Collision integrals of oxygen atoms and ions in electronically excited states

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Chemical Physics 344 (2008) 13–20 www.elsevier.com/locate/chemphys

Collision integrals of oxygen atoms and ions in electronically excited states A. Laricchiuta

a,*

, D. Bruno a, M. Capitelli

b,a

, R. Celiberto c, C. Gorse

b,a

, G. Pintus

b

a CNR IMIP Bari, Italy Dipartimento di Chimica, Universita` di Bari, Italy Dipartimento di Ingegneria Civile ed Ambientale, Politecnico di Bari, Italy b

c

Received 27 July 2007; accepted 31 October 2007 Available online 13 November 2007

Abstract Diffusion and viscosity-type collision integrals for interactions between atoms and ions of oxygen in ground and electronically excited states have been evaluated from the relevant potentials of the molecular and molecular ion states, respectively. The inelastic contribution to the diffusion collision integrals has been also estimated, by using calculated resonant excitation and charge-transfer cross sections.  2007 Elsevier B.V. All rights reserved. Keywords: Elastic scattering; Excited states; Transport processes in plasma

1. Introduction A great effort [1–5] is nowadays devoted to the refinement of high-temperature air component transport cross sections, representing the input in transport equations for the calculation of the transport properties (viscosity, thermal and electrical conductivity) of LTE (local thermodynamic equilibrium) and non-LTE air plasmas. In general tabulated results refer to ground electronic state of neutral and ionized atomic air species, i.e. N(4S), N+(3P) and O(3P), O+(4S) respectively. However a renewed interest in interactions involving species in excited states is found in the literature [7,8]. A pioneering work in the field was the calculation of diffusion cross sections of N(2P) and N(2D) states colliding with N(4S) [6], followed by a complete tabulation [9] of diffusion and viscosity-type collision integrals for all the interactions involving O(3P), O(1D) and O(1S) states. Later the diffusion-type transport cross sections for N(4S)–N+(3P,1D) [10] and O(3P,1D,1S)–O+(4S,2D,2P) [11] collisions have been calculated, including the inelastic con*

Corresponding author. E-mail address: [email protected] (A. Laricchiuta).

0301-0104/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.10.032

tribution of the charge-exchange process; the corresponding cross sections have been obtained by using the gerade–ungerade splitting of the relevant states of the molecular ion. However g–u splittings were estimated in a range of atom–ion internuclear separations too close to the equilibrium distance of the molecular ion so that the resulting cross sections appeared too small. Recently a new set of chargetransfer cross sections have been calculated, in the framework of the asymptotic approach, for the so-called low-lying excited states of atomic oxygen O(3P,1D,1S)–O+(4S,2D,2P) [12] and nitrogen N(4S,2D,2P)–N+(3P,1D,1S) [13] (i.e. states with the same principal quantum number of the ground state), obtaining more reliable results. A recent study on high-temperature–high-pressure LTE atomic hydrogen plasmas [14,15] has shown that also highly electronically excited states (n 6 12) can significantly affect the plasma transport properties. Charge-transfer cross sections for atom–ion collisions involving nitrogen and oxygen atoms in highly excited states [16] have been calculated and results show a n4–n5 dependence, thus confirming the role of electronically excited states in affecting the transport properties of LTE air plasmas. The aim of the present work is to present tables of diffusion and viscosity-type collision integrals for interactions involving the

14

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

low-lying and the high-lying excited states of O and O+ species by using • more reliable potential energy curves from quantum theory and • recent charge-transfer cross sections (for the estimation of inelastic contribution to diffusion-type collision integrals). Particular attention will be devoted to the comparison of new results with old ones, where possible. 2. Method of calculation Collision integrals can be obtained by averaging the contributions coming from the different potentials curves arising in the atom–atom or atom–ion interaction P ð‘;sÞ ð‘;sÞ n wn Xn Xav ¼ P ð1Þ n wn where wn is the statistical weight of the nth molecular state. These quantities enter in the transport equations for viscosity and heat conductivity of a multi-component mixture. Potentials of purely repulsive states have been fitted according to the exponential form uðrÞ ¼ A expðarÞ

ð2Þ

where r is the internuclear distance and A and a are the potential parameters. Corresponding collision integrals can be calculated by the following equation: Xð‘;sÞ ¼

8b2 I ð‘;sÞ ðbÞ h i ‘ a2 ðs þ 1Þ! 1  12 1þð1Þ 1þ‘

ð3Þ

where b = ln (A/kT). The integrals I‘,s are tabulated in Ref. [17], but can be evaluated through the approximate formula [18], with parameters ai given in Table 1 I ð‘;sÞ ðbÞ ¼ a1

1 þ a2 ba3 1 þ a4 ba5

ð4Þ

The potential curves representing bound molecular states have been fitted by Morse functions      2C C uðrÞ ¼ De exp  ðr  re Þ  2 exp  ðr  re Þ r r ð5Þ with the constrain rre ¼ 1 þ lnð2Þ . For this potential reduced C collision integrals are tabulated as a function of Tw = kT/e for different C values [19], thus requiring a double interpolation. The dependence on the C parameter, at fixed Tw, can be described with analytical expression of the form Table 1 Parameters ai entering in Eq. (4) I(‘,s)

a1

a2

a3

a4

a5

I(1,1) I(2,2)

0.08174 0.21816

1.46296 1.24837

1.00908 2.37222

0.47814 0.49958

1.01003 2.39267

X ¼ b1 þ b2 C b3 þ b4 C

ð6Þ

while collision integrals of bound states (Morse curves) as a function of Tw, once defined the C value (characteristic of the molecular state), have been fitted with formula c

X¼

1 þ ðT H Þ 1 c c2 þ c3 ðT H Þ 4

ð7Þ

bi coefficients have been determined fitting the tabulated reduced collision integrals for given values of C parameter, while ci coefficients depend on the peculiar C value and, in turn, on the considered electronic state. Tabulated values for diffusion and viscosity-type collision integrals are mul˚ 2). tiplied by r2 in order to get dimensional quantities (A The inelastic contribution to diffusion-type collision integrals has been calculated from the cross sections for the corresponding resonant excitation/charge-transfer processes, Qex. Dynamical results for the excitation exchange have been derived from g–u splitting functions using a semiclassical approach, according to which Qex is given by P ðnÞ n p n Qex ð8Þ ðQex Þ ¼ P n pn 1 2 ðnÞ ðQex Þ ¼ pðbcðnÞ Þ ð9Þ 2 where bnc is the solution of the transcendental equation !1=2 An pbðnÞ 1 c ð10Þ expðan bcðnÞ Þ ¼ p hg 2an An and an, entering in Eq. (10), are the exponential fitting parameter of the g–u splitting function jugðnÞ  uuðnÞ j ¼ An expðan rÞ

ð11Þ

Corresponding parameters have been obtained from relevant potential energy gerade–ungerade couples and tabulated in Section 3. Cross sections for charge-exchange processes obtained in the asymptotic approach are available in the literature [12,16]. As suggested by Devoto [20], fitting the exchange cross ˚ 2) dependence on the relative velocity, g (cm/s) section (A with relation 1 2 ðnÞ Qex ¼ ðC n  Dn ln gÞ ð12Þ 2 an analytic expression of collision integrals can be derived in terms of the parameters C and D  2 1 1 Dx þ DnðDx  2CÞ Xð‘;sÞ ¼ C 2  CDx þ 2 2 ! sþ1 2 2 X 1 D p þ  þ n2 2 n 4 6 n¼0  2 D T D T ln þ fDðx þ nÞ  2C g ln þ ð13Þ 2 M 2 M

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

where M is the molecular weight, x = ln(4R) and n ¼ P sþ1 1 c. n¼0 n   The diffusion-type collision integral results from the elastic and inelastic contributions [2] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1;1Þ 2 ð1;1Þ 2 ð14Þ Xð1;1Þ ¼ ðXel Þ þ ðXex Þ

Table 3 Parameters of g–u splittings fitting for excitation exchange in oxygen– oxygen asymetric collisions ˚ 1) Interaction O2 states A (eV) a (A O(3P) + O(1D)

43Pg,u 3 Rg,u 3 Ug,u 33Pg,u 3 Rg,u 3 Dg,uI 3 Dg,uII 3 Rg,u 23Pg,u

608.14 500 1500 2000 210 1793.6 550 275 102320

3.863 3.225 3.860 4.048 3.189 4.583 3.239 3.560 6.364

O(1D)+O(1S)

1

Rg,u Pg,u 1 Dg,u

329.66 120 750

2.700 2.528 3.671

3

2000 900

4.540 3.760

The adopted approach gives reliable results in the considered temperature range. 3. Results 3.1. Atom–atom interactions

15

1

In Table 2 the valence states arising in the interaction of ground and low-lying excited states of atomic oxygen (3P,1D,1S), according to the Wigner–Withmer rule, are reported. For these interactions a complete revision of old results [9] has been done, using accurate ab initio potential energy curves for the valence states of O2 molecule [21]. The corresponding fitting parameters for repulsive and bound (Morse) molecular states are also reported. In Table 3 the parameters for the exponential fitting of g–u splittings,

O(3P)+O(1S)

3

Rg,u Pg,u

for the evaluation of the inelastic contribution to X(1,1) due to excitation exchange in asymmetric collisions (i.e. collision of atoms in different electronic states), are presented. In Table 4 the diffusion and viscosity-type collision integrals for the different interactions are presented as a

Table 2 O2 valence molecular states Interaction

Repulsive states

Bound states ˚ 1) a (A

Term

234.626 703.855 1005.876 1354.729 607.635 120.280

3.054 3.766 4.594 4.539 3.0207 1.9697

23 Rþ u 25 Rþ g 5 Dg 5 Pg 5 Pu 5  Ru

210.424 391.738 342.104 1531.468 310.021 921.738

2.3326 2.4469 2.6308 4.2405 2.5987 3.7102

Cg Ug 1 Uu 21Pu 21 R u 31Dg

159.884 165.185 153.375 467.087 259.718 351.465

1.954 2.758 2.157 3.575 2.810 2.923

31Pg 31Pu 31 Rþ g 31 R u 41 Rþ g 1 þ 5 Rg

116.207 174.867 694.576 154.728 449.182 258.616

2.408 2.382 3.606 2.080 3.115 2.488

71 Rþ g

407.71

2.977

O( P)–O( S)

3 R u 43 R g 53Pg 3 5 Pu

221.87 203.97 921.26 271.38

2.488 2.364 3.846 2.707

O(1D)–O(1S)

21Du 41Dg 41Pg 41Pu 61 Rþ g

130.97 183.40 1286.46 171.75 409.57

2.248 2.288 4.291 2.484 2.949

O(3P)–O(1D)

13Dg 23Dg 23Du 23Pg 23Pu 23 R u 33Du 3 3 Pg

82.9 176.7 370.7 230.5 288.7 289.1 178.6 210.6

1.836 2.053 3.035 3.019 3.027 2.772 2.106 2.820

Term 3

3

1

O( P)–O( P)

1 Pg 11Pu 13Pg 13Pu 15 Rþ g 21 Rþ g

O(1D)–O(1D)

1 1

O(1S)–O(1S) 3

1

3

A (eV)

33Pu 33 R g 33 Rþ u 3 Ug 3 Uu 3 þ Rg 43Pg 43Pu

A (eV)

219.6 372.6 277.1 169.9 178.8 83.1 627.0 188.6

˚ 1) a (A

2.569 2.944 2.522 2.606 2.213 1.827 3.384 2.291

De (eV)

˚) re (A

C

˚) r (A

a Dg A3 Rþ u c1 R u X3 R g b1 Rþ g C3Du

3.845 0.737 1.055 4.943 3.167 0.813

1.27 1.59 1.59 1.22 1.27 1.59

2.573 3.555 3.164 2.707 2.808 3.439

1.000 1.328 1.304 0.968 1.018 1.321

11Du 21Dg 21Pg

0.905 0.437 1.111

1.63 1.96 1.61

3.239 3.068 2.978

1.343 1.599 1.313

11 Rþ u

1.648

1.59

2.799

1.272

23 R g B3 R u

0.503 1.123

2.12 1.59

2.805 3.185

1.697 1.304

Term 1

16

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

Table 4 ˚ 2) for interactions of atomic oxygen in ground and excited (low-lying) states Collision integrals (A O(3P)–O(3P)

T (K)

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

O(1D)–O(1D)

O(1S)–O(1S)

O(3P)–O(1D)

O(3P)–O(1S)

X(2,2)

X(1,1)

X(2,2)

X(1,1)

X(2,2)

X(1,

7.059 5.854 5.200 4.759 4.430 4.170 3.956 3.774 3.618 3.481

11.967 10.660 9.982 9.540 9.217 8.968 8.766 8.596 8.452 8.327

8.581 6.904 6.002 5.400 4.956 4.608 4.324 4.086 3.883 3.705

11.040 9.997 9.441 9.070 8.795 8.578 8.401 8.251 8.122 8.009

7.255 5.975 5.282 4.816 4.469 4.195 3.970 3.780 3.616 3.472

15.655 14.258 13.499 12.989 12.609 12.308 12.062 11.854 11.673 11.514

X(1,1)

X(2,2)

X(1,1)

X(2,2)

X(1,

1)

6.014 4.880 4.257 3.837 3.525 3.280 3.080 2.913 2.770 2.645

6.967 5.742 5.059 4.592 4.243 3.966 3.740 3.549 3.385 3.243

7.019 5.433 4.594 4.048 3.654 3.352 3.109 2.910 2.741 2.596

8.214 6.529 5.614 5.003 4.555 4.207 3.924 3.689 3.489 3.316

5.845 4.788 4.219 3.838 3.556 3.333 3.150 2.996 2.864 2.748

Table 5 Diffusion and viscosity-type collision integrals for O(3P)–O(3P) interaction from different authors T (K)

X(1,1)a

X(1,1)b

X(1,1)c

X(1,1)d

X(2,2)a

X(2,2)b

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

6.014 4.880 4.257 3.837 3.525 3.280 3.080 2.913 2.770 2.645

4.689 3.977 3.576 3.300 3.093 2.928 2.791 2.676 2.579 2.494

4.837 4.003 3.567 3.270 3.046 2.866 2.717 2.591 2.481 2.385

5.271 4.386 3.905 3.575 3.335 3.148 2.996

6.967 5.742 5.059 4.592 4.243 3.966 3.740 3.549 3.385 3.243

5.455 4.657 4.210 3.904 3.674 3.490 3.337 3.208 3.099 3.001

a

Present work. Capitelli and Ficocelli [9]. Levin et al. [22]. Yun and Mason [23].

b c d

Table 6 O2 Rydberg molecular states Interaction 5

3

O( P)–O( P) O(3S)–O(3P) O(5S)–O(3P)

Bound states Term

De (eV)

˚) re (A

C

˚) r (A

R g 3  Ru 1 Pg 1 þ Rg 3 Pg

4.527 5.754 5.882 3.576 5.309

1.129 1.123 1.164 1.164 1.124

3.109 2.734 2.511 3.188 2.961

0.922 0.895 0.912 0.956 0.911

3

O(1D)+O(1S) 1)

X(2,2) 7.948 6.453 5.635 5.082 4.669 4.344 4.078 3.853 3.661 3.493

function of temperature. It should be noted that the X(1,1) for asymmetric interactions are higher than the corresponding ones for the symmetric case, reflecting the dominant contribution of inelastic process. The accuracy of the present results can be tested by comparison with theoretical data available in the literature for the interaction of ground state oxygen atoms (see Table 5). In general a satisfactory agreement has been found with the results from different authors, with discrepancies within 20% in the low-temperature region and decreasing with temperature, reaching about 10% at 20,000 K. Differences found with data of Ref. [9], obtained with the same approach, are due to the different potential energy curves adopted (the older electronic structure calculations of Schaefer et al. [24]). This point is confirmed considering the O(1S)–O(1S) interaction, characterized through a single molecular state ð71 Rþ g Þ, whose potential curves from Refs. [21,24] differ, especially in the high collision energy region. On the other hand the disagreement between the current results and those of Levin et al. [22] can be explained since Levin’s collision integrals are based on quantum mechanically derived potential energy surfaces rather than analytic fits. Interaction of a ground state oxygen atom with atomic partners in higher electronic excited states have been also considered, referring to the ab initio quantum calculations of Ref. [25] of few O2 Rydberg states (see Table 6 for fitting parameters).

Table 7 Diffusion and viscosity-type collision integrals for interactions involving high-lying excited states of atomic oxygen (n=3) T (K)

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

O(5P)–O(3P)

O(3S)–O(3P)

ð1;1Þ Xel

ð1;1Þ Xex

X

6.591 5.640 5.022 4.556 4.183 3.874 3.610 3.383 3.183 3.007

13.446 12.349 11.730 11.300 10.972 10.707 10.486 10.296 10.131 9.983

14.975 13.576 12.760 12.184 11.742 11.386 11.090 10.838 10.619 10.426

(1,1)

O(5S)–O(3P)

(2,2)

ð1;1Þ Xel

X

6.462 5.492 4.930 4.526 4.208 3.946 3.722 3.527 3.354 3.200

6.247 5.331 4.741 4.298 3.946 3.653 3.406 3.192 3.006 2.841

14.826 13.451 12.652 12.090 11.660 11.313 11.025 10.779 10.568 10.379

X

(1,1)

w

X

(2,2)

6.162 5.233 4.694 4.306 4.002 3.751 3.538 3.352 3.188 3.041

ð1;1Þ

Xel

X(1,1)w

X(2,2)

6.949 5.945 5.292 4.799 4.404 4.075 3.795 3.553 3.342 3.155

15.136 13.706 12.868 12.277 11.823 11.456 11.152 10.892 10.668 10.470

6.787 5.767 5.176 4.752 4.417 4.141 3.904 3.698 3.516 3.353

ð1;1Þ w Data reported in these columns have been obtained assuming the excitation exchange component, Xex , the same of O(5P)–O(3P) interaction.

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

Results, obtained including only the available interaction potentials, are reported in Table 7 and show that corresponding viscosity-type collision integrals present similar values compared with the O(3P)–O(3P) system. In the case of O(5P)–O(3P) interaction the excitation exchange cross section has been estimated through the splitting g–u, confirming the dominant contribution of inelastic processes to the diffusion collision integrals. Adopting a similar contribution for O(3S)–O(3P) and O(5S)–O(3P), we have estimated also the diffusion-type collision integrals for these interactions. 3.2. Atom–ion interactions In the case of O–O+ interactions, considering the very recent results [12] for charge-exchange cross sections in

17

collision processes between atom–(parent)ion in the ground or in the first excited states, i.e. O(3P,1D,1S)–O+(4S,2D,2P), inelastic contribution to diffusion-type collision integrals have been evaluated, while both viscosity-type collision integrals and the elastic contribution to X(1,1) have been calculated from potential curves of the molecular ion, obtained by quantum mechanical calculations [26]. In Tables 8 and 9 the fitting parameter for the potential curves of the relevant O–O+ interactions have been reported. A different approach has been considered in the case of the O(1S)–O+(4S) interaction due to the strong potentialshape-dependence of the corresponding collision integral values, actually determined only by two slightly bound 4  electronic terms, i.e. 4 R g and Ru not well reproduced using a Morse function. In fact, in this case, Morse-based

Table 8 Oþ 2 valence molecular states Interaction

Repulsive states Term

A (eV)

O( P)–O ( S)

2

Rþ g 4 þ Ru 6 þ Rg 6 Pg 6 Pu

6396.358 2620.368 1657.64 4689.647 1795.856

O(1D)–O+(4S)

4

6903.3

3

+ 4

Pg

Bound states ˚ 1) a (A

Term

De (eV)

re (a.u.)

C

˚) r (A

6.626 5.0621 3.2631 4.9688 3.7796

2

Rþ u 4 þ Rg 6 þ Ru 2 Pg 2 Pu 4 Pg 4 Pu

1.2810 1.7844 1.980 6.1846 2.4321 0.7585 3.509

3.4180 3.484 3.38 2.257 2.7304 3.4557 2.6352

2.7225 2.8118 3.1447 2.7426 3.0422 3.4966 2.9059

1.441 1.479 1.466 0.953 1.177 1.526 1.126

5.225

4

R g R u 4 Pu 4 Dg 4 Du

3.604 0.436 0.809 1.685 1.473

2.4624 3.6613 3.6160 3.446 3.0617

3.0266 4.2316 3.1575 2.7986 3.1345

1.060 1.665 1.569 1.461 1.327

12 Rþ g 12 Rþ u 14 Rþ u 24 Rþ u 2  Rg 2  Ru 4  Rg 12Pg 22Pg 12Pu 22Pu 32Pu 14Pg 12Dg 22Dg 12Du 22Du 14Du 2 Ug 2 Uu

2.583 0.572 2.164 0.444 2.588 2.166 1.693 2.659 0.711 3.590 1.666 0.109 0.566 3.344 0.839 0.872 0.501 2.316 1.755 4.550

3.3132 3.1787 3.4053 4.6893 2.4594 3.0235 3.3576 3.1062 3.3992 2.6789 3.0426 4.2639 4.0559 2.5456 3.506 3.2986 3.5764 3.3156 3.1425 2.6212

2.6005 3.7781 3.0884 3.1946 3.5214 2.5885 2.9295 2.6252 3.3676 2.7974 3.0009 4.2438 3.3198 2.8795 3.5655 2.9933 4.5107 3.229 3.104 2.6384

1.384 1.421 1.471 2.039 1.087 1.262 1.437 1.300 1.492 1.136 1.308 1.939 1.775 1.085 1.553 1.417 1.640 1.444 1.359 1.098

12 Rþ u 12 R u 12 R g 2 1 Ug 12Dg 32 R u 12Du 2 1 Uu

1.088 0.076 1.077 0.125 0.478 0.100 0.700 0.310

4.389 4.852 4.630 4.273 4.693 7.036 3.929 4.772

2.587 4.226 2.077 4.505 2.751 3.214 3.144 3.534

1.832 2.206 1.837 1.960 1.983 3.063 1.703 2.110

Term

A (eV)

˚ 1) a (A

4

O(3P)–O+(2D)

22 Rþ g 22 Rþ u 14 Rþ g 24 Rþ g 4  Ru 32Pg 24Pg 34Pg 14Pu 24Pu 34Pu 14Dg 24Dg 24Du 4 Ug 4 Uu

942.092 27211.6 288.007 544.232 3233.28 27603.4 2454.949 548.341 3772.888 1940.05 951.372 650.87 691.33 2405.124 3957.655 2159.458

3.507 6.570 2.749 2.764 5.111 5.990 4.300 2.986 4.385 3.878 3.234 3.108 2.907 4.016 4.753 3.957

O(1D)–O+(2D)

22 R g 22 R u 22 Rþ g 2  3 Rg 12 Rþ g 22 Rþ u 22Uu 2 2 Ug

371.057 974.61 419.575 3001.983 425.344 1958.854 2706.629 4063.780

2.998 3.487 2.662 4.112 2.871 4.125 3.707 4.204

32Du 22Du 22Dg

1946.064 4489.641 1223.270

3.994 4.851 3.583

18

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

Table 9 Oþ 2 valence molecular states (continued from previous table) Interaction

Repulsive states Term

3

Rþ g Pu(2) 4 Pu(1) 4 Pg(2) 4 Pg(1) 4 Dg 2 þ Ru 2 Pg 2 Du

+ 2

4

O( P)–O ( P)

4

A (eV)

Bound states ˚ 1) a (A

Term

A (eV)

˚ 1) a (A

Term

De (eV)

re (a.u.)

C

˚) r (A

2274.835 1272.822 8973.841 677.106 5896.753 1409.860 1688.289 7073.383 2284.740

3.604 3.266 4.610 2.952 4.627 3.311 4.241 5.340 4.565

Rþ g Pu 4 Du 4 þ Ru 12 R g 22 R g 12 R u 22Dg 14 R g 24 R u 14 R u 24 R g

0.228 1.799 0.361 0.200 1.384 1.079 0.968 0.230 0.301 0.164 0.156 0.141

4.560 4.982 4.953 4.781 3.456 3.433 3.797 3.613 4.952 6.183 6.170 6.260

4.426 1.774 3.687 4.183 2.918 3.377 2.845 4.566 3.326 3.375 3.014 3.436

2.086 1.896 2.206 2.170 1.477 1.507 1.616 1.660 2.168 2.714 2.655 2.756

2

2

O(1D)–O+(2P)

22 Rþ u 2 Uu 2 Ug

13629.47 2808.237 2660.695

5.140 3.702 3.673

12 Rþ u 22 Rþ g 2  Rg 2  Ru 22Du 12 Rþ g

0.469 0.055 0.120 0.025 0.272 0.0523

4.283 6.290 5.193 6.397 5.228 6.114

3.460 4.098 3.929 4.693 3.507 4.041

1.888 2.847 2.335 2.949 2.309 2.761

O(1S)–O+(2D)

2

R g

4156.57

4.009

2

R u Du

0.120 0.076

5.597 6.443

3.979 3.796

2.522 2.883

2

Table 10 Hulburt Hirschfelder interpolation coefficients of Oþ 2 valence molecular states correlating with O(1S)–O+(4S) ˚) Term De (eV) re (A aHH bHH cHH 4 4

R g R u

0.789 0.571

2.070 2.239

4.613 5.551

16.380 31.400

1.460 1.633

calculated collision integrals present very high values, not compatible with the general framework. For these reasons potential energy curves, characterized by small potential wells and, in the case of the 4 R u state, by a short-range relative minimum, have been fitted using the more suitable Hulbert Hirschfelder potential function [27]        r r u ¼ De exp 2aHH  1  2 exp aHH 1 re re  3    r r þbHH 1 1 þ cHH 1 re re    r  exp 2aHH 1 ð15Þ re

Potential parameters have been presented in Table 10 and the corresponding collision integrals calculated by using a novel algorithm able to handle any kind of potential type, regardless the number of extrema [28]. Concerning the inelastic contribution to X(1,1), in Table 11 the parameters C and D, entering in Eq. (13), derived by interpolation of charge-exchange cross sections of Ref. [12], are reported. In Table 12 collision integrals for all the interactions are presented. Note that in the case of charge-exchange processes only transitions characterized by a variation of the total and spin angular momenta equal to DL = 0, 1 and DS = 1 are taken into account, i.e. elementary processes involving the exchange of more than one electron are forbidden. Due to this selection rules the exchange cross sections for the interactions O(1D)–O+(4S), O(1S)–O+(4S) and O(1S)–O+(2D) have been considered negligibly small and so the corresponding collision integrals, Xð1;1Þ ex , not included in Table 12. ð1;1Þ Concerning the O(1S)–O+(2P) interaction, only the Xex contribution has been calculated. However considering

Table 11 C and D parameters, entering in Eq. (13), for the estimation of Xð1;1Þ ex , obtained by interpolation of charge-exchange cross sections from different authors Interaction

Ref. [12] C

3

+ 4

O( P)–O ( S) O(3P)–O+(2D) O(3P)–O+(2P) O(1D)–O+(2D) O(1D)–O+(2P) O(1S)–O+(2P)

22.468 20.038 18.824 22.703 20.101 22.695

Interaction

Ref. [16]

D 0.924 0.909 0.855 1.015 0.924 1.003

4 3

+ 4

O(2p P)–O ( S) O(2p33s 5S)–O+(4S) O(2p33s 3S)–O+(4S) O(2p34s 5S)–O+(4S) O(2p35s 5S)–O+(4S)

C

D

22.938 49.133 52.499 110.47 159.16

1.059 1.844 1.989 3.758 5.173

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

19

Table 12 ˚ 2) for interactions of atomic oxygen in ground and Diffusion-type (with elastic and charge-exchange contributions) and viscosity-type collision integrals (A excited (low-lying) states T (K)

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

O(3P) O+(4S) ð1;1Þ Xel

Xð1;1Þ ex

7.385 5.977 5.151 4.580 4.153 3.818 3.545 3.319 3.127 2.961

36.757 34.599 33.367 32.506 31.846 31.312 30.864 30.479 30.141 29.840

O(3P)–O+(2D)

O(3P)–O+(2P)

X

X

(2,2)

ð1;1Þ Xel

Xð1;1Þ ex

X

X

37.492 35.111 33.762 32.827 32.116 31.544 31.067 30.659 30.303 29.987

7.775 6.408 5.632 5.089 4.677 4.348 4.076 3.847 3.650 3.478

7.461 5.937 5.080 4.501 4.075 3.743 3.476 3.255 3.069 2.909

23.024 21.350 20.400 19.739 19.234 18.826 18.485 18.192 17.935 17.707

24.203 22.160 21.023 20.246 19.661 19.194 18.809 18.481 18.196 17.944

8.035 6.549 5.711 5.137 4.707 4.367 4.090 3.857 3.658 3.485

(1,1)

(1,1)

(2,2)

ð1;1Þ

Xel

ð1;1Þ Xex

X(1,1)

X(2,2)

9.186 6.696 5.441 4.661 4.123 3.726 3.420 3.175 2.975 2.807

20.264 18.788 17.950 17.367 16.922 16.562 16.261 16.002 15.776 15.575

22.249 19.946 18.757 17.982 17.417 16.976 16.617 16.314 16.054 15.826

10.030 7.611 6.341 5.519 4.933 4.490 4.142 3.860 3.626 3.429

O(1D)–O+(4S)

O(1D)–O+(2D)

O(1D)–O+(2P)

ð1;1Þ Xel

X(2,2)

ð1;1Þ Xel

ð1;1Þ Xex

X(1,1)

X(2,2)

ð1;1Þ Xel

O(1S)–O+(4S)

ð1;1Þ Xex

X(1,1)

X(2,2)

Xel

ð1;1Þ

X(2,2)

8.805 6.741 5.559 4.768 4.193 3.755 3.407 3.125 2.890 2.691

8.947 7.106 6.032 5.289 4.732 4.294 3.940 3.645 3.396 3.183

8.168 5.902 4.809 4.146 3.694 3.362 3.107 2.902 2.734 2.593

30.773 28.612 27.385 26.530 25.876 25.348 24.906 24.526 24.194 23.898

31.839 29.214 27.804 26.852 26.138 25.570 25.099 24.697 24.348 24.038

9.067 6.816 5.673 4.954 4.451 4.075 3.780 3.543 3.346 3.179

7.139 5.086 4.134 3.565 3.178 2.894 2.676 2.501 2.357 2.236

22.382 20.707 19.757 19.096 18.592 18.185 17.844 17.551 17.295 17.068

23.493 21.322 20.185 19.426 18.862 18.414 18.044 17.728 17.455 17.214

8.172 5.983 4.923 4.273 3.825 3.493 3.236 3.029 2.858 2.715

9.082 6.808 5.474 4.643 4.085 3.683 3.378 3.138 2.942 2.780

9.857 7.773 6.358 5.439 4.822 4.382 4.048 3.784 3.567 3.384

O(1S)–O+(2D)

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

O(1S)–O+(2P)

Xel

ð1;1Þ

X(2,2)

Xð1;1Þ ex

7.453 4.908 3.822 3.201 2.793 2.500 2.279 2.105 1.964 1.847

8.759 5.886 4.611 3.871 3.380 3.027 2.760 2.549 2.378 2.235

31.636 29.468 28.236 27.378 26.721 26.190 25.746 25.364 25.029 24.732

Table 13 Diffusion and viscosity-type collision integrals for O(3P) O+(4S) interaction from different authors X(1,1)a

X(1,1)b

X(1,1)c

X(2,2)a

X(2,2)b

X(2,2)c

2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000 20,000

36.757 34.599 33.367 32.506 31.846 31.312 30.864 30.479 30.141 29.840

24.88 23.11 22.21 21.60 21.13 20.76 20.44 20.17 19.93 19.72

25.49 23.87 22.95 22.31 21.82 21.42 21.09 20.80 20.55 20.33

7.736 6.397 5.623 5.083 4.671 4.343 4.072 3.843 3.646 3.473

8.72 6.94 5.95 5.26 4.75 4.36 4.05 3.80 3.58 3.41

7.40 6.09 5.29 4.73 4.31 3.96 3.73 3.51 3.33 3.18

a b c

Present work. Stallcop et al. [3]. Capitelli et al. [1].

n=5

Ω(1,1) [Å2] O+-O(n)

T (K)

104

n=3 102

n=2 101 1.70

ð1;1Þ Xel does not 1 + 2

reliable the assumption that differ significantly from the values for the O( S)–O ( D) interaction, the diffusion-type collision integral derived on the base of

n=4

103

1.75

1.80

1.85

1.90

1.95

2.00

2-(1/n2) Fig. 1. Diffusion-type collision integrals as function of the principal quantum number at T = 10,000 K for O+(4S)–Ow interactions.

20

A. Laricchiuta et al. / Chemical Physics 344 (2008) 13–20

the charge-exchange contribution should be considered a good approximation. At the same time the viscosity-type collision integrals, not calculated due to the lack of corresponding potentials, can be considered equal, in a first approximation, to the corresponding ones for the interaction O(1S)–O+(2P). Collision integral accuracy can be assessed by comparison with other theoretical results for the O(3P)–O+(4S) collisions (Table 13). The present diffusion-type collision integrals, based on recent and accurate cross sections [12], are systematically higher, up to 30%, than all other calculations, including that based on experimental charge-transfer cross sections. On the other hand the present viscosity-type collision integrals for the same interaction differ from the accurate values of Stallcop et al. [3] from 11% to 1.8% in the temperature range 2000– 20,000 K. The diffusion-type collision integrals for interactions involving excited states, compared with results of Ref. [11], are at least a factor 2 higher than the old calculations, due to the differences in the corresponding charge-transfer cross sections. Finally in Fig. 1 diffusion-type collision integrals for the interaction of a ground state oxygen ion, O+(4S), with highly excited atomic partners, at fixed temperature, is displayed (for n = 3 values of both O(5S)–O+(4S) and O(3S)– O+(4S) interactions are reported). Collision integrals have been estimated, through the inelastic contribution (Table 11), from the corresponding charge-transfer cross sections, derived in Ref. [16] by an asymptotic approach. The collision integral value strongly depend on the principal quantum number of the atomic valence shell, according to the case of electronically excited atomic hydrogen colliding with protons [14]. 4. Conclusions In the present work a complete set of diffusion, X(1,1), and viscosity-type, X(2,2), collision integrals have been derived for interactions between oxygen atoms and oxygen atom with parent ion in electronically excited states. The accuracy has been estimated by comparison with available theoretical results in the literature for ground state interactions, finding, in general, a satisfactory agreement. Collision integrals for electronically excited states can be regarded as a further step in the creation of a complete database for the interactions occurring in air plasmas and an improvement in respect with the scarce and fragmentary results existing in the literature. Work is in progress for the estimation of the role of lowlying and high-lying electronically excited states in affecting the transport properties of LTE and non-LTE oxygen plasmas, the present data representing the input data of the relevant transport equations.

Acknowledgements The present work has been partially supported by MIUR PRIN 2005 (Project 2005039049_005), MIUR PRIN 2004 (Project 2004092040_003), University of Bari (Project No. 63396 III/11 2007) and MIUR FIRB (Project RBAU01H8FW_003).

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