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Doubly excited resonance states of two-electron systems in exponential cosine-screened Coulomb potentials
Computer Physics Communications 182 (2011) 122–124
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Doubly excited resonance states of two-electron systems in exponential cosine-screened Coulomb potentials Arijit Ghoshal ∗,1 , Y.K. Ho Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106
a r t i c l e
i n f o
a b s t r a c t e
We have made an investigation on the 2s21 S resonance states in two-electron systems interacting with exponential cosine-screened Coulomb potentials (ECSCP) within the framework of stabilization method. Highly correlated wave functions are used to take into account the correlations of the charge particles. Results for resonance energies and widths are reported for Z = 1, 2 and 3. For the unscreened case, our reported results are in nice agreement with the results so far available in the literature. © 2010 Elsevier B.V. All rights reserved.
Article history: Received 28 January 2010 Accepted 27 May 2010 Available online 4 June 2010 Keywords: Doubly excited resonance states Stabilization method Density of the resonance states Exponential cosine-screened Coulomb potential Correlated wave function
1. Introduction The study of atomic processes for which particles are interacting with exponential-cosine screened Coulomb potential (ECSCP) with screening parameter μ,
V (r ) = (1/r )e −μr cos (μr )
(in a.u.),
(1)
is currently an interesting topic of research [1–5] (and further references therein). The screening parameter μ shortens the range of the potential in comparison to the Coulomb potential. Exponential cosine-screened Coulomb potentials are widespread in many different areas of physics, such as, solid-state physics, nuclear physics, plasma physics [2–5]. These potentials differ from the well-known screened Coulomb potentials (SCP) with screening parameter μ,
V (r ) = (1/r )e −μr
(in a.u.),
(2)
from the existence of the cos (μr ) term, as a result an ECSCP exhibits stronger screening effect than an SCP. In this paper our objective is to investigate the 2s21 S e autoionization resonance states of a system consisting of a nucleus of charge Z e and two electrons interacting via ECSCP. Investigation of the autoionization states of an atomic system is of utmost importance in several respects [6] and indeed there is a continuous
*
Corresponding author. E-mail address: [email protected] (A. Ghoshal). 1 Permanent affiliation: Department of Mathematics, Burdwan University, Golapbag, Burdwan 713 104, West Bengal, India. 0010-4655/$ – see front matter doi:10.1016/j.cpc.2010.05.010
© 2010
Elsevier B.V. All rights reserved.
interest to investigate autoionization states of two-electron systems [3–5,7]. Moreover, investigation on two-electron systems in ECSCP is also important role owing to the role played by the correlation effects between the charged particles. We have designed this paper as follows. Describing the underlying theory and calculations of our investigation in Section 2 we present and discuss our computed results in Section 3. Finally, in Section 4 we give our concluding remarks. The atomic units (a.u.) are used throughout the present work, and all calculations are performed in quadruple precision (32 significant figures) on IBM–AMD workstations in UNIX environment. 2. Theory and calculations The non-relativistic Hamiltonian of a system consisting of a nucleus of charge Z e and two electrons interacting via ECSCP is given by
−μr1 1 1 e e −μr2 H = − ∇12 − ∇22 − Z cos (μr1 ) + cos (μr2 ) 2 2 r1 r2 +
e −μr12 r12
cos (μr12 ),
(3)
where r1 and r2 are the coordinates of the two electrons relative to the nucleus (assumed to be at rest), and r12 is their relative distance. In order to determine 1 S e states, we have employed the wave function
A. Ghoshal, Y.K. Ho / Computer Physics Communications 182 (2011) 122–124
123
e
Fig. 1. (a) Stabilization plots of the 2s2 1 S states of Li+ in ECSCP for μ = 0.1, and ω 14 (N = 372). The number in the parentheses next to the solid line indicates the e order of appearance of the eigenvalues. (b) Calculated density (circles) and the fitted Lorentzian (solid line) for the 12th eigenvalue corresponding to the 2s2 1 S state of Li+ in ECSCP for μ = 0.1. Table 1 The resonance energies ( E r ) (a.u.) and widths (Γ ) (a.u.) along with the energies of Li++ (2S ), E Li++ (2S ) , in ECSCP for various values of the screening parameter H − ( Z = 1)
− E r (a.u.)
μ
Γ (a.u.)
− E r (a.u.)
μ.
Li+ ( Z = 3)
He( Z = 2)
Γ (a.u.)
0.0 (a) 0.05 0.1 0.15 0.2 0.3 0.45
0.14876 0.148777 0.09770 0.04686 0.00647
0.001732 0.001733 0.001764 0.001605 0.000534
Present results in ECSCP 0.77784 0.004503 0.77787 0.00454 0.62848 0.004487 0.48306 0.004378 0.34554 0.004085 0.22004 0.003559
0.05 0.1 0.2
0.10182 0.06311
0.001638 0.001309
Results of Kar and Ho [8] in SCP 0.63683 0.004450 0.51279 0.004159 0.31105 0.003191
− E r (a.u.)
Γ (a.u.)
− E Li++ (2S )
1.90582 1.905845 1.65648 1.41098 1.17264 0.94449 0.53008 0.06739
0.005667 0.00566 0.005658 0.005604 0.005470 0.005228 0.004318 0.001545
1.25
1.666661 1.448150 1.067568
0.005613 0.005432 0.004796
0.975546504 0.829104144 0.688046366 0.554244676 0.314471807 0.047531
(a): Results of this row are taken from Ref. [7], obtained by using method complex coordinate rotation and precision calculation.
Ψ (r1 , r2 ) =
N i =1
C i ψi =
N
ρn ( E ) = y 0 +
C i (1 + P 12 )e − A (r1 +r2 )α r1i r2 i r12i , l m n
i =1
l i , m i , n i = 0, 1, 2, . . . , l i m i ,
(4)
where A is a non-linear variational parameter, C i (i = 1, 2, 3, . . . , N) are linear expansion coefficients, α is a scaling constant to be discussed later and P 12 is an exchange operator such that P 12 f (r1 , r2 ) = f (r2 , r1 ) for an arbitrary function f . This wave function is expanded by generating the powers of r1 , r2 and r12 following the relation ω = li + mi + mi = 0, 1, 2, . . . . In the present work we use stabilization method similar to Ho and co-workers [3,8] to determine the resonance energies and widths. This method is a slight generalization of the stabilization method proposed by Mandelshtam et al. [9], and consists of three steps. In the first step, one obtains the energy levels E (α ) by diagonalizing the Hamiltonian (3) for a number of values of α within a certain range. The graph of E (α ) versus α is called a stabilization plot (as shown in Fig. 1(a)). If there is a resonance at energy E, stabilized or slowly decreasing energy levels will appear in the stabilization plateau. Once the existence of a resonance is assured, in the second step, density of resonance states for a single energy level is calculated using the following formula:
ρn ( E ) =
E n (αi +1 ) − E n (αi −1 )
α i +1 − α i −1
− 1 ,
(5)
E n (αi )= E
where the index i is the ith value for α and the index n is for the nth resonance. Finally, to obtain resonance energy E r and width Γ , we fit it to the following Lorentzian form:
Γ 2
π ( E − E r )2 + ( Γ2 )2
,
(6)
where y 0 is the baseline offset, is the total area under the curve from the base line, E r is the centre of the peak and Γ denotes the full width of the peak of the curve at half height. 3. Results and discussion Using the wave function (4) we determine the resonance energies and widths for H − ( Z = 1), He( Z = 2) and Li+ ( Z = 3) in ECSCP. These results have been computed using 372 terms in the wave function (4). The convergence of the results has also been checked by increasing the number of terms in the wave function (4). Results for H − ( Z = 1), He( Z = 2) in ECSCP have been reported in our earlier works [3,4]. The results for Li+ ( Z = 3) in ECSCP are new. Here we have considered these three together in order to make an overall discussion on the nature of the autoionization states of two-electron systems. The stabilization diagram, in Fig. 1(a), corresponding to μ = 0.1 for Li+ is constructed by covering the range 0.2 α 0.5 with 241 points in the mesh size of 0.00125. This figure shows a stabilization character near E ≈ −1.4. The density of resonance states for the individual energy levels in the range 0.2–0.5, with one energy level at a time, is calculated and then fitted to Eq. (6). The Lorentz fitting for 12th eigenvalue is shown in Fig. 1(b). This gives the best fit (with the least χ 2 ), from which resonance energy and width are computed. Results are shown for Z = 1, 2 and Z = 3 in Table 1. From this table we first note that for the unscreened case our results
124
A. Ghoshal, Y.K. Ho / Computer Physics Communications 182 (2011) 122–124
effect the width first slowly increases and then decreases rapidly. This is special feature of H − interacting with ECSCP. Such a behaviour is not observed for H − in SCP, as is evident from Fig. 2. But Ps− in SCP/ECSCP follows the same behaviour as that of the H − in SCP/ECSCP [5]. It seems that this is due to the combined effect of the oscillatory nature of ECSCP and the equality of the attractive and repulsive forces in H − and Ps− . A rigorous explanation of such a behaviour needs a further extensive study of these two factors, and is of interest for future investigations. 4. Conclusions e
Fig. 2. Resonance widths for H − , He and Li+ in ECSCP and SCP.
agree nicely with the accurate results in the literature. The computed resonance energies and widths are also in good agreement with the observed data [10]. The overall uncertainty in our present investigation is estimated to about 4 × 10−6 a.u. each for the resonance energy and width. From Table 1 it is seen that resonance energy for each Z increases with increasing screening effect and ultimately becomes very close to 2S energy of the corresponding one-electron system. The trend is same for ECSCP and SCP, except ECSCP results are larger than SCP due to the stronger screening effect of ECSCP. Also for a fixed μ resonance energy decreases steadily with increasing Z . The values of the 2S energies have been calculated by using a wave function of the form:
Φ(r) =
i
C i φi =
C i e − A i r r li ,
l i = 0, 1, 2, . . . ,
i
where A i is a non-linear variational parameter, and r denotes the coordinates of the electron relative to the nucleus, within the framework of Ritz’s variational principle. On the other hand, resonance width decreases rapidly with increasing screening effect in both ECSCP and SCP, except for the case in ECSCP for Z = 1. Fig. 2 clearly shows the behaviour of the resonance width with increasing screening effect in SCP and ECSCP for Z = 1, 2 and 3. For Z = 1 in ECSCP, with increasing screening
The 2s21 S autoionization resonance states of the two-electron systems interacting with ECSCP have been determined within the framework of stabilization method by employing highly correlated wave functions. This method is a computational powerful and efficient method to compute the resonance parameters. It has been found that the resonance width for H − follows a different behaviour from the other elements. We hope that our present investigation will provide useful information to the researchers in the field of few-body physics, plasma physics, solid state physics, nuclear physics and astrophysics. Acknowledgements This work has been sponsored by the National Science Council of Taiwan. References [1] S.-C. Na, Y.-D. Jung, Phys. Lett. A 372 (2008) 5605; S.-C. Na, Y.-D. Jung, Phys. Scr. 78 (2008) 035502. [2] A. Ghoshal, Y.K. Ho, Phys. Plasmas 16 (2009) 073302; A. Ghoshal, Y.K. Ho, J. Phys. B 42 (2009) 075002; A. Ghoshal, Y.K. Ho, Few Body Syst. 46 (2009) 249–256. [3] A. Ghoshal, Y.K. Ho, Phys. Rev. A 79 (2009) 062514. [4] A. Ghoshal, Y.K. Ho, J. Phys. B 42 (2009) 175006. [5] A. Ghoshal, Y.K. Ho, Eur. Phys. J. D 56 (2009) 151–156. [6] V.I. Lengyel, V.T. Navrotsky, E.P. Sabad, Resonance Phenomena in Electron–Atom Collisions, Springer-Verlag, New York, 1992. [7] S. Chakraborty, Y.K. Ho, Chem. Phys. Lett. 438 (2007) 99; Y.K. Ho, A.K. Bhatia, A. Temkin, Phys. Rev. A 15 (1977) 1423; Y.K. Ho, Phys. Rev. A 23 (1981) 2137; Y.K. Ho, Phys. Rev. A 34 (1986) 4402. [8] S. Kar, Y.K. Ho, J. Phys. B 38 (2005) 3299; S. Kar, Y.K. Ho, Chem. Phys. Lett. 402 (2005) 544. [9] V.A. Mandelshtam, T.R. Ravuri, H.S. Taylor, Phys. Rev. Lett. 70 (1993) 1932. [10] P.J. Hicks, J. Comer, J. Phys. B 8 (1975) 1866.
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Doubly excited resonance states of two-electron systems in exponential cosine-screened Coulomb potentials Arijit Ghoshal ∗,1 , Y.K. Ho Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106
a r t i c l e
i n f o
a b s t r a c t e
We have made an investigation on the 2s21 S resonance states in two-electron systems interacting with exponential cosine-screened Coulomb potentials (ECSCP) within the framework of stabilization method. Highly correlated wave functions are used to take into account the correlations of the charge particles. Results for resonance energies and widths are reported for Z = 1, 2 and 3. For the unscreened case, our reported results are in nice agreement with the results so far available in the literature. © 2010 Elsevier B.V. All rights reserved.
Article history: Received 28 January 2010 Accepted 27 May 2010 Available online 4 June 2010 Keywords: Doubly excited resonance states Stabilization method Density of the resonance states Exponential cosine-screened Coulomb potential Correlated wave function
1. Introduction The study of atomic processes for which particles are interacting with exponential-cosine screened Coulomb potential (ECSCP) with screening parameter μ,
V (r ) = (1/r )e −μr cos (μr )
(in a.u.),
(1)
is currently an interesting topic of research [1–5] (and further references therein). The screening parameter μ shortens the range of the potential in comparison to the Coulomb potential. Exponential cosine-screened Coulomb potentials are widespread in many different areas of physics, such as, solid-state physics, nuclear physics, plasma physics [2–5]. These potentials differ from the well-known screened Coulomb potentials (SCP) with screening parameter μ,
V (r ) = (1/r )e −μr
(in a.u.),
(2)
from the existence of the cos (μr ) term, as a result an ECSCP exhibits stronger screening effect than an SCP. In this paper our objective is to investigate the 2s21 S e autoionization resonance states of a system consisting of a nucleus of charge Z e and two electrons interacting via ECSCP. Investigation of the autoionization states of an atomic system is of utmost importance in several respects [6] and indeed there is a continuous
*
Corresponding author. E-mail address: [email protected] (A. Ghoshal). 1 Permanent affiliation: Department of Mathematics, Burdwan University, Golapbag, Burdwan 713 104, West Bengal, India. 0010-4655/$ – see front matter doi:10.1016/j.cpc.2010.05.010
© 2010
Elsevier B.V. All rights reserved.
interest to investigate autoionization states of two-electron systems [3–5,7]. Moreover, investigation on two-electron systems in ECSCP is also important role owing to the role played by the correlation effects between the charged particles. We have designed this paper as follows. Describing the underlying theory and calculations of our investigation in Section 2 we present and discuss our computed results in Section 3. Finally, in Section 4 we give our concluding remarks. The atomic units (a.u.) are used throughout the present work, and all calculations are performed in quadruple precision (32 significant figures) on IBM–AMD workstations in UNIX environment. 2. Theory and calculations The non-relativistic Hamiltonian of a system consisting of a nucleus of charge Z e and two electrons interacting via ECSCP is given by
−μr1 1 1 e e −μr2 H = − ∇12 − ∇22 − Z cos (μr1 ) + cos (μr2 ) 2 2 r1 r2 +
e −μr12 r12
cos (μr12 ),
(3)
where r1 and r2 are the coordinates of the two electrons relative to the nucleus (assumed to be at rest), and r12 is their relative distance. In order to determine 1 S e states, we have employed the wave function
A. Ghoshal, Y.K. Ho / Computer Physics Communications 182 (2011) 122–124
123
e
Fig. 1. (a) Stabilization plots of the 2s2 1 S states of Li+ in ECSCP for μ = 0.1, and ω 14 (N = 372). The number in the parentheses next to the solid line indicates the e order of appearance of the eigenvalues. (b) Calculated density (circles) and the fitted Lorentzian (solid line) for the 12th eigenvalue corresponding to the 2s2 1 S state of Li+ in ECSCP for μ = 0.1. Table 1 The resonance energies ( E r ) (a.u.) and widths (Γ ) (a.u.) along with the energies of Li++ (2S ), E Li++ (2S ) , in ECSCP for various values of the screening parameter H − ( Z = 1)
− E r (a.u.)
μ
Γ (a.u.)
− E r (a.u.)
μ.
Li+ ( Z = 3)
He( Z = 2)
Γ (a.u.)
0.0 (a) 0.05 0.1 0.15 0.2 0.3 0.45
0.14876 0.148777 0.09770 0.04686 0.00647
0.001732 0.001733 0.001764 0.001605 0.000534
Present results in ECSCP 0.77784 0.004503 0.77787 0.00454 0.62848 0.004487 0.48306 0.004378 0.34554 0.004085 0.22004 0.003559
0.05 0.1 0.2
0.10182 0.06311
0.001638 0.001309
Results of Kar and Ho [8] in SCP 0.63683 0.004450 0.51279 0.004159 0.31105 0.003191
− E r (a.u.)
Γ (a.u.)
− E Li++ (2S )
1.90582 1.905845 1.65648 1.41098 1.17264 0.94449 0.53008 0.06739
0.005667 0.00566 0.005658 0.005604 0.005470 0.005228 0.004318 0.001545
1.25
1.666661 1.448150 1.067568
0.005613 0.005432 0.004796
0.975546504 0.829104144 0.688046366 0.554244676 0.314471807 0.047531
(a): Results of this row are taken from Ref. [7], obtained by using method complex coordinate rotation and precision calculation.
Ψ (r1 , r2 ) =
N i =1
C i ψi =
N
ρn ( E ) = y 0 +
C i (1 + P 12 )e − A (r1 +r2 )α r1i r2 i r12i , l m n
i =1
l i , m i , n i = 0, 1, 2, . . . , l i m i ,
(4)
where A is a non-linear variational parameter, C i (i = 1, 2, 3, . . . , N) are linear expansion coefficients, α is a scaling constant to be discussed later and P 12 is an exchange operator such that P 12 f (r1 , r2 ) = f (r2 , r1 ) for an arbitrary function f . This wave function is expanded by generating the powers of r1 , r2 and r12 following the relation ω = li + mi + mi = 0, 1, 2, . . . . In the present work we use stabilization method similar to Ho and co-workers [3,8] to determine the resonance energies and widths. This method is a slight generalization of the stabilization method proposed by Mandelshtam et al. [9], and consists of three steps. In the first step, one obtains the energy levels E (α ) by diagonalizing the Hamiltonian (3) for a number of values of α within a certain range. The graph of E (α ) versus α is called a stabilization plot (as shown in Fig. 1(a)). If there is a resonance at energy E, stabilized or slowly decreasing energy levels will appear in the stabilization plateau. Once the existence of a resonance is assured, in the second step, density of resonance states for a single energy level is calculated using the following formula:
ρn ( E ) =
E n (αi +1 ) − E n (αi −1 )
α i +1 − α i −1
− 1 ,
(5)
E n (αi )= E
where the index i is the ith value for α and the index n is for the nth resonance. Finally, to obtain resonance energy E r and width Γ , we fit it to the following Lorentzian form:
Γ 2
π ( E − E r )2 + ( Γ2 )2
,
(6)
where y 0 is the baseline offset, is the total area under the curve from the base line, E r is the centre of the peak and Γ denotes the full width of the peak of the curve at half height. 3. Results and discussion Using the wave function (4) we determine the resonance energies and widths for H − ( Z = 1), He( Z = 2) and Li+ ( Z = 3) in ECSCP. These results have been computed using 372 terms in the wave function (4). The convergence of the results has also been checked by increasing the number of terms in the wave function (4). Results for H − ( Z = 1), He( Z = 2) in ECSCP have been reported in our earlier works [3,4]. The results for Li+ ( Z = 3) in ECSCP are new. Here we have considered these three together in order to make an overall discussion on the nature of the autoionization states of two-electron systems. The stabilization diagram, in Fig. 1(a), corresponding to μ = 0.1 for Li+ is constructed by covering the range 0.2 α 0.5 with 241 points in the mesh size of 0.00125. This figure shows a stabilization character near E ≈ −1.4. The density of resonance states for the individual energy levels in the range 0.2–0.5, with one energy level at a time, is calculated and then fitted to Eq. (6). The Lorentz fitting for 12th eigenvalue is shown in Fig. 1(b). This gives the best fit (with the least χ 2 ), from which resonance energy and width are computed. Results are shown for Z = 1, 2 and Z = 3 in Table 1. From this table we first note that for the unscreened case our results
124
A. Ghoshal, Y.K. Ho / Computer Physics Communications 182 (2011) 122–124
effect the width first slowly increases and then decreases rapidly. This is special feature of H − interacting with ECSCP. Such a behaviour is not observed for H − in SCP, as is evident from Fig. 2. But Ps− in SCP/ECSCP follows the same behaviour as that of the H − in SCP/ECSCP [5]. It seems that this is due to the combined effect of the oscillatory nature of ECSCP and the equality of the attractive and repulsive forces in H − and Ps− . A rigorous explanation of such a behaviour needs a further extensive study of these two factors, and is of interest for future investigations. 4. Conclusions e
Fig. 2. Resonance widths for H − , He and Li+ in ECSCP and SCP.
agree nicely with the accurate results in the literature. The computed resonance energies and widths are also in good agreement with the observed data [10]. The overall uncertainty in our present investigation is estimated to about 4 × 10−6 a.u. each for the resonance energy and width. From Table 1 it is seen that resonance energy for each Z increases with increasing screening effect and ultimately becomes very close to 2S energy of the corresponding one-electron system. The trend is same for ECSCP and SCP, except ECSCP results are larger than SCP due to the stronger screening effect of ECSCP. Also for a fixed μ resonance energy decreases steadily with increasing Z . The values of the 2S energies have been calculated by using a wave function of the form:
Φ(r) =
i
C i φi =
C i e − A i r r li ,
l i = 0, 1, 2, . . . ,
i
where A i is a non-linear variational parameter, and r denotes the coordinates of the electron relative to the nucleus, within the framework of Ritz’s variational principle. On the other hand, resonance width decreases rapidly with increasing screening effect in both ECSCP and SCP, except for the case in ECSCP for Z = 1. Fig. 2 clearly shows the behaviour of the resonance width with increasing screening effect in SCP and ECSCP for Z = 1, 2 and 3. For Z = 1 in ECSCP, with increasing screening
The 2s21 S autoionization resonance states of the two-electron systems interacting with ECSCP have been determined within the framework of stabilization method by employing highly correlated wave functions. This method is a computational powerful and efficient method to compute the resonance parameters. It has been found that the resonance width for H − follows a different behaviour from the other elements. We hope that our present investigation will provide useful information to the researchers in the field of few-body physics, plasma physics, solid state physics, nuclear physics and astrophysics. Acknowledgements This work has been sponsored by the National Science Council of Taiwan. References [1] S.-C. Na, Y.-D. Jung, Phys. Lett. A 372 (2008) 5605; S.-C. Na, Y.-D. Jung, Phys. Scr. 78 (2008) 035502. [2] A. Ghoshal, Y.K. Ho, Phys. Plasmas 16 (2009) 073302; A. Ghoshal, Y.K. Ho, J. Phys. B 42 (2009) 075002; A. Ghoshal, Y.K. Ho, Few Body Syst. 46 (2009) 249–256. [3] A. Ghoshal, Y.K. Ho, Phys. Rev. A 79 (2009) 062514. [4] A. Ghoshal, Y.K. Ho, J. Phys. B 42 (2009) 175006. [5] A. Ghoshal, Y.K. Ho, Eur. Phys. J. D 56 (2009) 151–156. [6] V.I. Lengyel, V.T. Navrotsky, E.P. Sabad, Resonance Phenomena in Electron–Atom Collisions, Springer-Verlag, New York, 1992. [7] S. Chakraborty, Y.K. Ho, Chem. Phys. Lett. 438 (2007) 99; Y.K. Ho, A.K. Bhatia, A. Temkin, Phys. Rev. A 15 (1977) 1423; Y.K. Ho, Phys. Rev. A 23 (1981) 2137; Y.K. Ho, Phys. Rev. A 34 (1986) 4402. [8] S. Kar, Y.K. Ho, J. Phys. B 38 (2005) 3299; S. Kar, Y.K. Ho, Chem. Phys. Lett. 402 (2005) 544. [9] V.A. Mandelshtam, T.R. Ravuri, H.S. Taylor, Phys. Rev. Lett. 70 (1993) 1932. [10] P.J. Hicks, J. Comer, J. Phys. B 8 (1975) 1866.