The study of the pressure broadening phenomena in the far wings of the emission and absorption spectra of the gaseous mixture has in recent years gained a lot of attention from theoretical physicist, spectroscopisist and astrophysicist. In fact, such broadening is arrise from the interactions of radiator atom, molecule or ion with the neighboring particles of the gas or plasma. Accurate simulations or measurements of the collisionally broadened spectra lines and the knowledge of the possible satellite peak positions can provide informations on the atom-atom interaction, temperature, density and composition of the gas studied.^[1,2] Furthermore after the discovery of brown dwarfs and extrasolar giant planets which their atmosphere are dominated by alkli-metal,^[3–6] the pressure broadened line profile of alkli-metal atoms perturbed by parent or foreign atoms, have been of particular in the last two decades extensively studied by theoretical^[5–10] and experimental^[11–14] groups.

The aim of this investigation is to determine quantum mechanically the sodium–lithium (NaLi) photoabsorbtion spectra, while the radiating Na ( ) atoms are perturbed by the ground lithium Li (2s) atoms. For this purpose we will first construct carefully the Na (3s) + (2s) and Na (3p) + Li (2s) potential–energy curves (PECs) and the required transition dipole moments (TDMs). We will thereafter attempt to evaluate the quality of the constructed (PECs) and (TDMs) through the computation of the NaLi ro-vibrational levels of each electronic molecular states and the radiative lifetimes of the excited states. The constructed NaLi (PECs) and (TDMs) will be then used to carry out the reduced-photoabsorption coefficients to analyze the satellite futures (intensity and position) in the far wings, and to look at the effect of temperature on the shape of the spectra.

The pressure broadening of the sodium ( ) resonance absorption line, when the radiating Na (3s/3p) interacting with ground lithium Li (2s) atoms, is attribute to the singlet transitions from the NaLi ground state to the first and excited molecular symmetries, and to the triplet transitions from the ground state to the excited and states. From molecular spectroscopy^[1,2] it is well known that during any photoabsorption process, the transitions from one ground molecular symmetry to the corresponding excited state are governed, in general by four types of transitions namely, bound–bound (bb), bound–free (bf), free–bound (fb), and free–free (ff). For the case of the NaLi quasimolecule, as it will be seen in the next section, the singlet , , and states are quite bound, both transitions and are then expected to contribute with the all possible transitions. Concerning the triplet transitions, the ground potential curve has a deeper well, whereas the excited and curves are purely repulsive we then consider solely the bound–free and free–free transitions of the and contributions. The pressure broadening phenomena in the far wing photoabsorption spectra of sodium Na ( ) caused by lithium perturber, is described for an absorbed frequency ν and at a given temperature T, by the reduced-photoabsorption coefficient which should be the sum of the four absorption coefficients corresponding to different types of transitions. In this study, we adopt the reduced-photoabsorption coefficients formula of Chung et al.^[15,16] which have been given in detail in our previous works.^[17,18] If we denote by double and single prime the initial and final states, and assuming that where J is the total angular momentum, the free–bound reduced-absorption coefficient for transitions from the all lower continuum to a set of ro-vibrational levels of the upper states is^[16]

Likewise, the reduced absorption coefficients corresponding to the remaining type of transitions are given by the following set of expressions

The function of the form appearing in Eq. (4) is approximated in this computation by the inverse of the frequency bin size and is the transition frequency defined by

In the matrix elements the symbols , defined each internuclear separation R, means the electronic transition dipole moment that connect the lower and upper states involved in the allowed transitions, and are respectively the energy- and space-normalized radial wave functions, determined by solving numerically the radial wave equation

To be able to carry out the reduced-absorption coefficients for the NaLi system we must construct carefully the potential-energy curves for the considered molecular states and the transition dipole moments for allowed transitions.

3.1. Potential energy curves (PECs)

The molecular symmetries through which a ground sodium atom Na (3s) interacts with a ground Li (2s) are the singlet or the triplet states. When the sodium atom is in the first excited state Na (3p) and interact with the ground Li (2s), the two atoms form a quasimolecule in one of the four , , and excited molecular states. To build the above six states in the entire range of interatomic distance R we used, in the intermediate region going from to the accurate ab-initio data points provided to us by Aubert–Frécon M.^[19] We completed the potential energy curves analytically at the short-range and the long-range by using the Born Mayer Potential ^[20] and the dispersion potential , respectively. Where α and β are two constant parameters that have been computed by fitting and listed in Table 1, and the C_n (n=6, 8, 10) are the dispersion coefficients given by Marinescu and Sadeghpour.^[21] These coefficients are for the ground states, C₆=1.459×10³, C₈= 9806×10⁴ and C₁₀=91276×10⁶, while for the excited molecular states C₆=10648×10³ and C₈= 13333×10⁴ and for the ^1,3Π ones C₆=10849×10³ and C₈=1199×10⁴. The constructed Na (3s/3p) + Li (2s) (PECs) are plotted and compared with computed data of Schmidt et al.^[22] in Fig. 1.

Fig. 1.

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	Fig. 1. Na (3s) + Li (2s) and Na (3p) + Li (2s) potential–energy curves V(R) (upper graphes) and potential-differences (lower graphes) versus internuclear separation (R). All singlet and triplet curves are compared with ab-initio data from Schmidt-Mink I et al.[22] The potential differences are converted into wavelengths.

Table 1.

Table 1.

Table 1. Potentials short-range constant parameters (in unit a.u. The unit a.u. is short for atomic unit). . Constant parameters NaLi molecular states A 1Σ α 47.879 1.6245 2.4471 1.86563 1.7312 1.86569 β 2.22765 0.87182 1.0411 0.93505 0.863743 0.878159

Table 1. Potentials short-range constant parameters (in unit a.u. The unit a.u. is short for atomic unit). .

The overall agreement is seen to be good. The spectroscopic constants such as dissociation energy D_e, the equilibrium position R_e, and the minimum-to-minimum electronic excitation energy T_e, derived from our constructions are assembled in Table 2 and compared with theoretical data of Schmidt–Mink I et al.,^[22] Mabrouk N and Berriche H,^[23] and Petsalakis I D et al.,^[24] and measured values of Kappes M M et al.^[25] In the present constructions the triplet excited states and are both found to be predominantly repulsive. Two another theoretical investigations^[23,24] revealed that these two states are mainly repulsive. Furthermore, we have in particular plotted in panel (b) of Fig. 1 the potential differences , , , and . According to the semiclassical theory,^[2] a satellite structure is expected to be observed in the absorption spectra when the potential-difference curves presents an extremum. The position of such extremum, converted in terms of the wavelengths, predicted the position of the possible satellites. Therefore, the curve in Fig. 1(b), predicts that one satellite future should appear in the red wing from the transitions close to the wavelength . Two others satellites are expected to be occurred in the blue wing, one from the transitions near the wavelength λ ∼569 nm and the second from the transitions at approximately .

Table 2.

Table 2.

Table 2. Comparison of the spectroscopic constants of the ground and excited NaLi potentials with experimental and theoretical values. . Molecular states De/cm−1 Re/a0 Te/cm−1 References 7005.83 2.883 — this work 7057 2.89 — Ref. [22]theor. 7054 2.873 — Ref. [23]theor. 237.761 4.720 6768 this work 220 4.720 6834 Ref. [22]theor. 216 4.754 6841 Ref. [23]theor. 4361.32 4.166 19613.825 this work 4377 4.133 Ref. [22]theor. 3867 — — Ref. [24]theor. 4341 4.164 — Ref. [23]theor. 5424 3.59 Ref. [25]exp. 1726.94 3.729 22248.205 this work 1724 3.758 Ref. [22]theor. 1450 — — Ref. [24]theor. 1722 3.725 22300 Ref. [23]theor. 2.78 Ref. [25]exp.

Table 2. Comparison of the spectroscopic constants of the ground and excited NaLi potentials with experimental and theoretical values. .

3.2. Transition dipole moments (TDMs)

Following the same way that used early for constructing the potential–energy curves, we constructed the singlets and triplets transition dipole moments D(R) related to the transition from the ground states to the excited ones. In fact we used in the intermediate range of interatomic distances the ab-initio data points provided for us by Aubert-Frecon M.^[19] The foregoing data are smoothly connected in the long range to the formula proposed by Chu and Dalgarno^[27] . Here, the constant A and the asymptotic value are determined by fitting the long range (TDMs) data points to a form analogous to the last equation. Our fitting give , A=−2790.76 for the Σ–Σ transitions and A=−2790.76 for the Σ–Π transitions. In the short range region the transition dipole moments D(R) follow the linear form . Both parameter a and b are given by the continuity conditions of the D(R) function. The found values are compiled in Table 3.

Table 3.

Table 3.

Table 3. Constant parameters adopted for the constructed singlets and triplets transition dipole moments D(R). . Transitions ΣΣ ΣΠ Singlets Triplets Singlets Triplets a 2.192 0.964 1.781 0.831 b −0.210 −6.494 −0.262 0.266

Table 3. Constant parameters adopted for the constructed singlets and triplets transition dipole moments D(R). .

Transition dipole moments D(R) against internuclear separation (R) are shown in Fig. 2.

Fig. 2.

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	Fig. 2. Transition dipole moments D(R) against internuclear separation (R). The singlet DΣΣ (TDMs) are compared with ab-initio data from Mabrouk N and Berriche H.[23]

We assess the quality and accuracy of the constructed potential–energy curves and transition dipole moments by calculating the ro-vibrational energy levels and their radiative lifetimes.

4.1. Vibrational levels

The ro-vibrational energy levels of the singlet and triplet Na (3s/3p) + Li (2s) states, are determined by solving numerically the the radial wave equation (Eq. (6)). To achieve the goal, we adapted, with some important modifications, the subcode ALF (Automatic Levels Funder) of LeRoyʼs package Level version 7.4.^[28] Our calculation yields to the 48 levels for both of the and states, 16 for the state and 13 for the triplet ground state . The corresponding values of the binding energies E(v,J) with J = 0 for all the Σ⁺ states and J = 1 for the Π ones are compared with theoretical data of Schmidt-Mink I et al.^[22] as listed in Table 4. We should note here that the present computation discloses 1 quasibound level for the state and 7 for the molecular state.

Table 4.

Table 4.

Table 4. Rotationl-vibrational energy levels E(v,J), in unit cm−1, for the NaLi molecular states calculated with respect to their minimum. J = 0 and J = 1 for Σ and Π states respectively. . a v This work Ref. [22] Ref. [26] This work Ref. [22] This work Ref. [22] This work Ref. [22] 0 126.30 128 128.1 53.68 53 77.10 75 20.43 20 1 376.76 381 381.7 161.20 159 228.41 224 58.50 56 2 624.07 632 632.0 269.10 266 376.40 370 92.84 89 3 868.12 879 878.9 377.32 373 520.10 511 123.41 119 4 1108.89 1122 1122.4 485.83 481 659.24 648 150.15 144 5 1346.38 1362 1362 594.55 590 792.70 781 172.93 166 6 1580.53 1599 1598.9 703.40 698 922.20 908 191.76 184 7 1811.30 1832 1831.9 812.32 807 1044.75 1029 206.60 198 8 2038.64 2062 2061.3 921.23 916 1160.80 1145 217.63 207 9 2262.50 2288 2287.2 1030.04 1025 1269.10 1253 225.63 10 2483.0 2511 2509.4 1138.70 1134 1371.10 1354 231.58 11 2670.0 2730 2728.0 1247.10 1243 1463.80 1447 231.60 12 2913.0 2945 2942.9 1355.20 1351 1546.95 1531 235.83 13 3122.04 3156 3154.0 1462.88 1459 1618.61 1604 237.63 14 3327.61 3363 3361.3 1570.13 1567 1678.70 1666 15 3529.29 3567 3564.2 1676.90 1674 1673.90 1714 16 3727.01 3766 3764.2 1783.10 1781 — 1737 17 3920.66 3961 3959.7 1888.56 1888 18 4110.15 4152 4151.1 19993.40 1993 19 4295.36 4338 4338.2 2097.50 2098 20 4476.20 4520 4521.1 2200.70 2202 21 4652.50 4698 4699.5 2303.10 3206 22 4824.16 4870 4873.3 2404.50 2408 23 4991.01 5038 5042.4 2504.96 2510 24 5153.00 5201 5206.6 2604.40 2610 25 5309.65 5365.7 2702.60

Table 4. Rotationl-vibrational energy levels E(v,J), in unit cm−1, for the NaLi molecular states calculated with respect to their minimum. J = 0 and J = 1 for Σ and Π states respectively. .

4.2. Radiative lifetimes

The purpose here is to calculate the radiative lifetimes of the both exited states and of the NaLi dimer. In the framework of the quantum theory, the radiative lifetime τ of the transition from the upper bound level of the exited state to all the lower continuum and bound levels of the corresponding ground states, is defined as being the inverse of the sum of the spontaneous bound–free and bound–bound emission rates ,^[29,30] then

with

Table 5 gives some of and radiative lifetime values obtained by computations. To our knowledge, no measured or computed lifetime values are available in literature for comparison. Normally the lifetime of the highest rotationless vibrational states near to the dissociation limit should be close to the atomic lifetime of isolated Na atom. Therefore, we have evaluated the sodium atomic 3s–3p lifetime by using the following weighted sum

Table 5.

Table 5.

Table 5. Radiative lifetimes (in unit ns) of some ro-vibratoinal levels of the NaLi and states. . 0 10 20 30 40 50 60 0 445.06 458.20 446.60 478.84 492.52 504.47 511.64 10 916.86 258.19 258.469 301.69 267.50 273.24 281.13 20 578.95 184.67 183.56 206.02 148.16 183.88 183.52 30 337.04 126.19 124.44 133.40 120.42 116.98 112.35 40 176.76 73.06 71.01 73.14 64.17 58.69 51.11 42 150.027 62.59 60.48 56.93 46.86 46.29 36.23 44 119.38 51.98 49.75 40.24 40.51 30.87 46 92.23 4640.97 38.40 23.96 33.01 47 69.88 34.84 31.86 0 21.19 21.06 20.701 20.13 19.381 18.49 17.50 5 64.60 16.26 16.06 15.75 15.34 14.83 14.24 10 53.58 13.56 13.41 13.17 12.84 12.40 11.77 11 51.74 13.12 12.98 12.74 12.41 11.94 12 49.92 12.72 12.57 12.33 11.98 13 48.12 12.34 12.19 11.95 11.56 14 45.66 11.99 11.85 11.60 15 12.31 11.76

Table 5. Radiative lifetimes (in unit ns) of some ro-vibratoinal levels of the NaLi and states. .

Our value 16.80 ns does not exceed a difference of 0.5 ns from the experimental values 16.3 ± 0.1 ns found by Reho et al.,^[31] 16.35 ns of Carlsson and Sturesson,^[32] 16.3 ns given by Volz and Schmoranze,^[33] and 13.31 ns of Tiemann, et al.^[34]

To determinate the NaLi photoabsoption spectra and the effect of temperature, we have carried out the fb, bf, ff, and bb reduced-absorption coefficients by using the above equations (1)–(4). By adopting the Numerov algorithm,^[35] we solved numerically the radial wave equation Eq. (6), which allowed us to determine the normalized wave functions and the corresponding energies required in the calculation of the matrix elements .

For the computation of the free–free integral appearing in Eq. (3) we used the Gauss–Laguerre quadratureu^[36] with 100 weighted points. In the aim to simulate accurately the NaLi photoabsorption profile we have performed the calculations for wavelengths interval going from 350 nm to 750 nm with a frequency step and by taking the maximum values of the rotational quantum number J_max=500 for the free–free transitions and for the remaining transitions. Moreover all the bound and quasibound levels are including in the calculations.

The profile spectra, around the Na ( ) resonance line λ₀=589 nm, in the presence of Lithium atoms, obtained by our full quantum mechanical computation for four temperatures, namely T=2.0×10³, 6.0×10³, 1.0×10⁴, and 1.4×10⁴ K are plotted in Fig. 3.

Fig. 3.

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	Fig. 3. NaLi full quantum-mechanical reduced-photoabsoption coefficients for temperature range 2000 K–1.4×104 K. Part (a) shows the details of the , , , and transitions. Part (b) represents the sum of all the previous contributions. The insert displays the bound–bound and free–free transitions around the red wing satellite positions.

The column (a) in Fig. 3 shows the , , , and reduced-absorption coefficients. Column(b) presents the sum of all the contributions. From these curves, one may observed, in particular, that the both singlet transitions and dominate the general shape of the total spectra. In fact the bands are the principal component in the red wing and contribute also to the blue branches, whereas the transitions contribute solely and strongly to the blue wing.

One may notice that both and reduced-absorption coefficients decrease considerably in magnitude with increasing temperatures. The intensity decrease of the blue wing of the transitions is more severe. The transitions generate one satellite structure in far-red wing around the wavelengths λ=685 nm, this red satellite result from the bound–bound and free–free transitions, as it can be seen in the insert of Fig. 3, and it is present in the total spectra for all studied temperatures.

The column (a) in Fig. 3 revealed that the triplet and transitions constitute a minor contributions compared with the singlet spectra. In other words their contribution is restricted to the creation of the satellites in the blue wing. Indeed the transitions participate with one satellite close to the λ =574 nm which become visible in the total spectra beyond K. Likewise the transitions generate one satellite in the vicinity of the wavelength λ =490 nm. The latter satellite do not appears on the total spectra for all temperature previously cited. According to the Fig. 3 column (a) the intensity of the satellite under consideration at T = 2000 K is about a factor of 10⁴ less than of the and transitions intensities. As stated above, the intensity spectrum of the transitions diminishes when the temperature increases and becomes, at T= 1.4×10⁴ K, of the same order of magnitude than the intensity corresponding to the transitions, whereas the transitions intensity remains high compared with the transitions ones.

We have also carried out the NaLi reduced-absorption coefficients for more high temperatures. We represent in Fig. 4 the total spectra and those corresponding to the four partial transitions at T=1.6×10⁴ K and 1.8×10⁴ K. At such temperatures the spectrum intensity decreases again and becomes comparable with the transitions intensity the satellite at 490 nm begins then to emerge. From the above discussion, We recap that the NaLi photoabsorption spectra exhibit in the red wing an apparent satellite, at all temperatures, in the vicinity of the wavelength , while the blue wing shows beyond a second satellite located at λ =574 nm and a third one at approximately λ =490 nm which starts to appear in the global spectra at T=1.8×10⁴ K. All the positions of the above sattelites agree quite well with the wavelength values found earlier from the classical method.

Fig. 4.

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	Fig. 4. NaLi full quantum-mechanical reduced-photoabsoption coefficients for temperatures 1.6×104 K and 1.8×104 K. The upper row shows the details of the different transitions and the lower row represents the sum of the all contributions.

In order to examine the behavior of the satellite intensity and positions with temperature, we present in Fig. 5 the satellite peaks appearing at the wavelengths 685, 574, and 490 nm for temperature range 2000 K–1.4×10⁴ K. Figure 5(a) reveals that the amplitude of satellite peak that appear at 685 nm diminishes, almost, with a factor of 10² when the temperature goes from 2000 K to 6000 K. Beyond this temperature the peak intensity change without reacheing 10%. We think that the reason for this strong lowering is due to the fact that this satellite arises bassically from bound–bound transitions, at higher temperature the lower ro-vibrational levels are less populated. In the range of 6000 K to 1.4×10⁴ K, the temperature has no influence on the intensity satellite peaks occurring in the positions 574 nm and 490 nm resulting from the triplet and transitions respectively, as we can easily seen in Fig. 5(b) and Fig. 5(c). As well, from Fig. 5 one may notice that is no influence of temperature on the position of the satellites.

Fig. 5.

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	Fig. 5. Temperature effect on the position and intensity of the satellite peaks. Panel (a) shows the red wing satellite near 685 nm. Panels (b) and (c) represent the blue wing peaks located around 574 nm and 490 nm respectively.

We have determined theoretically the pressure-broadening spectrum of radiator sodium atom interacting with perturbed lithium. The needed potential–energy curves and transition dipole moments are constructed upon purely ab-initio data points. Their reliability is assessed by determining the ro-vibrational levels and the radiative lifetimes of the NaLi excited states. The results showed an important effect of temperature on the far wings and the profile is found dominate by the singlet transitions. Our calculations revealed that the photoabsorbtion spectra display, three satellite structures close to 685 nm in the red wing and 574 nm and 490 nm in the blue wing. The first satellite generated by the transitions occurred at all interested temperatures. Whereas the second one given by the bands appears at beyond K and the third peak due to the transitions emerges at approximately T=1.8×10⁴ K.

The authors are very grateful to Aubert-Frécon Monic for all the data points she provided us.