Relativistic electron-impact ionization of hydrogen atom from its metastable 2S-state in the symmetric/asymmetric coplanar geometries
RRelativistic electron-impact ionization of hydrogen atom from itsmetastable 2S-state in the symmetric/asymmetric coplanar geometries
M. Jakha, S. Mouslih,
2, 1
S. Taj, and B. Manaut ∗ Université Sultan Moulay Slimane, Faculté Polydisciplinaire,Équipe de Recherche en Physique Théorique etMatériaux (ERPTM), Béni Mellal, 23000, Morocco. Faculté des Sciences et Techniques, Laboratoire de Physiquedes Matériaux (LPM), Béni Mellal, 23000, Morocco. (Dated: August 21, 2020) a r X i v : . [ phy s i c s . a t o m - ph ] A ug bstract We analytically compute, in the first Born approximation for symmetric and asymmetric coplanargeometries, the triple differential cross sections for electron-impact ionization of hydrogen atom in themetastable 2S-state at both low and high energies. The process is investigated by using the relativisticDirac-formalism and it is also shown that the nonrelativistic limit is accurately reproduced when usinglow incident kinetic energies. At high energies, relativistic and spin effects significantly affect the tripledifferential cross sections. Our analytical approach which seems exact is compared to some other resultsin the nonrelativistic regime for asymmetric coplanar geometry. For this particular process and in theabsence of any experimental data and theoretical models at high energies, we are not in a position to vali-date our model. We hope that the present study will provide significant contribution to future experiments.Keywords : Relativistioc ionization ; Relativistic Coulomb wave functions ; Analytical calculationsof integrals.
I. INTRODUCTION
Electron-impact ionization is the removal of one or more electrons from the target resulting fromthe collision between it and an electron. We can distinguish different types of ionization; singleionization, called (e, 2e) process, which occurs when the resulting ion leaves the collision regionwith a single positive charge, multiple ionization where several electrons in the electronic cortegeare ejected and the ion can have multiple positive charges. In this work, however, we will onlydeal with the case of single ionization of the hydrogen atom from its metastable 2S-state, whenis bombarded by an electron of energy E i greater than the ionization potential. In the collisionzone, two electrons emerge with energies E f and E B . Even though these two electrons cannot bedistinguished, it is convenient to call the faster electron "scattered electron" and the slower one"ejected electron". All ionization reactions are studied in two geometric frameworks. The firstis called asymmetric geometry and the second is symmetric geometry, and each of them may becoplanar or noncoplanar. In asymmetric geometries, a fast electron of energy E i is incident on thetarget atom, and a fast scattered electron is detected in coincidence with a slow ejected electron.This kind of experiment was first performed by Ehrhardt et al. [1]. Symmetric geometries, which ∗ [email protected] E f (cid:39) E B and θ f (cid:39) θ B ), was introduced by Amaldi etal. [2]. In coplanar geometry, the three momenta p i , p f and p B are in the same plane, whereas innoncoplanar geometry the momentum p B is out of the ( p i , p f ) reference plane. Electron-impactionization of atomic, ionic or molecular systems is one of the important processes of collisionalphysics, in particular for the study of the structure of matter. It also finds its application in vari-ous fields such as astrophysics and plasma physics. Especially, the electron-impact single ionizationhas proved to be a powerful tool for studying the structure of atoms and their dynamics. Ionizationof hydrogen atoms by electron impact is the fundamental and simplest ionization process. Thehydrogen atom is an ideal target due to its analytically known wave functions, although it is aparticularly difficult target for experimentalists. At present, there are many theoretical modelsto compute the cross sections of hydrogen-atoms ionization in both the ground and metastablestates at various incident kinetic energies and under different kinematic conditions. Unfortunately,ionization from metastable states has not been investigated to the same extent, especially in therelativistic regime, as ionization from the ground state; and this is mainly due to the lack of anyexperimental studies on this type of ionization. The investigation of the ionization from metastablestates of hydrogen atoms by charged particles is now equally interesting and experimental resultswill soon be available in this field. In particular, the fully triple differential cross sections (TDCS)for the (e, 2e) process have been extensively studied for the ground state hydrogen atom boththeoretically and experimentally, while for the ionization from the metastable state no such mea-surement of TDCS is yet available in the literature, although the absolute total cross sections havebeen measured much earlier [3, 4]. However, on the theoretical side, quite a few calculations havebeen performed on the TDCS of the metastable (2S) hydrogen atom using electron impact [5–11]and significant differences were observed in the TDCS structures when compared to the cross sec-tion of ground-state ionization. All these theoretical calculations available in the literature to datehave been done within the framework of the asymmetric geometry and at low energies. To thebest of our knowledge, there is no study available to the ionization of the hydrogen atom from itsmetastable 2S-state using relativistic formalism at high energies. This work addresses, for the firsttime, a theoretical study and an analytical calculation of the ionization of the hydrogen atom fromits metastable 2S-state at high energies in both symmetric and asymmetric coplanar geometriestaking into account the effects of spin and relativity. In the asymmetric coplanar geometry, wepresent a theoretical semirelativistic Coulomb Born approximation (SRCBA) for the description3f the ionization of hydrogen atom by electron impact in the first Born approximation. In thisapproximation, the incident and scattered electrons are described by Dirac plane relativistic wavefunctions while the ejected electron is described by a Sommerfeld-Maue semirelativistic Coulombwave function and the hydrogen atom, in its metastable state, is described by Darwin’s semirela-tivistic wave function. The TDCS obtained in SRCBA will be compared with the correspondingone in the nonrelativistic Coulomb Born approximation (NRCBA). In the symmetric coplanar ge-ometry, we present the relativistic formalism of the (e, 2e) reaction in the relativistic plane waveBorn approximation (RPWBA), where the incident, scattered and ejected electrons are describedby relativistic plane waves, and the hydrogen atom in its metastable 2S-state is described by therelativistic exact function, and it will be compared, in the nonrelativistic domain, with the non-relativistic plane wave Born approximation (NRPWBA). We have found that the relativistic andspin effects, become more and more important by increasing the energy of the incident electron.All the appropriate numerical tests to verify the validity of the analytical results we found wereperformed with a very good degree of accuracy. The paper is constructed as follows. In section 2,we deliver the different theoretical models in the asymmetric and symmetric coplanar geometriesand give, for each model, a detailed account of the techniques which we have used to evaluate theTDCS. In section 3, we discuss the numerical results we have obtained in each geometry. Finally,section 4 is devoted to the conclusions. Atomic units (cid:126) = m = e = 1 are used throughout thiswork. II. THEORETICAL MODELS
Let us consider a collision between a hydrogen atom in its metastable 2S-state and an incidentelectron moving along the z -axis. As a result of this collision, the hydrogen atom becomes ionizedand the projectile electron changes its four-momentum from p i to p f . In the final state, twoelectrons (scattered and ejected) emerge with four-momenta p f and p B . This reaction can bedescribed, symbolically, as follows:e − ( p i ) + H(2S) −→ H + + e − ( p B ) + e − ( p f ) . (1)During this work, we will study the process (1) under two different geometries. We will start firstwith asymmetric coplanar geometry and then secondly with symmetric coplanar geometry. Thedetailed calculation of each TDCS in each geometry will be presented.4 . Asymmetric coplanar geometry We remember that in the case of the Ehrhardt coplanar asymmetric geometry, a fast electron ofkinetic energy T i is incident on the hydrogen target, and a fast scattered electron of kinetic energy T f is detected in coincidence with a slow ejected electron of kinetic energy T B . Additionally, thethree momenta p i , p f , and p B are in the same plane and the scattering angle θ f of the scatteredelectron is fixed and small, while the angle θ B of the ejected electron is varied. In this geometry,we calculate step by step the exact analytical expression of the semirelativistic spin-unpolarizedTDCS in the SRCBA appriximation corresponding to the electron-impact ionization of atomichydrogen in its metastable 2S-state.
1. The S-matrix element
We begin with the first Born ionization S-matrix element for the process (1) in the direct channelin which the exchange effects are neglected. It can be written as S fi = − ic (cid:90) + ∞−∞ dx (cid:104) ψ p f ( x ) φ f ( x ) | V d | ψ p i ( x ) φ i ( x ) (cid:105) , = − i (cid:90) + ∞−∞ dt (cid:90) d r ψ † p f ( t, r ) ψ p i ( t, r ) (cid:104) φ f ( x ) | V d | φ i ( x ) (cid:105) . (2)Here, the potential V d = 1 /r − /r presents the direct interaction between the incident electronand the hydrogen atom, where r = | r − r | and r = | r | . The nucleus of the target atom,which is assumed to be infinitely massive, is chosen to be the origin of the coordinate system. Thecoordinates of the incident and atomic electrons are labeled by r and r , respectively. ψ p i and ψ p f are the wave functions describing, respectively, the incident and scattered electrons given bya free Dirac solution normalized to the volume Vψ p i ( x ) = u ( p i , s i ) √ E i V e − ip i .x ,ψ p f ( x ) = u ( p f , s f ) (cid:112) E f V e − ip f .x , (3)where E i and E f are, respectively, the total energies of the incident and scattered electrons. φ i ( x ) = φ i ( t, r ) is the semirelativistic Darwin wave function of atomic hydrogen in its metastable2S-state, which is accurate to the order Z/c in the relativistic corrections. It is given by φ i ( t, r ) = exp[ − i E b (2 S ) t ] ϕ ( ± )2 S ( r ) , (4)5here E b (2 S ) is the binding energy of the metastable 2S-state of atomic hydrogen given by E b (2 S ) = c √ (cid:113) √ − α − c , (5)where α = 1 /c is the fine structure constant. For spin up, ϕ (+)2 S ( r ) is expressed by ϕ (+)2 S ( r ) = N D − r i (4 − r )4 c cos( θ ) i (4 − r )4 c sin( θ ) e iφ √ π e − r / , (6)where θ and φ are the spherical coordinates of r and N D = 4 c √ c + 10 (7)is the normalization constant. The wave function φ f ( x ) = φ f ( t, r ) in Eq. (2) is the Sommerfeld-Maue wave function for continuum states [12], also accurate to the order Z/c in the relativisticcorrections. We have φ f ( t, r ) = e − iE B t ψ ( − ) p B ( r ) , (8)where ψ ( − ) p B ( r ) is given, in its final compact form normalized to the volume V, by ψ ( − ) p B ( r ) = e πη B / Γ(1 + iη B ) e i p B . r (cid:26) F ( − iη B , , − i ( p B r + p B . r )) + i cp B ( α. p B + p B α. ˆ r ) F ( − iη B + 1 , , − i ( p B r + p B . r )) (cid:27) u ( p B , s B ) √ E B V . (9) η B is the Sommerfeld parameter given by η B = E B c p B , (10)where E B is the total energy of the ejected electron and p B = | p B | is the norm of the ejectedelectron momentum. In Eq. (9), the operator [ α. p B ] acts on the free spinor u ( p B , s B ) and theoperator [ α. ˆ r ] acts on the spinor part of the Darwin wave function.The integral over the time coordinate in Eq. (2) can be separated yielding (cid:90) dt exp[ i ( E f + E B − E i − E b (2 S )) t ] = 2 πδ ( E f + E B − E i − E b (2 S )) , (11)while the integration over d r can be performed by using the well-known following Bethe integral (cid:90) d r e i ( p i − p f ) . r (cid:18) r − r (cid:19) = 4 π ∆ ( e i ∆ . r − , (12)6here the quantity ∆ = p i − p f is the momentum transfer.The direct S-matrix element in Eq. (2) becomes S fi = − i (cid:90) d r ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V (cid:26) F ( iη B , , i ( p B r + p B . r )) − i cp B ( α. p B + p B α. ˆ r ) F ( iη B + 1 , , i ( p B r + p B . r )) (cid:27) ¯ u ( p B , s B ) γ √ E B V ϕ (+)2 S ( r ) e − i p B . r ( e i ∆ . r − × π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) . (13)This S-matrix element contains two terms S (1) fi , S (2) fi . The first one is given by S (1) fi = − i (cid:90) d r ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V ¯ u ( p B , s B ) γ √ E B V (cid:8) F ( iη B , , i ( p B r + p B . r )) (cid:9) ϕ (+)2 S ( r ) × e − i p B . r ( e i ∆ . r −
1) 8 π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) . (14)This first term can be reformulated in the following form S (1) fi = − i [ H ( q = ∆ − p B ) − H ( q = − p B )] ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V ¯ u ( p B , s B ) γ √ E B V × π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) , (15)where H ( q ) is given by H ( q ) = (cid:90) d r e i q . r F ( iη B , , i ( p B r + p B . r )) ϕ (+)2 S ( r ) . (16)According to the expression of ϕ (+)2 S ( r ) given in Eq. (6), H ( q ) can be written as H ( q ) = N D √ π ( I , , I , I ) T , (17)and one has to evaluate I = (cid:90) d r (2 − r ) e − r / e i q . r F ( iη B , , i ( p B r + p B . r )) . (18)In this integral, we are confronted with the task of evaluating two types of integrals, one of whichis I (cid:48) = 2 (cid:90) d r e − r / e i q . r F ( iη B , , i ( p B r + p B . r )) , (19)and the other one is I (cid:48)(cid:48) = (cid:90) d r r e − r / e i q . r F ( iη B , , i ( p B r + p B . r )) . (20)7n order to evaluate the first integral I (cid:48) , we take recourse to the well-known integral [13] I ( λ ) = (cid:90) d r e i q . r e − λr r F ( iη B , , i ( p B r + p B . r )) , = 4 πq + λ exp (cid:20) iη B ln (cid:18) q + λ q + λ + 2 q . p B − iλp B (cid:19)(cid:21) , (21)where λ is a real variable. Then, the integral I (cid:48) reads I (cid:48) = 2 (cid:18) − ∂I ( λ ) ∂λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / . (22)For the integral I (cid:48)(cid:48) , there are two methods to perform it. The first one is by using the integral I ( λ ) (21). This yields I (cid:48)(cid:48) = (cid:18) ∂ I ( λ ) ∂λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / . (23)The second method that we prefer to use here is through the direct application of an analyticalformula given by Gravielle in [14] I (cid:48)(cid:48) = 8 πD A − iη B (cid:20) L + L A + L A (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / , (24)where D = λ + q , A = 1 − q . p B + iλp B ) D ,L = (1 − iη B )[2(2 − iη B ) λ − D ] ,L = iη B [4 λ (1 − iη B )( λ − ip B ) − D ] ,L = 2 iη B (1 + iη B )( λ − ip B ) . (25)The other integrals I and I in (17) can be obtained by noting that cos( θ ) e i q . r = − ir ∂∂q z e i q . r , (26)and sin( θ ) e iφ e i q . r = − ir (cid:18) ∂∂q x + i ∂∂q y (cid:19) e i q . r . (27)Thus we finally get I = 1 c ∂∂q z (cid:20) I ( λ ) + 14 ∂I ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / ,I = 1 c (cid:20) ∂∂q x + i ∂∂q y (cid:21)(cid:20) I ( λ ) + 14 ∂I ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / . (28)8he second term in the S-matrix element given in Eq. (13) is S (2) fi = S (2) , fi + S (2) , fi , (29)with S (2) , fi = − (cid:90) d r ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V cp B ¯ u ( p B , s B ) γ √ E B V (cid:20) γ E B c − /p B (cid:21) ϕ (+)2 S ( r ) × F ( iη B + 1 , , i ( p B r + p B . r )) e − i p B . r ( e i ∆ . r − × π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) , (30)and S (2) , fi = − (cid:90) d r ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V c ¯ u ( p B , s B ) γ √ E B V ϕ (cid:48) (+)2 S ( r ) F ( iη B + 1 , , i ( p B r + p B . r )) × e − i p B . r ( e i ∆ . r −
1) 8 π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) . (31)The operator [ α. p B ] in Eq. (30) is replaced by (cid:2) γ E B c − /p B (cid:3) , and in Eq. (31) ϕ (cid:48) (+)2 S ( r ) is given by ϕ (cid:48) (+)2 S ( r ) = [ α. ˆ r ] ϕ (+)2 S ( r ) = N D √ π e − r / i (4 − r )4 c − r ) cos( θ )(2 − r ) sin( θ ) e iφ . (32) S (2) , fi can be recasted in the following form S (2) , fi = − [ H ( q = ∆ − p B ) − H ( q = − p B )] ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V cp B ¯ u ( p B , s B ) γ √ E B V × (cid:20) γ E B c − /p B (cid:21) π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) , (33)where H ( q ) is the integral expressed by H ( q ) = (cid:90) d r e i q . r F ( iη B + 1 , , i ( p B r + p B . r )) ϕ (+)2 S ( r ) . (34)Replacing the Darwin function in Eq. (34) by its expression (6) leads to H ( q ) = N D √ π ( J , , J , J ) T , (35)where J = (cid:90) d r e i q . r e − r / (2 − r ) F ( iη B + 1 , , i ( p B r + p B . r )) ,J = i c (cid:90) d r e i q . r e − r / (4 − r ) cos( θ ) F ( iη B + 1 , , i ( p B r + p B . r )) ,J = i c (cid:90) d r e i q . r e − r / (4 − r ) sin( θ ) e iφ F ( iη B + 1 , , i ( p B r + p B . r )) . (36)9o evaluate this three integrals, we introduce a new integral that has been calculated analyticallyby Attaourti et al in [15] J ( λ ) = (cid:90) d r e i q . r e − λr r F ( iη B + 1 , , i ( p B r + p B . r )) , = 4 π ( q + λ ) F (cid:18) iη B + 1 , , , − q . p B − iλp B ) q + λ (cid:19) , (37)where λ is a real variable.In the same way as before and after some manipulations, one gets J = − (cid:18) ∂J ( λ ) ∂λ (cid:19) − ∂ J ( λ ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 / ,J = 1 c ∂∂q z (cid:20) J ( λ ) + 14 ∂J ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / ,J = 1 c (cid:20) ∂∂q x + i ∂∂q y (cid:21)(cid:20) J ( λ ) + 14 ∂J ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / . (38)For the term S (2) , fi , it can be written as S (2) , fi = − [ H ( q = ∆ − p B ) − H ( q = − p B )] ¯ u ( p f , s f ) (cid:112) E f V γ u ( p i , s i ) √ E i V c ¯ u ( p B , s B ) γ √ E B V × π ∆ δ ( E f + E B − E i − E b (2 S )) e πη B / Γ(1 − iη B ) . (39)The quantity H ( q ) is given by H ( q ) = N D √ π ( K , , K , K ) T , (40)where K , K and K are three integrals whose solutions are K = − ic (cid:20) ∂J ( λ ) ∂λ + 14 ∂ J ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / ,K = − i ∂∂q z (cid:20) J ( λ ) + ∂J ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / ,K = − i (cid:20) ∂∂q x + i ∂∂q y (cid:21)(cid:20) J ( λ ) + ∂J ( λ ) ∂λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / . (41)
2. Spin-unpolarized TDCS in the SRCBA
Using the standard procedures of QED [16], we obtain for the spin-unpolarized TDCS d ¯ σ ( SRCBA ) dE B d Ω B d Ω f = 116 π c | p f || p B || p i | e πη B ∆ (cid:12)(cid:12) Γ(1 − iη B ) (cid:12)(cid:12) (cid:12)(cid:12) (cid:98) S (1) fi + (cid:98) S (2) , fi + (cid:98) S (2) , fi (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) E f = E i + E b (2 S ) − E B , (42)10ith (cid:98) S (1) fi = 12 (cid:88) s i ,s f (cid:88) s B (cid:2) ¯ u ( p f , s f ) γ u ( p i , s i ) (cid:3)(cid:2) ¯ u ( p B , s B ) γ (cid:3)(cid:2) i ( H ( q = ∆ − p B ) − H ( q = − p B )) (cid:3) , (43) (cid:98) S (2) , fi = 12 (cid:88) s i ,s f (cid:88) s B (cid:2) ¯ u ( p f , s f ) γ u ( p i , s i ) (cid:3)(cid:2) ¯ u ( p B , s B ) γ (cid:3)(cid:20) γ E B c − /p B (cid:21) cp B × (cid:2) H ( q = ∆ − p B ) − H ( q = − p B ) (cid:3) , (44) (cid:98) S (2) , fi = 12 (cid:88) s i ,s f (cid:88) s B (cid:2) ¯ u ( p f , s f ) γ u ( p i , s i ) (cid:3)(cid:2) ¯ u ( p B , s B ) γ (cid:3) c (cid:2) H ( q = ∆ − p B ) − H ( q = − p B ) (cid:3) . (45)In Eq. (42), | p i | and | p f | are, respectively, the norms of the initial and final electron momenta. Allthe calculations in Eq. (42) can be done analytically and only five terms out of nine are nonzero,the diagonal terms (cid:12)(cid:12) (cid:98) S (1) fi (cid:12)(cid:12) , (cid:12)(cid:12) (cid:98) S (2) , fi (cid:12)(cid:12) , (cid:12)(cid:12) (cid:98) S (2) , fi (cid:12)(cid:12) , and (cid:98) S (1) † fi (cid:98) S (2) , fi , as well as (cid:98) S (2) , † fi (cid:98) S (1) fi . In Eqs. (43)-(45),the different sums over spin states give the following results: (cid:88) s i ,s f (cid:12)(cid:12) ¯ u ( p f , s f ) γ u ( p i , s i ) (cid:12)(cid:12) = 2 c (cid:18) E i E f c − ( p i .p f ) + c (cid:19) , (46) (cid:88) s B (cid:12)(cid:12)(cid:12)(cid:12) ¯ u ( p B , s B ) γ (cid:20) γ E B c − /p B (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = 4 E B (cid:18) E B c − c (cid:19) , (47) (cid:88) s B (cid:12)(cid:12) ¯ u ( p B , s B ) γ (cid:12)(cid:12) = 4 E B , (48) (cid:88) s t ( ... ) = 1( ... ) , (49)where ( p i .p f ) in Eq. (46) is the scalar product of initial and final four-momentum, and (cid:80) s t ( ... ) / denotes the averaged sum over the spin states of the target atomic hydrogen.We have to compare the TDCS in Eq. (42) with the corresponding one in the Non-RelativisticCoulomb Born Approximation (NRCBA), where the incident and scattered electrons are describedby non-relativistic plane waves: ψ p i,f ( r ) = (2 π ) − / e i p i,f . r , (50)whereas the ejected electron is described by a Coulomb wave function: ψ c,p B ( r ) = (2 π ) − / e i p B . r e π/ p B Γ (cid:16) ip B (cid:17) F (cid:16) − ip B , , − ( p B r + p B . r ) (cid:17) , (51)and the hydrogen atomic in its metastable 2S-state is described by the non-relativistic (NR) wavefunction [17] ψ NR S ( r ) = 14 √ π (2 − r ) e − r / . (52)11hus, the TDCS in the NRCBA is given by: d ¯ σ ( NRCBA ) dE B d Ω B d Ω f = p f p B p i (cid:12)(cid:12) f CBAion (cid:12)(cid:12) , (53)where f CBAion is the first Coulomb-Born amplitude corresponding to the ionization of metastable2S-state hydrogen atom by electron impact which is given by: f CBAion = − e π/ p B π ) Γ (cid:18) − ip B (cid:19)(cid:20) − (cid:18) ∂I ( q = ∆ − p B ) ∂λ − ∂I ( q = − p B ) ∂λ (cid:19) − (cid:0) I (cid:48)(cid:48) ( q = ∆ − p B ) − I (cid:48)(cid:48) ( q = − p B ) (cid:1)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 / , (54)where the integral I ( q ) is the same as that given previously in Eq. (21), and I (cid:48)(cid:48) ( q ) is the sameintegral taken from [Gravielle] (24), but here η B = 1 /p B . B. Symmetric coplanar geometry
The symmetric coplanar geometry can be considered as a particular case of asymmetric coplanargeometry. Let us first remind that the symmetric geometry, also called binary geometry, is definedby the requirement that the kinetic energies of the scattered and ejected electrons are nearly thesame, and the scattered and ejected electron angles with respect to the incident beam directionare equal to each other. In this section, we present the relativistic formalism of the ( e, e ) reactionin the Relativistic Plane Wave Born Approximation (RPWBA), where the incident, scattered, andejected electrons are described by relativistic plane waves, and the hydrogen atom in its metastable2S-state is described by the relativistic exact function given by: φ i ( t, r ) = exp[ − i E b (2 S ) t ] ϕ ( ± ) , SExact ( r ) , (55)where E b (2 S ) is the binding energy of the metastable 2S-state of atomic hydrogen given in (5). Forspin up, ϕ (+) , SExact ( r ) is expressed by: ϕ (+) , SExact ( r ) = 12 √ π (2 Z ) γ H +1 / a γ H +12 S (cid:115) γ H + 1( a S + 1)Γ(2 γ H + 1) r γ H − e − Zr /a S × ig S / ( r )0 f S / ( r ) cos( θ ) f S / ( r ) sin( θ ) e iφ , (56)12here θ and φ are the spherical coordinates of r . The two quantities g S / ( r ) and f S / ( r ) aresuch as: g S / ( r ) = (cid:115) Zα (cid:112) − γ H ) (cid:20)(cid:18) − Zr a S (2 γ H + 1) (cid:19)(cid:0) a S + 1 (cid:1) − (cid:21) ,f S / ( r ) = (cid:115) − Zα (cid:112) − γ H ) (cid:20)(cid:18) − Zr a S (2 γ H + 1) (cid:19)(cid:0) a S + 1 (cid:1) + 1 (cid:21) , (57)where Z is the atomic number, and the two parameters γ H and a S are given by: γ H = √ − Z α ,a S = (cid:112) γ H + 1) , (58)with α = 1 /c is the fine structure constant.Substituting all these expressions into the first Born S-matrix element (2) and after some manip-ulations, one gets d ¯ σ ( RP W BA ) dE B d Ω B d Ω f = 12 p f p B c p i ∆ (cid:18) (cid:88) s i ,s f (cid:12)(cid:12) ¯ u ( p f , s f ) γ u ( p i , s i ) (cid:12)(cid:12) (cid:19) (cid:88) s B (cid:12)(cid:12) ¯ u ( p B , s B ) γ (cid:12)(cid:12) × (cid:12)(cid:12) Φ , / , / ( q = ∆ − p B ) − Φ , / , / ( q = − p B ) (cid:12)(cid:12) . (59)The different sums over spin states s i , s f and s B are given before in Eqs. (46-48). The functions Φ , / , / ( q ) are the Fourier transforms of the relativistic atomic hydrogen wave functions Φ n =2 ,j =1 / ,m =1 / ( q ) = (2 π ) − / (cid:90) d r e i q . r ϕ (+) , SExact ( r ) , (60)and ∆ = p i − p f is the momentum transfer. Replacing the exact function ϕ (+) , SExact ( r ) by itsexpression (56) yields Φ n =2 ,j =1 / ,m =1 / ( q ) = (2 π ) − / √ π (2 Z ) γ H +1 / a γ H +12 S (cid:115) γ H + 1( a S + 1)Γ(2 γ H + 1) × (cid:82) d r e i q . r r γ H − e − Zr /a S ig S / ( r )0 (cid:82) d r e i q . r r γ H − e − Zr /a S f S / ( r ) cos( θ ) (cid:82) d r e i q . r r γ H − e − Zr /a S f S / ( r ) sin( θ ) e iφ . (61)The expression of the TDCS in the Semi-Relativistic Plan Wave Born Approximation (SRPWBA)remains similar to that given in the RPWBA (59), except the expression of the Fourier transformwhich changes since the wave function describing the hydrogen atom in the SRPWBA is replaced by13he Darwin wave function that we have previously expressed in Eq. (6). This TDCS in Eq. (59) is tobe compared with the corresponding one in the Non-Relativistic Plane Wave Born Approximation(NRPWBA), where the incident, scattered, and ejected electrons are described by non-relativisticplane waves: d ¯ σ ( NRP W BA ) dE B d Ω B d Ω f = 2 π ∆ p f p B p i (cid:20) q − q ) − q − q ) (cid:21) , (62)where q = ∆ − p B and q = − p B . III. RESULTS AND DISCUSSION
In this paper, we develop an exact relativistic model, in the first Born approximation, to studythe ionization of the metastable 2S-state hydrogen atom by electrons impact at high energies in theasymmetric and symmetric coplanar geometries. The required derivatives of hypergeometric func-tions and all integrals resulting from the Fourier transforms of the relativistic and semirelativisticatomic hydrogen wave functions are computed in closed analytic forms using the programminglanguage MATHEMATICA, which is also used to plot the various figures of the present work. Inthis section, we will present all the numerical results obtained in both asymmetric and symmetricgeometries; during that, we will follow the same arrangement that we adopted in the previoussection. We will start first with the results obtained in the case of asymmetric geometry and thensymmetric geometry. All the TDCSs are given in atomic units.
A. Asymmetric coplanar geometry
We will begin our discussion, in this case, by comparing our results with those obtained byHafid et al. [5] in the nonrelativistic domain. Hafid’s results were obtained using the well-knownapproximation BBK model of Brauner et al. [7], and when Hafid presented his results, he alsocompared with those obtained by Coulomb wave functions and second born calculations of Vucic et al. [6] with respect to the incoming electron kinetic energy of eV and the ejected electronkinetic energy of eV. In the following figures (Fig. (1), Fig. (2) and Fig. (3)), which containthe comparison with other theoretical calculations, the angular choice is as follows: p i is alongthe z -axis and θ i = 0 ◦ , φ i = 0 ◦ . For the scattered electron, we choose φ f = 0 ◦ and θ f is fixed inFigs. (1) and (2), respectively, to the values θ f = 3 ◦ and θ f = 5 ◦ , while in Fig. (3) θ f varies from − ◦ to ◦ . For the ejected electron, we choose φ B = 0 ◦ and θ B varies from ◦ to ◦ in Figs. (1)14nd (2) and it is fixed to the value θ B = 20 ◦ in Fig. (3).Figure 1: The TDCS of the (e, 2e) ionization of hydrogen 2S in terms of the ejection angle θ B .The incident and the ejected electron kinetic energies are eV and eV respectively and thescattering angle θ f = 3 ◦ . The solid red line gives our results (NRCBA) given in Eq.(53), the solidblack line those of Hafid et al. [5] and the dashed line results obtained by Coulomb wavefunctions.We compare, in Fig (1), our results in the NRCBA (Eq.(53)) with those of Hafid et al. and thoseobtained by Coulomb wave functions (where the product of two Coulomb wave solutions is used forthe scattered and ejected electrons) for the incident kinetic energy of eV, ejection kinetic energyvalue of eV and the scattering angle of θ f = 3 ◦ . Our results in the NRCBA model were obtained,as we have seen in the theoretical calculations in the previous section, by using a Coulomb wavefunction to describe only the ejected electron, whereas the fast incident and scattered electronsare described by non-relativistic plane wave functions, thus neglecting the Coulomb interaction ofthe fast scattered electron with the system. Comparing these results with those of other case inwhich the scattered and ejected electrons are described together by a Coulomb wave functions, wefind that the results are very close, both in the shape of the curve and the location of the peaks,as well as in the order of magnitude. The slight difference between the differents approaches,in terms of magnitude, may be due to the Coulomb wave function that was used not only todescribe the ejected electron, as we did in our model (NRCBA), but even to describe also thefinal electron. These two results obtained using the Coulomb wave functions remain different inmagnitude, as well as in the height of the binary peak from the result obtained by Hafid et al. using BBK approximation. Figure (2) represents similar parameterization as in Fig. (1), but withthe scattering angle θ f = 5 ◦ . We have also included here the second Born results of Vucic et al. θ B .The incident and the ejected electron kinetic energies are eV and eV respectively and thescattering angle θ f = 5 ◦ . The solid red line gives our results (NRCBA) given in (53), the solidblack line those of Hafid et al. [5], the dotted line those of Vucic et al. [6] and the dashed lineresults obtained using Coulomb wave functions.and it is clearly seen, from this figure for θ f = 5 ◦ , that there is an apparent differences between theresults obtained from the various calculations. The binary peak value of the present calculation islowest among all calculations and is about half of that which uses Coulomb wave functions. In therecoil region, all results remain close in form and magnitude. It is interesting to see that the resultsof Hafid et al. reveal a peak which is present also in the second Born calculation of Vucic et al. andabsent in the other calculations, where the Coulomb wave functions are used. Comparing Figs. (1)and (2), we note that the magnitude of the two peaks decreases with increasing the scatteringangle θ f . We study in Fig. (3), for the same kinetic energy, the variation of the NRCBA in termsof the scattering angle θ f for the ejection angle θ B = 20 ◦ . The comparison with the results ofHafid et al. is also included. This actually gives a sharp peak, higher than the other peaks inthe previous figures. We observe also that the scattered electron, which is relatively faster thanthe ejected one, goes out with small angles. Figure (4) depicts the TDCS in the SRCBA and thecorresponding one in the NRCBA for the scattering angle θ f = 3 ◦ . The incident electron kineticenergy is T i = 250 eV and the ejected electron kinetic energy is T B = 5 eV. We see that the twocurves are identical and have two peaks, one in the interval between − ◦ and ◦ (recoil peak)and the other in the range between ◦ and ◦ (binary peak). However, even in the nonrelativisticregime, small effects are presented; due to the semirelativistic treatment of the wave functions thatwe have used in the SRCBA, and these can only be related to the spin effect.16igure 3: The TDCS of the (e, 2e) ionization of hydrogen 2S in terms of the scattering angle θ f .The incident and the ejected electron kinetic energies are eV and eV respectively and theejection angle θ B = 20 ◦ . The solid red line gives our results (NRCBA) given in (53) and the solidblack line those of Hafid et al. [5]Figure 4: The two TDCSs as a function of the ejection angle θ B . The incident and the ejectedelectron kinetic energies are eV and eV respectively and the scattering angle θ f = 3 ◦ . Theother angles are chosen as follows: θ i = 0 ◦ , φ i = 0 ◦ , φ f = 0 ◦ and φ B = 180 ◦ . B. Symmetric coplanar geometry
In symmetric geometry, as we mentioned previously, the kinetic energies of both scattered andejected electrons are required to be approximately equal. The TDCS, in all models studied inthe previous section, depends explicitly on the kinetic energy values of the scattered and ejectedelectrons, in addition to the different spherical coordinates related to each electron. Therefore, caremust be taken when choosing the values of these kinetic energies, so that the above-mentioned17eometry condition is fulfilled. We remind the reader here of the relation that allows us to obtainthese values without violating the requirement of symmetric geometry. Using the kinetic energyconservation T f = T i + ε S − T B , we find that, according to the condition T f = T B , T B = ( T i + ε S ) / ,where ε S = − . eV = − . a.u. is the nonrelativistic binding energy of atomic hydrogen in itsmetastable 2S-state. Thus, every kinetic energy of the incoming electron corresponds to a kineticenergy of the scattered electron determined from that relation so that the condition of symmetricgeometry always remains true. For the symmetric coplanar geometry, we choose the followingangular situation where p i is along the z -axis ( θ i = 0 ◦ , φ i = 0 ◦ ). For the scattered electron, wechoose ( θ f = 45 ◦ , φ f = 0 ◦ ) and for the ejected electron we choose φ B = 180 ◦ and the angle θ B varies differently from a figure to another. First of all, we will try to clarify the limit between therelativistic and non-relativistic domains in the case of the ionization of the hydrogen atom from itsmetastable 2S-state. Because, compared to the results of the ground state, we found that there is asignificant difference between the two non-relativistic limit values. If the hydrogen atom is ionizedfrom its ground 1S-state, the non-relativistic limit value is defined by the relativistic parameter( γ = [1 − ( β/c ) ] − / ) value of . which corresponds to an incident electron kinetic energyof eV. We recall here that, in atomic units, the kinetic energy is related to γ parameter bythe following relation: T i = c ( γ − . It means that when the value of the relativistic parameter γ is greater than . , a difference between the relativistic and non-relativistic kinetic energieswill appear. In the case of the ionization of the hydrogen atom from its metastable 2S-state, wefound that the non-relativistic limit changed and increased slightly from eV until it reachedthe value of eV. In Fig. (5), it can be seen that there is no difference at all between theTDCSs (RPWBA, SRPWBA and NRPWBA) in the non-relativistic limit, since all the curvesof the three TDCSs are almost equal and identical. This figure represents the first check of ourmodels in particular in the non-relativistic limit ( T i = 4250 eV, T f = T B = 2123 . eV). But, wenote that when we pass this limit by raising the kinetic energy of the incoming electron to keVand keV, the non-relativistic TDCS begins to differ from the other two TDCSs that remainequal as depicted in Fig. (6). Thus, the agreement between the relativistic and nonrelativisticmodels is good from the nonrelativistic limit and below ( T i ≤ eV), but the disagreementincreases at high energies. It appears from Fig. (6) that at the relativistic domain, the effects ofthe spin terms and the relativity begin to be noticeable and that the non-relativistic formalismis no longer valid. We notice from Fig. (5) that there is a parfait symmetry around the value θ B = 45 ◦ and the three TDCSs are all peaked in the vicinity of the same value. We also note18igure 5: The Three TDCSs of the (e, 2e) ionization of hydrogen 2S as a function of the ejectionangle θ B . The incident and the ejected electron kinetic energies are eV and . eVrespectively and the scattering angle θ f = 45 ◦ . (a) (b) Figure 6: Same as in Fig. (5) but for the incident and the ejected electron kinetic energies of (a) eV and . eV and (b) eV and . eV respectively.from Fig. (6) that the binary peak position in the relativistic domain begins with a shift towardssmaller values than ◦ . Comparing Figs. (5) and (6), it is clearly seen that the magnitude of thebinary peak decreases with increasing the kinetic energy of the incident electron, which is the usualbehavior in charged particle-impact ionization of an atom. By the way, these two relativistic andsemirelativivstic TDCSs (RPWBA and SRPWBA) remain the same and equal, regardless of thekinetic energy value of the incoming electron. For example, we give in Fig. (7) a representation ofthe RPWBA and SRPWBA at high incident kinetic energy of eV. It appears to us through19ig. (7) that the two TDCSs (RPWBA and SRPWBA), despite the different wave functions usedto describe the hydrogen atom in each of them, give the same results even at high energies. Thisfact was proven and applied in more than one place when studying the excitation or ionizationof the hydrogen atom where it is sufficient to use only the Darwin wave function, instead of theexact analytical wave function, as a semirelativistic state to represent the atomic hydrogen, and itwas found that this gives nearly the same results as the exact description only when the condition Zα (cid:28) is fulfilled. This is precisely the reason why, when studying theoretically asymmetricgeometry in the previous section, we were satisfied with only the treatment of the SRCBA modelwithout the corresponding one in the Relativistic Coulomb Born Approximation (RCBA), so thereis no need to complicate the calculation more as long as both give the same results. For the sake ofFigure 7: The two TDCSs in the relativistic regime as a function of the ejection angle θ B . Theincident and the ejected electron kinetic energies are eV and . eV respectively andthe scattering angle θ f = 45 ◦ .illustration, in a similar way to the 2D-plot, the contour plot in Fig. (8) exhibits more informationson the variation and the shape of the TDCS in the RPWBA versus both incident electron kineticenergy and angle θ B in the binary coplanar geometry. For the variation with respect to θ B , weobserve that the TDCS decreases at small and large angles. We see also that the TDCS presentsa maximum only at the particular point of θ B = θ f = 45 ◦ , and its magnitude at this particularpoint decreases as the electron kinetic energy increases. Figure (9) shows that when the incidentelectron kinetic energy increases provided that θ B = θ f = 45 ◦ , the peak of the TDCS decreasesand remains nearly around φ B = 180 ◦ . From Fig. (9), we see that as the energy increases, theprobability to observe the ejected electron in the direction φ B = 180 ◦ diminishes progressively.20igure 8: The TDCS, in RPWBA, as a function of the angle θ B of the ejected electron and theincident electron kinetic energy T i varying from eV to eV. We have used the condition ◦ ≤ θ f = θ B ≤ ◦ .Figure 9: The TDCS, in RPWBA, as a function of the angle φ B of the ejected electron and theincident electron kinetic energy T i varying from eV to eV for θ B = θ f = 45 ◦ .Figure (9) also shows that for all figures in the symmetric geometry, in which we choose φ B to beconstant, it must be equal to ◦ since the pick is clearly located at the same value. Figure (10)represents the variations of TDCS in the RPWBA in terms of the scattered and ejected electronangles at the energies T i = 5000 eV and T B = 2498 . eV. The purpose of including this figureis to show how important the condition on both angles to be verified in the symmetric coplanargeometry. Through this figure, it becomes clear to us that the TDCS represents a maximum valueat θ f = θ B = 45 ◦ and begins to decrease directly in the areas where this condition is broken. From21igure 10: The TDCS, in RPWBA, as a function of the angle θ B of the ejected electron and theangle θ f of the scattered electron for T i = 5000 eV and T B = 2498 . eV.here it becomes evident that we must always respect this requirement and take it into accountwhen working within the symmetric coplanar geometry. In Fig. (11), we plot the two TDCSs(SRCBA and NRCBA) with respect to the symmetric coplanar geometry at relativistic energies.In the relativistic regime, by increasing the value of the incident kinetic energy ( keV, keV, keV), we notice the shift of the maximum of the TDCS in the SRCBA towards smaller valuesthan θ B = 45 ◦ , as well as the fact that the SRCBA is always lower than the NRCBA. Recall thatin the non-relativistic domain, we have already compared our results for the two asymmetric andsymmetric coplanar geometries. In Fig. (12), we plot the RPWBA and the NRPWBA with thetwo TDCSs (SRCBA and NRCBA). We see that the NRCBA and the SRCBA are the same andthey give a lower TDCS due to the fact that the ejected electron still feels Coulomb effect of theresidual ion as much as its kinetic energy ( T B = 123 . eV in this case) is insufficient to cross theCoulomb barrier imposed by the residual ion. By increasing the kinetic energy of the incident andejected electrons simultaneously (always checking the symmetric coplanar geometry), we find thatthe ejected electron begins slowly to escape from the Coulomb effect until it completely crosses itat kinetic energy T B = 1623 . eV. In Fig. (12b), there is a very good agreement between the fourmodels and they produce the same results as the use of the Coulomb wave function is no longernecessary. 22 a) (b)(c) Figure 11: The two TDCSs in symmetric coplanar geometry as a function of the ejection angle θ B for the scattering angle θ f = 45 ◦ . The incident and the ejected electron kinetic energies are(a) T i = 10000 eV and T B = 4998 . eV, (b) T i = 15000 eV and T B = 7498 . eV and (c) T i = 20000 eV and T B = 9998 . eV. IV. CONCLUSION
In this work, we have calculated the triple differential cross sections (TDCS) for the ionizationof hydrogen atom by electrons impact in the metastable 2S-state for asymmetric and symmetriccoplanar geometries. In the asymmetric coplanar geometry, we have compared our nonrelativisticresults with those of other theories and found that the present model is close to that obtainedby Coulomb wave functions only at the scattering angle θ f = 3 ◦ . Outside this value, there issignificant difference between the different theoretical approaches in which the present resultsbeing the minimum. In the symmetric coplanar geometry, a new nonrelativistic limit value is23 a) (b) Figure 12: The four TDCSs in symmetric coplanar geometry as a function of the ejection angle θ B for the scattering angle θ f = 45 ◦ . The incident and the ejected electron kinetic energies are(a) T i = 250 eV and T B = 123 . eV and (b) T i = 3250 eV and T B = 1623 . eV.determined theoretically to be eV, which is very different from that known for the ground state( eV). Relativistic triple differential cross section have been evaluated within the relativisticmodel (RPWBA) in the first Born approximation. The consistency of this theoretical model ischecked by taking the nonrelativistic limit. Semirelativistic TDCS in the SRPWBA gives nearlythe same results, regardless of the kinetic energy of the incoming electron, as the RPWBA if thecondition Zα (cid:28) is satisfied. It is shown that the nonrelativistic formalism is no longer valid, inboth geometries, for incident kinetic energies higher than keV, due to the spin and relativisticeffects which begin to appear at high energies. Comparing our results for the two asymmetric andsymmetric coplanar geometries, we found that the use of the Coulomb wave function to describethe ejected electron is no longer necessary as long as its kinetic energy T B ≥ . eV. Thevalidation of this work requires an experimental study. We hope that our results should serve asa motivation to perform such collisions experiments in the future. [1] H. Ehrhardt, M. Schulz, T. Tekaat, and K. Willmann, Phys. Rev. Lett. , 89 (1969).[2] U. Amaldi, A. Egidi, R. Marconero, and G. Pizzella, Rev. Sci. Instrum. , 1001 (1969).[3] A. J. Dixon, A. Von Engel, and M. F. A. Harrison, Proc. R. Soc. Lond. A. , 333 (1975).[4] P. Defrance, W. Clays, A. Cornet, and G. Poulaert, J. Phys. B: At. Mol. Phys. , 111 (1981).
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