The radiative processes induced by interaction of metastable Cd (53P2) atoms with Ar and Kr atoms

Chemical Physics Letters xxx (2013) xxx–xxx

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The radiative processes induced by interaction of metastable Cd ð53 P 2 Þ atoms with Ar and Kr atoms O.S. Alekseeva a,b,⇑, A.Z. Devdariani b,c, M.G. Lednev a, A.L. Zagrebin a,b a

Department of Physics, Baltic State Technical University, St. Petersburg 190005, Russia V.Fock’s Institute of Physics, St. Petersburg State University, Peterhof 198504, Russia c Department of Theoretical Physics, Herzen State Pedagogical University of Russia, St. Petersburg 191186, Russia b

a r t i c l e

i n f o

Article history: Received 23 January 2013 In final form 9 April 2013 Available online xxxx

a b s t r a c t We report on the theoretical study of qusimolecular absorption and emission near the forbidden atomic line Cd ð53 P2  51 S0 Þ induced by interaction with rare gas atoms (argon and krypton). With the use of the semiempirical method of quasimiolecular term analysis and the available experimental data, the potential curves for the Cd⁄ + Kr and Cd⁄ + Ar systems and the radiative widths were obtained. In the calculation the full semiempirical procedure was used for the first time. Also the probabilities of the m0 1ð3 P2 Þ  m00 0þ ð1 S0 Þ transitions and the radiative lifetimes of the metastable 1ð3 P2 Þ states of Cd–Kr and Cd–Ar quasimolecules as functions of the vibrational excitation degree were calculated. Based on these semiempirical results the processes of the collision-induced quasimolecular absorption and emission near the forbidden atomic line Cdð51 S0  53 P2 Þ in mixtures of Cd vapor with Ar and Kr atoms have been considered and the absorption coefficients, emission spectra and the total rate constants of radiative depopulation of the metastable state of Cd have been determined. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction In this Letter the radiative characteristics of the excimer states produced by the interaction of the atomic metastable states Cdð53 P2;0 Þ with rare gas atoms Kr or Ar in their ground state are considered. Although the underlying physical idea of collision induced radiative depopulation of the metastable states of the Group II atoms due to collisions with rare gas atoms was formulated in [1,2] the lack of quantum chemical information related to the process has been retarding rigorous calculations of spectral profiles. Such calculations requires that the potential curves of the interaction of atoms in the ground and the excited states and the square of the dipole moment of the radiative transitions between these states as a function of the interatomic distance be known. In what follows the spectral profile calculations are based on the semi-empirical approach to the calculations of the potential energy curves and dipole moments proposed in [2,3] and developed in [4]. Therefore the interaction potential curves for asymptotically metastable Cdð53 P 2;0 Þ–RG (where RG = Kr or Ar) in X = 0,1,2 states and the probabilities of the quasi-molecular radiative transitions 1ð3 P2 Þ  0þ ð1 S0 Þ were obtained with the available experimental interaction potential curves for asymptotically radiating 1(1,3P1) and 0+(1,3P1) states [5–9]. The semi-empirical calculations have revealed the existence of a few vibrational states in the 1ð3 P2 Þ quasi⇑ Corresponding author at: Department of Physics, Baltic State Technical University, St. Petersburg 190005, Russia. E-mail address: [email protected] (O.S. Alekseeva).

molecular electronic states. Therefore in addition to the spectral profile calculations the radiative lifetimes of the vibrational 1ð3 P2 Þ states of Cd–Kr and Cd–Ar quasimolecules have been calculated for the first time. 2. Potential curves and probabilities of quasi-molecular radiative transitions To calculate the potential curves of interatomic interaction and the probabilities of quasi-molecular radiative transitions we used the effective Hamiltonian method [3] in the formulation [10]. The nonzero matrix elements of the effective Hamiltonian in the basis j1;3 P J Xic i of diabatic quasi-molecular wave functions of coupling case c, which are the products of the atomic wave functions 1 j1;3 PJ Xic i ¼ jCdð51;3 PJ XÞiat ic jRGð S0 Þi 1;3

Þiat ic

ð1Þ

where (jCdð5 P J X – is the atomic wave function for intermediate coupling case), are presented in [4,10]. It must be said that being the products of the atomic wave functions the diabatic functions (1) do not diagonalize the effective Hamiltonian in contrast to adiabatic functions that do diagonalize. The adiabatic potential curves for the excited states were obtained within the framework of the semiempirical method of quasimiolecular term analysis [4]. The method is based on the comparison of eigenvalues of the matrix of the effective Hamiltonian with interaction potential curves reconstructed from the experimental data. The point is, all matrix elements of the effective Hamiltonian can be expressed through

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O.S. Alekseeva et al. / Chemical Physics Letters xxx (2013) xxx–xxx

atomic constants and the quantities 1,3Hr(R), 1,3Hp(R), which represent the potentials of the interaction in 1,3R and 1,3P quasi-molecular states without regard for the spin–orbit interaction in the Cd atom. The recent experiments with supersonic beams described in [5–9] yielded the reliable interaction potential curves both in singlet ð0þ ð1 P1 Þ and 1ð1 P1 ÞÞ and triplet ð0þ ð3 P 1 Þ and 1ð3 P 1 ÞÞ quasimolecular states which are produced by the interaction with the radiating states of the Cd atom. The use of these curves lets us to apply the full semiempirical procedure for the first time. Thus the four functions 1,3Hr(R), 1,3Hp(R) have been determined from the available four experimental potential curves for four states. The potential curves for the remaining four states have been restored from these functions by diagonalization of the matrix of the effective Hamiltonian with regard to the spin–orbit interaction. The semiempirical interaction potential curves of the triplet states and the experimental interaction potential curves for the ground states are shown in Figures. 1 and 2. The Morse potential parameters for the obtained semiempirical and the experimental [5,9] potential curves which are essential for the considered transitions are listed in Table 1. Now the probabilities Cð1ð3 P 2 Þ; RÞ of the quasimolecular radiative transitions 1ð3 P 2 Þ ! 0þ ð1 S0 Þ per unit time at a fixed interatomic distance R can be determined. These values can be also named the Einstein’s coefficients or the radiative widths. The adiabatic quasimolecular wave functions jXð3 P J Þi (the eigenfunctions of the effective Hamiltonian) can be presented as linear combinations of the diabatic functions. In particular,

j1ð3 P2 Þi ¼ cl ðRÞj1 P 1 1ic i þ c2 ðRÞj3 P2 1ic i þ c3 ðRÞj3 P1 1ic i;

ð2Þ

where the coefficients ci (R) are determined by diagonalizing the effective Hamiltonian matrix. The atomic wave functions for the intermediate coupling case in turn can be presented as linear combination of the wave functions for the LS coupling case with amplitudes a and b:

Figure. 2. The interaction potential curves of the triplet states for CdAr obtained in the frame of the present semi-empirical approach (full curves) and valence ab initio result [19] for 1(3P2) (dotted line). Table 1 Parameters for the Morse potential curves for the 1ð3 P2 Þ and 0þ ð1 S0 Þ states of the CdAr and CdKr molecules. CdKr

CdAr

3

Re, a.u. De, cm1 xe, cm1 xexe, cm1 a b

þ 1

1ð P 2 Þ

0 ð S0 Þ

9.25 107.86 10.7 0.3

8.07 165.0 18.1 0.5

a

1ð3 P 2 Þ

0þ ð1 S0 Þb

9.48 56.87 11.8 0.6

8.14 106.8 25.3 1.5

The experimental parameters obtained in [9] are used. The experimental parameters obtained in [5] are used.

at at 1 3 jCdð1 P1 XÞiat ic ¼ ajCdð P 1 XÞiLS þ bjCdð P 1 XÞiLS ; 3

Þiat ic

jCdð P1 X

1

Þiat LS

¼ bjCdð P1 X

3

Þiat LS :

þ ajCdð P1 X

ð3Þ ð4Þ

In the calculations the approximate values of the amplitudes a = 0.998 and b = 0.062 were used, which, as in [11], were determined from the energies of atomic levels using the semiempirical method of the atomic spectra analysis [12]. When calculating the probabilities CðXð3 P J Þ; RÞ of the quasimolecular radiative transitions with adiabatic wave functions we took into account the contribution from the diabatic functionsj1 P 1 0þ LS i and j1 P1 1LS i. The reason is that in the diabatic molecular basis j1;3 P J XLS i corresponding to the atomic LS coupling case the dipole moments of the transitions to the ground state are nonzeros only 1 for the 1 P 1 0þ LS and P 1 1LS states. In particular,

1

Figure. 1. The interaction potential curves of the triplet states for CdKr obtained in the frame of the present semi-empirical approach (full curves) and valence ab initio result [19] for 1(3P2) (dotted line).

P1 1LS j1ð3 P 2 Þi ¼ ac1  bc3 ;

ð5Þ

 ^ 1 S0 0 can be expressed finally and the dipole moment 1 P1 1LS jdj 3 ^ 1 S0 0 of the allowed atomic through the dipole moment P 1 1ic jdj transition. Thus the radiative width Cð1ð3 P2 Þ; RÞ of the quasimolecular radiative transition can be expressed in terms of the probability Cð3 P 1 Þ and frequency xð3 P1 Þ of the atomic intercombination electric dipole transition, the calculated amplitudes c1, c3 and the amplitudes a and b:

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O.S. Alekseeva et al. / Chemical Physics Letters xxx (2013) xxx–xxx

Cð1ð3 P 2 Þ; RÞ ¼ Cð3 P 1 Þ

 2 xð1ð3 P2 Þ; RÞ 3 a  c : c 1 3 b xð 3 P 1 Þ

3



ð6Þ

In Figures. 3 and 4 calculated values of reduced widths

cð1ð3 P2 Þ; RÞ ¼



3

Cð1ð3 P2 Þ; RÞ xð 3 P 1 Þ 3 Cð P 1 Þ xð1ð3 P2 Þ; RÞ

ð7Þ

are presented. These values are almost independent of the quasi-molecular ground state, which affects the magnitude of xð1ð3 P2 Þ; RÞ, and is proportional to the square of the ratio of the dipole moment of the quasi-molecular transition 1ð3 P2 Þ  0þ ð1 S0 Þ to the dipole moment of the 3 P 1  1 S0 atomic transition. The dips observed in the Figures. 3 and 4 (lgðcð1ð3 P 2 ÞÞÞ ! 1) correspond to the dipole moment’s sign change and consequently the zero value of the radiative width Cð1ð3 P 2 Þ; RÞ. The interaction potential curves for the ground U0 and excited U⁄ states and the difference potentials DU are also shown in Figures. 3 and 4. For the ground state the potential curves determined in studies [5,9] have been used. 3. The radiative lifetimes of the m0 1ð3 P 2 Þ states and the probabilities of the m0 1ð3 P 2 Þ  m00 0þ ð1 S 0 Þ transitions Given the proximity of the frequencies of the transitions

m0 1ð3 P2 Þ  m00 0þ ð1 S0 Þ to the frequency of the forbidden transition 53 P 2  51 S0 , the probability of the molecular transitions can be calculated without regard for the dependence of the transition frequency on the interatomic distance, by setting it constant and

Figure. 4. The interaction potential curves (1) U⁄, (2) U0, and (3) DU (upper panel) and the reduced probability c (lower panel) for the Cd + Ar system.

equal to the frequency of the 53 P2  51 S0 transition. In this approximation, the lifetime of the state m0 1ð3 P2 Þ is given by [13]

s1 ðm0 Þ ¼ hm0 jCð1ð3 P2 Þ; RÞjm0 i;

ð8Þ

where jm i is the wave function of the vibrational state, which sets identical to the corresponding Morse function. The probability of the radiative transition m0 1ð3 P 2 Þ  m00 0þ ð1 S0 Þ is defined as 0

Aðm0 ; m00 Þ ¼





2 xð3 P2 Þ 3 3

0 ac1

Cð P1 Þ hm j  c3 jm00 i : 3 xð P 1 Þ b

ð9Þ

The calculated lifetimes and transition probabilities are presented in Table 2. The calculations were performed for lower vibrational states for which the approximation of the potential curves by the Morse function is accurate. The obtained probabilities vary in a wide range that can be explained by some reasons. The most essential of them are the strong dependence of the dipole moments of the transitions on the interatomic distance and the quantities similar to the Franck–Condon factors. For reference, the radiative lifetime of the atomic state Cdð53 P 1 Þ is equal to 2.4  106 s, the lifetime of the metastable Cdð53 P2 Þ state is 130 s [14]. 4. Quasi-molecular absorption and emission near the forbidden atomic line Cdð51 S 0  53 P 2 Þ induced by collisions with Kr and Ar atoms

Figure. 3. The interaction potential curves (1) U⁄, (2) U0, and (3) DU (upper panel) and the reduced probability c (lower panel) for the Cd + Kr system.

For a mixture of cadmium vapor with inert gases, the optical absorption and emission spectra near the forbidden atomic line Cdð51 S0  53 P2 Þ are determined by the processes

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O.S. Alekseeva et al. / Chemical Physics Letters xxx (2013) xxx–xxx

Table 2 The probabilities A(m0 , m00 ) (in s1) of the m0 1ð3 P 2 Þ  m00 0þ ð1 S0 Þ transitions and the radiative lifetimes sðm0 Þ (in s) of the m0 1ð3 P 2 Þ states of the CdAr and CdKr molecules.

m00

RG

A(m0 ,

m Ar

0

0 1 0 1

Kr

m00 )

sðm0 Þ

0

1

2

3

4

5

0

1

74 408 110 408

102 750 321 750

4 237 242 237

51 0.7 19 0.7

36 2.4 28 2.4

0 74 41 74

3.7  103

1.6  103

1.3  103

6.2  104

( 21

Cdð5s S0 Þ þ RG þ hx ! ( Cdð5s5p3 P 2 Þ þ RG !

Cdð5s5p3 P2 Þ þ RG; CdRGð1ð3 P2 ÞÞ

Cdð5s21 S0 Þ þ RG þ hx; CdRGðX 1 RÞ þ hx

ðabsorptionÞ ðemissionÞ ð10Þ

and

( CdRGðX 1 RÞ þ hx ! ( 3

CdRGð1ð P2 ÞÞ !

Figure. 5. The spectral distributions of the absorption coefficients of mixtures of Cd vapor with Kr and Ar near the forbidden line Cdð51 S0  53 P 2 Þ for T = 300 K (curves 1 and 3, respectively) and for T = 700 K (curves 2 and 4, respectively).

3

Cdð5s5p P2 Þ þ RG; CdRGð1ð3 P2 ÞÞ

ðabsorptionÞ

Cdð5s21 S0 Þ þ RG þ hx; CdRGðX 1 RÞ þ hx

ð11Þ ðemissionÞ

The first of them corresponds to free–free and free-bound transitions and the second corresponds to bound-free and bound– bound transitions. The spectral distribution of absorption coefficient K abs ðT; kÞ of a mixture of cadmium vapor with an inert gas in the short-wavelength (with respect to the forbidden atomic line) spectral region, where the role of bound–bound transitions is insignificant, and the spectral distribution IðT; DxÞ of photons emitted in the processes R (10) and (11) normalized by the condition IðT; DxÞdDx ¼ 1, are determined within the framework of the well-known quasi-static approximation [15]

  k2 g  4pR2C CðRC Þ U 0 ðRC Þ ; exp  kT 8p g 0 j dDUðRÞ jR¼R dhR C   2 4pR2C CðRC Þ U  ðRC Þ ; exp  IðT; DxÞ ¼ 5 KðTÞj dDUðRÞ jR¼R kT dhR C

K abs ðT; kÞ ¼

ð12Þ ð13Þ

where k is the wavelength of quasi-molecular absorption, T is the temperature of the gas mixture, k is the Boltzmann constant, g  =g 0 ¼ 2 is the ratio of the statistic weights of the electron states 1ð3 P 2 Þ and 0þ ð1 S0 Þ , U0 and U⁄ are the interaction potential for atoms on the ground and excited states respectively, DU is the difference potential, CðRÞ is the probability of the quasi-molecular radiative transition 1ð3 P2 Þ ! 0þ ð1 S0 Þ, RC is the Condon point. The quantity

KðTÞ ¼

2 4p 5

Z 0

1



CðRÞ exp 

 U  ðRÞ 2 R dR kT

ð14Þ

characterizes the integrated (over the spectrum) rate constant of radiative decay of the metastable state. The values of K(T) for T = 300 K and T = 700 K are presented in Table 3. The results of calculations of the spectral distributions of the absorption coefficients Table 3 Integrated (over the spectrum) rate constants K(T) (in 1018 cm3 s1) of radiative decay of the metastable state in the processes (10) and (11) for CdKr and CdAr. T, K

CdKr CdAr

300

700

2.4 1.1

3.3 1.9

Figure. 6. The normalized spectral distribution for the quasimolecular emission of mixtures of Cd vapor with Kr and Ar near the forbidden line Cdð51 S0  53 P 2 Þ for T = 300 K (curves 1 and 3, respectively) and for T = 700 K (curves 2 and 4, respectively).

K abs ðT; DxÞ and emission spectrum of mixtures of cadmium vapor with krypton and argon for T = 300 K and T = 700 K are presented in the Figures. 5 and 6. The main quantitative conclusion which follows from the calculations is that the increase of temperature leads to a minor shift of the spectral distribution maximum for absorption and emission toward the shorter wave lengths and to the decrease of its value. In general, the spectral distribution broadens with increasing temperature. 5. Conclusion In this Letter the interaction potential curves for the quasimolecular states with X = 0,1,2 produced by the interaction of the metastable atomic state Cdð53 P 2 Þ with the Ar and Kr atoms were obtained from the available experimental interaction potential curves for 1(1,3P1) and 0+(1,3P1) states [5–9] by making use of the semiempirical method of quasimolecular term analysis [2–4]. With the use of the semi-empirical method we have expressed the R-dependent probabilities CðXð3 PJ Þ; RÞ of the quasimolecular

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radiative transitions Xð3 P J Þ  0þ ð1 S0 Þ through the corresponding amplitudes of the expansions of adiabatic wave functions and the probabilities of atomic transitions. It has given us a direct relationship between the probabilities of the considered quasimolecular transitions and the known probabilities of the single-atom transitions, and as a consequence the lifetimes of the m0 1ð3 P 2 Þ vibronic quasimolecular states and the probabilities per unit time of the m0 1ð3 P2 Þ  m00 0þ ð1 S0 Þ transitions. In general, the interaction with a rare-gas atom reduces the lifetime by 5–6 orders of magnitude. We have also considered the processes of collision-induced absorption and emission near the forbidden line Cdð51 S0  53 P2 Þ in mixtures of cadmium vapor with krypton and argon. The emission spectra represents continuous bands shifted to the shortwavelength region with respect to the position of the forbidden atomic line. According to the present calculations the depth of the potential well of the state 1ð3 P2 Þ is smaller than kT for T P 300 K. Therefore the relative population of bound states is small and states of continuum are preferentially populated. The major contribution to the emission of mixtures of cadmium vapor with Kr and Ar near the forbidden atomic line is produced by quasi-molecular radiative transitions in the region of the closest approach of atoms in the collision process, i.e. the spectrum is formed by the collision induced radiative depopulation of the atomic metastable state. Absorption spectra also represents continuous bands. As one can see from the behavior of the interaction potentials and the reduced width the process of absorption is most efficient in the short-wavelength (with respect to the forbidden line) spectral region. The main result of the process of the absorption is the formation of metastable atoms Cd(3P2). The long-lived excited molecules Cd–RG in the state 1ð3 P2 Þ can be formed only for relatively small shifts, therefore the processes of optical absorption lead to the selective population of the metastable state Cd(3P2). It can be remarked that the interaction potentials of the Cd–Ar, Kr quasimolecules had been calculated previously by Czuchaj et al. [16–19] in the frame of pseudopotential approach as well as in the frame of some synthetic approaches combining pseudopotentials and ab initio calculations. We point out rather good agreement

5

between the present results and the last version of the interaction potential 1ð3 P2 Þ in [19], see dotted lines in Figures. 1 and 2 plotted on the basis of Figures. 3 and 4 in [19]. The disagreement between results obtained by two different approaches does not exceed 50 cm1 at interatomic distances more than 7.5. However more elaborate calculations in the region of the potential minimum would be desired to define more exactly the position of the vibrational levels. Experimental data on the spectral profiles and on the lifetimes of the molecular states produced by the interaction Cd (5s5p) – Ar, Kr could also result in the improvement of the potential energy curves and the probabilities of the radiative transitions. Tentative results of this letter were presented at ICSLS 21 [20]. References [1] A.Z. Devdariani, A.L. Zagrebin, Opt. Spectrosc. (USSR) 58 (6) (1985) 752. [2] A.Z. Devdariani, Spectral Line Shapes, in: R. Stamm, B. Talin (Eds.), Nova Publishes, 1993, p. 235. [3] A.Z. Devdariani, A.L. Zagrebin, K.B. Blagoev, Ann. Phys. Fr. 14 (1989) 467. [4] O.S. Alekseeva, A.Z. Devdariani, M.G. Lednev, A.L. Zagrebin, Russ. J. Phys. Chem. B 5 (2011) 946. [5] D.J. Funk, A. Kvaran, W.H. Breckenridge, J. Chem. Phys. 90 (1989) 2915. [6] M. Ruszczak, M. Strojecki, J. Koperski, Chem. Phys. Lett. 416 (2005) 147. [7] A. Kvaran, D.J. Funk, A. Kowalski, W.H. Breckenridge, J. Chem. Phys. 88 (1989) 6069. [8] J. Koperski, Sz.M. Kiełbasa, M. Czajkowski, Spectrochim. Acta 56A (2000) 1613. [9] J. Koperski, M. Łukomski, M. Czajkowski, Spectrochim. Acta 58A (2002) 2709. [10] A.L. Zagrebin, M.G. Lednev, Opt. Spectrosc. 77 (1994) 481. [11] A.L. Zagrebin, M.G. Lednev, S.I. Tserkovnyi, Opt. Spectrosc. 74 (1993) 24. [12] I.I. Sobel’man, Introduction to the Theory of Atomic Spectra, Pergamon Press, Oxford, New York, 1972. [13] M. Krauss, F.H. Mies, Topics in Applied Physics, in: C.K. Rodes (Ed.), vol. 30, Springer, Berlin, Heidelberg, 1984, p. 5. [14] A.A. Radzig, B.M. Smirnov, Reference Data on Atoms, Molecules, and Ions, Springer-Verlag, Berlin, 1985. [15] A. Gallagher, Topics in Applied Physics, in: C.K. Rodes (Ed.), vol. 30, Springer, Berlin, Heidelberg, 1984, p. 139. [16] E. Czuchaj, J. Sienkiewicz, J. Phys. B: At. Mol. Phys. 17 (1984) 2251. [17] E. Czuchaj, H. Stoll, Chem. Phys. 248 (1999) 1. [18] E. Czuchaj, M. Krosnicki, J. Czub, Eur. Phys. J. D 13 (2001) 345. [19] E. Czuchaj, M. Krosnocki, H. Stoll, Theor. Chem. Acc. 105 (2001) 219. [20] O. S. Alekseeva, A. Z. Devdariani, M. G. Lednev, A. L. Zagrebin, ICSLS-21 International Conference on Spectral Line Shapes, Saint-Petersburg, Russia, June 3–9, 2012, Book of Abstracts, Saint-Petersburg: VVM Publishing Ltd (2012) 18. ISBN 978-5-9651-0649-3.

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