1D3V PIC simulation of propagation of relativistic electron beam in an inhomogeneous plasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a y Chandrashekhar Shukla , ∗ Amita Das , and Kartik Patel Institute for Plasma Research, Bhat , Gandhinagar - 382428, India and Bhabha Atomic Research Centre, Trombay, Mumbai - 400 085, India (Dated: September 12, 2018)
Abstract
A recent experimental observation has shown efficient transport of Mega Ampere of electroncurrents through aligned carbon nanotube arrays [Phys. Rev Letts. , 235005 (2012)]. Theresult was subsequently interpreted on the basis of suppression of the filamentation instability inan inhomogeneous plasma [Phys. Plasmas , 012108 (2014)]. This inhomogeneity forms as aresult of the ionization of the carbon nanotubes. In the present work a full 1 D V Particle-in-Cell(PIC) simulations have been carried out for the propagation of relativistic electron beams (REB)through an inhomogeneous background plasma. The suppression of the filamentation instability,responsible for beam divergence, is shown. The simulation also confirms that in the nonlinearregime too the REB propagation is better when it propagates through a plasma whose density isinhomogeneous transverse to the beam. The role of inhomogeneity scale length, its amplitude andthe transverse beam temperature etc., in the suppression of the instability is studied in detail. ∗ [email protected] . INTRODUCTION The generation of relativistic electron beams (REB) through the interaction of a highpower laser (I ≥ W/ cm ) with a solid target [1, 2] and it’s collimated propagation througha plasma is a topic of extensive studies due to its applications in cutting edge technologies,such as fast ignition scheme (FIS) [3] of laser fusion, compact particle acceleration [4], fastplasma switches [5] and laser induced radiation sources [6].The transportation of an electron beam which carries a current much higher than theAlfven current limit I = ( mc /e ) γ = 17 γ b k A, ( m is the electron mass, e is the electroniccharge, c is speed of light and γ b is the Lorentz factor of the beam) is not permitted due toits own strong magnetic field. However, in plasmas a spatially overlapping return shieldingcurrent by the background electrons inhibits the magnetic field generation and permits thepropagation of beams with such high forward currents. The combination of the forwardand return currents in plasmas are, however, susceptible to several instabilities viz., two-stream [7], Weibel [8], filamentation [9] and oblique modes [10]. The two-stream instabilityis a purely longitudinal electrostatic mode which generate density stripes perpendicular tothe beam flow direction. The Weibel instability is purely transverse electromagnetic modewhich occurs in the system due to the temperature anisotropy. This instability generatesa magnetic field in the unmagnetized plasma system. A similar instability which generatesa magnetic field in unmagnetized beam-plasma systems is known as the filamentation in-stability or beam-Weibel instability [11]. These instabilities have a detrimental influenceon the propagation of the relativistic energy electron beam through a plasma and henceput practical constraints. In order to suitably utilize the relativistic electron beams theseconstraints on the propagation of beam through a plasma needs to be overcome. A lot ofexperimental, theoretical and simulation work has been devoted to control and suppressthese instabilities [12–15]. The effect on the filamentation instability of beam density [16],beam velocity [16], transverse temperature [17] and collisions [18] has been widely studied.A recent experimental [19]result, has shown many fold improvement in the propagation ofa hot relativistic electron beam through an array of carbon nanotubes (CNTs). An expla-nation of this has been put forth by Mishra et al. [20] suggesting that the inhomogeneousplasma created by the ionization of the CNTs by the front of the beam is responsible forthe stabilization of transverse instabilities, thereby aiding the collimated propagation of the2eam through longer distances compared to a homogeneous plasma. Here we carry outthe PIC simulations to study the role of inhomogeneity with arbitrary amplitude, on thepropagation of electron beam through plasma.The manuscript has been organized as follows. Section II contains the details of the modelconfiguration and governing equations. In section III, the analytical results are presented.The details of PIC simulations are given in section IV. The observation of results from PICsimulation are presented in section V and it’s interpretation are given in sec.VI. In last sec.VII, we conclude our results. II. MODEL CONFIGURATION AND GOVERNING EQUATIONS
The model configuration chosen for our studies has been shown in Fig. 1. The beam andthe return current are chosen to flow along the ± ˆ y respectively. The ion density is chosento have sinusoidal inhomogeneity along ˆ x riding on a constant density of n . The amplitudeof the inhomogeneity is ǫ and the spatial variations are characterized by a wavenumber k s = 2 π/λ s as shown in Fig. 1. The ions are considered to be at rest at such fast electrontime scales and provide for charge neutralization as a background species. The density,velocity and temperature are denoted by n α , v α and T α , with suffix α standing for b (beam)and p ( background) plasma electrons respectively. The combination of the beam - plasmasystem can be represented by a coupled fluid Maxwell equations, with electrons contributingto forward and return currents treated as two distinct fluids. The normalized governingequation in such a case is: ∂n α ∂t + ∇ · ( n α ~v α ) = 0 (1) ∂ ~p α ∂t + ~v α · ∇ ~p α = − (cid:16) ~E + ~v α × ~B (cid:17) − ∇ P α n α (2) ∂ ~B∂t = −∇ × ~E (3) ∂ ~E∂t = ∇ × ~B − Σ α ~J α (4)with ~v α = ~p α / (1 + p α ) , ~J α = − n α ~v α . The pressure P α is provided by the equation of state.In the above equations, velocity is normalized by speed of light c, density by n , frequencyby ω = 4 πn e /m e and electric and magnetic field by E = B = m e cω /e where m e iselectron rest mass and e is electron charge. 3n the equilibrium there is no electric and magnetic field, so there is complete charge aswell as current neutralization. This is achieved by balancing the forward and return electroncurrents at each spatial location. The total charge density due to electrons is balanced bythe background ion charge density. The background plasma has been chosen cold ( T p = 0)in all our analytical as well as in simulation studies. The profile of transverse temperature T α in beam is chosen in such a way that gradient of pressure is zero in equilibrium. Thetemperature parallel to beam propagation direction is chosen to be zero. Thus, for aninhomogeneous beam plasma system considered by us in this work we have the followingconditions for equilibrium: n i ( x ) = n b ( x ) + n p ( x ) (5)Σ α n α ( x ) ~v α = 0 (6)In equilibrium beam pressure P b is chosen to be independent of x. This is achieved bychoosing the beam temperature T b ( x ) to satisfy the following condition P b = T b ( x ) n b ( x ) = constant = k. (7) T b ( x ) = k/n b ( x ) (8)The suffix 0 indicates the equilibrium fields. This choice is very artificial construct, Howeverit has been chosen to satisfy the equilibrium conditions so that comparisons with simulationcan be made. We linearize the equations (1)-(5) to obtain linear growth rate of instability.For the inhomogeneous case the equation cannot be Fourier analyzed in x and the eigenvalue can be determined by solving following coupled set of differential equations: ωγ α v ′′ lαy + ω γ α v lαy − ω Σ α n α v lαy + i Σ α v αy ( n α v lαx ) ′ = 0 (9) ηT α v ′′ lαx + (cid:18) ηT α n α n ′ α + ηT ′ α (cid:19) v lαx ′ + (cid:18) ηT α n α n ′′ α + η T ′ α n α n ′ α + ω γ α (cid:19) v lαx − Σ α n α v lαx + iωγ α v αy v ′ lαy = 0 (10)Where η is ratio of specific heat. For the case of homogeneous plasma, eq. (9) and eq. (10)reduce to ωγ α v ′′ lαy + ω γ α v lαy − ω Σ α n α v lαy + i Σ α v αy n α v ′ lαx = 0 (11) ηT α v ′′ lαx + ω γ α v lαx − Σ α n α v lαx + iωγ α v αy v ′ lαy = 0 (12)4y taking Fourier transform in x, we obtain the following standard dispersion relation ω − k x − X α n α γ α ! ω γ b γ p − ω γ b γ p X α n α γ α − ηT b k x ( ω γ p − n p ) ! − k x ω γ b γ p X α n α v α γ α − n p n b X α v α + 2 n p n b v p v b − ηT b k x n p v p ! = 0 (13)This equation contains two oscillatory mode and one purely growing electromagnetic modewhich is known as filamentation or Weibel instability. III. ANALYTICAL STUDIES
For analytical tractability of the inhomogeneous problem a choice of sinusoidal variationriding on a homogeneous background density such as n i = [1 + εcos ( k s x )] (14)is chosen. Here ε is the inhomogeneity amplitude and k s = 2 πm/L x (m is an integer) is theinhomogeneity wave number with L x as the system length. To satisfy the quasi-neutralitycondition the equilibrium beam and plasma density is chosen as n b = β (1 + εcos ( k s x )) n p = (1 − β )(1 + εcos ( k s x )) (15)where β is a fraction. Now choosing the perturbed fields as f α = Σ ...., ± , ± , j f αj e i (( k + jk s ) x − ωt ) and assuming ε to be small so that retaining only the first order terms (as done in the paperby Mishra et al. [20]), we can evaluate the growth rate. We evaluate the growth rate Γ gr asa function of k , using the same method. The plots shown in Fig. 2 compare the growth ratesfor the homogeneous case with inhomogeneous cases. Fig. 2 (a) , we plot the growth rate Γ gr versus wave vector k for a homogeneous (solid line) and inhomogeneous cold beam plasmasystem for ε =0.1, for k s = π (–), 2 π (-.-) and 3 π (*). The other parameters are n b /n p = 1 / n b /n e = 0 . v b = 0 . c , v p = − . c , T b = 0, T p = 0. From this plot we can seethat for the homogeneous case, the Γ gr increases with k for small values of k and saturatesat k ≃
3. However, for the inhomogeneous case ( ε =0.1) Γ gr for small values of k is largecompared to the homogeneous one. The increase in Γ gr with k is very mild and ultimatelythere is a saturation at higher k values. The maximum value of the growth rate Γ gr increases5ith k s for a cold system. The effect of transverse beam temperature ( T b ⊥ = 10 keV) overthe growth rate can be seen in Fig. 2(b). The growth rate Γ gr of the homogeneous hot beamand cold background plasma system (solid line) increases with k for small values of k butstarts decreasing after k > . k > k s over the Γ gr at fixed value of ε . We see that for ε = 0 . k s = π (which is λ s > c/ ω p ),the instability domain of Γ gr is much larger compared to the homogeneous case. It is alsoobserved that when the inhomogeneity scale length is further reduced to half, i.e. k s = 2 π (-o-) for ε = 0 .
1, the instability is suppressed for all values of k . In Fig. 2(c), we plot thegrowth rate Γ gr versus wave vector kc/ω p to see the effect of ε on Γ gr at T b ⊥ = 10 keV. Wesee that Γ gr reduces with increasing ε . Thus the study shows that at finite transverse beamtemperature the effect of increasing k s as well as ε is to stabilize the Weibel instability. InTable - I we have tabulated the value of maximum growth rate for various cases. TABLE I
The maximum growth rate of filamentation instability evaluated analytically under theapproximation of weak inhomogeneity amplitude. T b ⊥ ( keV ) ε k s Γ gr (max.)0.0 0.0 0.0 0.20060.0 0.1 π π π π π π π k s as well as ε . We feel that the reduction in growth rate6s responsible for transport over long distances observed in targets that had carbon nan-otubes attached to them in experiments. The analytical inference is further corroborated byParticle - In- Cell (PIC) simulations. The results of PIC studies are presented in the nextsection. IV. PIC SIMULATION DETAILS
We have performed Particle-In-Cell simulation to study the propagation of beam in bothhomogeneous and inhomogeneous plasma systems. The PIC code PICPSI3D used for thissimulation has been developed indigenously by one of the authors (Kartik patel) [21] andhas been used in several studies in the past [22]. The PICPSI3D was generalized by usto study the propagation of electron beam in an inhomogeneous plasma system. Only onedimensional variation in space (perpendicular to the beam propagation) has been chosen forour numerical studies here, for the purpose of comparison with recently carried out analyticallinear studies.The ions are not allowed to move in the simulations and merely provide a neutralizingbackground for the plasma. This makes the simulation faster and is a valid approximationto study the fast electron time scale phenomena of interest. The uniform plasma density n is chosen to be 10 n c where n c = 1 . × cm − is the critical density for 1 µm wavelength oflaser light. The spatial simulation box length L x = 60 c/ω , where c/ω = d e = 5 . × − µm is the skin depth corresponding to the density n . The one dimensional simulation box isdivided in 6000 cells. The grid size is therefore equal to 0 . d e . Thus, scales shorter thanthe skin depth can be resolved. The total number of electrons and ions chosen for thesimulations are 1800000 each. This number represents the sum of background n p and beam n b electrons. The choice of inhomogeneous ion density and the separation between the twoelectron species of beam and background are made as per Eq.(14) and Eq.(15)The beam temperature T b ⊥ is chosen to be finite in perpendicular direction according toEq.(8). The time step is decided by the Courant condition. The charge neutrality as wellas the null value of total current density is ensured initially. The considered system is alsofield free initially as required by equilibrium configuration. the system has an equilibriumconfiguration initially. 7 . SIMULATION OBSERVATIONS For the homogeneous plasma density case (e.g. ǫ = 0 . k s = 0 .
0) the system is plaguedby the usual Weibel instability. This causes spatial separation between the forward andreverse electron currents. The separation leads to finite current density in space resultingin the growth of magnetic field energy. The evolution of box averaged magnetic field energynormalized by E = ( m e cω /e ) energy of system is shown in Fig. 3 (b), Fig. 4 (b) and Fig. 5(b). After an initial transient the curve settles down to a linear regime and subsequentlyshows saturation.The slope of the linear portion of the main curve has been employed for the evaluationof the growth rate of the maximally unstable mode in the simulation. The growth rate hasbeen tabulated in Table - II for various cases of parameters. TABLE II:
The maximum growth rate of filamentation instability calculated from PIC simulation. T b ⊥ ( keV ) ε k s Γ gr (max.)0.0 0.0 0.0 0.20000.0 0.1 2 π π π π π π ǫ and k s the growth rate decreases when the temperature ofthe beam and background electrons is finite. This trend is similar to the behavior of growthrate evaluated analytically, shown in Table - I. When the plasma density is homogeneousthe value of the growth rates evaluated analytically and through simulations are in goodagreement. However, in the presence of inhomogeneity there is a small disagreement between8he quantitative values. This can be attributed to the approximate nature of the analyticaltreatment, wherein the inhomogeneity amplitude was assumed to be weak.From Figures 3 (a), 4 (a) and 5 (a) it can also be seen that along with the growth ofmagnetic field energy, electrostatic field energy also grows. The development of an electricfield directed along x during the course of simulation is responsible for this electrostaticenergy. This electrostatic field develops as a result of the redistribution and bunching ofelectron charges in physical x space. It can be seen from the phase space plots of Figs. 6 and9 that the electrons do reorganize in physical space. Furthermore, the locations where theseelectrons get accumulated are the regions with maximal currents and negligible magneticand electric field as can be seen from Fig. 7Finally we provide a comparison between the cases of homogeneous and inhomogeneousplasmas. It should be noted that the typical scale length of the magnetic field developedduring the initial phase in the homogeneous case (Fig. 8 (a)) is of the order of the backgroundplasma skin depth (e.g. 5 . × − µm ). For the inhomogeneous case, the scale length of themagnetic field matches initially with the inhomogeneity scale length defined by the choice of k s (Fig. 8 (b))(provided the scale length of inhomogeneity is smaller than the skin depth) elseit is determined by the typical value of skin depth. At later stages (the nonlinear phase ofthe instability), however, the magnetic structures coalesce and acquire long scales typicallycomparable to simulation box size in both homogeneous (Fig. 8 (c)) and in-homogeneous(Fig. 8 (d)) (provided the growth rate remains finite in this case) cases.To summarize the main observations are: (i) the inhomogeneous density causes no sig-nificant difference in the growth rate when the transverse temperature is chosen to be zero.(ii) the growth rate of the Weibel instability in the inhomogeneous density case is reducedcompared to the homogeneous case when the transverse beam temperature is finite, (iii)The momentum p x is typically quite large for beam electrons compared to the backgroundplasma electrons. (iv) in the nonlinear regime the typical profile of electrostatic field createddue to electron bunching in x is similar to that of the magnetic field. The zeros of boththe fields coincide with each other in space and it is these very locations where electronbunching is observed. 9 I. INTERPRETATION OF NUMERICAL OBSERVATIONS
We now provide a simplified understanding of the observations made by PIC simulationslisted out in previous section. In order to understand these results we consider the 1-Dlimit (with only variations along x being permitted) of the two fluid system of beam andbackground electrons described in section II. In 1-D the momentum equations of the twoelectron species from Eqs.(1) are: dp xb dt = − (cid:20) − ∂φ∂x + v yb B z (cid:21) − n b ∂P b ∂x (16) dp xp dt = − (cid:20) − ∂φ∂x + v yp B z (cid:21) (17) dp yb dt = − (cid:20) − ∂A y ∂t − v xb B z (cid:21) (18) dp yp dt = − (cid:20) − ∂A y ∂t − v xp B z (cid:21) (19)(20)Here p iα for i = x and i = y corresponds to the x and y component of momentum respectivelyfor the beam α = b or plasma α = p electrons. Also v iα = p iα /γ α (with γ α being therelativistic factor) is the corresponding velocity. Here P b represents the transverse pressurewhich is zero for the case when the system is cold. The scalar and vector potentials arerepresented by φ and ~A respectively. In 1-D only A y component is finite. Thus the onlyfinite component of magnetic field is along ˆ z and B z = ∂A y /∂x .The continuity equation can be written as ∂n α ∂t + ~v xα ∂n α ∂x + n α ∂v xα ∂x = 0 (21)The Maxwell’s equation become ∂ φ∂x = ( δn b + δn p ) (22) ∂ A y ∂t = ∂B x ∂x − [ n b v yb + n p v yp ] (23)Here n b and n p are the total densities of the beam and plasma electrons and δn b and δn p is the difference between the total and equilibrium densities respectively. If one considersthe transverse temperature to be zero, the linearization of the above set of equations has noother term dependent on the electron densities except for E x = − ∂φ/∂x in the momentum10quation. However, Weibel being primarily an electromagnetic instability the electrostaticfield is very weak. Thus, the predominant term in the momentum equation is due to sec-ond term of v y b B z , which is not influenced by the electron density. Thus, in the limit ofzero temperature the homogeneous and inhomogeneous cases do not show any significantdifference.When the transverse temperature is finite our simulations show reduction in the Weibelgrowth rate. In this case the pressure term in Eq.(13) is effective and depends on the densityinhomogeneity. It has been shown by an approximate analytical studies in [20] repeated andpresented by us in Fig.2 that the growth rate indeed decreases in the inhomogeneous case.This is the main result of our simulations which qualitatively verifies the approximate resultsof [20]. This can also be relevant for the observed propagation of electrons over long distancesin the presence of nanowires/nanotubes in experiments[19].As we have stated earlier along with the development of magnetic field an electrostaticfield also develops. This happens due to the bunching of electron densities at the location ofzero magnetic field as shown in Fig. 7. At the location of zero magnetic field the perturbeddensity shows a maxima and the electric field also passes through zero. This arrangementis self consistent. The Lorentz force at these locations vanishes and hence a particle hasa greater probability to accumulate over there. The location of maximum accumulation ofelectron density in turn results in the vanishing of the second derivative of electric field andfor a Fourier spectrum this location should correspond to the zero of electric field. VII. SUMMARY AND CONCLUSION
We have shown through 1D3V PIC simulations that the growth rate of Weibel instabilitygets reduced in the presence of density inhomogeneity. This has relevance to a recent ex-perimental observation of efficient transport of Mega Ampere of electron currents throughaligned carbon nanotube arrays. The ionization of the carbon nanotubes by the front of laserpulse produces the plasma which has inhomogeneous density. Since the Weibel instabilitygets suppressed in such a inhomogeneous plasma, the current separation is reduced leadingto the propagation of beam electrons over large distances.This mechanism of efficient electron transport was earlier invoked by Mishra et al. [20]wherein it was shown analytically using two fluid description that the Weibel instability11ets suppressed. The present work supplements it with PIC studies. Our PIC simulationssupport the analytical observations qualitatively. The quantitative values of the growth ratediffer slightly showing the approximate nature of the analysis.
Acknowledgement:
We thank S. Mishra for many useful discussions.12
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Physics of Plasmas , 20(7), 2013. n b /n e = 0 . v b = 0 . c , v p = − . c and for inhomogeneous, ε =0.1 at k s = π (–), k s = π (-.)and k s = π (*). (b) attransverse beam temperature T b ⊥ =10 keV, homogeneous(solid black), inhomogeneous for ε =0.1 at k s = π (- -), 2 π (-.) and 3 π (-*) (c) at transverse beam temperature T b ⊥ =10 keV,homogeneous(solid black), inhomogeneous at k s =3 π for ε =0.1(- -) and 0.2 (-.)16IG. 3: Temporal evolution of the field energy densities for cold homogeneous andinhomogeneous beam-plasma simulation. (a) normalized electrostatic x-component ofelectric field energy (b) normalized z-component of magnetic field energy −11 −10 −9 ω t B z / B T ⊥ b =10k(cid:13) v ε =0, k s =0 ε =0.1, k s =2 πε =0.2, k s =2 π −11 −10 −9 E x / E ω t T ⊥ b =10k(cid:13) v ε =0, k s =0 ε =0.1, k s =2 πε =0.2, k s =2 π FIG. 4: Temporal evolution of the field energy densities for hot ( T b ⊥ =10 keV )homogeneous and inhomogeneous beam-plasma simulation. (a) normalized electrostaticx-component of electric field energy E x for ε =0, k s =0 and ε =0.1 and 0.2 for k s =2 π (b)normalized z-component of magnetic field energy B z for ε =0, k s =0 and ε =0.1 and 0.2 for k s =2 π
50 100 15010 −11 −10 −9 ω t B z / B T ⊥ b =10k(cid:13) v ε =0, k s =0 ε =0.1, k s = πε =0.2, k s =3 π −11 −10 −9 E x / E ω t T ⊥ b =10k(cid:13) v ε =0, k s =0 ε =0.1, k s =3 πε =0.2, k s =3 π FIG. 5: Temporal evolution of the field energy densities for hot ( T b ⊥ =10 keV )homogeneous and inhomogeneous beam-plasma simulation. (a) normalized electrostaticx-component of electric field energy E x for ε =0, k s =0 and ε =0.1 and 0.2 for k s =3 π (b)normalized z-component of magnetic field energy B z for ε =0, k s =0 and ε =0.1 and 0.2 for k s =3 π FIG. 6: projection of f (x , p x ) for homogeneous cold beam plasma at ω t = 25.1789 and74.055518 E Z X FIG. 7: Bunching of perturbed electron19IG. 8: spatial configuration of normalized magnetic field homogeneous andinhomogeneous hot beam plasma (a) ε =0.0, k s =0 at ω t =49.3703 (b) ε =0.2, k s =3 π at ω t =49.3703 (c) ε =0.0, k s =0 at ω t =74.0555 (d) ε =0.2, k s =3 π at ω t =74.055520 a) (b) FIG. 9: projection of f (x , p x ) for warm system (a) beam with ε =0.0, k s =0 at ω t =49.3703(b) beam with ε =0.2, k s =3 π at ω tt