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The bus stop effect revisited
Nuclear Instruments and Methods in Physics Research B 214 (2004) 126–130 www.elsevier.com/locate/nimb
The bus stop effect revisited John F. Reading *, Jun Fu Center for Theoretical Physics, Texas A&M University, College Station, TX 77840-4242, USA
Abstract We revisit the question as to how cross sections for direct reactions, caused by a projectile of positive charge interacting with an atom at asymptotic velocities, heal to the first Born approximation. Finite Hilbert basis calculations indicate that for protons the cross section approaches the Born result from above. We can confirm this result for shorter range Yukawa potentials by using Response Theory but not for the Coulombic problem of direct interest. Ó 2003 Elsevier B.V. All rights reserved. PACS: 25.43.+t; 34.50.Bw Keywords: Bus stop effect; Barkas effect
1. Introduction When a bare ion, charge Zp , collides with an atom it can excite, ionize or capture the struck electrons. To theoretically determine the cross sections for these processes is straightforward. The ion can be accurately treated as moving classically on a straight line path; its velocity, v, is minimally altered by the actual events realized in the collision. The Coulombic interaction is known precisely. And further we can select simple target species such as hydrogen molecules or helium atoms. Our hope has been that by confronting theory with experiments we will not just reproduce digitally measured cross sections; rather we look for new developments in theory and new understanding of collisions. The building of ion sources in the sixties coincided with the development of modern computers
*
Corresponding author.
and their introduction to this field. Before that most theorists focused on using analytical techniques. Cross sections, r, for direct reactions approach the Born result, rB , at asymptotic projectile velocities. One analytical prediction was that proton cross sections rþ were less than rB , and antiproton cross sections r were greater than rB in this asymptotic region; a result called the bus stop effect [1–6]. As we shall see, this is not confirmed by most finite Hilbert basis set calculations, and the bus stop effect was consigned to a quiet death. But just recently it has been reported that antiproton stopping powers appear to be greater than proton stopping powers in the 600 keV region for both molecular hydrogen and helium [7]. Assuming no experimental difficulty, there are three possible explanations for this result. The first is that there is a bus stop effect for protons in collision with a single electron isolated atom. The second is that the other electron on a hydrogen molecule or helium atom target influences the outcome through pair correlation. The third is that
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.08.009
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
the environment of the target atoms plays a role. We have fairly good numerical calculations of the relevant cross sections for the first two situations. We do not believe there is an atomic bus stop effect. And while there definitely is a difference in proton and antiproton ionization cross sections in the right direction caused by pair correlation it appears too small to give the whole effect observed [8]. In this paper we revisit the atomic bus stop effect. Our aim was to produce an analytical theory which would confirm the numerical results. We are successful in this with the Response Theory for short ranged potentials, but fail for the Coulombic potential to get a satisfactory result.
2. The single electron Born series We will use atomic units. The charge of the projectile is Zp , its velocity is v. One unique feature of an ion–atom collision is the possibility of varying the coupling constant, n ¼ Zp =v. This strategic advantage is especially realized in comparing proton and antiproton projectiles. In a Born type perturbation series we can easily study the interplay of the odd terms and the even terms. Werner Brandt and his coworkers [9,10] explored the interference of the first and second Born terms in effective single electron systems for ionization; they discovered at low energies that for positive projectiles the interference was negative. Cross sections, rþ , fall below the Born result, rB . They explained this as an increased binding effect. As the energy of the projectile is raised a positive interference is found which they interpreted as a polarization effect; the electron cloud is thought to be attracted to the proton before the sharp binary collision that results in ionization. The increase in density near the proton enhances rþ . This cancels the binding effect, and the cross section is slightly bigger than rB . The situation is complicated somewhat for protons by charge transfer, which also enhances the total proton ionization cross section. But at high enough velocities we assume charge transfer is negligible. As the velocity of the projectile is raised further still it is expected that all direct cross sections,
127
excitation and ionization, will heal to the Born result. A mystery has always been how quickly as a function of the coupling constant, n ¼ Zp =v, this occurs, e.g. the Born approximation starts to give good results for protons and antiprotons on hydrogen at 400 keV. But the Glauber [1] approximation, based on a simple closure idea, supposedly relevant to high projectile velocities, gives the result r ¼ rB jCð1 þ inÞj2 enp :
ð1Þ
This gives a large correction to the Born result at this energy, about 50%; and importantly would predict that for protons, cross sections heal to the Born result from below. This result fueled speculation that a third projectile velocity region existed. The Born limit had not been reached at 400 keV; the cross sections were just crossing the Born results. The true cross sections were thought to approach the Born result above 1 MeV. This result was called the bus stop effect. The idea was that even though the electron was attracted to the proton, it was ejected in a binary encounter at some distance from the proton, where the density was perforce smaller.
3. The two electron Born series It was BrandtÕs hope that his studies would lead to a deeper understanding of the Barkas effect, the Zp3 term that appears in stopping powers as a correction to the dominant Born term, which is proportional to Zp2 [11]. And indeed as a function of energy the Barkas term seems to follow the binding and polarization effects. All these effects of course are reversed, and hence exposed, if one uses antiprotons as projectiles. The immediate leap from a single electron Born series to accurately describing the interaction of an ion with a correlated many electron atom buried in a dense solid is tempting but, in our opinion, too optimistic. Brandt did not have the advantage of an antiproton beam; he could use only protons, deuterons and alpha-particles. Experiments at LEAR above 400 keV where the single electron Born term was supposedly an accurate description of the binary encounter showed
128
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
cross sections for doubly ionizing helium were much higher for antiprotons than protons. This was a clear signal of an interference between first and second Born terms but the effect did not arise from when the projectile interacted with a single electron twice! In the first Born term, a Zp -process, one electron is struck by the projectile while it is interacting with the second electron. The target forces between the two electrons, i.e. internal dynamic correlation, ionize the second electron. The second Born term, a Zp2 -process, certainly has contributions from when the projectile and the struck electron interacts twice. However there is also a term the projectile, interacts once with each electron. This term interferes destructively with the first term for protons, constructively for antiprotons. Experiments on molecules showed similar results [8]. In terms of the Barkas effect in the energy region stated these effects are thought to be small, at the two percent level. But the sign of this electron correlation phenomena is such that the stopping power for protons would be less than for antiprotons. And would thus mimic a bus stop effect. The announcement that a bus stop like effect had in fact been discovered in the stopping power of antiprotons and protons in hydrogen and helium at 600 keV revised this question again. Is there really a bus stop effect in the single electron Born series, or are we dealing with a correlation effect possibly enhanced by the environment of the molecules and atoms? Single collision type experiments, [12,13], in this region demonstrate a polarization effect, no bus stop effect, for ionization (see Fig. 1). This experimental result is confirmed by a single center finite Hilbert basis state (FHBS) calculation presented here. And in Fig. 2 we calculate that there is a corresponding polarization result in the excitation to the 2s state, and hardly any difference between rþ and r for excitation to the 2p state. This is exactly in accord with dipole dominance, where it is asserted that the Coulomb interaction mainly changes the parity of the initial state. Thus the second Born term for s-excitation is large, as we allow intermediate p-states with two dipole interactions. And the second Born term for p-excitation
2
1.5
1
experiment theory
0.5
0 10
100
1000
10000
ENERGY (keV/amu) Fig. 1. Proton and antiproton ionization cross section ratios, rþ , as a function of energy of the projectiles (keV/amu). The r experimental results [12,13] are the closed circles. The finite Hilbert basis set (FHBS) results are the solid line. Note the presence of a polarization effect in both the experimental data and theory.
2
1.5
1
2s 2p (m = 0) 2p (m = 1)
0.5
0 10
100
1000
10000
ENERGY (keV/amu) Fig. 2. FHBS theoretical calculation of the differential cross section ratio rrþ for 1s–2s, 2p (m ¼ 0) and 2p (m ¼ 1). Note that the dipole forbidden 2s transition demonstrates a polarization effect. But the dipole allowed transition shows no interference from a second Born term.
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
is small because of the necessity for including a mono pole interaction. We find this evidence for absence of a bus stop effect compelling. But the most satisfactory situation would be that we has a straight forward analytical calculation of asymptotic cross sections which agreed with the numerical calculations; we could then settle the question once and for all. Further if we had such a theory perhaps it could be usefully applied to many electron systems.
4. The Response Theorem In the following we closely follow Hall et al. [14]. The second order scattering amplitude for excitation of an electron in state vn , energy n , to state vm , energy m by an ion, if we ignore charge transfer, can be written as * Tm;n ðQÞ ¼ mp vm ðrÞeiQr Vp ðQÞ þ
1 2p2
Z
129
effect. For the Coulomb problem as c is set to zero we deduce a ¼ p=v:
ð4Þ 02
A second approximation is replace h by Q . This is called the Cheshire approximation. For the Coulomb problem this gives a ¼ p=v sgnð2v Q þ Q2 Þ:
ð5Þ
At high velocities this gives a smaller bus stop effect for ionization [1]. These results come directly from off energy shell properties of the Coulomb t-matrix. An alternative procedure to these approximations is to chose to replace h by a quantity which is guaranteed to give zero to the next perturbative correction. i.e. replace 1 1 ðh hhiÞ ¼ þ ðD þ hÞ ðD þ hhiÞ ðD þ hhiÞ2 2
þ
ðh hhiÞ
ðD þ hhiÞ
3
þ
ð6Þ
This leads to the propagator
0
dQ0 eiðQQ Þr
D þ hhi ¼ 2ðv pÞ Q0 þ Q02 :
1 Vp ðjQ Q0 jÞ 0 2v Q þ 2ðHe n Þ + 0 iQ0 r Vp ðQ Þvn ðrÞ : ð2Þ e
ð7Þ
Here the momentum transfer is Q. And
The interpretation of this result is that the struck target electron is not considered to be fixed as in the Glauber method; nor is it assumed to be free and at rest as in the Cheshire method. Rather it is approximated to be free but moving with a definite momentum p. The method defines
v Q ¼ ðn m Þ:
p ¼ hvm jeiQr j½irr vn i:
We will assume a Yukawa type potential, hence
It is straightforward to show that for any quantum eigenstates
V ðQÞ ¼ 2Zp =ðQ2 þ c2 Þ: If we use Eq. (2) to calculate a cross section differential in Q we can write keeping leading order terms 2
2
jT j ¼ V ðQÞ ð1 þ Zp aÞ:
ð3Þ
The problem with simplifying Eq. (2) is the presence of h ¼ 2ðHe n Þ, the target Hamiltonian in the denominator. It may be argued that at high projectile velocities we can neglect this term. This allows closure; it is called the Glauber approximation. It leads to a prediction of a large bus stop
2ðv pÞ Q þ Q2 ¼ 0:
ð8Þ
ð9Þ
This implies that the corresponding t-matrix is on the energy shell. In Fig. 3 we show the results for a Yukawa potential with c ¼ 0:1. We show the excitation of the 1s–2s state. The Response Theory answer is in fair agreement with a corresponding FHBS calculation when compared say to the Cheshire result. One can include the next higher order term in Eq. (6). This is shown in Fig. 3 as Hall. This so called variance term does not improve the answer.
130
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
zero. The continued fraction expression, Eq. (10), has to be evaluated numerically. For the 1s–2s transition it gives a 0:10=v2 . This is much too small to reproduce the FHBS result of Fig. 2. Ishii et al. [3] have suggested an ad hoc addition of a polarization term which would solve the problem but it is disappointing that the straight forward Response Theory cannot give the desired result directly. Hopefully some way may be found around this difficulty in the future.
1 γ = 0.1 0.5
Hall
Response
2 α vp
0
-0.5
Cheshire -1
Acknowledgements -1.5 10
15
20 vp
25
30
Fig. 3. The second Born interference term plotted as function of projectile velocity in atomic units for a Yukawa potential of cr the form e r where c ¼ 0:1. The present work is the solid line and compares well with the FHBS calculation given by the closed circles. Other results are described in the text.
But if instead we write the second order correction as a continued fraction i.e. ! 2 hh2 i hhi ðD þ hÞ ¼ D þ hhi ; ð10Þ D þ hhi we obtain satisfactory agreement between the analytical theory and the FHBS calculation shown. This is a new result. The result for the pure Coulomb force, of most interest to us, is disappointing. The first order Response Theory gives no correction, a is zero. This result is independent of the particular reaction to be studied, as it follows from the on-shell behavior of the Coulomb t-matrix. The higher order Born terms merely add a phase to the first Born amplitude. The straightforward variance term, Eq. (6), gives a divergence as c is set down to
This work is supported by grants from the US National Science Foundation PHY-9605217.
References [1] J.F. Reading, E. Fitchard, Phys. Rev. A 10 (1974) 168. [2] J. Binstock, J.F. Reading, Phys. Rev. A 11 (1975) 1205. [3] K. Ishii, K. Sera, A. Yamdera, M. Sebata, H. Arai, S. Morita, Phys. Rev. A 25 (1982) 2511. [4] T. Quinteros, J.F. Reading, Nucl. Instr. and Meth. B 53 (1991) 363. [5] A.M. Ermolaev, J. Phys. B 23 (1990) L45. [6] H. Knudsen, J.F. Reading, Phys. Rep. 212 (1992) 107. [7] E. Lodi Rizzini et al., Phys. Rev. Lett. 89 (2002) 183201. [8] A.L. Ford, J.F. Reading, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 4215. [9] G. Basbas, W. Brandt, R. Laubert, Phys. Rev. A 7 (1973) 1983. [10] G. Basbas, W. Brandt, R. Laubert, A. Ratkowsky, A. Schwarzchild, Phys. Rev. Lett. 27 (1971) 171. [11] W.H. Barkas, J.N. Dyer, H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [12] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, J.O.P. Pedersen, S. Tang-Petersen, E. Uggerhoj, K. Elsener, E. Morenzoni, J. Phys. B 23 (1990) L395. [13] M.B. Shah, P. McCallion, H.B. Gilbody, J. Phys. B 22 (1989) 3983. [14] K.A. Hall, J.F. Reading, A.L. Ford, J. Phys. B 26 (1993) 1697.
The bus stop effect revisited John F. Reading *, Jun Fu Center for Theoretical Physics, Texas A&M University, College Station, TX 77840-4242, USA
Abstract We revisit the question as to how cross sections for direct reactions, caused by a projectile of positive charge interacting with an atom at asymptotic velocities, heal to the first Born approximation. Finite Hilbert basis calculations indicate that for protons the cross section approaches the Born result from above. We can confirm this result for shorter range Yukawa potentials by using Response Theory but not for the Coulombic problem of direct interest. Ó 2003 Elsevier B.V. All rights reserved. PACS: 25.43.+t; 34.50.Bw Keywords: Bus stop effect; Barkas effect
1. Introduction When a bare ion, charge Zp , collides with an atom it can excite, ionize or capture the struck electrons. To theoretically determine the cross sections for these processes is straightforward. The ion can be accurately treated as moving classically on a straight line path; its velocity, v, is minimally altered by the actual events realized in the collision. The Coulombic interaction is known precisely. And further we can select simple target species such as hydrogen molecules or helium atoms. Our hope has been that by confronting theory with experiments we will not just reproduce digitally measured cross sections; rather we look for new developments in theory and new understanding of collisions. The building of ion sources in the sixties coincided with the development of modern computers
*
Corresponding author.
and their introduction to this field. Before that most theorists focused on using analytical techniques. Cross sections, r, for direct reactions approach the Born result, rB , at asymptotic projectile velocities. One analytical prediction was that proton cross sections rþ were less than rB , and antiproton cross sections r were greater than rB in this asymptotic region; a result called the bus stop effect [1–6]. As we shall see, this is not confirmed by most finite Hilbert basis set calculations, and the bus stop effect was consigned to a quiet death. But just recently it has been reported that antiproton stopping powers appear to be greater than proton stopping powers in the 600 keV region for both molecular hydrogen and helium [7]. Assuming no experimental difficulty, there are three possible explanations for this result. The first is that there is a bus stop effect for protons in collision with a single electron isolated atom. The second is that the other electron on a hydrogen molecule or helium atom target influences the outcome through pair correlation. The third is that
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.08.009
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
the environment of the target atoms plays a role. We have fairly good numerical calculations of the relevant cross sections for the first two situations. We do not believe there is an atomic bus stop effect. And while there definitely is a difference in proton and antiproton ionization cross sections in the right direction caused by pair correlation it appears too small to give the whole effect observed [8]. In this paper we revisit the atomic bus stop effect. Our aim was to produce an analytical theory which would confirm the numerical results. We are successful in this with the Response Theory for short ranged potentials, but fail for the Coulombic potential to get a satisfactory result.
2. The single electron Born series We will use atomic units. The charge of the projectile is Zp , its velocity is v. One unique feature of an ion–atom collision is the possibility of varying the coupling constant, n ¼ Zp =v. This strategic advantage is especially realized in comparing proton and antiproton projectiles. In a Born type perturbation series we can easily study the interplay of the odd terms and the even terms. Werner Brandt and his coworkers [9,10] explored the interference of the first and second Born terms in effective single electron systems for ionization; they discovered at low energies that for positive projectiles the interference was negative. Cross sections, rþ , fall below the Born result, rB . They explained this as an increased binding effect. As the energy of the projectile is raised a positive interference is found which they interpreted as a polarization effect; the electron cloud is thought to be attracted to the proton before the sharp binary collision that results in ionization. The increase in density near the proton enhances rþ . This cancels the binding effect, and the cross section is slightly bigger than rB . The situation is complicated somewhat for protons by charge transfer, which also enhances the total proton ionization cross section. But at high enough velocities we assume charge transfer is negligible. As the velocity of the projectile is raised further still it is expected that all direct cross sections,
127
excitation and ionization, will heal to the Born result. A mystery has always been how quickly as a function of the coupling constant, n ¼ Zp =v, this occurs, e.g. the Born approximation starts to give good results for protons and antiprotons on hydrogen at 400 keV. But the Glauber [1] approximation, based on a simple closure idea, supposedly relevant to high projectile velocities, gives the result r ¼ rB jCð1 þ inÞj2 enp :
ð1Þ
This gives a large correction to the Born result at this energy, about 50%; and importantly would predict that for protons, cross sections heal to the Born result from below. This result fueled speculation that a third projectile velocity region existed. The Born limit had not been reached at 400 keV; the cross sections were just crossing the Born results. The true cross sections were thought to approach the Born result above 1 MeV. This result was called the bus stop effect. The idea was that even though the electron was attracted to the proton, it was ejected in a binary encounter at some distance from the proton, where the density was perforce smaller.
3. The two electron Born series It was BrandtÕs hope that his studies would lead to a deeper understanding of the Barkas effect, the Zp3 term that appears in stopping powers as a correction to the dominant Born term, which is proportional to Zp2 [11]. And indeed as a function of energy the Barkas term seems to follow the binding and polarization effects. All these effects of course are reversed, and hence exposed, if one uses antiprotons as projectiles. The immediate leap from a single electron Born series to accurately describing the interaction of an ion with a correlated many electron atom buried in a dense solid is tempting but, in our opinion, too optimistic. Brandt did not have the advantage of an antiproton beam; he could use only protons, deuterons and alpha-particles. Experiments at LEAR above 400 keV where the single electron Born term was supposedly an accurate description of the binary encounter showed
128
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
cross sections for doubly ionizing helium were much higher for antiprotons than protons. This was a clear signal of an interference between first and second Born terms but the effect did not arise from when the projectile interacted with a single electron twice! In the first Born term, a Zp -process, one electron is struck by the projectile while it is interacting with the second electron. The target forces between the two electrons, i.e. internal dynamic correlation, ionize the second electron. The second Born term, a Zp2 -process, certainly has contributions from when the projectile and the struck electron interacts twice. However there is also a term the projectile, interacts once with each electron. This term interferes destructively with the first term for protons, constructively for antiprotons. Experiments on molecules showed similar results [8]. In terms of the Barkas effect in the energy region stated these effects are thought to be small, at the two percent level. But the sign of this electron correlation phenomena is such that the stopping power for protons would be less than for antiprotons. And would thus mimic a bus stop effect. The announcement that a bus stop like effect had in fact been discovered in the stopping power of antiprotons and protons in hydrogen and helium at 600 keV revised this question again. Is there really a bus stop effect in the single electron Born series, or are we dealing with a correlation effect possibly enhanced by the environment of the molecules and atoms? Single collision type experiments, [12,13], in this region demonstrate a polarization effect, no bus stop effect, for ionization (see Fig. 1). This experimental result is confirmed by a single center finite Hilbert basis state (FHBS) calculation presented here. And in Fig. 2 we calculate that there is a corresponding polarization result in the excitation to the 2s state, and hardly any difference between rþ and r for excitation to the 2p state. This is exactly in accord with dipole dominance, where it is asserted that the Coulomb interaction mainly changes the parity of the initial state. Thus the second Born term for s-excitation is large, as we allow intermediate p-states with two dipole interactions. And the second Born term for p-excitation
2
1.5
1
experiment theory
0.5
0 10
100
1000
10000
ENERGY (keV/amu) Fig. 1. Proton and antiproton ionization cross section ratios, rþ , as a function of energy of the projectiles (keV/amu). The r experimental results [12,13] are the closed circles. The finite Hilbert basis set (FHBS) results are the solid line. Note the presence of a polarization effect in both the experimental data and theory.
2
1.5
1
2s 2p (m = 0) 2p (m = 1)
0.5
0 10
100
1000
10000
ENERGY (keV/amu) Fig. 2. FHBS theoretical calculation of the differential cross section ratio rrþ for 1s–2s, 2p (m ¼ 0) and 2p (m ¼ 1). Note that the dipole forbidden 2s transition demonstrates a polarization effect. But the dipole allowed transition shows no interference from a second Born term.
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
is small because of the necessity for including a mono pole interaction. We find this evidence for absence of a bus stop effect compelling. But the most satisfactory situation would be that we has a straight forward analytical calculation of asymptotic cross sections which agreed with the numerical calculations; we could then settle the question once and for all. Further if we had such a theory perhaps it could be usefully applied to many electron systems.
4. The Response Theorem In the following we closely follow Hall et al. [14]. The second order scattering amplitude for excitation of an electron in state vn , energy n , to state vm , energy m by an ion, if we ignore charge transfer, can be written as * Tm;n ðQÞ ¼ mp vm ðrÞeiQr Vp ðQÞ þ
1 2p2
Z
129
effect. For the Coulomb problem as c is set to zero we deduce a ¼ p=v:
ð4Þ 02
A second approximation is replace h by Q . This is called the Cheshire approximation. For the Coulomb problem this gives a ¼ p=v sgnð2v Q þ Q2 Þ:
ð5Þ
At high velocities this gives a smaller bus stop effect for ionization [1]. These results come directly from off energy shell properties of the Coulomb t-matrix. An alternative procedure to these approximations is to chose to replace h by a quantity which is guaranteed to give zero to the next perturbative correction. i.e. replace 1 1 ðh hhiÞ ¼ þ ðD þ hÞ ðD þ hhiÞ ðD þ hhiÞ2 2
þ
ðh hhiÞ
ðD þ hhiÞ
3
þ
ð6Þ
This leads to the propagator
0
dQ0 eiðQQ Þr
D þ hhi ¼ 2ðv pÞ Q0 þ Q02 :
1 Vp ðjQ Q0 jÞ 0 2v Q þ 2ðHe n Þ + 0 iQ0 r Vp ðQ Þvn ðrÞ : ð2Þ e
ð7Þ
Here the momentum transfer is Q. And
The interpretation of this result is that the struck target electron is not considered to be fixed as in the Glauber method; nor is it assumed to be free and at rest as in the Cheshire method. Rather it is approximated to be free but moving with a definite momentum p. The method defines
v Q ¼ ðn m Þ:
p ¼ hvm jeiQr j½irr vn i:
We will assume a Yukawa type potential, hence
It is straightforward to show that for any quantum eigenstates
V ðQÞ ¼ 2Zp =ðQ2 þ c2 Þ: If we use Eq. (2) to calculate a cross section differential in Q we can write keeping leading order terms 2
2
jT j ¼ V ðQÞ ð1 þ Zp aÞ:
ð3Þ
The problem with simplifying Eq. (2) is the presence of h ¼ 2ðHe n Þ, the target Hamiltonian in the denominator. It may be argued that at high projectile velocities we can neglect this term. This allows closure; it is called the Glauber approximation. It leads to a prediction of a large bus stop
2ðv pÞ Q þ Q2 ¼ 0:
ð8Þ
ð9Þ
This implies that the corresponding t-matrix is on the energy shell. In Fig. 3 we show the results for a Yukawa potential with c ¼ 0:1. We show the excitation of the 1s–2s state. The Response Theory answer is in fair agreement with a corresponding FHBS calculation when compared say to the Cheshire result. One can include the next higher order term in Eq. (6). This is shown in Fig. 3 as Hall. This so called variance term does not improve the answer.
130
J.F. Reading, J. Fu / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 126–130
zero. The continued fraction expression, Eq. (10), has to be evaluated numerically. For the 1s–2s transition it gives a 0:10=v2 . This is much too small to reproduce the FHBS result of Fig. 2. Ishii et al. [3] have suggested an ad hoc addition of a polarization term which would solve the problem but it is disappointing that the straight forward Response Theory cannot give the desired result directly. Hopefully some way may be found around this difficulty in the future.
1 γ = 0.1 0.5
Hall
Response
2 α vp
0
-0.5
Cheshire -1
Acknowledgements -1.5 10
15
20 vp
25
30
Fig. 3. The second Born interference term plotted as function of projectile velocity in atomic units for a Yukawa potential of cr the form e r where c ¼ 0:1. The present work is the solid line and compares well with the FHBS calculation given by the closed circles. Other results are described in the text.
But if instead we write the second order correction as a continued fraction i.e. ! 2 hh2 i hhi ðD þ hÞ ¼ D þ hhi ; ð10Þ D þ hhi we obtain satisfactory agreement between the analytical theory and the FHBS calculation shown. This is a new result. The result for the pure Coulomb force, of most interest to us, is disappointing. The first order Response Theory gives no correction, a is zero. This result is independent of the particular reaction to be studied, as it follows from the on-shell behavior of the Coulomb t-matrix. The higher order Born terms merely add a phase to the first Born amplitude. The straightforward variance term, Eq. (6), gives a divergence as c is set down to
This work is supported by grants from the US National Science Foundation PHY-9605217.
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