Comment on Mott Scattering in Strong Laser
Field
Y. Attaourti
∗, B. Manaut
†, and M. Chabab
‡LPHEA, Physics Department, Faculty of Science-Semlalia, P. O. Box 2390, Marrakesh, Morocco.
Abstract
The first Born differential cross section for the Mott scattering of a Dirac-Volkov electron is reviewed. The expression (26) derived by Szymanowski et al. [Physical Review A56, 3846, (1997)] is corrected. In particular, we disagree with the expression of
dσ dΩ
they obtained and we give the exact coefficients multiplying the various Bessel functions appearing in the differential cross section, in natural units as well as in atomic units. PACS number(s): 34.80.Qb, 12.20.Ds ∗[email protected] †[email protected] ‡[email protected] 1
1
Introduction
In a pionneering paper, Szymanowski et al. [1] have studied the Mott scattering process in a strong laser field. The main purpose was to show that the modifications of the Mott scattering differential cross section for the scattering of an electron by the Coulomb poten-tial of a nucleus in the presence of a strong laser field, can yield interesting physical insights concerning the importance and the signatures of the relativistic effects. Their spin depen-dent relativistic description of Mott scattering permits to distinguish between kinematics and spin-orbit coupling effects. They have compared the results of a calculation of the first Born differential cross section for the Coulomb scattering of the Dirac-Volkov electrons dressed by a circularly polarized laser field to the first Born cross section for the Coulomb scattering of spinless Klein Gordon particles and also to the non relativistic Schrodinger-Volkov treatment. The aim of this comment is to provide the correct expression for the first-Born differential cross sections corresponding to the Coulomb scattering of the Dirac-Volkov electrons. On the one hand, we show that the terms proportional to sin(2φ0) are missing in [1], where φ0 is the angle entering the expression of the circularly polarised electromagnetic field. The claim of [1] that they vanish is not true. These terms do not depend on the chosen description of the circular polarisation in cartesian components. On the other hand, we also correct all the coefficients that intervene in the expression of dσdΩ. Moreover, we perform the calculations with some details and provide our results in natural units (¯h = c = 1) as well as in atomic
units (¯h = e = m = 1) where m denotes the electron mass.
The organization of this paper is as follows: in Section 2, we review the formalism and establish the expression of the S-matrix transition amplitude both in natural units and atomic units. In Section 3, we derive the correct expression of the differential scattering cross section associated to the exchange of a given number of laser photons. Section 4 is devoted to the conclusion.
2
The Coulomb Scattering of a Dirac-Volkov electron
Exact solutions of relativistic wave equations [2] are very difficult to obtain. However, in a seminal paper, Volkov [3] obtained the formal solution of the Dirac equation for the relativistic electron with 4-momentum p inside a classical monochromatic electromagnetic field A. These solutions are called the relativistic Volkov states. The plane wave electromagnetic field A of
4-momentum k (k.k = k2 = 0) depends only on the argument φ = kx and therefore A is such that :
A = A(k.x) = A(φ) (2.1)
The 4-vector A satisfies the Lorentz gauge condition ∂A = 0 or kA = 0 . The Dirac equation for an electron in an external field A (for the electron e < 0) is :
n
(ˆp− eA)2− m2− 1/2ieFνσνoψ(x) = 0 (2.2) where Fν is the electromagnetic field tensor Fν = ∂Aν− ∂νAµ and σν = 12[γ , γν]
where γ are the anticommuting Dirac matrices such that γ γν + γνγµ = 2gν.1
4, where the
metric tensor is gν = diag(1,−1, −1, −1). Putting ψ = exp(−ip.x)F (φ) into Eq(2.2) one
obtains : F (φ) = exp ( −iZ kx 0 " e(p.A) (k.p) − e2 2(k.p)A 2 # dφ + e k/A/ 2(k.p) ) u(p, s) √ 2p0 (2.3) where u(p, s) represents a bispinor for the free electron which satisfies the first order Dirac equation without field and is normalized according to uu = 2m. In equation(2.3), one has the term exp2(k.p)e k/A/. Because k is a pure light vector k2 = 0 and due to the transversality condition k.A = 0, it is straightforward to see that this exponential factor reduces to two terms, exp2(k.p)ek/A/ = 14 + 2(k.p)e k/A/ + 0. Indeed, a quick inspection of the second order term
proportional to A = k/A/k/A/ yields A =−k/k/A/A/ = −k2A2 = 0. So all terms of order higher that the first one vanish.
Therefore, the relativistic Volkov wave functions are given by :
ψp(x) = " 1 + e 2(k.p)k/A/ # u(p, s) √ 2p0 exp(is(x)) (2.4) where : s(x) =−p.x − Z kx 0 e (k.p) p.A− e 2A 2dφ
These wave functions are normalized according to : Z
ψp†0(x)ψp(x)dx =
Z
ψp0(x)γ0ψp(x)dx = (2π)3δ(p0− p) (2.5)
The 4-current is j = ψp0γ ψp. Using the Gordon identity:
u(p)γ u(p0) = 1 2m(u(p) [(p + p 0)σν(p− p0) ν] u(p0)) one obtains : j = 1 p0 ( p− eA + " e(p.A) (p.k) − e2A2 2(k.p) # k ) (2.6)
If one assumes that A is quasi-periodic so that its time average is zero A = 0, the averaged 4-current is therefore given by :
j = 1 p0 ( p− e 2 2(k.p)A 2k ) (2.7) Setting : q = p− e 2A2 2(k.p)k (2.8) we note that : q.q = m2 1−e 2A2 m2 ! = m2∗ (2.9)
is Lorentz invariant. One often calls the averaged 4-momentum q a quasi impulsion. The quantity m∗ plays the role of an effective mass of the electron inside the electromagnetic field. We consider a circularly polarized field :
A = a1cos(φ) + a2sin(φ)
where φ = k.x. We choose a21 = a22 = a2 = A2 and a1.a2 = a2.a1 = 0. The Lorentz condition
k.A = 0 implies a1.k = a2.k = 0. For the study of the Mott scattering in presence of a laser
field, one can use Volkov wave functions [3] normalized according to :
ψp(x) = " 1 + e 2(k.p)k/A/ # u(p, s) √ 2q0V exp(is(x)) = " 1 + e 2(k.p)(k/a/1cos(φ) + k/a/2sin(φ)) # u(p, s) √ 2q0V exp (is(x)) (2.10) with : s(x) =−q.x − e a1.p (k.p)sin(φ) + e a2.p (k.p)cos(φ)
2.1
The S-matrix element in natural units.
We treat the interaction of the dressed electrons with the Coulomb field :
Acoul= −Ze
|x|, 0, 0, 0
!
(2.11) as a first order perturbation. This is well justified if Zα << β = vc. The appropriate S-matrix element is given by :
Sf i=−ie Z
In complete analogy to the free-electron case, the calculation of the transition amplitude associated to this process can be carried out using the Feynman rules with the requirement that the free-electron wave functions are replaced by the Volkov wave functions. the S-matrix element (2.12) becomes then,
Sf i = iZe2q 1 2Qi2Qf 1 V Z d4xu(pf, sf) ( 1 + e
2(k.pf){a/1k/ cos(φ) + a/2k/ sin(φ)} ) γ0 |x| × ( 1 + e 2(k.pi)[a/1k/ cos(φ) + a/2k/ sin(φ)] )
u(pi, si) exp (i(qf − qi).x− iz sin(φ − φ0)) (2.13) where : z = q α21+ α22 cos(φ0) = α1 z sin(φ0) = α2 z (2.14)
and α1 and α2 are respectively given by :
α1 = e ( a1.pi k.pi − a1.pf k.pf ) (2.15) α2 = e ( a2.pi k.pi − a2.pf k.pf ) (2.16) Therefore : Sf i = iZe2q 1 2Qi2Qf 1 V Z
d4xu(pf, sf){C0+ C1cos(φ) + C2sin(φ)} 1
|x|u(pi, si) (2.17)
× exp (i(qf − qi).x− iz sin(φ − φ0)) (2.18)
with : C0 = γ0− 2k0a2k/c(pi)c(pf), C1 = c(pi)γ0k/a/1+ c(pf)a/1k/γ0, and C2 = c(pi)γ0k/a/2+ c(pf)a/2k/γ0 (2.19) where : c(pi) = 2(k.pe i), and c(pf) = e 2(k.pf).
Next step, we invoke the well-known identities involving ordinary Bessel functions Jn(z) : 1 cos(φ) sin(φ) exp (− iz sin(φ − φ0)) = +∞X s=−∞ Bs(z) B1s(z) B2s(z) e −isφ (2.20)
where : Bs(z) = Js(z)eisφ B1s(z) = Js+1(z)ei(s+1)φ0 + J s−1(z)ei(s−1)φ0 /2 B2s(z) = Js+1(z)ei(s+1)φ0 − J s−1(z)ei(s−1)φ0 /2i (2.21)
Then the S matrix simplifies to
Sf i = iZ(4πe 2 q 2Qi2Qf 1 V +∞X s=−∞ u(pf, sf)Γ(s)u(pi, si)2πδ(Qf − Qi− sω) |qf − qi− sk|2 (2.22) where Γ(s) is given by : Γ(s) = C0Bs(z) + C1B1s(z) + C2B2s(z) (2.23) with : c(pi) = 2(k.pe i), and c(pf) = e 2(k.pf)
2.2
The transition amplitude in atomic units
The relativistic Volkov states are now given by:
ψq(x) = " 1 + k/A/ 2c(k.p) # u(p, s) √ 2QVe is(x) (2.24) where : s(x) =−qx − a1.p c(k.p)sin(φ) + a2.p c(k.p)cos(φ)
u(p, s) represents a bispinor for the free electron normalized by uu = 2c2. These wave functions are normalized in the volume V . The four-vector q = (Qc, q) labels the Volkov state|q >.
q = p− A 2
2(k.p)c2k (2.25)
such that q.q = m2∗c2 with :
m2∗ = 1−A2
c2
!
(2.26) Proceeding along the same lines as we did before, first the interaction of the dressed electrons with the central Coulomb field :
A = −Z
|x|, 0, 0, 0
is considered as a first-order perturbation. Then, the appropriate S-matrix element is given by: Sf i = iZ c Z d4xψf γ 0 |x|ψi = iZ4π cq2Qi2Qf 1 V +∞X s=−∞ u(pf, sf)Λ(s)u(pi, si)2πδ(Qf − Qi− sω) |qf − qi− sk|2 (2.27) where : Λ(s)= C0Bs(z) + C1B1s(z) + C2B2s(z) (2.28) The coefficients C0, C1 and C2 have been defined in Eq.(2.19). We also have : c(pi) = 2c(k.p1
i),
and c(pf) = 2c(k.p1
f)
3
The differential cross section
We now turn to the calculation of the first-Born differential cross-section.
3.1
Final expression in natural units
We first evaluate the transition probability per particle into the final states within the range of momentum dqf : dWf i = |Sf i|2V dqf (2π)3 = Z 2(4πα)2 2Qi2Qf X l |Mf i(s)|2 |qf − qi− sk|4 dqf (2π)3T 2πδ(Qf − Qi− sω) (3.1) Here we have set e2 = (4πεe2
0¯hc) = α the fine-structure constant and used the rule of replacement:
[2πδ(Qf − Qi− sω)]2 = 2πδ(0)2πδ(Qf − Qi− sω)
= T.2πδ(Qf − Qi − sω) (3.2) The transition probability (3.1) per unit time
dRf i= dωf i T = Z2α2 QiQf +∞X s=−∞ |M(s)f i|2 |qf − qi− sk|4 δ(Qf − Qi− sω)dqf
Dividing dRf i by the flux of incoming particles :
|Jinc| = |qi|c2
Qi = |qi|
We then obtain the differential cross section : dσ = Z 2α2 QiQf Qi |qi| X s |Mfi(s)|2 |qf − qi− sk|4 dqfδ(Qf − Qi− sω) = Z2α2X s |Mf i(s)|2δ(Qf − Qi− sω) |qf − qi− sk|4 dqf Qf|qi| (3.3)
where : Mfi(s) =|u(pf, sf)Γ(s)u(pi, si)|2 Noting that m2∗ = Q2f−q2f, one has|qf|d|qf| = QfdQf
so that (dqf =|qf|2d|qf|dΩf): dσ(s) dΩf = Z 2α2|qf| |qi| |Mf i(s)|2 |qf − qi− sk|4 Qf=Qi+sω (3.4)
The calculation is now reduced to the computation of traces of products of γ matrices. This is routinely done using Reduce [4]. We calculate the unpolarized differential cross-section. Therefore, the various possible polarization states have the same probability and the actually measured cross-section is given by summing over the final polarization sf and averaging over the initial polarization si. Therefore :
dσ(s) dΩf = Z 2α2|qf| |qi| 1 |qf − qi − sk|4 1 2 X si X sf |Mf i(s)|2 Qf=Qi+sω (3.5) where : 1 2 X si X sf |Mf i(s)|2 = 1 2T r (p/f + m)Γ(s)(p/i+ m)Γ(s) (3.6) = 2nJs2(z)A + (Js+12 (z) + Js−12 (z))B + (Js−1(z)Js+1(z))C + Js(z)(Js+1(z) + Js−1(z))Do with : A = m2− (qi.qf) + 2QiQf − a 2e2 2 (k.qf) k.qi + (k.qi) (k.qf) ! + a 2e2ω2 (k.qi)(k.qf)((qi.qf)− m 2) + (a2)2e4ω2 (k.qi)(k.qf) + a 2e2ω(Q f − Qi) 1 (k.qi)− 1 (k.qf) ! (3.7) B = − (a 2)2e4ω2 2(k.qf)(k.qi) + e2ω2 2 " (a1.qi) (k.qi) (a1.qf) (k.qf) + (a2.qi) (k.qi) (a2.qf) (k.qf) # − 1 2a 2e2+ 1 4a 2e2((k.qi) (k.qf) + (k.qf) (k.qi))− 1 2 a2e2ω2 (k.qi)(k.qf)((qi.qf)− m 2) + 1 2a 2e2ω(Q f − Qi) 1 k.qf − 1 k.qi ! (3.8)
C = e
2ω2
(k.qf)(k.qi){cos(2φ0) [(a1.qf)(a1.qi)− (a2.qf)(a2.qi)] +
sin(2φ0) [(a1.qf)(a2.qi) + (a1.qi)(a2.qf)]} (3.9)
D = e 2((qi.˚A) + (qf.˚A))− e 2 " (k.qf) (k.qi) ! (˚A.qi) + (k.qi) (k.qf) ! (˚A.qf) # + eω (˚A.qi)Qf (k.qi) + (˚A.qf)Qi (k.qf) ! (3.10) where we have introduced the vector : ˚A = a1cos(φ0) + a2sin(φ0). At this stage, we notice that the coefficient appering in dσ(s)
dΩf
derived by Szymanowski et al. is different from our shown in Eq.(3.9). Indeed, in the expression of dσdω(s)
f
, the term (sin(2φ0)) is absent. The justification of Ref. [1] is that this term does not describe a physical situation but arises from the chosen description of the circular polarization in cartesian components. Our claim is different. In a fully covariant calculation, the term proportional to sin(2φ0) has to be taken into account since its net contribution is :
e2ω2a2sin(2φ0)
(k.qi)(k.qf) (qiyqf x+ qixqf y) (3.11) and the circular polarization chosen has a dynamical effect (on the components of qi and qf that have to be considered ) and not a geometrical effect. The fact that a1 and a2 appear in scalar products does not mean that these scalar products are zero since : a1.a2 = 0. To make the point more clear, let us remark that in Equation (30) of [1] concerning the scattering of a spinless particle both terms proportional to sin(2φ0) and cos(2φ0) occur in the expression of
dσ(s) dΩf
.
To be more confident on our claim and in order to have a direct comparison between our result and the result of [1], we rederive in the next section the final expression of
dσ(s) dΩf in atomic units.
3.2
Final expression in atomic units
Proceeding along the same lines as we did before, we have for the differential cross section :
dσ = |Sf i| 2 T . V dqfQiV (2π)3c2|qi| (3.12) so that : dσ = Z 2 c2 +∞ X s=−∞ |Mf i(s)|2δ(Qf − Qi− sω) |qf − qi − sk|4 dqf Qf|qi| (3.13)
and noting that dqf = |qf|QfcdQ2 fdΩf, we obtain : dσ dΩf = +∞X s=−∞ dσ(s) dΩf Qf=Qi+sω (3.14) where : dσ(s) dΩf = Z2|qf| c2|qi| 1 |qf − qi− sk|4 1 2.c2T r (cp/f + c2)Λ(s)(cp/i+ c2)Λ(s) (3.15) The calculation of the trace gives :
dσ(s) dΩf = Z2|qf| c2|qi| 1 |qf − qi− sk|4 × (3.16) 2 c2 n Js2(z)A + (Js+12 (z) + Js−12 (z))B + (Js+1(z)Js−1(z))C + Js(z)(Js−1(z) + Js+1(z))Do with: A = c4− (qf.qi).c2+ 2Qf.Qi− a 2 2 (k.qf) (k.qi) + (k.qi) (k.qf) ! + a 2.ω2 c2(k.qf)(k.qi)((qf.qi)− c 2) + (a2)2.ω2 c4(k.qf)(k.qi)+ a2.ω c2 (Qf − Qi) 1 (k.qi)− 1 (k.qf) ! (3.17) B = − (a 2)2ω2 2c4(k.qf)(k.qi)+ ω2 2c2 (a1.qf) (k.qf) (a1.qi) (k.qi) + (a2.qf) (k.qf) (a2.qi) (k.qi) ! −a2 2 + (3.18) a2 4( (k.qf) (k.qi) + (k.qi) (k.qf))− a2ω2 2c2(k.qf)(k.qi)((qf.qi)− c 2) + a2ω 2c2 (Qf − Qi) 1 (k.qf) − 1 (k.qi) ! C = ω 2
c2(k.qf)(k.qi)( cos(2φ0){(a1.qf)(a1.qi)− (a2.qf)(a2.qi)} +
sin(2φ0){(a1.qf)(a2.qi) + (a1.qi)(a2.qf)}) (3.19)
D = c 2 (˚A.qi) + (˚A.qf)− c 2 (k.qf) (k.qi)(˚A.qi) + (k.qi) (k.qf)(˚A.qf) ! + ω c Qi.(˚A.qf) (k.qf) + Qf.(˚A.qi) (k.qi) ! (3.20) where ˚A = a1cos(φ0) + a2sin(φ0). Once again the obvious difference between the result of [1] and our is contained in the coefficient C. We have a term proportional to sin(2φ0) that is absent in Equation(26) of [1]. Moreover, in their expression multiplying the product
2Jn(ξ)2, the single term c2(k.q)(k.q(a2)2ω20) should come with a coefficient 12. Also, the term multiplying (Jn+1(ξ)2+ Jn−1(ξ)2) is incorrect since it contains quantities proportional to (a2)3, which is not possible because the highest power of a2 in the expression of the transition amplitude is 1, so when squaring this amplitude, the highest power of a2 must be 2. The 4-vectors q and q0 of [1] correspond respectively to the 4-vectors qi and qf of the present work. We have already mentionned that in the term multiplying the product Jn+1(ξ)Jn−1(ξ), an expression containing
sin(2φ0) is missing. Finally, the last expression multiplying Jn(ξ) (Jn+1(ξ) + Jn−1(ξ)) must be linear in the electromagnetic potential a. If one ignores the term quadratic in a in that expression, one recovers the result we have obtained. We have also noticed that the intro-duction of the 4-vector ˜q used in [1] makes the calculations rather lengthy and gives rise to
complicated expressions. As a consistency check of our procedure, we have reproduced the result of the differential cross section corresponding to the Compton scattering in an intense elctromagnetic field given by Berestetzkii, Lifshitz and Pitaevskii [5].
4
Conclusion
In this comment, we derived the correct expression of the first Born differential cross section for the scattering of a Dirac-Volkov electron by a Coulomb potential of a nucleus in the presence of a strong laser field. We have given the correct relativistic generalization of the Bunkin and Fedorov treatment [6] that is valid for an arbitrary geometry. In a forthcoming paper, we shall carry out numerical simulations and focus on the angular distributions of dΩdσ
f
.
Aknowledgement
This work was partially supported by the Portugal-Morocco Convention, under the grant CNR/681/02 and by the France-Morocco Cooperation Program, project MA/02/38.
References
[1] C. Szymanowski, V. Vniard, R. Taieb, A. Maquet, and C. H. Keitel, Physical Review A
56, 3846, (1997).
[2] V. G. Bagrov, and D. M. Gitman, Exact Solutions of Relativistic Wave Equations, Kluwer
[3] D. M. Volkov, Z. Phys, 94, 250 (1935).
[4] A. C. Hearn, Reduce User’s and Contributed Packages Manual, Version 3.7, Konrad-Zuse-Zentrum fur Informationstechnik, Berlin, 1999.
[5] V. B. Berestetzkii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon Press, Oxford, 1982).
[6] F. V. Bunkin and M. V. Fedorov, Zh. Eksp. Teor. Fiz. 49, 1215 (1965) [Sov. Phys. JETP
