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Filamentation instability in a quantum magnetized plasma ∗ A. Bret ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain and Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario de Ciudad Real, 13071 Ciudad Real, Spain (Dated: February 3, 2008) The filamentation instability occurring when a non relativistic electron beam passes through a quantum magnetized plasma is investigated by means of a cold quantum magnetohydrodynamic 8 model. It is proved that the instability can be completely suppressed by quantum effects if and 0 only if a finite magnetic field is present. A dimensionless parameter is identified which measures 0 thestrength of quantumeffects. Strongquantumeffectsallow for amuch smaller magnetic field to 2 suppressthe instability than in the classical regime. n a J 5 I. INTRODUCTION 2 ] Thedevelopmentofquantumhydrodynamicandmagnetohydrodynamicequations[1,2]madeitpossibletoquickly h evaluate quantum effects connected to the physics of microelectronic devices and laser plasmas interaction (see Ref. p [3]andreferencestherein). Plasmaphysics hasalsogainedfromthese progressesas quantumeffects appearinFusion - m settings or Astrophysics. The behavior of waves in quantum plasmas [3, 4, 5, 6, 7] magnetized or not, as well as turbulence is such environments [8] has thus received attention. Another very classical topic of plasma physics, s a namely plasma instabilities, needs to be revisited from the quantum point of view. The quantum theory of the pl two-stream instability has already been developed [9, 10] while quantum effects on the filamentation instability were . recently evaluated [11] for a non magnetized plasma. Due to the importance of magnetized plasmas, especially in s c astrophysics,wedevotethepresentpapertothe evaluationofquantumeffectsonthe filamentationinstabilityinsuch i setting. Since the relativistic quantum magnetohydrodynamic equations are yet to be defined, the present analysis s y restricts to the non-relativistic regime. On the other hand, we do not make any approximation on the beam density h so that present theory remains valid even when the beam density equals the plasma electronic one. p The paper is structured as follow: we start explaining the formalism and derive the dispersion equation. We then [ turn to the investigation of the marginal stability and derive some exact relations satisfied in this case. We finally 1 study the maximum growth rate and the most unstable wave vector before we reach our conclusions. v Let us then consider an infinite and homogenous cold non-relativistic electron beam of velocity Vbz and density nb 0 entering a cold plasma along the guiding magnetic field B0 = B0z. The plasma has the electronic density np and 5 ions form a fixed neutralizing backgroundofdensity n +n . The beam prompts a return currentin the plasma with b p 9 velocity V z such as n V = n V . We use the fluid conservation equations for the beam (j = b) and the plasma p p p b b 3 (j =p), . 1 0 ∂n j + (n v )=0 (1) 8 j j ∂t ∇· 0 v: and the force equation in the presence of the static magnetic field B0 with a Bohm potential term [2], i arX ∂∂vtj +(vj ·∇)vj =−mq (cid:18)E+ vj ×c B(cid:19)+ 2~m22∇ ∇√2√njnj!, (2) where q > 0 and m are the charge and mass of the electron, n the density of species j, p its momentum, and B j j equals B0 plus the induced magnetic field. We now study the response of the system to density perturbations with k V , varying like exp(ıkr ωt) with k = kx, and linearize the equations above. With the subscripts 0 and 1 b ⊥ − denoting the equilibrium and perturbed quantities respectively, the linearized conservation equation (1) yields k v j1 nj1 =nj0ω ·k vj0, (3) − · ∗Electronicaddress: [email protected] 2 and the force equation (2) gives i(k·vj0−ω)vj1 =−mq (cid:18)E1+ vj0×B1+c vj1×B0(cid:19)−i4~mk22nnjj10k. From the linearized equations above, we derive the perturbed density and velocity fields in terms of E1 and B1 and eventually express the current through, J=q nj0vj1+nj1vj0. (4) j=p,b X Finally,weexpressB1 intermsofE1 throughB1 =(c/ω)k E1 andclosethesysteminsertingthecurrentexpression × in a combination of Maxwell Ampère and Faraday’s equations, c2 4iπ ω2k×(k×E1)+E1+ ω J=0⇔T(E1)=0. (5) ThetensorThasherebeencalculatedsymbolicallyusinganadaptedversionoftheMathematica Notebookdescribed in Ref. [12]. It takes the form ∗ T11 T12 0 T= T12 T22 0 , (6)   0 0 T33   where the superscript * refers to the complex conjugate and (1+α) 2 T11 = x 1− (x2 Ω2) ΘZ4 , (cid:18) − B − (cid:19) Z2 (1+α)(ΘZ4 x2) 2 T22 = x − β2 − (x2 Ω2) −ΘZ4 , − B − 2 2 Z α(1+α)β 2 T33 = x −1−α− β2 1+ (x2 Ω2) ΘZ4 , (cid:18) − B − (cid:19) x(1+α)Ω B T12 = ı(x2 Ω2) ΘZ4, (7) − B − in terms of ω kVb Vb nb qB0 x= , Z = , β = , α= , Ω = , (8) B ω ω c n mcω p p p p whereω istheelectronicplasmasfrequency. Quantumeffectsappeartobemeasuredthroughaparameterpreviously p highlighted [9, 10, 11], Θ ~ω 2 c p Θ= , with Θ = . (9) β4 c 2mc2 (cid:18) (cid:19) Numerically, −33 −3 Θ =1.3 10 n [cm ], (10) c p × so that this parameter will hardly be larger than 1, even when dealing with the densest space plasmas. Let us finally emphasized a point regarding the beam to plasma density ratio α defined in Eq. (8). If the ground state which stability is investigated consisted of the plasma only, the beam representing the perturbation, then this parameter would have to remain much smaller than 1 within the framework of a linear response theory. In turns out thatthe dispersionequationwhichhas justbeen derivedis the dispersionequationofthe beam+plasmasystem. The perturbed ground state is therefore the sum of the beam and the plasma. It is thus perfectly possible to investigate the linear response of the whole system even when α=1 so that we do need to make any assumption regarding this parameter. 3 (a) Classical, non magnetized 0.03 e t a r h t 0.02 w o (b) Classical, magnetized r G 0.01 (c) Quantum, magnetized 0.2 0.4 0.6 0.8 1 Z FIG. 1: Classical (non-quantum) growth rate of the filamentation instability in terms of the reduced wave vector Z without (a) and with (b) magnetic field. Curve (c) includes quantum effects. Parameters are α = 0.1, β = 0.1 for (a,b,c), ΩB = 0.03 for (a,b) and Θc =1.3×10−7 for (c), which corresponds to theplasma density np =1026 cm−3. II. CLASSICAL MAGNETIZED PLASMA Beforeweturntothequantumcase,letusquicklyremindsomebasicfeaturesofthe coldmagnetizedfilamentation instabilities [13] in the classical (non-quantum) regime. To this extent, the dispersion equation, which is just the determinant ofthe tensor we just defined, is solvednumericallyand Figure 1 displays the growthratesobtained with and without the magnetic field (curves a and b). The stabilizing effect of the magnetic field is twofold. On one hand, the smallest unstable wave vector switches from Z =0 to βΩ √1+α b Z1 = . (11) α(1+α)β2 Ω2 − B On the other hand, the growth rate saturation valupe δ∞ for large Z is lower with δ∞ = α(1+α)β2 Ω2, (12) − B q which vanishes exactly for Ω =Ω β α(1+α) (13) B Bc ≡ Noteworthily, this value of the magnetic field also makes thepquantity Z1 diverge. The physical interpretation of this threshold is simple as β α(1+α) is just the maximum growth rate of the instability in the non-magnetized case [14]. Filamentation instability is thus inhibited when the electron response to the magnetic field is quicker. p III. QUANTUM MAGNETIZED PLASMA A. Marginal stability analysis Figure 1c displays the growth rate in terms of Z accounting for quantum effects. As in the non-magnetized case [11], quantum effects introduce a cut-off at large Z so that we now have to characteristic wave vectors Z1 and Z2 determining the instability range. Both of them can be investigated directly from the dispersion equation. Since the growth rate vanishes for these wave vector while the root yielding the filamentation instability has no real part, we can write detT(x=0,Z =Z1,2)=0. (14) 2 It turns out that this equation can be simplified. After replacing Z and eliminating x=0 as a double root of →Z the dispersion equation, we find that the equation above is equivalent to P( )Q( )=0 with Z Z 2 4 2 2 6 P( ) = (Θ +β Ω )( +(1+α)β ) α(1+α)β Z cZ B Z −Z 4 2 2 2 4 Q( ) = β Ω +( +(1+α)β )(Θ +(1+α)β ). (15) Z Z B Z cZ 4 F, G G F F(0) 1 2 FIG. 2: Schematicrepresentation of thefunctions F and G definedby Eqs. (16). Every term of the second polynomial is clearly positive so that it yields only negative roots <0, implying some complex wave vector. Because we seek real wave vectors, we can conclude that 1 =(Z1)2 andZ 2 =(Z2)2 are both Z Z zero’s of P( ). This function being a polynomial of the third order, it is possible to find the exact solutions. We Z nevertheless use some graphical method for a more intuitive approach. Let us then define 2 4 2 2 F( ) = (Θ +β Ω )( +(1+α)β ), Z cZ B Z 6 G( ) = α(1+α)β , (16) Z Z so that P = 0 is equivalent to F = G. F is a third order polynomial, monotonically increasing for > 0, and starting from F(0) = (1+α)Ω2β6 with an initial slope F′(0) = Ω2β4. G is a first order monotonicallZy increasing B B polynomialwith G(0)=0andslope α(1+α)β6. We cannowconductthe graphicalanalysisofthe problemfollowing the guidelines set by the schematic representation of F and G on Figure 2. When increasing Ω or Θ , the curve G B c is not modified because neither Ω nor Θ appear in its expression. Meanwhile, F(0) increases with Ω , and F( ) B c B Z increases all the more than Θ and Ω are large. This allows us to draw the following conclusions: c B ′ Intheabsenceofmagneticfield,F(0)=F (0)=0whilethepreviousanalysisremainsunchanged. Theequation • F = G thus has two positive solutions regardless of the other parameters. One solution is 1 = 0, i.e. Z = 0, Z and we label the other 2 > 0. We recover the existence of a quantum cut-off [11] at large wave vector, and Z prove here that the instability is never completely stabilized since 2 never vanishes. Z ′ For any finite magnetic field, one has F(0) > 0 and F (0) > 0, and the typical resulting situation is the one • represented on Fig. 2. As long as F(0), or the growth of F( ), are “not too high”, the equation F = G has two positive roots 1,2 = (Z1,2)2. But it is obvious that as ΩZB or Θc increase, the too real roots become one Z before they vanish. We thus come to conclusion that the instability can be completely suppressed by quantum effectsif, andonlyif,the systemismagnetized,regardlessofthe strengthofthe magneticfield. Itisgraphically obviousthatsinceanincreaseofbothΩ orΘ contributetothecollapseofthetworealsroots,thestabilization B c condition should result in a balance between these quantities. It should be possible to stabilize the system at low Ω with an high Θ , or vice versa. B c In the magnetized case, the instability is marginal when the equation F = G has one double root 1 = 2 = L. Z Z Z Here, L stands for Last because Z =√ is eventually the last unstable wave vector before complete stabilization. L L Z If is double root of F for the parameters defining the marginalstability, then for these very parameters F can be L Z 2 castunder the formF( )=( a)( ) where a is the thirdroot. Bydevelopingthis lastformandidentifying L Z Z− Z−Z the coefficients of the polynomial with the ones extracted from Eq. (16), we can write the following equations, 2 a+2 = (1+α)β , (17) L Z − β4Ω∗2 α(1+α)β6 2a + 2 = B − , (18) ZL ZL Θ∗ c (1+α)β6Ω∗2 a 2 = B , (19) ZL − Θ∗ c 5 0 10 -1 10 α = -2 1 10 B α= - 3 0. Ω 10 1 α = -4 0.0 10 1 α = 0. 0 10-5 01 α =0.00 α=0.0 α=0. α= 1 1 1 1 -6 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10 10 10 10 Θ c FIG. 3: (Color online) Values of Θ∗ and Ω∗ implicitly defined by Eqs. (20,21) for various beam to plasma density ratios α c B andβ =0.1(redbold curves)andβ =10−3 (bluethincurves). Givenαandβ,thesystemismarginally stableforparameters (Θ∗c,Ω∗B) located on the corresponding curve, and stable above. The value of ΩBc (see Eq. 13) for β = 0.1 and α = 1 is represented by thehorizontal dashed red line, and theoblique dashed curvecorresponds to Eq. (25) with thesame α,β. where the superscript * refers to the values at marginal stability. By eliminating a between the first and the second equation, one finds a second order equation for which positive solution can be cast under the form, L Z β2(1+α) Ω2 Ω∗2 = 1+3 Bc− B 1 . (20) ZL 3 s Θ∗c(1+α)2 − ! ∗ ∗ Then, eliminating a between the first and the third yields an implicit relation between , Θ and Ω at marginal ZL c B stability, 6 ∗2 2 ∗ 2 (1+α)β Ω = Θ (2 +(1+α)β ). (21) B ZL c ZL ∗ ∗ Equations (20,21) therefore define Θ and Ω in terms of each other, and of the others parameters of the problem. c B The curvesthus definedappearonFigure 3 forvariousα’s andβ’s. Parameters(Θ ,Ω )locatedabovea givencurve c B ∗ ∗ (Θ ,Ω ) define a completely stabilized system. c B IV. ANALYTICAL EXPRESSIONS FOR MARGINAL STABILITY A. Classical limit ∗ ∗ We observe on Fig. 3 that Ω reaches a finite value when Θ 0. This classical limit is obviously the marginal B c → ∗ ∗ magnetic parameter Ω given by Eq. (13). We thus assume a leading term in the development of Ω for small Θ Bc B c of the form Ω∗ =(1 κΘ∗ξ)Ω . Inserting this expressionin Eqs. (20,21) and expanding the results in series of Θ∗, B − c Bc c 6 we find 3(1+α)1/3 Ω∗(Θ∗ 0) 1 Θ∗1/3 Ω , (22) B c → ∼ − 25/3α1/3β2/3 c Bc (cid:18) (cid:19) and α1/3(1+α)2/3β8/3 ∗ (Θ 0) . (23) ZL c → ∼ 21/3Θ∗1/3 c Inaccordancewiththe classicalcasewherethe smallestunstable wavevectordivergesformarginalstability(see Eqs. 11,12), the last unstable wave vector Z behaves like 1/Θ∗1/6 in the weak quantum regime since =Z2. L c ZL L B. Strong quantum limit Havingelucidatedthe weakquantumregime,we nowturnto thestrongquantumone. Figure 3makesitclearthat marginal stability behaves differently within each regime. In order to discuss this point, let us consider expression ∗ (20) of in terms of the marginal classical magnetic parameter Ω . In the “large” Θ regime, the ratio under the ZL Bc ∗ c squarerootbecomes smallcomparedto unity, andFig. 3 showsthatΩ Ω . Developing the squareroot,we find B ≪ Bc directly 2 2 β Ω = Bc . (24) ZL 2Θ∗(1+α) c In this strongly quantum regime, the last unstable wave vector thus tends to zero like 1/ Θ∗. Inserting the former c expression in Eq. (21) yields the magnetic parameter required to stabilize the system p Ω2 αβ2 Ω∗ Bc = . (25) B ∼ 2(1+α)Θ∗ 2 Θ∗ c c This limit is plotted on Fig. 3 for β =0.1 and α=1 and perfectlypfits the numerical evaluation for large Θ∗. c It is now possible to exhibit the dimensionless parametermeasuring the strength ofquantum effects. The equation ∗ ∗ above indicates that Ω Ω if Ω 2(1+α)Θ , andthe curves plotted on Fig. 3 demonstrate that a reduction B ≪ Bc Bc ≪ c of the stabilizing parameter is the signature of the strong quantum regime. Because we think here in terms of orders of magnitudes, we drop the 2(1+α) factor and finally define Ω Bc Λ= , (26) Θ c as the parameter determining the strength of quantum effects. These are weak for Λ 1 and strong in the opposite ≫ limit Λ 1. ≪ V. UNSTABLE SYSTEMS Having elucidated how the system can be completely stabilized by quantum magnetic effects, we now turn to unstable systems in order to investigate the growth rate of the instability and the most unstable wave vector for a given configuration. A. Maximum growth rate In the weakly quantum regime with Λ 1, the maximum quantum growth rate is very close to its classical ≫ counterpart all the way down to complete stabilization which, as we just mentioned, occurs for similar magnetic parameters Ω (see Eq. 22 above). Figure 4 present a plot of the maximum growth rates along the Z axis, in terms B of Ω in the classical and quantum cases for β =0.1 and various α’s. Parameters have been chosen to illustrate the B present weak quantum regime. Such a system can thus be viewed as basically magnetized with some weak quantum effects, and stabilization mainly comes from the magnetic field. 7 δmax 0.03 0.02 0.01 0.01 0.02 0.03 Ω B FIG.4: Maximumgrowth ratesintermsofΩB intheclassical andquantumcases forβ =0.1andα=10−1,10−2,10−3. With Θc = 10−10 (np = 7.6×1022 cm−3), the parameter Λ given by Eq. (26) is always larger than 3×107 and the classical and quantumcurves are hardly distinguishable although thequantumgrowth rate is a little bit smaller. When Λ 1 (strong quantum regime), stabilization is reachedearlier with respect to Ω (see Fig. 3 and Eq. 25). B ≪ Here, stabilization comes from a combination of quantum and magnetic effects, as indicated by the oblique slope of the curves in Fig. 3. We plot on Figure 5 the maximum growth rate in terms of Ω . We recognize the kind of curve B obtained for a classical magnetized plasma δ2 =Ωcut off 2 Ω2 with a “cut-off” magnetic parameter αβ2/2 Θ∗. max B − B c This is why we plotted together the numerical evaluation of the maximum growth rate together with the function, p 2 αβ2 δ = Ω2. (27) max vu 2 Θ∗c! − B u t p It can be checked that this function fits the result all the more than Λ is small. With Eq. (25), we then come to the conclusion that as far as the maximum growth rate is concerned, strong quantum effects are equivalent to the substitution, Ω Bc Ω Λ . (28) Bc ⇔ 2(1+α) Because this new quantum cut-off is much smaller than the classical one, the maximum growth rate is reduced accordingly. B. Most unstable wave vector The most unstable wave vector is, together with the maximum growth rate, the most relevant information about the unstable system. In the classical case, the growth rate just saturates at large Z yielding a continuum of most unstable modes. But quantum effects stabilize the large Z modes, so that there is always one mode growing faster that the others. Forsystemsnearmarginalstability,thelastunstablewavevectorZ =√ ,givenexactlybyEq. (20), andinthe L L Z weakand strongquantumlimits by Eqs. (23,24) respectively,is by definition a verygoodapproximationof this most ∗ unstablewavevectorwhenreplacingthemarginalparameterΘ byitsactualvalueΘ . Indeed,wefoundnumerically c c that expressions(23,24) arestill quite accurate,evenfor systems far fromstabilization. This can be understood from Fig. 2: on one hand, the most unstable wave vector for a given configuration is necessarily between Z1 and Z2. On the other hand, the last unstable wave vector belongs to the same interval because Z1 increases while Z2 decreases as the system moves towards stabilization. For a typical situation such as the one represented on Fig. 1c, Z1 and 8 δmax 10 δmax 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.00 0.01 α = 1, Λ=14 0.01 α = 0.1, Λ=3 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 Ω 10 Ω B B 2 3 10 δmax 10 δmax 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 α = 10−2, Λ=1 α = 10−3, Λ=0.3 0.01 0.01 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 2 3 10 Ω 10 Ω B B FIG. 5: Maximum growth rates in termsof ΩB for β=0.1 and Θc =10−2. The thin curveshavebeen computed numerically, and the bold ones (when distinguishable from thethin) correspond toEq. (27). Agreement improves with Λ decreasing. Z2 are eventually quite close to each other so that ZL, which is in between, cannot be far from the most unstable 5 wavevector. With the parameters chosenfor this plot, we find Λ=2.5 10 indicating a weak quantum regime. We × thereforeturnto Eq. (23)andfindthe mostunstable wavevectorforZ 0.4,whichfits accuratelywhatis observed. ∼ VI. DISCUSSION Quantum effects have been assessed with respect to the filamentation instability in a magnetized plasma. As far as the unstable wave vector range is concerned, magnetic effects set it a finite lower bound, while quantum effects introduce a cut-off at large k. As a result, the unstable domain takes the form [k1,k2] and can eventually vanish for some parameters configurations which were elucidated. We also found that the dimensionless parameter Λ = Ω /Θ determines the strength of quantum effects. When Bc c Λ 1,the instabilitycanbe describedinclassicalterms,andeventuallyvanishes whenincreasingthe magneticfield, ≫ while the unstable wave vector range shifts towards infinity. When quantum effects are strong, namely Λ 1, the ≪ instability still vanishes with the magnetic field, but the unstable wave vector range tends to zero. Furthermore, the magneticfieldrequiredto stabilizethe systemisdividedby Λ/2(1+α) 1withrespecttoits classicalvalue,sothat ≪ filamentation can be suppressed by a much smaller magnetic field than in the non-quantum case. These results may have important consequences when dealing with dense space plasmas. Finally, it will be necessary to assess both relativistic effects, which tend to enlarge the instability domain while reducingthemaximumgrowthrate[15],andkineticeffectswhichusuallyhaveastabilizingeffect[16]. Tothisextent, relativistic quantum kinetic theory will be required, or the relativistic form of the quantum Euler equation (2) will havetobeelaborated. 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