The band system of ethane around 7 micron: Frequency analysis of the ν6 band

Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

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The band system of ethane around 7 micron: Frequency analysis of the ν6 band N. Moazzen-Ahmadi n, J. Norooz Oliaee Department of Physics and Astronomy, University of Calgary, 2500 University Drive North West, Calgary, Alberta, Canada T2N 1N4

a r t i c l e i n f o

abstract

Article history: Received 14 February 2016 Received in revised form 28 March 2016 Accepted 29 March 2016 Available online 12 April 2016

High quality line parameters of the band systems of ethane are required for accurate characterization of spectral features observed in the atmospheres of Jovian planets and their satellites. To date, experimental characterization of the excited vibrational states lying below 1300 cm
1 has been made. This includes the torsional bands around 35 mm, ν9 (820 cm
1), ν3 (990 cm
1), ν12
ν9 (380 cm
1) and ν9 þ ν4
ν4 (830 cm
1) bands. These earlier high resolution ro-vibrational analyses were made to experimental accuracy. Here, we report a detailed analysis of the weak ν6 band in the 1340–1410 cm
1 region using a spectrum recorded at a resolution of 0.003 cm
1 and temperature of 200 K. The Hamiltonian model included couplings between ν6 and ν9 (in particular with ν9 þ 2ν4 with which it is resonantly coupled) as well as couplings between ν6 and ν8. An excellent fit to within experimental accuracy was obtained. Taking the results of this 5-state fit, together with earlier results on lower lying vibrations, we now have experimental characterization for torsion–vibration states of ethane lying below 1400 cm
1. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Ethane ν6 band Infrared spectrum

1. Introduction The interest in laboratory spectroscopy of ethane stems from the desire to understand the methane cycle in planetary atmospheres and their moons and the importance of ethane as a trace species in the terrestrial atmosphere. Solar decomposition of methane in the upper part of these atmospheres followed by a series of reactions leads to a variety of hydrocarbon compounds among which ethane is often the second most abundant species [1–7]. Because of its high abundance, ethane spectra have been measured by Voyager and Cassini in the regions around 30, 12, 7, and 3 mm. Therefore, a complete knowledge of line parameters of ethane is crucial for spectroscopic remote sensing of planetary atmospheres.

n

Corresponding author. Tel.: 1 403 220 5394; fax: 1 403 289 3331. E-mail address: [email protected] (N. Moazzen-Ahmadi).

http://dx.doi.org/10.1016/j.jqsrt.2016.03.041 0022-4073/& 2016 Elsevier Ltd. All rights reserved.

In Ref. [8], we reported a comprehensive analysis of ethane vibrational states located below 1300 cm
1. As shown in Fig. 1, the torsional states {ground torsional state, 41(ν4), 42(2ν4)} were fitted separately (1-state fit, the first block in Fig. 2) [9], while {ground torsional state, 41(ν4), 42(2ν4), 91(ν9), 43(3ν4)} was analyzed in a 2-state fit (the first 3  3 blocks in Fig. 2) [10] and {ground torsional state, 41(ν4), 42(2ν4), 91(ν9), 43(3ν4), 31, 44, 9141, 121} was studied in a 4-state fit (the first 6  6 blocks in Fig. 2) [8]. The 1, 2, and 4-state studies were made to within experimental uncertainty. As discussed below, the 5-state analysis reported here uses the entire matrix in Fig. 2 and includes frequencies between or within torsional stacks from five different vibrational states. Here, we extend the analysis to the 1340–1600 cm
1, shown in the box at the top of Fig. 1. These four levels constitute a complicated manifold with many local and global perturbations. This spectral region corresponds to the excitation of two infrared active fundamentals, ν6  1379 cm
1 and ν8 1473 cm
1, and two combination bands, ν9 þ2ν4

8

N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

and ν12 þ ν4. The combination bands do not carry significant infrared intensity but become partially observed through intensity borrowing from the strong ν8 fundamental. In this work, we focus our attention to the weaker ν6 parallel band (1340–1410 cm
1). A total of 561 lines from the ν6 band with 0 r K r 9 and 0 rJ r30 were identified and analysed by varying 77 parameters and including a total of 5408 frequencies. The rms deviation of this 5-state fit for the ν6 band was 37  10
5 cm
1 which is comparable to the experimental uncertainty of 5  10
4 cm
1. Previous studies of the band system of ethane in the 7 μm region were made by Lattanzi et al. [11,12]. In these analyses the states shown in the box at the top of Fig. 1 were considered as four interacting states but separate from all lower lying torsional and vibrational states which were ignored. For example, the 9142 (ν9 þ2ν4) was treated as a separate state and not as a member of the torsional stack of 91 as is done in the present work. Furthermore, the RMS deviation of the fit in the work by Lattanzi et al. was 3.68  10
3 cm
1 which is more than ten times the experimental uncertainty.

Fig. 1. Vibrational levels of ethane. The four levels in the uppermost enclosed box are the subject of the present paper.

2. Spectra and analysis A spectrum at a resolution of 0.003 cm
1 was recorded using a modified Bomem Fourier transform spectrometer at the National Research Council (NRC) of Canada. The ethane vapor pressure of 80 mTorr at temperature of 200 K was used in a 2 m multiple-traversal gas cell set for a total path of 40 m. The ν6 fundamental, whose upper state symmetry is A4s, is an infrared active parallel band, but much weaker than the ν8 fundamental. A room temperature spectrum of this band is completely dominated by the much stronger lines from the neighboring ν8 fundamental. This point is well illustrated in Fig. 1 of Ref. [11]. However, in a 200 K spectrum such as that obtained in this work, the ν6 band is clearly visible. The overview of the spectrum from 1340– 1410 cm
1 in Fig. 3 shows a clear Q-branch around 1379 cm
1 and blue shaded K structures in both P-and Rbranches. Fig. 4 shows close-up of three potions of the spectrum, one in the P-branch (the top trace), one in the central region of the ν6 band (the middle trace which includes part of the Q-branch), and one in the R-branch (the bottom trace). Rotational assignment of the ν6 band was straight forward. Only lines with 0 r K r9 had sufficient signal to noise to be included in the fit. In the absence of any perturbations the torsional fine structure is not resolved, indicating that the barrier heights in the ground state and v6 ¼ 1 are not significantly different. One of the main interactions affecting low values of K in ν6 is the interaction with ν9 þ2ν4, leading to measurable torsional fine structure for K ¼ 3; 4; and 5: As can be seen from Fig. 5, the closest approach for J ¼ 15 is 0.44 cm
1 with ðK ¼ 4; σ ¼ 0; v6 ¼ 1Þ above its interacting partner (i.e., ðK ¼ 3; l9 ¼ 1; σ ¼ 0; v4 ¼ 2; v9 ¼ 1Þ) and 1.13 cm
1 for ðK ¼ 4; σ ¼ 2; v6 ¼ 1Þ which is below its interacting partner (i.e., ðK ¼ 3; l9 ¼
1; σ ¼ 2; v4 ¼ 2; v9 ¼ 1Þ). The fact that ðK ¼ 4; σ ¼ 0; v6 ¼ 1Þ and ðK ¼ 4; σ ¼ 2; v6 ¼ 1Þ are located on the opposite sides of their interacting levels leads to the largest observed torsional splitting for K ¼ 4: Here, σ ¼ 0; 7 1; 7 2; 3 labels the torsional sublevels. A second and much larger Coriolis-type interaction affecting the ν6 band is with ν8. For low values of K, this

Fig. 2. Hamiltonian matrix used to obtain the total energy. Here, K g represents the value of K for the ground vibrational state. The non-degenerate states are represented by a single block, including the lowest 9 torsional states. The degenerates states are represented by two diagonal blocks each, one for lv ¼ þ 1 and the other for lv ¼
1. Each of these blocks also includes 9 torsional states, except for the last two blocks which comprise of the lowest 4 torsional states. The off-diagonal blocks are marked by their torsion mediated intervibrational couplings. See Table 7.

N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

9

Fig. 3. Overview of the observed spectrum of ethane, showing the ν6 fundamental recorded with a vapor pressure of 80 mTorr, temperature of 200 K and absorption path length of 40 m. The Q-branch is clearly visible around 1378 cm
1. The R-branch is dominated by the much stronger lines from the ν8 fundamental.

Fig. 4. From top to bottom: portions of P, Q, and R-branch of the ν6 fundamental. Transitions with K¼ 3 and 4 show torsional doubling as a result of interaction with 9142. Transitions with K¼5 show torsional doubling at higher J.

interaction is non-resonant with an energy gap ranging from 100 cm
1 (with ν8 level being at higher energy) for K ¼ 0 to about 50 cm
1 for K ¼ 9: This interaction becomes resonant around K ¼ 16 where level crossing between ν6 and ν8 occurs. For given values of J, K g (where K g represents the value of K for the ground vibrational state), and σ the Hamiltonian matrix was
evaluated in a
two-step process using the 
 
 basis set jv6 ijv3 i
v9 ; l9
v12 ; l12
v8 ; l8
J; k jv4 ; σ i. Here jv6 i and jv3 i are the harmonic oscillator wave for the
functions 


ν
6 and ν 3 non-degenerate modes and v9 ; l9 , v12 ; l12 and 
v8 ; l8 are the corresponding functions for the ν9, ν12 and ν8 degenerate modes. The rotational wavefunction is
 represented by
J; k , where k is the signed quantum number associated with projection J z of the overall angular momentum J on the molecular symmetry axis, while



K ¼
k
. The torsional wave function was expanded in terms of free rotor basis given by þ 10 X 1 jv4 ; σ i ¼ pffiffiffiffiffiffi Av4 eið6ks þ σ Þγ 2π ks ¼
10 6ks þ σ

ð1Þ

In the first step, the torsional problem was solved. This provides torsional energy and the expansion coefficients Av6k4 s þ σ which were then used to obtain the matrix elements of the torsional operators. In the second step, the total energy was obtained from diagonalization Hamiltonian matrix shown in Fig. 2. The first, fourth and seventh diagonal blocks include the lowest 9 torsional states for the gs, ν3 and ν6, respectively. The degenerates states are represented by two diagonal blocks each, one for lv ¼ þ 1 and the other for lv ¼
1. Each of these blocks also

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N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

Table 1 Characteristics of the 5-state fit. Type of data

Na

Previous fit [8] b

s  10 ν6 fundamental band (1)J r 30 and 0r Kr 9 Torsional transitions (2) ν4 ¼1’0 (3) ν4 ¼2’1 (4) ν4 ¼1’0 (low res.) ν9 fundamental (5) low T spectrum (6) Room T spectrum (7) ν9 resonant and 3ν4 ν3 fundamental (8) Raman spectrum ν9 þ ν4
ν4 hot band (9)
15r KΔK r 15 ν12
ν9 difference band (10)
10r KΔK o 11 Total a b

Fig. 5. This energy level diagram shows
the level crossing between 61  1 2

and 9 4 . A local resonance between J; K ¼ 2; σ; v4 ¼ 0; v6 ¼ 1 and
J; K ¼ 0; σ þ 3; v4 ¼ 2; v9 ¼ 1 with selection rules ΔK ¼ 7 2; Δl ¼ 8 1; Δσ ¼ 3 was modeled by a 2  2 matrix. A second interaction with selection rules ΔK ¼ 7 1; Δl ¼ 7 1; Δσ ¼ 0 is responsible for torsional doubling for a limited range of K (i.e., rotational series with K ¼ 3; 4; and 5).

includes 9 torsional states, except for the last two which comprise of the lowest 4 torsional states. The off-diagonal blocks are marked by their torsion mediated intervibrational couplings. The dimensions of the Hamiltonian matrix in the second step were 89 and 89 with ν8 being treated as a dark state.

3. Results, discussion and conclusions The coupling between ν9 and ν12 states which was introduced for the first time in connection with the 4-state analysis [8] is given by  i h H νint9 ;ν12 ¼ C 12;9 þC J12;9 J 2 þ C K12;9 J 2z ; 2   J þ p9 þ q12 þ
J
p9
q12
þ cos 3γ ð2Þ The subscript þ on the square bracket indicates the anticommutator. This coupling is required to explain the perturbation observed for rotational series with K ΔK r
5 in the ν9 þ ν4
ν4 band. Since the terms with J
p9
q12
affect only K ΔK Z 0, these were ignored in the 4-state analysis. However, in this work these non-resonant terms were included for completeness. This resulted in a slight simplification of the Hamiltonian model for the gs, namely, H 0;JK which was required in the 4-state analysis is no longer needed here. Another point worthy of mention is the presence of the minus sign in Eq. (2) between J þ p9 þ q12 þ and J
p9
q12
. This is

5

χ

2

Present work s  105

χ2

561

…

…

37

214

770 302 215

30 37 82

130 71 16

30 37 82

127 73 16

1178 234 24

8 9 21

61 36 2

9 9 22

69 34 3

166

71

65

73

69

1262

13

191

12

167

696 5408

85

127

81

115 887

The number of frequencies in each component. Here, s (in cm
1) represents standard deviation.

consequence of the fact that the matrix representation of   q9 þ ; q9
with E1d symmetry are chosen to be identical to the matrix representation of ðT þ ; T
Þ and the repre  sentation for q12 þ ; q12
with E2d symmetry the same as   those for J þ ; J
except for the symmetry elements in the CD; B4 CD; and D classes [13] where they differ by a minus sign. When these representations are combined with the phase convention in Ref. [14] for the matrix elements of the vibrational coordinates (real) and momenta (purely imaginary), we obtain matrix elements which are real for Eq. (2). Finally an equally valid coupling between ν9 and ν12 states is H νint9 ;ν12 ¼

   i n 0 2 C 12;9 þC 0;J J 2 þ C 0;K 12;9 J z ; J þ p9 þ q12 þ
q9 þ p12 þ 12;9 4

  o
J
p9
q12

q9
p12


þ

cos 3γ

ð3Þ

Although Eq. (3) is more symmetrical with respect to the vibrational coordinates and momenta, its matrix elements are identical to those of Eq. (2), hence the parameters in Eqs. (2) and (3) are 100% correlated. To treat the data from the ν6 fundamental, three torsion-mediated vibrational couplings were used. The first of these treated a local resonance between

 
J; K ¼ 2; σ ; v4 ¼ 0; v6 ¼ 1 and
J; K ¼ 0; σ þ 3; v4 ¼ 2; v9 ¼ 1 with an interaction Hamiltonian given by   ν9 ;ν6 H 1; ¼ iC 19;6 q6 J 2þ q9 þ
J 2
q9
sin 3γ ð4Þ int This interaction was modeled as a 2  2 matrix with an
 effective parameter C~ 9;6 ¼ iC 9;6 σ ; v4 ¼ 0; v6 ¼ 1
sin 3γ jσ þ 3; v4 ¼ 2; v9 ¼ 1i ¼ 181:2ð28Þ Hz whose value is not sensitive to σ . The second coupling also treated a resonance between ν9 and ν6. As mentioned in Section 2, this interaction is responsible for torsional fine structure in the ν6 fundamental which is observed for a limited range of K (i.e., rotational series with K ¼ 3; 4; and 5). It has the form

N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

Table 2 Molecular parameters for the gs of ethane from the 5-state fita. i

Parameter

Operator Oi


1

1 2 ð1 þ

cos 6γ Þ

Values 1047.787(15)

1

V0,3/cm

2

A0/MHz

J 2z

80384.9(21)

3

B0/MHz

J 2
J 2z

19913.646(22)

4

D0,J/kHz


J4

31.020(19)

5

D0,JK/kHz


J 2 J 2z

78.39(24)

6

D0,K/kHz

283

7

D0,m/MHz

8

D0,Jm/MHz

9

D0,Km/MHz


J 4z
J 4γ
J 2 J 2γ
J 2z J 2γ J 2 O1 J 2z O1 J 2 12ð1
J 2z 12ð1
J 4 O1 J 4z O1 J 2 J 2z O1 J 2 J 2z J 2γ

10

F0,3J/MHz

11

F0,3K/MHz

12

F0,6J/MHz

13

F0,6K/MHz

14

F0,3JJ/kHz

15

F0,3KK/kHz

16

F0,3JK/kHz

17

H0,JKm/Hz

Table 4 Molecular parameters for v3 ¼ 1 state of ethane from the 5-state fita,b. i

Parameter
1

Operator Oi

Value

44 45 46 47

v~ 3 /cm V3,3/cm
1 V3,6/cm
1 ΔA3/MHz

unity O1 O20 J 2z

993.019(22) 1017.94(30) 11c
87.52(29)

48

ΔB3/MHz

J 2
J 2z


183.846(87)

b

0.4508(79) 0.3028(11) 2.0833(80)
396.4(20)

a A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator (column 3), Oi. b D3,J, D3,JK, D3,K, D3,m, D3,Jm, D3,Km, F3,3J, F3,3K, F3,6J, F3,6K and all sextic H parameters were fixed to their gs values. c Fixed. See Ref. [8].

1014.1(19) cos 12γ Þ

7.95 (58)

cos 12γ Þ

8.25(36) 1.725(67) 23.9(13)
15.3(36) 47 (12)

a

A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator (column 3), Oi. b Fixed. See Ref. [15].

Table 3 Molecular parameters for v9 ¼ 1 state of ethane from the 5-state fita,b. i

Parameter

Operator Oi

Value

18 19 20

ν~ 9 /cm
1 V9,3/cm
1 V9,6/cm–1

unity O1 1 2ð1
cos 12γÞ

973.97353(40)c 1081.539(49) 13.880(25)

21

A9,R/MHz

J 2γ

79171.9(26)

22

ΔA9/MHz

247.36(37)

23

ΔB9/MHz

J 2z 2

J
J 2z


36.434(30)

24

ΔD9,K/kHz


J 4z

5.51(15)

25

ΔD9,m/MHz


J 4γ


1.646(11)

26

ΔD9,Jm/MHz

0.1020 (39)

27

ΔF9,3J/MHz

28

ΔF9,3K/MHz

29

ΔF9,6K/MHz

30

F9,3JJ/kHz

31

F9,3KK/kHz

32 33 34 35

F9,3JK/kHz Aςz9 /MHz η9,J/MHz η9,K/MHz

2
J 2 J~ γ 2 J O1 J 2z O1 J 4z O20 J 4 O1 J 4z O1 J2J2z O1


29.9(17) 71(15)
20.2(61) 3.193(60) 43.9(17)

J 3z l9


44.0(39) 20917.6(21)
0.04127(97) 0.7275(31)


2Jzl9 J2Jzl9

36

η9,m/MHz

J z l9 J 2γ


8.97(13)

37 38

η9,3/MHz η9,3J/MHz

J z l9 O1 J O37


2207(45) 0.8937(59)

39

η9,3K/MHz

J 2z O37


1.466(18)

40

η9,3JK/kHz

41

q9/MHz

J 2 J 2z O37 h i 2 2 1 2 2 4 J þ q12 þ þ J
q12


53.050(45)

42

q9,J/kHz

43

q9,6/MHz

2

J 2 O41 O41 O1

11

0.303(15)


0.133(21)
46.4(20)

a A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator (column 3), Oi. b D9,J, D9,JK, D9,Km, F9,6J, sextic H parameters were fixed to their gs values. c This gives a band center of 821.29 cm
1.

Table 5 Molecular parameters for v12 ¼1 state of ethane from a 5-state fita,b. i

Parameter
1

49 50 51

v~ 12 /cm V12,3/cm
1 ΔA12/MHz

52

ΔB12/MHz

53 54

ΔD12,J/kHz ΔD12,JK/kHz

55

ΔD12,K/kHz

56 57 58

Aςz12 /MHz η12,J/MHz η12,K/MHz

59

q12/MHz

Operator Oi

Value

unity O1

1349.66106 (63)c 972.8(41) 124.86(36)

J 2z J 2
J 2z
J4
J 2 J 2z
J 4z
2J z l12 J2Jzl12 J 3z l12 h i J 2þ q212 þ þ J 2
q212


1 4


73.730(67)
0.680(72) 15.48(55)
11.7(25) 32430.6(42) 0.4364(34) 0.666(14) 25.596(92)

a A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator (column 3), Oi. b D12,m, D12,Jm, D12,Km, F12,3J, F12,3K, F12,6J, F12,6K and all sextic H parameters were fixed , F3,3J, F3,3K to their gs values. c This gives a band center of 1196.98 cm
1.

given by    1  ν9 ;ν6 1 þ cos 6γ ¼ i C 29;6 þ J 2 C 2;J H 2; 9;6 p6 J
q9 þ
J þ q9
int 2 ð5Þ The third torsion-mediated coupling is between ν6 and ν8. This interaction affect the entire range of K (0 r K r9) studied here. It is large in magnitude and becomes resonant near K¼ 16, well outside the range of K observed here. This interaction has the same form as Eq. (5) H νint6 ;ν8 ¼ i



1    C 6;8 þ J 2 C J6;8 1 þ cos 6γ þ P 6;8 J 2γ p6 J
q8 þ
J þ q8
2

ð6Þ A total of 561 lines from the ν6 band with 0 r K r 9 and 0 r J r 30 were included in the 5-state fit. The overall data set included 5408 frequencies. These were fitted to within experimental error using a 82-parameter Hamiltonian, 77 of which were varied. The standard deviation for lines in the ν6 band was 0.00037 cm
1, well within the minimum experimental uncertainty of 0.0005 cm
1. The standard deviations for the remaining data are comparable to the 4state fit. These are reported in Table 1. The molecular parameters thus obtained are listed in Tables 2–7. The detailed output of the best fit is given in Table A of

12

N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

Table 6 Molecular parameters for v6 ¼1 and v8 ¼ 1 states of ethane from a 5-state fita,b,c. i

Parameter
1

Operator Oi

Value

unity O1

60 61 62

v~ 6 /cm V6,3/cm
1 ΔA6/MHz

J 2z

1378.53019(24) 1047.787d 196.82(10)

63

ΔB6/MHz

J 2
J 2z


60.20(13)

64

ΔD6,J/kHz

65 66 67

ν~ 8 /cm
1 V8,3/cm
1 Aςz8 /MHz


J4 unity O1
2J z l8

1467.791(90) 1047.787d
24219e


2.55(32)

a

A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator, Oi. D6,JK, D6,K, D6,m, D6,Jm, D6,Km, F6,3J, F6,3K, F6,6J, F6,6K and all sextic H parameters were fixed to their gs values. D8,J, D8,JK, D8,K, D8,m, D8,Jm, D8,Km, F8,3J, F8,3K, F8,6J, F8,6K and all sextic H parameters were fixed to their gs values. d Fixed at the gs value. e Held fixed. b c

Table 7 Interstack coupling parameters for ethane from the 5-state fita. i

Parameter

Operator Oi

gs/v9 Coriolis interactions 68

  Bςx4;9a =KHz

1 2

h i J γ ; J þ q9
þ J _q9 þ

Value

336.17(22) þ


2λς sin 6γ J þ P 9
þ J _ p9 þ

69

  Bςx;J 4;9a =KHz

70

P yz 12;t a =MHz

J2O68

 
iJ γ J z J þ þ J þ J z p9

J z J
þ J
J z p9 þ 

gs/v3 Fermi interactions 71

C 33;0 =cm
1

q3 12ð1 þ cos 6γÞ

258.17(13)

C 63;0 =cm
1 C 3;J 3;0 =MHz

q3 12ð1
cos 12γÞ

3.408(39)

J 2 O71


171.7(75)

C 12;9 =MHz

  i J þ p9 þ q12 þ
J
p9
q12
cos ð3γÞ

1554 (39)

76

C J12;9 =KHz C K12;9 =MHz

J 2 O74   2 1 2 2 J z O74 þ O74 J z

v6/v9 interactions 77

C 19;6 =Hz

72 73 v9/v12 interactions 74 75

C 29;6 =MHz C 2;J 9;6 =KHz

78

  iq6 J 2þ q9 þ
J 2
q9
sin 3γ  1 ip6 J
q9 þ
J þ q9
2ð1þ cos 6γ Þ


3.65(24)
9.882(33)


53.3 (45) 11.83(40) 181.1(28) 3091(28)

J O78


823(63)

C 6;8 =GHz

 
ip6 J þ q8

J
q8 þ 12ð1 þ cos 6γ Þ

201.1(47)

81

C J6;8 =MHz

J 2 O80

82

P 6;8 =MHz

79 v6/v8 interactions 80

a

2

 
ip6 J þ q8

J
q8 þ J 2γ


2.51(30) 202(13)

A term in the Hamiltonian is given by ith paremter (column 2) multiplied the ith operator (column 3), Oi.

Appendix. For each line included in the analysis, this output lists the identification, the measured frequency, the assigned experimental uncertainty, and the difference between the observed and calculated frequencies. In summary, a frequency analysis of the ν6 band of ethane in the 1340–1410 cm
1 region was made using a spectrum recorded at a resolution of 0.003 cm
1 and temperature of 200 K. All lines in the ν6 band with significant intensity were identified and analysed in a

5-state frequency analysis which also included frequencies from ν4, 2ν4
ν4, 3ν4, ν9, ν9 þ ν4
ν4, ν3, ν12
ν9 bands. The analysis of the data from the ν6 band required two couplings with the torsional stack of the v9 ¼ 1 and a coupling with the nearby state v8 ¼ 1, with v8 ¼ 1 treated as a dark state. With the successful conclusion of the 5-state analysis we now have experimental characterization for torsion–vibration states of ethane lying below 1400 cm
1.

N. Moazzen-Ahmadi, J.N. Oliaee / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 7–13

Acknowledgment The financial support of the Canadian Space Agency is gratefully acknowledged. We also like to thank Dr. A.R. W. McKellar for assistance with the experiment and J.T. Hougen for fruitful group theoretical discussions.

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j. jqsrt.2016.03.041.

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