Eur. Phys. J. D 53, 189–191 (2009) DOI: 10.1140/epjd/e2009-00110-1
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Low energy o-Ps–o-Ps elastic scattering using a simple model Himanshu Sharma1 , Kiran Kumari2 , and Sumana Chakraborty3,a 1 2 3
University Department of Physics, Veer Kunwar Singh University, Arrah-802301, Bihar, India P. G. Department of Physics, R N College, Hajipur (Vaishali)-844101, Bihar, India Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India Received 21st January 2009 / Received in final form 4 March 2009 c EDP Sciences, Societ` Published online 26 March 2009 – a Italiana di Fisica, Springer-Verlag 2009 Abstract. A simple model is employed to investigate o-Ps–o-Ps scattering at low energies. This model contains the effect of exchange explicitly and a model long range potential in the framework of staticexchange model. These two physical features are of key importance in Ps-Ps (atom-atom) scattering system. S-wave triplet-triplet and singlet-singlet scattering lengths and corresponding phase shifts up to the incident momentum k = 0.5 a.u. are in excellent agreement with those yielded by most elaborate and theoretically sound predictions. PACS. 36.10.Dr Positronium – 34.50.-s Scattering of atoms and molecules
1 Introduction In recent years, the interactions of positrons and positroniums (Ps) with atoms and molecules are of topical interest. Properties and dynamics of a positronium atom containing a positron and an electron have received considerable attention both experimentally and theoretically. These studies necessarily include the formation of positronium molecule (Ps2 ) and the Bose-Einstein condensate (BEC) with spin polarized positronium atoms. Since Ps is the lightest atom, its critical temperature for BoseEinstein transition is expected to be high. Knowledge of Ps-Ps scattering will be helpful in tuning an experimental procedure to trap Ps and form a stable BEC. The physical extent of an atom is characterized by the s-wave scattering length. For the formation of a stable BEC, the triplet scattering length has to be positive. Experimentally, Harvard group [1] is trying to obtain the BEC of the Ps atoms. Due to complicacies in handling a fourfermionic system at low energies [2], very few calculations have been carried out for this system. In an attempt, Oda et al. [3] employed semi-empirical van der Waals and Lenard-Jones potentials tuned to Ps2 binding energy. They predicted the value of scattering length and effective range for the total spin S = 0. Three theoretically sound models have been applied to the system: Ivanov et al. [4,5] investigated the system using the stochastic variational method (SVM) which is restricted to elastic process only; Shumway and Ceperley [6] employed quantum Monte Carlo (QMC) model for the s-wave elastic scattering; the most recent calculation was carried out by a
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Chakraborty et al. [2,7] using ab-initio variationally consistent close coupling approximation (CCA) model with different basis sets. All the s-wave triplet-triplet scattering lengths (total spin S = 2) are in good agreement. The above mentioned three methodologies are complicated and laborious. Here we consider a simple model to reinvestigate o-Ps–o-Ps scattering system. In Ps-Ps scattering, the effect of electron (positron) exchange and lowest order long range van der Waals force are of key importance [2]. In our model, we pay attention to these physical aspects. We employ here a static-exchange model. This model takes into account the effect of exchange explicitly. The total wave function for the static-exchange model is written as = √1 (1 ± Pr12 ) (1 ± Px12 ) ψ pm ( ρ1 , ρ 2 , R) 4 ρ1 )η1s ( ρ2 )F11 (R). ×ω1s (
(1)
Here Pr12 and Px12 are the electron and positron exchange operators, respectively. ω1s and η1s are the ground state wave functions describing the bound states of the colliding Ps atoms. From the symmetry considerations, equation (1) can be recasted as: = 1±Pr (x ) ω1s ( (2) ψ pm ( ρ1 , ρ 2 , R) ρ1 )η1s ( ρ2 )F11 (R). 12 12 The corresponding Schr¨ odinger equations are expressed as 1 ± B± (k , k) = f1s1s,1s1s (k , k) − 2 dk f1s1s,1s1s 2π B± ± f1s1s,1s1s (k , k )f1s1s,1s1s (k , k) × 2 k1s1s − k 2 + iε (3)
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Fig. 1. Comparison of the s-wave triplet-triplet phase shifts of present model with those of CCA, SVM and QMC; curves: solid, present; dashed, CCA [2]; dotted, SVM [5]; dash-dotted, QMC [6].
Fig. 2. Comparison of the s-wave singlet-singlet phase shifts of present model with those of CCA, SVM and QMC; curves: solid, present; dashed, CCA [2]; dotted, SVM [5]; dash-dotted, QMC [6].
where
quantum number. Our main motivation is to find out the reliability of our model by comparing the present predictions with existing theoretically sound results. The van der Waals coefficient C6 obtained by Chakraborty et al. [2] is 203.4 a.u. and value used by Oda et al. [3] is 192 a.u. Au and Drachman [8] obtained the van der Waals coefficient C6 = 207.96886 a.u. Assuming the value of van der Waals coefficient calculated by Au and Drachman is most accurate; we use this value in the present study. Now we have one parameter, rc is to be determined. We have taken rc = 1.71 a0 so that the present s-wave triplet-triplet (S = 2) phase shift coincides with the CCA value [2] at k = 0.1 a.u. (0.136 eV). The present value of the phase shift with rc = 1.71 a0 is found to be in the vicinity of other two theoretically sound values of phase shifts (SVM and QMC). We have also observed that the value of rc is sensitive in predicting scattering parameters. With these values of C6 and rc we solve the one dimensional two sets of coupled integral equations (Eq. (6)) numerically. Details of numerical methods are given by Chakraborty and Ghosh [7].
± B B (k , k) = f1s1s,1s1s (k , k) ± g1s1s,1s1s (k , k). (4) f1s1s,1s1s
Here f B and g B are the plane wave direct and exchange scattering amplitudes, respectively. It is well known that plane wave direct scattering amplitude (f B ) vanishes for the even parity transitions of each Ps atom. Findings of CCA (comparing the results of model (c) and model (d) of Chakraborty et al. [2]) indicate that the lowest order van der Waals potential is sufficient to predict low energy scattering parameters. Considering this we include only the lowest order long range van der Waals force in the direct channel by a model potential which is of the form ⎡ 6 ⎤ − r 1 − e rc ⎢ ⎥ V6 = −C6 ⎣ (5) ⎦ r6 where rc the cut-off radius, r is the distance between the centers of mass of the two atoms and C6 is the van der Waals coefficient. However, we have neglected the long range effect in the exchange channel. In the present case, plane wave direct scattering amplitudes survive due to the presence of van der Waals interaction. After partial wave analysis one dimensional two sets of coupled integral equations (for total spin S = 0 and S = 2) are obtained. The resulting equations take the form [2,7] 1 T J± (τ k ; τ k) = B J± (τ k ; τ k) − 2 dk k 2π B J± (τ k ; τ k )T J± (τ k ; τ k) × 2 k1s1s − k 2 + iε (6) where τ ≡ (1, 0, 1, 0, L, J1), L is the orbital angular momentum of the continuum wave and J is the good
2 Results and discussion Here we present the results up to the incident energy 3.4 eV which corresponds to the incident momentum 0.5 a.u. In Figures 1 and 2 we show s-wave triplet-triplet (S = 2) and singlet-singlet (S = 0) phase shifts respectively from k = 0.01 a.u. to k = 0.5 a.u. Both the phase shifts are compared with those of three theoretically sound CCA, SVM and QMC calculations. Our singlet-singlet phase shifts are in good agreement with the other results. At the highest incident momentum k = 0.5 a.u., our results deviate slightly from the other results. On the other hand, the triplet-triplet phase shifts are in very good agreement with the other elaborate theoretical predictions
H. Sharma et al.: Low energy o-Ps–o-Ps elastic scattering using a simple model Table 1. The s-wave elastic scattering lengths for the o-Ps–o-Ps scattering for total spin S = 0 and S = 2. Models
Scattering length (a.u.)
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a suitable long range forces. As our incident wave is very accurate we plan to calculate single and double excitation cross sections of o-Ps–o-Ps scattering using distorted wave model in the near future.
Total spin S = 0 Total spin S = 2 Present model
8.59
2.99
Chakraborty et al. [2]
9.32
2.95
Oda et al. [3]
8.26
3.02*
SVM [5]
8.443
2.998
The authors are thankful to Prof A.S. Ghosh for his keen interest during the progress of this investigation. One of the authors, K.K. is thankful to UGC, Government of India for partial financial support (UGC Project No: PSB-021/06-07 ERO dated 23 Feb. 2007).
9.148 ± 0.042 3.024 ± 0.058 QMC [6] * Estimated on physical insight about the nature of collision.
References throughout the energy range considered here. In Table 1 we exhibit the present singlet-singlet and triplet-triplet s-wave scattering lengths along with those reliable predictions available. It is apparent that our triplet-triplet scattering length (total spin S = 2) is in excellent agreement with others and corresponding singlet-singlet scattering length (total spin S = 0) is also in close agreement with those of others. In this work we have presented a model which is very easy to apply and computer time consumed is equivalent to the time taken to solve static-exchange model. Our model predicts reliable scattering parameters of o-Ps–o-Ps system at low energies. Our long range van der Waals force contains only one adjustable parameter that is cut-off radius rc . Present phase shifts and scattering lengths are in excellent agreement with those of existing elaborate and theoretically sound results. This model can be used also in other atom-atom scattering systems with
1. D.B. Cassidy, J.A. Golovchenko, New Directions in Antimatter Chemistry and Physics, edited by C.M. Surko and F.A. Gianturco (Kluwer Academic Publishers, Dordrecht, 2001) 2. S. Chakraborty, A. Basu, A.S. Ghosh, Nucl. Instrum. Meth. B 221, 112 ( 2004) 3. K. Oda, T. Miyakawa, Y. Yabu, T. Suzuki, J. Phys. Soc. Jpn 70, 1549 (2001) 4. I.A. Ivanov, J. Mitroy, K. Varga, Phys. Rev. Lett. 87, 063201 (2001) 5. I.A. Ivanov, J. Mitroy, K. Varga, Phys. Rev. A 65, 022704 (2002) 6. J. Shumway, D.M. Ceperley, Phys. Rev. B 63, 165201 (2001) 7. S. Chakraborty, A.S. Ghosh, Phys. Rev. A 72, 052508 (2005) 8. C.K. Au, R. Drachman, Phys. Rev. Lett. 56, 324 (1986)
