Quantum Interference and Phase Mixing in Multistream Plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Quantum Interference and Phase Mixing in Multistream Plasmas
M. Akbari-Moghanjoughi Faculty of Sciences, Department of Physics,Azarbaijan Shahid Madani University, 51745-406 Tabriz, Iran (Dated: February 12, 2021)
Abstract
In this paper the kinetic corrected Schr¨odinger-Poisson model is used to obtain the pseudoforcesystem in order to study variety of streaming electron beam-plasmon interaction effects. The nonin-teracting stream model is used to investigate the quantum electron beam interference and electronfluid Aharonov-Bohm effects. The model is further extended to interacting two-stream quantumfluid model in order to investigate the orbital quasiparticle velocity, acceleration and streamingpower. It is shown that quantum phase mixing in the two-stream model is due to quasiparticleconduction band overlap caused by the Doppler shift in streaming electron de Broglie wavenumbers,a phenomenon which is also known to be a cause for two-stream plasma instability. However, inthis case the phase mixing leads to some novel phenomena like stream merging and backscattering.To show the effectiveness of model, it is used to investigate the electron beam-phonon and electronbeam-lattice interactions in different beam, ion and lattice parametric configurations. Current den-sity of beam is studied in spatially stable and damping quasiparticle orbital for different symmetricand asymmetric momentum-density arrangements. These basic models may be helpful in betterunderstanding of quantum phase mixing and scattering at quantum level and can be elaborated tostudy electromagnetic electron beam-plasmon interactions in complex quantum plasmas.
PACS numbers: 52.30.-q,71.10.Ca, 05.30.-d . INTRODUCTION Plasmons are high frequency elementary quantized excitations of plasma electron oscilla-tions [1, 2]. They play inevitable role in many fundamental properties of plasmas semicon-ductors and metallic nanoparticles from electric and heat transport phenomena to opticaland dielectric response, etc. [3, 4]. Dynamics of these quantized electromagnetic quasiparti-cles make an ideal platform for miniaturization of ultrafast terahertz device communications[5], where conventional integrated circuits fail to operate. They also have numerous otherinteresting applications in nanotechnology [6], plasmonics [7–9], optoelectronics [10], etc. forengineering low-dimensional nano-fabricated semiconductor industry [11–13]. Energy con-version by plasmons is an new way of solar power extraction due to its high efficiency inphotovoltaic and catalytic devices. Use of the collective oscillations of electrons instead ofsingle particles makes huge amount of energy extraction in an operation step in plasmonicsolar devices [14, 15]. Collective oscillations in local surface plasmon resonance (LSPR)[16].process by surface electrons which are so called hot electrons are collectively excited byelectromagnetic radiations in UV-VIS range. The hot electron current are collected in anappropriate contacts of nanoparticle surfaces by an efficient electron collecting material like
T iO in Schottky configuration [17].Collective charge screening effect which manifests itself as the characteristic optical edgein metallic surfaces already have may applications in metallic alloys making then opticallyunique among other solids. Collective electron excitations rule almost every aspect of solidfrom optical to dielectric response in plasmas [18–20] and condensed matter. Recent infraredspectroscopic techniques shows that Low dimensional semiconductors [21] such as gappedgraphene also demonstrate interesting surface plasmon effects. The collective electron trans-port property of graphene makes it an ideal element for multilayer composite devices suchas compact ultrafast switches, optical modulators, optical lattices, photodetectors, tandemsolar cells and biosensors [22–24]. The first theoretical development of the idea of collec-tive electron excitations by Bohm and Pines dates back to mid-nineties, when they coinedplasmon name for such excitations due to the long-range electromagnetic nature of interac-tions [25–29]. The theoretical as well as experimental aspects of collective electron dynamicsin quantum level has been the subject of intense investigations over the past few decades[30–35], due to its fundamental importance in many field of physics and chemistry.2ioneering developments of quantum statistical and kinetic theories [36–39] had a longtradition furnishing a pavement for modern theories of quantum plasmas [40 ? –43]. Manyinteresting new aspects of collective quantum effects in astrophysical and laboratory plas-mas has been recently investigated using quantum plasma theories [44–57]. The quantumkinetic theories like time-dependent density functional theories (TDFT) are, however, lessanalytic as compared to the quantum hydrodynamic analogues, due mostly to mathemati-cal complexity which require large scale computational programming. Recent investigationreveals [58] that quantum hydrodynamic approaches based on the density functional formal-ism [43] can reach beyond the previously thought kinetic limitations, such as the collisionlessdamping if accurately formulated. One of the most effective hydrodynamic formalism forstudying the quantum aspects of plasmas is the Schr¨odinger-Poisson model [59, 60], basedon the Madelung quantum fluid theory which originally attempted for the single-electronquantum fluid modeling [61]. It has been recently shown that the analytic investigation oflinearized Schr¨odinger-Poisson system for arbitrary degenerate electron gas provides routesto some novel quantum feature of collective plasmon excitations [62, 63]. In current studywe use this model in order investigate the quantum interference and phase mixing in elec-tron gases with arbitrary degree of degeneracy, based on the collective quasiparticle orbitalconcept based on pseudoforce formulation. II. THE QUANTUM HYDRODYNAMIC MODEL
In order to investigate the dynamics of quantum plasma we consider the following quan-tum hydrodynamic equations consisting of the continuity, force balance including the quan-tum force and the Poisson’s equation [42] ∂n s ∂t + ∂n s v s ∂x = 0 , (1a) ∂v s ∂t + v s ∂v s ∂x = q s m s ∂φ∂x − m s ∂µ s ∂x + γ s ¯ h m s ∂∂x (cid:18) √ n s ∂ √ n s ∂x (cid:19) , (1b) ∂ φ∂x = − π X s q s n s , (1c)where the index s refers to streams, q s is the charge state of given stream, n s denotesthe stream number density, v s the fluid velocity, φ , the electrostatic potential and µ s , thechemical potential of given stream in the quantum fluid. Also, ¯ h is the reduced Planck3onstant and m s is the stream species mass. The chemical potential-temperature dependentparameter γ s denotes the kinetic correction [54, 62, 64, 65] to plasmon excitations in longwavelength and low phase speed plasma regime, which is given as γ s = Li / [ − exp( βµ s )] Li − / [ − exp( βµ s )]3Li / [ − exp( βµ s )] , (2)in which β = 1 /k B T , µ being the equilibrium chemical potential and the polylog functionLi is defined through the Fermi integrals F k ( η ) = Z ∞ x k exp( x − η ) + 1 d = − Γ( k + 1)Li k +1 [ − exp( η )] x, (3)where Γ is the gamma function. The fundamental thermodynamic quantities depend throughthe equation of state (EOS) which is used to close the hydrodynamic system. Here we choosethe isothermal EOS which is appropriate in the low phase speed plasmon phenomenon [66] n s ( η s , T ) = 2 / πm / s h F / ( η ) = − / ( πm s k B T ) / h Li / [ − exp( η s )] , (4a) P s ( η s , T ) = 2 / πm / s h F / ( η ) = − / ( πm s k B T ) / ( k B T ) h Li / [ − exp( η s )] , (4b)in which P s is the statistical pressure of given stream satisfying, ∂P s /∂µ s = n s , and η s = βµ s .The hydrodynamic system (1) is used to formulate an effective Schr¨odinger-Poisson model[59] for the system using the Madelung transformation [61] i ¯ h √ γ s ∂ N s ∂t = − γ s ¯ h m s ∂ N s ∂x + qφ N s + µ s ( n s , T ) N s , (5a) ∂ φ∂x = − π X s q s |N s | , (5b)where N s = p n s ( x, t ) exp[ iS s ( x, t ) / ¯ h √ γ s ] is the state-function characterizing the collectivequasiparticle excitations of each stream in the gas. The equivalence between (1) and (5)may be readily examined by separating the real and imaginary parts for each stream, whichfollows m s ∂n s ∂t + ∂n s ∂x ∂S s ∂x + n s ∂ S s ∂x = 0 , (6a) ∂ S s ∂t∂x + 1 m s ∂S s ∂x ∂ S s ∂x = − q s ∂φ∂x − ∂µ s ∂x + ∂B s ∂x , (6b) B s = γ s ¯ h m s n s " n s ∂ n s ∂x − (cid:18) ∂n s ∂x (cid:19) , (6c)4hich reduce to first two equations in (1) by the definition, v s = (1 /m s ) ∂S s /∂x . Notealso that, the fluid velocity of plasma species satisfy the relation v s ( x, t ) = j s ( x, t ) /n s ( x, t ),where, j s ( x, t ) = i ¯ h √ γ s / (2 m )[ ∂ N s ( x, t ) /∂x × N ∗ s ( x, t ) − ∂ N ∗ s ( x, t ) /∂x × N s ( x, t )] is the cur-rent density of given specie. The fluid velocity is briefly given by the relation v s ( x, t ) =(¯ h √ γ s /m )Im[ ∂ N s ( x, t ) /∂x/ N s ( x, t )], which is identical in the form to the velocity of par-ticles in pilot-wave guiding equation. Let us first consider monokinetic streams withthe action S s ( x, t ) = p s x + f s ( t ) with p s = m s v s and f s ( t ) being an arbitrary func-tion of time. The space-time separated multistream Schr¨odinger-Poisson system, assuming N s ( x, t ) = ψ s ( x ) T s ( t ) exp[ i ( p s x − f s t ) / ¯ h √ γ s ], then reads (cid:26) γ s ¯ h m s d dx − q s φ ( x ) + ǫ − µ s (cid:27) ψ s ( x )exp (cid:18) ik s x √ γ s (cid:19) = 0 , (7a) d φ ( x ) dx = − π X s q s ψ s ( x ) , (7b) i ¯ h √ γ s dT s ( t ) dt = ǫT s ( t ) e − iω s ( t ) , (7c)where ǫ is the energy eigenvalue of the multistream quantum system, k s = p s / ¯ h being the s -th stream de Broglie’s wavenumber and ω s ( t ) = − f s ( t ) / ¯ h √ γ . III. ELECTRON BEAM INTERFERENCE EFFECTS
For our illustration purpose we would like to study the two stream electron interferenceeffect with γ = γ = γ . To this end, we linearize the system (7) using the perturbations ψ s = ψ + ψ , φ = 0 + φ and p = 0 + p which leads to the following driven coupledpseudoforce system after linearization [67] γ d Ξ( x ) dx + Φ( x ) + E Ξ( x ) = (8a) α ( k − E ) exp (cid:18) ik x √ γ (cid:19) + (1 − α )( k − E ) exp (cid:18) ik x √ γ (cid:19) , (8b) d Φ( x ) dx = Ξ( x ) , (8c)where Ξ( x ) = α Ψ ( x ) + (1 − α )Ψ ( x ) is the complete wavefunction with α = ω p /ω p beingthe fractional plasmon frequency of streams. We have used the Thomas-Fermi assumptionin which the chemical potential variations are neglected ( µ = µ = µ ) and the temperatureis fixed in agreement with the single-electron Fermi-Dirac distribution [68, 69]. Further more5e have used the normalization scheme in which E = ( ǫ − µ ) /E p with E p = ¯ hω p being theplasmon energy and ω p = p πe n /m the plasmon frequency. Also, Ψ ( x ) = ψ ( x ) / √ n ,Ψ ( x ) = ψ ( x ) / √ n and Φ( x ) = eφ ( x ) /E p . The space, x and time t variables are, respec-tively, normalized with the inverse of the plasmon wavenumber, k p = p mE p / ¯ h and inverseof plasmon frequency. Note that the quasineutrality condition n + n = n holds. Notethat while the model (8) is appropriate for resonant electron-plasmon interactions to studythe beam-interference effect it ignores the interaction between the electron beams. In theproceeding sections we will use a generalized model to study such effects. Assuming thenontransient solution to the system (8) readsΞ( x ) = αB e ik x/ √ γ + (1 − α ) B e ik x/ √ γ , B j = − k j ( k j − E ) γ + k j ( k j − E ) , (9a)Φ( x ) = αA e ik x/ √ γ + (1 − α ) A e ik x/ √ γ , A j = γ ( k j − E ) γ + k j ( k j − E ) ., (9b)Note that the kinetic correction in the normalized form reads γ = Li / [ − exp( σ/θ )] Li − / [ − exp( σ/θ )]3Li / [ − exp( σ/θ )] , (10)in which θ = T /T p with T p = E p /k B being the characteristic plasmon temperature. Thenumber density distribution corresponding to quasiparticle orbital, E , is given by n ( E ) = Ξ( x )Ξ ∗ ( x ) = α B + (1 − α ) B + 2 α (1 − α ) B B cos (cid:20) ( k − k ) x √ γ (cid:21) . (11)The corresponding current density, on the other hand, reads j ( E ) = i ¯ h √ γ m (cid:20) Ξ( x ) ∂ Ξ ∗ ( x ) ∂x − Ξ ∗ ( x ) ∂ Ξ( x ) ∂x (cid:21) (12a) j ( E ) = ¯ hm (cid:26) α B k + (1 − α ) B k + α (1 − α ) ( k + k ) B B cos (cid:20) ( k − k ) x √ γ (cid:21)(cid:27) . (12b)The quasiparticle velocity corresponding to the energy orbital level, E , may be written inthe following form v ( E ) = j ( x ) n ( x ) = i ¯ h √ γ m (cid:20) ∗ ( x ) ∂ Ξ ∗ ( x ) ∂x − x ) ∂ Ξ( x ) ∂x (cid:21) = ¯ h √ γm Im (cid:20) d Ξ( x ) /dx Ξ( x ) (cid:21) , (13)Note that the quasiparticle orbital velocity is the same as the velocity of guiding equationin pilot wave theory in Bohmian quantum mechanics [70, 71]. For our two stream electrongas model, the quasiparticle orbital velocity reads v ( E ) = (cid:18) ¯ hm (cid:19) α B k + (1 − α ) B k + α (1 − α ) ( k + k ) B B cos h ( k − k ) x √ γ i α B + (1 − α ) B + 2 α (1 − α ) B B cos h ( k − k ) x √ γ i . (14)6or the case of single stream with the plane-wave wavefunction solution, ( α = 0 , k = k , the result also reduces to the single stream case with constant beam velocity.Equations (11) and (12) clearly indicate that the quantum interference of two electron beamsoccur as sinusoidal patterns in the electron number density and current density profiles.The average quantities over all possible orbitals from E = E = 2 √ γ (for ground statequasiparticle level) up to infinity may be obtained as follows J t = ∞ Z E j ( E ) D ( E ) f ( E ) dE, v t = ∞ Z E v ( E ) D ( E ) f ( E ) dE, (15)where f ( E ) = 1 / [1 + exp( E/θ )] is the occupation number for fermion quasiparticles and D ( E ) is the 1D density of states (DOS) given by [67] D ( E ) = ( E − γ ) ( L + γ ) + E ( L − γ ) p E − γ πL √ E − γ ) q γ ( L − γ ) p E − γ − L γ + E ( L + γ ) . (16)in which L is the normalized size of the quantum plasma. In all our simulations we havechosen L = 100.Figures 1(a) and 1(b) depicts the current density and velocity of collective excitations atdifferent quasiparticle orbital. It is remarked that, while the orbital current density has asinusoidal form the velocity has a nonlinear shape. Moreover, Figs. 1(c) and 1(d) show thetotal average current density and velocity over all posible quasiparticle orbital. It is seenthat the main contribution to the average quantity comes from the ground state orbital, E = E . However, due to important phase contribution from different orbital they may notbe neglected in statistical summation.The two stream model (8) may be generalized by including the vector magnetic potentialin the effective Schr¨odinger-Poisson model. Using the generalized action ∇ S s = m s v s + q s A s ,we find the generalized electron beam wavenumber, k s = ( p s − eA s ) / ¯ h for given stream.Therefore, the Aharonov-Bohm effect may be visualized for single-momentum narrow elec-tron beams splitting around the current carrying solenoid [72, 73]. In this case we have k = ( p − eA ) / ¯ h and k = ( p + eA ) / ¯ h , where A is the constant magnetic vector poten-tial and p is the electron beam momentum. The phase shift between the clockwise andcounterclockwise electron beams amounts to δϕ = ( e/ ¯ h ) H A · dl , as in the Aharonov-Bohmconfiguration. 7 - -
10 0 10 20 30020406080100120 x ( a ) j E = E , 1.5, 2, σ = θ = k = k = - - -
10 0 10 20 300.150.200.250.300.350.400.450.50 x ( b ) v E = E , 1.5, 2, σ = θ = k = k = - - -
10 0 10 20 30102.0102.5103.0103.5104.0 x ( c ) j t σ = θ = k = k = - - -
10 0 10 20 300.360.380.400.420.440.46 x ( d ) v t σ = θ = k = k = FIG. 1: (a) Current density of different quasiparticle orbitals for given values of normalized chem-ical potential, σ , temperature θ and stream wavenumbers. (b) Quasiparticle velocity of differentquasiparticle orbitals for given values of normalized chemical potential, σ , temperature θ andstream wavenumbers. (c) Total current density averaged over all orbitals for parameters given in(a). (d) Total quasiparticle velocity averaged over all orbitals for parameters given in (a). V. ELECTRON BEAM-PHONON INTERACTIONS
In order to study the plasmon-phonon interaction effects, let us first consider the energydispersion of the following system with only one electron stream γ d Ψ( x ) dx ± i √ γk d d Ψ( x ) dx + Φ( x ) + ( E − k d )Ψ( x ) = 0 , (17a) d Φ( x ) dx − Ψ( x ) = 0 , (17b)with k d being the de Broglie’s wavenumber of the stream. The system (22) which can beobtained by direct expansion of (7) admits the following dispersion relation E ± = 1 k + ( √ γk ± k d ) , (18)where the plus/minus signs refer to the right/left going streams, respectively. Note thatintroduction of the electron stream leads to a Doppler shift in the particle-like branch of theenergy dispersion and destroys the quasiparticle wavenumber symmetry. Such a Dopplershift is familiar from two-stream instability in plasma environments [74].Figure 2 shows the effect of Doppler shift on the energy dispersion of collective free elec-tron excitations. The effect of fractional chemical potential σ = µ /E p on the dispersioncurve is shown in Fig. 2(a), in the absence of electron stream. It is remarked that, in fixedtemperature θ , the quasiparticle conduction minimum ( E = 2 √ γ ) is lowered by increasein the σ . Also, the conduction minimum wavenumber k = ± γ − / shifts slightly to higherwavenumbers. Figure 2(b) shows that for fixed chemical potential the increase in the nor-malized temperature θ has the converse effect by shifting the conduction minimum to higherenergy and lower wavenumbers. Effect of the electron stream is shown in Figs. 2(c) and2(d). It is clearly remarked that the introduction of stream destroys the despersion symme-try shifting the conduction band energy minimum, making collective excitation orbital moreavailable/unavailable to quasiparticles in the counterstream/stream direction. Such effectmay be the origin of well-known two-stream instability. Note that ǫ is the exact quasiparticlenormalized energy eigenvalue, while, in forthcoming analysis we use the scaled eigenvalue E = ǫ − σ , which depends on the chemical potential of the electron beam, for simplicity.In the model (22) neutralizing ion background are static (jellium mdel). This model may9 - - k ( a ) E σ = θ = k d = - - - k ( b ) E σ = θ = k d = - - - k ( c ) E σ = θ = k d = - - - k ( d ) E σ = θ = k d = - - - FIG. 2: (a) Variations in energy dispersion zero-stream plasmon excitations due to changes in thenormalized chemical potential. (b) Variations in energy dispersion zero-stream plasmon excitationsdue to changes in the normalized temperature. (c) Variations in energy dispersion noninteractingtwo-stream plasma excitations due to changes in the positive stream wavenumber. (d) Variationsin energy dispersion noninteracting two-stream plasma excitations due to changes in the negativestream wavenumber.
10e generalized considering ion dynamics as follows γ d Ψ e ( x ) dx ± i √ γk d d Ψ e ( x ) dx + Φ( x ) + ( E − k d )Ψ e ( x ) = 0 , (19a) ζ d Ψ i ( x ) dx − Φ( x ) + ( E + µ )Ψ i ( x ) = 0 , (19b) d Φ( x ) dx − Ψ e ( x ) + Ψ i ( x ) = 0 , (19c)where ζ = m e /m i . The system (19) admits the following eigenvalue system Ψ e Ψ i Φ = γk − √ γkk − ( E − k ) 0 − ζ k − ( E + µ ) − k α − α Ψ e Ψ i Φ , (20)which admits the following energy dispersion relation E ± = 2 + ξk ± p ξ k k , ξ = ( k d − k √ γ ) − ζ k + σ, (21)where the plus/minus sign refers to the upper/lower energy band.Figure 3 shows the plasmon energy dispersion for quasineutral electron beam-ion plasmawith various plasma parameters. Figure 3(a) indicated that in the absence of electron streamthere are symmetric upper and lower energy bands. However, the presence of electronstream, as shown in Fig. 3(b), disturbs the symmetry of bands making more quasiparticlelevels available to counterstream particles. Figure 3(c), on the other hand, depicts theeffect of higher normalized electron chemical potential. It is remarked that, Increase in thisparameter lowers the values of both upper and lower energy bands. Finally, Fig. 3(d) revealsthat increase in the normalized electron temperature does not have a significant effect onthe energy band of collective excitations, as compared to Fig. 3(b).Figure 4 shows the spatial electron drift velocity variations in electron-ion plasma withdynamic ions for different plasma and drift parameters. Figure 4(a) shows the quasiparticlevelocity at orbital E = 0 . , . σ = θ = 0 . k d = 0 .
5. It is clearlyremarked that the electron beam backscatters at orbital E = 0 .
2. The later is evidently dueto presence of phonon excitations by dynamic ions. Figure 4(b) depicts the quasiparticlevelocity at high energy orbital E = 1 . , E = 2 the velocity is in a11 - - k ( a ) E σ = θ = k d = - - - k ( b ) E σ = θ = k d = - - - - k ( c ) E σ = θ = k d = - - - k ( c ) E σ = θ = k d = FIG. 3: Energy dispersion of electron plasmon-phonon interction with (a) zero electron drift (b)non-zero electron drift, (c) increased electron normalized chemical potential and (d) increasedelectron normalized temperature. periodic oscillatory form. Figures 4(c) and 4(d) show the velocity for the same orbital as4(a) and 4(b), respectively, for lower electron beam drift wavenumber. It is seen that fororbital E = 0 . k d does not radically change the orbital velocity12
10 20 30 40 - - x ( a ) v e ( x ) E = γ = k d = x ( b ) v e ( x ) E = γ = k d = - x ( c ) v e ( x ) E = γ = k d = x ( d ) v e ( x ) E = γ = k d = FIG. 4: The quasiparticle velocity at different energy orbital for electron-ion plasma with driftingelectrons. (a) E = 0 . , . σ = θ = 0 . k d = 0 . E = 1 . , σ = θ = 0 . k d = 0 .
5, (c) E = 0 . , . σ = θ = 0 . k d = 0 .
3, (d) E = 1 . , σ = θ = 0 . k d = 0 . pattern. 13 . PLASMON-PLASMON INTERACTIONS AND PHASE-MIXING Next we consider the following pseudoforce system of interacting two-stream model γ d Ψ dx + 2 i √ γk d Ψ dx + Φ + ( E − k )Ψ = 0 (22a) γ d Ψ dx + 2 i √ γk d Ψ dx + Φ + ( E − k )Ψ = 0 (22b) d Φ dx − α Ψ − (1 − α )Ψ = 0 , (22c)where k and k are the generalized momentums, as normalized to the plasmon momentum k p = p mE p / ¯ h . The coupled system (22) does not have straightforward analytical solutionand must be solved numerically. Valuable information may be extracted from the energy dis-persion relation which is obtained by assuming plane wave expansions, Ψ ( x ) = Ψ exp( ikx ),Ψ ( x ) = Ψ exp( ikx ) and Φ ( x ) = Φ exp( ikx ), leading to the following eigenvalue system Ψ Ψ Φ = γk − √ γkk − ( E − k ) 0 − / γk + 2 √ γkk − ( E − k ) − / k α − α Ψ Ψ Φ , (23)from which one obtains the following energy dispersion relations E ± = 14 k h ± p βk ( k − k ) + 2 k (cid:0) k + k − ηk √ γ (cid:1)i , (24a) β = η (cid:2) η ( k − k ) k + 2 α − (cid:3) , η = k + k − k √ γ. (24b)Note that we assumed γ = γ = γ and σ = σ = σ , hence E = E = E , for simplic-ity. However, a more general dispersion relation can be obtained for different streamingconditions. A more general model may also include the screening (pseudodamping) effect[75].Figure 5 shows the two-stream energy dispersion which consists of two distinct lower andupper branches. It is clearly evident that appearance of lower/upper branches is due tomixing of the collective excitation modes caused by dual streams. Dispersion of symmetricstream momentum with asymmetric density is shown in Fig. 5(a) indicating a Dopplershift towards positive stream direction which is less dense, i.e., α = 0 .
1. In Fig. 5(b)the symmetric momentum case with slightly lower wavenumber is depicted. It is remarkedthat more quasiparticle orbital are available for lower stream momentum. The case with14 - k ( a ) E γ = k = k = - α = - - k ( b ) E γ = k = - k = α = - - - k ( c ) E γ = k = - k = α = - - - k ( d ) E γ = k = - k = α = FIG. 5: Energy dispersion of interacting two-stream plasma excitations with (a) symmetric mo-mentum with asymmetric density configuration (b) symmetric momentum asymmetric densityconfiguration with different stream wavenuber than in (a), (c) symmetric momentum and densityconfiguration and (d) asymmetric momentum with symmetric density configuration. symmetric stream momentum and density is shown in Fig. 5(b) resulting in a completelysymmetric dispersion curves. However, despite the fact that the total current is zero inthis case, the mixing of modes is still present. The case of asymmetric momentum streamwith symmetric density in Fig. 5(d) indicates that dispersion curve is broken due to stream15 - - x ( a ) j ( x ) E = γ = k = - k = α = - - x ( b ) j ( x ) E = γ = k = - k = α = - - - - - - - x ( c ) j ( x ) E = γ = k = - k = α = - x ( d ) j ( x ) E = γ = k = - k = α = FIG. 6: Current density of orbital E = 1 . density distribution imbalance.We have numerically evaluated the two stream system (22) for initial condition Ψ = α ,Ψ = (1 − α ), Φ = 0 and Ψ ′ = Ψ ′ = Φ ′ = 0. Figure 6 depicts the current density ifcounter streaming electron gas at orbital E = 1 .
5. The case of complete symmetric streamis shown in Fig. 6(a) where the total current density vanishes. However, Fig. 6(b) shows16hat for asymmetric momentum configuration phase mixing occurs, where current of lowmomentum stream reverses at some points of its current. The latter is cause by the fact themore orbital become available to low momentum beam at the other side of stream leadingto backscattering of the low momentum stream. While similar in concept, this is however adifferent phenomenon than stream instability. The phase mixing effect due to the densityimbalance of counter streaming electrons is shown in Fig. 6(c), for α = 0 .
1. Note thatstrong oscillations in current density of dense stream as compared to the other one. Figure6(d) reveals that the phase mixing is absent for symmetric momentum and α = 0 . E = 1 . E = 0 . E = 0 . E = 1 and E = 1 . a ( x ) = v ( x ) dv ( x ) /dx , anddissipated power p ( x ) = a ( x ) v ( x ) for unstable orbital E = 1 . E = 0 .
7. The force actingon each beam particles, in orbital E = 1 .
2, for slightly asymmetric unstable counter-streamis depicted in Fig. 9(a). It is seen that the force acting on the low and momentum negativestream is stronger and rapidly grows over distance. Figure 9(b) show the orbital velocityof quasiparticle corresponding to Fig. 9(a). The grow and damping of oscillations are alsopresent in the velocity pattern. Figures 9(c) and 9(d) show the acceleration and dissipatedpower of orbital E = 0 . - - x ( a ) v ( x ) E = γ = k = k = - α = - - x ( b ) v ( x ) E = γ = k = k = - α = - - - - - - x ( c ) v ( x ) E = γ = k = - k = α = - - x ( d ) v ( x ) E = γ = k = k = - α = FIG. 7: Quasiparticle speed for unstable orbital E = 1 . periodic energy exchange between streams as they merge into single beam. The plasmonstopping power of an electron beam may be readily calculated using ∆ E/ ∆ x = δ R a ( x ) dx inwhich δ is the penetration depth. 18 x ( a ) v ( x ) E = γ = k = k = α = x ( b ) v ( x ) E = γ = k = k = α = - - x ( c ) v ( x ) E = γ = k = - k = α = - - - x ( d ) v ( x ) E = γ = k = - k = α = FIG. 8: Quasiparticle speed for different stable and unstable orbital with different two-streamwavenumber and density parameters. - - x ( a ) a ( x ) E = γ = k = - k = α = - - - x ( b ) v ( x ) E = γ = k = - k = α = - - x ( c ) a ( x ) E = γ = k = - k = α = - - x ( d ) p ( x ) E = γ = k = - k = α = FIG. 9: (a) The quasiparticle acceleration and (b) speed for unstable quasiparticle orbital E = 1 . E = 0 . I. ELECTRON BEAM-LATTICE INTERACTION
In order to prove the effectiveness of current model we would like to study the beam latticeinteractions. To this end, we consider the following driven coupled pseudoforce system γ d Ψ dx + 2 i √ γk d d Ψ dx + Φ + ( E − k d )Ψ = 0 (25a) d Φ dx − Ψ + U cos( Gx ) = 0 , (25b)in which G = 2 π/a is the reciprocal lattice vector with atomic spacing a and U is the latticepotential strength. Note that we use a simple toy model in which the atomic potential issinusoidal, for our illustration purpose. However, current model may be readily generalizedto fourier components of a real Columbic potential [76]. The Doppler shift in the energydispersion is same as in single stream model. However, in this case the electron streamwavenumber can resonantly interact with intrinsic periodic structure of the plasmonic lattice.In Fig. 10 we have shown the quasiparticle velocity corresponding to the orbital E = 1 . = 1, Φ = 0 and Ψ ′ = Φ ′ = 0. The orbital with k d = 0 . G = 5, as shown in Fig. 10(a). The periodic nondampingvelocity profiles is an indication of the orbital stability. However, as the drift wavenumberincreases to the value k d = 0 . k d = 0 . G = 2, as shown in Fig. 10(c). It is remarked that the electronbeam backscatters due to resonant beam-lattice interaction. The effect becomes even morepronounced and forms a regular pattern as the lattice potential increases to the value U = 2,as is evident from Fig. 10(d).Figure 11(a) shows the current density of stable orbital E = 2 for given beam and latticeparameters. There are strong fluctuations in the current density due to beam-plasmoninteractions. Moreover, Fig. 11(b) reveals that increase of the beam momentum relativelyincreases the amplitude of average current density, as expected. Decrease of the reciprocallattice vector to G = 3 with other parameter same as in Fig. 11(a) leads to negative currentfor this orbital, as shown in Fig. 11(c). Finally, increase in the lattice potential strengthleads to strong backscattering of the electron beam for this stable beam, producing a chaoticcurrent density pattern in Fig. 11(d). Such a backscattering effect may also be related to21
10 20 30 4001234 x ( a ) v ( x ) E = γ = k d = U = G = x ( b ) v ( x ) E = γ = k d = (cid:4) = (cid:5) =
50 10 20 30 (cid:6)(cid:7) - - (cid:8) - (cid:9)(cid:10) x ( c ) v ( x ) E = (cid:11)(cid:12)(cid:13)(cid:14) γ = k d = (cid:15) = (cid:16) = (cid:17)(cid:18) - (cid:19) - (cid:20) - (cid:21)(cid:22) x ( d ) v ( x ) E = (cid:23)(cid:24)(cid:25)(cid:26) γ = k d = (cid:27) = (cid:28) = FIG. 10: (a) Stable quasiparticle velocity of electron beam-lattice interaction in orbital E = 1 . E = 1 .
8. (c) Backscatteringof electron beam from lattice at orbital E = 1 . G = 2 and latticepotential strength U = 1. (d) Backscattering of electron beam from lattice at orbital E = 1 . G = 2 and lattice potential strength U = 2. the well-known phenomenon, the Umklapp scattering [4] or U-process which is the dominantineffective electronic heat transport in metallic elements at low temperatures.22 x ( a ) j ( x ) E = γ = k d = U = (cid:29) = x ( b ) j ( x ) E = γ = k d = &’() U = * =
50 5 10 15 20 - +,- - ./2 - x ( c ) j ( x ) E = γ = k d = = = - - ;<= - >?@ - ABC - x ( d ) j ( x ) E = γ = k d = = D = FIG. 11: The current density of stable orbital E = 2 for different values of electron beam wavenum-ber, k d , reciprocal lattice vector, G = 2 π/a and lattice potential strength, U . VII. CONCLUSION
Using the kinetic corrected Schr¨odinger-Poisson model we developed a pseudoforce theoryto study the multistream quantum plasmas. The noninteracting electron two-stream modelwas used to study the quantum beam interference and Aharonov-Bohm-like effects in the twobeam electron gas. On the other hand, quasiparticle velocity, acceleration and power at given23rbital was used to investigate the stream interactions and quantum phase mixing in two-stream plasma model for different stream momentum and density configurations. We furtherextend the model to include the electron beam-lattice interactions and backscattering effect.Current research may be important in understanding of electron-plasmon and plasmon-plasmon interaction in quantum orbital level and may be further elaborated to study thequantum wave-particle interactions in complex plasmas.
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