1 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 < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1987 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
2 An J. Physique 48 (1987) OCTOBRE 1987, 1685 Classification Physics Abstracts K Description of the absorption spectrum of bromine recorded by means of Fourier transform spectroscopy : the (B 303A00+u ~ X 1 03A3+g) system S. Gerstenkorn, P. Luc, A. Raynal and J. Sinzelle Laboratoire Aimé Cotton (*), C.N.R.S. II, Bâtiment 505, Orsay Cedex, France (Reçu le 6 fgvrier 1987, revise le 20 mai 1987, accept6 le 29 mai 1987) Résumé L analyse du spectre d absorption de la molécule de brome représenté par le système (BX ) Br2 et enregistré par spectroscopie par Transformée de Fourier est présentée. On montre que les transitions enregistrées couvrant le domaine cm1 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 cm1 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 cm1 en accord avec l incertitude estimée des mesures expérimentales Abstract. in extenso analysis of the (BX ) Br2 bromine absorption spectrum recorded by means of Fourier Transform Spectroscopy is presented. It is shown that the recorded transitions covering the cm1 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 cm1 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 cm1 in agreement with the experimental uncertainties. 1. Introduction. The successful description of the absorption spectrum of iodine belonging to the (BX) system  encouraged us to undertake the same work on the 79Br2 molecule. Although numerous and extensive studies of the (BX) bromine system have already been made , in the paper of Barrow Clark, Coxon and Yee  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 data, as was found in the iodine case . 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 cm 1 where the (BX) system is located has been explored, and about 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. . Since the measurements of the wavenumbers given by the F.T.S. method are one order of magnitude more reliable  than those obtained with conventional spectroscopy techniques , 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) . Article published online by EDP Sciences and available at
3 Temperature 1686 Fortunately, thanks to Hutson s work  published in 1981 and program , 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 BornOppenheimer 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] (IPApotential) using C.R. Vidal program . 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 (± cm1 ). 2. Experimental. The experimental setup 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  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. ), 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 . 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 cm 1 to cm 1 with, however, a small discontinuity located around the wavenumber UL cm1. In this region, the local noise due to the fluctuations of the HeNe laser beam used for monitoring in the Fourier spectrometer  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 . However, at present, the absolute wavenumber values of only two transitions of the bromine spectrum are known with precision. They are the (P ± cm 1 transi (17, 7) 79Br2) tion and the (P 129 (12, 4)) cm 1 transition belonging to the 8IBr2 molecule species (Eng and Latourette ). Being situated in the region saturated by the UL 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. Table I. 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 .
4 Calibration non The lead In 1687 Fig. 1. of the bromine 7 Br, spectrum (A) by comparison with an absorption spectrum produced by 9Br2 and 127I2 molecules (B). The iodine line u cm 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 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 , 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 ± 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 to cm 1 in an atlas form (2 volumes) and are available by order from the Laboratoire Aime Cotton . 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 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 Assignments and precision of the measurements. 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. , 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 cm1 which corresponds to an average uncertainty of (0.0014/ J2) cm cm1 on the vertex position of the measured lines. A total of 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 , which is about In other words, the selection criteria blended and symmetrical lines 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 cm1 from the dissociation limit (at cm1 ) ; 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  Least Square AnalysisMethod. order to determine the vibrational and rotational molecular constants, the global fit of the 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 . The energy levels E (v, J) are given by  : where The fitting process concerns only the vibrational E(v,o) and rotational B(v) constants. Of course, the
5 . A2F 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. (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.
6 1689 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 cm 1 and 107 cm1 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 . 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 to cm1 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 (BX) system of 12 . 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 , c) differential equation method of Hutson , d) determination of the experimental polynomials for «compact» representation of the computed CDC according Le Roy , e) substraction of the quantities ( D,, K2+ H,, K3 + L,, K4+ M,, k5) yields to «distortionfree wavenumbers» , f) solution of the 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 Y Yif (v + 1/2)i . The input data for the leastsquare fit are the 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 op(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
7 Observed The 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 «distortionfree» wavenumbers . 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 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  curve may be constructed  and used to generate centrifugal distortion constants [9, 28]. An iterative approach is then necessary to obtain a selfconsistent set of vibrational, rotational and centrifugal distortion constants . However transitions connected to rotational levels with J values situated near the full lines MU K 5 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 constants, which take account of the neglected higher Nv, 0 v... constants (see Ref. ), 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 cml Fig. 4. data field of the X state. The contribution of the Lv" constants can be neglected, the data field being outside the full line LU K CM (K J(J + 1 )). Fig. 3. Observed data field of the B state. The data field analysed is limited by the full line Mv KS cm1. effective constants in the Including the Mv kmv 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 cm 1. The analysis of the residuals (u cal. 0" mes) leads to conclusions similar to those obtained in the case of the iodine spectrum , 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  Molecular constants an2i «compact representation». 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 (BX) system ; the CDC, are not independent parameters since they are deter
8 Dunham Dunham 1691 Table V. coefficients describing the vibrational (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 digits necessary to recalculate the wavenumbers of the transition belonging to the (BX ) Br2 system are the followings (given in parentheses) : Yio(12) and Yil(ll). Table VI. coefficients Yi 2 and Yi 3 describing 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
9 1692 mined from the values of G (v ) and B (v ) . Table V is central to our work ; it gives the most compact representation of the absorption (BX ) Br2 spectrum. Indeed, by means of the above 38 Dunham coefficients, the wavenumbers of more than the 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 (BX) 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  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 to the standard error of the differences , equal 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 wavenumbers of which were previously determined by Eng and Latourette . 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 cm1 ). Note that the molecular constants of the gibr2 molecule have been deduced from the classical isotopic relation  : Table IX. A test of the molecular constants. Comparison between calculated and absolute measured wavenumbers made by Eng and La Tourette . Remark : for the P 57 (17, 7) 79Br2 transition the contribution kmv (J (J + 1»5 Mv* K 5 is cm1 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.
10 i. the FranckCondon 1693 and with M where p (M79/M81)1/2; and M , p 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 . 3.2 FRANCKCONDON FACTORS AND INTENSITY MEASUREMENTS. Agreement between experimental and calculated relative intensities should also be valuable confirmation of the band assignments e. vibrational numbering as shown by R. N. Zare . The intensity labs (v, v ", J") of an absorbed line is given by the classical relation : where u v", J" is the wavenumber of the line, Sj, j,, the HollLondon rotational line strength , 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 strength I A, 1 2and a FranckCondon Factor : If we assume that the average electronic transition strength remains constant for all the recorded bands in the cm 1 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 FranckCondon 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, 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  directly usable in our UNIVAC machine , 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. Fig. 5. factors of the (v, v ") bands for 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 leastsquare 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.
11 Intensities Comparison In 1694 Fig. 6. of the recorded lines belonging to the (3, 10) band around J Intensity measurements and vibrational num bering. 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 at J 67. Moreover the observed intensities of the are weak bands in the cm 1 region (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 the F.C.F. values because the Boltzmann factor Nv,,j,, 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.  is 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 the (47, 0) band ; at room temperature the maximum of the intensity distribution should lay at J 31. between the measured relative in Fig. 7. tensities of the band progression (v, 10) where v 0, 1, 2, 3 and the calculated ones, the previous numbering  being adopted. (Open circles : mesured intensities ; full circles calculated intensities). 48, the band as shown in figure 8, but for v > 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.  and it is not an isolated feature, being also observed in the iodine absorption spectrum . Here, again, the J dependence on the calculated F.C.F., alone does not explain these observations. 4. Discussion. As in the study of the (BX) system of iodine  the recalculation of the whole absorption spectrum due
12 1695 Fig. 9. J dependence of the observed intensities in the (51, 0) band, the maximum of the intensities shifted towards low J values, occurs at J ~ 18. to the (BX) system of Br2 is now possible within the experimental uncertainties. «Experimental uncertainties» are the uncertainties with which the vertex position of the measured lines are estimated. For moderately intense lines these uncertainties are of the order of ± cm1 (see Sect ), 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. , Sect. 3.3). Therefore, as usual, a good estimation of the mean value of the «experimental uncertainties» is given by the standard deviation &, between recalculated and measured wavenumbers, provided that &, which was found to be equal to cm1, remains comparable to the accuracy (± cm1 ) 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 cm1) is somewhat smaller than that obtained in the iodine case where & was found to be equal to cm1. 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  than the iodine ones ; it follows that the widths resulting from the Doppler, hyperfine structure and apparatus contributions of the observed bromine lines are to the widths of the iodine lines! comparable Accordingly, the accuracy of the wavenumber measurements of the bromine spectrum is a little higher, ± cm 1 for lines of medium intensity, (Tab. III) instead of ± cm1 for lines of comparable intensities (see Tab. II, Ref. ), in the iodine case. Perturbations of different kinds , apparently exist in the bromine spectrum too. Indeed, it was not 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. . The shaded region represents the estimated uncertainties associated with the Fourier spectroscopic measurements. possible, with our data, to construct an IPA potential 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 Cn/R: (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 . Comparison between molecular constants determined in this work, and those obtained by B.C.C.Y. using classical photographic methods , 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 . 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 . 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 , 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 (BX) system of bromine remains unresolved. 5. Conclusion. As in the iodine case, the (BX) system of bromine has been analysed entirely by means of the simple
13 1696 oscillator model in the framework of the Bornwhich seems to be Oppenheimer approximation, adequate for the analysis of the data field represented in figures 3 and 4 ; in this case the (BX) 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 recorded lines  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 (BX ) 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]. This may be due to experimental errors owing to the lack of precise standards of bromine wavenumbers  or to the presence of hidden perturbations leading to the failure of the BornOppenheimer 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.  GERSTENKORN, S. and LUC, P., J. Physique 46 (1985)  BROWN, W. G., Phys. Rev. 38 (1931)  HORSLEY, J. A. and BARROW, R. F., Trans. Faraday Soc. 63 (1967) 32.  COXON, J. A., J. Mol. Spectrosc. 37 (1971)  COXON, J. A., J. Quant. Spectrosc. Radiat. Transfer. 12 (1972)  BARROW, R. F., CLARK, T. C., COXON, J. A. and YEE, K. K., J. Mol. Spectrosc. 51 (1974)  ALBRITTON, D. L., SCHMELTEKOPF, A. L. and ZARE, R. N., Molecular spectroscopy, Modern Research, ed. K. Narahari Rao (Academic Press, New York) 1976, Vol. II, p  GERSTENKORN, S. and LUC, P., Nouv. Rev. Opt. 7 (1976) 149.  HUTSON, J. M., J. Phys. B 14 (1981)  HUTSON, J. M., Program CDIST, référence 435, Quantum Chemistry Program Exchange, Indiana University, Bloomington, Indiana ; and private communication.  KosMAN, W. M. and HINZE, J., J. Mol. Spectrosc. 56 (1975)  VIDAL, C. R. and SCHEINGRABER, H., J. Mol. Spectrosc. 65 (1977)  VIDAL, C. R. (Private communication).  GERSTENKORN, S. and LUC, P., «Atlas du spectre d absorption de la molécule d iode», ISBN et «Complément», ISBN , (Edition du CNRS, Paris).  GERSTENKORN, S., LUC, P. and SINZELLE, J., J. Physique 41 (1980)  GERSTENKORN, S. and Luc, P., Laser Chem. 1 (1983).  GERSTENKORN, S., LUC, P. and VETTER, R., Revue Phys. Appl. 16 (1981)  ENG, R. S. and LA TOURETTE, J., J. Mol. Spectrosc. 52 (1974)  GERSTENKORN, S., LUC, P. and RAYNAL, A., Atlas du spectre d absorption de la molécule de Brome, References 2 vol. available at Aimé Cotton Laboratory, Bât. 505, Campus d Orsay, CNRS II, Orsay, France.  HERZBERG, G., Spectra of Diatomic Molecules (Van Nostrand, New York)  PIQUE, J. P., HARTMANN, F., CHURASSY, S. and BACIS, R., J. Physique 47 (1986)  LE ROY, J. P., in Semiclassical Methods in Molecular Scattering and Spectroscopy, M. S. Child (Ed) (D. Reidel Dordrecht, Netherlands 1980) ; and private communication.  BARWELL, M. G., Thesis (1976) University of Waterloo, Waterloo, Ontario, Canada and TROMP, J. W. and LE ROY, R. J., Can. J. Phys. 60 (1982)  DUNHAM, J. L., Phys. Rev. 41 (1932)  BROWN, J. D., BURNS, G. and LE ROY, R. J., Can. J. Phys. 51 (1973) and TELLINGHUISEN, J., MC KEEVER, M. R. and ABHA SUR, J., Mol. Spectrosc. 82 (1980)  RYDBERG, R., Z. Phys. 73 (1931) 376 ; 80 (1933) 514 ; KLEIN, O. Z. Phys. 76 (1932) 225 ; REES, A. L. G., Proc. Phys. Soc. 59 (1947) 948.  TELLINGHUISEN, J., Comput. Phys. Commun. 6 (1984)  ALBRITTON, D. L., HARROP, N. J., SCHMELTEKOPF, A. L. and ZARE, R. N., J. Mol. Spectrosc. 46 (1973) 25.  HUTSON, J. M., GERSTENKORN, S., LUC, P. and SINZELLE, J., J. Mol. Spectrosc. 96 (1982)  GERSTENKORN, S. and Luc, P., Molecular constants describing the (BX) system and dissociation limits of the three isotopic bromine species 79Br2, 79,81Br2, 79,81Br2 and 81Br2, to be published.  ZARE, R. N., J. Chem. Phys. 40 (1964)  HOUGEN, J., The calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules. Monograph n 115 (N.B.S.), Washington (1970).  HOUGEN, J. (Private communication).
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