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>
HAL Id: jpa-00210608
https://hal.archives-ouvertes.fr/jpa-00210608
Submitted on 1 Jan 1987
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1987,
1685
Description of the absorption spectrum of bromine recorded by
means
J.
Physique
48
(1987)
1685-1696
OCTOBRE
Classification
Physics Abstracts
32.20K
31.90
-
(B 303A00+u
of Fourier transform spectroscopy : the
S. Gerstenkorn, P. Luc, A.
Laboratoire Aimé Cotton
(Reçu
le 6
fgvrier 1987,
(*),
Raynal
~
X 1 03A3+g)
system
and J. Sinzelle
C.N.R.S. II, Bâtiment 505, 91405
revise le 20 mai 1987,
accept6 le 29
mai
Orsay Cedex,
France
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-1et 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
14 et de l’état B jusqu’à v’
52 (niveau situé à
vibrationnelles et rotationnelles des états X jusqu’à 03BD"
5,3 cm-1de 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.
An in extenso analysis of the (B-X ) Br2 bromine absorption spectrum recorded by means of
Abstract.
Fourier Transform Spectroscopy is presented. It is shown that the 80 000 recorded transitions covering the
11600-19 577 cm-1range 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-1from 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-1in agreement with the
experimental uncertainties.
2014
=
1. Introduction.
The successful
trum of iodine
description of the absorption specbelonging to the (B-X) system [1]
encouraged
to undertake the same work on the
79Br2
us
as was found in the iodine case [1]. Therefore
have recorded the bromine absorption spectrum
again by means of Fourier Transform Spectroscopy
molecule. The range
(F.T.S.) using the isotopic
11 600-19 600 cm-11 where the (B-X) system is located has been explored, and about 80 000 transitions 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
data,
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 polynomials if the whole range of observed v values is
considered. The origin of this difficulty may be
ascribed either to the existence of local perturbations, or to a lack of precision of the experimental
we
79Br2
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
1686
Fortunately, thanks to Hutson’s work [9] published 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 described in references [14, 15]. Three different absorption 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
52. From
0 to v’
whole B state from v’
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 measurements 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-11 to
19 577 cm-11 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 intense 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 spectrum are known with precision. They are the (P 57
(17, 7) 79Br2) 15 798.0037 ± 0.00013 cm-11 transition and the (P 129 (12, 4)) = 15 798.0247 +
0.00013 cm-11 transition belonging to the 8IBr2
molecule species (Eng and Latourette [18]).
Being situated in the region saturated by the
1
UL = 15 798.0 cm-1 laser radiation, the measurements 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.
=
=
Temperature of the absorption cells necessary to observe the (v’, v") bands connected to the
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].
Table I.
v" levels
-
1687
measurements and connected matter such as assignments 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.
and
precision
of the
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 differences A2F (J") is 0.0014 cm-1 which corresponds to
cm- 1 =
an average uncertainty of (0.0014/
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
4
lines
out
of
5.
The
about
156
reject
(v’, v") bands
all
the
well
of
encompass nearly
depth 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].
3.1.1
Assignments
measurements.
-
=
J2)
-
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-11 is taken as the reference line and the
bromine lines calibrated in this way are indicated by
-
=
dotted lines.
cell containing 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
published in extenso from
analysed in this work were
11 600 to 19 577 cm-11 in an atlas form (2 volumes)
and are available by order from the Laboratoire
Aime Cotton [19].
The first spectrum, A,
corresponds to
a
-
=
=
=
In order to
3.1.2. Least Square Analysis-Method.
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
3. Results.
3.1 POSITION MEASUREMENTS. - Section 3 is split
into two parts ; the first one (3.1) deals with position
The
E(v,o)
fitting process concerns only the vibrational
and rotational B(v) constants. Of course, the
1688
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.
accuracy
ð.2F (J")
A2F (J") differences and estimate of the
of the wavenumber measurements.
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.
-
1689
of this method is based on the assumption that
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-11 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 perturbations 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
use
the
exponential polynomials :
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 polynomials 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, withouth loss of precision. Indeed these constants increase 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 Yif (v + 1/2)i [24]. The input data
Y
obtained from the values calculated by Hutson’s
method. The exponential form provides an adequate
where
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).
=
and To, o is the distance between the ground levels
v"
J 0 of the X state and the level v’
0,
J 0 of the B state, the unknowns being the
=
=
0,
=
=
1690
molecular constants. If the CDC values of the B
state are known from theory, the centrifugal distortion 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
=
0.001 cm-l1 (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 conwhich
take
account
the
of
stants,
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 contribution 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 leastsquares 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 representation ».
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 coefficients are needed to recalculate the observed absorption spectrum of the (B-X) system ; the CDC, are
not independent parameters since they are deter-
Fig. 3. - Observed data field
field analysed is limited by
0.001 cm-1.
of the B state. The data
the full line - Mv KS
=
1691
Dunham coefficients describing the vib(Yi 0) and rotational (Yi 1) molecular constants of the X state (valid up to v" = 14) and of the B
state (valid up to v’
52). The number of significant
to
the wavenumbers of the
recalculate
digits necessary
transition belonging to the (B-X ) Br2 system are the
followings (given in parentheses) : Yio(12) and
Table V.
-
rational
=
Table VI.
Dunham coefficients Yi 2 and Yi 3 dethe
D"
and H" molecular constants, and
scribing
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) :
-
Yil(ll).
Table VII.
- Energies (Ev), rotational (Bv) and CDC values for the 13 and
X states
1692
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 spectrum. 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. However, 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 exponential 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 programs are available from us, at Aime Cotton Laboratory). 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].
the vibrational and rotational
of the uncertainties
in
the
aE (v )
G (v ) constants, or more precisely in
the E (v ) constants defined as :
3.1.4
Accuracy of
constants. - An upper limit
can
be taken,
[1], equal
as we
have shown in the iodine
case
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
and 9B (v) of the vibrational energies E(v) and of
rotational constants Bv, respectively.
(v)
the
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 wavenumbers 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-1and 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.
=
1693
and
(M79/M81)1/2;
where p
with M79
78.918332
and M81= 80.916306, p
0.987577.
The molecular constants describing the 81Br2 and
79,81Brz molecules, i.e. the vibrational, rotational
and CDC, constants, will be published in a separated
=
=
=
paper
[30].
3.2 FRANCK-CONDON
FACTORS AND INTENSITY
between experimental and calculated relative intensities should
also be valuable confirmation of the band assignments
i. e.
the vibrational numbering as shown
by R. N. Zare [31]. The intensity labs (v’, v ",
J") of an absorbed line is given by the classical
relation :
MEASUREMENTS. - Agreement
-
-
where u v", J" is the wavenumber of the line, Sj, j,, the
Holl-London rotational line strength [32], g’ and
g" are the electronic degeneracies, N v",J" the population of the (v",J") level proportional to the
Boltzmann factor. The last factor can be represented
by the product of an average electronic transition
a Franck-Condon Factor :
strength I A , 1 2and
If
that the average electronic transition
remains constant for all the recorded bands
in the 12 600-13 200 cm -11 region, then the ratio of
the intensities of two lines depends on only three
quantities, namely (T v", J", Nv,, j,, and the FCF values.
we assume
strength
3.2.1 Franck-Condon Factors (F.C.F.). - F. C. F.
values were calculated for 672 bands corresponding
to 0 v " -- 14 and 0 -- v’ -- 48, together with their
rotational dependence for J = 0, 25, 50... up to
J = 150. These values are listed in a separate volume
also available from the Aime Cotton Laboratory.
Figure 5 represents the F.C.F. values grouped according to v’ progressions and for J 0. These
F.C.F. values were calculated by means of F.C.F.
programs [28] directly usable in our UNIVAC 1 190
machine [33], the input data being the IPA potential
of the X state (Tab. II) and the RKR potential of the
B state. However, near the dissociation limit, above
v’
48, reliable F.C.F. values could not be calculated, and additional work in this region is needed
in order to improve the RKR potential
or, much
better, to determine the potential by means of the
IPA method.
=
=
-
Franck-Condon factors of the (v’, v ") bands for
5.
0 , v" =s;; 48 and 0 v" ,14 corresponding to the rotationless states. Full circles : bands listed in table II are taken
into account in the least-square fits. Open circles : weak
bands identified but not used as input data in the fits. For
the FCF dependence on J, see text. Three regions are
delimited by the two curves 2li and d2- In the regions A, B
and C, the temperature of the cells must be at least 20 °C,
250 °C and 750 °C respectively, in order to observe the
bands represented by full circles.
Fig.
-
1694
Fig.
6.
-
Intensities of the recorded lines
belonging to the (3, 10)
Intensity measurements and vibrational numbering. In our experiments, in order to observe
the (v’, 10) progression where v’
0, 1, 2. The
1.1 m cell (3 torr) was heated at 750 °C ; in this case
the maximum intensity of the bromine bands occurs
3.2.2
-
=
at J
67. Moreover the observed intensities of the
bands in the 12 600-13 600 cm -1 region are weak
(see for example the (3, 10) band, Fig. 6). The
absorption remains linear and it is possible to
determine the relative intensities of the (v’, 10)
bands without the knowledge of absorption coefficients. Comparison between experimental and calculated intensities requires only the knowledge of
=
band around J
=
67.
the F.C.F. values because the Boltzmann factor
is common to all the (v’, 10) bands. Figure 7
shows the comparison between computed and
measured intensities : good agreement is found
when the numbering established by B.C.C.Y. [6] is
Nv,,j,,
adopted.
On the other hand, at room temperature, the
maximum of the (v’, 0) bands should lie around
J 31. This is approximately the case for the (47, 0)
=
Fig. 8. - J dependence of the observed intensities in
(47, 0) band ; at room temperature the maximum of
intensity distribution should lay at J 31.
the
the
=
band as shown in figure 8, but for v’ &#x3E; 48, the
maximum is shifted to lower J values : for the (51, 0)
band the maximum occurs at J ~ 18 (Fig. 9). This
effect was also observed by B.C.C.Y. [6] and it is not
an isolated feature, being also observed in the iodine
absorption spectrum [1]. Here, again, the J dependence on the calculated F.C.F., alone does not
explain these observations.
,
7.
Comparison between the measured relative intensities of the band progression (v’, 10) where v’
0, 1,
2, 3 and the calculated ones, the previous numbering [6]
being adopted. (Open circles : mesured intensities ; full
circles calculated intensities).
Fig.
-
=
4. Discussion.
study of the (B-X) system of iodine [1] the
recalculation of the whole absorption spectrum due
As in the
1695
Fig. 10. Differences between the wavenumbers of origins of the bands a 0, v’ deduced from Fourier spectroscopy
measurements and from those published by Barrow et al.
[6]. The shaded region represents the estimated uncertainties associated with the Fourier spectroscopic measure-
Fig. 9. - J dependence of the
(51, 0) band, the maximum
ments.
towards low J values,
possible, with our data, to construct an IPA potential
to the
occurs
observed intensities in the
of the intensities shifted
at J ~ 18.
(B-X) system of Br2 is now possible within the
uncertainties. « Experimental uncer-
experimental
tainties
the uncertainties with which the vertex
the measured lines are estimated. For
intense lines these uncertainties are of
the order of ± 0.001 cm-1 (see Sect. 3.1.1), and of
course, these uncertainties depend, among others
parameters, mainly on the signal/noise ratio of each
line. The average value of the experimental uncertainties for the whole spectrum is not easy to be
determined, a priori, because, for example, at the
two extremities of the spectrum « hot » and « weak »
bands are present (for more details, see Ref. [1],
Sect. 3.3). Therefore, as usual, a good estimation of
the mean value of the « experimental uncertainties »
is given by the standard deviation &#x26;, between
recalculated and measured wavenumbers, provided
that &#x26;, which was found to be equal to 0.0016 cm-1,
remains comparable to the accuracy (± 0.001 cm-1 )
on the vertex position of moderately intense lines,
which is indeed, the case. But the standard deviation
6 between recalculated and measured wavenumbers
in the Br2 case (û = 0.0016 cm-1) is somewhat
smaller than that obtained in the iodine case where
&#x26; was found to be equal to 0.002 cm-1. This
discrepancy is easily explained by the fact that the
profiles of the Br2 lines are more symmetrical than
those of the 12 lines, because the hyperfine structure
of the bromine lines is about 1/3 narrower [18] than
the iodine ones ; it follows that the widths resulting
from the Doppler, hyperfine structure and apparatus
contributions of the observed bromine lines are
comparable to the widths of the iodine lines !
Accordingly, the accuracy of the wavenumber measurements of the bromine spectrum is a little higher,
1
± 0.001 cm-1 for lines of medium intensity, (Tab.
III) instead of ± 0.0016 cm-1 for lines of comparable
intensities (see Tab. II, Ref. [1]), in the iodine case.
Perturbations of different kinds [1], apparently exist
in the bromine spectrum too. Indeed, it was not
»
are
position of
moderately
for the B state, even limited to low J values.
Therefore the analysis of the long range potential of
the bromine B state should be postponed until
results of laser spectroscopy studies near the dissociation limit will be available. However, in mean time
the classical analysis (using the multipole expansion
expression G(v) = De (n 5, 6, 8 and 10)
where De and the four Cn values are considered as
free parameters) of the RKR long range potential
curve has been made and the results of this work will
be published, together with the molecular constants
of the glBr2 and 79,81 Brz molecules [30]. Comparison
between molecular constants determined in this
work, and those obtained by B.C.C.Y. using classical
photographic methods [6], must be restricted to the
values of the vibrational constants : firstly because
the models used in these two studies (five and three
term expression for the rotational levels) were
different and secondly, because E, (or Gv) constants
are practically free from correlation effects [7].
Figure 10 shows the differences of the wavenumbers
of band origins (0" v’, 0), deduced from Fourier Spectroscopy merasurements and those published in
B.C.C.Y.’s paper [6]. This plot shows that the two
sets of measurements are sometimes substantially
different, but if we add the fact that the Fourier
measurements allow the representation of the vibrational constants in Dunham expansion series, while
photographic measurements do not [6], then it
becomes obvious that Fourier measurements have to
be adopted. However, representation of molecular
constants by Dunham series is not sufficient to
assure that perturbations are not present in the
spectrum, as long as an IPA potential cannot be
constructed, and therefore the problem of the search
of perturbations on the (B-X) system of bromine
remains unresolved.
Cn/R:
=
5. Conclusion.
As in the iodine case, the (B-X) system of bromine
has been analysed entirely by means of the simple
1696
oscillator model in the framework of the BornOppenheimer approximation, which seems to be
adequate for the analysis of the data field represented in figures 3 and 4 ; in this case the
(B-X) Br2 system can be recalculated, within experimental uncertainties, provided that the vibrational
and rotational constants are known ; in other words
the recalculation of the wavenumbers of the 80 000
recorded lines [19] required the knowledge of only
38 Dunham coefficients, plus one empirical scaling
factor defining the effective Mv constants describing
both the X state (up to v"= 14) and the B state (up
to v= 52). The « compact » representation of the
(B-X ) Br2 system was obtained assuming that
« effective » vibrational and rotational constants are
sufficient to describe the B state ; however it was not
possible to compute a quantum potential of the B
state covering the entire potential well using the IPA
method
[11, 12].
owing to
This may be due to experimental
the lack of precise standards of
bromine wavenumbers [18] or to the presence of
hidden perturbations leading to the failure of the
errors
Born-Oppenheimer approximation.
Acknowledgments.
We wish to express our gratitude to all those, in
particular to .Mr J. B. Johanin and his operator team
of the UNIVAC 1190 of the computer centre of the
Faculte d’Orsay (PSI Paris IX), who helped us to
achieve the present work.
We are much obliged to Y. D’Aignaux for his
assistance in programming and data processing problems. We would also like to express our sincere
gratitude to H. Calvignac and B. D6marets who
have prepared the graphs, figures and tables for this
paper.
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