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Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer
JOUBNALOFMOLFXULAR
SPECTROSCOPY 73, 154-167 (1978)
Extensive Analysis of the Red System of the CN Molecule with a High Resolution Fourier Spectrometer D. CERNY,*
R. BACIS,* G. GUELACHVILI,**
AND
F. Roux*
*Unioersilt Claude Bernard (Lyon I), Luboratoire de SpectromCtrie lonique et Moltculaire (Associt au C.N.R.S.), 43, Boulevard du II Novembre 1918,69621 Villeurbanne, France, and *+Univtmitt de Paris&d, Laboratoire d'lnfra-rage,B&time& 350 Campus d’orsay, 91405 Orsay, CWex, France The red system of CN molecule emitted by a nitrous oxide-acetylene flame has been measured between 11000 and 4000 cm-r with a high resolution Fourier Spectrometer. Fourteen bands of the AV = +l, 0, - 1, and -2 sequences are reported and analysed. The molecular constants of the *II and W states are determined using a computer program written to directly reduce the data in a single fit of the whole set of lines, a standard deviation of 0.0025 cm-r was obtained. Some parameters reflecting small interactions are derived and discussed.
INTRODUCTION
The spectrum of the CN molecule is observed in the atmospheres of stars, comets, and sun. It appears in numerous chemical processes at high temperature: flames, chemical reactions, discharges, and postluminescence. Therefore, a precise knowledge of this spectrum is highly desirable for astrophysical and chemical process studies. A number of works have been done on the most intense electronic transitions B2Z+X22+ (violet system) and A%-X22+ (red system) (1-5). However, very high resolution techniques such as Fourier transform spectroscopy are more and more used for astrophysical observations which are extended through the infrared atmospheric windows. Then data laboratory of the same precision are needed for the molecules like CN. The red system of the CN molecule is emitted in a very wide region and extensive measurements mainly in the visible and near IR range can be found in the Davis and Phillips’ Atlas (3). But little is known for wavelengths greater than 1 pm. Moreover, the A211 state is perturbed by the B22+ state for the high vibrational levels and by the X22+ state for the lower ones. The study of the latter perturbation has already been extensively undertaken (6). A precise knowledge of the lower levels of the A211 state could help to complete this study. The present work is an extensive analysis with high resolution Fourier transform spectroscopy of the A211 -+ x22+ transition of the CN molecule in the 4000 to 11 000 cm-’ region. The spectrum is emitted in a C2H2/N20 flame. The experimental set-up and the recording procedure were described in a preliminary study of the (O-O) band (7). Let us only recall that the spectrum was recorded with 1450 000 spectral elements 1.54 0022-2852/78/0731-0154$02.00/O CopyrightQ 1978by Academic Press. Inc. All rights of reproduction in any form reserved.
ANALYSIS
OF THE
155
CN MOLECULE
0.2cmm’ t---4
8876
cm?
8878
cm- ’
cm-’
FIG. 1. Lines of the O-O (A% -+XZ+) transition of CN. The full-width lines (-0.080 cm-l) is mainly due to Doppler effect.
at half-maximum
of the
and a 16 X 1e3 cm-’ apparatus resolution. The linewidth is mainly due to the source broadening (T N 3000 K) as can be seen in Fig. 1. For the most intense bands all the expected satellite branches are observed (Fig. 1) but the related measurements are often not used in the final fit (overlapping with intense line, automatic elimination, . .). The absolute accuracy of the wavenumbers is one part in 10’ at 10 000 cm-‘. The explored spectral range corresponds to wavelengths lesser than 2.5 I.trn because for higher wavelengths there are strong overlappings with the very intense emission spectrum of water. Fourteen bands of AV = 1, 0, -1, -2 sequences have been analyzed. They come from the ZJ’= 0, 1, 2,3,4 levels of the upper state -4% and V” = 0, 1, 2,3, 4 of the lower state XQ+. Rotational lines until 11~CE 90 have been observed for the most intense bands. A whole fit of the 14 bands has been carried out in a one-step approach. It gives a weighted standard deviation (SD) of 2.5 X 1e3 cm-‘. Analysis of the Da&aand Experimental Results The ground state X*2+ is described Fl(,V) = BN(N F,(N)
with the usual spectral
+ 1) - D[N (N + l)]” + H[S!X
= BN(N + 1) - D[NCN + l)]’ + N[:V(Y
terms
+ I)]” + $rN + l)]” - fr(A’ + l),
where y represents the spin doubling of the *Z+ state and Fl(N) and Fz(IV)! respectively correspond to the components with N = J - $ and h'= J -I- a. As the measured lines reach high values of iV, centrifugal distortion terms are necessary to fit the data; so an effective spin-rotation parameter Yeff has been used with the usual expansion : Yeff = y + r.7N(N + 1) + rJJC.Vw
+ 1)P.
156
ET AL.
CERNY
TABLE Energy
I
Matrix
of Q State
7
P
Zn1
2n3 2
1 T,+;Ao-+~-;A
_Bfi+ZDXd?+;
+(B-A,)(Xt1)-D((X+l)2+X]
t
-
H fi(3
KVT X2 + X + 1) + A,,
6
211, + H [(X
2
+;p
+ 1)3 +X
(3 X + I)]
(lSG)+lq
-4
A,,
[ 3 (X + I)’ +X
1
q+ \r[-MT-Y]
-:*+fi+:
(x+z’zfi)+. 1
+ (B tA,)(X-1)
-D[(X-l)‘+X]
t +tl[(X-l)3+X(3X-I)]+l
2n3
A I
I +;q
X=(J-;)
[3(X-l)‘+X] ”
X
(J+;)
The A211 state is described with the 2 X 2 matrices reported in Table I. The interactions with YZ+ or 2x- states in a second order approximation lead to the A doubling. The closest states are two 2Z+ states and the 2Z- states are rather far (9). Then the A doubling is probably mainly due to 2Zf states and as usual we write p+ = pz+ - pz- ( = p) and 9+ = 9z+ - qx-(=q) and part of the 2Z- interactions are totally correlated to To, A, B, and y (20). We have taken into account of the centrifugal distortion using effective parameters such as Peff
=
p
+
PJJV
Qeff
=
q + qJ(J
+
1)
+
p.T.r[JV
+
+ 1) + QJJCJV +
l>l” l>P.
The whole set of lines was compared directly to this theoretical model. A global fitting program has been set up for the study of a few bands of the iodine spectrum (II) according to the Athenour’s method (12). It has been generalized to the case of doublet transitions for the present work. In a preliminary study every band is fitted separately. This allows the elimination of badly defined lines, mainly when there are important asymmetry or unusual intensities (overlapping of lines, proximity of ghosts, . . . ). Faint lines or branches can be calculated and then easily picked out. After this first study we determine for every branch a SD and a root mean square deviation (RMS = 1((Observed-calculated)2). This RMS has been used for weighting in the global fitting (Wbrancr, = l/ (RMS)2h,,c$. The Table II gives for every band the origine ~0 (cm-l) determined with the 14 1Use of Wbranch so calculated for a fit of the same band separately, gives new (RMS)z slightly different from the first ones, but we have checked with a few bands that the corresponding changes in the parameters are non significant.
ANALYSIS
OF THE
157
CN MOLECULE
TABLE
II
3
2 505a.wJ92
4
WL?)
mQ _, I.1065 x 10 270 linea 2.8x 10&m-' 6846.55189 (120)
10345.25114 (377)
MS.75599
(143)
3.8 x m-3,,-
2.8 x x)'
8355.4594
6392.029:
(400)
240 -2 9.9mO x a 141 linea 7.3 x 10Jcm-'
I
4
I
band global fitting, the maximum relative intensity (directly from the recording) of the lines in the Q branches, the Franck-Condon factors [quoted from Whiting et al. (13)] the number of lines used in the fit and the SD (cm-‘) in the separate fit of the band. In Tables III and IV are collected the parameters of the A*II and X22+ states of the TABLE
III
Parameters* of The CN A211 + XzZ+ Transition (cm-‘)
“=O T
Y
A
Y=
I
+ 9117.39268
(35)
- 52.6MM
(68)
- a.57602
+ 0.177 - 0.74
A,x104 AJYD9 B
+ 109a5.10355
v=2 (53)
+ 126.57.24425
(98)
- 52.X+26
(194)
111)
+ 0.176
(13)
+
(13)
(321
- 0.49
(35)
0.180
"=4
“=3 1105)
+ WO3.8029 - 52.4289 + 0.2m
(31)
+ 161U.7716
(54)
- 52.3705
(96) (155)
(30)
+ 0.473
182)
+ 1.7073145 143)
+ 1.69m415
(44)
+ 1.6727238 149)
+ 1.6553535(100)
+ I.637902 (29)
D *lo5
+ 0.61497
(13,
+ 0.61613
(13)
* 0.6,746
(I81
+ 0.61877
(66)
+0.6166
1221
HxlO"
+ 0.406
(11)
+ 0.375
(12)
+ 0.354
(21)
+ 0.372
(124)
-0.51
(46)
px IO2
+ o.e4c9
I421
+0.8400
(34)
to.5347
(58)
+ 0.858
(25)
+ 0.759
(34)
p,x IO6
- 0.2708
WI
- 0.2916
(98)
- 0.315
(22)
- 0.386
(170)
q x103
- 0.38961
(52)
- 0.39772
(95)
-0.40&1
(21)
- 0.4160
(56)
-0.4112
(69)
,,x107
+ 0.1082
(221
+ 0.1146
(50)
+ O.l3&?
(144)
+ 0.135
(31)
- 0.97
122)
- 0.95
(64)
- 0.38
(23)
13 qJJXD ' Th. numben
in porenthnn
C
or. th. unc.rtaintyin tba lastdigit,that corrnpdr
line pooitians weight par bmnch
1
Th. standad devidon
to two ~sndard deviationr computd
of the mid,,.,,~is 2 5 x 1O-3 cm-'
for the 3682 linn
using 011 the
158
CERNY
ET
TABLE
AL. IV
of the CN A*II + X*2+ Transition
Parameter@
(cm-l)
x21+Lowsr st.ts “=O T
v=
+ 2042.42261
0.0
Y
B
1
“=2 (49)
+ 4058.55176
v=3 (67)
+ 6048.34766
v=4 PI)
+ 8011.7734
(21)
+ 1.8910627 (43)
+ l.87%591(44)
+ l..mlm2(44)
+ l.e3%440
(47)
+ 1.82w468
04)
D
x105
+ 0.64072
(12)
+ 0.64b2
(13)
+ 0.64263
(13)
+ 0.64367
(IS)
+ 0.64445
(51)
H
x IO'l
+ 0.636
(II)
+ 0.608
(II)
+ 0.585
(13)
+ 0.547
(25)
+ 0.380
(118)
Y. 102
+ 0.7417
(33)
+ 0.7367
(331
+ 0.7272
(46)
+ 0.7190
(45)
+ 0.72Q
(1441
YJ" 107
- 0.70
(18)
- 0.96
(19)
- 0.97
(34)
- 0.96
(20)
- 3.0
(19)
+ 0.16
(16)
+ 0.45
(20)
+ 0.45
(53)
+ 7.8
(621
lj,x 10
11
CN molecule. The Table V gives the constants at the equilibrium calculated from these parameters. The wavenumbers experimentally measured, used in the fit, and those calculated from the theoretical model with the previous parameters have been made available.* In this compilation one can obtain the result of the fitting of two groups of seven bands: the first group is dealing with the (O-O), (O-l), (O-2), (l-O), (l-2), and (l-3) bands, the second one with (2-l), (2-3), (24), (3-2), (3-3), (3-4), and (4-4) bands. In order to identify easily the different branches, Fig. 2 recalls the relative position of the levels and the different branches with their first line for the A211 + X22+ transition of CN. Moreover, no attempt has been made to analyze the perturbation of the A*II state by levels of the ground state. The very low SD for the whole set of lines (0.0025 cm-l) shows that they are surely very weak or far from the explored region for the vibrational levels studied. Actually a few discarded lines only could be assigned to a perturbed line and with shifts lower than 10 SD (6). RESULT
ANALYSIS
This paper is a first analysis with a global fitting of an emission spectrum and as we have a great number of observed lines with precise measurements the statistical significance of the standard deviations and the related problems have been examined in detail. The global fitting may allow to determine small parameters whose contribution may be non-significant when fitting each band separately. This happens for levels which are common with several bands. It is also the case for levels defined by one (or several) unprecise bands (weak bands-for instance v” = 4 in X9+) the other level being best * This material is on deposit in the Editorial data (line frequencies, non-corrected relative available on request to one of the authors.
Office of the Journal of Molecular Spectroscopy. Original intensities, widths, asymmetries of lines) could be made
150
ANALYSIS OF THE CN MOLECULE
(Fpi
2w2I r-
2n cl
lFll
--IN' J'
72 10,
‘R27 I
Q2
101
R 7
I 35
-e
f,
-f
F2
+e
f,
+f
f2
3 v
25
m 2
1
25 15
I v
81 /0.5/
‘42 P 10.51 Q,20
Qt
1151
P, (2.51
(251
R2 1051
P,2 12.51
‘R2r 0,
(051
1051 P2 /r5:
R
Q,, /05/
“P,y /15/
FIG. 2. Relative position of the levels and first lines in the ATt + X2+ transition Under
the arrow
is noted
of the CN molecule.
the first J” value of the branch.
known from other bands (this comes from the great correlation between a number of parameters in the two levels). We have checked the precision of our results by different tests. In the first test we have searched the influence of small contribution parameters. In the second one we have examined the different types of weighting process or of elimination of poor lines. The third test will compare different types of fittings with parameters corresponding to different definitions but undiscernable at least in the second order.
160
CERNY
ET
TABLE Vibrational
and Rotational
Constants”
AL. V
(cm-‘) of the A211 and XW
States of CN
A2n A.
=
aA
=
8.
=
4
%
= 1813.419 =
%Ke
(42)
=
- 52.6072 - 0.743
(27) (33) x 10-I
1 .7159372
(22)
0. v237
(55) x 10-I
pg
=
- 0.154
(35) x IO-4
v,
=
- 0.153
(67) x lO-5
'2.8'9 ('a)
D
=
0.61434
(14) x lO-5
"1
=
0.420
(28) x IO-ll
iYD 9, 9, aq
=
0.122
(10) x 10-7
=
0.&(27
(35) x 10-2
=
- 0.3855
(13) x 10-3
=
0.83
(2(0x 10-5
= 1.232983 (4) x lo-'
'e
x21+
6
= 2068.67860
%
l&Jan,= W*Y* =
(97)
13.11735 (37) - 0.6543
(b-4).
lO-2
=
I.89977481 (23)
ag
=
0.1737135
Pe
=
*
'.
G,
=
we
Y,
=
- 0.459
=
0.641
(31’ x 10-5
Ii.
=
0.6488
(44) x 10-l’
aH
=
0.264
(28) x 10~‘~
= =
0.7463
(20) x 10-2
0.74
(11) x IO-4
l.l7lmb
(v + ;)-qxe(“+;)2
A,
=
B ”
= Be-aa(v+;)+&
A.-aA
(45) x 10-l (22) x IO-4
De
% a” =
- 0.2543
(32' x IO*
(4) x n-a
1 3 + + WJc (V + 2 )
(v+;)+.....
DV = De + aD
6
(v + y
+ ;
n"
=He-*'+("+;j
P,
= P, - ap(v*;)+_
4,
= q* - uq
(v +;
+ YB”+l’
I 3
+....
1+ ._._
)+
y,= Ve-ap(Y+;k+~Y(“+;)2 First test. The introduction of parameters with a small contribution is necessary since lines with high values of J (up to 90) are measured. But what is the higher order of the distortion parameters we have to use in the fit? We have considered that a parameter is at the limit of being significant: (1) when its magnitude is within once and twice its SD,3 and/or (2) when its maximum contribution is within one and two SD of the whole fit. Figure 3 shows the variations of the parameters y, yJ, and yJJ (0” = 0 of X’Z+) and of their SD (AT,AYJ, ATJJ) according to the maximum value of N used in the fit. Then we see that when we do not introduce a parameter at the limit of being significant in
* A simple empirical rule allows a direct estimate of the standard deviation the case of B or q: (AB/B)I(AD/D) s (AD/D)/(AH/H) = (AH/H)/(AL/L)
of
missing
..
parameters
ANALYSIS
‘5
OF THE
@
Q 0.75000
161
CN MOLECULE
378
-;. 0.7400
-
0.7300
_
0.7200
_
0.7100
-
0
% rcz
_
0 - 0.60 ‘(
_
2
20
30
40
60
60
70
80
91
1
1
I
I
I
I
I
I
I
-0.80
_
[
-
1
1
!
-“.20@
- 0.40
J
10
f
-
f
I I 0 3 I :
- LOO _ I I
- 1.20 _
I - 1.40 _
; I I
- 1.60_
2.00
_ 1
1.50 1 1.00 _
0.50
_
1
! i
0.00 0
I
I
I
I
I
1
I
10
20
30
40
50
‘60
70
f
1 80
N
91
maximum
FIG. 3. Values of y, YJ, and ~JJ f two standard deviations (in zr” = 0 of XB+) according to the maximum value of N (= N,,,) used in the fit. All the lines with N > N mar are discarded and the number of lines remaining in the fit is indicated in a. For N,, = 60 the dotted lines for 7 (in a ) and ye (in b ) are value determined with ~JJ used in the fit (value of y,, = dotted line in c ) and the full line for these two parameters is with YJJ set to zero (~JJN& cv 2.8 X 10-a cm-r, SD = 1.9 x lo+ cm-’ for both fits). For N,, = 44 the dotted line for -r (in a ) has been determined using yJ = dotted line (in b ) and the full line is the value of y with YJ set to zero (~JN~ = -3.8 x 10-3, SD = 2.5 X 10e5 cm-i for both fits).
162
CERNY ET AL.
it is absorbed in the previous order (+rJ in 7, YJJ in ?J) giving an apparent decrease of the SD and a shift of the magnitude of the remaining parameter which can get out of its expected interval. On the other hand using a non-significant parameter might give unexpected change in other parameters. For instance using PJJ in 21’= 0 or 21’= 1 (A% state) give PJJ < %PJJ and 10’ X YJJ = 6 f 16 Cm-’ (~&JJ = 16 X 10-T cm-*) instead of 10’ X ~JJ = 16 f 16 when PJJ is set to zero. In conclusion when a parameter is at the limit of being significant [condition (1) or (2)] it has been used in the fit except when it gives aberrant fitting. Second test. A certain number of lines have a difference between observed and calculated frequencies greater than three and even four SD. We have developed fits with automatic elimination of lines whose difference is greater than n SD, n being equal to three in a band by band fit. In that way the worse lines are eliminated and we have seen that the parameters are not modified in a significant way. As to the global fit (weight per branch) of 14 bands with n = 4 about 200 lines are eliminated out of 3700. The SD are obviously lowered but all the parameters defined within f2 SD are in agreement in the two fits (with and without elimination). The differences in the SD of the parameters are very often less than 10%. Since a great number of lines have differences greater than three SD, we have tried a different line weighting process. Indeed a close examination of the distribution of the lines when fitting without weighting (or with a branch weight) shows that the large deviations come for the main part from low intensity lines. Moreover with a sufficiently high signal-to-noise ratio (S/N) the uncertainty in the position of the lines is expected to be inversely proportional to S/N (14-15). Then as the noise is the same in the explored range, every line has been weighted with a weight proportional to its intensity. But it appears that the rather intense lines are defined with nearly the same precision and it is necessary to give a maximum weight from a certain value of the intensity (S/N),,,. (All the lines having greater S/N than (S/N),,, are attributed this maximum weight). Because through the central-limit theorem the tendency of the errors occurring is to be normally distributed (16), this maximum weighting value is determined by the best agreement between the experimental distribution of the residual and the theoretical Gaussian distribution (x” test) when varying (S/N),,,. In that way we found (S/N),, - 10. The different distributions are shown in Fig. 4. With a weight per line the distribution is closest to the normal one but the low intensity lines are still overweighted because they appear too often above f 1.5 SD. Nevertheless, with a weight per branch or a weight per line we found a good agreement for the parameters (within two SD). It is easy to see that the weight per branch decreases the SD for the small parameters to a normal distribution it is possible to infer that (H, PJ, qJ, . . .). By extrapolation this underestimation is of the order of -20% of the SD determined with a weight per branch. This difference is not very important and finally the branch weighting process, quicker and easier to handle, was used in the final global fit. The previous difference can give the order of magnitude of the correction we can add to the SD of the centrifugal distortion parameters determined in Tables IV and V. This non-normal distribution (mainly for weak lines) does not come from the way of measuring the position of the lines. It has been found in absorption for Iz (11) and NO
ANALYSIS
0
OF THE
163
CN MOLECULE
I 0 + 0 ,
-
- 3.570
- ZSSTD
15 STD
.?STD
FIG. 4. 0, Distribution of the residuals for 2027 lines (5 bands); A, normal distribution; mental distribution (weight per branch); -, experimental distribution (weight per (S/N),, = 10.
(10) and more recently being noted in different Delouis
in emission for P2 (17) and YO (18-19) ways from program
the position of the line
by J. Chauville
(15, 20) or H.
(22).
Third test. The centrifugal correlated
written
0, experiline) with
to the spin-rotation
distortion
of the spin-orbit
parameter
constant
-y(n) in a 211 state.
AJ is nearly totally
Then it is impossible
to
164
CERNY
ET AL.
determine A_, and y separately but it is possible to fit the data either with AJ (and y = 0) or with y (and AJ = 0) and the results must be the same if AJ and y are totally equivalent. Actually comparing the two fits a little difference appears in the two values of the same other parameters (lO_ZOyo of their SD); but for a few parameters this difference is greater than 207o and is 65% for q of v’ = 3. This non-rigourous equivalence of the two fits is probably due to the slight mathematical difference between the two models. Indeed the usual way of introducing distortion parameters like ye, PJ, qJ . . . is an approached form. Using this approximation we correlate small terms to other parameters. For instance, it is easy to see that using A.H = A + ~AJJ(J + 1) instead of the true AJ matrix elements of Table I we would correlate AJ/4 to A (negligible here with respect to AA). More complex correlations could be expected with approximate expressions for D or H. Then it is interesting to examine if the approached effective parameters like '?J, pJ, qJ . . . could give significant differences. With that aim we have tried as in the case of NO (10) to fit the data in two different ways correlating to T, A, B, and y either the Z- interactions, or the Z+ interactions. These interactions are totally correlated and only their difference is attainable. But the two related models are not exactly the same because higher order interactions are expressed in an approached form which is not exactly the same when correlating 2- or Z+. These tests have been done with a few isolated bands or group of bands and give good agreements ; but it has been seen that the parameters YJJ which are significant in the final fit are sometimes insignificant (correlation of Z+ interactions). Differences of the order of four SD for T, and D of the 211 states have been noted. Then the previous tests have shown, that the two SD given for the different parameters in the Tables IV, V, . . . do not correspond always to a confidence interval of about 9.5% probability. These SD are in general a little too low mainly for centrifugal distortion parameters, and for some other parameters like q, T, and probably r(“Z+).
DISCUSSION
The precision of the Fourier transform spectroscopy allows the determination small parameters corresponding to weak interactions. The simple model adopted describe the transition can be easily checked with usual relations.
TABLE Molecular Constants D
211
meowd Gdc"lated
*‘E+
medurd s.lcul.ted
VI
of AaD and XW
0.61434
an
He
a
0.614668
States (cm-l)
(14) x 10
-5
x 10 -5
0.641
(31) x 10-5
0.641
x 10-5
0.420 O.Mo
-11 (28) x 10 x 10-l'
0.17237 (55) x IO-' 0.16851
x 10-1
-II 0.6488 (44) x 10
0.1737135 (45) x 10
0.7366
0.1703847
I 10-1'
-1
x IO_I
of to
ANALYSIS
In Table VI are compared
165
OF THE CN MOLECULE
the measured
and calculated
values of
and 6 &&Bd ~3
Pekeris’ formula
=
--
6B,2 We >-
We
The excellent agreement shows that the usual approximations expressing the vibrationrotation interactions are valuable. Then it is possible to calculate with the same accuracy the expected value of the centrifugal distortion parameter at the equilibrium AJ(,J in the 211 state using the Merer’s relation (22) (or the nearly equivalent Veseth’s relation (8)) -D&A
A J(e) =
= 0.1684 X 10v4 cm-‘. 6B,2 aB
+
~ We
It is possible to see in Table III that this value is very close to the value determined for v = 0, since taking account of the experimental uncertainty A ~~~~~~~ = AJ eff(~ = 0) = 0.177(11) This indicates A J eff(e) - AJW
that the value of Tefr, correlated -
C(e)Yeff(e)
X 10-4 cm-‘.
to A J eff, is very small. Indeed using
with B, C(e)
=
A eff(e)
-
2B,
we have Yerr(s) = -2.6
X 10v6 cm-l.
This is orders of magnitude smaller that the same parameter in NO (10) or YO (18, 19). Then Yerr(e) could be close to the “true” spin-rotation interaction parameter (23), the other correlated interactions (Z-, A, . . . ) being very small or cancelling each other. The relation of Zare el al. (24) and Veseth (8) related to the centrifugal distortion of the A doubling constants p and q give the expected order of magnitude. For instance, for v = 0 we found (cm-r) qs = p,,:
= -2.73
X lOA (measured
= -3.8961
(52) X 10m4)
D pJ=2p(~_cJ~_
6.6 X lO-‘I (measured
QJ = -4qt
= 5.6 X 10es (measured
= -27.08
(46) X lo-*)
= 10.82 (22) X 1O-8)
v Some constants of the ground state can be compared to the very precise measurements of T. A. Dixon and R. C. Woods with microwave spectroscopy (25). This is shown in
CERNY
166
ET AL.
TABLE
VII
Comparison of Measurements of Rotational and Spin-Rotation in the XZ+ Ground State of CN Fourier transform
8”
I
spctrorc~py
Microwvs
rpctracopy
B\
56 693.241
(130)
MHz
56 693.097
( 4)
B’i
56
(130)
MHr
56
(30)
_ 8”
0
v;
170.695
522.346
222.5
2.22
170.743
MHZ
522.354
(IO)
MHZ
217.486
(33)
MHZ
“Y
‘;-vi
=
2.46
Parameters
(25)
MHz
MM
MHZ
(D)
(9)
MHZ
MHZ
Table VII. We note the very good agreement of B “1 - B”o which is known with a better accuracy than B”O or B”l in our measurements (high correlation between B”o and B”l). As ~“1 - ~“0 is at the limit of being significant we have reported the value of c+ (Table V) which is known better. But we cannot clearly explain why our value of ~“0 seems too high-about ten SD. CONCLUSION
The molecular constants of the A211 and X*Zf states of the CN molecule have been determined with precision and possible systematic errors have been extensively discussed. Very small interactions have been measured leading to a good check of the rotating oscillator model used. The whole set of lines is recalculated with a SD of 0.0025 cm-‘. So these precise data covering the 1 to 2.5 pm region can help in the identification or for the reconstitution of high resolution spectra obtained for instance in astrophysics. RECEIVED:
March
15, 1978 REFERENCES
1. N. H. KIESS AND H. P. BROIDA,J. Moi. Speclrosc. 7, 197 (1970). 2. A. E. DOUGLASAND P. M. ROUTLY, Astrophys. J. Suppl. 1,295 (1954). of the CN molecule,” Berkeley, 3. S. P. DAVIS AND J. G. PHILLIPS, “The red system (AQ-XZ) University of California, 1963. 4. G. POLEITO AND M. RIGUTTI, NUOVO Cimendo 39, 2, 519 (1965). 5. J. M. WEINBERG,E. S. FISHBURNE,AND K. NARAHARIRAO, J. Mol. Spectrosc. 22, 406 (1967). 6. T. KOTLAR AND R. W. FIELD, private communication. 7. R. BACIS, D. CERNY, J. D’INCAN, G. GUELACHVILI,AND F. Roux, Aslrophys. J. 214, 946 (1977) 8. L. VESETH,J. Phys. B. 3, 1677 (1970). 9. H. F. SCHAEFERIII AND T. G. HEIL, J. Chem. Phys. 54, 6,2573 (1971). 10. C. AMIOT, R. BACIS, ANDG. GUELACWILI, Can. J. Phys. (1978). 11. A. PERRIN, These 3e cycle, Orsay (1977). 12. C. ATEENOUR,“These Doctorat d’Etat ts Sciences Physiques no C.N.R.S.,” A.O. 22.500, Universite de Nice, 1975. 13. E. E. WHITING, J. 0. ARNOLD,AND G. C. LYLE, NASA TN D-5088, 1969. 14. J. P. MAILLARD, “These Doctorat d’Etat es Sciences Physiques no 1157,” Facult6 des Sciences d’orsay, 1973.
ANALYSIS
OF THE
CN MOLECULE
15. S. GERSTENKORN,P. Luc, A. PERRIN, AND J. CHAUVILLE,Astron.
167
Astrophys. 58, 255 (1977).
16. D. L. ALBRITTON,A. L. SCHMELTEKOPF, ANDR. N. ZARE, Molecular Spectroscopy, Modern Research, II (K. Narahari Rao, Ed.), Academic Press, New York, 1976. C. EFFANTIN,R. BACIS, C. AXIOT, ANDJ. VERGES,J. Mol. Spectrosc. 69, 79 (1978). A. BERNARD,“These de Doctorat d’Etat es Sciences Physiques,” Universite de Lyon 1, 1977. A. BERNARD,R. BACIS, AND P. Luc, to be published. J. CHAUVILLE,private communication. H. DELOUIS,These Orsay, 1973. A. J. MERER, Mol. Phys. 23, 309 (1972). S. GREEN AND R. N. ZARE, J. Mol. Spectrosc. 64, 217 (1977). R. N. ZARE, A. L. SCHMELTEKOPF, W. J. HARROP, ANDD. I,. ALBRITTON,J. Mol. Spectrosc. 46, 37 (1973). 25. T. A. DIXON ANDR. C. WOODS,J. Chem. Phys. 67, 9, 3956 (1977).
17. 18. 19. 20. 21. 22. 23. 24.
SPECTROSCOPY 73, 154-167 (1978)
Extensive Analysis of the Red System of the CN Molecule with a High Resolution Fourier Spectrometer D. CERNY,*
R. BACIS,* G. GUELACHVILI,**
AND
F. Roux*
*Unioersilt Claude Bernard (Lyon I), Luboratoire de SpectromCtrie lonique et Moltculaire (Associt au C.N.R.S.), 43, Boulevard du II Novembre 1918,69621 Villeurbanne, France, and *+Univtmitt de Paris&d, Laboratoire d'lnfra-rage,B&time& 350 Campus d’orsay, 91405 Orsay, CWex, France The red system of CN molecule emitted by a nitrous oxide-acetylene flame has been measured between 11000 and 4000 cm-r with a high resolution Fourier Spectrometer. Fourteen bands of the AV = +l, 0, - 1, and -2 sequences are reported and analysed. The molecular constants of the *II and W states are determined using a computer program written to directly reduce the data in a single fit of the whole set of lines, a standard deviation of 0.0025 cm-r was obtained. Some parameters reflecting small interactions are derived and discussed.
INTRODUCTION
The spectrum of the CN molecule is observed in the atmospheres of stars, comets, and sun. It appears in numerous chemical processes at high temperature: flames, chemical reactions, discharges, and postluminescence. Therefore, a precise knowledge of this spectrum is highly desirable for astrophysical and chemical process studies. A number of works have been done on the most intense electronic transitions B2Z+X22+ (violet system) and A%-X22+ (red system) (1-5). However, very high resolution techniques such as Fourier transform spectroscopy are more and more used for astrophysical observations which are extended through the infrared atmospheric windows. Then data laboratory of the same precision are needed for the molecules like CN. The red system of the CN molecule is emitted in a very wide region and extensive measurements mainly in the visible and near IR range can be found in the Davis and Phillips’ Atlas (3). But little is known for wavelengths greater than 1 pm. Moreover, the A211 state is perturbed by the B22+ state for the high vibrational levels and by the X22+ state for the lower ones. The study of the latter perturbation has already been extensively undertaken (6). A precise knowledge of the lower levels of the A211 state could help to complete this study. The present work is an extensive analysis with high resolution Fourier transform spectroscopy of the A211 -+ x22+ transition of the CN molecule in the 4000 to 11 000 cm-’ region. The spectrum is emitted in a C2H2/N20 flame. The experimental set-up and the recording procedure were described in a preliminary study of the (O-O) band (7). Let us only recall that the spectrum was recorded with 1450 000 spectral elements 1.54 0022-2852/78/0731-0154$02.00/O CopyrightQ 1978by Academic Press. Inc. All rights of reproduction in any form reserved.
ANALYSIS
OF THE
155
CN MOLECULE
0.2cmm’ t---4
8876
cm?
8878
cm- ’
cm-’
FIG. 1. Lines of the O-O (A% -+XZ+) transition of CN. The full-width lines (-0.080 cm-l) is mainly due to Doppler effect.
at half-maximum
of the
and a 16 X 1e3 cm-’ apparatus resolution. The linewidth is mainly due to the source broadening (T N 3000 K) as can be seen in Fig. 1. For the most intense bands all the expected satellite branches are observed (Fig. 1) but the related measurements are often not used in the final fit (overlapping with intense line, automatic elimination, . .). The absolute accuracy of the wavenumbers is one part in 10’ at 10 000 cm-‘. The explored spectral range corresponds to wavelengths lesser than 2.5 I.trn because for higher wavelengths there are strong overlappings with the very intense emission spectrum of water. Fourteen bands of AV = 1, 0, -1, -2 sequences have been analyzed. They come from the ZJ’= 0, 1, 2,3,4 levels of the upper state -4% and V” = 0, 1, 2,3, 4 of the lower state XQ+. Rotational lines until 11~CE 90 have been observed for the most intense bands. A whole fit of the 14 bands has been carried out in a one-step approach. It gives a weighted standard deviation (SD) of 2.5 X 1e3 cm-‘. Analysis of the Da&aand Experimental Results The ground state X*2+ is described Fl(,V) = BN(N F,(N)
with the usual spectral
+ 1) - D[N (N + l)]” + H[S!X
= BN(N + 1) - D[NCN + l)]’ + N[:V(Y
terms
+ I)]” + $rN + l)]” - fr(A’ + l),
where y represents the spin doubling of the *Z+ state and Fl(N) and Fz(IV)! respectively correspond to the components with N = J - $ and h'= J -I- a. As the measured lines reach high values of iV, centrifugal distortion terms are necessary to fit the data; so an effective spin-rotation parameter Yeff has been used with the usual expansion : Yeff = y + r.7N(N + 1) + rJJC.Vw
+ 1)P.
156
ET AL.
CERNY
TABLE Energy
I
Matrix
of Q State
7
P
Zn1
2n3 2
1 T,+;Ao-+~-;A
_Bfi+ZDXd?+;
+(B-A,)(Xt1)-D((X+l)2+X]
t
-
H fi(3
KVT X2 + X + 1) + A,,
6
211, + H [(X
2
+;p
+ 1)3 +X
(3 X + I)]
(lSG)+lq
-4
A,,
[ 3 (X + I)’ +X
1
q+ \r[-MT-Y]
-:*+fi+:
(x+z’zfi)+. 1
+ (B tA,)(X-1)
-D[(X-l)‘+X]
t +tl[(X-l)3+X(3X-I)]+l
2n3
A I
I +;q
X=(J-;)
[3(X-l)‘+X] ”
X
(J+;)
The A211 state is described with the 2 X 2 matrices reported in Table I. The interactions with YZ+ or 2x- states in a second order approximation lead to the A doubling. The closest states are two 2Z+ states and the 2Z- states are rather far (9). Then the A doubling is probably mainly due to 2Zf states and as usual we write p+ = pz+ - pz- ( = p) and 9+ = 9z+ - qx-(=q) and part of the 2Z- interactions are totally correlated to To, A, B, and y (20). We have taken into account of the centrifugal distortion using effective parameters such as Peff
=
p
+
PJJV
Qeff
=
q + qJ(J
+
1)
+
p.T.r[JV
+
+ 1) + QJJCJV +
l>l” l>P.
The whole set of lines was compared directly to this theoretical model. A global fitting program has been set up for the study of a few bands of the iodine spectrum (II) according to the Athenour’s method (12). It has been generalized to the case of doublet transitions for the present work. In a preliminary study every band is fitted separately. This allows the elimination of badly defined lines, mainly when there are important asymmetry or unusual intensities (overlapping of lines, proximity of ghosts, . . . ). Faint lines or branches can be calculated and then easily picked out. After this first study we determine for every branch a SD and a root mean square deviation (RMS = 1((Observed-calculated)2). This RMS has been used for weighting in the global fitting (Wbrancr, = l/ (RMS)2h,,c$. The Table II gives for every band the origine ~0 (cm-l) determined with the 14 1Use of Wbranch so calculated for a fit of the same band separately, gives new (RMS)z slightly different from the first ones, but we have checked with a few bands that the corresponding changes in the parameters are non significant.
ANALYSIS
OF THE
157
CN MOLECULE
TABLE
II
3
2 505a.wJ92
4
WL?)
mQ _, I.1065 x 10 270 linea 2.8x 10&m-' 6846.55189 (120)
10345.25114 (377)
MS.75599
(143)
3.8 x m-3,,-
2.8 x x)'
8355.4594
6392.029:
(400)
240 -2 9.9mO x a 141 linea 7.3 x 10Jcm-'
I
4
I
band global fitting, the maximum relative intensity (directly from the recording) of the lines in the Q branches, the Franck-Condon factors [quoted from Whiting et al. (13)] the number of lines used in the fit and the SD (cm-‘) in the separate fit of the band. In Tables III and IV are collected the parameters of the A*II and X22+ states of the TABLE
III
Parameters* of The CN A211 + XzZ+ Transition (cm-‘)
“=O T
Y
A
Y=
I
+ 9117.39268
(35)
- 52.6MM
(68)
- a.57602
+ 0.177 - 0.74
A,x104 AJYD9 B
+ 109a5.10355
v=2 (53)
+ 126.57.24425
(98)
- 52.X+26
(194)
111)
+ 0.176
(13)
+
(13)
(321
- 0.49
(35)
0.180
"=4
“=3 1105)
+ WO3.8029 - 52.4289 + 0.2m
(31)
+ 161U.7716
(54)
- 52.3705
(96) (155)
(30)
+ 0.473
182)
+ 1.7073145 143)
+ 1.69m415
(44)
+ 1.6727238 149)
+ 1.6553535(100)
+ I.637902 (29)
D *lo5
+ 0.61497
(13,
+ 0.61613
(13)
* 0.6,746
(I81
+ 0.61877
(66)
+0.6166
1221
HxlO"
+ 0.406
(11)
+ 0.375
(12)
+ 0.354
(21)
+ 0.372
(124)
-0.51
(46)
px IO2
+ o.e4c9
I421
+0.8400
(34)
to.5347
(58)
+ 0.858
(25)
+ 0.759
(34)
p,x IO6
- 0.2708
WI
- 0.2916
(98)
- 0.315
(22)
- 0.386
(170)
q x103
- 0.38961
(52)
- 0.39772
(95)
-0.40&1
(21)
- 0.4160
(56)
-0.4112
(69)
,,x107
+ 0.1082
(221
+ 0.1146
(50)
+ O.l3&?
(144)
+ 0.135
(31)
- 0.97
122)
- 0.95
(64)
- 0.38
(23)
13 qJJXD ' Th. numben
in porenthnn
C
or. th. unc.rtaintyin tba lastdigit,that corrnpdr
line pooitians weight par bmnch
1
Th. standad devidon
to two ~sndard deviationr computd
of the mid,,.,,~is 2 5 x 1O-3 cm-'
for the 3682 linn
using 011 the
158
CERNY
ET
TABLE
AL. IV
of the CN A*II + X*2+ Transition
Parameter@
(cm-l)
x21+Lowsr st.ts “=O T
v=
+ 2042.42261
0.0
Y
B
1
“=2 (49)
+ 4058.55176
v=3 (67)
+ 6048.34766
v=4 PI)
+ 8011.7734
(21)
+ 1.8910627 (43)
+ l.87%591(44)
+ l..mlm2(44)
+ l.e3%440
(47)
+ 1.82w468
04)
D
x105
+ 0.64072
(12)
+ 0.64b2
(13)
+ 0.64263
(13)
+ 0.64367
(IS)
+ 0.64445
(51)
H
x IO'l
+ 0.636
(II)
+ 0.608
(II)
+ 0.585
(13)
+ 0.547
(25)
+ 0.380
(118)
Y. 102
+ 0.7417
(33)
+ 0.7367
(331
+ 0.7272
(46)
+ 0.7190
(45)
+ 0.72Q
(1441
YJ" 107
- 0.70
(18)
- 0.96
(19)
- 0.97
(34)
- 0.96
(20)
- 3.0
(19)
+ 0.16
(16)
+ 0.45
(20)
+ 0.45
(53)
+ 7.8
(621
lj,x 10
11
CN molecule. The Table V gives the constants at the equilibrium calculated from these parameters. The wavenumbers experimentally measured, used in the fit, and those calculated from the theoretical model with the previous parameters have been made available.* In this compilation one can obtain the result of the fitting of two groups of seven bands: the first group is dealing with the (O-O), (O-l), (O-2), (l-O), (l-2), and (l-3) bands, the second one with (2-l), (2-3), (24), (3-2), (3-3), (3-4), and (4-4) bands. In order to identify easily the different branches, Fig. 2 recalls the relative position of the levels and the different branches with their first line for the A211 + X22+ transition of CN. Moreover, no attempt has been made to analyze the perturbation of the A*II state by levels of the ground state. The very low SD for the whole set of lines (0.0025 cm-l) shows that they are surely very weak or far from the explored region for the vibrational levels studied. Actually a few discarded lines only could be assigned to a perturbed line and with shifts lower than 10 SD (6). RESULT
ANALYSIS
This paper is a first analysis with a global fitting of an emission spectrum and as we have a great number of observed lines with precise measurements the statistical significance of the standard deviations and the related problems have been examined in detail. The global fitting may allow to determine small parameters whose contribution may be non-significant when fitting each band separately. This happens for levels which are common with several bands. It is also the case for levels defined by one (or several) unprecise bands (weak bands-for instance v” = 4 in X9+) the other level being best * This material is on deposit in the Editorial data (line frequencies, non-corrected relative available on request to one of the authors.
Office of the Journal of Molecular Spectroscopy. Original intensities, widths, asymmetries of lines) could be made
150
ANALYSIS OF THE CN MOLECULE
(Fpi
2w2I r-
2n cl
lFll
--IN' J'
72 10,
‘R27 I
Q2
101
R 7
I 35
-e
f,
-f
F2
+e
f,
+f
f2
3 v
25
m 2
1
25 15
I v
81 /0.5/
‘42 P 10.51 Q,20
Qt
1151
P, (2.51
(251
R2 1051
P,2 12.51
‘R2r 0,
(051
1051 P2 /r5:
R
Q,, /05/
“P,y /15/
FIG. 2. Relative position of the levels and first lines in the ATt + X2+ transition Under
the arrow
is noted
of the CN molecule.
the first J” value of the branch.
known from other bands (this comes from the great correlation between a number of parameters in the two levels). We have checked the precision of our results by different tests. In the first test we have searched the influence of small contribution parameters. In the second one we have examined the different types of weighting process or of elimination of poor lines. The third test will compare different types of fittings with parameters corresponding to different definitions but undiscernable at least in the second order.
160
CERNY
ET
TABLE Vibrational
and Rotational
Constants”
AL. V
(cm-‘) of the A211 and XW
States of CN
A2n A.
=
aA
=
8.
=
4
%
= 1813.419 =
%Ke
(42)
=
- 52.6072 - 0.743
(27) (33) x 10-I
1 .7159372
(22)
0. v237
(55) x 10-I
pg
=
- 0.154
(35) x IO-4
v,
=
- 0.153
(67) x lO-5
'2.8'9 ('a)
D
=
0.61434
(14) x lO-5
"1
=
0.420
(28) x IO-ll
iYD 9, 9, aq
=
0.122
(10) x 10-7
=
0.&(27
(35) x 10-2
=
- 0.3855
(13) x 10-3
=
0.83
(2(0x 10-5
= 1.232983 (4) x lo-'
'e
x21+
6
= 2068.67860
%
l&Jan,= W*Y* =
(97)
13.11735 (37) - 0.6543
(b-4).
lO-2
=
I.89977481 (23)
ag
=
0.1737135
Pe
=
*
'.
G,
=
we
Y,
=
- 0.459
=
0.641
(31’ x 10-5
Ii.
=
0.6488
(44) x 10-l’
aH
=
0.264
(28) x 10~‘~
= =
0.7463
(20) x 10-2
0.74
(11) x IO-4
l.l7lmb
(v + ;)-qxe(“+;)2
A,
=
B ”
= Be-aa(v+;)+&
A.-aA
(45) x 10-l (22) x IO-4
De
% a” =
- 0.2543
(32' x IO*
(4) x n-a
1 3 + + WJc (V + 2 )
(v+;)+.....
DV = De + aD
6
(v + y
+ ;
n"
=He-*'+("+;j
P,
= P, - ap(v*;)+_
4,
= q* - uq
(v +;
+ YB”+l’
I 3
+....
1+ ._._
)+
y,= Ve-ap(Y+;k+~Y(“+;)2 First test. The introduction of parameters with a small contribution is necessary since lines with high values of J (up to 90) are measured. But what is the higher order of the distortion parameters we have to use in the fit? We have considered that a parameter is at the limit of being significant: (1) when its magnitude is within once and twice its SD,3 and/or (2) when its maximum contribution is within one and two SD of the whole fit. Figure 3 shows the variations of the parameters y, yJ, and yJJ (0” = 0 of X’Z+) and of their SD (AT,AYJ, ATJJ) according to the maximum value of N used in the fit. Then we see that when we do not introduce a parameter at the limit of being significant in
* A simple empirical rule allows a direct estimate of the standard deviation the case of B or q: (AB/B)I(AD/D) s (AD/D)/(AH/H) = (AH/H)/(AL/L)
of
missing
..
parameters
ANALYSIS
‘5
OF THE
@
Q 0.75000
161
CN MOLECULE
378
-;. 0.7400
-
0.7300
_
0.7200
_
0.7100
-
0
% rcz
_
0 - 0.60 ‘(
_
2
20
30
40
60
60
70
80
91
1
1
I
I
I
I
I
I
I
-0.80
_
[
-
1
1
!
-“.20@
- 0.40
J
10
f
-
f
I I 0 3 I :
- LOO _ I I
- 1.20 _
I - 1.40 _
; I I
- 1.60_
2.00
_ 1
1.50 1 1.00 _
0.50
_
1
! i
0.00 0
I
I
I
I
I
1
I
10
20
30
40
50
‘60
70
f
1 80
N
91
maximum
FIG. 3. Values of y, YJ, and ~JJ f two standard deviations (in zr” = 0 of XB+) according to the maximum value of N (= N,,,) used in the fit. All the lines with N > N mar are discarded and the number of lines remaining in the fit is indicated in a. For N,, = 60 the dotted lines for 7 (in a ) and ye (in b ) are value determined with ~JJ used in the fit (value of y,, = dotted line in c ) and the full line for these two parameters is with YJJ set to zero (~JJN& cv 2.8 X 10-a cm-r, SD = 1.9 x lo+ cm-’ for both fits). For N,, = 44 the dotted line for -r (in a ) has been determined using yJ = dotted line (in b ) and the full line is the value of y with YJ set to zero (~JN~ = -3.8 x 10-3, SD = 2.5 X 10e5 cm-i for both fits).
162
CERNY ET AL.
it is absorbed in the previous order (+rJ in 7, YJJ in ?J) giving an apparent decrease of the SD and a shift of the magnitude of the remaining parameter which can get out of its expected interval. On the other hand using a non-significant parameter might give unexpected change in other parameters. For instance using PJJ in 21’= 0 or 21’= 1 (A% state) give PJJ < %PJJ and 10’ X YJJ = 6 f 16 Cm-’ (~&JJ = 16 X 10-T cm-*) instead of 10’ X ~JJ = 16 f 16 when PJJ is set to zero. In conclusion when a parameter is at the limit of being significant [condition (1) or (2)] it has been used in the fit except when it gives aberrant fitting. Second test. A certain number of lines have a difference between observed and calculated frequencies greater than three and even four SD. We have developed fits with automatic elimination of lines whose difference is greater than n SD, n being equal to three in a band by band fit. In that way the worse lines are eliminated and we have seen that the parameters are not modified in a significant way. As to the global fit (weight per branch) of 14 bands with n = 4 about 200 lines are eliminated out of 3700. The SD are obviously lowered but all the parameters defined within f2 SD are in agreement in the two fits (with and without elimination). The differences in the SD of the parameters are very often less than 10%. Since a great number of lines have differences greater than three SD, we have tried a different line weighting process. Indeed a close examination of the distribution of the lines when fitting without weighting (or with a branch weight) shows that the large deviations come for the main part from low intensity lines. Moreover with a sufficiently high signal-to-noise ratio (S/N) the uncertainty in the position of the lines is expected to be inversely proportional to S/N (14-15). Then as the noise is the same in the explored range, every line has been weighted with a weight proportional to its intensity. But it appears that the rather intense lines are defined with nearly the same precision and it is necessary to give a maximum weight from a certain value of the intensity (S/N),,,. (All the lines having greater S/N than (S/N),,, are attributed this maximum weight). Because through the central-limit theorem the tendency of the errors occurring is to be normally distributed (16), this maximum weighting value is determined by the best agreement between the experimental distribution of the residual and the theoretical Gaussian distribution (x” test) when varying (S/N),,,. In that way we found (S/N),, - 10. The different distributions are shown in Fig. 4. With a weight per line the distribution is closest to the normal one but the low intensity lines are still overweighted because they appear too often above f 1.5 SD. Nevertheless, with a weight per branch or a weight per line we found a good agreement for the parameters (within two SD). It is easy to see that the weight per branch decreases the SD for the small parameters to a normal distribution it is possible to infer that (H, PJ, qJ, . . .). By extrapolation this underestimation is of the order of -20% of the SD determined with a weight per branch. This difference is not very important and finally the branch weighting process, quicker and easier to handle, was used in the final global fit. The previous difference can give the order of magnitude of the correction we can add to the SD of the centrifugal distortion parameters determined in Tables IV and V. This non-normal distribution (mainly for weak lines) does not come from the way of measuring the position of the lines. It has been found in absorption for Iz (11) and NO
ANALYSIS
0
OF THE
163
CN MOLECULE
I 0 + 0 ,
-
- 3.570
- ZSSTD
15 STD
.?STD
FIG. 4. 0, Distribution of the residuals for 2027 lines (5 bands); A, normal distribution; mental distribution (weight per branch); -, experimental distribution (weight per (S/N),, = 10.
(10) and more recently being noted in different Delouis
in emission for P2 (17) and YO (18-19) ways from program
the position of the line
by J. Chauville
(15, 20) or H.
(22).
Third test. The centrifugal correlated
written
0, experiline) with
to the spin-rotation
distortion
of the spin-orbit
parameter
constant
-y(n) in a 211 state.
AJ is nearly totally
Then it is impossible
to
164
CERNY
ET AL.
determine A_, and y separately but it is possible to fit the data either with AJ (and y = 0) or with y (and AJ = 0) and the results must be the same if AJ and y are totally equivalent. Actually comparing the two fits a little difference appears in the two values of the same other parameters (lO_ZOyo of their SD); but for a few parameters this difference is greater than 207o and is 65% for q of v’ = 3. This non-rigourous equivalence of the two fits is probably due to the slight mathematical difference between the two models. Indeed the usual way of introducing distortion parameters like ye, PJ, qJ . . . is an approached form. Using this approximation we correlate small terms to other parameters. For instance, it is easy to see that using A.H = A + ~AJJ(J + 1) instead of the true AJ matrix elements of Table I we would correlate AJ/4 to A (negligible here with respect to AA). More complex correlations could be expected with approximate expressions for D or H. Then it is interesting to examine if the approached effective parameters like '?J, pJ, qJ . . . could give significant differences. With that aim we have tried as in the case of NO (10) to fit the data in two different ways correlating to T, A, B, and y either the Z- interactions, or the Z+ interactions. These interactions are totally correlated and only their difference is attainable. But the two related models are not exactly the same because higher order interactions are expressed in an approached form which is not exactly the same when correlating 2- or Z+. These tests have been done with a few isolated bands or group of bands and give good agreements ; but it has been seen that the parameters YJJ which are significant in the final fit are sometimes insignificant (correlation of Z+ interactions). Differences of the order of four SD for T, and D of the 211 states have been noted. Then the previous tests have shown, that the two SD given for the different parameters in the Tables IV, V, . . . do not correspond always to a confidence interval of about 9.5% probability. These SD are in general a little too low mainly for centrifugal distortion parameters, and for some other parameters like q, T, and probably r(“Z+).
DISCUSSION
The precision of the Fourier transform spectroscopy allows the determination small parameters corresponding to weak interactions. The simple model adopted describe the transition can be easily checked with usual relations.
TABLE Molecular Constants D
211
meowd Gdc"lated
*‘E+
medurd s.lcul.ted
VI
of AaD and XW
0.61434
an
He
a
0.614668
States (cm-l)
(14) x 10
-5
x 10 -5
0.641
(31) x 10-5
0.641
x 10-5
0.420 O.Mo
-11 (28) x 10 x 10-l'
0.17237 (55) x IO-' 0.16851
x 10-1
-II 0.6488 (44) x 10
0.1737135 (45) x 10
0.7366
0.1703847
I 10-1'
-1
x IO_I
of to
ANALYSIS
In Table VI are compared
165
OF THE CN MOLECULE
the measured
and calculated
values of
and 6 &&Bd ~3
Pekeris’ formula
=
--
6B,2 We >-
We
The excellent agreement shows that the usual approximations expressing the vibrationrotation interactions are valuable. Then it is possible to calculate with the same accuracy the expected value of the centrifugal distortion parameter at the equilibrium AJ(,J in the 211 state using the Merer’s relation (22) (or the nearly equivalent Veseth’s relation (8)) -D&A
A J(e) =
= 0.1684 X 10v4 cm-‘. 6B,2 aB
+
~ We
It is possible to see in Table III that this value is very close to the value determined for v = 0, since taking account of the experimental uncertainty A ~~~~~~~ = AJ eff(~ = 0) = 0.177(11) This indicates A J eff(e) - AJW
that the value of Tefr, correlated -
C(e)Yeff(e)
X 10-4 cm-‘.
to A J eff, is very small. Indeed using
with B, C(e)
=
A eff(e)
-
2B,
we have Yerr(s) = -2.6
X 10v6 cm-l.
This is orders of magnitude smaller that the same parameter in NO (10) or YO (18, 19). Then Yerr(e) could be close to the “true” spin-rotation interaction parameter (23), the other correlated interactions (Z-, A, . . . ) being very small or cancelling each other. The relation of Zare el al. (24) and Veseth (8) related to the centrifugal distortion of the A doubling constants p and q give the expected order of magnitude. For instance, for v = 0 we found (cm-r) qs = p,,:
= -2.73
X lOA (measured
= -3.8961
(52) X 10m4)
D pJ=2p(~_cJ~_
6.6 X lO-‘I (measured
QJ = -4qt
= 5.6 X 10es (measured
= -27.08
(46) X lo-*)
= 10.82 (22) X 1O-8)
v Some constants of the ground state can be compared to the very precise measurements of T. A. Dixon and R. C. Woods with microwave spectroscopy (25). This is shown in
CERNY
166
ET AL.
TABLE
VII
Comparison of Measurements of Rotational and Spin-Rotation in the XZ+ Ground State of CN Fourier transform
8”
I
spctrorc~py
Microwvs
rpctracopy
B\
56 693.241
(130)
MHz
56 693.097
( 4)
B’i
56
(130)
MHr
56
(30)
_ 8”
0
v;
170.695
522.346
222.5
2.22
170.743
MHZ
522.354
(IO)
MHZ
217.486
(33)
MHZ
“Y
‘;-vi
=
2.46
Parameters
(25)
MHz
MM
MHZ
(D)
(9)
MHZ
MHZ
Table VII. We note the very good agreement of B “1 - B”o which is known with a better accuracy than B”O or B”l in our measurements (high correlation between B”o and B”l). As ~“1 - ~“0 is at the limit of being significant we have reported the value of c+ (Table V) which is known better. But we cannot clearly explain why our value of ~“0 seems too high-about ten SD. CONCLUSION
The molecular constants of the A211 and X*Zf states of the CN molecule have been determined with precision and possible systematic errors have been extensively discussed. Very small interactions have been measured leading to a good check of the rotating oscillator model used. The whole set of lines is recalculated with a SD of 0.0025 cm-‘. So these precise data covering the 1 to 2.5 pm region can help in the identification or for the reconstitution of high resolution spectra obtained for instance in astrophysics. RECEIVED:
March
15, 1978 REFERENCES
1. N. H. KIESS AND H. P. BROIDA,J. Moi. Speclrosc. 7, 197 (1970). 2. A. E. DOUGLASAND P. M. ROUTLY, Astrophys. J. Suppl. 1,295 (1954). of the CN molecule,” Berkeley, 3. S. P. DAVIS AND J. G. PHILLIPS, “The red system (AQ-XZ) University of California, 1963. 4. G. POLEITO AND M. RIGUTTI, NUOVO Cimendo 39, 2, 519 (1965). 5. J. M. WEINBERG,E. S. FISHBURNE,AND K. NARAHARIRAO, J. Mol. Spectrosc. 22, 406 (1967). 6. T. KOTLAR AND R. W. FIELD, private communication. 7. R. BACIS, D. CERNY, J. D’INCAN, G. GUELACHVILI,AND F. Roux, Aslrophys. J. 214, 946 (1977) 8. L. VESETH,J. Phys. B. 3, 1677 (1970). 9. H. F. SCHAEFERIII AND T. G. HEIL, J. Chem. Phys. 54, 6,2573 (1971). 10. C. AMIOT, R. BACIS, ANDG. GUELACWILI, Can. J. Phys. (1978). 11. A. PERRIN, These 3e cycle, Orsay (1977). 12. C. ATEENOUR,“These Doctorat d’Etat ts Sciences Physiques no C.N.R.S.,” A.O. 22.500, Universite de Nice, 1975. 13. E. E. WHITING, J. 0. ARNOLD,AND G. C. LYLE, NASA TN D-5088, 1969. 14. J. P. MAILLARD, “These Doctorat d’Etat es Sciences Physiques no 1157,” Facult6 des Sciences d’orsay, 1973.
ANALYSIS
OF THE
CN MOLECULE
15. S. GERSTENKORN,P. Luc, A. PERRIN, AND J. CHAUVILLE,Astron.
167
Astrophys. 58, 255 (1977).
16. D. L. ALBRITTON,A. L. SCHMELTEKOPF, ANDR. N. ZARE, Molecular Spectroscopy, Modern Research, II (K. Narahari Rao, Ed.), Academic Press, New York, 1976. C. EFFANTIN,R. BACIS, C. AXIOT, ANDJ. VERGES,J. Mol. Spectrosc. 69, 79 (1978). A. BERNARD,“These de Doctorat d’Etat es Sciences Physiques,” Universite de Lyon 1, 1977. A. BERNARD,R. BACIS, AND P. Luc, to be published. J. CHAUVILLE,private communication. H. DELOUIS,These Orsay, 1973. A. J. MERER, Mol. Phys. 23, 309 (1972). S. GREEN AND R. N. ZARE, J. Mol. Spectrosc. 64, 217 (1977). R. N. ZARE, A. L. SCHMELTEKOPF, W. J. HARROP, ANDD. I,. ALBRITTON,J. Mol. Spectrosc. 46, 37 (1973). 25. T. A. DIXON ANDR. C. WOODS,J. Chem. Phys. 67, 9, 3956 (1977).
17. 18. 19. 20. 21. 22. 23. 24.