A quasifree model for the absorption effects in positron scattering by atoms

Transcription

1 J. Phys. B: At. Mol. Opt. Phys. 29 (1996) L127 L133. Printed in the UK LETTER TO THE EDITOR A quasifree model for the absorption effects in positron scattering by atoms David D Reid and J M Wadehra Department of Physics and Astronomy, Wayne State University, Detroit, MI 48202, USA Received 18 August 1995, in final form 28 December 1995 Abstract. We present a model absorption potential for the calculation of cross sections for the scattering of positrons from various atomic targets. Our model, which is free of any adjustable parameters, is analogous to the quasifree scattering model which has been used by other investigators for the scattering of electrons from rare gases. As a test of the proposed model we use it to calculate total and differential cross sections for positrons scattered from various rare gas targets. Investigations of the scattering of positrons by atomic and molecular gases have gained significant importance because the positron, being a positively-charged probe, offers a more sensitive test of our ability to understand atomic interactions than the electron does. Additional importance of positron scattering derives from the fact that it involves interactions of matter with antimatter which have possible applications in the astrophysical arena. A method which has been quite successful for calculating the scattering cross sections is the model potential approach which seeks to use a complex local potential, generally represented by analytical expressions, to model the actual projectile target interaction. This approach had its origins in nuclear physics and was adapted for atomic physics applications by Mittleman and Watson (1959). The imaginary part of the interaction potential, V abs, which is non-zero for projectile energies above the lowest inelastic threshold, accounts for the absorption effects, that is, absorption of incident projectile flux from the purely elastic channel. For positron scattering the real part of the interaction potential is partitioned into the static and correlation polarization parts. The imaginary (or absorption) part takes into account various inelastic processes such as positronium formation as well as excitation and ionization of the target by positron impact. In the past, models of various parts of the total interaction potential have been proposed which contain adjustable parameters that are fitted to provide theoretical cross sections that match the experimental results. In our work we focus on obtaining model interaction potentials which are free of any adjustable parameters. In the present paper we propose such a model for the absorption potential for positron scattering. Our model is analogous to the quasifree model for electron scattering which was introduced previously in this journal (Staszewska et al 1983). The absorption potential is obtained using a modification of a method originally proposed in nuclear physics for nucleon nucleon scattering (Goldberger 1948). Present address: Department of Physics and Astronomy, Eastern Michigan University, Ypsilanti, MI 48197, USA /96/ $19.50 c 1996 IOP Publishing Ltd L127

2 L128 Letter to the Editor During the last few years a number of calculations for positron scattering cross sections have been carried out which employ model potentials of increasing sophistication. In the earliest calculations the absorption effects were completely omitted by choosing real potentials for the positron target interaction, even for positron energies above the lowest inelastic threshold (Khan et al 1984, Nakanishi and Schrader 1986, Basu et al 1987, Hasenburg et al 1987, Nahar and Wadehra 1987a, Khurana et al 1988). A few attempts to use complex model potentials for positron scattering were made; however, in these calculations either the absorption potential for electrons was used (Khare et al 1986) or an arbitrarily modified form of the electron absorption potential was used (Baluja et al 1991, Nahar and Wadehra 1991, Baluja and Jain 1992). Using an eikonal analysis of the second Born term, Joachain and Potvliege (1987) numerically demonstrated a strong effect of the absorption channel on elastic scattering cross sections for incident positron energies above the inelastic threshold. Having realized the need to take into account the effect of the absorption interaction on cross sections, a number of recent positron scattering calculations, using purely real interaction potentials, were confined so that the incident positron energy remained below the lowest inelastic threshold (Sienkiewicz and Baylis 1989, Jain 1990, Gianturco et al 1993, DeFazio et al 1994, Reid and Wadehra 1994, Bhattacharyya et al 1995). For electron scattering a number of model absorption potentials have been proposed. For positron scattering we are making such an attempt here. To begin with, one notes that the imaginary part of the interaction potential V abs represents an absorption probability per unit time of 2V abs / h. This result is compared with the corresponding result from classical kinetic theory for a projectile with energy E = p 2 /2m in a free electron gas of density ρ. For this latter case, the absorption probability per unit time is ρ σ b v where v is the local speed of the projectile and σ b, the average cross section for absorption producing events, is constructed from the binary collisions between the projectile and the target electrons. Thus, we can write V abs = h 2 ρ σ bv. (1) An explicit and general expression for the average binary collision cross section σ b is given by Goldberger (1948) as σ b = 1 dkn(k F ) p k dg dσ b 1 p d p0 2 δ(p 0 p f ) (p,k,k F ) (2) where k and p are the laboratory frame momenta of the target electron and the projectile particle before the collision and k and p are the same quantities after the collision. p 0 and p f are the initial and final momenta of the projectile in the binary collision frame and g is the momentum transfer of the projectile. dσ b /d is the differential scattering cross section for binary collisions and is the Pauli-blocking factor that is unity for Pauli-allowed final states and zero for Pauli-blocked final states in the binary collision (Staszewska et al 1984). N(k F ), the momentum state density per target electron, is 3/(4π h 3 kf 3) for k hk F and is 0 for k> hk F. The quantities k F = (3π 2 ρ) 1/3 and E F = h 2 kf 2 /2m are the Fermi wavenumber (or momentum) and the Fermi energy corresponding to the target electron density ρ. In the case of electron impact collisions, the average binary collision cross section σ b for electron electron scattering was calculated (Staszewska et al 1984) subject to the constraints k 2 h 2 kf 2 + 2m (3a) p 2 h 2 kf 2. (3b)

3 Letter to the Editor L129 The constraints provided by equation (3) on k and p were chosen to account for the Pauli exclusion principle. In particular, the energy of the initially bound electron is required to exceed the Fermi energy by an amount equal to the lowest inelastic threshold which is taken as the energy gap,, between the ground state and the lowest excited state. Similarly, the final energy of the incident unbound electron is required to exceed the Fermi energy of the target. Therefore, the Pauli-blocking factor can be expressed, in the case of electron impact, by a product of two Heaviside unit-step functions representing the constraints of equations (3a) and (3b) as =H(k 2 h 2 kf 2 2m )H (p hk F ). In the case of positron impact collisions we have to modify the above model in two ways. First, due to the absence of the Pauli exclusion constraint, condition (3b) does not apply in the case of the incident positron. Therefore, we have to remove the constraint imposed by equation (3b) from the Pauli blocking factor, allowing the positron to emerge from the binary collision with any momentum. Evaluation of the six-dimensional integral of equation (2), with = H(k 2 h 2 kf 2 2m ) only, is somewhat laborious but the final result is rather compact. For the convenience of showing this result, we define δ = E ε = E F E F and f(x)= 2 ( ) ε x δ x3 +6x+3εln. ε + x Then, for positron scattering the absorption potential is given by equation (1) with σ b expressed as ( ) ao R 2 f(1) ε 2 δ 1 σ b = 4π f( ε εe F 2 δ) 1 ε 2 δ 0 f(0) 0 ε 2 δ. Here, a 0 and R are the Bohr radius and the Rydberg constant, respectively. Second, since the lowest inelastic threshold for positron scattering in rare gases is determined by the energy, E Ps, at which positronium formation begins, we take the energy gap,, to be the positronium formation threshold. In order to test this model absorption potential for positron scattering we use it to calculate the elastic differential and the total cross sections for the scattering of positrons from rare gases. The five rare gas atoms from helium to xenon are chosen as the target atoms. The real part of the complex interaction potential is taken as the sum of the static and correlation polarization potentials. The static potential is obtained by using the analytical Hartree Fock wavefunctions of the target atom (Clementi and Roetti 1974). The parameterfree correlation polarization potential, which is based upon the model of Reid and Wadehra (1994), is of the form V cp (r) = α dr 2 +α q (r 3 + d 3 ). 2 The non-adjustable parameter d is obtained by setting V cp equal to the positron correlation potential of Jain (1990) at the charge density peak of the outermost occupied orbital. The above potentials are placed in the radial Schrödinger equation and integrated out to a distance of 60 au from the nucleus via the Numerov technique. Depending upon the incident (4)

4 L130 Letter to the Editor positron energy, the first few (typically 25) complex phase shifts are calculated exactly. The contributions of higher partial waves are included by using the Born approximation for the phase shifts (Nahar and Wadehra 1987b). For each value of the positron energy, a sufficient number of partial waves are included to ensure that the scattering cross sections are well converged. Figure 1. Total cross sections for the scattering of positrons by helium and neon. The thick full curve shows the cross sections obtained using the present positron absorption potential. The thin curve, obtained using the real potential only, shows the result of neglecting absorption effects in the calculations. Symbols used for the experimental points are as follows. For helium: open circles, Stein et al (1978); full circles, Kauppila et al (1981); open triangles, Coleman et al (1976); full triangles, Canter et al (1973); full rectangles, Brenton et al (1977); full diamonds, Griffith et al (1979). For neon: open circles, Stein et al (1978); full circles, Kauppila et al (1981); open triangles, Coleman et al (1976); full triangles, Tsai et al (1976); full diamonds, Griffith et al (1979). Figure 1 shows the total cross sections for the scattering of positrons from helium and neon. In each case the energy of the positron varies from the lowest inelastic threshold to 1000 ev. The thick full curves show the cross sections which are obtained using the complex local potential whose imaginary part is the present model absorption potential. In order to observe the effect of including the absorption potential in the calculations, we show in figure 1, using thin curves, the total cross sections which are obtained by using only the real part of the interaction. Also shown in figure 1, for comparison, are the experimental values of the total cross sections for the scattering of positrons by both helium and neon measured by Coleman et al (1976), Stein et al (1978), Griffith et al (1979) and Kauppila

5 Letter to the Editor L131 et al (1981). In addition, the figure shows the experimental data of Canter et al (1973) and of Brenton et al (1977) for helium and of Tsai et al (1976) for neon. In all the figures, the error bars, indicating the estimated experimental errors, are shown only if they are numerically provided in the corresponding experimental work. As expected, in each case at the lowest inelastic threshold (17.8 ev for helium and 14.8 ev for neon) the theoretical cross section curve using the purely real potential merges into the cross section curve using the complex potential. On comparing these two curves, for each atomic target, we observe that the inclusion of the absorption potential has a significant effect on the cross sections and it provides cross sections that agree quite favourably with the experimental data. In fact, the use of the real potential alone does not correctly reveal even the shape of the cross section curve. Figure 2. Same as in figure 1 except for argon, krypton and xenon. Symbols used for the experimental points are as follows. For argon: open circles, Kauppila et al (1976); full circles, Kauppila et al (1981); open triangles, Coleman et al (1976); full triangles, Canter et al (1973); full rectangles, Brenton et al (1978); full diamonds, Griffith et al (1979). For krypton and xenon: full circles, Dababneh et al (1982); open circles, Dababneh et al (1980). Figure 2 shows, in three frames, the total cross sections for the scattering of positrons from argon, krypton and xenon. Again, the energy of the positron is varied from the lowest inelastic threshold (9.0, 7.2 ev and 5.3 ev for argon, krypton and xenon, respectively) to 1000 ev. In each frame of this figure, the thick full curve and the thin curve show the cross

6 L132 Letter to the Editor sections which are obtained using the complex and the real model potential, respectively. The experimental cross sections shown in these figures are measured by Canter et al (1973), Kauppila et al (1976, 1981), Coleman et al (1976), Brenton et al (1978) and Griffith et al (1979) for argon and by Dababneh et al (1980, 1982) for krypton and xenon. Once again, we note that above the inelastic threshold inclusion of absorption potential in the calculations is paramount in obtaining cross sections that show agreement with the experimental data. For the targets considered, a purely real interaction potential, when used in the cross section calculations, provides an incorrect shape of the curve completely missing the peak in the cross sections. The present calculations underestimate the experimental cross section values at energies below the energy of the maximum total cross section. This is the energy region where the process of positronium formation has its greatest effect on the total cross sections. Figure 3. Elastic differential cross sections for the scattering of positrons by helium, neon and argon. The relative experimental data of Smith (1989) are normalized to the calculated cross sections at 90. Finally, figure 3 shows the differential cross sections for the elastic scattering of 200 ev positrons by helium and neon and 300 ev positrons by argon. The presently calculated cross sections are compared in the figure with the relative experimental data of Smith (1989). A good agreement is obtained when the relative experimental measurements are normalized to the present cross sections at 90. For krypton and xenon the measurements of differential

7 Letter to the Editor L133 cross sections are available only for low values of positron energy. To summarize, we have presented a parameter-free model absorption potential for positron scattering by atomic and molecular targets. Use of this potential in the scattering calculations provides total cross sections which show significant improvement, in agreement with the corresponding experimental data, over the use of purely real potentials. We gratefully acknowledge the support of the US National Science Foundation (grant No PHY ) for this work. References Baluja K L, Jain A, Jones H W, Weatherford C A and Karim K R 1991 J. Phys. B: At. Mol. Opt. Phys. 24 L93 8 Baluja K L and Jain A 1992 Phys. Rev. A Basu D, Datta S K, Khan P and Ghosh A S 1987 Phys. Rev. A Bhattacharyya S, Talukdar B and Mandal P 1995 Phys. Rev. A Brenton A G, Dutton J and Harris F M 1978 J. Phys. B: At. Mol. Phys. 11 L15 19 Brenton A G, Dutton J, Harris F M, Jones R A and Lewis D M 1977 J. Phys. B: At. Mol. Phys Canter K F, Coleman P G, Griffith T C and Heyland G R 1973 J. Phys. B: At. Mol. Phys. 6 L201 3 Clementi E and Roetti C 1974 At. Data Nucl. Data Tables Coleman P G, Griffith T C, Heyland G R and Twomey T R 1976 Appl. Phys Dababneh M S, Kauppila W E, Downing J P, Laperriere F, Pol V, Smart J H and Stein T S 1980 Phys. Rev. A Dababneh M S, Hsieh Y F, Kauppila W E, Pol V and Stein T S 1982 Phys. Rev. A DeFazio D, Gianturco F A, Rodriguez-Ruiz J A, Tang K T and Toennies J P 1994 J. Phys. B: At. Mol. Opt. Phys Gianturco F A, Jain A and Rodriguez-Ruiz J A 1993 Phys. Rev. A Goldberger M L 1948 Phys. Rev Griffith T C, Heyland G R, Lines K S and Twomey T R 1979 Appl. Phys Hasenburg K, Bartschat K, McEachran R P and Stauffer A D 1987 J. Phys. B: At. Mol. Phys Jain A 1990 Phys. Rev. A Joachain C J and Potvliege R M 1987 Phys. Rev. A Kauppila W E, Stein T S and Jesion G 1976 Phys. Rev. Lett Kauppila W E, Stein T S, Smart J H, Dababneh M S, Ho Y K, Downing J P and Pol V 1981 Phys. Rev. A Khan P, Datta S K, Bhattacharyya D and Ghosh A S 1984 Phys. Rev. A Khare S P, Kumar A and Lata K 1986 Phys. Rev. A Khurana I, Srivastava R and Tripathi A N 1988 Phys. Rev. A Mittleman M H and Watson K M 1959 Phys. Rev Nahar S N and Wadehra J M 1987a Phys. Rev. A b Phys. Rev. A Phys. Rev. A Nakanishi H and Schrader D M 1986 Phys. Rev. A Reid D D and Wadehra J M 1994 Phys. Rev. A Sienkiewicz J E and Baylis W E 1989 Phys. Rev. A Smith S J 1989 PhD Thesis Wayne State University, Detroit, USA Staszewska G, Schwenke D W, Thirumalai D and Truhlar D G 1983 J. Phys. B: At. Mol. Phys. 16 L281 7 Staszewska G, Schwenke D W and Truhlar D G 1984 Phys. Rev. A Stein T S, Kauppila W E, Pol V, Smart J H and Jesion G 1978 Phys. Rev. A Tsai J S, Lebow L and Paul DAL1976 Can. J. Phys