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Positronium formation cross section for positron–lithium collisions
Nuclear Instruments and Methods in Physics Research B 241 (2005) 257–261 www.elsevier.com/locate/nimb
Positronium formation cross section for positron–lithium collisions S.J. Ward a
a,*
, J. Shertzer
b
University of North Texas, P.O. Box 311427, Denton, TX 76203, USA b College of the Holy Cross, Worcester, MA 01610, USA Available online 18 August 2005
Abstract The hyperspherical hidden crossing method (HHCM) has been used to calculate the positronium formation cross section for positron–lithium collisions in the energy range 0–1.8 eV. Results for the s-, p- and d-wave positronium formation cross sections were previously reported; here we present new results for the f-wave. In energy range studied, we found that the Stu¨ckelberg phase DL varied only slightly with incident positron momentum ki and decreased in a systematic way with increasing partial wave L. The HHCM cross section for positronium formation (summed over the four partial waves) lies between experimental measurements of the lower and upper limits of this cross section; the f-wave contribution to the cross section dominates near the lithium excitation threshold. Ó 2005 Elsevier B.V. All rights reserved. PACS: 34.85.+x; 36.10.Dr Keywords: Hyperspherical hidden crossing method; Positron–lithium collisions; Positronium
1. Introduction Recently, experimental measurements have been made of the positronium formation cross section for low-energy e+–Li and e+–Na collisions down to a few tenths of an electron volt [1]. In the experiments, both a lower limit (LL) and an upper limit *
Corresponding author. Tel.: +1 940 565 4739; fax: +1 940 565 2515. E-mail address: [email protected] (S.J. Ward).
(UL) to the positronium formation cross section were measured. At very low energies, the LL measurements are thought to be more reliable than the UL measurements and thus, the actual positronium formation cross section is expected to lie closer to the LL. Calculations using the close-coupling approximation (CCA) have been performed for e+–Li collisions [2–4] and e+–Na collisions [4,5]. There is good agreement between the CCA calculations and the LL experimental measurements for e+–Li collisions. However, for e+–Na collisions,
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.07.086
258
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
there is significant difference in magnitude and shape near threshold between the CCA calculations and the experimental measurements. The CCA calculations lie below the LL measurements for energies less than 2 eV. We have begun a series of investigations of lowenergy positron collisions with the alkali–metal atoms using the hyperspherical hidden crossing method (HHCM). Previously, we reported HHCM calculations of s-, p- and d-wave cross sections for positronium formation for e+–Li collisions up to the first excitation threshold of lithium (ki = 0.36, E = 1.8 eV) [6]. We also computed these cross sections including a correction term to the HHCM that emerges from the oneSturmian theory; we refer to these calculations as HHCM+cor. The HHCM and HHCM+cor calculations provided an interpretation of the main features of the cross sections. In these calculations, we used the same model potential that was used by Watts and Humberson in their Kohn variational calculation [7]. Recently, we investigated the effect on the positronium formation cross section of including the core polarization term in the model potential [8]. In this paper, we report our HHCM and HHCM+cor calculations of the f-wave cross section for positronium formation in e+–Li collisions in the energy range 0–1.8 eV (without the core polarization term). We compare the sum of the s-, p-, dand f-wave positronium formation cross sections with the experimental measurements. With results for four partial waves, we are able to study the Stu¨ckelberg phase as a function of both the incident momentum ki and the partial wave L. Such studies are expected to play an important role in interpreting the shapes and magnitudes of the positronium formation cross section. We use atomic units throughout this paper unless explicitly stated.
2. Method The formulation of the HHCM is given in [9] and the derivation of the correction term to the HHCM is given in [10]. The details of the application of the HHCM to e+–Li collisions are found in
[6]. We present here only the equations that show how the cross section for positronium formation depends on the one-way transition probability and the Stu¨ckelberg phase. The Lth partial wave cross section (in units of pa20 ) is given by rLij ¼
ð2L þ 1Þ L 4P ij ð1 P Lij Þsin2 DLij ; k 2i
ð1Þ
where ki is the incident positron momentum, i is the initial level [e+ + Li(2s)] and j is the final level [Ps(1s) + Li+]. P Lij is the one-way transition probability hR i 2I KðRÞdR L c P ij ¼ e ð2Þ and DLij is the Stu¨ckelberg phase Z DLij ¼ R KðRÞdR .
ð3Þ
c
In Eqs. (2) and (3), R is the hyperradius and the wave vector K(R) is obtained from e(R). The function e(R) is single valued on a multi-sheeted Riemann surface and its branches are the adiabatic energy eigenvalues el(R). The wave vector defined for a branch el(R) is given by 1 ¼ 2½E e0l ðRÞ; ð4Þ 4R2 where E is the total energy and l = i, j, . . . The contour c in the integral of Eqs. (2) and (3) is from the classical turning Rti on the branch of the Riemann surface corresponding to level e+ + Li(2s), around the branch point Rb, to the classical turning point Rtj on the branch corresponding to level Ps(1s) + Li+. In the HHCM calculation, the wave vector given by Eq. (4) is used for all R, real and complex. Using the one-Sturmian theory [9], a correction to the wave vector was derived for large real R e 2 ðRÞ is given by [10]. The corrected wave vector K l * + 2 o ul e 2 ðRÞ ¼ 2½E el ðRÞ þ ul K l oR2
K 2l ðRÞ ¼ 2½E el ðRÞ
¼ 2½E ~e0l ðRÞ;
ð5Þ
where ul(R) is the adiabatic basis function. The potential ~e0l ðRÞ in Eq. (5) agrees asymptotically
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
with the close-coupling potential through order 1/R2. In the HHCM+cor calculation, the wave vector given by Eq. (4) is used for complex R; the corrected wave vector given by Eq. (5) is used for real R.
80 TOTAL
HHCM
60
σPs (πa02)
3. Results
259
d-wave
40 p-wave
We located the f-wave branch point Rb that connects eigenvalues ej(R) and ei(R) in a square with sides 0.1 centered at R = 10.55 + i5.15. The position of the f-wave branch point is close to the positions of the s-, p- and d-wave branch points. Fig. 1 shows the HHCM and HHCM+cor f-wave positronium formation cross sections for e+–Li collisions. The potential ~e0i ðRÞ is more repulsive than the potential e0i ðRÞ for values of real R in the contour integral. The classical turning point Rti associated with the more repulsive potential occurs at a larger value of R. As a result, the HHCM+cor cross section is smaller than the HHCM cross section, and the maximum is shifted to higher energy. Fig. 2 shows the s-, p-, d- and f-wave HHCM cross sections for positronium formation, as well as the sum of the four partial waves. Fig. 3 gives
20 s-wave
0
0.00
0.05
f-wave
0.10
0.15
0.20
0.25
0.30
0.35
ki (a.u.) Fig. 2. The s-, p-, d- and f-wave cross sections for positronium formation computed by the HHCM; the solid line represents the sum of the four partial waves.
80 HHCM+cor
TOTAL
60
σPs (πa20)
40
40
d-wave
HHCM
30
p-wave
20
σPs3 (πa02)
f-wave
HHCM+cor
20
s-wave
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ki (a.u.) 10
Fig. 3. The s-, p-, d- and f-wave cross sections for positronium formation computed by the HHCM+cor; the solid line represents the sum of the four partial waves. 0 0.0
0.1
0.2
0.3
ki (a.u.) Fig. 1. The f-wave positronium formation cross section for e+– Li collisions computed by the HHCM (solid line) and the HHCM+cor (dashed line).
the same results for the HHCM+cor calculation. In both HHCM and HHCM+cor, the contribution of the f-wave is small for ki < 0.1, but it accounts for at least half of the summed cross section for
260
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
creases in a systematic way with increasing L. As discussed in [6], the small s-wave cross section for positronium formation in e+–Li collisions is due to the Stu¨ckelberg phase being close to p. When the Stu¨ckelberg phase is p, the two amplitudes that correspond to different paths leading to positronium formation destructively interfere. For the f-wave, the Stu¨ckelberg phase approaches p/2 with increasing ki, which means the two amplitudes interfere constructively. Fig. 5 compares the HHCM and HHCM+cor cross sections for positronium formation (summed over the four partial waves) with the LL and UL experimental measurements. Both calculations lie between the LL and UL measurements, while the HHCM+cor lies closer to the more reliable LL measurements.
s-wave
3.0
p-wave
2.5
2.0 d-wave
∆
L
1.5 f-wave
1.0
0.5
0.0 0.0
0.1
0.2
0.3
ki (a.u.) Fig. 4. The s-, p-, d- and f-wave Stu¨ckelberg phase computed by the HHCM (solid line) and the HHCM+cor (dashed line).
4. Conclusion ki > 0.29. Fig. 4 shows the Stu¨ckelberg phase DL for the s-, p-, d- and f-waves. For a given L, the phase varies slowly with incident positron momentum ki. In the energy range considered here (E 6 1.8 eV, ki 6 0.36) the Stu¨ckelberg phase de-
100
80
HHCM
60
HHCM+cor
We have extended our original HHCM analysis [6] to include the f-wave contribution to the cross section for positronium formation in e+–Li in the energy range 0–1.8 eV. The f-wave cross section appears to make the dominant contribution to the total positronium formation cross section at energies near the first excitation threshold of lithium. A comparison of the Stu¨ckelberg phase for the four partial waves suggests that DL varies systematically as a function of ki and L. Having established that the HHCM is a promising method for studying positron–alkali metal collisions, we plan to calculate the positronium formation cross section for e+ + Na collisions.
σ
2 Ps (πa 0 )
40
References
20
0 0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
E (eV) Fig. 5. The HHCM (solid line) and HHCM+cor (dashed line) cross sections for positronium formation (summed over the first four partial waves) are compared with lower (+) and upper ( ) experimental limits.
[1] E. Surdutovich, J.M. Johnson, W.E. Kauppila, C.W. Kwan, T.S. Stein, Phys. Rev. A 65 (2002) 032713; and private communication. [2] A.A. Kernoghan, M.T. McAlinden, H.R.J. Walters, J. Phys. B 27 (1994) L625. [3] M.T. McAlinden, A.A. Kernoghan, H.R.J. Walters, J. Phys. B 30 (1997) 1543. [4] C.P. Campbell, M.T. McAlinden, A.A. Kernoghan, H.R.J. Walters, Nucl. Instr. and Meth. B 143 (1998) 41. [5] G. Ryzhikh, J. Mitroy, Phys. Rev. Lett. 79 (1997) 5545. [6] S.J. Ward, J. Shertzer, Phys. Rev. A 68 (2003) 032720.
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261 [7] M.S.T. Watts, J.W. Humberston, J. Phys. B 25 (1992) L491. [8] S.J. Ward, J. Shertzer, Nucl. Instr. and Meth. B 221 (2004) 206.
261
[9] J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 54 (1996) 544. [10] S.J. Ward, J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 59 (1999) 4418.
Positronium formation cross section for positron–lithium collisions S.J. Ward a
a,*
, J. Shertzer
b
University of North Texas, P.O. Box 311427, Denton, TX 76203, USA b College of the Holy Cross, Worcester, MA 01610, USA Available online 18 August 2005
Abstract The hyperspherical hidden crossing method (HHCM) has been used to calculate the positronium formation cross section for positron–lithium collisions in the energy range 0–1.8 eV. Results for the s-, p- and d-wave positronium formation cross sections were previously reported; here we present new results for the f-wave. In energy range studied, we found that the Stu¨ckelberg phase DL varied only slightly with incident positron momentum ki and decreased in a systematic way with increasing partial wave L. The HHCM cross section for positronium formation (summed over the four partial waves) lies between experimental measurements of the lower and upper limits of this cross section; the f-wave contribution to the cross section dominates near the lithium excitation threshold. Ó 2005 Elsevier B.V. All rights reserved. PACS: 34.85.+x; 36.10.Dr Keywords: Hyperspherical hidden crossing method; Positron–lithium collisions; Positronium
1. Introduction Recently, experimental measurements have been made of the positronium formation cross section for low-energy e+–Li and e+–Na collisions down to a few tenths of an electron volt [1]. In the experiments, both a lower limit (LL) and an upper limit *
Corresponding author. Tel.: +1 940 565 4739; fax: +1 940 565 2515. E-mail address: [email protected] (S.J. Ward).
(UL) to the positronium formation cross section were measured. At very low energies, the LL measurements are thought to be more reliable than the UL measurements and thus, the actual positronium formation cross section is expected to lie closer to the LL. Calculations using the close-coupling approximation (CCA) have been performed for e+–Li collisions [2–4] and e+–Na collisions [4,5]. There is good agreement between the CCA calculations and the LL experimental measurements for e+–Li collisions. However, for e+–Na collisions,
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.07.086
258
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
there is significant difference in magnitude and shape near threshold between the CCA calculations and the experimental measurements. The CCA calculations lie below the LL measurements for energies less than 2 eV. We have begun a series of investigations of lowenergy positron collisions with the alkali–metal atoms using the hyperspherical hidden crossing method (HHCM). Previously, we reported HHCM calculations of s-, p- and d-wave cross sections for positronium formation for e+–Li collisions up to the first excitation threshold of lithium (ki = 0.36, E = 1.8 eV) [6]. We also computed these cross sections including a correction term to the HHCM that emerges from the oneSturmian theory; we refer to these calculations as HHCM+cor. The HHCM and HHCM+cor calculations provided an interpretation of the main features of the cross sections. In these calculations, we used the same model potential that was used by Watts and Humberson in their Kohn variational calculation [7]. Recently, we investigated the effect on the positronium formation cross section of including the core polarization term in the model potential [8]. In this paper, we report our HHCM and HHCM+cor calculations of the f-wave cross section for positronium formation in e+–Li collisions in the energy range 0–1.8 eV (without the core polarization term). We compare the sum of the s-, p-, dand f-wave positronium formation cross sections with the experimental measurements. With results for four partial waves, we are able to study the Stu¨ckelberg phase as a function of both the incident momentum ki and the partial wave L. Such studies are expected to play an important role in interpreting the shapes and magnitudes of the positronium formation cross section. We use atomic units throughout this paper unless explicitly stated.
2. Method The formulation of the HHCM is given in [9] and the derivation of the correction term to the HHCM is given in [10]. The details of the application of the HHCM to e+–Li collisions are found in
[6]. We present here only the equations that show how the cross section for positronium formation depends on the one-way transition probability and the Stu¨ckelberg phase. The Lth partial wave cross section (in units of pa20 ) is given by rLij ¼
ð2L þ 1Þ L 4P ij ð1 P Lij Þsin2 DLij ; k 2i
ð1Þ
where ki is the incident positron momentum, i is the initial level [e+ + Li(2s)] and j is the final level [Ps(1s) + Li+]. P Lij is the one-way transition probability hR i 2I KðRÞdR L c P ij ¼ e ð2Þ and DLij is the Stu¨ckelberg phase Z DLij ¼ R KðRÞdR .
ð3Þ
c
In Eqs. (2) and (3), R is the hyperradius and the wave vector K(R) is obtained from e(R). The function e(R) is single valued on a multi-sheeted Riemann surface and its branches are the adiabatic energy eigenvalues el(R). The wave vector defined for a branch el(R) is given by 1 ¼ 2½E e0l ðRÞ; ð4Þ 4R2 where E is the total energy and l = i, j, . . . The contour c in the integral of Eqs. (2) and (3) is from the classical turning Rti on the branch of the Riemann surface corresponding to level e+ + Li(2s), around the branch point Rb, to the classical turning point Rtj on the branch corresponding to level Ps(1s) + Li+. In the HHCM calculation, the wave vector given by Eq. (4) is used for all R, real and complex. Using the one-Sturmian theory [9], a correction to the wave vector was derived for large real R e 2 ðRÞ is given by [10]. The corrected wave vector K l * + 2 o ul e 2 ðRÞ ¼ 2½E el ðRÞ þ ul K l oR2
K 2l ðRÞ ¼ 2½E el ðRÞ
¼ 2½E ~e0l ðRÞ;
ð5Þ
where ul(R) is the adiabatic basis function. The potential ~e0l ðRÞ in Eq. (5) agrees asymptotically
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
with the close-coupling potential through order 1/R2. In the HHCM+cor calculation, the wave vector given by Eq. (4) is used for complex R; the corrected wave vector given by Eq. (5) is used for real R.
80 TOTAL
HHCM
60
σPs (πa02)
3. Results
259
d-wave
40 p-wave
We located the f-wave branch point Rb that connects eigenvalues ej(R) and ei(R) in a square with sides 0.1 centered at R = 10.55 + i5.15. The position of the f-wave branch point is close to the positions of the s-, p- and d-wave branch points. Fig. 1 shows the HHCM and HHCM+cor f-wave positronium formation cross sections for e+–Li collisions. The potential ~e0i ðRÞ is more repulsive than the potential e0i ðRÞ for values of real R in the contour integral. The classical turning point Rti associated with the more repulsive potential occurs at a larger value of R. As a result, the HHCM+cor cross section is smaller than the HHCM cross section, and the maximum is shifted to higher energy. Fig. 2 shows the s-, p-, d- and f-wave HHCM cross sections for positronium formation, as well as the sum of the four partial waves. Fig. 3 gives
20 s-wave
0
0.00
0.05
f-wave
0.10
0.15
0.20
0.25
0.30
0.35
ki (a.u.) Fig. 2. The s-, p-, d- and f-wave cross sections for positronium formation computed by the HHCM; the solid line represents the sum of the four partial waves.
80 HHCM+cor
TOTAL
60
σPs (πa20)
40
40
d-wave
HHCM
30
p-wave
20
σPs3 (πa02)
f-wave
HHCM+cor
20
s-wave
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ki (a.u.) 10
Fig. 3. The s-, p-, d- and f-wave cross sections for positronium formation computed by the HHCM+cor; the solid line represents the sum of the four partial waves. 0 0.0
0.1
0.2
0.3
ki (a.u.) Fig. 1. The f-wave positronium formation cross section for e+– Li collisions computed by the HHCM (solid line) and the HHCM+cor (dashed line).
the same results for the HHCM+cor calculation. In both HHCM and HHCM+cor, the contribution of the f-wave is small for ki < 0.1, but it accounts for at least half of the summed cross section for
260
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261
creases in a systematic way with increasing L. As discussed in [6], the small s-wave cross section for positronium formation in e+–Li collisions is due to the Stu¨ckelberg phase being close to p. When the Stu¨ckelberg phase is p, the two amplitudes that correspond to different paths leading to positronium formation destructively interfere. For the f-wave, the Stu¨ckelberg phase approaches p/2 with increasing ki, which means the two amplitudes interfere constructively. Fig. 5 compares the HHCM and HHCM+cor cross sections for positronium formation (summed over the four partial waves) with the LL and UL experimental measurements. Both calculations lie between the LL and UL measurements, while the HHCM+cor lies closer to the more reliable LL measurements.
s-wave
3.0
p-wave
2.5
2.0 d-wave
∆
L
1.5 f-wave
1.0
0.5
0.0 0.0
0.1
0.2
0.3
ki (a.u.) Fig. 4. The s-, p-, d- and f-wave Stu¨ckelberg phase computed by the HHCM (solid line) and the HHCM+cor (dashed line).
4. Conclusion ki > 0.29. Fig. 4 shows the Stu¨ckelberg phase DL for the s-, p-, d- and f-waves. For a given L, the phase varies slowly with incident positron momentum ki. In the energy range considered here (E 6 1.8 eV, ki 6 0.36) the Stu¨ckelberg phase de-
100
80
HHCM
60
HHCM+cor
We have extended our original HHCM analysis [6] to include the f-wave contribution to the cross section for positronium formation in e+–Li in the energy range 0–1.8 eV. The f-wave cross section appears to make the dominant contribution to the total positronium formation cross section at energies near the first excitation threshold of lithium. A comparison of the Stu¨ckelberg phase for the four partial waves suggests that DL varies systematically as a function of ki and L. Having established that the HHCM is a promising method for studying positron–alkali metal collisions, we plan to calculate the positronium formation cross section for e+ + Na collisions.
σ
2 Ps (πa 0 )
40
References
20
0 0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
E (eV) Fig. 5. The HHCM (solid line) and HHCM+cor (dashed line) cross sections for positronium formation (summed over the first four partial waves) are compared with lower (+) and upper ( ) experimental limits.
[1] E. Surdutovich, J.M. Johnson, W.E. Kauppila, C.W. Kwan, T.S. Stein, Phys. Rev. A 65 (2002) 032713; and private communication. [2] A.A. Kernoghan, M.T. McAlinden, H.R.J. Walters, J. Phys. B 27 (1994) L625. [3] M.T. McAlinden, A.A. Kernoghan, H.R.J. Walters, J. Phys. B 30 (1997) 1543. [4] C.P. Campbell, M.T. McAlinden, A.A. Kernoghan, H.R.J. Walters, Nucl. Instr. and Meth. B 143 (1998) 41. [5] G. Ryzhikh, J. Mitroy, Phys. Rev. Lett. 79 (1997) 5545. [6] S.J. Ward, J. Shertzer, Phys. Rev. A 68 (2003) 032720.
S.J. Ward, J. Shertzer / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 257–261 [7] M.S.T. Watts, J.W. Humberston, J. Phys. B 25 (1992) L491. [8] S.J. Ward, J. Shertzer, Nucl. Instr. and Meth. B 221 (2004) 206.
261
[9] J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 54 (1996) 544. [10] S.J. Ward, J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 59 (1999) 4418.