- Email: [email protected]
Hydrogen–antihydrogen interactions
Nuclear Instruments and Methods in Physics Research B 143 (1998) 218±224
Hydrogen±antihydrogen interactions E.A.G. Armour *, J.M. Carr Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, UK Received 24 October 1997; received in revised form 26 January 1998
Abstract A small number of antihydrogen (AH) atoms have recently been prepared at CERN and at Fermilab. However, these atoms were travelling at speeds close to that of light. It is intended to carry out experiments on AH by trapping it at very low temperature (<1 K) in an inhomogeneous magnetic ®eld. The main cause of loss of AH is due to collisions with H2 and He with energies up to room temperature. However, these reactions are not easy to treat theoretically. In this paper we consider the interaction between AH and H. This has already received some attention. Initially, in a collision between AH and H the electron is bound to the proton and the positron is bound to the antiproton. Clearly, if the proton and antiproton coincide they cannot bind the two light particles. There exists a critical value, Rc , of the internuclear distance, probably not very much below a0 , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. As a ®rst stage in our work on the H±AH interaction, we are carrying out variational calculations of the energy of the H±AH system for internuclear separations a short distance above Rc . The aim is to determine Rc as accurately as possible. The basis set used is similar to that of a previous calculation by W. Koøos, D.L. Morgan, D.M. Schrader, L. Wolniewiez, Phys. Rev. A 11 (1975) 1792. However, it also contains a function to represent weakly bound positronium. Initial results suggest that Rc < 0.8a0 . Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 3430; 3610 Keywords: Antihydrogen; Hydrogen; Collision; Annihilation; Positronium; Protonium
1. Introduction A small number of antihydrogen (AH) atoms have recently been prepared at CERN [1] and at Fermilab. However, these atoms were travelling at speeds close to that of light. The aim of present
* Corresponding author. Tel.: +44 115 9514922; fax: +44 115 9514951; e-mail: [email protected].
experimental work is to produce AH atoms in suf®cient quantities and at suciently low temperatures to make it possible to use high precision laser spectroscopy to probe the spectrum of AH. Comparison of the results with corresponding results for H would make possible a detailed test of the CPT invariance of relativistic quantum mechanics and also of Einstein's principle of equivalence, which is the cornerstone of the general theory of relativity.
0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 2 1 3 - 4
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
It is intended to carry out the experiments with AH by trapping the AH at very low temperature (<1 K) in an inhomogeneous magnetic ®eld [2]. The main cause of loss of AH is due to collisions with H2 and He with energies up to room temperature. In view of this, there is considerable interest in obtaining accurate theoretical values for cross sections for these collision processes in order to determine annihilation rates under various experimental conditions. This should make it possible to choose the simplest method for maximizing the AH lifetime [3]. However, the H2 and He reactions are not easy to treat theoretically. In this paper we consider the interaction between AH and H. This has already received some attention [4±8]. Initially, in a collision between AH and H the electron is bound to the proton and the positron is bound to the antiproton. Clearly, if the proton and antiproton coincide they cannot bind the two light particles. There exists a critical value, Rc , of the internuclear distance, probably not much below a0 , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. This is similar to the situation in the case of an electron or a positron in the ®eld of a proton and an antiproton. It is well known that in this case no bound state of this system exists if the internuclear distance is less than 0.639a0 . See, for example, Refs. [9±14].
2. Calculation As a ®rst stage in our work on the H±AH interaction, we are carrying out variational calculations of the energy of the H±AH system for internuclear separations a short distance above Rc . The aim is to determine Rc as accurately as possible. The basis set used is similar to that of Koøos et al. [6] in that it contains functions of the form, 1 mi ni ji ki ÿa1 k1 ÿa2 k2 b1 l1 b2 l2 k k l l e Wi 2p 1 2 1 2 ÿ1ji ki kn1i km2 i lk1i lj2i eÿa2 k1 ÿa1 k2 ÿb2 l1 ÿb1 l2 s12 pi : 1
219
However, it also contains a function of the form ÿjq e WPs g qUPs r12 2 q to represent weakly bound positronium. k, l and / are prolate spheroidal coordinates. k
rA rB ; R
l
rA ÿ rB : R
A is the proton and B is the antiproton. rA , for example, is the distance from the proton to the point under consideration. / is the usual azimuthal angle. The z-axis is taken to be in the direction of AB. Particle 1 is the electron and particle 2 is the positron. UPs r12 is the positronium groundstate wave function. q is the distance of the centre of mass of the positronium from the centre of mass of the nuclei. 8 1 pi 0; > > < 2 r pi 1; s12 pi R 12 > > : M12 cos /1 ÿ /2 pi 2; 1=2 M12 k21 ÿ 1 1 ÿ l21 k22 ÿ 1 1 ÿ l22 ; where R is the internuclear distance, j a variational parameter and g q a shielding function to ensure good behaviour of WPs at q 0. The wave function for the groundstate of the H±AH system is of R symmetry. It must be even or odd under the operation of re¯ection in the plane containing the perpendicular bisectors of the internuclear axis, followed by interchange of the electron and the positron. We have followed Koøos et al. [6], and assumed it to be even. The spin wave function need not be considered as interactions involving spin are neglected. The calculations are carried out in the Born± Oppenheimer approximation in which the interactions between the motion of the light particles and the nuclei are neglected. For a ®xed value of the internuclear distance R, the non-relativistic Hamiltonian for the light particles is of the form, ^ ÿ 1 r2 ÿ 1 r2 ÿ 1 1 1 ÿ 1 ÿ 1 H 2 1 2 2 rA1 rB1 rA2 rB2 r12 1 3 ÿ : R
220
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
The calculation using the basis functions (1) was carried out by doing all integral evaluations in terms of prolate spheroidal coordinates. This is a well-established procedure [15,16,6]. r12 can be expressed in terms of the prolate spheroidal coordinates using the relation R r12 k21 k22 l21 l22 ÿ 2 ÿ 2k1 k2 l1 l2 2 1=2 ÿ2M12 cos /1 ÿ /2 :
4
Matrix elements of the 1=r12 term in the Hamiltonian (3) were evaluated using the Neumann expansion [15] 1 X s 1 2X Dm P m k< Qms k> Psm l1 Psm l2 r12 R s0 m0 s s
cos m /1 ÿ /2 ;
5
where Dms is a coecient and Psm and Qms are the ®rst and second solutions, respectively, to the associated Legendre equation. k< denotes the lesser of k1 and k2 . Evaluation of such matrix elements is more complicated than in the case of calculations of the energy of the hydrogen molecule in the vicinity of its nuclear equilibrium position [15,16] as, in contrast to the H2 calculations, the b parameters in Eq. (1) cannot be set to zero. This is because the basis set must allow in a direct way for the fact that the electron, for example, is attracted by the proton but repelled by the antiproton. Techniques for such evaluations are well established. See, for example, R udenberg [17]. If the b parameters are zero, the series in the integral evaluation terminates after a ®nite, usually quite small, number of terms [15]. If these parameters are nonzero, the series in the integral evaluation is in®nite but converges quite rapidly for small values of jbj. Nevertheless, care has to be taken with the evaluation of the k integrals for higher terms in the series. An initial set of calculations was carried out for R a0 using only basis functions (1) of a form similar to those used by Koøos et al. [6]. An approximation to the groundstate energy of the system was obtained using the Rayleigh±Ritz variational method and solving the secular equations
H ÿ ESc 0;
6
where H is the Hamiltonian matrix and S is the overlap matrix. 3. Results using basis functions of type (1) The results were sensitive to the values of the non-linear a and b parameters in the basis functions (1). Thus these were optimised by carrying out a series of calculations with 20 basis functions with a range of values of these parameters. As a check on the accuracy of the Hamiltonian matrix elements in Eq. (6), H was evaluated in two ways, ®rstly by evaluating the lower triangle of H and secondly by evaluating the upper triangle of H. In each case, the remaining matrix elements were obtained using the fact that H is a symmetric matrix. The resulting values of the energy were found to agree in every case to at least six decimal places. The results for 20, 26 and 32 basis functions are compared in Table 1 with the corresponding results obtained by Junker and Bardsley [4] and Koøos et al. [6]. It can be seen that our results are lower than those of Junker and Bardsley, and therefore more accurate. This is to be expected. As Koøos et al. point out, Junker and Bardsley's calculation is a con®guration interaction (CI) calculation using basis functions similar to those in (1) but excluding Hylleraas-type functions with pi 1. This is inappropriate for systems containing positrons as well as electrons. This is because the attraction between a positron and an electron makes it necessary for the wave function to describe accurately the region where they are close Table 1 Calculated values of E R, the energy of H±AH for R a0 Number of basis functions N 20 26 32 75 77 a b c
a a a b c
This calculation. Junker and Bardsley [4]. Koøos et al. [6].
E R (a.u.) )1.266774 )1.267871 )1.268225 )1.257076 )1.271095
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
to each other and this requires the inclusion in the basis set of Hylleraas-type functions [18,19]. It can be seen that our results are only converging very slowly and are above the energy value obtained by Koøos et al. using 77 basis functions. This is to be expected as their basis set only diers from ours in that they replace M12 cos /1 ÿ /2 by
2 r12 R
2 :
4. Inclusion of virtual positronium At R a0 , we are not very far above the critical value, Rc , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium and thus no bound state of the system exists. In such a situation the basis set needs to be able to represent weakly bound positronium, i.e. virtual positronium. The wave function for this is of the form (2). Including a function of this type is similar to the procedure used by Rotenberg and Stein [20] to obtain an upper bound on the mass a `positron' would have to form a bound state with an H atom. They included a basis function in their variational calculation that represents a weakly bound `positron' and this enabled them to obtain an improved upper bound. The problem with including virtual positronium is the usual one with including a rearrangement channel: the coordinates it is convenient to use in the rearrangement channel are very dierent from those used in the entrance channel. Matrix elements involving only the virtual positronium function (2) can easily be evaluated analytically by integrating over r12 and q. The diculty arises in evaluating matrix elements involving both basis functions (1) and (2). In terms of prolate spheroidal coordinates, r12 in Eq. (2) is given by Eq. (4) and q by q
R 2 k k22 l21 l22 ÿ 2 2k1 k2 l1 l2 4 1 1=2 2M12 cos /1 ÿ /2 :
7
221
On account of the R symmetry of the ground state, the wave function is a function of u /1 ÿ /2 but not of v /1 /2 . Thus, one dimension of the integration can be removed by transforming to the variables u and v, and integrating over v [21]. However, it is not clear how integration over the remaining ®ve variables could be carried out analytically. Thus, we have carried this out using Gaussian quadrature. We used 32 points for each dimension. Tests using 64 points showed only a change of 0.0003 in the energy. As a test of the accuracy of the numerical integration procedure, we evaluated the energy and overlap matrix elements involving only the virtual positronium function (2) exactly by analytical integration and also by 5-dimensional Gaussian quadrature. The results are shown in Table 2. It can be seen that the results are accurate to three signi®cant ®gures. The variable parameter j in Eq. (2) was chosen to have a value appropriate to weakly bound positronium. In an exact calculation it would satisfy the relation E
1 1 1 ÿ j2 ÿ ; R 4 4
8
where E is the energy of the system and ÿ 14 is the groundstate energy of positronium. g q in Eq. (2) was taken to be of the form n
g q 1 ÿ eÿcq :
9
After some numerical trials, c was taken to be 0.5 and n to be 3.
Table 2 Comparison of the results obtained using exact integration and Gaussian quadrature WPs eÿjq =q 1 ÿ eÿcq n UPs r12
Exact Gaussian
a
quadrature
hWPs jWPs i
hWPs jH^ jWPs i
2.4140 2.4155
)0.5408 )0.5412
Units are atomic units. a Five dimensional Gaussian quadrature using 32 points per dimension.
222
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
5. Results with the inclusion of the virtual positronium function (2) The results obtained for 0:85a0 6 R 6 a0 are shown in Table 3. It can be seen that, for R a0 and 0:95a0 , the energy values obtained by including the virtual positronium function (2) are below the results obtained by Koøos et al. using 72 basis functions of a form very similar to Eq. (1). This dierence is in the third decimal place, which our tests with 64 points indicate should be reliable. Note that the dierence increases as R decreases and we move closer to Rc . These results demonstrate the importance of including a basis function representing virtual positronium in the basis set if accurate results are to be obtained for the energy of the system for R values, greater than, but close to Rc . In the case of an electron or a positron in the ®eld of a proton and an antiproton, the binding energy of the electron or positron is very small over quite a wide range above the critical value of 0:639a0 below which no bound state exists [10± 14]. See Fig. 1. It can be seen that this feature is not present in the results Koøos et al. [6] obtained for H±AH. If this feature is, in fact, present for this system, it can be expected to appear if the virtual positronium function (2) is included in the basis set. It can be seen that inclusion of the virtual positronium function also brings about a signi®cant
decrease in the energy of the system for R 0:9a0 and 0:85a0 . For R 0:85a0 , it brings the energy below the critical value for the existence of a bound state. Preliminary calculations suggest that this is also the case for R 0:8a0 , but this conclusion must be regarded as tentative at present. If con®rmed, this would indicate that the energy of the system makes a gradual approach to the critical value as R ! Rc from above. 6. Semi-classical treatment of H±AH scattering [22] Consider the internal motion of the proton (p) in their centre of mass and the antiproton (p) frame. The motion is planar and can thus be taken to be in the x ÿ y plane. The Lagrangian for the system is of the form 1 1 L T ÿ V lR_ 2 lR2 /_ 2 ÿ V R; 2 2 where l is the reduced mass of p and p is 918 a.u., / the azimuthal R the distance between p and p, angle of p (say), V R E R 1 and E(R) energy of H±AH as calculated using quantum mechanics. 1 is added so that V R ! 0 as R ! 1. The Euler±Lagrange equations are d @L @L ÿ 0 w R; /: dt @ w_ @w
Table 3 Calculated values of E R, the energy of H±AH, to illustrate the eect of including WPs in the basis set Number of basis functions N 32 33 77 32 33 77 32 33 32 33 a b c
a a;b c a a;b c
b
b
This calculation. Basis set includes WPs . Koøos et al. [6].
Internuclear distance R a0
E R a.u.
Positronium binding energy in a.u.
1.0 1.0 1.0 0.95 0.95 0.95 0.9 0.9 0.85 0.85
)1.268225 )1.2723 )1.271095 )1.311495 )1.3174 )1.314522 )1.362262 )1.3682 )1.420076 )1.4289
0.018 0.022 0.021 0.009 0.015 0.012 0.001 0.007 Unbound 0.002
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
223
ÿ e ; (B) Fig. 1. Light particle (i.e. electron + positron) energy as a function of R: (A) Curve obtained by Koøos et al. [6] for H±AH ppe ÿ Exact curve for ppe and ppe [10±14].
It follows from the fact that V R is a central potential that lR2 /_ C constant:
10
C is the angular momentum of the internal motion. It can be shown that lR1 v C;
11
where R1 is the impact parameter and v is the asymptotic velocity of approach of p. The energy, 1 1 12 Ec T V lR_ 2 lR2 /_ 2 V R 2 2 is a constant of the motion. This implies that 1 Ec lv2 as V R ! 0 as R ! 1: 13 2 It can easily be shown using Eqs. (10)±(13) that R0 , the distance of closest approach of the p to the p, is the largest value of R for which R2 Ec ÿ Ec 12 V R 0; R i.e. V R ÿ Ec R2 ÿEc R21 :
14
7. Calculation of the rearrangement cross section Morgan and Hughes [5] consider that it is highly probable that rearrangement into protonium + positronium will occur if, and only if, (pp) R0 < Rc where Rc is the critical internuclear distance below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. For a given Ec , R0 decreases as R1 decreases. In some cases R0 is a discontinuous function of R1 . The value of R1 at which R0 becomes equal to Rc or suddenly drops below it, is designated R1c [5]. The rearrangement cross section, r, is then given by r pR21c . As the protonium produced annihilates to form c-rays, this is also the cross section for annihilation. If Ec < 0:1 a:u: 2:7 eV; R0 varies discontinuously with R1 in such a way that it is reasonable to assume that at the discontinuity, R0 jumps from above the unknown value Rc to below it. R1c is then the value of R1 at which this discontinuity occurs [6]. At higher Ec values, the situation is not so clear.
224
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
In the low energy range, ÿ5
Ec < 4 10 a:u: 1 meV; kT 1 meV at T 10 no more than two partial waves would contribute to a quantum mechanical treatment of the scattering. The semi-classical treatment is certainly unlikely to be valid in this region. 8. Future plans In the second stage of this work, we will adapt our calculation of positronium formation in positron-H 2 scattering [23] so that it can be applied to positronium scattering by protonium in an excited state. In this calculation, positronium is represented by open-channel functions of the form Wopen Ps f jqg qUPs r12 ; where f jq is a Bessel or a Neumann function. The virtual positronium function (2) changes into some linear combination of these two functions as R decreases below Rc . We plan to link the variational calculation of the interaction between H and AH described above with the positronium scattering by protonium calculation and carry out an accurate quantum mechanical calculation of AH±H scattering and annihilation. This will involve taking into account the nuclear motion. The prolate spheroidal coordinates are derived from Cartesian coordinates with z-axis along the internuclear axis AB. This frame is not inertial. The resulting terms in the Hamiltonian that couple the electron and positron motion with the motion of the proton and the antiproton are likely to be important in the region in which R < Rc , where the Born±Oppenheimer approximation breaks down [5].
Acknowledgements We are grateful to the Engineering and Physical Sciences Research Council (UK) for support for this research. References [1] A. Watson, Science 271 (1996) 147. [2] M. Charlton, J. Eades, D. Horv ath, R.J. Hughes, C. Zimmermann, Phys. Rep. 241 (1994) 65. [3] M. Charlton, private communication. [4] B.R. Junker, J.N. Bardsley, Phys. Rev. Lett. 28 (1972) 1227. [5] D.L. Morgan, V.W. Hughes, Phys. Rev. A 7 (1973) 1811. [6] W. Koøos, D.L. Morgan, D.M. Schrader, L. Wolniewicz, Phys. Rev. A 11 (1975) 1792. [7] R.I. Campeanu, T. Beu, Phys. Lett. A 93 (1983) 223. [8] G.V. Shlyapnikov, J.T.M. Walvern, E.L. Surkov, Hyper®ne Interactions 76 (1993) 31. [9] E. Fermi, E. Teller, Phys. Rev. 72 (1947) 399. [10] A.J. Wightman, Phys. Rev. 77 (1950) 521. [11] M.H. Mittleman, V.P. Myerscough, Phys. Lett. 23 (1966) 545. [12] J.M. Levy-Leblond, Phys. Rev. 153 (1967) 1. [13] W.B. Brown, R.E. Roberts, J. Chem. Phys. 46 (1967) 2006. [14] C.A. Coulson, M. Walmsley, Proc. Phys. Soc. Lond. 91 (1967) 31. [15] H.M. James, A.S. Coolidge, J. Chem. Phys. 1 (1933) 825. [16] W. Koøos, C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 205. [17] K. R udenberg, J. Chem. Phys. 19 (1951) 1459. [18] C.F. Lebeda, D.M. Schrader, Phys. Rev. 178 (1969) 24. [19] E.A.G. Armour, D.J. Baker, M. Plummer, J. Phys. B 23 (1990) 3057. [20] M. Rotenberg, J. Stein, Phys. Rev. 182 (1969) 1. [21] E.A. G Armour, Mol. Phys. 26 (1973) 1093. [22] E.M. Purcell, G.B. Field, Astrophys. J. 124 (1956) 542. [23] J.M. Carr, Ph.D. Thesis, University of Nottingham, 1997.
Hydrogen±antihydrogen interactions E.A.G. Armour *, J.M. Carr Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, UK Received 24 October 1997; received in revised form 26 January 1998
Abstract A small number of antihydrogen (AH) atoms have recently been prepared at CERN and at Fermilab. However, these atoms were travelling at speeds close to that of light. It is intended to carry out experiments on AH by trapping it at very low temperature (<1 K) in an inhomogeneous magnetic ®eld. The main cause of loss of AH is due to collisions with H2 and He with energies up to room temperature. However, these reactions are not easy to treat theoretically. In this paper we consider the interaction between AH and H. This has already received some attention. Initially, in a collision between AH and H the electron is bound to the proton and the positron is bound to the antiproton. Clearly, if the proton and antiproton coincide they cannot bind the two light particles. There exists a critical value, Rc , of the internuclear distance, probably not very much below a0 , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. As a ®rst stage in our work on the H±AH interaction, we are carrying out variational calculations of the energy of the H±AH system for internuclear separations a short distance above Rc . The aim is to determine Rc as accurately as possible. The basis set used is similar to that of a previous calculation by W. Koøos, D.L. Morgan, D.M. Schrader, L. Wolniewiez, Phys. Rev. A 11 (1975) 1792. However, it also contains a function to represent weakly bound positronium. Initial results suggest that Rc < 0.8a0 . Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 3430; 3610 Keywords: Antihydrogen; Hydrogen; Collision; Annihilation; Positronium; Protonium
1. Introduction A small number of antihydrogen (AH) atoms have recently been prepared at CERN [1] and at Fermilab. However, these atoms were travelling at speeds close to that of light. The aim of present
* Corresponding author. Tel.: +44 115 9514922; fax: +44 115 9514951; e-mail: [email protected].
experimental work is to produce AH atoms in suf®cient quantities and at suciently low temperatures to make it possible to use high precision laser spectroscopy to probe the spectrum of AH. Comparison of the results with corresponding results for H would make possible a detailed test of the CPT invariance of relativistic quantum mechanics and also of Einstein's principle of equivalence, which is the cornerstone of the general theory of relativity.
0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 2 1 3 - 4
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
It is intended to carry out the experiments with AH by trapping the AH at very low temperature (<1 K) in an inhomogeneous magnetic ®eld [2]. The main cause of loss of AH is due to collisions with H2 and He with energies up to room temperature. In view of this, there is considerable interest in obtaining accurate theoretical values for cross sections for these collision processes in order to determine annihilation rates under various experimental conditions. This should make it possible to choose the simplest method for maximizing the AH lifetime [3]. However, the H2 and He reactions are not easy to treat theoretically. In this paper we consider the interaction between AH and H. This has already received some attention [4±8]. Initially, in a collision between AH and H the electron is bound to the proton and the positron is bound to the antiproton. Clearly, if the proton and antiproton coincide they cannot bind the two light particles. There exists a critical value, Rc , of the internuclear distance, probably not much below a0 , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. This is similar to the situation in the case of an electron or a positron in the ®eld of a proton and an antiproton. It is well known that in this case no bound state of this system exists if the internuclear distance is less than 0.639a0 . See, for example, Refs. [9±14].
2. Calculation As a ®rst stage in our work on the H±AH interaction, we are carrying out variational calculations of the energy of the H±AH system for internuclear separations a short distance above Rc . The aim is to determine Rc as accurately as possible. The basis set used is similar to that of Koøos et al. [6] in that it contains functions of the form, 1 mi ni ji ki ÿa1 k1 ÿa2 k2 b1 l1 b2 l2 k k l l e Wi 2p 1 2 1 2 ÿ1ji ki kn1i km2 i lk1i lj2i eÿa2 k1 ÿa1 k2 ÿb2 l1 ÿb1 l2 s12 pi : 1
219
However, it also contains a function of the form ÿjq e WPs g qUPs r12 2 q to represent weakly bound positronium. k, l and / are prolate spheroidal coordinates. k
rA rB ; R
l
rA ÿ rB : R
A is the proton and B is the antiproton. rA , for example, is the distance from the proton to the point under consideration. / is the usual azimuthal angle. The z-axis is taken to be in the direction of AB. Particle 1 is the electron and particle 2 is the positron. UPs r12 is the positronium groundstate wave function. q is the distance of the centre of mass of the positronium from the centre of mass of the nuclei. 8 1 pi 0; > > < 2 r pi 1; s12 pi R 12 > > : M12 cos /1 ÿ /2 pi 2; 1=2 M12 k21 ÿ 1 1 ÿ l21 k22 ÿ 1 1 ÿ l22 ; where R is the internuclear distance, j a variational parameter and g q a shielding function to ensure good behaviour of WPs at q 0. The wave function for the groundstate of the H±AH system is of R symmetry. It must be even or odd under the operation of re¯ection in the plane containing the perpendicular bisectors of the internuclear axis, followed by interchange of the electron and the positron. We have followed Koøos et al. [6], and assumed it to be even. The spin wave function need not be considered as interactions involving spin are neglected. The calculations are carried out in the Born± Oppenheimer approximation in which the interactions between the motion of the light particles and the nuclei are neglected. For a ®xed value of the internuclear distance R, the non-relativistic Hamiltonian for the light particles is of the form, ^ ÿ 1 r2 ÿ 1 r2 ÿ 1 1 1 ÿ 1 ÿ 1 H 2 1 2 2 rA1 rB1 rA2 rB2 r12 1 3 ÿ : R
220
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
The calculation using the basis functions (1) was carried out by doing all integral evaluations in terms of prolate spheroidal coordinates. This is a well-established procedure [15,16,6]. r12 can be expressed in terms of the prolate spheroidal coordinates using the relation R r12 k21 k22 l21 l22 ÿ 2 ÿ 2k1 k2 l1 l2 2 1=2 ÿ2M12 cos /1 ÿ /2 :
4
Matrix elements of the 1=r12 term in the Hamiltonian (3) were evaluated using the Neumann expansion [15] 1 X s 1 2X Dm P m k< Qms k> Psm l1 Psm l2 r12 R s0 m0 s s
cos m /1 ÿ /2 ;
5
where Dms is a coecient and Psm and Qms are the ®rst and second solutions, respectively, to the associated Legendre equation. k< denotes the lesser of k1 and k2 . Evaluation of such matrix elements is more complicated than in the case of calculations of the energy of the hydrogen molecule in the vicinity of its nuclear equilibrium position [15,16] as, in contrast to the H2 calculations, the b parameters in Eq. (1) cannot be set to zero. This is because the basis set must allow in a direct way for the fact that the electron, for example, is attracted by the proton but repelled by the antiproton. Techniques for such evaluations are well established. See, for example, R udenberg [17]. If the b parameters are zero, the series in the integral evaluation terminates after a ®nite, usually quite small, number of terms [15]. If these parameters are nonzero, the series in the integral evaluation is in®nite but converges quite rapidly for small values of jbj. Nevertheless, care has to be taken with the evaluation of the k integrals for higher terms in the series. An initial set of calculations was carried out for R a0 using only basis functions (1) of a form similar to those used by Koøos et al. [6]. An approximation to the groundstate energy of the system was obtained using the Rayleigh±Ritz variational method and solving the secular equations
H ÿ ESc 0;
6
where H is the Hamiltonian matrix and S is the overlap matrix. 3. Results using basis functions of type (1) The results were sensitive to the values of the non-linear a and b parameters in the basis functions (1). Thus these were optimised by carrying out a series of calculations with 20 basis functions with a range of values of these parameters. As a check on the accuracy of the Hamiltonian matrix elements in Eq. (6), H was evaluated in two ways, ®rstly by evaluating the lower triangle of H and secondly by evaluating the upper triangle of H. In each case, the remaining matrix elements were obtained using the fact that H is a symmetric matrix. The resulting values of the energy were found to agree in every case to at least six decimal places. The results for 20, 26 and 32 basis functions are compared in Table 1 with the corresponding results obtained by Junker and Bardsley [4] and Koøos et al. [6]. It can be seen that our results are lower than those of Junker and Bardsley, and therefore more accurate. This is to be expected. As Koøos et al. point out, Junker and Bardsley's calculation is a con®guration interaction (CI) calculation using basis functions similar to those in (1) but excluding Hylleraas-type functions with pi 1. This is inappropriate for systems containing positrons as well as electrons. This is because the attraction between a positron and an electron makes it necessary for the wave function to describe accurately the region where they are close Table 1 Calculated values of E R, the energy of H±AH for R a0 Number of basis functions N 20 26 32 75 77 a b c
a a a b c
This calculation. Junker and Bardsley [4]. Koøos et al. [6].
E R (a.u.) )1.266774 )1.267871 )1.268225 )1.257076 )1.271095
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
to each other and this requires the inclusion in the basis set of Hylleraas-type functions [18,19]. It can be seen that our results are only converging very slowly and are above the energy value obtained by Koøos et al. using 77 basis functions. This is to be expected as their basis set only diers from ours in that they replace M12 cos /1 ÿ /2 by
2 r12 R
2 :
4. Inclusion of virtual positronium At R a0 , we are not very far above the critical value, Rc , below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium and thus no bound state of the system exists. In such a situation the basis set needs to be able to represent weakly bound positronium, i.e. virtual positronium. The wave function for this is of the form (2). Including a function of this type is similar to the procedure used by Rotenberg and Stein [20] to obtain an upper bound on the mass a `positron' would have to form a bound state with an H atom. They included a basis function in their variational calculation that represents a weakly bound `positron' and this enabled them to obtain an improved upper bound. The problem with including virtual positronium is the usual one with including a rearrangement channel: the coordinates it is convenient to use in the rearrangement channel are very dierent from those used in the entrance channel. Matrix elements involving only the virtual positronium function (2) can easily be evaluated analytically by integrating over r12 and q. The diculty arises in evaluating matrix elements involving both basis functions (1) and (2). In terms of prolate spheroidal coordinates, r12 in Eq. (2) is given by Eq. (4) and q by q
R 2 k k22 l21 l22 ÿ 2 2k1 k2 l1 l2 4 1 1=2 2M12 cos /1 ÿ /2 :
7
221
On account of the R symmetry of the ground state, the wave function is a function of u /1 ÿ /2 but not of v /1 /2 . Thus, one dimension of the integration can be removed by transforming to the variables u and v, and integrating over v [21]. However, it is not clear how integration over the remaining ®ve variables could be carried out analytically. Thus, we have carried this out using Gaussian quadrature. We used 32 points for each dimension. Tests using 64 points showed only a change of 0.0003 in the energy. As a test of the accuracy of the numerical integration procedure, we evaluated the energy and overlap matrix elements involving only the virtual positronium function (2) exactly by analytical integration and also by 5-dimensional Gaussian quadrature. The results are shown in Table 2. It can be seen that the results are accurate to three signi®cant ®gures. The variable parameter j in Eq. (2) was chosen to have a value appropriate to weakly bound positronium. In an exact calculation it would satisfy the relation E
1 1 1 ÿ j2 ÿ ; R 4 4
8
where E is the energy of the system and ÿ 14 is the groundstate energy of positronium. g q in Eq. (2) was taken to be of the form n
g q 1 ÿ eÿcq :
9
After some numerical trials, c was taken to be 0.5 and n to be 3.
Table 2 Comparison of the results obtained using exact integration and Gaussian quadrature WPs eÿjq =q 1 ÿ eÿcq n UPs r12
Exact Gaussian
a
quadrature
hWPs jWPs i
hWPs jH^ jWPs i
2.4140 2.4155
)0.5408 )0.5412
Units are atomic units. a Five dimensional Gaussian quadrature using 32 points per dimension.
222
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
5. Results with the inclusion of the virtual positronium function (2) The results obtained for 0:85a0 6 R 6 a0 are shown in Table 3. It can be seen that, for R a0 and 0:95a0 , the energy values obtained by including the virtual positronium function (2) are below the results obtained by Koøos et al. using 72 basis functions of a form very similar to Eq. (1). This dierence is in the third decimal place, which our tests with 64 points indicate should be reliable. Note that the dierence increases as R decreases and we move closer to Rc . These results demonstrate the importance of including a basis function representing virtual positronium in the basis set if accurate results are to be obtained for the energy of the system for R values, greater than, but close to Rc . In the case of an electron or a positron in the ®eld of a proton and an antiproton, the binding energy of the electron or positron is very small over quite a wide range above the critical value of 0:639a0 below which no bound state exists [10± 14]. See Fig. 1. It can be seen that this feature is not present in the results Koøos et al. [6] obtained for H±AH. If this feature is, in fact, present for this system, it can be expected to appear if the virtual positronium function (2) is included in the basis set. It can be seen that inclusion of the virtual positronium function also brings about a signi®cant
decrease in the energy of the system for R 0:9a0 and 0:85a0 . For R 0:85a0 , it brings the energy below the critical value for the existence of a bound state. Preliminary calculations suggest that this is also the case for R 0:8a0 , but this conclusion must be regarded as tentative at present. If con®rmed, this would indicate that the energy of the system makes a gradual approach to the critical value as R ! Rc from above. 6. Semi-classical treatment of H±AH scattering [22] Consider the internal motion of the proton (p) in their centre of mass and the antiproton (p) frame. The motion is planar and can thus be taken to be in the x ÿ y plane. The Lagrangian for the system is of the form 1 1 L T ÿ V lR_ 2 lR2 /_ 2 ÿ V R; 2 2 where l is the reduced mass of p and p is 918 a.u., / the azimuthal R the distance between p and p, angle of p (say), V R E R 1 and E(R) energy of H±AH as calculated using quantum mechanics. 1 is added so that V R ! 0 as R ! 1. The Euler±Lagrange equations are d @L @L ÿ 0 w R; /: dt @ w_ @w
Table 3 Calculated values of E R, the energy of H±AH, to illustrate the eect of including WPs in the basis set Number of basis functions N 32 33 77 32 33 77 32 33 32 33 a b c
a a;b c a a;b c
b
b
This calculation. Basis set includes WPs . Koøos et al. [6].
Internuclear distance R a0
E R a.u.
Positronium binding energy in a.u.
1.0 1.0 1.0 0.95 0.95 0.95 0.9 0.9 0.85 0.85
)1.268225 )1.2723 )1.271095 )1.311495 )1.3174 )1.314522 )1.362262 )1.3682 )1.420076 )1.4289
0.018 0.022 0.021 0.009 0.015 0.012 0.001 0.007 Unbound 0.002
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
223
ÿ e ; (B) Fig. 1. Light particle (i.e. electron + positron) energy as a function of R: (A) Curve obtained by Koøos et al. [6] for H±AH ppe ÿ Exact curve for ppe and ppe [10±14].
It follows from the fact that V R is a central potential that lR2 /_ C constant:
10
C is the angular momentum of the internal motion. It can be shown that lR1 v C;
11
where R1 is the impact parameter and v is the asymptotic velocity of approach of p. The energy, 1 1 12 Ec T V lR_ 2 lR2 /_ 2 V R 2 2 is a constant of the motion. This implies that 1 Ec lv2 as V R ! 0 as R ! 1: 13 2 It can easily be shown using Eqs. (10)±(13) that R0 , the distance of closest approach of the p to the p, is the largest value of R for which R2 Ec ÿ Ec 12 V R 0; R i.e. V R ÿ Ec R2 ÿEc R21 :
14
7. Calculation of the rearrangement cross section Morgan and Hughes [5] consider that it is highly probable that rearrangement into protonium + positronium will occur if, and only if, (pp) R0 < Rc where Rc is the critical internuclear distance below which the electron and the positron can attain a lower energy by separating from the nuclei and forming positronium. For a given Ec , R0 decreases as R1 decreases. In some cases R0 is a discontinuous function of R1 . The value of R1 at which R0 becomes equal to Rc or suddenly drops below it, is designated R1c [5]. The rearrangement cross section, r, is then given by r pR21c . As the protonium produced annihilates to form c-rays, this is also the cross section for annihilation. If Ec < 0:1 a:u: 2:7 eV; R0 varies discontinuously with R1 in such a way that it is reasonable to assume that at the discontinuity, R0 jumps from above the unknown value Rc to below it. R1c is then the value of R1 at which this discontinuity occurs [6]. At higher Ec values, the situation is not so clear.
224
E.A.G. Armour, J.M. Carr / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 218±224
In the low energy range, ÿ5
Ec < 4 10 a:u: 1 meV; kT 1 meV at T 10 no more than two partial waves would contribute to a quantum mechanical treatment of the scattering. The semi-classical treatment is certainly unlikely to be valid in this region. 8. Future plans In the second stage of this work, we will adapt our calculation of positronium formation in positron-H 2 scattering [23] so that it can be applied to positronium scattering by protonium in an excited state. In this calculation, positronium is represented by open-channel functions of the form Wopen Ps f jqg qUPs r12 ; where f jq is a Bessel or a Neumann function. The virtual positronium function (2) changes into some linear combination of these two functions as R decreases below Rc . We plan to link the variational calculation of the interaction between H and AH described above with the positronium scattering by protonium calculation and carry out an accurate quantum mechanical calculation of AH±H scattering and annihilation. This will involve taking into account the nuclear motion. The prolate spheroidal coordinates are derived from Cartesian coordinates with z-axis along the internuclear axis AB. This frame is not inertial. The resulting terms in the Hamiltonian that couple the electron and positron motion with the motion of the proton and the antiproton are likely to be important in the region in which R < Rc , where the Born±Oppenheimer approximation breaks down [5].
Acknowledgements We are grateful to the Engineering and Physical Sciences Research Council (UK) for support for this research. References [1] A. Watson, Science 271 (1996) 147. [2] M. Charlton, J. Eades, D. Horv ath, R.J. Hughes, C. Zimmermann, Phys. Rep. 241 (1994) 65. [3] M. Charlton, private communication. [4] B.R. Junker, J.N. Bardsley, Phys. Rev. Lett. 28 (1972) 1227. [5] D.L. Morgan, V.W. Hughes, Phys. Rev. A 7 (1973) 1811. [6] W. Koøos, D.L. Morgan, D.M. Schrader, L. Wolniewicz, Phys. Rev. A 11 (1975) 1792. [7] R.I. Campeanu, T. Beu, Phys. Lett. A 93 (1983) 223. [8] G.V. Shlyapnikov, J.T.M. Walvern, E.L. Surkov, Hyper®ne Interactions 76 (1993) 31. [9] E. Fermi, E. Teller, Phys. Rev. 72 (1947) 399. [10] A.J. Wightman, Phys. Rev. 77 (1950) 521. [11] M.H. Mittleman, V.P. Myerscough, Phys. Lett. 23 (1966) 545. [12] J.M. Levy-Leblond, Phys. Rev. 153 (1967) 1. [13] W.B. Brown, R.E. Roberts, J. Chem. Phys. 46 (1967) 2006. [14] C.A. Coulson, M. Walmsley, Proc. Phys. Soc. Lond. 91 (1967) 31. [15] H.M. James, A.S. Coolidge, J. Chem. Phys. 1 (1933) 825. [16] W. Koøos, C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 205. [17] K. R udenberg, J. Chem. Phys. 19 (1951) 1459. [18] C.F. Lebeda, D.M. Schrader, Phys. Rev. 178 (1969) 24. [19] E.A.G. Armour, D.J. Baker, M. Plummer, J. Phys. B 23 (1990) 3057. [20] M. Rotenberg, J. Stein, Phys. Rev. 182 (1969) 1. [21] E.A. G Armour, Mol. Phys. 26 (1973) 1093. [22] E.M. Purcell, G.B. Field, Astrophys. J. 124 (1956) 542. [23] J.M. Carr, Ph.D. Thesis, University of Nottingham, 1997.