x E ( R 1 2 + 1 ) 2 exp ( - 4 x E ( R 1 2 + 1 ) 2 ) [ I 0 ( 4 x E ( R 1 2 + 1 ) 2 ) + I 1 ( 4 x E ( R 1 2 + 1 ) 2 ) ] + exp ( - 4 x E ( 1 + R - 1 2 ) 2 ) I 0 ( 4 x E ( 1 + R - 1 2 ) 2 ) - exp ( - 4 x E ( R 1 2 + 1 ) 2 ) I 0 ( 4 x E ( R 1 2 + 1 ) 2 ) } ,
β E = 4 1 ln R R 1 2 - 1 R 1 2 + 1 2 π b d
x E = 1 4 ( 1 + R - 1 2 ) 2 S M u / 2 π b .
( W ¯ / d ) / β E = g ( x E ) = 2 x E exp ( - 4 x E ) [ I 0 ( 4 x E ) + I 1 ( 4 x E ) ] + 1 4 exp ( - 4 x E ) I 0 ( 4 x E ) - 1 4 ,
S M , r S M , r 2 S M , r n S M             ( r < 1 ) ,
P ( S ) = 1 n + 1 m = 0 n δ ( S - r m S M ) .
W ¯ n = 1 n + 1 m = 0 n W ( r m S M , u ) ,
d n = d / ( n + 1 ) .
W ¯ n d n = 2 π b d m = 0 n f ( r m S M u 2 π b ) ,
W ¯ n / d n = β F ( x M , r ) ,
F ( x M , r ) = m = 0 n f ( r m x M )
x M = S M u / 2 π b .
β E F ( x E , r ) = β F ( x M , r ) ,
lim x E 0 F ( x E , r ) / x E = 1 ,
lim x E F ( x E , r ) / x E 1 2 = ( 2 / π ) 1 2 .
β E = [ ( 1 + r 1 2 ) / ( 1 - r 1 2 ) ] β
x E = ( 1 + r 1 2 ) - 2 x M .
F ( x E , r ) = 1 - r 1 2 1 + r 1 2 m = 0 f [ r m ( 1 + r 1 2 ) 2 x E ] .
¯ = 1 - exp [ - β E F ( x E , r ) ] .
exp ( - h c ω 0 / k T ) > 0.1
T ( in ° K ) > 0.6 ω 0 ( in cm - 1 )