Whether two neighboring infrared absorption lines, both in the square-root region of the curve of growth, are nearly free of mutual intrusion or so overlapped as to nearly coincide over a broad range of experimental conditions and line parameters is formulated. We also are able to estimate the unsullied equivalent width of the collision-broadened line pair at these limits. This should greatly extend the number of isolated infrared vibration–rotation band members that may be measured and tested for the validity of assigned line parameters and tested for the assumed Lorentz shape. The experimentally reported strong water lines at ν0 = 1429.96 cm−1, 1447.98 cm−1, and 1464.93 cm−1 in the 6.3-μ band are confirmed to be independent of their strong neighbors.
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Suitable Coefficients an for Use with Power Series
n
an for 1%
an for 2%
an for 5%
an for 10%
Coincident Case
0
1.739
2.037
2.790
4.024
1
−7.502
−8.235
−8.636
−10.859
2
20.996
25.330
18.401
18.682
3
−31.724
−42.682
−13.297
−11.213
4
24.627
36.392
−11.358
−1.128
5
−7.537
−12.002
23.811
2.624
6
0.0
0.0
−10.302
0.0
Independent Case
0
1.955
1.103
.066
−1.140
1
42.236
33.501
25.788
24.416
2
−96.273
−96.069
−83.202
−68.044
3
183.756
226.032
204.633
114.042
4
−201.283
−307.176
−294.091
−95.546
5
105.880
213.337
220.017
31.073
6
−18.770
−58.539
−65.821
0.0
Note: These polynomials predict the wavenumber separation at which the equivalent width of two spectrum lines departs from Wc or Wi by ∊* = 1%, 2%, 5%, and 10%. The series are applicable for 0.10 ≤ f ≤ 1.0, except for coincidence 5% and independence 10%, where the lower limit rises to 0.15, and coincidence 10%, which extends from 0.20 to 1.0.
Table II
Wavenumber Separations for Two Strong Absorption Lines as Their Equivalent Widths Approach Wc or Wi to Within 1%, 2%, 5%, and 10%
Note: These numbers are generated by the polynomial representations of Table I for four ratios of
. Column labels, followed by percent deviation from that precise condition, are I for independent and C for coincident.
The coincident polynomial of 10% does not apply at f = 0.15.
Table III
Corrections to Equivalent Width for Cases When Exact Conditions of Coincidence or Independence Do Not Quite Hold, for Three Different Ratios of W2/W1
γ1z1
C1
C2
C5
C10
I1
I2
I5
I10
f = 0.1
0.02
0.001
0.003
0.006
0.013
0.001
0.003
0.007
0.014
0.20
0.013
0.025
0.063
0.126
0.014
0.028
0.069
0.138
2.00
0.126
0.253
0.631
1.263
0.138
0.276
0.691
1.382
10.00
0.631
1.263
3.157
6.315
0.691
1.382
3.456
6.911
f = 0.5
0.02
0.001
0.003
0.007
0.014
0.002
0.004
0.009
0.019
0.20
0.014
0.028
0.070
0.140
0.019
0.038
0.094
0.188
2.00
0.140
0.281
0.702
1.405
0.188
0.377
0.942
1.885
10.00
0.702
1.405
3.512
7.025
0.942
1.885
4.712
9.425
f = 1.0
0.02
0.002
0.004
0.009
0.018
0.003
0.005
0.013
0.025
0.20
0.018
0.036
0.089
0.178
0.025
0.050
0.126
0.251
2.00
0.178
0.355
0.889
1.777
0.251
0.503
1.257
2.513
10.00
0.889
1.777
4.443
8.886
1.257
2.513
6.283
12.566
Note: I denotes independent, C being equivalent to coincident. Numbers following these letters give the percent deviation from that specific condition.
Table IV
Comparison With Plass’s Formulation of the Overlapped Equivalent Width
1 % Departure from coincidence
Z
π2
π3
π4
Patm
10−3
0.05
(0.00)
0.15
(0.01)
0.48
(0.03)
0.85
0.15
0.48
10−2
0.48
(0.03)
1.51
(0.10)
4.75
(0.32)
0.48
1.53
4.80
10−1
4.84
(0.33)
15.13
(1.02)
47.52
(3.20)
4.86
15.28
47.99
5% Departure from coincidence
10−3
0.05
(0.01)
0.15
(0.02)
0.48
(0.08)
0.05
0.16
0.50
10−2
0.49
(0.08)
1.52
(0.24)
4.75
(0.75)
0.51
1.59
4.99
10−1
4.94
(0.76)
15.17
(2.40)
43.53
(7.54)
5 06
15.88
49.90
Note: Numbers from Plass’s Eq. (9), Ref. 4, appear at the top of each block and are strictly valid only when line centers coincide. Present results appear thereunder. Parenthetically enclosed numbers are the critical wavenumber separations appropriate to these percentage deviations for two equal strong lines.
Table V
Application to Three Strong Lines in the 6.3-μ Water Band at p = 3.5 Torr and T = 287.7 K
Reference line frequency
Strong neighbor
|Δν|
OverlΔν criteria imply Ind to 1% if
1429.96
1423.72
6.24
Δν = 1.26
1424.12
5.84
Δν = 1.12
1428.26
1.70
Δν = 0.69
1432.05
2.09
Δν = 0.58
1433.29
3.33
Δν = 0.78
1447.98
1452.08
4.10
Δν = 1.06
1464.93
1456.85
8.08
Δν = 3.14
1458.24
6.69
Δν = 2.39
1459.27
5.66
Δν = 1.94
1471.72
6.79
Δν = 1.30
Tables (5)
Table I
Suitable Coefficients an for Use with Power Series
n
an for 1%
an for 2%
an for 5%
an for 10%
Coincident Case
0
1.739
2.037
2.790
4.024
1
−7.502
−8.235
−8.636
−10.859
2
20.996
25.330
18.401
18.682
3
−31.724
−42.682
−13.297
−11.213
4
24.627
36.392
−11.358
−1.128
5
−7.537
−12.002
23.811
2.624
6
0.0
0.0
−10.302
0.0
Independent Case
0
1.955
1.103
.066
−1.140
1
42.236
33.501
25.788
24.416
2
−96.273
−96.069
−83.202
−68.044
3
183.756
226.032
204.633
114.042
4
−201.283
−307.176
−294.091
−95.546
5
105.880
213.337
220.017
31.073
6
−18.770
−58.539
−65.821
0.0
Note: These polynomials predict the wavenumber separation at which the equivalent width of two spectrum lines departs from Wc or Wi by ∊* = 1%, 2%, 5%, and 10%. The series are applicable for 0.10 ≤ f ≤ 1.0, except for coincidence 5% and independence 10%, where the lower limit rises to 0.15, and coincidence 10%, which extends from 0.20 to 1.0.
Table II
Wavenumber Separations for Two Strong Absorption Lines as Their Equivalent Widths Approach Wc or Wi to Within 1%, 2%, 5%, and 10%
Note: These numbers are generated by the polynomial representations of Table I for four ratios of
. Column labels, followed by percent deviation from that precise condition, are I for independent and C for coincident.
The coincident polynomial of 10% does not apply at f = 0.15.
Table III
Corrections to Equivalent Width for Cases When Exact Conditions of Coincidence or Independence Do Not Quite Hold, for Three Different Ratios of W2/W1
γ1z1
C1
C2
C5
C10
I1
I2
I5
I10
f = 0.1
0.02
0.001
0.003
0.006
0.013
0.001
0.003
0.007
0.014
0.20
0.013
0.025
0.063
0.126
0.014
0.028
0.069
0.138
2.00
0.126
0.253
0.631
1.263
0.138
0.276
0.691
1.382
10.00
0.631
1.263
3.157
6.315
0.691
1.382
3.456
6.911
f = 0.5
0.02
0.001
0.003
0.007
0.014
0.002
0.004
0.009
0.019
0.20
0.014
0.028
0.070
0.140
0.019
0.038
0.094
0.188
2.00
0.140
0.281
0.702
1.405
0.188
0.377
0.942
1.885
10.00
0.702
1.405
3.512
7.025
0.942
1.885
4.712
9.425
f = 1.0
0.02
0.002
0.004
0.009
0.018
0.003
0.005
0.013
0.025
0.20
0.018
0.036
0.089
0.178
0.025
0.050
0.126
0.251
2.00
0.178
0.355
0.889
1.777
0.251
0.503
1.257
2.513
10.00
0.889
1.777
4.443
8.886
1.257
2.513
6.283
12.566
Note: I denotes independent, C being equivalent to coincident. Numbers following these letters give the percent deviation from that specific condition.
Table IV
Comparison With Plass’s Formulation of the Overlapped Equivalent Width
1 % Departure from coincidence
Z
π2
π3
π4
Patm
10−3
0.05
(0.00)
0.15
(0.01)
0.48
(0.03)
0.85
0.15
0.48
10−2
0.48
(0.03)
1.51
(0.10)
4.75
(0.32)
0.48
1.53
4.80
10−1
4.84
(0.33)
15.13
(1.02)
47.52
(3.20)
4.86
15.28
47.99
5% Departure from coincidence
10−3
0.05
(0.01)
0.15
(0.02)
0.48
(0.08)
0.05
0.16
0.50
10−2
0.49
(0.08)
1.52
(0.24)
4.75
(0.75)
0.51
1.59
4.99
10−1
4.94
(0.76)
15.17
(2.40)
43.53
(7.54)
5 06
15.88
49.90
Note: Numbers from Plass’s Eq. (9), Ref. 4, appear at the top of each block and are strictly valid only when line centers coincide. Present results appear thereunder. Parenthetically enclosed numbers are the critical wavenumber separations appropriate to these percentage deviations for two equal strong lines.
Table V
Application to Three Strong Lines in the 6.3-μ Water Band at p = 3.5 Torr and T = 287.7 K