Elementary Many-Particle Processes in Plasma Microfields

Transcription

1 Contrib. Plasma Phys. 46, No. 3, (2006) / DOI /ctpp Elementary Many-Particle Processes in Plasma Microfields M.Yu. Romanovsky 1 and W. Ebeling 2 1 General Physics Institute, Academy of Science of Russia, Vavilov str.38, R Moscow, Russia 2 Institut für Physik, Humboldt Universität zu Berlin, Newtonstr. 15, D Berlin, Germany Received 5 Januar 2006, accepted 19 Januar 2006 Published online 17 Februar 2006 Key words Electric microfields, magnetic microfields, elementary processes, many-particle processes, ionization, recombination, fusion, tunneling, radiation. PACS Nk, Vy, Gj, y, j, z, The effect of electric and magnetic plasma microfields on elementary many-body processes in plasmas is considered. As detected first by Inglis and Teller in 1939, the electric microfield controls several elementary processes in plasmas as transitions, line shifts and line broadening. We concentrate here on the many-particle processes ionization, recombination, and fusion and study a wide area of plasma parameters. In the first part the state of art of investigations on microfield distributions is reviewed in brief. In the second part, various types of ionization processes are discussed with respect to the influence of electric microfields. It is demonstrated that the processes of tunnel and rescattering ionization by laser fields as well as the process of electron collisional ionization may be strongly influenced by the electric microfields in the plasma. The third part is devoted to processes of microfield action on fusion processes and the effects on three-body recombination are investigated. It is shown that there are regions of plasma densities and temperatures, where the rate of nuclear fusion is accelerated by the electric microfields. This effect may be relevant for nuclear processes in stars. Further, fusion processes in ion clusters are studied. Finally we study in this section three-body recombination effects and show that an electric microfield influences the three-body electron-ion recombination via the highly excited states. In the fourth part, the distribution of the magnetic microfield is investigated for equilibrium, nonequilibrium, and non-uniform magnetized plasmas. We show that the field distribution in a neutral point of a non-relativistic ideal equilibrium plasma is similar to the Holtsmark distribution for the electrical microfield. Relaxation processes in nonequilibrium plasmas may lead to additional microfields. We show that in turbulent plasmas the broadening of radiative electron transitions in atoms and ions, without change of the principle quantum number, may be due to the Zeeman effect and may exceed Doppler and Stark broadening as well. Further it is shown that for optical radiation the effect of depolarization of a linearly polarized laser beams propagating through a magnetized plasma may be rather strong. Corresponding author: slon@kapella.gpi.ru

2 196 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields Contents 1 Introduction - short review on microfield distributions BasicconceptsandHoltsmarkdistribution Furtherworkonmicrofielddistributions Ionization processes in electric microfields Electron collisional ionization of ions Influence of electric microfields on the dynamics of electrons and on velocity distributions Corrections of electron impact ionization rates by electric microfields in weakly-coupled plasmas Estimation of mean impact ionization rates using Thompson and Lotzcross-sections Tunnel ionization of ions by laser fields, influenced by electric microfields Effectofelectricmicrofieldsontherescatteringprocess Recombination and fusion processes influenced by electric microfields Fusionratesinhomogeneousplasmas Dynamics of nuclei and microfields in nonideal plasmas Semiclassicalmicroscopictheoryoffusionrates Discussion of dynamic enhancement effects Fusionratesinlaser-ionizeddeuteriumclusters Distributionofelectricmicrofieldinsideandoutsideionclusters Semiclassical theory of fusion rates in finite charged objects Influenceoftwo-particleKeplermotion Nearest neighbor distribution of electrical microfield in two-particle Kepler motion Discussionoftheinfluenceonfusionrates Effect of an electric microfield on three-body electron-ion recombination via highly excitedstates Recombinationviahighlyexcitedstates Density effects of cold plasma bunch confinement by laser radiation

3 Contrib. Plasma Phys. 46, No. 3 (2006) Enhancementofdielectronicrecombination Magnetic microfield in plasmas and controlled processes Distribution of magnetic microfield in homogeneous and magnetized plasmas Distribution in a neutral point of an ideal equilibrium plasma Velocity-velocity correlations in space in nonequilibrium plasmas Distributionofmagneticmicrofieldsinmagnetizedplasmas Zeemanbroadeningofionlinesinturbulentplasmas Depolarizationoflaserradiationinplasmas The effect of magnetic microfields on three-body recombination Appendix Appendix Appendix Appendix Conclusion Acknowledgements References...258

4 198 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields 1 Introduction - short review on microfield distributions 1.1 Basic concepts and Holtsmark distribution The first investigation of plasma microfields was performed in 1919 by J. Holtsmark. He derived the distribution of an electric microfield at a neutral point of an ideal plasma [1]. Noteworthy, this was 10 years before Tonks and Langmuir [2] introduced the term plasma itself. When Lanczos found in 1930 that atomic hydrogen lines may disappear in a static electric field [3], astrophysics problems became the main focus of investigators. This is in close connection with the fact that stars consist of plasmas and their radiative properties may give valuable informations on the plasma state of the stars. Soon, it became clear that the electric microfield strongly affects the radiation processes of stars due to the Stark effect. This way the astrophysicists obtained a powerful new tool to study stars. The Holtsmark distribution gives the probability density to find a definite value E of the electric field at certain given location in an ideal plasmas. The electric microfield is determined by the sum of the elementary fields created by a very large number of elementary dot charges. As the distribution of an instant absolute value of an electric microfield follows a distribution which is quite different from the Maxwell distribution. The Holtsmark distribution W H provides the probability for the sum of the elementary fields E i generated by all N particles in the plasma at the origin of the coordinate system (which is assumed to be a neutral point): N N E = E i = i=1 i=1 e i r i ri 3. This distribution reads in dimensionless variables W H (β) = 2β π 0 x sin(βx)exp( x 3/2 )dx where β = E/E H, E H =2π(4/15) 2/3 en 2/3 ;heree = e i are the charges of the plasma particles; we consider here a one-component plasma (OCP) and suppose that all charges are equal, n is their density. The main feature of the Holtsmark distribution is the long tail, i.e. the slow decay of the fluctuations wtih increasing fields. Indeed, the asymptotic drop of the Holtsmark distribution is proportional to β 5/2. Such slow power-law decay gives rise to large field fluctuations, i.e. the fields with β 10 have still a probability 0.03 instead of for normal Gaussian distributions with the corresponding argument. The pioneering work about the disappearing of Rydberg lines in the ion spectra in a relatively dense plasma was performed by Inglis and Teller in 1939 [4]. This work attracted the interest of many researchers to the effect of plasma microfields on elementary processes like quantum emission and line broadening. Special interest was paid to the Stark effect, the line shift and/or broadening. This determined the main direction of microfield investigations as the cause of observable ionic lineshapes. Here we will follow another route, focussing mainly on the elementary processes ionization, recombination, fusion etc.. In all these processes several particles are directly involved, at least two in ionization and fusion, at least three in recombination. However the other particles in the plasma also may play an important role. In reality, ionization, recombination and fusion take place in a very complicated time-dependent plasma environment. The action of the plasma particles which are not directly involved in the studied elementary process, we will model here by the concept of a microfield. According to this concept, the elementary process (ionization, recombination, fusion etc.) is always embedded into a timeand space-dependent electrical and/or magnetic field generated by the environment. Of course, this assumption is of model character, it may be however a first step from a study of isolated elementary processes to a proper many-particle treatment. Our approach is based on the general assumption that the field-structure in a plasma has domain character as demonstrated in Fig. 1. The elementary processes take place in such field domains which are quickly changing in time and space, details will be explained in the 2nd section. A plasma has two types of charges, electrons and ions. Therefore there are two types of microfields: one is generated by ions, the second one by the electrons. No question about the total microfield arises in an ideal plasma, when the kinetic energy of charged particles is much larger than the energy of their interaction. This fact leads to the homogeneous distribution of charged particles in space: there are no shielded points in such plasmas.

5 Contrib. Plasma Phys. 46, No. 3 (2006) 199 Both electron and ion microfields have Holtsmark distribution, therefore the instant total microfield is also described by a Holtsmark distribution. The Holtsmark distribution is an example of a stable Levy distribution (see, for example [5]). The sum of two random values both distributed according to Levy is distributed also according to Levy. This is called stability of the distribution. On the other hand, the dynamics of both components of the field is quite different, since electrons move much faster than ions. Therefore the ionic field has a much longer life time (relative physical stability), the correlation time is about the inverse of the ionic plasma frequency. For this reason, the first-order Stark effect could be observed so far only in the ionic microfield, the electron microfield gives rise to second-order Stark effects. These results were summarized in a number of well-known studies; in particular the book of Griem [6] provided a clear and exhaustive summary. At the same time, it was known that the plasma interior of stars, the cores of the stars, is strongly-coupled. This was observed by spectroscopical measurements of lineshapes of ions in stars, which demonstrated deviations from the normal Holtsmak drop of line wing ( ω) 5/2 (ω is the frequency of ion transition) [7]. 1.2 Further work on microfield distributions In connection with the astrophysical observations it became necessary to determine corrections to the Holtsmark distribution in dense plasmas. However, since the electron microfield gave only a small correction to the wings of ionic lines (see, for example [8]), in the first time less attention was paid to these effects. When several researchers started to work on dense laser plasmas in the mid-sixties, many laboratory obtained plasmas were still weaklycoupled (close to ideal ones). This required the generalization of the simple Holtsmark s approach by including of (comparably small) charged particle interactions. The most fruitful approach was done by Baranger and Moser [9]. Their original formulation of electric microfield distributions leads to a series representation ordered according to the particles correlation functions for the plasma. Instead of the simple exponent x 3/2 under the integral of the Holtsmark distribution, there appears a series in the charge density n: F (K) =exp ( P =1 ) n P P! h P (K) where P is the number of particles in a definite plasma cluster [9]: W (E) = 2E K sin(ke)f (K)dK π 0 The functions h P are determined through the corresponding space correlation functions. Considering a hydrogen plasma we find h 1 (K) = 4 15 (2πeK)3/2 i.e. the above x =2π( 4 15 )2/3 ek = E H K. This way we get once more the Holtsmark distribution. The second function is determined through the two-particle correlation function. Introducing the two-particle interaction potential Φ 12 (r 1, r 2 ),wefind where the notation h 2 (K) = e2 T φ 1 φ 2 Φ 12 d 3 r 1 d 3 r 2 φ i =1 exp(ike i ) is used [9]. The third and higher terms have more complicated expressions through correlation functions. The difference between neutral points in plasmas and the points where ions ( radiators ) are located is expressed through the second-order correlations.

6 200 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields The Baranger-Moser formulation is very useful if the series can be truncated at first or second order. In the classical limit, when the electron degeneracy can be avoided, the main correlations are due to the Coulomb interaction among the particles, and the terms of the series are ordered according to powers of the plasma nonideality parameter Γ=e 2 /k B Tr 0 which is the ratio of the mean energy of Coulomb interaction between neighboring particle (at average distance r 0 ) to the mean kinetic particle energy k B T. We will not discuss here the specific problems of microfields in degenerate plasmas; these problems have been considered in [10 12]. Computing strongly coupled Coulomb sytems, the Baranger-Moser series required a partial resummation, since de Broglie wavelength could be large compared to the interparticle distance while Γ is small. At the same time, the case of strong coupling often corresponds to the opposite situation when Γ is large and the de Broglie wavelength is small. There are several methods to perform the resummation of Baranger-Moser series or to invent alternative methods to include correlations between the particles. The most sucessful of them is connected with the model of one component plasmas (OCP) where an electron part of plasmas is considered as the neutralized background of (interesting) ions. The first sucessful calculation of ion microfields for large plasma parameter Γ was presented by Iglesias et. al. [11, 13]. The idea of Iglesias et al. was to retain in the low order terms the structural simplicity of the Baranger-Moser series, while incorporating higher order correlations through an approprialely screened microfield. The screening can be large for large Γ, then the microfield is small outside the sphere of the screening radius r 0. In this case the series can be ordered according to powers of r 0 /l 0 where l 0 is the interparticle (ions) distance. Thus, the truncation in Baranger-Moser series is performed according to the screening effects but not to correlation functions. The theory leads to distributions of the following type W (β) = 2β π 0 x sin(βx)t (x)dx [ i T (x) =exp 2 ne 1 0 dλ dr Kr r 3 ] [g ei(r,λ) g ee (r,λ)] where g ea denote field-dependent correlation functions [14]. In the Iglesias theory the following approximation is used g ea (r,λ)=g ea (r, 0) exp [iλke ] Here g ea (r, 0) is the normal equilibrium pair distribution function of the pair e, a. The Iglesias series contains also a screened microfield E which replaces the bare Coulomb field E in the Baranger-Moser series: ) ) E (r) =E(r) (1+ rrs exp ( rrs where r s is an adjustable-parameter of the proposed method. This approximation yields good agreement with the Monte-Carlo method of microfield calculations [13]. The comparison of Monte-Carlo distributions with the calculated distribution including the screened field E permits to obtain expressions for the adjusted parameter r s. Form this considerations, it is easy to see that in the case of weakly-coupled plasmas r s r D is the Debye radius. On the other hand, the Iglesias series provides for large Γ a reasonable reduction of the microfield distribution to a Gauss form (see also [19]). A more precise representation of the procedure discussed above was presented in [15]. A general review of this approach for microfield distributions in strongly-coupled plasmas can be found in [16]. An alternative approach for the microfield calculation in semi-classical two-component plasmas has bewen proposed by Ortner et al. [14]. The basic assumption is a nonlinear Debye-Hückel-type approximation which reads [ [ ( ea g ea (r, λ) =g ea (r, 0) exp iλk r exp r )]] r D

7 Contrib. Plasma Phys. 46, No. 3 (2006) 201 where again r D is the Debye screening parameter. As shown in [14] this relatively simple approximation yields (without using any adjustable parameter) a good agreement with microfield simulations for quasi-classical twocomponent plasmas up to Γ 2. Time-dependent statistical properties of the electric microfield seen by a neutral radiator were considered in [17]. Here, the above effective-field E approach provides the simple and reliable expressions for the Fourier transform of joint probability density of the microfield at the time t with respect the microfield at the initial time. The mean-force version of this approach has a range of validity which is restricted to the small Γ. At the same time, the used adjustable-parameter exponential approximation (APEX) for effective-field theory overcomes this defect and remains reasonably accurate in the intermediate coupling regime 1 Γ 10. In particular, APEX reproduced also the oscillatory behavior of the Fourier transform of the joint probability density which is connected with plasmon oscillations. Electric microfield dynamics in a charged point was investigated in [18]. The simple model for electric (micro)field dynamics studied here is based on the exact representation for impurity ion velocity response as that for viscoelastic medium. The electric field autocorrelation function was calculated. The results discussed above refer to the case of strongly-coupled one-component plasmas (OCP). This approximation was quite sucessful in accounting the electron action on lineshapes through electron-ion collisions (for a more detailed discussion see [6]). We note also that this notation (OCP) was used first in the work of Hooper [19]. Hooper also demonstrated first the transition from Holtsmark microfield distributions in an ideal plasma to the Gauss distribution in a strongly coupled OCP. The OCP model was generalized by a consequent account of the electron quasistatic input to the microfield distribution inside a sphere with Weisskopf-radius [7]. The distribution obtained for the electric microfield in two component plasmas [20] leads to a good agreement of atomic line wing shapes with experiments. Similar results were derived later by several authors as [8] and [21]. The main effects obtained in these investigations refer to the enlargement of strong microfield fluctuations in the tail of the distribution. A theoretical approach to these and other effects in two-component plasmas was given in [14] taking into account spatial correlations between oppositely charged particles in strongly coupled plasmas. It was shown that these effects lead to rather large local microfields even in comparision with the predictions of the Holtsmark theory. Due to the fact that a third particle may be located near to this (correlated) system, the actual microfield acting on one particle may be very large. In comparison with the OCP-theory, the second moment of two-component plasmas (TCP) microfield distribution diverges. In the statistical theory this divergence may be eliminated based on quantum effects [14]. Further it was shown in [14] theoretically, and confirmed by molecular dynamical (MD) simulations, that for Γ-values in the range between 0.2 and 2, the maximum of TCP microfield distributions is shifted to lower microfield values while the tail of distributions becomes even more fat than the tail of the Holtsmark distribution [14]. The MD dynamics of microfields was studied also by several other groups [22, 23]. In [22] a simple polarization model of ionic microfields was developed for the TCP. This model provided semi-quantivative results in comparison with known Monte-Carlo method and contained also an interesting generalisation of the APEX method for the TCP. In [23], MD simulations of three-body recombination processes in plasmas demonstrated first a de-acceleration of this process with increasing plasma density. Unfortunately, in this work no explicite results for the electric microfield in the plasma were given. The efficiency of various methods to calculate the electric microfield acting on an embedded ion was discussed in [24]. The electron charge density and electron electric fields at the ion can be studied accurately by MD simulations even at strong electron-ion coupling up to Γ=0.5. Conditions of applicability of APEX and Baranger-Moser approximation have been derived also. Note that the problem of electric plasma microfield is closely connected with the problem of large scale gravitational field distribution in the Universe (see the early and now classical work of Chandrasekhar [25]). On smaller (but still large) scales one observed inhomogeneities of the mass distributions; correspondingly the gravitational field appear to be non-holtsmark -distributed. Several OCP methods were used to explain the generation of primordial cosmological perturbations [26]. In fact, this work clearly demonstrated deviations from the Holtsmark distribution for systems on mesoscopic scales (see also [27]). Related problems with electric microfield distributions arise when the plasma shows structures on mesoscopic scales. The best example for these effects are the turbulent plasmas [28]. Here charges are distributed uniformly in turbulent streams, which produce the mesoscopic scale. It has been demonstrated that this scale generates a

8 202 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields Rayleigh distribution of the microfield, which looks like a Maxwell distribution for the quantity E/E R (E R is the characteristic field of Rayleigh noise). The final microfield distribution is a convolution of the Holtsmark distribution and the Rayleigh distribution. This was experimentally observed in Stark lines of hydrogen [31]. We do not go here into further details of the problem of electric microfield distributions since the goal of this work is not the study of microfields by itself, but the study of various elementary processes affected by the plasma microfields [29,30]. Let us note the principal difference between spectroscopic problems of ion lineshapes and the mentioned (multi-particle) elementary processes which will be studied here. Most of the elementary processes we have in mind here, are realized by particles with quite large energies. With respect to particle distributions, these processes are determined by the long tails of the particle distributions (with respect to energy or velocity). The classical example of this is the Gamov-Thompson cross-section for nuclear fusion [32]. Thus we will be interested to study the tails of the particle energy distribution, and, since the energy is determined by an electric microfield, by tails of this microfield distribution. We return to the problem of microfield distributions for situations when this distribution was not investigated in detail before: namely for magnetic microfields and for microfields in finite bodies. The above mentioned works considered the one-body (Stark) effect. At the same time, the effect of plasma microfields on such many-particle elementary processes like collisional and tunnel ionizations, rescattering, threebody electron-ion recombination, nuclear fusion in plasmas etc. was not yet studied in detail. Moreover, not only effects in plasma magnetic microfield but even the plasma magnetic microfield itself were not yet widely discussed. At the same time, characteristic values of electric and magnetic microfields in relatively dense hot plasmas are not small. For ideal plasmas, their amplitudes distributions are, as it was noted, of Holtsmark type. This result for a magnetic microfield was obtained recently [27]. The Holtsmark mean value of the magnetic H H microfield distributions (for an electron componet of plasmas) is about 9.0 en 2/3 v T /c where e is an electron charge, n is an electron plasmas density, v T is the mean heat velocity mvt 2 /2=T e, m is an electron mass, T e is an electron plasmas temperature, c is the speed of light. For hot plasmas T e 1 kev with moderate densities n cm 3 one finds H H 50 CGS. The fields in the tail of the Holtsmark distribution with such mean values may provide various observable physical effects, as for example, the large magnetic microfield provides the effect of laser radiation depolarization while this radiation has a long distance propagation through hot dense plasmas [34]. Many elementary processes like ionization of ions, three-body recombination occur in plasmas in presence of a plasma electric microfield. Even the characteristic value of the Holtsmark microfield E H gives already a good estimate to correct the known results for these processes. For tunnel ionization of ions by (strong) laser field, the naive parameter while an action of plasma electric microfield can be considerable is the ratio E H /E 0 where E 0 is the so-called inner-atomic electric field E 0 = e/rb 2 (r B is the Bohr radius). It is clear that this naive parameter is very small for most experimental situations. Nevertheless, the effective parameter of an electric microfield influences the process of tunnel ionization by laser fields [33] where E H E 0 /El 2 instead of the above naive parameter E l /E 0 (E l is an amplitude of laser field) can achieve values of several units. This is one example why an action of electric microfield (as well as a magnetic one) should be taken into account in calculations of elementary many-particle processes in plasmas. Similar qualitative discussions have been given for other multi-particle effects. Thus, the problem of microfield effects on elementary processes in plasmas has to be considered systematically. Above the mentioned classical works of Holtsmark and Griem, one should consider many interesting results concerning three-body and dielectronic recombinations [35, 36]. A review of the field of plasma physics scetched above will be given in the following. We underline again, that we do not claim to give a full survey of the existing work on microfield distributions, there exist already excellent reviews on this matter [16]. Therefore we may concentrate here on multi-particle processes influenced by microfields. The short survey of microfield distributions given above should provide only the material we need in the following to study the effects of microfields on elementary processes in the plasma. The effects of microfields on radiative processes are are also not in the center of our interest, in spite of the fact that we will discuss them here on several places relevant in the context of elementary processes.

9 Contrib. Plasma Phys. 46, No. 3 (2006) Ionization processes in electric microfields The three main important ionization processes which may be strongly influenced by electric microfields are: the collisional ionization by electrons, the tunnel ionization, and the rescattering processes. These three processes have in common that already weak electric microfields (corresponding to moderate plasma densities) can provide measurable corrections to ionizations rates. The most widespread ionization process in plasmas is the ionization by impacts with free electrons [37 39]. Other relevant processes are the tunnel ionization by the laser field [40 43] and the rescattering ionization [44 48]. The latter effects are of interest mainly for laser-produced plasmas. The influence of electric microfields on tunnel ionization and rescattering was considered, for example, in [33]. 2.1 Electron collisional ionization of ions Here we will study the effect of microfields in domains on the impact ionization [49]. Looking at the local values of the electric field, the plasma can be at any time decomposed into domains with certain direction of the electrical field, as schematically shown in Fig.1. The domain structures changes typically in time intervals of τ d ω 1 pl where the value ω pl is the corresponding plasma frequency. The characteristic size of the domain is about the Debye radius. It is clear that the local fields strongly influence all processes on time scales τ < τ d, as e.g. ionization processes. The first study of microfield effects on ionization processes was due to Inglis and Teller [4]. In accordance to this work the influence of the electric microfield in low density plasmas on the impact ionization should be quite small. As we will show here there exist additional effects of ionization rate enhancement due to the electric field in domains leading to measurable phenomena. Fig. 1 The instantaneous state of the plasma as a set of domains with different constant electric microfields. Each electron e passing a domain will change its short-time velocity distribution (v 0 is the initial velocity at the entrance into a domain, v 1 is the final velocity of electron after passing a number of domains ). Indeed, the electron impact ionization rate is determined by the energy of electrons at the time of collision as well as by the ionization threshold. As shown already by Inglis and Teller [4], the threshold energy depends on the actual strength of the electric microfield at the electron position. Here we will study another effect of microfield action based on particle acceleration in domains. The microfield can be considered to be constant within a domain with a length of the order of the Debye radius (see, for example, [50] for the microfield correlation length). The actual microfield in the problem is mainly due to the ions. It can be considered as stable while an electron passes this domain of the microfield. The initial velocity of impact electron at the entrance of a microfield domain is a random value (and should be distributed normally). The additional velocity (see Fig.1) acquired by the microfield inside the domain is determined by two random values: the time of electron acceleration inside the domain, i.e. at the length of microfield stability (meanvalue of them is the correlation length), and the amplitude. The product of these values

10 204 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields corrects the normal distribution of the initial velocity if the length of the domain is much smaller than the electron mean free path. The last requirement is fulfilled in all ideal plasmas. Corrections to the Maxwell distribution due to external electric fields in plasmas are well-known [51, 52]. The ionic microfield can be considered for electron acceleration as external due to its above mentioned stability: the process of electron acceleration and the following electron-ion collision is much faster than the inverse ion plasma frequency. We will show that the change of (initial) velocity of an electron inside the domain induced by the ionic microfield within one domain is of the order of Γ 3/4, where the nonideality parameter Γ is the ratio of mean energy of Coulombic interaction between particles to the plasma temperature. At the same time, the accuracy of ionization rates calculations with respect to atomic spectroscopy is sufficiently high to check for effects of order Γ and Γ 1/2 [54, 55]. It means that even in hot plasmas the action of microfield on the collision ionization process can be distinguished on the base of modern experimental and theoretical works. In nonideal plasmas, the problem of impact ionization due to strong microfields is essential but can be considered only numerically. The partial problem how relatively microfields correct the process of impact ionization will be considered in the present work Influence of electric microfields on the dynamics of electrons and on velocity distributions Let us consider the electron dynamics inside a given domain of the plasma microfield. At the entrance into a domain where a collision occurs, the impact electron has the velocity v 0. This value is distributed normally, since the transition between two domains is in fact the collision (so-called Weizsäcker collision [52]), which normalizes the distribution. Then the electron is accelerated by the (nearly) constant ionic microfield, the length of acceleration coincides with the microfield correlation length, which is of the order of the Debye radius [50]. The electron part of the plasma microfield can be considered as constant only for the very rapid pass of the electrons through the domain. Nevertheless, we find electrons which are able to ionize the ion, i.e. their kinetic energy is T,whereT is a plasma temperature. It means that both electron and ionic parts of the electric microfield are of importance for the problem under consideration. Then the electron velocity at the time of the collision is v = v 0 + eeτ (1) m where E is the total microfield inside the domain, e is an electron charge, m is an electron mass. The time τ is the ratio of minimal correlation length of ionic and electron microfields to the electron velocity. Since the electron plasma temperature T is normally larger the the ion one T i, one can write m Ti τ = 4πe 2 n i T and v = v 0 + ee m Ti m 4πe 2 (2) n i T Thus, the actual electron velocity at the time of collision is the sum of two random values: the normally distrubuted v 0,andδv E. In fact, the above differentiation of electron velocity is the attempt to account the action of all remained bodies on the process of two-body collisions. It is valid only in the case that this action is comparably small, i.e. in rare hot plasmas. The above physical approach has been considered first by Griem (see, for example, [6]). It means that the distribution of the value δv is of Holtsmark type. To obtain the final distribution of the value v, one needs to calculate first the distribution of all projection v x,..., and the distribution of the absolute value v. The distribution function of v 0x is f ξ (v 0x )= 1 ( ) exp v2 0x πvt vt 2 where v T is the mean thermal velocity. The distribution function of δv x is f η (δv x )= 1 ( ) tδvx ( cos exp t 3/2) dt 2πv E v E

11 Contrib. Plasma Phys. 46, No. 3 (2006) 205 where v E = ee H m Ti m 4πe 2 n i T. Further E H =2π(4/15) 2/3 e(n 2/3 i + n 2/3 e ) is the characteristic Holtsmark microfield [1]. The distribution function of the sum of two stochastic quantities η and ξ (with v x = v 0x + δv x )is f ζ (v x )= f ξ (v x x)f η (x)dx = 1 ( ) ) tvx cos exp ( t 3/2 v2 T t2 2πv E v E 4vE 2 dt = = 1 ( ) cos(tv x )exp v 3/2 E π t3/2 v2 T t2 dt (3) 0 4 We note that x is the projection of local field direction. Following the standard procedure to obtain the distribution function of the vector mean value v from the projection distributions (3), one finds the distribution function f ζ (v): f ζ (v) = 2v ( ) t sin(tv)exp v 3/2 E π t3/2 v2 T t2 dt (4) 0 4 It is easy to see that the expression (4) becomes the Maxwell distribution if v E =0, and it becomes a Holtsmark distribution if v T =0. The goal of these calculation is to find the distribution of v as f ζ (v) =f 0ζ (v) +δf(v). Since v T v E, one can write exp( v 3/2 E t3/2 ) 1 v 3/2 E t3/2. Than the respective integral in (4) can be evaluated using the parabolic cylinder functions, and we get f ζ (v) 4 πv 3 T v 2 exp ( v2 v 2 T ) + 15v3/2 E 4 2πv 5/2 (5) Fig. 2 The short-time distribution function f ζ (β) (solid curve). The normalized velocity is β = v v T,thevalueΓ is the parameter of plasma nonideality Γ= e2 n 1/3 i,here T Γ= 1. The dashed curve represents the drop of a pure 78 Maxwell distribution. The long tail is due to the domain acceleration effect. The second term in the right part is the exact asymptotic value according to the Holtsmark distribution [1]. This distribution has a strong tail for large v (see Fig.2). Note, that the second correction term for the Maxwell distribution is proportional to E 3/2. It is easy to show that the first term in the Taylor expanson of f ζ (v) with respect to ve v T E is equal to zero since it is proportional to the angle between v and E. We mention finally that the distribution function (5) is valid only in time intervals smaller than the time of domains existence. In the average over long times, the field effects disappear and the Maxwell distribution is valid again.

12 206 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields Corrections of electron impact ionization rates by electric microfields in weakly-coupled plasmas We will study now the impact ionization rates in weakly coupled plasmas using the classical approach. The number of inelastic collision of one electron with velocity v with ions is: ν = σ(v)vn i (6) where σ(v) is the cross-section of the corresponding inelastic process (an ionization in our case), n i is the ion density. In order to obtain the inelastic process rate, eq.(6) has to be averaged over all velocities exeeding the velocity v tr : ν =< ν >= n i v trσ(v)vf ζ (v)dv (7) where ɛ tr = mvtr 2 /2 is the inelastic process threshold energy. Taking, for example, for f ζ(v) the standard Maxwell distribution function and for σ(v) the Thompson cross-section, we arrive to well-known Zel dovich results [56] ( ν = <ν >= 2 ( πe 4 n i exp ɛ tr ) ( T ɛtr ) ) m 2 ɛ tr E 1 T T ( ) = 2 πe 4 n i T 2 m 2 ɛ 2 exp tr ɛ tr T. This case corresponds to the limit of ideal plasmas Γ=0. The problem of collision ionization rates in non-ideal plasmas has been investigated in [14]. As it was shown above, the electric microfield changes the distribution f ζ (v) by acceleration effects, the value ɛ tr by Inglis-Teller effects, and, therefore the ionization rate ν. The depression of the Coulombic barrier of electron is approximately ɛ tr 2 Ze 3 E,whereZ is an ion charge, i.e. one should introduce v tr where mv 2 tr 2 = mv2 tr 2 ɛ tr instead of v tr as the lower limit of the integral in (7). The determination of inelastic cross-section is a complicated problem which can be attacked both by quantum and by classical approaches. Let us remain in frames of the classical theory and extract the unperturbed (i.e. for Γ=0) value of the impact ionization rate and microfield corrections from it. Then we can re-write (7) as: ν = n i v tr σ(v)vf 0ζ (v)dv+ vtr n i σ(v)vf 0ζ (v)dv+ v tr n i v tr σ(v)vδf(v)dv = ν 0 + ν 1 + ν 2 (8) where the first term ν 0 is the unperturbed impact ionization rate (see [54 56]). Two other terms represent the desired corrections. Since the area of integration in ν 1 is small and the value of the cross-section near the threshold of inelastic process (impact ionization) is [57]: σ(ɛ) ɛ ɛtr C(ɛ ɛ tr )=C ɛ where C is a constant and ɛ = mv 2 /2 is the electron kinetic energy in the moment of collision. Producing an appropriate expansion of ν 1 into the Taylor series over the small parameter v E /v T, one can find that ν 1 = Cn i ɛv tr (9) 4 The expression ν 2 depends on the real form of the cross-section σ(v) and will be determined in the next section.

13 Contrib. Plasma Phys. 46, No. 3 (2006) Estimation of mean impact ionization rates using Thompson and Lotz cross-sections Exact calculations of respective cross-section are very complicated. Therefore we use two standard approximations for cross-section: Thompson formula which describes the cross-section of energy tranfer between two free electrons, and the semi-empirical Lotz formula of the first order [58]. Since we considered the ionization of only one electrons, the Thompson formula gives the value C T = 4πrB 2 ɛ2 H /ɛ3 tr where r B is the Bohr radius, ɛ H is the ionization potential energy of hydrogen. Calculating the value ν 2T, one can see that this value loses the exponential factor exp( ɛ/t ). It is convenient to write the final expression of the considered effect through the parameter of plasmas nonideality Γ. This way we get for the Thompson cross-section for (ν T ν 0T )/ν 0T : ( ) 1/4 ν T ν 0T T Γ 3/4 exp ν 0T ɛ tr ( v 2 tr v 2 T ) +8.2 Γ 2. (10) The second term which is determined by the Inglis-Teller effect is the correction to the first term. Taking the parameters of plasmas with Z =2, i.e. we consider, for example, the ionization of He+, T = T i =10eV,n i = cm 3, Γ= 1 78,and ν T ν 0T ν 0T This value is large enough to be distinguished by modern spectroscopic methods. The semi-empirical first order Lotz formula for impact ionization cross-section can be written as σ L =2.76πrB 2 ɛ 2 ( ) H ɛ ln. ɛɛ tr This expression leads to better coincidence with experiment than the Thompson formula. The value C L is now 2.76πrB 2 ɛ2 H /ɛ3 tr. Using the above fact ɛ tr T, one can find ( ) 5/4 ( ) ν L ν L0 T v 0.32 Γ 3/4 2 exp tr ν L0 ɛ tr vt Γ 2. (11) This gives for the same estimation of the He+ ionization ν L ν L ν L0 This value is by a factor of 2 larger than the above simple formula obtained in the Thompson approximation. Both estimations of impact ionization rate corrections demonstrate the relatively large influence of an electric microfield of plasmas on rates. The larger values obtained in the Lotz approximation in comparison to the Thompson one is connected with the slower drop of the Lotz cross-section for large energy of the impact electrons. Let us remind that the Lotz formula for ionization cross-section is much more applicable for real calculations due to its semi-empirical character. Thus the electric microfield of a plasma may distinguishably influence the electron impact ionization process. There are no direct experiments known to us, which would allow to compare the developed theory for the ionization with the measured data. Indeed, the calculations performed in this paper show the dependence of the ionization rate from the parameter of plasma nonideality. Note, that the considered approach can be used in an ideal plasma only, where Γ < 1. Moreover, the rather large values of ionization rates corrections (10, 11) demonstrate that the effect can be observed even in a very rare hot plasmas, i.e. while Γ Tunnel ionization of ions by laser fields influenced by electric microfields The tunnel ionisation occurs usually in strong laser fields. Since the amplitude of the electric microfield is much smaller than the laser field amplitude even for a sub-critical dense plasma, one expects that the influence of the microfield on the ionization process should be small, too. Nevertheless, the opposite case holds. The parameter for the influence of the microfield on the tunnel ionization occurs to be the value E 0 E m /El 2 instead of the naive ɛ tr

14 208 M.Yu. Romanovsky and W. Ebeling: Elementary many-particle processes in plasma microfields parameter E m /E l which one expects on the first glance. The value E 0 E m /El 2 is about unity for an intense laser radiation with the intensity (I W/cm 2 ) and a not very dense plasma. For example for a He+ plasma we may estimate an electron and ion density n e = n i cm 3 at which the microfield action becomes relevant. Moreover, in the next section it will be shown that the long tailed Holtsmark distribution induces a still greater influence of the microfield such that the microfield affects the ionization process at still lower densities. At the same time in the standard experiments for the measuring of the ionisation rate the action of the microfield may be neglected. The experiment reported in [44] (as well as other similar experiments) has been performed for extremely rare plasmas in order to avoid any collective processes ( normal collisions, etc). But the density region, when the tunnel ionization is the fastest process, is not limited by such extremely rare plasmas. It means, that the microfield effect on a tunnel ionization might be considerable and should be observable in not so rare plasmas which at the same time are for from being called dense plasmas (see our estimations below). Study first the modification of the tunnel ionization rate by the action of the plasma microfield. In this section we consider the region of plasmas densities less than cm 3. Then the problem of the electric microfield distribution essentially simplifies. Any laser produced plasma is hot enough and at the above densities the plasma is weakly coupled. As a result the distribution for both - the electron and ion component of the electric microfield - is the Holtsmark one. The relevant value in the problem under consideration is the distribution of the projection of the microfield vector on the direction of the laser field. (For the sake of simplicity we consider the case of linear polarized laser light.) The distribution of this projection E m is given by the expression (see 2.1.1) W (β) = 1 cos(β p x)exp( x 3/2 )dx (12) π 0 where β p = E m /E H. The time of stability of the electric microfield is about the inverse value to the corresponding plasma frequency. The time of stability is large at our conditions even for the electron component (comparing with the time of electron tunnel barrier penetration). Therefore, the total actual electric microfield is the sum of the electron and ion microfields since both these random values are stable random values [5] (see below). Thus the tunnel ionization process in the presence of the plasma microfield can be modelled by an ionization process in a constant electric field with an amplitude being the sum of the laser field amplitude E l and the microfield E m. The latter splits into electronic and ionic part, E m = E mi + E me, and it is characterized by the distribution (12). The probability of the tunnel field ionization reads in the case of a linear polarized laser radiation [43]: ( ) 3 16eI 2 2n ( ) [ 3/2 E w t = 8 1 2I π 3 E Z 2n+1 exp 2 ] 3 2I, (13) 3 E where I is the energy necessary for the corresponding ionization. The value E in this expression represents the sum of the laser field amplitude E l and the projection of the microfield amplitude on the direction of laser field vector. Averaging (13) with (12), we get for the tunnel ionization rate from the ground state w tg (the ionization rates for the excited states can be obtained by differentiation of the final expression with respect to the parameters I and E l ): 1 <w tg >= 4 6 E0 π 2 E5/3 0 dz exp( z 3 B 3/2 az)cosz 2 [cos az sin az]. (14) E l 0 Where a = 8/3 E 0 /E l with E 0 =(2I) 3/2,andB = E l /E h. The parameter a is supposed to be large. The parameter B is large in a dilute plasmas and for sufficiently strong laser intensities, in the opposite case it is small. Generally, the equation (14) can be analyzed only numerically. However, we may obtain simple analytical formulas for the ionization rate in two limiting regimes. First, in a dilute plasma, the transition to the ADK rate 1 Strictly speaking we should use the total field amplitude as the vector sum of the total microfield and the laser field instead of the projection on the laser field vector only. However, even in the case of a vanishing laser field when the deviation between the Eq. (14) and the exact ionization rate becomes maximal the approximate ionization rate according to Eq. (14) differs from the exactexpression by a factor of the order unity. Therefore we use the more simple expression Eq. (14).

15 Contrib. Plasma Phys. 46, No. 3 (2006) 209 <w tg0 > in (14) can been done by performing the limiting procedure a, B, or by ommiting the value B 3/2 z 3 under the exponent in the integral (14). The integration without the magnitude B 3/2 z 3 just gives the ADK value w tg0 without the influence of the microfield, i.e., it gives Eq. (13) with E = E l. Second, in a dense plasma we may calculate the ionization rate induced by the effect of the microfield only. In this case only the large values of the electric microfields contribute to a significant ionization rate. Therefore it suffices to consider the tail of the distribution of the microfield projection Eq. (12) which reads W (β p ) π β 5/2 (15) for large β p. Averaging Eq. (13) over the distribution Eq. (15), we get the ionization rate by the microfield only (i.e. in the absence of an external laser field) <w tgm >= 27 6e 2 8π 2 Z 3 E5/3 0 ( Eh E 0 ) 3/2. (16) Notice that the ionization rate induced by the microfield only is proportional to the electron density n. The full ionization rate <w tg > (Eq. (14)) induced by the combined action of the external laser field and the internal microfield lies above both the pure ADK rate (Eq. (13) with E = E l ) and the microfield ionization rate <w tgm > (Eq. (16)). In Fig.3 we show the ionization rate for the process He + He ++ (E 0 =8)in dependence on the laser intensity and on the charge density of the surrounding plasma. It is seen that the electric microfield prevails over the laser field up to enough large intensities; in range of densities of cm 3 it occurs up to the intensities of W/cm 2. In this range, no collective processes provide such large rate of ionization (except the rescattering, see below). Thus the action of the electric microfield on the tunnel ionization by laser radiation should be taken into account for a wide class of experiments with laser-produced plasmas. ionization rate (in a.u.) e-05 1e-06 1e-07 1e-08 1e-09 ADK n=10 15 cm -3 n=10 17 cm -3 n=10 19 cm -3 rescattering 1e-10 1e-11 1e+14 1e+15 1e+16 Intensity (W/cm 2 ) Fig. 3 The ionization rates for the tunnel ionization of single ionized helium induced by the combined action of the plasma microfield and the laser field are shown versus the laser intensity and for different plasma densities. The figure also shows the ionization rate of the rescattering channel at the beginning of the laser pulse (dotted line) and ten cycles after the begin of the laser puls (dashed-dotted line for n e =10 19 cm 3 ). All curves are shown for a wavelength of linear laser radiation λ =0.78µm