Landau Damping and the Onset of Particle Trapping in Quantum Plasmas Jérôme Daligault

Landau damping and the onset of particle trapping in quantum plasmas Jérôme Daligault

Citation: Physics of Plasmas (1994-present) 21, 040701 (2014); doi: 10.1063/1.4873378 View online: http://dx.doi.org/10.1063/1.4873378 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/4?ver=pdfcov Published by the AIP Publishing

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Landau damping and the onset of particle trapping in quantum plasmas Jer ome^ Daligault Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 14 February 2014; accepted 14 April 2014; published online 22 April 2014) Using analytical theory and simulations, we assess the impact of quantum effects on non-linear wave-particle interactions in quantum plasmas. We more specifically focus on the resonant interaction between Langmuir waves and electrons, which, in classical plasmas, lead to particle trapping. Two regimespffiffiffiffiffiffiffiffiffiffiffiffiffiffi are identified depending on the difference between the time scale of 2 oscillation tBðkÞ¼ m=eEk of a trapped electron and the quantum time scale tqðkÞ¼2m=hk related to recoil effect, where E and k are the wave amplitude and wave vector. In the classical- like regime, tBðkÞ < tqðkÞ, resonant electrons are trapped in the wave troughs and greatly affect the evolution of the system long before the wave has had time to Landau damp by a large amount according to linear theory. In the quantum regime, tBðkÞ > tqðkÞ, particle trapping is hampered by the finite recoil imparted to resonant electrons in their interactions with plasmons. VC 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4873378]

The notion of wave-particle interactions, the couplings the one-dimensional quantum Liouville-Poisson system; between collective and individual particle behaviors, is funda- they found that quantum effects can disrupt the hole forma- mental to our comprehension of plasma phenomenology.1–4 tion in phase-space that, in the classical case, coincides with Landau damping, the collisionless damping of an electron that saturation of Landau damping due to particle trapping. plasma (Langmuir) wave, is the paradigmatic illustration of In this paper, we examine the onset of resonant particle trap- these interactions.5 Resonant electrons with velocities suffi- ping in small amplitude plasma waves across quantum ciently close to the wave phase velocity experience a nearly degeneracy regimes. constant force and so can effectively exchange energy with To discriminate between quantum mechanical and quan- the wave. For waves of small amplitude about a uniform equi- tum statistical effects, we systematically report the results librium, the energy transfers overall result in the exponential for three models of electrons: quantum (q), semi-classical decay of the wave amplitude in time at a rate cðkÞ,wherek is (sc), and classical (c) (a subscript is sometimes used to indi- the wave number. In general, this linear theory prediction cate a result pertaining to a specific model, e.g., A ¼scB). breaks down after a time OðtBðkÞÞ beyond which particles can The classical description is commonly used in plasma get trapped and oscillate at a bounce frequency xBðkÞ physics, where both dynamics and statistics are classical. For ¼ 1=tBðkÞ in the wave troughs. Landau damping is effective semi-classical electrons, the dynamics is classical (wave-me- provided cðkÞxBðkÞ, whereas when cðkÞxBðkÞ,the chanical effects like diffraction are overlooked), but the fer- damping saturates and the wave amplitude remains finite mionic (Fermi-Dirac) statistics substitutes the Boltzmann (neglecting collisions).6,7 statistics. The quantum description includes both quantum The question arises as to how wave-particle interactions statistics and quantum mechanics by ascribing wave-like are modified when the quantum nature of the electrons can attributes to the electrons. In each case, we assume that ions no longer be ignored. Such is the case when the electrons’ do not participate in the high-frequency plasma oscillations thermal energy kBT is of the order of or smaller than their and act as a uniform neutralizing background. Moreover, h2 2 1=3 Fermi energy EF ¼ 2m ð3p nÞ (n and m are the electron collisions between individual electrons are omitted from the density and mass). The physics of quantum plasmas (e.g., of analysis12 and the plasma is described within thep collective,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 the warm dense matter regime) is a frontier of high-energy mean-field approximation. We denote by xp ¼ e n=0m density physics with relevance to many laboratory experi- the electron plasma frequency, a ¼ð3=4pnÞ1=3 the Wigner- 8 ments and to astrophysics. This field of research is being Seitz radius, rs ¼¼ a=aB ¼ 1(aB Bohr radius) the reduced emboldened by new experimental facilities and high- density, h ¼ kBT=EF the degeneracy parameter, fFDðpÞ¼ 2 performance computing. Nonlinear effects such as wave- 1=½1 þ expððp =2m lÞ=kBTÞ the Fermi-Dirac distribution, 2 3=2 p =2mkBT particle couplings have been ignored and studies rely on lin- and fMBðpÞ¼nkðÞBT=2pm e the Maxwell- ear response theory to model the experimental measurements Boltzmann distribution. (e.g., the X-ray Thomson scattering cross section9). In view First, we evaluate the quantum effects on the basis of of their significance in traditional plasmas, it is compelling the usual trapping threshold criterion cðkÞ¼xBðkÞ for reso- to consider the nature and the role that wave-particle cou- nant electrons in a plasma wave created by an initial density plings take on in quantum plasmas. Few studies of non- perturbation dnðrÞ=n ¼ cosðkxÞ. In the linear regime, the 10 cðkÞt linear effects in the quantum regime have been reported. wave electric field Eðr; tÞ¼E0e sinðÞkx xðkÞt x^. Here, 11 Of particular significance is the work by Suh et al. who E0 ¼en=0k, and xðkÞ and cðkÞ satisfy the dispersion report simulations of the non-linear Landau damping using equation

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2 0 ðk; xðkÞicðkÞÞ ¼ 0 ; (1) where ks ¼e v ð0; 0Þ=0 is the inverse screening length. Also shown in Fig. 1 is the long wavelength (Bohm-Gross) e2 0 where ðk; zÞ¼1 2 v ðk; zÞ is the analytic continuation 0k limit of retarded dielectric function in the random-phase approxi- ÀÁ 0 Im k; x k mation (RPA), and v is the free-electron density response 2 2 2 2 ðÞ2 ðÞð Þ 13,14 x ðkÞ¼xp 1 þ a k þ ok ; cðkÞ¼@ Re ; function @x ðÞk; xðkÞ ð (4) 0 f0ðp þ hk=2Þf0ðp hk=2Þ v ðk; zÞ¼q dp (2) Ð hz hk p=m a 2 I3=2ðlÞkBT kBT 1 x where a ¼q;sc2 ¼c 3 ,whereIaðyÞ¼ dx I1=2ðlÞ m m 0 1þexpðxyÞ ð @ k f0ðpÞ is the Fermi integral. We see that Landau damping is less and ¼ dp @p ; (3) sc;c z k p=m less effective as one penetrates the quantum regime: for a given wave number k, cqðkÞ < cscðkÞ < cclðkÞ and the gap h with f0p¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;sc fFD¼cfMB. Theffiffi bounce frequency is between these rates increases with decreasing . Thus, in light p 6 xBðkÞ¼ eE0k=m ¼ xp . To determine cðkÞ, we solve of the usual trapping criterion cðkÞxBðkÞ, one expects that numerically the dispersion relation for each electron model. resonant electrons in quantum plasmas are more readily prone to trapping than in classical plasmas. Moreover, Fig. 1(b) The results for xðkÞ and cðkÞ for rs ¼ 1 and several degeneracy parameters h across degeneracy regimes are shown in shows that, while the quantum and semiclassical descriptions are equal at small k, they markedly differ at intermediate k: Fig. 1 as a function of the dimensionless wave-vector k=ksc, this suggests that the electron wave-like character signifi- cantly modifies the nature of traditional wave-particle interactions. To support the previous analysis, we perform numerical simulations for semi-classical electrons across the quantum degeneracy regimes. Like in traditional plasma physics, electrons are modeled by a phase-space distribution f(r, p,t) satisfying the Vlasov-Poisson equations

@f p ¼ r f þr v r f ; (5a) @t m r r p ð e r2vðr; tÞ¼ dpf ðr; p; tÞn : (5b) 0

But unlike in plasma physics, we use an initial condition consistent with the Pauli principle, namely,

f ðr; p; t ¼ 0Þ¼ðÞ1 þ cosðÞkx fFDðpÞ ; (6)

which describes a sinusoidal density perturbation (wave vector k ¼ kx^, amplitude ) around the homogenous Fermi- Dirac equilibrium.15 Note that the fermionic character is preserved since the Vlasov dynamics (5a) conserves phase- space volumes. Our numerical solution uses particle-in-cell techniques with parameters carefully chosen to ensure energy and entropy conservation (and hence the fermionic character), and to alleviate spurious noise due to particle dis- creteness.16 Figure 2 shows the results of simulations for the time evolution of the Fourier component of the electric field Eðk; tÞ¼ik~vðk; tÞ obtained for rs ¼ 1, ¼ 0:005; h ¼ 0:01; h ¼ 0:1 (degenerate), and 7.5 (classical regime). FIG. 1. Plasma wave dispersion relation (left) and linear Landau damping Results are shown for several wave-vectors k around the rates (right) for reduced density r 1 and degeneracy parameters between s ¼ threshold value k ðhÞ defined such as cscðk Þ¼xBðk Þ and h ¼ 7:5 (nearly classical) to h ¼ 0:1 (strongly degenerate), obtained by below which the usual trapping criterion c < xB is satisfied; numerically solving Eq. (1) for quantum (red lines) and semiclassical * (black). In both figures, the dashed lines corresponds to the Bohm-Gross in Fig. 1, k is determined by the intersection of the horizon- limit (4) for quantum electrons, the blue crosses to numerical semi-classical tal dashed line with the black line, giving k ðhÞ=ks PIC simulations, and the curves have been shifted vertically (left) and hori- ¼ 0:4; 0:97 and 2.9 for h ¼ 7:5, 0.1, and 0.01. For k > k, zontally (right) for clarity (in the right figure, a vertical arrow indicates the after a fast transient time due to phase mixing, the electric origin of the horizontal axis). For h ¼ 7:5, the quantum, semi-classical, and classical (not shown) results are indistinguishable to within the thickness of field oscillates and its envelope decays exponentially in time the curve. with frequency and damping time in excellent agreement

with linear theory (see blue crosses in Fig. 1). In contrast, k. For k k, in the time asymptotic limit, the oscillation in when k < k, the wave amplitude displays a preliminary ex- the electric ﬁeld envelope disappears and the wave goes on ponential Landau decay (again in remarkable accordance propagating at nearly constant amplitude (e.g., see Fig. 2 for with linear theory as shown in Fig. 1), but then the latter sat- h ¼ 0:01). These are signatures of trapping:6,17 nonlinear urates: the amplitude starts oscillating on a small frequency effects stop the damping at a threshold in agreement with the time scale, the amplitude of which increases with decreasing theory, even for small amplitude waves, long before the wave has had a chance to damp by a substantial amount according to linear theory. We now investigate whether quantum mechanics affects the usual trapping criterion. To facilitate the comparison between the quantum and classical descriptions, we work with the Wigner representation in which the system’s state is described by a phase-space function f(r, p, t), which, in 10,18 many ways, resembles the classicalÐ distribution: for example, the localÐ particle density n ¼ dp f , momen- tumÐ density P ¼ dpp f , and kinetic energy density p2 K ¼ dp 2m f . In the mean-ﬁeld approximation

@f p @f ¼ þ I½f ; (7) @t m @r

where ð ð i 0 dy iyp0 0 I½f ¼q dp e f ðr; p þ p ; tÞ h ð2pÞ3 ½vðr þ hy=2Þvðr hy=2Þ (8a)

@v @f ¼ ðcf:Eq:ð5aÞÞ; (8b) sc;c @r @p

where v satisﬁes Eq. (5b).19 Following O’Neil,6 we determine the onset of particle trapping by looking at the break- down of the linear approximation to the theory Eq. (8).We consider the case in which a monochromatic plasma wave is excited with Eðr; tÞ¼EðtÞsinðÞk r xðkÞt x^, as produced by the initial condition (6) with f0 ¼q;sc fFD ¼c fMB. To make analytical predictions, we assume that the wave amplitude remains constant on time scale t < 1=cðkÞ, i.e., E(t) ¼ E0 for t < 1=cðkÞ. Under these conditions, the linear solution of Eq. (7) is

f ðr; p; tÞ¼½1 þ cosðÞk ðr p t=mÞ f ðpÞ 0 k p cosðÞk r xðkÞt cos k r t eE0 m þ ; k k p xðkÞ 8 m > > 1 hk hk > f0 p þ f0 p ðÞq < h 2 2 > > @f0ðpÞ : k ðÞsc; c @p FIG. 2. Temporal evolution of the self-consistent electric field E(k, t) for (9) semi-classical plasmas at reduced density rs ¼ 1 and degeneracy h ¼ 7:5 (left), h ¼ 0:1 (middle), and h ¼ 0:01 (right), following an initial density perturbation nðr; tÞ=n ¼ 0:005 cosðkxÞ at wave-vectors k distributed around with f0 ¼q;sc fFD ¼c fMB; the first term describes the free- the trapping threshold value kðhÞ (see Fig. 1). Note that over the time dura- streaming of the initial perturbation, while the second term tion of the plot for h ¼ 7:5 and 0.1, the plasma oscillations are undistin- originates from the electron-wave couplings. For momenta guishable. Measured initial Landau decay rates and wave frequencies shown kp in Fig. 1 (blue crosses) are in excellent agreement with the dispersion rela- such that m ¼ xðkÞ, the linear solution (9) exhibits a secu- tion results. larity proportional to t and we find

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 192.12.184.6 On: Wed, 12 Nov 2014 17:34:43 040701-4 Jer ome^ Daligault Phys. Plasmas 21, 040701 (2014) ÀÁ I½f I½f f ðp þ hkÞ2f ðpÞþf ðp hkÞ eE sin hk 2t=2m 0 ðr; p ¼; tÞ¼ sinðÞk r xt 0 0 0 0 q 2 2 I½f0 f0ðp þ hk=2Þf0ðp hk=2Þ k h k =2m "#ÀÁ (10a) f ðp þ hkÞf ðp hkÞ eE cos hk2t=2m 1 þ cosðÞk r xt 0 0 0 2 2 f0ðp þ hk=2Þf0ðp hk=2Þ k h k =2m

2 2 2 @ f0=@ px eE0k t ¼ sc;csinðÞk r xt eE0t þ cosðÞk r xt : (10b) @f0=@px m 2

6 For classical electrons, Eq. (10b) exhibits the usual result: as t is the explicitÐ expression of c given in Eq. (4) with 1=2 approaches tBðkÞ¼ðÞm=ekE (see second term proportional gðpxÞ¼ dpydpzf ðpx; py; pzÞ. Moreover, energy conservation 2 to t ), dI ¼ I½f I½f0 outgrows the linear term and the linear implies dhKi=dt ¼dhEi=dt. By defining the number of approximation breaks down. For semi-classical electrons, Eq. energy quanta N k ¼hEi=hxðkÞ in the wave, the previous (10b) shows that the same result holds when quantum statistics rate equations become alone is accounted for. The quantum result (10a), on the other 2 dhKi dN dhPi dN hand, displays an additional time scale tqðkÞ¼2m=hk .For ¼hxðkÞ k ; ¼hk k ; (14) tqðkÞtBðkÞ, i.e., for density amplitudes dt dt dt dt 4 ðakÞ 4 and ðk; rsÞ/k =n; (11) 12rs dN k ¼2cðkÞN : (15) the quantum result (10a) reduces to the classical (10b) for dt k times t of order tB(k): the linear approximation breaks down as t approaches tB(k). For tqðkÞtBðkÞ or ðk; rsÞ, Thus, the quantum of plasma wave behaves as a quasi-particle, however, the term within brackets limits the growth of dI the plasmon: the increase or decrease in the wave energy by and the linear approximation does not break down as t one quantum is accompanied by the absorption or emission of approaches tB(k): the usual trapping criterion does not hold. electron energy hxðkÞ and momentum hk. Equation (15) In fact, as we shall explain, the recoil phenomenon due to describes the stimulated absorption and emission of plasmons the interaction of resonant electrons with plasmons prevents leading to the Landau damping of the plasma wave at the rate particle trapping from happening. cðkÞ (the factor 2 in Eq. (15) arises because N k is proportional To interpret physically the new time scale tq(k), we find it to the square of the wave amplitude). Note that in Eq. (15),we useful to go over from the wave description of plasma waves to have omitted the term describing the (spontaneous) plasmon the particle description in terms of plasmons.20 The latter is not emission by electrons in the form a wake behind them, which widely used in plasma physics, perhaps because typical electro- requires fast electrons because xðkÞ=k is usually large, and is static waves contain a huge number of these elementary excita- thus negligible at or near equilibrium.23 tions. Yet, as we argue here, plasmons provide an insightful We now return to our purpose in light of the plasmon alternative picture of Landau damping and particle trapping concept. From momentum and energy conservation, for an valid for both quantum and classical electrons. The notion of electron to absorb (þ)oremit() a plasmon ðhk; hxðkÞÞ, its plasmons, first derived from the collective coordinate method momentum p must satisfy of Bohm and Pines,21 is readily recovered within our approach as follows. We calculate the rate of change in kinetic energy k xðkÞ hk xðkÞ p ¼ p ¼q m 7 ¼sc;c m ; (16) hKi and momentum hPi per wavelength 2p=k from the linear k k 2 k solution (9); in the long-wavelength limit, we find22 in the process, its energy changes by dhKi dhPi k ¼ 2cðkÞhEi; ¼ 2cðkÞhEi ; (12) dt dt xðkÞ p h2k2 p 6 hxðkÞ¼q 6 hk þ ¼sc;c 6 hk : (17) 2 m 2m m where hEi ¼ 0E0=4 is the electrostatic wave energy per wavelength, and This simple calculation reveals two important effects of the wave-like, diffractive nature of electrons. First, the resonant 2pnk mxðkÞ hk mxðkÞ hk cðkÞ¼ g g þ momentum condition is shifted by 7 hk =2 with respect to q hx3 FD k 2 FD k 2 p the classical and semi-classical conditions; this in turn 2pnk2 mxðkÞ affects the relation of the Landau damping rate to the shape ¼ g0 sc x3 FD k of the distribution function around mxðkÞ=k as found in Eq. p 2 (13) and Fig. 1. Second, the energy transfer differs by the 2pnk mxðkÞ h2k2 0 recoil energy ErecðkÞ¼ associated with the momentum ¼c 3 gMB ; (13) 2m xp k transfer 6hk. The quantum time-scale found earlier is

4 intimately related to recoil since tqðkÞ¼h=ErecðkÞ and the D. S. Montgomery, J. A. Cobble, J. C. Fernandez, R. J. Focia, R. P. two regimes delineated by can be understood as follows. Johnson, N. Renard-LeGalloudec, H. A. Rose, and D. A. Russell, Phys. Plasmas 9, 2311 (2002). 5 In the wave frame, a trapped electron oscillates in a potential2 L. D. Landau, J. Phys (USSR) 10, 25 (1946). VtrapðkÞ tqðkÞ 6 well of height VtrapðkÞ¼2eE0=k such that ¼ . T. O’Neil, Phys. Fluids 8, 2255 (1965). ErecðkÞ tBðkÞ Thus, for amplitudes smaller than 7J. R. Danielson, F. Anderegg, and C. F. Driscoll, Phys. Rev. Lett. 92, ðk; r Þ; V ðkÞE ðkÞ, and the recoil of a resonant elec- 245003 (2004). s trap rec 8P. Drake, Phys. Today 63(6), 28 (2010). tron that absorbs or emits a plasmon, kicks it outside the 9S. H. Glenzer and R. Redmer, Rev. Mod. Phys. 81, 1625 (2009). potential well: quantum diffraction lowers the trapping life- 10F. Haas, Quantum Plasmas: An Hydrodynamic Approach (Springer, time of resonant particles and deteriorates trapping overall. 2011). 11N.-D. Suh, M. R. Feix, and P. Bertrand, J. Comp. Phys. 94, 403 (1991). When Eq. (11) is satisfied, VtrapðkÞ > ErecðkÞ, most elec- 12The validity of the collective, mean-field description requires that the trons can remain trapped by the wave and the usual theory mean collision time for electron collisions, which tend to disrupt the col- remains qualitatively valid. lective motion, be large compared to the Landau damping time scale In summary, our study reveals two distinct regimes for the 1=cðkÞ of interest here. 13S. Ichimaru, Statistical Plasma Physics (Westview Press, 2004). collisionless damping of a monochromatic plasma wave in 14D. Kremp, M. Schlanges, and W.-D. Kraeft, Quantum Statistics of quantum plasmas. First, a classical-like regime, where the non- Nonideal Plasmas (Springer, 2005). linear trapped-particle dynamics of resonant electrons, qualita- 15This semiclassical framework extends to dynamical phenomena, the cele- tively akin to that well-known in classical plasmas, sets in long brated Thomas-Fermi approximation. 16For similar calculations in classical plasmas, see F. Valentini, Phys. before the wave has had time to damp by a substantial amount. Plasmas 15, 022102 (2008) and references therein. Second, a truly quantum regime, where particle trapping is 17R. C. Davidson, Methods in Nonlinear Plasma Theory (Academic Press, hampered by the finite recoil imparted to resonant electrons in 1972). 18 their interactions with the wave. Given the importance of wave- H. Hillery, R. F. O’Connell, M. O. Scully, and E. R. Wigner, Phys. Rep. 106, 121 (1984). particle couplings in traditional plasmas, further work to com- 19Eq. (8a) with Eq. (5b) is simply the Hartree approximation, the quantum prehend them in quantum plasmas may be beneficial both from extension of the Vlasov approximation. One could add to v a term that a fundamental physics standpoint and as a practical matter. accounts for electron exchange (in the form of a density dependent potential) but since in the remaining of the present analysis we assume that the wave amplitude remains constant (v(r) is fixed) on the time scale of inter- This research was supported by the DOE Office of Fusion est, we do not discuss this here. Energy Sciences – NNSA joint program in High-Energy Density 20D. Pines and J. R. Schrieffer, Phys. Rev. 125, 804 (1962). Laboratory Physics. The author wishes to thank Dr. Zehua Guo 21D. Pines, Rev. Mod. Phys. 28, 184 (1956). 22 and Dr. Scott Baalrud for many useful conversations. In deriving Eq. (12), we dropped the free-streaming term that quickly phase mixes to zero, and set sinðÞt = pdðÞ with v ¼ xðkÞk p=m. 23 1 Accounting for spontaneous emission, Eq. (15) becomes T. M. O’Neil and F. V. Coroniti, Rev. Mod. Phys. 71, S404 (1999). dN k sp ¼ 2cðkÞN k þ C ðkÞ, where CspðkÞ is the rate of spontaneous plasmon 2B. T. Tsurutani and G. S. Lakhina, Rev. Geophys. 35, 491, dt emission. A traditional stopping-power calculation gives doi:10.1029/97RG02200 (1997). Ð 2 3 2 e2 2 kp e m xðkÞ xðkÞ 3 CspðkÞ¼ 2 xðkÞ dpf ðpÞdxðkÞ ¼ 3 g , which is P. H. Diamond, S.-I. Itoh, and K. Itoh, Modern Plasma Physics 0k m k k (Cambridge University Press, 2010). negligible for a thermal distributions.13