1. INTRODUCTION
When an intense laser pulse (>1018 W/cm2) irradiates a solid target, the light energy is deposited at the relativistic critical density, and then the relativistic electrons are produced (Perry & Mourou, Reference Perry and Mourou1994). These laser-driven fast electron beams constitute the mega-ampere currents and have been proposed that such a beam could be used in astrophysical plasmas (Gibbon, Reference Gibbon2005), radiography (Park et al., Reference Park, Chambers, Chung, Clarke, Eagleton, Giraldez, Goldsack, Heathcote, Izumi, Key, King, Koch, Landen, Nikroo, Patel, Price, Remington, Robey, Snavely, Steinman, Stephens, Stoeckl, Storm, Tabak, Theobald, Town, Wickersham and Zhang2006), and fusion physics (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994; Kodama et al., Reference Kodama, Norreys, Mima, Dangor, Evans, Fujita, Kitagawa, Krushelnick, Miyakoshi, Miyanaga, Norimatsu, Rose, Shozaki, Shigemori, Sunahara, Tampo, Tanaka, Toyama, Yamanaka and Zepf2001; Atzeni, Reference Atzeni and M.-t.-V.J.2003). When these relativistic electrons enter in a dense plasma, the return current is immediately generated in the background plasma for the demand of the charge neutrality. However, the return current is unstable for the Weibel instability, which will result in the filaments of the forward propagating fast electron beams and the fast growth of the magnetic field perturbation (Weibel, Reference Weibel1959; Sentoku et al., Reference Sentoku, Mima, Kojima and Ruhl2000). These generated strong magnetic fields can significantly increase the divergence of the forward fast electrons. Lots of experimental and simulation works have shown that the fast electron beams from laser–matter interaction are generally of large divergence (40°–50°) (Sentoku et al., Reference Sentoku, Mima, Kojima and Ruhl2000; Santos et al., Reference Santos, Amiranoff, Baton, Gremillet, Koenig, Martinolli, Le Gloahec, Rousseaux, Batani, Bernardinello, Greison and Hall2002; Kodama et al., Reference Kodama, Azechi, Fujita, Habara, Izawa, Jitsuno, Jozaki, Kitagawa, Krushelnick, Matsuoka, Mima, Miyanaga, Nagai, Nagatomo, Nakai, Nishimura, Norimatsu, Norreys, Shigemori, Shiraga, Sunahara, Tanaka, Tanpo, Toyama, Tsubakimoto, Yamanaka and Zepf2004a ; Stephens et al., Reference Stephens, Snavely, Aglitskiy, Amiranoff, Andersen, Batani, Baton, Cowan, Freeman, Hall, Hatchett, Hill, Key, King, Koch, Koenig, MacKinnon, Lancaster, Martinolli, Norreys, Perelli-Cippo, Le Gloahec, Rousseaux, Santos and Scianitti2004; Lancaster et al., Reference Lancaster, Green, Hey, Akli, Davies, Clarke, Freeman, Habara, Key, Kodama, Krushelnick, Murphy, Nakatsutsumi, Simpson, Stephens, Stoeckl, Yabuuchi, Zepf and Norreys2007; Green et al., Reference Green, Ovchinnikov, Evans, Akli, Azechi, Beg, Bellei, Freeman, Habara, Heathcote, Key, King, Lancaster, Lopes, Ma, MacKinnon, Markey, McPhee, Najmudin, Nilson, Onofrei, Stephens, Takeda, Tanaka, Theobald, Tanimoto, Waugh, Van Woerkom, Woolsey, Zepf, Davies and Norreys2008). To lower the fast electron divergence, the control of the relativistic fast electrons has been studied extensively (Borghesi et al., Reference Borghesi, Mackinnon, Bell, Malka, Vickers, Willi, Davies, Pukhov and Meyer-ter-Vehn1999; Campbell et al., Reference Campbell, DeGroot, Mehlhorn, Welch and Oliver2003; Danson et al., Reference Danson, Brummitt, Clarke, Collier, Fell, Frackiewicz, Hancock, Hawkes, Hernandez-Gomez, Holligan, Hutchinson, Kidd, Lester, Musgrave, Neely, Neville, Norreys, Pepler, Reason, Shaikh, Winstone, Wyatt and Wyborn2004; Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagatomo, Mima and Koga2007; Robinson & Sherlock, Reference Robinson and Sherlock2007; Robinson et al., Reference Robinson, Kingham, Ridgers and Sherlock2008a , Reference Robinson, Sherlock and Norreys b ; Kar et al., Reference Kar, Robinson, Carroll, Lundh, Markey, McKenna, Norreys and Zepf2009).
Recently, the experimental works performed on the carbon nanotube (CNT) array have shown the novel transportation of the fast electrons in CNT. The collimated fast electron transportation over a macroscopic (millimeter) distance (about 100 times longer than the typical filamentation length) has been observed in the CNT array (Ji et al., Reference Ji, Jiang, Wu, Wang, Gu and Tang2010, Chatterjee et al., Reference Chatterjee, Singh, Ahmed, Robinson, Lad, Mondal, Narayanan, Srivastava, Koratkar, Pasley, Sood and Kumar2012; Liao et al., Reference Liao, Wu, Gu, Zhou, Wang, Fu, Yang and Tang2013; Liao et al., Reference Liao, Wu, Wang, Zhou, Fu and Tang2014). These experimental results indicate that the CNT array can effectively suppress the Weibel instability of the fast electron beams. Otherwise, the mega-ampere intensity cannot sustain as long as a millimeter.
Recently, Mishra et al. (Reference Mishra, Kaw, Das, Sengupta and Kumar2014) present an approach to achieve the suppression of the Weibel instability by periodically modifying the electron density of the background with the equilibrium density ripples shorter than the skin depth. It is pointed out that the CNT target is exactly the periodic structure with the density ripples and the observed collimated fast electron transportations are the direct consequence of the weakened Weibel instability by the periodic density ripples of the CNT array. However, they give these results by a linearized two stream fluid model. For the electrons in the CNT array structure, the density variation is not a weak ripple but a huge density variation (from zero density of the vacuum to the electron density of the CNT in a period). It is not appropriate to apply the linearized perturbation model on the CNT array.
In the present work, we research the filamentation of the fast electron beams in a nanotube array and the growth of the magnetic field by a hybrid-particle-in-cell simulation. It is found that the CNT array can effectively weaken the Weibel instability and the explanation of the over millimeter distance of the fast electrons transportation recently found in CNT array has been given (Ji et al., Reference Ji, Jiang, Wu, Wang, Gu and Tang2010; Chatterjee et al., Reference Chatterjee, Singh, Ahmed, Robinson, Lad, Mondal, Narayanan, Srivastava, Koratkar, Pasley, Sood and Kumar2012; Liao et al., Reference Liao, Wu, Gu, Zhou, Wang, Fu, Yang and Tang2013, Reference Liao, Wu, Wang, Zhou, Fu and Tang2014). The self-generated magnetic field and electric field near the surface of the tubes play the essential role. The electric field draws the fast electrons into the tube, but the magnetic field pulls them out, and these two forces will reach equilibrium near the tube surface (Kodama et al., Reference Kodama, Sentoku, Chen, Kumar, Hatchett, Toyama, Cowan, Freeman, Fuchs, Izawa, Key, Kitagawa, Kondo, Matsuoka, Nakamura, Nakatsutsumi, Norreys, Norimatsu, Snavely, Stephens, Tampo, Tanaka and Yabuuchi2004b ). It makes most of the fast electrons propagate near the tube surfaces, and they could hardly move across the tubes. As the inner spaces between the successive tubes are very short, this effect makes the spaces play as “traps”. The fast electrons are confined into these inner spaces, and their transverse dispersion is impeded. In the absence of the transverse movement of the fast electrons, the filamentation loses the supplies of the fast electrons and cannot develop forward. As a result, the Weibel instability is stabilized.
2. THEORY AND DISCUSSION
The generation of the magnetic field during the propagation of the fast electrons through a solid target can be described by combining a simple Ohm's law
$\vec E = - {\rm \eta} \vec j_{\rm f} $
, with Faraday's law to yield
$$\displaystyle{{\partial \vec B} \over {\partial t}} = {\rm \eta} \nabla \times \vec j_{\rm f} + (\nabla {\rm \eta} ) \times \vec j_{\rm f}, $$
where η is the resistivity, and
$\vec j_{\rm f} $
is the fast-electron current density. The first term on the right-hand side generates a magnetic field that pushes the fast electrons toward to the region where possesses a higher fast-electron current density, while the second term pushes the fast electrons toward to the region where possesses the higher resistivity. Moreover, the first term is the origin of the magnetic instability. Giving a ripple of the fast-electron current density, normal to the streaming direction, the generated field by the first term would enhance the current density ripples. The fast electrons are pushed into the high-density region by the magnetic field, and the generated field gets further stronger. As a result, the significant filamentation of the fast electrons is formed.
For the tube array, the inner spaces between tubes are vacuum, whose η is infinitely large, but the η of plasma in tubes is finite. Therefore, at the surface of the tubes, that is, the interface between the vacuum and the plasma, the gradient of the resistivity is very large and the generated field by the second terms of Eq. (1) is dominated. During the propagation of intense fast electron flux along the tubes, the fast electron beams are bent into the vacuum region by the magnetic field as depicted in Figure 1. In the transverse direction (normal to the tubes), these inner spaces act as the magnetic traps and the fast electrons are confined in these regions to propagate forward. Without enough transverse kinetic energy or force on the fast electrons, these electrons cannot escape from the magnetic traps.
Fig. 1. The plot of the magnetic field at the CNT surfaces.
Considering there existing a magnetic perturbation in the transverse direction. Because of the second term of Eq. (1), the transverse movement of these fast electrons is impeded by the magnetic field near the surfaces and the fast electrons are difficult to move across one tube to another. If the perturbation is larger than the aligning period of the tubes, the growing rate of the perturbation would be suppressed. For the perturbations smaller than the tube width, it is a well-known fact that the transverse electromagnetic perturbations are also weakened if the perturbations are shorter than the skin depth (Mishra et al., Reference Mishra, Kaw, Das, Sengupta and Kumar2014) (i.e., ck/ω pc > 1). Therefore, if the tube width is not larger than the skin depth, the magnetic instability is also suppressed. For a tube array, the inner magnetic instability (with large k and small size) can be suppressed by decreasing the tube width, while the magnetic instability across the tubes (with small k and large size) can be suppressed by increasing the resistive magnetic field at the tube surface.
3. NUMERICAL RESULTS AND DISCUSSION
For a further quantitative investigation, the numerical calculation is required. For a comparison's purpose, the growth of the instability in a bulk carbon target and the CNT array structure are both numerically modeled by a hybrid algorithm that means a code using a particle-in-cell algorithm to describe the kinetic fast electrons, where the background electrons are treated as a Spitzer resistive conductor (Spitzer & Harm, Reference Spitzer and Harm1953; Robinson & Sherlock, Reference Robinson and Sherlock2007; Robinson et al., Reference Robinson, Kingham, Ridgers and Sherlock2008a , Reference Robinson, Sherlock and Norreys b ). Because of the intense background return current induced by the fast electrons, the backgrounds are ionized into plasma by the Ohm heating in a short time. The ionization process is thus not taken into consideration and we assume that the background is instantaneously ionized and heated to 100 eV on the fast electrons arriving. Hence, we use the Spitzer resistivity model of the hot plasma for the background (Spitzer & Harm, Reference Spitzer and Harm1953):
$${\rm \eta} = 10^{ - 4} Z\ln {\rm \Lambda} /T_{\rm c} ^{1.5}, $$
where T c is the temperature of the cold electrons (in eV). The Ohm heating of the background plasma gives:
$$\displaystyle{{\partial T_{\rm c}} \over {\partial t}} = \displaystyle{{{\rm \eta}\, j_{\rm c}^2} \over C},$$
where C is the heat capacity of the cold electron specified by the ideal gas heat capacity: C = 1.5n c where n c and j c is the density and the current of the cold electron. Here, j c gives by generalized the Ohm law ignoring the heat pressure and the inertia term:
$$\vec j_{\rm c} = (\vec E - \displaystyle{{\vec j_{\rm c} \times \vec B} \over {en}})/{\rm \eta}. $$
The simulations are carried out in a domain of 6.0 µm × 6.0 µm (x × y), represented by a 4096 × 4096 grids. The CNT target is modeled by a group of tubes periodically aligning in the y-direction (the tube width is 50 nm and the space between the successive tubes is also 50 nm, ck/ω pc ≈ 5 for one single tube) as shown in Figure 2. The fast electron density in the tubes is set to 1.0 × 1022 cm−3 but zero for the inner spaces. These fast electrons are uniformly injected from the bottom of the box. The initial fast-electron energetic distribution is set as the Maxwellian of temperature: kT = 2.0 MeV and the initial velocity is along the y-axis. The electron density inside the tubes is chosen to be 6.0 × 1023 cm−3. The cold electrons in the background are assumed to be pre-heated to 100 eV.
Fig. 2. The plot of the CNT array structure. The blue color presents carbon regions and the white color present the vacuum regions. The tube width is 50 nm and the space between the successive tubes is also 50 nm. The fast electrons are uniformly injected from the bottom of the box with an average energy of 2.0 MeV.
By this model, we first simulate the evolution of the fast electrons in a bulk carbon plasma. The evolution of the fast electron distribution is contained in Figure 3, which illustrates a significant filamentation of the fast electrons. Figure 4 shows the evolution of the self-generated magnetic field. The mechanism of the Weibel instability is well known. While the fast electrons propagate in the uniform plasma, a return cold electron beam with an equal current is created and the equilibrium with two counter beams is formed. Giving a density ripple in the fast electron beam, normal to the streaming direction, it leads to a magnetic field, which repulses the return current out of the high-density region and reinforces the magnetic field again, by which, more and more fast electrons are gathered in the high-density region, thus providing a positive feedback responsible for the current sheet separation as depicted in Figure 3 and a very strong self-generated magnetic field as shown in Figure 4. It is clear that the transverse movement of the electrons driven by the magnetic field plays an essential role. If the transverse movement of the electrons is stopped, the Weibel instability will be weakened.
Fig. 3. The log10 plot of the fast electron density in a bulk carbon target in: (a) at 25 fs and (b) at 100 fs.
Fig. 4. The plots of the self-generated magnetic field in the bulk carbon target in: (a) at 25 fs and (b) at 100 fs.
The self-generated field in the CNT array can effectively stop the transverse movement of the fast electrons. Thus, the stabilization of the Weibel instability in this structure can be expected. Figure 5 gives the calculated fast electron distribution in the CNT array and Figure 6 shows the generated magnetic field. As these fast electrons enter the CNT array, a pair of the self-generated magnetic fields will create in the inner spaces between the successive tubes because of the strong mismatch of matter-vacuum at tube surface. Both these magnetic fields act to push the fast electrons into the inner spaces between the successive tubes. As a result, the fast electrons mainly propagate in the inner spaces. These magnetic fields act as a magnetic “trap” to trap the fast electrons in the inner spaces, by which, the transverse movement of the fast electrons is significantly suppressed. If there the fast-electron density ripples exist, the induced magnetic field cannot conquer the magnetic “trap” to push more fast electrons into the high-density region. It weakens the positive feedback between the magnetic field perturbation and the electrons density perturbation. Consequently, the magnetic field cannot grow strong enough to separate the current sheet as in bulk carbon background. Moreover, because of the tube diameter shorter than the skin depth, the transverse electromagnetic perturbation is also weakened and there is no current sheet separation happens in the tubes. Consequently, either for the whole tube array or a single tube, the growth of the Weibel instability in the CNT is significantly weakened.
Fig. 5. The log10 plot of fast electron density in the CNT array in: (a) at 25 fs and (b) at 100 fs.
Fig. 6. The plot of the magnetic field in the CNT array at 25 fs. The inserted figure shows the details of the magnetic field in the tubes and the inner spaces.
4. CONCLUSION
In summary, we has researched the evolution of the fast electrons’ filamentation in the CNT array by a hybrid particle-in-cell simulation and found that the self-generated magnetic field act to trap the fast electrons in the inner spaces between the successive tubes. It prevents the fast electrons move from one tube to others in the transverse direction, and thus breaks the positive feedback loop between the magnetic field perturbation and the electrons density perturbation. As a result, the fast electron filamentation and the Weibel instability are weakened. We believe that other similar nanotube array structures such as ZnO, Cu, and Au nanorod arrays can also weaken the Weibel instability as well as the CNT array. Because of the high-energy density delivered by the MeV electrons, these results indicate some specific applications in the high-energy physics, such as radiography, fast-electron beam focusing, and perhaps fast ignition.
ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos 11404045 and 11304407).
