Radiative charge transfer and radiative association of protons colliding with Na at low energies

Physics Letters A 373 (2009) 3761–3763

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Physics Letters A www.elsevier.com/locate/pla

Radiative charge transfer and radiative association of protons colliding with Na at low energies C.H. Liu a , Y.Z. Qu a,b,∗ , J.G. Wang b,c , Y. Li d , R.J. Buenker d a

College of Physical Sciences, Graduate University of the Chinese Academy of Sciences, PO Box 4588, Beijing 100049, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collisions, Lanzhou 730000, China c Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China d Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany b

a r t i c l e

i n f o

a b s t r a c t Using the fully quantum-mechanical approach, the radiative charge transfer for H+ + Na(3s) collisions has been investigated. The charge transfer emission spectra are analyzed at resonant and non-resonant collision energies. The radiative association cross sections, obtained by subtracting the radiative charge transfer part from total radiative decay cross sections calculated by the optical potential method, are presented in the energy range 10−6 –1 eV. © 2009 Elsevier B.V. All rights reserved.

Article history: Received 18 May 2009 Received in revised form 26 July 2009 Accepted 13 August 2009 Available online 15 August 2009 Communicated by P.R. Holland PACS: 34.70.+e 34.20.-b Keywords: Atomic collision Charge transfer Radiative association

1. Introduction Collisions of atoms at ultracold temperatures have received considerable attention because of their importance in the cooling and trapping of atoms [1] and molecules [2,3]. In ultracold atomic systems, electric charges may play an important role [4,5]. The distribution of sodium between its ionized and neutral forms is important for interpreting the observations of the Na resonance lines in the atmospheres of planets and comets [6]. Charge transfer between neutral sodium atoms and protons may modify the ionization distribution in stellar winds [7]. The radiative decay including the radiative charge transfer process

Na(3s) + H+ → Na+ + H(1s) + hν ,

(1)

and the radiative association reaction





Na(3s) + H+ → NaH+ X 2 Σ + + hν

(2)

has been investigated by Watanabe et al. [8] using an optical potential method for which the molecular data were obtained by

a pseudopotential approach. In our previous work [9], we have reexamined the radiative decay cross sections using the optical potential and semiclassical method, and compared them with those of Watanabe et al. [8]. The discrepancies between the results of Watanabe et al. [8] from ours are thought to result from inaccuracies in their calculation as well as differences in transition probabilities [9]. 2. Theoretical method In the present work, we use the fully quantum-mechanical approach [10,11] to calculate the radiative charge transfer cross sections and the emission spectrum. The radiative association cross sections are obtained from the differences between the radiative decay and the radiative charge transfer cross sections. The radiative decay cross sections themselves are calculated by the optical potential method. Atomic units will be used in the remaining part of this article, unless explicitly indicated otherwise. In the fully quantum-mechanical approach [10,11], the radiative charge transfer cross section is given by

*

Corresponding author at: College of Physical Sciences, Graduate University of the Chinese Academy of Sciences, PO Box 4588, Beijing 100049, China. Tel.: +86 10 82640447. E-mail address: [email protected] (Y.Z. Qu). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.08.022

ω max

σ= ωmin

dσ dω

dω ,

(3)

3762

C.H. Liu et al. / Physics Letters A 373 (2009) 3761–3763

with

dσ dω

=



8 3

2

π

ω3 c3

kA

×





J M 2J , J −1 (k A , k X ) + ( J + 1) M 2J , J +1 (k A , k X ) ,

(4)

J

where ω is the angular frequency of the emitted photon, c is the speed of light, and J is the angular momentum quantum number

∞ dR f JA (k A R ) D ( R ) f JX (k X R ).

M J , J  (k A , k X ) =

(5)

0

In this expression, D ( R ) is the transition dipole moment connecting the two molecular electronic states, k A and k X are the entrance and exit momenta, respectively, given by

kA = kX =







2μ E − V A (∞) ,







2μ E − V X (∞) − h¯ ω ,

(6)

Fig. 1. Adiabatic potentials of (NaH)+ as a function of internuclear distance. The X 2 Σ + and A 2 Σ + correspond to Na+ + H(1s) and Na(3s) + H+ in the asymptotic regions, respectively.

where E is the collision energy in the entrance channel, and V A and V X are the adiabatic potential energies of the entrance and exit channels, respectively (R is the internuclear distance and μ is the collision system reduced mass). The partial wave f iJ (ki R ) (i = A , X ) is the regular solution of the homogeneous radial equation



d2 dR 2

J ( J + 1)

−

R2

 − 2μ V i ( R ) − V i (∞) + k2i f iJ (ki R ) = 0, (7)

and normalized asymptotically according to

f

i J (k i R )

=

2μ

π ki

 sin ki R −

Jπ 2

 + δ iJ ,

(8)

where δ iJ (i = A , X ) are the phase shifts. On the other hand, the optical potential method [10–12] can be adopted to obtain the total cross sections for radiative decay, including both the radiative charge transfer and radiative association processes. Total radiative decay cross sections can be written as

σ (E) =

∞ π 

 (2 J + 1) 1 − exp(−4η J ) ,

k2A

J

and the phase shift tion as

ηJ =

π

∞



η J is given in the distorted-wave approxima2

dR  f JA (k A R ) A ( R ),

2

(9)

(10)

0

where A ( R ) is the transition probability for the radiative transition given by

A(R ) =

4 3

Fig. 2. Radiative decay cross sections, radiative charge transfer cross sections and radiative association cross sections for H+ + Na(3s) collisions. Solid line: radiative decay cross sections calculated by the optical potential method; solid points: radiative charge transfer cross sections from the fully quantum-mechanical approach; solid line with open circles: radiative association cross sections.

D2(R)

| V A ( R ) − V X ( R )|3 c3

.

(11)

3. Results and discussion The adiabatic potentials and dipole matrix elements are calculated in the internuclear distance range from R = 1.4–30 a.u. [9] by using the multireference single- and double-excitation configuration interaction (MRD-CI) method [13,14]. Fig. 1 shows that the adiabatic potentials X 2 Σ + and A 2 Σ + correspond to Na+ + H(1s) and Na(3s) + H+ states in the asymptotic regions, respectively.

The radiative charge transfer cross sections and the radiative decay cross sections [9], calculated by the fully quantummechanical approach and the optical potential method, respectively, are shown in Fig. 2. The radiative association results (obtained by subtracting the radiative charge transfer part from the radiative decay cross sections) are also displayed in Fig. 2. In the present system of interest, since the lower X 2 Σ + state only has a very shallow well (∼ 0.125 eV) at short range, the radiative charge transfer is responsible for most of the radiative decay process [9,12]. The radiative association cross sections are about one order of magnitude smaller than the radiative charge transfer cross sections at energies from 10−6 eV to 10−3 eV. As the collision energies increase, the differences between the radiative decay and charge transfer cross sections decreases. The radiative association cross sections decrease more rapidly than the radiative decay ones, so that they becomes more than two orders of magnitude smaller than the radiative decay cross sections at 1 eV. This is because of the fact that the effective angular momentum quantum numbers increase with collision energy, and this results in a shallower 2 well of the effective potential V eff J ( R ) = V ( R ) + J ( J + 1)/2μ R for the lower state, causing the number of the quasi-bound vibra-

C.H. Liu et al. / Physics Letters A 373 (2009) 3761–3763

(a)

3763

the total charge transfer emission spectra are obtained by summing the contributions of each partial wave until a convergence of dσ /dω is achieved. At the collision energy of 1 meV, near resonant energy, the charge transfer spectrum is mainly due to the contribution of the J = 20 partial wave as shown in Fig. 3(a). While at 10 meV, the non-resonant energy, there is no dominant partial wave, as shown in Fig. 3(b), the larger partial waves ( J = 21–35) make more contributions, but the smaller ones ( J = 0–20) cannot be neglected. Also, the magnitude of the spectrum at the resonance energy is about one order larger than that at nonresonance energy, resulting in the peaks of the cross section. The oscillatory structure in Fig. 3 reflects the phase of the outctp going wave of momentum k X at R A of the incoming partial wave of momentum k A . The pronounced peak near 1510 Å is similar to that reported in Ref. [11] for the Li + H+ system, and does not correspond to the wavelength of the asymptotic energy defect (1442 Å) as seen in the He+ + H system [17]. The attractive portion of the potential well results in a larger phase than the asymptotic phase for f JA . Therefore, the optimization of M J , J  requires k A < k X giving h¯ ω ∼ V A ( R e ) − V X (∞), where R e is the equilibrium distance [11]. The resonance-like structures at short wavelengths near threshold are due to the orbiting effects in the ground state X 2 Σ + [11,17] because of the attractive nature of its potential curve. If the outgoing energy is near the maximum of the effective potential for the lower state, the wave function has a resonance-like structure. 4. Conclusion

(b) Fig. 3. (a) The total charge transfer emission spectrum (solid line) and the spectrum for the J = 20 partial wave (dotted line) at the resonance energy of 1 meV. (b) The total emission spectrum (solid line), the spectrum of the J = 0–20 partial wave (dashed line) and the J = 21–35 (dotted line) partial wave at the non-resonance energy of 10 meV.

tional levels to become smaller. As a result, the interaction time for emitting the radiation is reduced [15], and this results in the reduction of the cross sections for the radiative association processes. Since the radiative association cross sections are obtained as the differences between the radiative decay and charge transfer cross sections in our calculation, the accuracy is not sufficient in the energy region E > 10−3 eV. Consequently, the radiative association cross section become increasingly less smooth, especially at the position of resonances. The resonant structures that appear in the energy region 10−5 –0.2 eV are attributed to the presence of quasi-bound or virtual rotational–vibrational levels in the entrance channel [9,11,12]. Regarding the calculation the calculation accuracy of radiative association cross section obtained by subtracting the radiative charge transfer part from the radiative decay cross sections, Zygelman and Dalgarno had pointed out that at energy of 1 meV for the He+ + H system [16], the inaccuracy produced by subtracting is less than 1%. At high collision energies the accuracy is also quite good for the cross sections for J < 10, but for high value of J there is a discrepancy of between 20% and 30%. So, we estimated that the inaccuracy produced by subtracting is less than 30%. The resonant behaviour can be elucidated with the total charge transfer emission spectra as shown in Fig. 3. By using Eq. (4),

We have calculated the radiative charge-transfer cross sections in collisions of protons with Na(3s) atoms for collision energies from 10−6 eV to 1 eV using the fully quantum-mechanical approach. The radiative association cross sections are obtained from the differences between the radiative decay and the radiative charge transfer cross sections. The charge transfer emission spectrum at resonant and non-resonant energies are also presented and analyzed. These results will be useful in simulations of the ultracold collision processes. Acknowledgements This work was supported in part by National Science Foundation of China under Grant Nos. 10774186, 10876043 and 10676014, and the National Key Laboratory of Computational Physics Foundation (No. 9140C6904030808). References [1] J. Weiner, V.S. Bagnato, S. Zilio, P.S. Julienne, Rev. Mod. Phys. 71 (1999) 1. [2] A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, P. Pillet, Phys. Rev. Lett. 80 (1998) 4402. [3] N. Balakrishnan, R.C. Forrey, A. Dalgarno, Phys. Rev. Lett. 80 (1998) 3224. [4] R. Côté, V. Kharchenko, M.D. Lukin, Phys. Rev. Lett. 89 (2002) 093001. [5] T.C. Killian, S. Kulin, S.D. Bergeson, L.A. Orozco, C. Orzel, S.L. Rolston, Phys. Rev. Lett. 83 (1999) 4776. [6] C. Courbin, R.J. Allan, P. Salas, P. Wahnon, J. Phys. B 23 (1990) 3909. [7] T. Royer, D. Dowek, J.C. Houver, J. Pommier, N. Andersen, Z. Phys. D 10 (1988) 45. [8] A. Watanabe, C.M. Dutta, P. Nordlander, M. Kimura, A. Dalgarno, Phys. Rev. A 66 (2002) 044701. [9] C.H. Liu, Y.Z. Qu, Y. Zhou, J.G. Wang, Y. Li, R.J. Buenker, Phys. Rev. A 79 (2009) 042706. [10] B. Zygelman, A. Dalgarno, Phys. Rev. A 38 (1988) 1877. [11] P.C. Stancil, B. Zygelman, Astrophys. J. 472 (1996) 102. [12] L.B. Zhao, J.G. Wang, P.C. Stancil, J.P. Gu, H.-P. Liebermann, R.J. Buenker, M. Kimura, J. Phys. B 39 (2006) 5151. [13] R.J. Buenker, R.A. Phillips, J. Mol. Struct.: THEOCHEM 123 (1985) 291. [14] S. Krebs, R.J. Buenker, J. Chem. Phys. 103 (1995) 5613. [15] F.A. Gianturco, P.G. Giorgi, Astrophys. J. 479 (1997) 560. [16] B. Zygelman, A. Dalgarno, Astrophys. J. 365 (1990) 239. [17] B. Zygelman, A. Dalgarno, M. Kimura, N.F. Lane, Phys. Rev. A 40 (1989) 2340.