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 Articles you may be interested in Verification of particle simulation of radio frequency waves in fusion plasmas Phys. Plasmas 20, 102515 (2013); 10.1063/1.4826507 Electron and ion kinetic effects on non-linearly driven electron plasma and ion acoustic waves Phys. Plasmas 20, 032107 (2013); 10.1063/1.4794346 Nonlinear beam generated plasma waves as a source for enhanced plasma and ion acoustic lines Phys. Plasmas 18, 052107 (2011); 10.1063/1.3582084 Reduced kinetic description of weakly-driven plasma wavesa) Phys. Plasmas 15, 055911 (2008); 10.1063/1.2907777 Nonlinear plasma maser driven by electron beam instability Phys. Plasmas 6, 994 (1999); 10.1063/1.873340 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 PHYSICS OF PLASMAS 21, 040701 (2014) 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 À lÞ=kBTÞ the Fermi-Dirac distribution, 2 3=2 Àp =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 ÀcðkÞt linear effects in the quantum regime have been reported. wave electric field Eðr; tÞ¼E0e sinðÞkx À 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 1070-664X/2014/21(4)/040701/5/$30.00 21, 040701-1 VC 2014 AIP Publishing LLC 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 040701-2 Jer ome^ Daligault Phys. Plasmas 21, 040701 (2014) 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 À 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 À hk=2Þ v ðk; zÞ¼q À dp (2) Ð hz À hk Á p=m a 2 I3=2ðlÞkBT kBT 1 x where a ¼q;sc2 ¼c 3 ,whereIaðyÞ¼ dx I1=2ðlÞ m m 0 1þexpðxÀyÞ ð @ k Á f0ðpÞ is the Fermi integral. We see that Landau damping is less and ¼ À dp @p ; (3) sc;c z À k Á 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 ÀeE0k=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 degener- acy 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 vec- tor 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.