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Strong-field behaviour of laser-modified collision frequencies

F. MORALES(1), G. FERRANTE(1) and R. DANIELE(2)

(1) INFM and Dipartimento di Energetica ed Applicazioni di Fisica Parco d’Orleans, 90128 Palermo, Italy

(2) INFM and Istituto di Fisica - via Archirafi 36, 90123 Palermo, Italy

(ricevuto il 17 Maggio 1996; revisionato il 10 Ottobre 1996; approvato il 22 Ottobre 1996)

Summary. — Superelastic and momentum transfer cross-sections and collision

frequencies for the case when electrons are scattered in the presence of a very intense radiation field are calculated. The interacting system is treated within different approximations, taking into account, among others, the electron relativistic dynamics, the field spatial dependence as well as the possibility that the radiation field exhibits some statistical distribution of its parameters. A wide range of the field intensity values is considered, but the emphasis is put on the domain where the peak quiver velocity veis equal to or larger than the initial electron velocity vi. As a rule, the collision frequencies decrease with intensity when veD vi, but the details of such a decrease depend in an important way on the specific properties of the assisting field. Changing the field model, expected trends may result inverted. It applies to «parallel» and «perpendicular» momentum transfer collision frequencies. The reported results are expected to improve the understanding of anisotropic heating and velocity distribution function shaping when plasma electrons interact with very strong fields. Due to the importance of the ponderomotive energy and some subtle features of the energy conservation relation for this kind of elementary processes in the relativistic domain, the detailed multiphoton picture is found to lose, to some extent, its significance.

PACS 52.40.Nk – Laser-plasma interactions (e.g., anomalous absorption, back-scattering magnetic field generation, fast particle generation).

1. – Introduction

In the experiments concerning plasma formation by lasers and laser-plasma interactions, with increasing frequency physical situations are created, when the laser radiation is very strong, with values of intensity well beyond that of the intratomic field. To understand and to describe the plasma characteristics and behaviour in such situations, traditional items like the determination of the plasma electron velocity distribution function, or of the electron collision frequencies need to be reconsidered taking adequately into account the effects due to the presence of a strong laser field.

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For instance, it is now largely known that the electron velocity distribution of a collisional plasma embedded in a strong laser field may significantly be different as compared to a Maxwellian distribution [1, 2]. A rather common feature of the calculations dealing with laser-modified electron velocity distribution is the use of field-free electron collision frequencies when solving the pertinent kinetic equation. When instead field-modified expressions of the collision frequencies are used, the latter, as a rule, are obtained by asymptotic behaviour considerations. In such cases, however, the domain of the actual physical parameters covered is poorly defined, and, more to the point, the domains are not covered where the key physical parameters can be considered neither very small nor very large. As a contribution towards the detailed investigation of a group of items, which need to be elucidated for the understanding of plasma behaviour in the presence of strong laser fields, in this paper we calculate the frequencies of superelastic collisions and of the momentum transfer.

The collision process is that of an electron which is scattered by a screened Coulomb potential, mimicking a structureless massive ion surrounded by a cloud of plasma electrons. The scattering event takes place in the presence of a strong laser field. Adopting such a model for the collision event, and following well-established procedures, the collision frequencies may be calculated with accuracy over a very wide range of the laser intensity.

The modifications to the collision frequency are expected to be significant only for relatively intense fields. As characteristic parameter discriminating the domains of small and large modifications the ratio is usually taken

R 4 ve vi , (1) with ve4 eEL m v (2)

the peak quiver velocity in a field of the form E(t) 4ELcos vt , and vi the initial electron velocity. Sizeable modifications of the collision frequency are expected when

R F1. As values of R larger than unity may require very intense fields, we address, as

a preliminary item, the questions of the applicability, for our purposes, of the ubiquitously used dipole approximation for the external field, and of the non-relativistic theory to treat the collision event. The two questions are connected, and cannot be ignored in the case of intense external fields able to accelerate electrons to very high velocities.

Thus, in the first part of our paper, we treat and compare the pertinent cross-sections in three different approximations; namely

1) non-relativistic theory in the dipole approximation (NR ODA); 2) non-relativistic theory without dipole approximation (NR OWDA); 3) relativistic theory (RT).

In the second part of the paper, taking advantage of the results concerning the cross-sections, we calculate the collision frequencies related to the superelastic collisions and to the momentum transfer.

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2. – Outline

As discussed before, we are interested to calculate the superelastic and the momentum transfer collision frequencies of an electron-ion scattering process. To this end we need to know the cross-sections which are involved in the process and enter into the definition of the collision frequencies as

nx(U) 4Nvisx(U) ,

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where N is the target ions density, vi is the incoming electron velocity and sx(U)

stands for the superelastic or momentum transfer total cross-section for a given angle

U between the incident electron and laser field directions. These cross-sections are

calculated using a three- or four-dimensional S-matrix formalism, depending on the relativistic or non-relativistic treatment considered. Further, in both cases the unperturbed states are given by the so-called Volkov solutions of the corresponding Schrödinger or Dirac equation while the perturbation responsible for the transition is a screened static potential treated in first Born approximation. In both cases the cross-sections are found as infinite summations over the number n of the exchanged photons sx(U) 4

!

n s (n) x (U) 4

!

n

_F_x_(w)

g

ds(U) dV

h

n dV , (4)

where s(n)x are the single-channel cross-sections

s(n) x (U) 4

Fx(w)

g

ds(U) dV

h

n dV (5)

and the function Fx(w) is

Fx(w) 4 . / ´ 1

(

_{1 2 ( p}f(n) Opi) cos w

)

x 4S superelastic collisions ,

x 4T momentum transfer collisions .

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In eqs. (4), (5) and (6), w is the scattering angle, while pi and pf(n) are the initial and final electron three-dimensional momenta, and

(

_{ds(U) OdV}

)

n is the field-free

differential cross-section evaluated at the value of the final energy corresponding to the exchange of n photons. The value of pf(n) also is depending on the number of exchanged photons as results from the energy conservation relation.

Together with nx(U), eq. (3), we calculate also collision frequencies averaged over

the laser field directions with respect of the incident electron direction

nx4 1 p

0 p nx(U) dU . (7)

The averaged collision frequencies nx are expected to give a first piece of

information on the effects due to a strong external field acting on a collisional ensemble of electrons and ions moving in all directions.

In other words, eq. (7) may prove informative for a fully ionized plasma as well. At the moment we do not consider the question of averaging eq. (3) over a specific electron velocity distribution. First, we note that for a plasma embedded in a strong laser field,

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the electron velocity distribution is expected to depart significantly from a Maxwellian; second, for a given plasma model in a laser field, the proper electron velocity distribution must result from the ad hoc self-consistent solution of the Boltzmann equation.

One of the reasons why we investigate here the momentum transfer collision frequency, besides its interest in its own right, stems from the known circumstance that it appears when the collision integral entering the Boltzmann equation in the case of weak fields is treated approximately within procedures like Legendre polynomial expansion of the unknown electron distribution function. However, it is an open question whether or not the collision integral of the Boltzmann equation may be still expressed through the momentum transfer collision frequency when duly account is taken of the field effects on the collision event.

3. – The cross-sections

Now we will briefly give the cross-sections to be used in the calculations of collision frequencies. In all the three different approximations, we consider the process consisting of an electron scattered by a massive ion represented as a static screened Coulomb potential in the presence of an electromagnetic radiation field. The collisional model is that of electron states embedded in the laser field, assumed as unperturbed states, while the perturbation responsible for the transition is a static screened Coulomb potential.

The NR ODA differential cross-section of the collisional process which is considered there, for a purely coherent single-mode laser field is [3, 4]

g

ds(U) dV

h

n 4

g

Z 2_e4_m2 (ˇ)3

h

g

pf(n) pi

h

J2 n(lfi)

g

1 Q2 fi1 b2

h

, (8)

in which e and Ze are, respectively, the incoming electron and ion charges, m the electron mass; Qfi4 ( 1 O ˇ)

(

pf(n) 2pi

)

is the momentum transfer; b the screening factor of the potential, Jn(lfi) the Bessel function of integer order n and real argument

lfi; the argument of the Bessel function, which gives the laser field coupling with the electron transferred momentum is

lfi4

eE mcv2 Q Qfi,

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where E 4 (1Oc)(ˇAOˇt) is the electric laser field, and A the laser vector potential which, in dipole approximation, is defined as A 4nA cos vt, with amplitude A, photon frequency v and polarization direction n.

The energy conservation law is

E_fNR_{4 E}_iNR_{1 nˇv ,}

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a 4 p2aO 2 m with a 4 i , f the initial and final electron energy states.

The derivation of the relativistic cross-section requires a 4-dimensional formalism, and a redefinition, in terms of four vectors, of the scattering and laser potentials, according to the prescription: V(r , t) is changed into V_m_{4 (V , 0) and A(r , t) is changed} into Am4 ( 0 , A), with A 4 nA cos W , W 4 (vt 2 kgQ r) and kg the photon wave

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In the Born approximation, using relativistic Volkov states, the RT differential cross-section is obtained as

g

p2ac24 (EaR)22 m2c44 (EaR)22 E02 (a 4i, f ) , (15)

with E0and m, respectively, the rest energy and mass. The energy conservation law is now

E_fR_{4 E}iR1 nˇv 1 2 vˇR2. (16)

For relativistic processes in the presence of strong laser fields, the relations, like eq. (16), giving the conservation of energy exhibit, as a rule, a behaviour, which is more complicated and rich as compared with the non-relativistic case. For this reason, below we briefly discuss the behaviour of eq. (16).

We note also that, to the best of our knowledge, no numerical calculations of the RT cross-sections are available in the literature. Different derivations, instead, have been reported [5-7]. An independent, simple derivation is reported in appendix.

Below, among others, we will report calculations, based upon an approximation, which is non-relativistic as far as the electron velocities are concerned, but retains the space dependence of the laser field. In other words, we essentially remove the dipole approximation and introduce the photon momentum. It is believed that the removal of the dipole approximation is necessary when very intense fields are used, or when the differential cross-section is a very rapidly varying function of the scattering angle. For this approximation, which in the Introduction has been termed non-relativistic theory without dipole approximation (NR OWDA), formally the differential cross-section does not differ from eq. (11) of the relativistic case, but all the quantities appearing in eq. (11) need to be given in their non-relativistic limit. In particular, the non-relativistic

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definition of the energy ENR a as EaNR4 p2 a 2 m (a 4i, f)

with m the rest electron mass, does not alter the formal RT definitions of la, R1, R2,

Mfi and energy conservation law, but produces different values.

Thus, in a non-relativistic treatment, maintaining the laser field spatial dependence, the cross-section keeps a form significantly departing from the familiar form given by eq. (8). We anticipate that numerical calculations too will show significant differences in the high-intensity domain.

The cross-sections reported above are meant to offer different approximations in treating the dynamics of the collision event, when a very strong field assists the process. In all the cases the field model is that of a coherent monochromatic plane wave, when all the characteristics parameters are well defined and fixed.

It is by now well known that very strong fields are never purely coherent, and that statistical fluctuations of field parameters may be present; besides, space and time inhomogeneities are frequently present, due to pulsed operation and focussing. Statistical and operation properties of real laser systems may affect significantly elementary processes in strong radiation fields [8-10].

In our context, concerned with very strong laser fields, it is important to investigate how a laser system with statistically or deterministically distributed parameters may change the results as compared to the case of a single-mode field with well-fixed parameters. As an instance of a laser field with distributed parameters we consider here the chaotic-field model, with zero bandwidth corresponding to a field in which both amplitude and phase fluctuate. Due to its simplicity this model has received considerable attention in contexts similar to the present ones, and in the Born approxi-mation the differential cross-section within the NR ODA treatment is given by

g

ds(U) dV

h

CH n 4

g

Z 2_e4_m2 (ˇ)3

h

g

pf(n) pi

h

exp

k

₂l 2 fi 2

l

In

g

l2 fi 2

hg

1 Q2fi1 b2

h

, (17)

In being the Bessel function of real order n and imaginary argument lfi, and lfi is defined by eq. (9), with the only difference that the amplitude of the electric field ELis substituted by the variance

e04

k

aE2(t)b 2 aE(t)b24

k

aE2(t)b . (18)

(For this laser model aE(t)b 40.)

Below, whenever comparison will be carried out between results obtained with single mode and chaotic fields, it is understood that the chaotic field has a value of the variance e0 equalling the constant value of the electric-field amplitude of the single-mode field (e04 E). As far as the influence of the laser properties on the cross-sections and collision frequencies is concerned, below we restrict to eq. (17) only. Here we note that to obtain the «chaotic field» version of eq. (8), a way to proceed is by

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averaging eq. (8) over the distribution function of the field realizations f (E , W) 4 1 pe0 E exp

y

₂E 2 e2 0

z

; (19)

in other words, to evaluate the integral

g

ds(U) dV

h

CH n 4

0 Q E d E

0 2 p dW f(E , W)

g

ds(U) dV

h

n . (20)

4. – Conservation of energy in relativistic collision processes in strong fields

In collisions in the presence of an external field for the incident electron one has the following picture. At t 42Q the particle has the initial asymptotic free energy Ei. Further, inside the field, but just before the scattering event, the particle has the quasi-energy ei4 Ei1 Qiwith Qithe average quiver energy. During the collision event the exchange of energy D E 4nˇv takes place. After the collision, still inside the field, the particle has the quasi-energy which may be written as

ef4 ei1 nˇv 4 Ei1 Qi1 nˇv 4 Ef1 Qf. (21)

Outside the field, when reaching the detector at t 41Q, the scattered particle is detected with the asymptotic final energy Ef.

In the non-relativistic case, in dipole approximation, the initial and final states quiver energy is the same Qi4 Qf, and accordingly we will have

Ef4 Ei1 nˇv . (22)

In the relativistic case, not only the relativistic expressions for the particle energy come into play, but also Qic Qf. It gives a complicated equation to relate the asymptotic energies outside the field to the energies inside the field. In the particular case in which the radiation field is taken linearly polarized and with the field vector directed along the incident electron momentum (AV pi), the final energy Ef outside the field is related to the corresponding initial energy by the equation

(23) Ef2 Ei1 nˇv 1 e2_A2_ˇ 4

u

1 Ef2 cos w

k

Ef22 E 2 0 2 1 Ei2

k

Ei22 E 2 0

v

4 0 ,

E_a_{(a 4i, f ) being the relativistic particle energy which must satisfy eq. (15). Equations}

similar to (23) are obtained for other elementary processes in strong radiation fields, such as, for instance, field ionization, electron-positron pair production, Compton scattering and so on. Different authors [11-14] have pointed out to and discussed the unusual and complicated features of equations like eq. (23) as conservation of energy relation, but no conclusive nor quantitative results seems to have never been reported. We have addressed the solution of eq. (23) numerically.

The determination of the final energy Ef reduces to a solution of an equation of fourth degree which can have generally one or more real solutions, depending on the problem parameters. In the case at hand we have found that two imaginary and two

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Fig. 1. – a) Final energy Ef(in eV) as a function of the laser intensity I (in W Ocm2_{) for one-photon} emission and scattering angle w 4307 and U40. The radiation field is taken linearly polarized with the electric-field vector directed along the incident electron momentum (AV pi). The potential screening factor is b 41Or0; r04 100 a0(a0being the Bohr radius). The incoming electron energy is Ei4 100 eV, the laser photon energy ˇv 4 1 .17 eV. Solid line, relativistic values. Dashed line, non-relativistic values. b) Relativistic final energy Ef(in eV) as a function of the initial energy Ei (in eV) for one-photon emission and scattering angle w 4307. Other parameters as in a). Solid line, I 41015 W Ocm2 . Dashed line, I 41017 W Ocm2 . Dotted line, I 45Q1018 W Ocm2_.

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real solutions are present. The two real solutions correspond to the values of w and

p 2w

(

we adopted a procedure, in which we solve squared equations and then impose the solutions to satisfy eq. (23)

)

. Thus, the final energy depends not only on the usual energy parameters, such as initial energy and numbers of exchanged photons, but also on the scattering angle. It is evident that this circumstance makes the calculation of total cross-sections a much more difficult task. A sample calculation of the final energy is reported in fig. 1a) and b).

In fig. 1a) the final energy is shown as a function of the field intensity for one photon emission with energy ˇv 41.17 eV. The scattering angle is w4307. As expected, at relatively low intensities, where the non-relativistic treatment is expected to hold, the results coincide with the simple radiation, eq. (22). Increasing the intensity, the final energy grows over the non-relativistic limit by about 30%. In fig. 1b) the final energy is shown as a function of the initial one for different values of the field intensity. As in fig. 1a), w 4307 and one-photon emission are considered. Figure 1b) shows that, irrespective of the number of photons exchanged during the collision event, the final energy is highly dependent on the field intensity. According to the above results, in the energy balance of the process, the detailed multiphoton picture seems to lose most of its significance. Larger scattering angles are expected to give a much more significant growth of the final energy.

5. – Calculations

Below we report several sets of numerical calculations aimed to show the consequences of the different approximations considered in the paper, and to provide information on cross-sections and collision frequencies. In all the calculations, the charge of the scattering potential is Z 41.

5.1. Differential cross-sections. – Figures 2 and 3 report differential cross-sections in the parallel geometry (U 40) calculated in different approximations. Basically, the exact relativistic (RT) treatment is found to give largely the same results as the non-relativistic one in dipole approximation (NR ODA) only in the case of small-angle scattering and low energies, but considerable lower values for w F407.

The non-relativistic treatment without dipole approximation (NR OWDA) is found to give results very close to the relativistic ones in all the scattering angles. As the differential cross-sections are wildly oscillating functions, fig. 3 reports a fit to them to show their averaged behaviour. Altogether including also several results not reported here, the conclusion is that in very strong fields and Oor when one has fast incident electrons, the NR ODA treatment may give significant overestimates, while the NR OWDA treatment performs very well up to when relativistic effects are not particularly important. When relativistic effects are sizeable, the NR OWDA treatment too gives overestimates, but of smaller values.

5.2. Total cross-sections. – Figures 4-6 report total cross-sections calculated within the relativistic treatment. We have confined only to these cross-sections as similar calculations within the NR ODA treatment are familiar in the literature [5, 8, 9], and the NR OWDA treatment is expected to give largely the same results as the RT treatment, except in domains where relativistic behaviour is important. The cross-sections are generally smaller than in the non-relativistic case, except at very high intensities. Figure 6 showing the total cross-sections summed over all the photon

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Fig. 2. – Differential cross-sections (ds(U) OdV)₂₁ (in atomic units) as a function of the scattering w (in degrees) for the process of one-photon absorption. The laser intensity is I 41017

W Ocm2

. Other parameters as in fig. 1a). Curve 1 is a NR ODA differential cross-section, eq. (8), while curve 2 is the RT one, eq. (11).

exchanges may be taken as an instance of the cross-sections behaviour vs. intensity. It shows two principal maxima. The first one, larger and smaller, takes place as in the non-relativistic case and corresponds to the situation when the incident particle energy

Fig. 3. – Fitting of differential cross-sections(_{ds(U) OdV})₂₁(in atomic units) as a function of the scattering angle w (in degrees) for the process of one-photon absorption. The laser intensity is I 41016

W Ocm2_{, the electron initial energy Ei}

4 1000 eV. Other parameters as in fig. 1a). Solid line, NR OWDA values. Dashed line, RT values.

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Fig. 4. – RT superelastic total cross-sections s(n)

s (U) (in atomic units) eq. (5) as a function of the laser intensity I (in W Ocm2_{). Other parameters as in fig. 1a). Solid lines stand for absorption} processes, while other lines stand for emission processes. Curves A: 1 photon exchanged. Curves B: 5 photons exchanged. Curves C: 40 photons exchanged.

Fig. 5. – RT superelastic total cross-sections s(21)

s (U) (in atomic units) eq. (5) as a function of the laser intensity I (in W Ocm2_{) for one-photon absorption. Other parameters as in fig. 1a). Curve 1 is} obtained for electron energy Ei4 100 eV, curve 2 for Ei4 1 keV, and curve 3 for Ei4 10 keV.

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is approximately equal to the peak quiver velocity (parallel geometry is understood).

Fig. 6. – RT superelastic total cross-sections ss(U) (in atomic units) eq. (4) as a function of the laser intensity I (in W Ocm2_{). Other parameters as in fig. 1a).}

Fig. 7. – Superelastic collision frequencies nS(U) (in arbitrary units) eq. (3) as a function of the laser intensity I (in W Ocm2_{) for a single-mode purely coherent field with parallel geometry AV pi}_. Other parameters as in fig. 1a). Solid line, NR ODA values. Dashed line, RT values.

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The second maximum, occurring at high intensities, is higher and narrower, and is not observed in the non-relativistic case. At the moment, we do not have an explanation for the physical origin of this second maximum, as clear as that of the first maximum. We observe only that the second maximum occurs when vecvi, and the electron mass starts to depend significantly on the velocity.

5.3. Collision frequencies. – Figures 7 and 8 report superelastic and momentum transfer collision frequencies, calculated within the non-relativistic treatment in dipole approximation (NR ODA) and the relativistic theory (RT), in parallel geometry and for very high values of the field. Except for the second narrow maximum, the RT nS(fig. 7) has largely the same shape as the NR ODA one, but the numerical values are smaller by about 20%. Considering the behaviour of the corresponding cross-sections, noted above, this result was to be expected. The RT momentum transfer collision frequency

nT (fig. 8) exhibits two features worthy to be remarked. First, it is smaller by about 20 –25 % than the corresponding NR ODA frequency, where significant growth is exhibited up to the principal maximum. Afterwards the two frequencies decrease rapidly almost in the same way up to 1015 _{W Ocm}2_{. Second, the NR ODA n}T shows oscillatory behaviour (or secondary small maxima), which is the consequence of using the single-mode model for the laser field with well-fixed values of the parameters. Any other model, containing some kind of the field parameters distribution, is expected do not reproduce this behaviour. Such behaviour is in fact absent in the relativistic collision frequency, thanks to the circumstance that the removal of the dipole approximation amounts to introduce the space dependence of the laser electric field. For values of the intensities larger than 1015 _{W Ocm}2(for the used value of field photon,

Fig. 8. – Momentum transfer collision frequencies nT(U) (in arbitrary units) eq. (3) as a function of the laser intensity I (in W Ocm2_{) for a single-mode purely coherent field with parallel geometry}

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Fig. 9. – NR ODA superelastic collision frequencies nS(U) (in arbitrary units) eq. (3) as a function of the laser intensity I (in W Ocm2_{). The upper scale reproduces the values of the ratio R of eq. (1).} Other parameters as in fig. 1a). Solid line, single-mode purely coherent field in parallel geometry

AV pi; dotted line, single-mode purely coherent field in perpendicular geometry A » pi; dashed line, chaotic field in parallel geometry AV pi; point-dashed line, chaotic field in perpendicular geometry A » pi.

ˇ_{v 41.17 eV) the relativistic n}_T _{is considered much more realistic than the} non-relativistic one, and it is very small. Considering that it is a result concerning the parallel geometry, and that it may be considered representative of the electrons of an ensemble flying parallel to the laser electric field, the conclusion is that as soon as veD

vithe momentum transfer «parallel» collision frequency decreases in an essential way. In its turn, it is expected to have important consequences in the applications.

Due to the presence of the so-called generalized Bessel functions and to the peculiarity of energy conservation, the calculations of the derived collision frequencies for strong fields are very time-consuming. Below, accordingly, we will restrict ourselves to calculations within the non-relativistic treatment in dipole approximation. However, keeping in mind the remark concerning the calculations reported in fig. 7 and 8, from the results reported below on NR ODA collision frequencies useful information on the expected behaviour of relativistic collision frequencies too may be inferred. Figure 9 and 10 report non-relativistic superelastic and momentum transfer collision frequencies for two geometries (AV pi and A » pi), and two laser models (single-mode field and chaotic field). Concerning the superelastic collision frequency (fig. 9), we observe that: a) Up to intensities of 1013

W Ocm2 _{(with the photon energy} ˇ_{v 41.17 eV) n}_S _{calculated within the single-mode model in both the geometries is} little affected by the field. Instead, calculations based on the chaotic-field model, in

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Fig. 10. – NR ODA momentum transfer collision frequencies nT(U) (in arbitrary units) eq. (3) as a function of the laser intensity I (in W Ocm2_{). The upper scale reproduces the values of the ratio R} of Eq. (1). Other parameters as in fig. 1a). Solid line, single-mode purely coherent field in parallel geometry AV pi; dotted line, single-mode purely coherent field in perpendicular geometry A » pi; dashed line, chaotic field in parallel geometry AV pi; point-dashed line, chaotic field in perpendicular geometry A » pi.

which the field parameters are statistically distributed, show that the field begins to affect the collision frequency at values of the intensities smaller than 1013

W Ocm2_{. In} particular, nSin parallel geometry is larger than the field-free collision frequency while in perpendicular geometry is smaller. b) In general, a very important domain of the process parameters is that when veB vi; in the case of nS, we have that in the parallel geometry the collision frequency exhibits the maximum growth, while in the perpendicular one the most rapid decrease; besides, the single-mode model predicts values larger than those of the chaotic-field model. c) For veD vi, the chaotic-field model predicts values of the collision frequency larger than the single-mode model; the

nS in parallel geometry tends to the field-free value, while nS in perpendicular geometry decreases appreciably.

Altogether, the calculations of fig. 9 (and of fig. 10 as well) show that a laser field assumed to have distributed parameters affects the collision frequencies in a smoother way as compared to a field with well-fixed parameters. Besides, very strong fields are likely to be poorly described by the single-mode model.

More interesting and significant are the results concerning the momentum transfer collision frequencies nT, in particular for veF vi. Namely, nT in parallel geometry calculated within the single-mode model is found to increase by a factor of 4 when veBvi and then, for veD vito decrease drastically to very small values (as remarked in more

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Fig. 11. – NR ODA collision frequencies nS(U) eq. (3) and nS eq. (7) (in arbitrary units) as a function of the laser intensity I (in W Ocm2_{) for a single-mode purely coherent field. The upper} scale reproduces the values of the ratio R of eq. (1). Other parameters as in fig. 1a). Solid line, nS values. Dashed line, nS(U) values in parallel geometry AV pi. Dotted line, nS(U) values in perpendicular geometry A » pi.

than one place above, we consider the oscillatory behaviour exhibited by nT in parallel geometry, single-mode model and very high fields, to be too much model dependent).

nTin perpendicular geometry is a monotonically, but slowly, decreasing function of the intensity; in such a way that, as soon as veD vi, nT in perpendicular geometry becomes larger than nT in parallel geometry.

Imagining applications of these results to laser heating of electron ensembles (like in a plasma), we have that when veD vi (more precisely, when veD vT, with vT same appropriately defined effective thermal velocity), energy from the laser field flows preferentially into perpendicular (to the field direction) degrees of freedom. With the consequence that anisotropic heating and electron velocity distribution functions result.

Interesting and intriguing are the results for nTin the case of a chaotic field. Apart from the smooth behaviour of the pertinent curves, the basic point is the following: for all values of the intensity considered (up to 1018 _{W Ocm}2) nT in parallel geometry remains larger than nT in perpendicular geometry, with the consequence that heating and shape of the resulting electron distribution function (in a plasma) may well be quite different as compared to those of the previous field model. The precise explanation of these results lies in the circumstance that according to the chaotic-field model, the electric field experiences statistical fluctuations from zero to infinity. Obviously, this is

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Fig. 12. – NR ODA collision frequencies nT(U) eq. (3) and nT eq. (7) (in arbitrary units) as a function of the laser intensity I (in W Ocm2_{) for a single-mode purely coherent field. The upper} scale reproduces the values of the ratio R of eq. (1). Other parameters as in fig. 1a). Solid line, nT values. Dashed line, nT(U) values in parallel geometry AV pi. Dotted line, nT(U) values in perpendicular geometry A » pi.

not a general feature of any laser system, being restricted to systems with a large number of uncorrelated modes. However, the present results may well be considered representative also of the cases, when the laser system exhibits in general some distribution of its parameters. Such distribution may be statistical, or deterministic in space and time due to the way the laser system is operated. It is likely that any laser system performing at high intensities exhibits some kind of parameters distribution. Finally, fig. 11 and 12 report angle-integrated superelastic and momentum transfer collision frequencies vs. intensity. For comparison, shown are also the same frequencies in parallel and perpendicular geometries. Having in mind an ensemble of electrons moving in all directions, angle-integrated collision frequencies are expected to give information on the average, cumulative effect the external field has on the process considered in all its realizations. Angle-integrated collision frequencies are found to differ little from field-free results up to at least 1014

W Ocm2_{, to become} decreasing functions of intensity for higher values of the latter.

6. – Conclusions

We have reported calculations in different treatments of superelastic and momentum transfer cross-sections and collision frequencies for the case when electrons are acted upon by a strong laser field. The differences in the theoretical

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treatments concern several aspects of the scattering process. They dynamics of the collision event is treated in both a relativistic and non-relativistic way; the radiation field is treated with and without the dipole approximation; besides the single-mode model, the chaotic-field model too has been used to mimick a strong field with statistically distributed parameters. In the calculations, emphasis has been put on values of intensity such that the peak quiver velocity veis equal to or larger than the incoming particle velocity vi. We have tried to get information on the behaviour of the collision frequencies under different physical conditions. As a rule, the collision frequencies decrease when veD vi, but the details of a such decrease are found to depend in an important way on the specific properties of the interacting system. The reported results suggest that anisotropic heating and shaping of the electron velocity distribution functions (in a plasma) may be a process much more complex as expected. In particular, the distribution function may acquire a prolate or an oblate shape, or both, during its time evolution. Under this respect, a very significant role is played by the radiation field properties. In conclusion, we have contributed new information and quantitatively accurate data on a number of items, such as: a) the characteristics of final energy distribution in strong field-assisted relativistic processes; b) the comparison of electron scattering cross-sections and collision frequencies in the different approximations over a wide range of radiation field intensity. For the two most accurate approximations (relativistic case, and inclusion of the field space dependence), the reported results are the only ones available at the date; c) the role of the laser field statistical properties. The reported results suggest that expected features in laser-plasma interaction may be inverted changing the description of the laser field; d) the quantitative investigation of the collision frequencies behaviour in domains inaccessible to asymptotic field considerations; and assessment of asymptotic field predictions.

* * *

The authors express their thanks to the University of Palermo Computational Centre for the computer time generously provided to them. This work was supported by the Italian Ministry of University and Scientific Research, the National Institute of Physics of the Matter (INFM), the National Group of Structure of Matter of the National Research Council (GNSM-CNR) and the Regional Committee for Nuclear and Structure of Matter Researches (CRRNSM).

AP P E N D I X

Derivation of the relativistic cross-section

For the scattering process considered in this work, the non-relativistic cross-section is rather familiar. It is not so for the relativistic case, though it has received some attention in the past literature [5-7, 13]. In what follows, we outline the basic steps which lead to the cross-sections used in our calculations.

The relativistic equation of motion of an electron, of rest mass m and charge e, in the presence of single mode, linearly polarized, electromagnetic field defined by its

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four-vector potential Am4 ( 0 , A) is the Dirac equation

g

!

gm

g

2iˇ ˇ ˇx 2 eA_m c

h

2 imc

h

C(x) 40 . (A.1)

In eq. (A.1) we have that A 4nA cos W with the field phase W4 (vt2kgQ r), with kg4

n(v Oc) and the photon four-wave vector km4 (v , kg); the metric (c2, 21, 21, 21, ) is

chosen and the four-momentum of the electron in the field is given as P 4 (E, p) and

x 4 (t, r) the four-coordinate. The solution of eq. (A.1) is given by

(A.2) _{C(x) 4}

g

1 2 p

h

3 O2

o

m 2 E [ 1 1g(t) ] exp [2i(PQx) ]Q Q exp

y

2i

h

g

A Q P 2 P Q K 2 A2 2 P Q K

h

dh

z

U with h 4KQx; P4 (E, p) the four-momentum and U the free-particle spinor that satisfies the equation

(gmQ P 2m) U40

(A.3)

and gm4 (b , 2iba) with

a 4

u

0 s

2s 0

v

(A.4)

and s the Pauli matrices. g(t) is a matrix function arising from the coupling between the spin and the electromagnetic field. The relativistic cross-sections are calculated in the first Born approximation in the S-matrix four-dimensional formalism. The perturbation is the screened Coulomb potential described before while the unperturbed initial and final states are those given by eq. (A.2).

The S-matrix is given by

Sfi4 2 i ˇ d 4 x Cf(x) Vm(x) Ci(x) . (A.5)

In eq. (A.5) the scattering four-dimensional potential V_m_{(x) 4 (V, 0) is assumed to have} the scalar part

V(t , r) 4 Ze

2

r exp [2br]

(A.6)

with Ze the ion charge and b the screening factor. In what follows the sole approximations which are done are the neglect of the spin effects and the first Born approximation for the scattering S-matrix

(

eq. (A.5)

)

. Neglecting spin effects the wave function (A.2), normalized to the volume V, becomes

C(x) 4

o

mc

2

2 EV exp [2i(PQx) ] exp

y

2i

h

g

A Q P 2 P Q K 2 A2 2 P Q K

h

dh

z

. (A.7)

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In (A.7) the particle energy and the momentum are related by the formula

p2_c2

4 E22 m2c4

4 c44 E22 E02. (A.8)

Inserting in (A.7) the value of the radiation field four-potential and carrying out the integration we obtain for the wave function the form

C(t , r) 4

o

mc 2 2 EV exp

k

ˇ ( pf2 pi) Q r

l

exp [ 2 iR2W] exp [inW] Jn 22m(R1) Jm(R2) . Substituting the explicit value of the field phase W and performing both time and spatial integrations the matrix element assumes the final form

Sfi4

!

n S n

fi (A.13)

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with Sfin4 2 p

g

2iZe 2_mc2 2 ˇVkEiEf

h

!

m Jn 22m(R1) Jm(R2)

g

4 p M2fi1 b2

h

d(Ef2 Ei1 nˇv 1 2 vˇR2) and Mfi given by eq. (14) of the main text.

(A.13) gives the S-matrix as a summation of contributions concerning the exchange of a given number of photons of frequency v (n D0 for emission, nE0 for absorption).

Proceeding now in the usual way, we arrive at the differential cross-section, eq. (11) of the main text, together with the energy conservation relation eq. (16).

R E F E R E N C E S

[1] For theoretical investigations, see, for instance, LANGDONA. B., Phys. Rev. Lett., 44 (1980) 575; BALESCUR., J. Plasma Phys., 27 (1982) 553; JONESR. D. and LEEK., Phys. Fluids, 25 (1982) 2307; CHICHKOVB. N., SHUMSKYS. A. and URYUPINS. A., Phys. Rev. E, 45 (1992) 7475; PORSHNEV P., FERRANTEG. and ZARCONEM., Phys. Rev. E, 48 (1993) 2081; JORNAS. and WOODL., Phys. Rev. A, 36 (1987) 397; PORSHNEVP., BIVONAS. and FERRANTEG., Phys. Rev. E, 50 (1994) 3943; PORSHNEVP., KHANEVICHE., BIVONAS. and FERRANTEG., Phys. Rev. E, 53 (1996) 1100; PULSIFERP. E. and WHITNEYK. G., Phys. Rev. E, 50 (1995) 4926; TOWNR. P. J., BELLA. R. and ROSES. J., Phys. Rev. Lett., 74 (1995) 924; DECKD., Laser Part. Beams, 5 (1987) 49.

[2] For recent experimental investigations, see, for instance, AMENDTP., EDERD. C. and WILKS S. C., Phys. Rev. Lett., 66 (1991) 2589; MOHIDEEN V. et al., Phys. Rev. Lett., 71 (1993) 509; MEVELE. et al., Phys. Rev. Lett., 75 (1995) 445; DONNELYT. D., LEER. W. and FALCONER. W., Phys. Rev. A, 51 (1995) R2691.

[3] BUNKINF. V. and FEDOROV M. V., Zˇ. E.ksp. Teor. Fiz., 49 (1965) 31(Sov. Phys. JETP, 22 (1966) 844).

[4] KROLL N. M. and WATSON K. M., Phys. Rev. A, 8 (1973) 804. [5] BREHMEH., Phys. Rev. C, 3 (1971) 837.

[6] KAMINSKI J. Z., J. Phys. A, 18 (1985) 3365.

[7] EHLOTZKYF., Nuovo Cimento, LXIX (1970) 73; Opt. Commun., 66 (1988) 265.

[8] DANIELER., TROMBETTAF., FERRANTEG., CAVALIEREP. and MORALESF., Phys. Rev. A, 36 (1987) 1156 and references therein.

[9] MORALESF., DANIELER., TROMBETTAF. and FERRANTEG., Phys. Rev. A, 40 (1989) 3681 and references quoted therein.

[10] MORALES F., DANIELE R. and FERRANTE G., Laser Phys., 3 (1993) 431 and references therein.

[11] KIBBLET. W., Phys. Rev., 138 (1965) B740.

[12] YAKOVLEV V. P., Zˇ. E.ksp. Teor. Fiz., 49 (1965) 318 (Sov. Phys. JETP, 22 (1966) 223). [13] DENISOVM. M. and FEDOROVM. V., Zˇ. E.ksp. Teor. Fiz., 53 (1967) 1340(Sov. Phys. JETP, 26

(1968) 779).

fulltext

Strong-field behaviour of laser-modified collision frequencies

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