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 ν_{0} = 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.

Darrell E. Burch and Dudley Williams Appl. Opt. 3(1) 55-61 (1964)

References

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Suitable Coefficients a_{n} for Use with Power Series
$\mathrm{\Delta}{{\nu}_{\u220a}}^{*}={\gamma}_{1}{\text{z}}_{1}\sum _{n=0}{\mathbf{a}}_{n}{\mathbf{f}}^{n}$

n

a_{n} for 1%

a_{n} for 2%

a_{n} for 5%

a_{n} 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 W_{c} or W_{i} 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 W_{c} or W_{i} to Within 1%, 2%, 5%, and 10%

Note: These numbers are generated by the polynomial representations of Table I for four ratios of
${({S}_{2}{\gamma}_{2}/{S}_{1}{\gamma}_{1})}^{{\scriptstyle \frac{1}{2}}}$. 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 W_{2}/W_{1}

γ_{1}z_{1}

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

π^{2}

π^{3}

π^{4}

P_{atm}

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 a_{n} for Use with Power Series
$\mathrm{\Delta}{{\nu}_{\u220a}}^{*}={\gamma}_{1}{\text{z}}_{1}\sum _{n=0}{\mathbf{a}}_{n}{\mathbf{f}}^{n}$

n

a_{n} for 1%

a_{n} for 2%

a_{n} for 5%

a_{n} 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 W_{c} or W_{i} 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 W_{c} or W_{i} to Within 1%, 2%, 5%, and 10%

Note: These numbers are generated by the polynomial representations of Table I for four ratios of
${({S}_{2}{\gamma}_{2}/{S}_{1}{\gamma}_{1})}^{{\scriptstyle \frac{1}{2}}}$. 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 W_{2}/W_{1}

γ_{1}z_{1}

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

π^{2}

π^{3}

π^{4}

P_{atm}

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