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Description of the absorption spectrum of bromine recorded by means of Fourier transform spectroscopy :
the (B 3Π0+ u ← X 1 Σ+g) system
S. Gerstenkorn, P. Luc, A. Raynal, J. Sinzelle
To cite this version:
S. Gerstenkorn, P. Luc, A. Raynal, J. Sinzelle. Description of the absorption spectrum of bromine recorded by means of Fourier transform spectroscopy : the (B 3Π0+ u← X 1 Σ+g) system. Journal de Physique, 1987, 48 (10), pp.1685-1696. �10.1051/jphys:0198700480100168500�. �jpa-00210608�
Description of the absorption spectrum of bromine recorded by means
of Fourier transform spectroscopy : the
(B 303A00+u
~X 1 03A3+g)
systemS. Gerstenkorn, P. Luc, A. Raynal and J. Sinzelle
Laboratoire Aimé Cotton (*), C.N.R.S. II, Bâtiment 505, 91405 Orsay Cedex, France (Reçu le 6 fgvrier 1987, revise le 20 mai 1987, accept6 le 29 mai 1987)
Résumé. 2014 L’analyse du spectre d’absorption de la molécule de brome représenté par le système (B-X ) Br2 et enregistré par spectroscopie par Transformée de Fourier est présentée. On montre que les 80 000 transitions enregistrées couvrant le domaine 11600-19 577 cm-1 et publiées sous forme d’un atlas peuvent être recalculées au moyen de 39 constantes : 38 étant les coefficients de Dunham servant à décrire les constantes vibrationnelles et rotationnelles des états X jusqu’à 03BD" = 14 et de l’état B jusqu’à v’ = 52 (niveau situé à 5,3 cm-1 de la limite de dissociation) plus un coefficient empirique permettant de tenir compte des constantes de distorsions négligées (supérieures à M03BD). L’erreur quadratique moyenne entre les nombres d’ondes recalculés et mesurés est trouvée égale à 0,0016 cm-1 en accord avec l’incertitude estimée des mesures
expérimentales.
Abstract. 2014 An in extenso analysis of the (B-X ) Br2 bromine absorption spectrum recorded by means of
Fourier Transform Spectroscopy is presented. It is shown that the 80 000 recorded transitions covering the
11600-19 577 cm-1 range and published in an atlas form may be recalculated by means of only 39 constants : 38 are Dunham coefficients describing the vibrational and rotational constants of both X state (up to
03BD" = 14) and B state (up to 03BD’ = 52, situated only at 5.3 cm-1 from the dissociation limit of the B state), and
one empirical scaling factor which takes account of neglected centrifugal constants higher than M03BD. The overall standard error between computed and measured wavenumbers is equal to 0.0016 cm-1 in agreement with the
experimental uncertainties.
Classification
Physics Abstracts
31.90 - 32.20K
1. Introduction.
The successful description of the absorption spec-
trum of iodine belonging to the (B-X) system [1]
encouraged us to undertake the same work on the
79Br2 molecule. Although numerous and extensive studies of the (B-X) bromine system have already
been made [2-6], in the paper of Barrow Clark,
Coxon and Yee [6] referred as B.C.C.Y. in this text,
some points still needed to be improved ; for example their experimental vibrational G(v’) and
rotational B(v’) cannot be fitted by simple polyno-
mials if the whole range of observed v values is considered. The origin of this difficulty may be ascribed either to the existence of local pertur- bations, or to a lack of precision of the experimental
data, as was found in the iodine case [1]. Therefore
we have recorded the bromine absorption spectrum again by means of Fourier Transform Spectroscopy (F.T.S.) using the isotopic 79Br2 molecule. The range 11 600-19 600 cm-1 1 where the (B-X) system is lo- cated has been explored, and about 80 000 transi- tions belonging to 156 bands have been identified.
The reduction and the analysis of the data were
performed according to the recommendations of D.L. Albritton et al. [7]. Since the measurements of the wavenumbers given by the F.T.S. method are one order of magnitude more reliable [8] than those
obtained with conventional spectroscopy techniques [6], the correct model capable of representing the
rotational levels must be, at least, a five term
expression Bv K - Dv K2 + Hv K3 + Lv K4 + Mv K5
(K = J(J + 1 )), instead of a three term expression as
used in B.C.C.Y.’s paper (1974) [6].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100168500
Fortunately, thanks to Hutson’s work [9] pub-
lished in 1981 and program [10], we now have the possibility of making an a priori calculation of the
high Centrifugal Distorsion Constants (CDC,), DU, Hv, Lv and MU, provided that accurate potentials
curves are available and that the Born-Oppenheimer approximation remains valid in the studied regions.
In addition, throughout this work a quantum
mechanical potential curve describing the X state up to v" = 14 was determined by means of the
« inverted. perturbation approach » method [11, 12]
(IPA-potential) using C.R. Vidal program [13].
Thus, in section 3, the molecular constants of the X state deduced from a quantum mechanical potential
are given, while, for the B state, only effective
molecular constants can be given.
Finally, it will be shown that, according to theory, only vibrational and rotational constants are needed to recalculate the whole observed spectrum within the experimental uncertainties (± 0.0016 cm-1 ).
2. Experimental.
The experimental set-up is similar to the one de- scribed in references [14, 15]. Three different absorp-
tion cells were built, the length of which were 0.25 m, 1.0 m and 1.1 m, respectively. While the
first two were filled with bromine (79Br2) at a
pressure of 1 torr, the pressure was adjusted to
3 torr in the last one. In all the experiments one single pass was sufficient to observe the absorption
spectrum and table I gives the temperatures at which the cells had to be used [4] in order to cover the
whole B state from v’ = 0 to v’ = 52. From v’ = 53 up to the dissociation limit, (the last bound
vibrational level being v’ = 59 according to
B.C.C.Y. [6]), absorption spectroscopy by means of
the F.T.S. method failed and fluorescence measure- ments have to be used, as in the case of the iodine
studies [16]. For the X state, only the low levels v" of this state can be observed, and even when the 3 torr, 1.1 m cell was heated at a temperature of 750 °C, the highest vibrational level observed was
v" = 14. The difficulties in reaching the low levels of the B states are due to the rapid decrease of the Franck Condon Factors (F. C. F. ) of the (v’, 0) bands
when v’ decreases, while it is the impossibility of populating appreciably all the levels of the X state,
by thermal heating alone, which is responsible for
the limitation to v " = 14, encountered in the study
of the bromine absorption spectrum.
Briefly, the absorption spectrum of the 79Br2
molecule was recorded from 11600 cm-1 1 to 19 577 cm-1 1 with, however, a small discontinuity
located around the wavenumber UL = 15 798.0 cm-1. In this region, the local noise due to the fluctuations of the He-Ne laser beam used for
monitoring in the Fourier spectrometer [14] is in-
tense enough to obscure the bromine absorption spectrum. Therefore, this region will be studied in the future by laser spectroscopy techniques as was
done for iodine [17].
However, at present, the absolute wavenumber values of only two transitions of the bromine spec- trum are known with precision. They are the (P 57 (17, 7) 79Br2) = 15 798.0037 ± 0.00013 cm-1 1 transi- tion and the (P 129 (12, 4)) = 15 798.0247 +
0.00013 cm-1 1 transition belonging to the 8IBr2
molecule species (Eng and Latourette [18]).
Being situated in the region saturated by the
UL = 15 798.0 cm-1 1 laser radiation, the measure-
ments of Eng and Latourette cannot be used directly
to calibrate our Fourier spectra, but they will be
used later to test the molecular constants determined in this work, by comparing the measured and
« calculated » wavenumbers of these two transitions.
Accordingly, it was only possible to calibrate the bromine spectrum indirectly, as shown in figure 1.
Table I. - Temperature of the absorption cells necessary to observe the (v’, v") bands connected to the
v" levels of the X state and to the v’ levels of the B state, together with the corresponding explored spectral regions. Temperature, length and pressure of the cells are also given in details in the bromine atlases [19].
Fig. 1. - Calibration of the bromine 7’Br, spectrum (A) by comparison with an absorption spectrum produced by
’9Br2 and 127I2 molecules (B). The iodine line u =
19 194.6090 cm-1 1 is taken as the reference line and the bromine lines calibrated in this way are indicated by
dotted lines.
The first spectrum, A, corresponds to a cell contain-
ing only 79Br2 molecules, while the second spectrum
B corresponds to the absorption of 79Br2 and 127I2 molecules. Given the absolute values of the wavenumbers of transitions belonging to iodine [1],
the absolute values of the wavenumbers of some
bromine transitions were deduced. The accuracy of the absolute values of the wavenumbers of bromine determined in this manner are estimated to be of the order of ± 0.005 cm- 1 ; but the errors in the relative values of the wavenumbers are estimated to be much lower (see next section).
Finally, the recordings of the absorption spectrum analysed in this work were published in extenso from
11 600 to 19 577 cm-1 1 in an atlas form (2 volumes)
and are available by order from the Laboratoire Aime Cotton [19].
3. Results.
3.1 POSITION MEASUREMENTS. - Section 3 is split
into two parts ; the first one (3.1) deals with position
measurements and connected matter such as assign-
ments and accuracy of the measurements, least square analysis and determination of molecular constants ; the second one (3.2) is devoted to F.C.F.
calculations and to intensity measurements.
3.1.1 Assignments and precision of the
measurements. - The number of assigned lines for
each of the 156 selected (v’, v") bands, as well as the
minimum and the maximum J values observed in each R and P branch, are given in table II. Estimates of the uncertainties of the measured wavenumbers
can be obtained in several ways, but as explained previously (Ref. [1], Part III, page X) we prefer to
consider the A2F(J") differences. Table III gives an example of a series of 22 pairs of moderately intense lines, where the 02F (J" ) = UR (I - 1) - ap (J + 1 )
difference has been calculated from 22 levels with 17 J’ , 46. The standard deviation of the differ-
ences A2F (J") is 0.0014 cm-1 which corresponds to
an average uncertainty of (0.0014/ J2) cm- 1 =
0.001 cm-1 on the vertex position of the measured
lines.
A total of 16 914 lines were assigned to the
selected 156 (v’, v") bands represented in table II ;
this number is to be compared with the total number of lines recorded in the bromine atlas [19], which is
about 80 000. In other words, the selection criteria
- non blended and symmetrical lines - lead us to reject about 4 lines out of 5. The 156 (v’, v") bands
encompass nearly all the well depth of the B state
from v’ = 0 to v’ = 52, the last observed v’ = 52 vibrational level being situated only 5.3 cm-1 from
the dissociation limit (at 19 579.6 cm-1 [6]) ; only
fifteen vibrational levels belonging to the ground
state are involved in our absorption study, which
covers about 1/4 of the well depth of the X state [6].
3.1.2. Least Square Analysis-Method. - In order to
determine the vibrational and rotational molecular constants, the global fit of the 16 914 assigned lines
was done following the method described in recent papers. We recall that in this method the values of
D, H, L, and M which are needed to calculate the
energies of the rovibrational levels E (v, J ) are not
« experimental » values but are those obtained from
theory [9]. The energy levels E (v, J) are given by [20] :
where
The fitting process concerns only the vibrational
E(v,o) and rotational B(v) constants. Of course, the
Table II. - Minimum (Jmin)’ maximum (Jmax), J
values and number of assigned lines selected in the
R (N1 ) and P(N2) branches of the 156 analysed
bands (N3 = N, + N2)-
Table II (continued).
Table III. - A2F (J") differences and estimate of the
accuracy of the wavenumber measurements.
ð.2F (J") = UR (J" - 1) - O’P(J" + 1 ) ; last column :
.
Table IV. - IPA potential of the X state : eigen
values G(v), expectation (B(v)) values, Rmin and Rmax for v" = 0 to v" =14.
use of this method is based on the assumption that
the analysed X and B states can be described in
terms of a single rotationless potential curve. This requirement can be considered to be fulfilled for the lower part of the rotationless potential of the X state containing the fifteen first vibrational levels ; indeed,
the IPA potential (Table IV) up to v" = 14 is in excellent agreement with the preliminary RKR
curve. The eigenvalues G (v") and the expectation
values B (v") reproduce the experimental ones within
0.001 cm-1 1 and 10-7 cm-1 respectively. But in the
case of the B state the situation is different : near the dissociation limit perturbations of different origins
can be present as in the iodine spectrum [21].
However, in the global fit of the data, these pertur-
bations will be ignored : indeed, in the iterative procedure the vibrational and rotational constants
are essentially considered as free parameters ; the
principal aim of the global fit of the data being to attempt to describe the whole observed spectrum
from 11 600 to 19 577 cm-1 with a minimum number of parameters. Accordingly, the vibrational G(v")
and rotational B (v") constants belonging to the first
fifteen levels of the ground state can be considered
as « true » molecular constants while the G (v’ ) and B (v’ ) constants of the B state, must be considered as
« effective » constants.
A flow diagram of the iteration procedure is
shown in figure 2. This procedure is essentially the
same as that used in the analysis of the (B-X) system of 12 [1]. The values of CDC, taken for the B state (0 v’ 52 ) in the least squares fits came from
exponential polynomials :
obtained from the values calculated by Hutson’s
method. The exponential form provides an adequate
Fig. 2. - The iterative procedure : a) origin of the data
(Fourier Transform Spectroscopy), b) RKR program of J.
Tellinghuisen [27], c) differential equation method of
Hutson [9], d) determination of the experimental polyno-
mials for « compact » representation of the computed
CDC according Le Roy [22], e) substraction of the
quantities ( - D,, K2+ H,, K3 + L,, K4+ M,, k5) yields to
« distortion-free wavenumbers » [25], f) solution of the 16 914 simultaneous equations with 106 unknowns (53 Gp
and 53 B,).
representation of the calculated CDC values, with-
outh loss of precision. Indeed these constants in-
crease rapidly at high v and finally diverge at the
dissociation limit ([22, 23]).
The vibrational and rotational constants (as well
as the CDC of the ground state up to v" = 14) are accurately represented by the classical Dunham
expansion series Y Yif (v + 1/2)i [24]. The input data
for the least-square fit are the 16 914 measured wavenumbers which obey equation (3) or (4). These expressions are derived from equations (1) and (2),
with AJ = -1 for UR(J) and OJ = + 1 for o-p(J).
where and To, o is the distance between the ground levels
v" = 0, J = 0 of the X state and the level v’ = 0,
J = 0 of the B state, the unknowns being the
molecular constants. If the CDC values of the B state are known from theory, the centrifugal distor-
tion contributions (quantities in square bracket in
equations (3) and (4)) can be substracted from the
« raw » measured wavenumbers crp(7) and ap(J) leading to « distortion-free » wavenumbers [25]. A
further simplification of the system is obtained by assuming that the grand state constants are well known and equal to those deduced from the IPA
potential (Tab. IV). Finally it remains to fit a system
containing 16 914 corrected wavenumbers associated with 106 parameters : the 53 vibrational Ev’ constants
and the 53 rotational constants Bv, belonging to the
B state with 0 , v’ , 52. The principal problem
consists in determining good initial Ev,, and BU,
values in order to start the iterative procedure (Fig. 2). For this purpose a preliminary least square fit was made with the raw measured wavenumbers where only the molecular constants Ev,, Bv,, Dv’ and Hv, are taken into account. The centrifugal distortion
constants Lv, and Mv, are too small for empirical determination, hence they were set equal to zero in
the preliminary fit.
Once a set of Ev, and Bv, constants are known,
their Dunham expansion parameters are determined and a RKR [26] curve may be constructed [27] and
used to generate centrifugal distortion constants [9, 28]. An iterative approach is then necessary to obtain a self-consistent set of vibrational, rotational
and centrifugal distortion constants [29].
However transitions connected to rotational levels with J values situated near the full lines - MU K 5 =
Fig. 3. - Observed data field of the B state. The data field analysed is limited by the full line - Mv KS =
0.001 cm-1.
0.001 cm-l 1 (K = J(J + 1) in Fig. 3) require higher
distortion constants than Mv, if they are be accurately
recalculated. By means of effective Mv = kMv con-
stants, which take account of the neglected higher Nv, 0 v ... constants (see Ref. [29]), in which k is an
empirical scaling factor found to be equal to 4.4, it
was possible to handle the whole field of data (Tab.
II and Figs. 3 and 4) in one sweep.
Fig. 4. - Observed data field of the X state. The contri- bution of the Lv" constants can be neglected, the data field
being outside the full line - LU K4 = 0.001 CM- (K = J(J + 1 )).
Including the Mv = kMv effective constants in the fits, the procedure represented in figure 2 converges
rapidly and only two iterations were required. The resulting overall standard error û between the
computed wavenumbers and the measured ones was
0.0016 cm- 1.
The analysis of the 16 914 residuals (u cal. - 0" mes)
leads to conclusions similar to those obtained in the
case of the iodine spectrum [1], which do not
therefore need to be repeated here ; briefly, the analysis of the data made by unweighted least-
squares fits does not introduce noticeable bias and the molecular constants can be considered as MVLU
(minimum variance linear unbiased) estimates [7].
3.1.3 Molecular constants an2i « compact represen- tation ». - The final Dunham coefficients for the
G, and Bv expansion of the B for 0 v ’ * 52 state and those describing the X state for 0 v 14 are
given in table V. Briefly, only 38 Dunham coeffi- cients are needed to recalculate the observed absorp-
tion spectrum of the (B-X) system ; the CDC, are
not independent parameters since they are deter-
Table V. - Dunham coefficients describing the vib-
rational (Yi 0) and rotational (Yi 1) molecular con-
stants of the X state (valid up to v" = 14) and of the B
state (valid up to v’ = 52). The number of significant digits necessary to recalculate the wavenumbers of the
transition belonging to the (B-X ) Br2 system are the
followings (given in parentheses) : Yio(12) and Yil(ll).
Table VI. - Dunham coefficients Yi 2 and Yi 3 de- scribing the D" and H" molecular constants, and
expansion coefficients of the polynomial exponent describing the CDC, constants of the B state. The
number of significant digits are the following : (given
in parentheses) :
Table VII. - Energies (Ev), rotational (Bv) and CDC values for the 13 and X states
mined from the values of G (v ) and B (v ) [9]. Table
V is central to our work ; it gives the most compact
representation of the absorption (B-X ) Br2 spec- trum. Indeed, by means of the above 38 Dunham coefficients, the wavenumbers of more than the 80 000 recorded lines contained in the bromine atlas,
can be recalculated within experimental error. How-
ever, these recalculations involve the use of Huston’s program which gives access to the necessary CDC, ; but, a posteriori, CDC, can also be represented in a
« compact » form by the coefficients of their expo- nential polynomials : they are given in table VI
(which contains also the Dunhan coefficients for
Dv,, and Hv")’ Finally table VII presents, in extenso, the molecular constants appearing in equations (3)
and (4) for 0 v’ 52 and 0 , v" ,14.
Thus the calculation of the molecular constants by
means of a simple computer program can be done by
use of the coefficients of table V and table VI which in turn permits the recalculation of the wavenumbers of the whole (B-X) Br2 system. (Such simple prog-
rams are available from us, at Aime Cotton Labora-
tory). Table VII is useful for people who need to identify a few transitions as frequently occurs in laser spectroscopy, and also enables one to check the calculated molecular constants deduced either from the use of the coefficients given in tables V and VI or
from the use of Hutson’s program [9].
3.1.4 Accuracy of the vibrational and rotational
constants. - An upper limit of the uncertainties
aE (v ) in the G (v ) constants, or more precisely in
the E (v ) constants defined as :
can be taken, as we have shown in the iodine case
[1], equal to the standard error of the differences between the recalculated wavenumbers (O’cal.) by
the molecular constants and the measured ones
(u mes.)’ encountered in the analysed bands.
Similarly the 9B(v) uncertainties correspond to
Table VIII. - Estimates of the uncertainties 9E (v)
and 9B (v) of the vibrational energies E(v) and of the
rotational constants Bv, respectively.
changes in B (v ) values which induce variations of the order of aE (v ) on the rotational energies. The
uncertainties given in table VIII appear to us to be much more realistic than the associated uncertainties
resulting from the global fits of the data which are
deemed small, as usual [7, 25].
A test of the accuracy of the molecular constants derived in this work can be made by computing the
wavenumbers of the two transitions P 57 (17, 7)
79Br2 and P 129 (12, 4) 81Br2, the absolute wavenum-
bers of which were previously determined by Eng
and Latourette [18]. Table IX compare measured and calculated wavenumbers. The agreement is
quite good, i.e., the differencies are within two standard deviations ( ± (1.6 x 2) x 10- 3 cm-1 ) .
Note that the molecular constants of the giBr2
molecule have been deduced from the classical
isotopic relation [20] :
Table IX. - A test of the molecular constants. Comparison between calculated and absolute measured wavenumbers made by Eng and La Tourette [18]. Remark : for the P 57 (17, 7) 79Br2 transition the contribution kMv (J (J + 1»5 == Mv* K 5 is 0.00004 cm-1 and is completely negligible. The transition P 129 (12, 4) 8’Br2 lay outside the explored data field (see Figs. 3 and 4) : the value of the empirical coefficient k is probably no longer valid. The calculated wavenumbers quoted in the table was computed with a value of
k = 4.4.
