C Dal Cappello

A Haddadou

F Menas

A C Roy

C Dal Cappello
email: cappello@univ-metz.fr

The second Born approximation for the single and double ionization of atoms by electrons and positrons

Keywords: 34.80.Dp single ionization, double ionization, second Born approximation

The second Born approximation for the single and double ionization of atoms by electrons and positrons

C Dal Cappello, A Haddadou, F Menas, a C Roy

Introduction

The ionization of atoms and molecules by electrons or positrons is of great interest in astrophysics, plasma physics and radiation physics. The knowledge of single and double ionization is also needed in other sciences such as life science. Recent sophisticated theories such as exterior complex scaling (Rescigno et al 1999), time-dependent close coupling (Colgan and Pindzola 2006), convergent close coupling [START_REF] Bray | R2818 Bray I[END_REF], R matrix theory (Reid et al 1998) are mostly successful in predicting cross sections for the ionization of hydrogen or helium but are not yet able to describe fully differential cross section for the double ionization of helium (Durr et al 2007, Pindzola et al 2008). Moreover, these theories can be applied, up to now and only for numerical reasons, to light atoms and to the simplest molecule H 2 . In the other cases it is necessary to apply perturbation theories like the Born series.

Ionization of the atomic hydrogen by fast projectiles is a good check for a perturbation theory because experimental results exist, particularly for the triple differential cross sections (Weigold et al 1979, Lohmann et al 1984, Ehrhardt et al 1985, Ehrhardt et al 1986). In such an experiment the ejected electron is detected in coincidence with the scattered electron and it is well known (Ehrhardt et al 1986) that this kind of experiment, called (e, 2e) experiments, is very sensitive to the details of the theory. It is due to the fact that in the case of triple differential cross section, the theory is directly comparable to the experiments without averaging over unobserved parameters (unlike for the double or single differential cross sections, for instance). Weigold et al (1979) performed the first (e,2e) experiment on atomic hydrogen at incident energies of 100, 113.6, 250 and 413.6 eV while the energy of the ejected electron varied from 25 eV to 200 eV. Lohmann et al (1984) used more asymmetric conditions (250 eV for the incoming and 5, 10 and 14 eV respectively, for the ejected electron). Finally Ehrhardt et al (1985) reported the first absolute triple differential cross sections for asymmetric geometries (250 eV for the incident electron and 5 eV for the ejected electron). In all these experiments the kinematics is coplanar. Byron et al (1980) were the first to apply the second Born approximation for the ionization of atomic hydrogen. Pathak and Srivastava (1981) used the second Born approximation for the ionization of the atomic hydrogen and compared their results to the first (e,2e) experiments of Weigold et al (1979) (at the incident energy of 250 eV and ejected electrons of 50 eV). Then they also made calculations for the ionization of helium by using the closure approximation and compared their results to those of Ehrhardt et al (1982). Byron et al (1982) also calculated triple differential cross sections for the ionization of helium by using the closure approximation and found results which disagreed with those of Pathak and Srivastava. Later on, Byron et al (1985) applied the second Born approximation by using very few discrete states as intermediate states and by taking into account the closure approximation and adding the contribution of the third Born approximation calculated with the Glauber approximation. It is time now to answer to several questions about the second Born approximation. The first is the number of discrete states which are necessary to be included in the calculation of the second Born approximation. A second question is the role played by the continuum states. We also must see what happens when we add pseudo-states as intermediate states.

In the present investigation we have performed a calculation by using the second Born approximation which includes 100 exact discrete states corresponding to n=1 to n=10. Then we also perform another calculation which uses both exact and pseudo-states. Finally we shall compare our results to those given by the 3C model (or BBK model) (Brauner et al 1989) which is considered as of infinite order when treating the interaction between the two outgoing electrons but in an approximate way.

We have also applied the second Born approximation to the double ionization of helium and compared our results to those of Lahmam-Bennani et al (2010) and Lahmam-Bennani et al (2002).

Atomic units are used throughout unless otherwise stated.

Theory

We first consider the ionization of atomic hydrogen by electrons e -+ H → 2 e -+ H + , (1) In the second Born approximation, the triple differential cross section (TDCS) is given by The first Born term 1 B f is given by

) ( ) . exp( ) , ( ) . exp( 2 1 1 0 1 0 1 r r k i V r k r k i f i i e C s B r r r r r r r Φ Ψ - = - π , (3) where
) ( 1 r i r Φ
is the exact wave function of the initial state of the atomic hydrogen and ) , (

1 r k e C r r - Ψ
is also the exact wave function for the continuum state of the atomic hydrogen given by

)) . ( , 1 , ( ) 2 exp( ) 1 ( ) . exp( ) 2 ( 1 ) , ( 1 1 1 1 1 2 / 3 1 r k r k i i F i r k i r k e e e e C + - - - Γ = Ψ - r r r r r r α α π α π , with e k Z / - = α , and 1 = Z .
The potential V represents the coulomb interaction between the incoming electron and the target and is written as

0 01 1 1 r r V - = (4)
We neglect the exchange effects by considering the scattered electron as the fast electron and the ejected electron as the slow electron. We can easily perform the integration over 0 r r analytically and get

) ( 1 ) . exp( ) , ( 2 1 1 1 2 1 r r K i r k K f i e C B r r r r r Φ - Ψ - = - , (5)
where

s i k k K r r r - =
is the momentum transfer.

Following [START_REF] Joachain | Quantum Collision Theory 3rd edn[END_REF] the second Born term 2

B f is given by ) ( ) . exp( ) ( ) . exp( ) ( ) . exp( ) , ( ) . exp( 8 1 1 0 1 0 1 0 1 0 2 2 4 2 r r k i V r r q i r r q i V r k r k i i k q q d f i i n n e C s n n B r r r r r r r r r r r r r r Φ Φ Φ Ψ - - = - ∑ ∫ ε π , (6)
with

+ → 0 ε and ) ( 2 2 2 2 i n i n E E k k - - = (7)
if we only consider the contributions of the n exact discrete states of the atomic hydrogen (n varying from n=1 to n ∞ → ), n E being the eigenvalue of the atomic hydrogen Hamiltonian corresponding to the eigenfunction n Φ .

We also perform the integration over 0 r r and get

) ( 1 ) . exp( ) ( ) ( 1 ) . exp( ) , ( 1 2 1 1 1 1 1 1 2 2 2 2 2 2 r r K i r r r K i r k K K i k q q d f i i n n f e C f i n n B r r r r r r r r r r Φ - Φ Φ - Ψ - - = - ∑ ∫ ε π (8) where q k K i i r r r - = and s f k q K r r r - = , f i K K K r r r + =
. When we consider n different from n=1 we have the orthogonality of the initial state and other states (final or intermediate) and the term -1

(corresponding to the interaction of the electron with the target nucleus) present in ( 5) and ( 8) does not contribute.

When the contribution of the final continuum state is considered in the second Born term we must add the following term (Byron et al 1983)

) ( ) . exp( ) , ( ) . exp( ) , ( ) . exp( ) , ( ) . exp( 8 1 1 0 1 0 1 0 1 0 2 2 4 2 r r k i V r p r q i r p r q i V r k r k i i k q q d p d f i i C C e C s C B r r r r r r r r r r r r r r r r r Φ Ψ Ψ Ψ - - = - - - ∫ ∫ ε π (9) with + → 0 ε and 2 1 2 2 2 2 2 2 - - = p k k i (10)
As pointed out by Byron et al (1983) the calculation of the off-shell free-free matrix element is a difficult task. Two ways are now possible to avoid the direct calculation of C B f 2 : the first is to consider the closure approximation and the second is to only use (8) but with pseudo-states.

The closure approximation consists in replacing the target energy difference

= Φ Φ ∑ ∫ n n n . Consequently we get ) ( 1 ) . exp( 1 ) . exp( ) , ( 1 2 1 1 1 1 2 2 2 2 2 2 r r K i r K i r k K K i p q q d f i i f e C f i B r r r r r r r r Φ - - Ψ - - = - ∫ ε π (11) with + → 0 ε and w k p i - = 2 2 2 2 (12)
We can rewrite (11) as Byron and Joachain (1966) have also proposed another formulation of the closure approximation: they exactly include some contributions from the low-lying discrete states (generally n=1 and n=2).

) ( 1 ) . exp( ) . exp( ) . exp( ) , ( 1 2 1 1 1 1 1 2 2 2 2 2 2 r r K i r K i r K i r k K K i p q q d f i i f e C f i B r r r r r r r r r r Φ + - - Ψ - - = - ∫ ε π (13)
To perform this task they have to calculate

.... ) ( ) ( 2 2 2 2 1 2 1 2 2 2 + + - + - = = = = = n B n B n B n B B BJ B f w f f w f f f (14)
In ( 14) ) ( 12 w f n B = corresponds to (8) with n=1 (it is an elastic collisions as a first collision) with

) ( 2 2 2 2 1 w k k i n - = = (15) The term ) ( 2 2 w f n B = means ) ( ) ( ) ( ) ( ) ( 1 1 0 2 2 2 2 2 2 2 2 2 2 w f w f w f w f w f p B p B p B s B n B - + + + = = , as for 2 2 = n B f .
Secondly we consider the single ionization of the helium atom by electrons e -+ He → 2 e -+ He + , (16) for which the final state of the ion He + is in its ground state (1s).

The first Born term 1

B f is now given by ) , ( ) . exp( ) , ( ) . exp( 2 1 2 1 0 2 1 0 1 r r r k i V r r r k i f i i f s B r r r r r r r r Φ Ψ - = π , (17) where )
, ( 2 1 r r i r r Φ
is the wave function of the initial state of the helium atom and ) , ( 21 r r f r r Ψ is the wave function of the final target system consisting of a He + (1s) ion and an unbound electron. The potential V represents the Coulomb interaction between the incoming electron and the target and is written as

0 02 01 2 1 1 r r r V - + = (18)
For the initial state we use an analytical fit to the Hartree-Fock wave function (Byron and Joachain 1966)

) ( ) ( ) , ( 2 0 1 0 2 1 r r r r i r r r r φ φ = Φ where )) exp( ) exp( ( ) 4 ( ) ( 2 1 0 r B r A r β α π φ - + - = - r with 60505 . 2 = A , 08144 . 2 = B , 41 . 1 = α and 61 . 2 = β .
The final state wave function is a symmetrised product of the He + ground-state wave function for the bound electron multiplied by a Coulomb wave function orthogonalized to the helium ground state as in Byron et al (1986) [ ]

) ( ) , ( ) ( ) , ( 2
1 ) , ( 1 2 2 1 2 1 r r k r r k r r n e C n e C f r r r r r r r r φ φ ⊥ - ⊥ - Ψ + Ψ = Ψ with 0 0 φ φ - - ⊥ - Ψ - Ψ = Ψ C C C .
Performing the integration on 0 r r in equation ( 17) one can write the first Born term as

[ ] 0 0 0 0 0 2 2 / 3 1 . exp( ) ( ) . exp( ) ( 2 φ φ φ φ φ φ n e C e C B r K i k r K i k K f r r r r r r - - Ψ - Ψ - = , (19)
Following [START_REF] Joachain | Quantum Collision Theory 3rd edn[END_REF]

1 2 1 0 2 1 0 2 1 0 2 1 0 2 2 4 2 r r r k i V r r r q i r r r q i V r r r k i i k q q d f i i n n f s n n B r r r r r r r r r r r r r r r r r Φ Φ Φ Ψ - - = ∑ ∫ ε π , ( 20
)
with

+ → 0 ε and ) ( 2 2 2 2 i n i n E E k k - - = (21)
where n E is the eigenvalue of the helium atom Hamiltonian corresponding to the eigenfunction n Φ .

Performing the integration over 0 r r in equation ( 20) one can write the second Born term as

) , ( 2 ) . exp( ) . exp( ) , ( ) , ( 2 ) . exp( ) . exp( ) , ( 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 2 r r r K i r K i r r r r r K i r K i r r K K i k q q d f i i i n n f f f f i n n B r r r r r r r r r r r r r r r r r Φ - + Φ Φ - + Ψ - - = ∑ ∫ ε π , ( 22
)
Now, by applying the closure approximation we get

) , ( 2 ) . exp( ) . exp( 2 ) . exp( ) . exp( ) , ( 1 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 r r r K i r K i r K i r K i r r K K i p q q d f i i i f f f f i B r r r r r r r v r r r r r Φ - + - + Ψ - - = ∫ ε π (23) with + → 0 ε and w k p i - = 2 2 2 2 (24)
w being the average excitation energy. After some straightforward algebra the expression (23) can be reduced to

[ ] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 /
φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ ε π r K i r K i r K i r K i r K i r K i r K i r K i r K i r K i r K i r K i r K i r K i K K i p q q d f i n f C f n i C n f C n i C n C n i C n f C f n i C i n f C n C f i B r r r r r r r r r r r r r r r r r r r r r r r r r r r r r - - - - - - - - - - Ψ - Ψ - Ψ + Ψ + Ψ - Ψ - Ψ - Ψ + Ψ + Ψ - - = ∫ (25)
This last expression is simpler than that found by Byron et al (1986).

Finally we consider the double ionization of helium by electrons e -+ He → 3 e -+ He ++ .

(26)

In the second Born approximation the fivefold differential cross section (FDCS) is given by The first Born term 1

b a s b a dE dE d d d d Ω Ω Ω = σ σ 5 ) 5 ( 2 2 1 B B i b a s f f k k k k + = , (27)
B f is given here by ) , ( ) . exp( ) , , , ( ) . exp( 2 1 2 1 0 2 1 0 1 r r r k i V r k r k r k i f i i b a f s B r r r r r r r r r r Φ Ψ - = ⊥ π , (28) where ) , ( 2 1 r r i r r Φ
is the wave function of the initial state of the helium atom and ) , , , (

2 1 r k r k b a f r r r r ⊥ Ψ
is the wave function for the double continuum state of the helium atom which is orthogonalized to the initial state. The potential V represents the coulomb interaction between the incoming electron and the target and is given by ( 18).

For the initial state we use an accurate Hylleraas-type wave function given by Bonham and Kohl (1966) [ ]

14 2 1 12 1 2 2 1 0 2 1 ) ( exp( ) exp( ) exp( ) exp( ) , ( BK i i r r r br ar br ar N r r Φ = + - - + - - + - - = Φ α γ β
[ ] 7 1 ' 2 ' 2 ' 1 ' ' 0 2 1 ) exp( ) exp( ) , ( BK i i r b r a r b r a N r r Φ = - - + - - = Φ r r where 1885 . 1 ' = a , 1832 . 2 ' = b
and which is often used actually (Ciappina et al 2008). The energy of the initial state given by this wave function is E=-2.875661 au.

The final state wave function is the approximate BBK wave function (Dal Cappello et al 1993) such

that i f i f f Φ Ψ Φ - Ψ = Ψ ⊥ [ ] ) ( ) , ( ) , ( ) , ( ) , ( 2
1 ) , , , ( 1 2 2 1 2 1 b a b C a C b C a C b a f k k r k r k r k r k r k r k r r r r r r r r r r r r r r - Ψ Ψ + Ψ Ψ = Ψ - - - - ϕ where ) ( b a k k r r - ϕ is the repulsive Gamow factor ) 1 ( ) 2 exp( ) ( ab ab b a i k k χ πχ ϕ - Γ - = - r r and b a ab k k r r - = 1 χ .
If we neglect the exchange effects by considering the scattered electron as the fast electron and the ejected electrons as the slow electrons we can easily perform the integration over 0 r r analytically and get

i b a f B r K i r K i r k r k K f Φ - + Ψ - = ⊥ 2 ) . exp( ) . exp( ) , , , ( 2 2 1 2 1 2 1 r r r r r r r r , (29)
The second Born term 2

B f is given by ) , ( ) . exp( ) , ( ) . exp( ) , ( ) . exp( ) , , , ( ) . exp( 8 1 2 1 0 2 1 0 2 1 0 2 1 0 2 2 4 2 r r r k i V r r r q i r r r q i V r k r k r k i i k q q d f i i n n b a f s n n B r r r r r r r r r r r r r r r r r r r Φ Φ Φ Ψ - - = ⊥ ∑ ∫ ε π (30)
where the summation over n means that we take into account all the contributions of the n discrete and continuum states of the helium atom. It means that the incident electron collides twice with the target and corresponds to the well-known two-step 2 mechanism (TS2) (Carlson and Krause 1965).

Performing the integration on 0 r r in equation ( 30) and applying the closure approximation we get

(Grin et al 2000) ) , ( 2 ) . exp( ) . exp( 2 ) . exp( ) . exp( ) , , , (
1 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 r r r K i r K i r K i r K i r k r k K K i p q q d f i i i f f b a f f i B r r r r r r r v r r r r r r r Φ - + - + Ψ - - = ⊥ ∫ ε π (31) and w k p i - = 2 2 2 2 (32)
w being the average excitation energy.

It is important to note that all integrals on q d r must be performed numerically with great care (see appendix).

Results and discussion a) Ionization of atomic hydrogen by electrons

We investigate the ionization of atomic hydrogen by electrons by comparing the results of our second Born approximation with the absolute data of Ehrhardt et al (1985Ehrhardt et al ( , 1986)). The incident energy is 250 eV and the ejected electron has an energy of 5 eV. First, we consider the case where the incoming electron is scattered at 0 3 = s θ which corresponds to a low momentum transfer (K=0.27 au). Next, we need to know if it is necessary to include many discrete states as intermediate states in the second Born approximation. In figure 1a we compare the results of the first Born treatment with those of the second Born by including successively n=1, n=2 and n=3 as intermediate states. It is clear that the contribution of the elastic case (n=1) plays no important role here. We only notice a small rise of the amplitude of the recoil peak. When we add the contribution of the excited states corresponding to n=2 (2s and 2p) we observe a fall of the binary lobe amplitude and a rise of the recoil lobe amplitude. As a matter of fact the most important contribution is due to the 2p excited state (see table 1). This result is easy to understand because it corresponds to the

dipolar transition p s 2 1 →
which is important for low momentum transfer. This result was also noticed by Byron et al (1985). Now if we add the contribution of the excited states corresponding to n=3 we clearly see that the increasing of the recoil lobe amplitude continues as the decreasing of the binary peak amplitude. It is true that the contribution of n=2 is the most important but not sufficient, and that the contribution from n=3 must be added.

Figure 1b shows the contributions of the excited states corresponding to n=4 and n=5. We only see a small contribution from these excited states. Clearly, the n=5 excited state gives no contribution for the binary peak and shows a very small decreasing of the recoil peak amplitude. It seems that we have reached the convergence limit of the contribution of the discrete states. In order to check this point we draw in figure 1c the contributions of n=6, n=7, n=8, n=9, and n=10 (which correspond to 100 discrete states). It is conspicuous that the contributions from n=7 to n=10 play no significant role here and can be neglected (see table 2).

If we consider another case where the momentum transfer is bigger (for example 0 8 = s θ which corresponds to a momentum transfer K=0.61 au ) we find the same results as before and conclude that the convergence is reached by adding contributions up to n=6. Now we investigate the closure approximation with the same kinematical conditions. Figure 2 shows the results of the closure approximation. We observe that the closure approximation using equation ( 13) or equation ( 14) gives close results. In our case equation ( 14) is calculated with n=1 to n=6 (Byron et al (1985) only include n=1 and n=2). From this figure it appears that the results given by the closure approximation are different from those given by the contributions of the discrete states (from n=1 to n=6). It means that we cannot neglect the contributions of the continuum states. It was also the conclusion of Byron et al (1985). It was written in the paper of Byron et al (1985) that their results of their second Born calculation were insensitive to the variation of w . We find that it is not the case with our calculations: figure 3 shows that the change of the value of the parameter changes the values of the results of our second Born calculation. We also perform, like Byron et al (1985), calculations with w =0.5 au and w =1 au but by including more discrete states. We can conclude from the results of the closure approximation that the contributions due to the continuum states are necessary and that, unhappily the closure approximation depends on the choice of the parameter w . This result was also found by Lahmam-Benanni et al (2003) for the double ionization of helium by electrons. Another way to include these contributions from the continuum states is to consider a basis with pseudo-states. Hence we consider two pseudo-states basis which include 31 (Callaway 1978) and 32 (Callaway 1993) states, respectively. The eigenvalues vary from -0.5 au to 1.02 au (Callaway 1978) and from -0.5 au to 4.8 au (Callaway 1993). Further details of these two basis can be found in [START_REF] Rouet | Thèse de doctorat[END_REF].

Figure 4 shows the results of the second Born approximation calculated with pseudo-states basis.

We see that the contributions due to the continuum states are important for the binary peak as well for the recoil peak. A basis which only includes discrete excited states and pseudo-states is never able to fully reproduce the decrease of the binary peak and the increase of the recoil peak. The two basis of Callaway (1978 and1993) including positive eigenvalues give close results particularly for the binary peak.

From the results of the closure approximation and those of the pseudo-states basis it is now clear that the continuum states play a role in the ionization of the atomic hydrogen for these particular kinematical conditions (large incident energy compared to the energy of the ejected electron).

It is now time to compare our theoretical results to the well-known BBK model (Brauner et al 1989) and to the absolute data of Ehrhardt (1985Ehrhardt ( , 1986)). We have followed the recommendation of Jones and Madison (2002) and multiplied all the data by a scaling factor of 0.88. Jones and Madison (2000) find a very good agreement between their CDW-EIS model (Crothers andMc Cann 1983, Jones and[START_REF] Jones | [END_REF], which contains the same BBK wave function for the final state with an eikonal initial state instead of the plane wave for the incident electron, with the nonperturbative Convergent Close Coupling model (CCC) (Bray et al 1994, Jones andMadison 2000). From this good agreement between the two theories (CDW-EIS and CCC) they conclude that the experimental data of Ehrhardt et al (1985Ehrhardt et al ( , 1986) must be multiplied by a scaling factor of 0.88. For the incident energy considered here (250 eV) the BBK model and CDW-EIS yield results which are in close agreement with those of the most sophisticated Convergent Close Coupling model (CCC) (Jones and Madison 2000). Figure 5 . These experiments are interesting because Pathak and Srivastava (1981) conclude that the second Born approximation is not able to bring about a general improvement in the values of the TDCS over the first Born results. These authors use the closure approximation with a high average excitation energy w =1 au. It was the start of a controversy with Byron et al (1982) who claimed that the closure approximation can only be justified if the energy of the ejected electron is small (and the scattering angle is small too). Figure 7a . Once again we notice that the contribution of n=1 is very small but, in this case, the other contributions (n=2 and n=3) are small too. Figure 7b shows that the second Born approximation calculated by including only the contributions of discrete states (n=1 to n=6) is insufficient to describe the experiments. The closure approximation (calculated with w =0.5 au) completely fails as claimed by Byron et al (1982). The shift of the binary peak is not reproduced contrary to the BBK model which predicts a significant shift. In figure 7c we see that the second Born approximation calculated by including discrete states and pseudo-states brings no improvement. We also notice that the exchange effects are small here (BBK model with exchange) and can be neglected. Figure 8a for 0 25 = s θ confirms that the closure approximation fails. The second Born approximation calculated by including only the contributions of discrete states (n=1 to n=6) gives a magnitude which is less than those given by the first Born approximation (contrary to the closure approximation). In figure 8b we notice that the second Born approximation calculated by including discrete states and pseudo-states overestimate the experiments. Finally the BBK model or the BBK model with exchange give the best agreement with experiments (the exchange effects are small too). Pathak and Srivastava (1981) and Byron et al (1982) were right: the second Born approximation is not working when the energy of the ejected electrons is not small. If the BBK model gives a good agreement it means that we need to take into account higher order of the interaction between the scattered electron and the ejected electron.

b) Ionization of atomic hydrogen by positrons

The ionization of atomic hydrogen by positrons is a more difficult problem because rearrangement collisions are possible due to positronium formation. Hence the ionization of atomic hydrogen by positrons is a two-center collision system while the ionization of atomic hydrogen is a single-center one. Although much progress has been made (see for instance Kadyrov et al 2007) we observe that the problem of the ionization of atomic hydrogen by positrons is not yet solved. Brauner et al (1989) have applied their BBK model to the ionization of atomic hydrogen by positrons and obtain the TDCS. Bandyopadhyay et al (1994) and Fiol and Olson (2002) also used the BBK model for the calculations of the triple differential cross sections and double differential cross sections. But, up to now, none has applied the second Born approximation for the ionization of atomic hydrogen by positrons (Sharma and Srivastava (1988) have applied the second Born approximation for the ionization of helium by electrons and positrons). In our Born approximation plane waves are used for the incident and scattered particles. It may be noted that the first Born amplitude term has opposite signs for positron and electron impacts. If we only consider the first Born approximation we find the same cross section for electron impact or positron impact because the triple differential cross section is directly connected to the square of the first Born term (see equation ( 2)). The sign of the second Born term does not depend on the charge of the particle but now the sign of the first Born term plays a role because the first Born term is added to the second Born term, explaining the change of behaviour when the second Born approximation is applied for electron impact and positron impacts.

c) Single ionization of helium by electrons

We now investigate the single ionization of helium by electrons by comparing the results of our second Born approximation with the relative data of Ehrhardt et al (1982) and a first theoretical calculation of Pathak and Srivastava (1981) using the closure approximation with w =1.3 au and a second theoretical calculation of Byron et al (1982) using the same closure approximation but with w =0.9 au. These two groups use the same theoretical expression (25). Figure 10 shows the results of the first and second Born approximation with the two values of the parameter w . We only notice small differences between the two calculations and find results close to those of Byron et al (1982): the first and second Born triple differential cross sections differ by only 17% at the binary peak, with the second Born result being smaller than the first Born result (as in the case of atomic hydrogen).We also find that the ratio of the binary peak to the recoil peak is reduced from the first Born value of 8.43 to the second Born value of 5.63 in better agreement with experiment. Thus, as written by Byron et al (1982), the results of Pathak and Srivastava (1981) in this case are incorrect.

The results of the BBK model are also drawn in Figure 10. Binary to recoil peak ratio of 4.21 given by the BBK model is approaching the experimental value. Once again we can conclude that the contributions of the continuum states are necessary because these contributions are assumed in the BBK model.

Figure 11 shows that all the terms of equation ( 3) in appendix contribute to the second Born approximation: if we neglect, for instance (like Fang and Bartschat 2001), the most difficult term (the triple numerical integral, equation (4) in appendix) we see practically no shift for the recoil lobe. We find that the maximum of the recoil lobe is located at 4) and can never be deleted.

d) Double ionization of helium by electrons and positrons

We finally investigate the double ionization of helium by electrons (or by positrons) by comparing the results of our second Born approximation with the relative data of Lahmam-Bennani et al (2002) and Lahmam-Bennani et al (2010). In these experiments the slow ejected electron is not detected and we must integrate the equation ( 27) over the solid angle b dΩ . The large angular shifts of the forward and backward lobes with respect to the momentum transfer direction were questioned by Götz et al (2003). Figure 12 shows that the second Born approximation is now able to reproduce the shift of the binary peak and describe partially the shift of the recoil peak. We also see that the results of the second Born approximation depend on the choice of the wave function used: the BK14 wave function reproduces a minimum around 310 0 as the experiments, contrary to the BK7 wave function which exhibits a maximum. But no model has yet been able to reproduce all the data. Furthermore, we notice that even the most sophisticated model DS6C of Götz et al (2006) fails to describe the strong violation of the first Born symmetry seen in the experiment. The relative good agreement between our model and the experiment means that the TS2 mechanism (which is implicitly included in the second Born approximation) plays an important role here. Figure 13 shows the results of our second Born approximation (by using the closure approximation) with the latest results of Lahmam-Bennani et al (2010). We notice a good agreement, particularly for the large angular shift of the binary peak. We also study the double ionization of helium by positrons.

In this case the amplitude of the binary peak increases while the amplitude of the recoil peak decreases. This is exactly what we observe when we investigate the single ionization of atomic hydrogen by electrons and positrons. In figure 14 we study the role played by the average excitation energy w . As for the case of the ionization of atomic hydrogen we find that the results depend on this value: we notice a change of the shape for w =30 eV. In this case the maximum at 210 0 disappears and becomes a minimum. These results confirm those found by Lahmam-Benanni et al (2003). It means that it is here important to include the contributions of all states before using the closure approximation but it is a difficult problem, particularly when we want to take into account of the single and double continuum states and the resonances.

Conclusion

The second Born approximation has been first studied for the ionization of atomic hydrogen using a large number of intermediate states. The results show that the contributions from the discrete states are insufficient and the continuum states must be added. We confirm the first results of Byron, Joachain and Piraux: the contribution of the n=2 as an intermediate state is the most important one.

Nevertheless, it is necessary to add the n=3, n=4, n=5 and n=6 contributions and those of the continuum states. The closure approximation, which is often used (Marchalant et al 1999, Marchalant et al 2000, Kheifets 2004, Grin et al 2000, Choubisa et al 2003), works very well for small energy of the ejected electrons. For higher ejection energies this approximation completely fails. The second Born approximation using discrete states in conjunction with pseudo-states gives better agreement in some cases but is not always sufficient to describe experiments. This result proves again that the contributions of the continuum states must be added and it is certainly necessary to consider more and more pseudo-states. The second Born approximation works for the ionization of targets by positrons as the BBK model. Although the BBK model gives here a good agreement with experiments it was noticed by Dey et al (2008) that the recent experiments of the ionization of helium by electrons (Catoire et al 2006, Stevenson et al 2007) are not well described by this model which overestimates the differential cross sections in the recoil region. Moreover Dal Cappello et al (2008) show that the ionization-excitation of helium cannot be described by the BBK model, contrary to a model using the second Born approximation (Watanabe et al 2007). The second Born approximation works quite well for the double ionization of helium by using the closure approximation with particular values of the average excitation energy w . The big shift of the binary peak found experimentally by Lahmam-Bennani et al (2010) is well reproduced showing that the TS2 mechanism is very important in this case.

The last two singularities are not difficult for numerical integration. Interestingly, as written by Marchalant et al (1998), these singularities are only apparent and can be overcome by using prolate spheroidal coordinates. We can also use the well-known integration formulas of Gaussian type to avoid these two particular singularities. The first singularity is more difficult for numerical integration. As proposed by Piraux (1983) the general integral ) , , (

2 2 q q q W i p q q d I ϕ θ ε ∫ - - = r (1)
with

+ → 0 ε can be performed by using ) ( ) 1 ( 1 2 2 2 2 2 2 p q i p q P i p q - + - = - - πδ ε (2)
where P stands for the principal value. Then we rewrite I as

∫ ∫ + - = ) , , ( 2 ) , , ( 2 2 q q q q p W q d p i q W p q q d P I ϕ θ π ϕ θ r (3)
The second term is easy to calculate contrary to the principal value which is sometimes neglected (Fang and Bartschat 2001) or approximated by using a peaking approximation (Franz and Altick 1995).

Following Piraux (1983) the principal part is written as

)) (cos( )) cos( , ( 1 1 2 2 2 q q d q G p q dq q P J θ θ ∫ ∫ + - - = (4)
and

∫ = π ϕ ϕ θ θ 2 0 ) , , (
)) cos( , (

q q q q d q W q G (5)
Finally, after some algebra (Piraux 1983)

∫ ∫ ∫ + - ∞ + -         + + -         + = 1 1 1 ' 0 ' 2 2 2 0 2 )) (cos( )) cos( , ( ' ' J p J d q G p q dq q J q q p p ε ε θ θ ε ε (6) with )) (cos( )) cos( , ( 1 1 0 q q d p H J θ θ ∫ + - = , ( 7
) ∫ + - = ∂ ∂ = 1 1 1 )) cos( , ( )) (cos( p q q q q H q d J θ θ (8) and )) cos( , ( )) cos( , ( q q q qG q H θ θ = (9)
Marchalant et al (1998) adopt a subtraction procedure to evaluate (3) and introduce an exponential factor to make the integrals convergent as

∞ → q : ∫ ∫ ∫ - - - - +         + - - - = ∞ q d p q p W p q q W q d p W p i p q dq q p q P I q q q q q q r 2 2 2 0 2 2 2 2 ) , , ( ) ) ( exp( ) , , ( ( ) , , ( 2 )
) ( exp( ϕ θ α ϕ θ ϕ θ π α (10)
The parameter α in ( 10) is free of choice and the results should be independent of it (Marchalant et al 1998).

As a check on the accuracy of our numerical procedures we have calculated the most singular integral contained in the above expressions following Byron et al (1985) 2 2 2 2 0

1 f i K K i p q q d I ∫ - - = ε r , (11)
with

+ → 0 ε .
This integral can be exactly evaluated by using the results of Lewis (1956) if

i s k p k 〈 〈 , then         - - ∆ + ∆ = 2 2 2 2 2 2 0 ) ( s i k p p k pK Ln K i I π , (12) where
) )( ( 2 2 2 2 2 2 s i k p k p p K - - + = ∆ . If s k p〈 , then ∆ +         - - ∆ + ∆ = K k p p k pK Ln K i I s i 3 2 2 2 2 2 2 0 ) ( π π , (13)
If s k p = , then the imaginary part of 0 I tends to infinity and the real part is

2 3 0 2 ) ( pK I R e π = , (14)
And if i k p = , the imaginary part of 0 I tends to the infinity and the real part is 13)) by a dotted line and those of the second Born approximation calculated by using the closure approximation (equation ( 14)) by a dash-dotted line. (Callaway 1978) by a dashed line and by a short-dashed line (Callaway 1993), those of the second Born approximation calculated by including all the eigenstates and pseudo-states (Callaway 1978) by a dotted line and by a short-dotted line (Callaway 1993). 1 Triple differential cross section (in au) for the ionization of atomic hydrogen as obtained from the second Born expressions (8), ( 13) and ( 14), using various approximations for 2 Table 2 Triple differential cross section (in au) for the ionization of atomic hydrogen as obtained from the second Born expression (8), using various approximations for 2

2 3 0 2 ) ( pK I R e π - = , (15)

  respectively, the elements of solid angles for the scattered and the ejected electron whereas the energy interval of the ejected electron is represented by e dE . The momenta of the incident, scattered, and ejected electrons are denoted by i k

  excitation energy w and summing over all the intermediate states 1

  respectively, the elements of solid angles for the scattered and the ejected electrons a and b whereas the energy intervals of the ejected electrons are represented by

.

  The energy of the initial state given by this wave function is E=-2.903115 au. We also use a wave function which only includes a part of the radial correlation

.

  The agreement is very good between the results of the BBK model, those of the second Born approximation with the closure approximation and the experiments. As seen above the second Born approximation calculated only with the discrete states (n=1 to n=6, or n=1 to n=10) is insufficient: the contribution of the continuum states is necessary. This contribution is included in the BBK model because the interaction between the scattered electron and the ejected electron is treated to an infinite order but in an approximate way.

Figure 6

 6 Figure6shows a comparison between our second Born treatment with pseudo-states, the BBK

Figures 7

 7 Figures 7 and 8 show the results of the second Born approximation and the BBK model for the

  shows our results of the second Born approximation with the contributions of the n=1, n=2 and n=3 intermediate states

Figure 9

 9 Figure9shows the results of our BBK model (which yields the same values as those found by

  approximation without the triple numerical integral) while the second Born approximation and the BBK model gives the maximum close to 0 130 = e θ . As a matter of fact the large shift of the recoil peak is generally due to the triple numerical integral in equation (

Figure captions Figure 1a :

 captions1a Figure captions

Figure 1b :

 1b Figure 1b: Same as figure 1a but the results of the second Born approximation calculated by

Figure 1c :

 1c Figure 1c: Same as figure 1a but the results of the second Born approximation calculated by

Figure 2 :

 2 Figure 2: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 3 :

 3 Figure 3: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 4 :

 4 Figure 4: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 5 :

 5 Figure 5: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 6 :

 6 Figure 6: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 7a :

 7a Figure 7a: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 7b :

 7b Figure 7b: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 7c :

 7c Figure 7c: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 8a :

 8a Figure 8a: Same as figure 7b except for s θ =25 0 .

Figure 8b :

 8b Figure 8b: Same as figure 7c except for s θ =25 0 .

Figure 9 :

 9 Figure 9: Triple differential cross section (TDCS) for ionization of atomic hydrogen by 250 eV

Figure 10 :

 10 Figure 10: Triple differential cross section (TDCS) for the single ionization of helium by 500 eV

Figure 11 :

 11 Figure 11: Triple differential cross section (TDCS) for ionization of helium by 500 eV electron

Figure 12 :

 12 Figure 12: Fourfold differential cross section (4DCS) for the (e,3-1e) double ionization of helium

Figure 13 :

 13 Figure 13: Fourfold differential cross section (4DCS) for the (e, 3-1e) double ionization of helium

Figure 14 :

 14 Figure 14: Fourfold differential cross section (4DCS) for the (e, 3-1e) double ionization of helium

  including only the contribution of the target ground state (1s) as an intermediate state; B2(2s) (B2(2p)), same calculation, in which only the contribution of the 2s(2p) state is included in 2 B f ; B2(0.5), full second Born approximation results in which BJ B f 2 is calculated by including exactly the 1s, 2s and 2p intermediate target states, and the closure approximation (with 5 . 0 = w au) is used to include the other target states; B2(1.0), full second Born approximation results in which BJ B f 2 is calculated by including exactly the 1s, 2s and 2p intermediate target states, and the closure approximation (used to include the other target states.B1: results of the first Born approximation (5). The incident electron energy is 250 eV, the ejected electron energy is 5 eV and the scattering angle s θ =3 0 .

Bf

  Figure 1a

Table 1

 1

	θ	e	(deg)	B1	B2(1s)	B2(2s)	B2(2p)	SB2(0.5) SB2(1.0) B2(0.5) B2(1.0)
	10		1.363	1.409	1.323	1.362	0.597	0.974	0.599	1.019
	30		0.247	0.261	0.235	0.250	0.296	0.443	0.289	0.433
	50		0.666	0.660	0.670	0.620	0.808	0.885	0.790	0.897
	70		1.581	1.576	1.573	1.495	1.577	1.695	1.552	1.758
	90		2.400	2.427	2.340	2.334	2.489	2.578	2.446	2.619
	110	2.906	2.996	2.834	3.037	3.467	3.354	3.401	3.129
	130	3.050	3.210	2.985	3.418	4.248	3.895	4.281	3.701
	150	2.830	3.026	2.712	3.229	4.456	4.010	4.618	4.225
	170	2.253	2.438	2.106	2.556	3.830	2.438	4.030	2.106
	190	1.387	1.521	1.270	1.589	2.455	2.355	2.605	2.758
	210	0.510	0.567	0.458	0.630	0.932	1.010	1.000	1.207
	230	0.313	0.286	0.314	0.370	0.446	0.442	0.489	0.538
	250	1.885	1.800	1.932	2.100	2.265	1.876	2.369	1.978
	270	5.823	5.747	5.995	6.272	6.139	5.502	6.113	5.077
	290	10.54	10.53	10.64	10.44	9.302	9.185	9.183	8.982
	310	12.41	12.46	12.39	11.74	9.058	9.811	9.034	10.19
	330	9.675	9.756	9.600	9.129	5.934	6.948	5.948	7.376
	350	4.847	4.922	4.779	4.665	2.504	3.234	2.511	3.445

Table 2

 2

	θ	e	(deg)	B1	B2(2)	B2(3)	B2(4)	B2(5)	B2(6)	B2(8)	B2(10)
	10		1.363	1.366	1.349	1.352	1.347	1.349	1.346	1.346
	30		0.247	0.252	0.250	0.249	0.248	0.248	0.247	0.247
	50		0.666	0.618	0.598	0.588	0.583	0.579	0.576	0.574
	70		1.581	1.482	1.433	1.410	1.394	1.386	1.377	1.374
	90		2.400	2.302	2.283	2.296	2.307	2.297	2.282	2.291
	110	2.906	3.056	3.099	3.131	3.152	3.153	3.148	3.150
	130	3.050	3.515	3.609	3.640	3.662	3.676	3.692	3.697
	150	2.830	3.312	3.474	3.561	3.609	3.640	3.672	3.686
	170	2.253	2.598	2.736	2.810	2.852	2.878	2.907	2.921
	190	1.387	1.608	1.703	1.755	1.784	1.801	1.820	1.829
	210	0.510	0.638	0.692	0.723	0.742	0.756	0.768	0.774
	230	0.313	0.346	0.396	0.423	0.438	0.447	0.456	0.460
	250	1.885	2.068	2.172	2.194	2.196	2.204	2.214	2.220
	270	5.823	6.370	6.408	6.411	6.424	6.416	6.376	6.388
	290	10.54	10.52	10.54	10.55	10.56	10.56	10.56	10.55
	310	12.41	11.77	11.46	11.29	11.20	11.15	11.09	11.06
	330	9.675	9.135	8.879	8.755	8.685	8.651	8.611	8.590
	350	4.847	4.670	4.573	4.530	4.507	4.493	4.477	4.469

Acknowledgments

We would like to thank B Piraux and Yu V Popov for helpful discussions. We also thank the PMMS (Pôle Messin de Modélisation et de Simulation) for computer time.

Appendix

As pointed out by Byron et al (1985) great care must be taken while carrying out the integration on q d r since the integrand is singular at p q = (equations ( 11), ( 13), (23), and (25)) or n k q = (equations ( 6), ( 8), (20), and ( 22)) or k q = (equation ( 9)) and is also singular at i k q r r = and s k q r r = .

Generally we have to deal with the expression (12) and we get an accuracy of 10 -5 by using typically 500000 quadrature points for the triple numerical integration. 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 2 1 0 2 4 0 2 7 0 3 0 0 3 3 0 3 6 0