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Two-stream instabilities in degenerate quantum plasmas
Physics Letters A 378 (2014) 2505–2508
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Two-stream instabilities in degenerate quantum plasmas Seunghyeon Son Princeton Plasmonics, 18 Caleb Lane, Princeton, NJ 08540, United States
a r t i c l e
i n f o
Article history: Received 23 March 2014 Received in revised form 14 June 2014 Accepted 30 June 2014 Available online 3 July 2014 Communicated by F. Porcelli Keywords: Quantum diffraction and degeneracy Two-stream instability
a b s t r a c t The quantum effects on the plasma two-stream instability are studied by the dielectric function approach. The analysis suggests that the instability condition in a degenerate dense plasma deviates from the classical theory when the electron drift velocity is comparable to the Fermi velocity. Specifically, for a high wave vector comparable to the Fermi wave vector, a degenerate quantum plasma has larger regime of instability than predicted by the classical theory. A regime is identified, where there are unstable plasma waves with frequency 1.5 times of a normal Langmuir wave. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The physics of dense plasmas becomes increasingly important [1–3]. Especially, an accurate theory for the two-stream instability is critical in order to understand the dense electron beams in the inertial confinement fusion [1,4,5] and astrophysical events such as the gamma ray burst [6–9]. It is noteworthy to realize that many classical theories need to be revised in dense plasmas because the quantum effects cannot be ignored; a few physical processes deviating from the classical prediction have been identified [2,3,10–18]. The main goal of this paper is to study the electron quantum effects on the two-stream instabilities. There has been a general theoretical attempt to study the quantum effects on the two-stream instabilities by utilizing fluid-type equations [10,11]. In this paper, the author instead utilizes the Lindhard dielectric function [19] with a view of including the kinetic effects. The relative advantage of the current approach over the quantum fluid theory will be discussed. Two cases are studied; the first case when a plasma has two different groups of electrons and the second case when the electrons have a drift velocity different from the ions. The analysis in this paper suggests the following. When the electron drift velocity is comparable to or lower than the Fermi velocity, the quantum effect cannot be ignored; the instability regime is larger than the classical prediction. Also, a regime is identified where the unstable Langmuir wave can have higher frequency than a normal Langmuir wave by 1.5 times. This paper is organized as follows. In Section 2, the dielectric function approach in the two-stream instability is introduced. In Section 3, the case, when there are two different groups of degenerate electrons with different drift velocities, is considered. In
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2014.06.048 0375-9601/© 2014 Elsevier B.V. All rights reserved.
Section 4, the case, when the degenerate electrons have a drift velocity different from the ions, is studied. In Section 5, the summary and discussion are provided. 2. Dielectric function for the two-stream instability analysis The longitudinal dielectric function of a plasma is given as
4π e 2
(k, ω) = 1 +
k2
χi ,
(1)
where the summation is over the groups of particle species and χi is the susceptibility. Given the dielectric function (k, ω), the analysis of the two-stream instability can be performed by finding the roots of (k, ω) = 0; If the dielectric function has a root (k, ω) = 0 in the complex upper-half plane, the plasma is unstable to the two-stream instabilities. In classical plasmas, the susceptibility is given as
χiC (k, ω) =
ni Z i2
mi
k · ∇v fi
ω−k·v
d3 v
(2)
where mi ( Z i , ni ) is the particle mass (charge, density) and f i is the distribution with the normalization f i d3 v = 1. For degenerate electrons, the susceptibility χe by Lindhard [19] is given as
χeQ (k, ω) =
3ne me v 2F
√
h( z, u ),
(3)
where v F = 2E F /me is the Fermi velocity, E F = h¯ 2 k2F /2me (k F = (3π 2 ne )1/3 ) is the Fermi energy (Fermi wave vector), z = k/2k F , u = ω/kv F , and h = hr + ih i . The real part of h is given from Lindhard [19] as
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hr =
S. Son / Physics Letters A 378 (2014) 2505–2508
| z − u + 1| 2 8z | z − u + 1|
| 1 z + u + 1| . + 1 − ( z + u )2 log 8z | z + u + 1| 1
+
1
1 − ( z − u )2 log
(4)
The imaginary part h i is from [19]
hi =
π 2
u,
for | z + u | < 1,
π
u , for | z − u | < 1 < | z + u |, 8z = 0, if | z − u | > 1.
=
(5)
In the next two sections, the author considers two cases. The first case is when there are two groups of electrons, where each group is the Maxwellian distribution (the degenerate Fermi distribution) with different drift velocities. The second case is when the Maxwellian (completely degenerate) electrons have a different drift velocity from the ions. In the first case, the dielectric function is given as
= 1 + 4π e2 /k2 χeC (ω, k) + χeC (ω − k · v0 , k) or 1 + 4π e 2 /k2
χeQ (ω, k) + χeQ (ω − k · v0 , k) ,
(6)
Fig. 1. The real part of the dielectric function R = Re[ (k, ω)] as a function of ω . The x-axis is U = ω/ω pe and the y-axis is R = Re[ ]. In this example, ne = 1024 /cc, T e = 21 eV, kλde = 0.26, λde = ( T e /4π ne e 2 )1/2 is the Debye length and the drift velocity has the electron kinetic energy of 720 eV so that kv 0 /ω pe ∼ = 2.29. The local maximum of the real part at ω = 1.5ω pe is less than 0 so that the plasma is unstable to the two-stream instability.
where v0 is the drift velocity. In the second case, the dielectric function is given as
= 1 + 4π e2 /k2 χeC (ω, k) + χiC (ω − k · v0 , k) or 1 + 4π e 2 /k2
χeQ (ω, k) + χiC (ω − k · v0 , k) .
(7)
In deriving Eqs. (6) and (7), the author uses Eqs. (2) and (3). In the classical plasmas, it is trivial to add two susceptibilities of two different groups of electrons; the first equations in Eq. (6) and Eq. (7), which are the sum of the two susceptibilities, is obvious. On the other hand, it is not clear for the second one in Eq. (6) because the existence of another group of electrons would affect the electron transition due to the degeneracy. In this paper, the author assumes that this interference is negligible or h¯ k/me < v 0 . If h¯ k/me ∼ = v0, the treatment of the degeneracy would be more complicated and beyond the scope of this paper. In the example as illustrated in Fig. 2, the condition, h¯ k/me < v 0 , is satisfied. The validity of the second equation in Eq. (7) is straightforward as the ions does not affect the degeneracy of the electron transition. 3. When there are two different groups of (degenerate) electrons In this section, the author analyzes Eq. (6). In the conventional classical two-stream instability, the threshold condition for the two-stream instability is that there should be a local maximum between ω pe < ω < kv 0 and that the local maximum should be less than 0. In Fig. 1, the author plots the classical dielectric function as a function of ω for a particular k. The hump at ω ∼ = 1.3ω pe is the local maximum less than 0; the plasma Langmuir wave is unstable to the two-stream instability. For a fixed electron density and temperature (or completely degenerate), the unstable regimes are estimated by plotting (k, ω) as a function of ω and using the two criteria given above. The analysis for various v 0 and various wave vector k suggests that the plasma is unstable when k < kC ( v 0 ) where kC ( v 0 ) is a threshold value. In Fig. 2, the author plots the instability boundary for a plasma with ne = 1024 /cc. In the figure, for the classical plasmas (Classical 21 eV), the instability begins to emerge when v 0 / v F ≥ 3.5. As the electron temperature of the classical plasma gets lower, the cutoff velocity decreases. For the complete degenerate plasma, the instability begins to emerge when v 0 / v F ≥ 2.75, which is the minimal drift for the existence of the two-stream instability for any temperature. From Fig. 2, it can be concluded
Fig. 2. The boundary wave-vector k C as a function of the drift velocity. In this example, ne = 1024 /cc and E F = 36 eV. The x-axis is V = v 0 / v F and the y-axis is K = k C /k F . In this figure, the author plots the zero temperature quantum plasma (Quantum Zero). In addition, the author plots the dielectric function of the classical plasmas, where the electron temperature is assumed to be T e = 0.6E F = 21 eV since the average kinetic energy of the electron in the completely degenerate case is 0.6E F .
that the regime of the two-stream instability is larger in the quantum prediction than the classical prediction. As the drift velocity becomes larger than v 0 > 3.5v F , the difference between the quantum plasma and the classical plasma is negligible. For more dense plasma, the regime of the quantum deviation, where the degenerate plasma (classical plasma) is unstable (stable), widens further. In the case considered, the imaginary part of the dielectric function hr is zero so that the Landau damping is ignored. The cutoff at v 0 = 3.5v F (v 0 = 2.7v F ), or the end point of the left of each curve in Fig. 2 is due to the finite electron temperature effect; the cutoff is the upper limit where the collective waves can be sustained. The higher the temperature is, the higher the cutoff becomes. There is no such cutoff in the classical plasma (quantum plasma) with the zero electron temperature (no degeneracy), which suggests that the quantum fluid theory cannot predict such cutoff accurately [10,11]. For this reason, the kinetic approach in
S. Son / Physics Letters A 378 (2014) 2505–2508
Fig. 3. The real and imaginary part of the dielectric function as a function of the frequency for the classical (degenerate) plasmas. The x-axis is W = ω/ω pe and the y-axis is R = Re[ ] or R = Im[ ]. In this example, ne = 1026 /cc and E F = 784 eV, k = 0.26k F and v 0 = 1.3v F so that kv 0 /ω pe = 1.43. The author plots the completely degenerate plasma. In addition, the author plots the classical plasma with T e = 0.6E F = 470 eV using Eq. (2). The local maximum of the real part of in the degenerate (at ω = 1.4ω pe ) is less than 0 and thus the plasma Langmuir wave is unstable for the quantum plasma. In the figure, the ‘Imag Q Zero’ (’Q Zero’) stands for the imaginary (real) part of the dielectric function for the completely degenerate plasma, the ‘Imag Cla 470 eV’ (‘Cla 470 eV’) stands for the imaginary (real) part of the dielectric function for the classical plasma. There is no local maximum for the classical plasmas and it is stable to the two-stream instability.
this paper is superior to the quantum fluid theory in the prediction of the two-stream instability for a high wave vector. 4. When the (degenerate) electrons have different drift velocity from the ions In this section, the author analyzes Eq. (6). The authors employs the same criteria used in Section 2. In Fig. 3, the author plots the real part of the dielectric function of a classical plasmas and a degenerate plasma. In this section, it is assumed that the ion temperature is zero and the ion is treated as the classical particle. As well-known in the conventional Lindhard function [19], the degenerate plasma can support the collective Langmuir waves whose wave length is much smaller than the classical Debye length. The regime of the two-stream instability is larger in the degenerate plasma than the classical plasmas due to this fact. The example in Fig. 3 is the case when the classical plasmas are stable regardless of the wave vector. As shown in the figure, the frequency of the unstable Langmuir wave is 1.4ω pe , which cannot be sustained in the classical plasmas and can only be supported via the electron diffraction and degeneracy effects. In the example, the imaginary part of the quantum plasma is zero at the resonance, but it is not negligible for the classical plasmas. Since the classical plasma is already stable to the two-stream instability, the Landau damping will further stabilize it. The exact computation of the stability when the plasma is subject to the two-stream instability but the Landau damping is not negligible is beyond the scope of this paper. In Fig. 4, the author plots the threshold condition for the twostream instability in the degenerate plasmas and classical plasmas; the instability condition is k < kC ( v 0 ). For the classical plasmas, the regime of the instabilities is smaller than the degenerate plasma. As in the previous section, the cutoff at v 0 = 1.35v F (v = 1.2v F ) for the classical plasma (the degenerate plasma) in Fig. 4 is due to the finite electron temperature effect; The cutoff is the upper limit where the collective waves can be sustained. The higher the temperature, the higher the cutoff becomes. In the classical plasma (quantum plasma) with the zero electron temperature
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Fig. 4. The boundary wave-vector k C as a function of the drift velocity. In this example, ne = 1026 /cc, E F = 784 eV. The x-axis is V = v 0 / v F and the y-axis is K = k C /k F . The author plots the completely degenerate plasma using Eq. (2). In addition, the author plots the classical plasma with the electron temperature T e = 0.6E F using Eq. (3). For the classical plasmas, the instability begin to emerge when v 0 / v F ≥ 1.35 and the cutoff in the figure is due to this. As the electron temperature decreases, the cutoff drift velocity decreases. For the degenerate plasmas, the instability begin to emerge when v 0 / v F ≥ 1.20, which is the minimum drift velocity for the two-stream instability in any electron temperature. The classical plasma and the quantum plasma has the same instability boundary for k C /k > 1.48.
(no degeneracy), there is no such cutoff, resulting that the quantum fluid theory cannot predict such cutoff accurately [10,11]. 5. Summary and conclusion In this paper, the author analyzes the electron quantum effects on the two-stream instability, focusing on the plasmas relevant to the inertial confinement fusion and the astrophysical system with the electron density ranging from ne = 1024 /cc to ne = 1028 /cc. The Lindhard random phase approximation is employed. The analysis suggests that the quantum effects are more pronounced as the density gets higher. If the drift velocity is smaller than 3.5 times of the Fermi velocity, the quantum mechanical prediction for the two-stream instability differs from the classical calculation. Specifically, for both cases the author considers in Sections 2 and 3, the regime of the instability predicted by the Lindhard approach is considerably larger than the classical one as illustrated in Figs. 2 and 4. The discussion on the cutoff in Figs. 2 and 4 suggests that the kinetic approach on the quantum plasma is superior to the quantum fluid theory. The author also has shown that the unstable Langmuir wave due to the two-stream instability can have the frequency as large as 1.5ω pe , which is impossible in the classical plasmas. Theses findings in this paper can have major implications on the Backward Raman scattering [20–26] and the beam stopping by the dense background plasmas. In the backward Raman scattering, the momentum conservation and energy conservation between the photons and the plasmon will be modified especially when the plasmon wave vector is high. The degeneracy will probably enhance or weaken the backward Raman scattering depending on the physical regime; some quantum aspects of the backward Raman scattering have been studied by the author [27]. On the beam stopping, the two-stream instability has larger regime of instability from the quantum theory than the classical theory; the probability of the electron beam or the ion beam exciting the Langmuir waves via the two-stream instability is higher so that the quantum theory of the two-stream instability implies the stronger electron or ion stopping than the classical theory. On the other hand, the electron degeneracy normally reduces the stopping power from the electron
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S. Son / Physics Letters A 378 (2014) 2505–2508
collisions. The above consideration suggests that there could be a physical regime where the beam stopping by the electrons could be dominated by the two-stream instability rather than the electron collisions. Especially, this is possible when the drift velocity is comparable to the Fermi energy. Those exciting questions should be addressed in the future researches. One important limitation of this paper is the fact that the electron degeneracy is short lived; the consideration in this paper would be only relevant to the initial phase of the plasma evolution because the heating from the two-stream instability will reduce the electron degeneracy eventually. References [1] M. Tabak, J. Hammer, M.E. Glinsky, W.L. Kruerand, S.C. Wilks, J. Woodworth, E.M. Campbell, M.J. Perry, R.J. Mason, Phys. Plasmas 1 (1994) 1626–1634. [2] S. Son, N.J. Fisch, Phys. Rev. Lett. 95 (2005) 225002. [3] S. Son, N.J. Fisch, Phys. Lett. A 329 (2004) 76–82. [4] S.P.D. Mangles, C.D. Murphy, Z. Najmudin, A.G.R. Thomas, J.L. Collier, A.E. Dangor, E.J. Divall, P.S. Foster, J.G. Gallacher, C.J. Hooker, et al., Nature 431 (2004) 535–538.
[5] M. Tatarakis, F.N. Beg, E.L. Clark, A.E. Dangor, R.D. Edwards, R.G. Evans, T.J. Goldsack, K.W.D. Ledingham, P.A. Norreys, M.A. Sinclair, M.S. Wei, M. Zepf, K. Krushelnick, Phys. Rev. Lett. 90 (2003) 175001. [6] T.P. Fleming, J.M. Stone, J.F. Hawley, Astrophys. J. 530 (2000) 464. [7] D.E. Innes, B. Inhester, W.I. Axford, K. Wilhelm, Nature 386 (1997) 811–813. [8] E. Nakar, Phys. Rep. 442 (2007) 166–236. [9] D. Biskamp, Phys. Fluids 29 (1986) 1520–1531. [10] F. Haas, G. Manfredi, M. Feix, Phys. Rev. E 62 (2000) 2763–2772. [11] G. Manfredi, F. Hass, Phys. Rev. B 64 (2001) 075316. [12] P.K. Shukla, B. Eliasson, Phys. Usp. 53 (2010) 51–76. [13] I. Zeba, W.M. Moslem, P.K. Shukla, Astrophys. J. 570 (2012) 71. [14] S.H. Glenzer, L.M. Divol, R.L. Berger, C. Geddes, R.K. Kirkwood, J.D. Moody, E.A. Williams, P.E. Young, Phys. Rev. Lett. 86 (2001) 2565–2568. [15] R.P. Drake, S.H. Batha, Phys. Fluids B 3 (1991) 2936–2938. [16] D.A. Horner, J.D. Kress, L.A. Collins, Phys. Rev. B 81 (2010) 214301. [17] K.P. Driver, B. Militzer, Phys. Rev. Lett. 108 (2012) 115502. [18] S. Son, N.J. Fisch, Phys. Lett. A 356 (2006) 65–71. [19] J. Lindhard, K. Dan, Vidensk. Sels. Mat. Fys. Medd. 28 (1954) 8. [20] S. Son, S. Ku, S.J. Moon, Phys. Plasmas 17 (2010) 114506. [21] S. Son, S. Ku, Phys. Plasmas 17 (2010) 010703. [22] S. Son, Sung Joon Moon, Appl. Phys. Lett. 98 (2011) 081501. [23] C.J. McKinstrie, A. Simon, Phys. Fluids 29 (1986) 1959–1970. [24] V.M. Malkin, N.J. Fisch, Phys. Rev. Lett. 99 (2007) 205001. [25] S. Son, Sung Joon Moon, Phys. Plasmas 18 (2011) 084504. [26] S. Son, S. Ku, Sung Joon Moon, Phys. Plasmas 17 (2010) 112709. [27] S. Son, Phys. Plasmas 21 (2014) 034502.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Two-stream instabilities in degenerate quantum plasmas Seunghyeon Son Princeton Plasmonics, 18 Caleb Lane, Princeton, NJ 08540, United States
a r t i c l e
i n f o
Article history: Received 23 March 2014 Received in revised form 14 June 2014 Accepted 30 June 2014 Available online 3 July 2014 Communicated by F. Porcelli Keywords: Quantum diffraction and degeneracy Two-stream instability
a b s t r a c t The quantum effects on the plasma two-stream instability are studied by the dielectric function approach. The analysis suggests that the instability condition in a degenerate dense plasma deviates from the classical theory when the electron drift velocity is comparable to the Fermi velocity. Specifically, for a high wave vector comparable to the Fermi wave vector, a degenerate quantum plasma has larger regime of instability than predicted by the classical theory. A regime is identified, where there are unstable plasma waves with frequency 1.5 times of a normal Langmuir wave. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The physics of dense plasmas becomes increasingly important [1–3]. Especially, an accurate theory for the two-stream instability is critical in order to understand the dense electron beams in the inertial confinement fusion [1,4,5] and astrophysical events such as the gamma ray burst [6–9]. It is noteworthy to realize that many classical theories need to be revised in dense plasmas because the quantum effects cannot be ignored; a few physical processes deviating from the classical prediction have been identified [2,3,10–18]. The main goal of this paper is to study the electron quantum effects on the two-stream instabilities. There has been a general theoretical attempt to study the quantum effects on the two-stream instabilities by utilizing fluid-type equations [10,11]. In this paper, the author instead utilizes the Lindhard dielectric function [19] with a view of including the kinetic effects. The relative advantage of the current approach over the quantum fluid theory will be discussed. Two cases are studied; the first case when a plasma has two different groups of electrons and the second case when the electrons have a drift velocity different from the ions. The analysis in this paper suggests the following. When the electron drift velocity is comparable to or lower than the Fermi velocity, the quantum effect cannot be ignored; the instability regime is larger than the classical prediction. Also, a regime is identified where the unstable Langmuir wave can have higher frequency than a normal Langmuir wave by 1.5 times. This paper is organized as follows. In Section 2, the dielectric function approach in the two-stream instability is introduced. In Section 3, the case, when there are two different groups of degenerate electrons with different drift velocities, is considered. In
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2014.06.048 0375-9601/© 2014 Elsevier B.V. All rights reserved.
Section 4, the case, when the degenerate electrons have a drift velocity different from the ions, is studied. In Section 5, the summary and discussion are provided. 2. Dielectric function for the two-stream instability analysis The longitudinal dielectric function of a plasma is given as
4π e 2
(k, ω) = 1 +
k2
χi ,
(1)
where the summation is over the groups of particle species and χi is the susceptibility. Given the dielectric function (k, ω), the analysis of the two-stream instability can be performed by finding the roots of (k, ω) = 0; If the dielectric function has a root (k, ω) = 0 in the complex upper-half plane, the plasma is unstable to the two-stream instabilities. In classical plasmas, the susceptibility is given as
χiC (k, ω) =
ni Z i2
mi
k · ∇v fi
ω−k·v
d3 v
(2)
where mi ( Z i , ni ) is the particle mass (charge, density) and f i is the distribution with the normalization f i d3 v = 1. For degenerate electrons, the susceptibility χe by Lindhard [19] is given as
χeQ (k, ω) =
3ne me v 2F
√
h( z, u ),
(3)
where v F = 2E F /me is the Fermi velocity, E F = h¯ 2 k2F /2me (k F = (3π 2 ne )1/3 ) is the Fermi energy (Fermi wave vector), z = k/2k F , u = ω/kv F , and h = hr + ih i . The real part of h is given from Lindhard [19] as
2506
hr =
S. Son / Physics Letters A 378 (2014) 2505–2508
| z − u + 1| 2 8z | z − u + 1|
| 1 z + u + 1| . + 1 − ( z + u )2 log 8z | z + u + 1| 1
+
1
1 − ( z − u )2 log
(4)
The imaginary part h i is from [19]
hi =
π 2
u,
for | z + u | < 1,
π
u , for | z − u | < 1 < | z + u |, 8z = 0, if | z − u | > 1.
=
(5)
In the next two sections, the author considers two cases. The first case is when there are two groups of electrons, where each group is the Maxwellian distribution (the degenerate Fermi distribution) with different drift velocities. The second case is when the Maxwellian (completely degenerate) electrons have a different drift velocity from the ions. In the first case, the dielectric function is given as
= 1 + 4π e2 /k2 χeC (ω, k) + χeC (ω − k · v0 , k) or 1 + 4π e 2 /k2
χeQ (ω, k) + χeQ (ω − k · v0 , k) ,
(6)
Fig. 1. The real part of the dielectric function R = Re[ (k, ω)] as a function of ω . The x-axis is U = ω/ω pe and the y-axis is R = Re[ ]. In this example, ne = 1024 /cc, T e = 21 eV, kλde = 0.26, λde = ( T e /4π ne e 2 )1/2 is the Debye length and the drift velocity has the electron kinetic energy of 720 eV so that kv 0 /ω pe ∼ = 2.29. The local maximum of the real part at ω = 1.5ω pe is less than 0 so that the plasma is unstable to the two-stream instability.
where v0 is the drift velocity. In the second case, the dielectric function is given as
= 1 + 4π e2 /k2 χeC (ω, k) + χiC (ω − k · v0 , k) or 1 + 4π e 2 /k2
χeQ (ω, k) + χiC (ω − k · v0 , k) .
(7)
In deriving Eqs. (6) and (7), the author uses Eqs. (2) and (3). In the classical plasmas, it is trivial to add two susceptibilities of two different groups of electrons; the first equations in Eq. (6) and Eq. (7), which are the sum of the two susceptibilities, is obvious. On the other hand, it is not clear for the second one in Eq. (6) because the existence of another group of electrons would affect the electron transition due to the degeneracy. In this paper, the author assumes that this interference is negligible or h¯ k/me < v 0 . If h¯ k/me ∼ = v0, the treatment of the degeneracy would be more complicated and beyond the scope of this paper. In the example as illustrated in Fig. 2, the condition, h¯ k/me < v 0 , is satisfied. The validity of the second equation in Eq. (7) is straightforward as the ions does not affect the degeneracy of the electron transition. 3. When there are two different groups of (degenerate) electrons In this section, the author analyzes Eq. (6). In the conventional classical two-stream instability, the threshold condition for the two-stream instability is that there should be a local maximum between ω pe < ω < kv 0 and that the local maximum should be less than 0. In Fig. 1, the author plots the classical dielectric function as a function of ω for a particular k. The hump at ω ∼ = 1.3ω pe is the local maximum less than 0; the plasma Langmuir wave is unstable to the two-stream instability. For a fixed electron density and temperature (or completely degenerate), the unstable regimes are estimated by plotting (k, ω) as a function of ω and using the two criteria given above. The analysis for various v 0 and various wave vector k suggests that the plasma is unstable when k < kC ( v 0 ) where kC ( v 0 ) is a threshold value. In Fig. 2, the author plots the instability boundary for a plasma with ne = 1024 /cc. In the figure, for the classical plasmas (Classical 21 eV), the instability begins to emerge when v 0 / v F ≥ 3.5. As the electron temperature of the classical plasma gets lower, the cutoff velocity decreases. For the complete degenerate plasma, the instability begins to emerge when v 0 / v F ≥ 2.75, which is the minimal drift for the existence of the two-stream instability for any temperature. From Fig. 2, it can be concluded
Fig. 2. The boundary wave-vector k C as a function of the drift velocity. In this example, ne = 1024 /cc and E F = 36 eV. The x-axis is V = v 0 / v F and the y-axis is K = k C /k F . In this figure, the author plots the zero temperature quantum plasma (Quantum Zero). In addition, the author plots the dielectric function of the classical plasmas, where the electron temperature is assumed to be T e = 0.6E F = 21 eV since the average kinetic energy of the electron in the completely degenerate case is 0.6E F .
that the regime of the two-stream instability is larger in the quantum prediction than the classical prediction. As the drift velocity becomes larger than v 0 > 3.5v F , the difference between the quantum plasma and the classical plasma is negligible. For more dense plasma, the regime of the quantum deviation, where the degenerate plasma (classical plasma) is unstable (stable), widens further. In the case considered, the imaginary part of the dielectric function hr is zero so that the Landau damping is ignored. The cutoff at v 0 = 3.5v F (v 0 = 2.7v F ), or the end point of the left of each curve in Fig. 2 is due to the finite electron temperature effect; the cutoff is the upper limit where the collective waves can be sustained. The higher the temperature is, the higher the cutoff becomes. There is no such cutoff in the classical plasma (quantum plasma) with the zero electron temperature (no degeneracy), which suggests that the quantum fluid theory cannot predict such cutoff accurately [10,11]. For this reason, the kinetic approach in
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Fig. 3. The real and imaginary part of the dielectric function as a function of the frequency for the classical (degenerate) plasmas. The x-axis is W = ω/ω pe and the y-axis is R = Re[ ] or R = Im[ ]. In this example, ne = 1026 /cc and E F = 784 eV, k = 0.26k F and v 0 = 1.3v F so that kv 0 /ω pe = 1.43. The author plots the completely degenerate plasma. In addition, the author plots the classical plasma with T e = 0.6E F = 470 eV using Eq. (2). The local maximum of the real part of in the degenerate (at ω = 1.4ω pe ) is less than 0 and thus the plasma Langmuir wave is unstable for the quantum plasma. In the figure, the ‘Imag Q Zero’ (’Q Zero’) stands for the imaginary (real) part of the dielectric function for the completely degenerate plasma, the ‘Imag Cla 470 eV’ (‘Cla 470 eV’) stands for the imaginary (real) part of the dielectric function for the classical plasma. There is no local maximum for the classical plasmas and it is stable to the two-stream instability.
this paper is superior to the quantum fluid theory in the prediction of the two-stream instability for a high wave vector. 4. When the (degenerate) electrons have different drift velocity from the ions In this section, the author analyzes Eq. (6). The authors employs the same criteria used in Section 2. In Fig. 3, the author plots the real part of the dielectric function of a classical plasmas and a degenerate plasma. In this section, it is assumed that the ion temperature is zero and the ion is treated as the classical particle. As well-known in the conventional Lindhard function [19], the degenerate plasma can support the collective Langmuir waves whose wave length is much smaller than the classical Debye length. The regime of the two-stream instability is larger in the degenerate plasma than the classical plasmas due to this fact. The example in Fig. 3 is the case when the classical plasmas are stable regardless of the wave vector. As shown in the figure, the frequency of the unstable Langmuir wave is 1.4ω pe , which cannot be sustained in the classical plasmas and can only be supported via the electron diffraction and degeneracy effects. In the example, the imaginary part of the quantum plasma is zero at the resonance, but it is not negligible for the classical plasmas. Since the classical plasma is already stable to the two-stream instability, the Landau damping will further stabilize it. The exact computation of the stability when the plasma is subject to the two-stream instability but the Landau damping is not negligible is beyond the scope of this paper. In Fig. 4, the author plots the threshold condition for the twostream instability in the degenerate plasmas and classical plasmas; the instability condition is k < kC ( v 0 ). For the classical plasmas, the regime of the instabilities is smaller than the degenerate plasma. As in the previous section, the cutoff at v 0 = 1.35v F (v = 1.2v F ) for the classical plasma (the degenerate plasma) in Fig. 4 is due to the finite electron temperature effect; The cutoff is the upper limit where the collective waves can be sustained. The higher the temperature, the higher the cutoff becomes. In the classical plasma (quantum plasma) with the zero electron temperature
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Fig. 4. The boundary wave-vector k C as a function of the drift velocity. In this example, ne = 1026 /cc, E F = 784 eV. The x-axis is V = v 0 / v F and the y-axis is K = k C /k F . The author plots the completely degenerate plasma using Eq. (2). In addition, the author plots the classical plasma with the electron temperature T e = 0.6E F using Eq. (3). For the classical plasmas, the instability begin to emerge when v 0 / v F ≥ 1.35 and the cutoff in the figure is due to this. As the electron temperature decreases, the cutoff drift velocity decreases. For the degenerate plasmas, the instability begin to emerge when v 0 / v F ≥ 1.20, which is the minimum drift velocity for the two-stream instability in any electron temperature. The classical plasma and the quantum plasma has the same instability boundary for k C /k > 1.48.
(no degeneracy), there is no such cutoff, resulting that the quantum fluid theory cannot predict such cutoff accurately [10,11]. 5. Summary and conclusion In this paper, the author analyzes the electron quantum effects on the two-stream instability, focusing on the plasmas relevant to the inertial confinement fusion and the astrophysical system with the electron density ranging from ne = 1024 /cc to ne = 1028 /cc. The Lindhard random phase approximation is employed. The analysis suggests that the quantum effects are more pronounced as the density gets higher. If the drift velocity is smaller than 3.5 times of the Fermi velocity, the quantum mechanical prediction for the two-stream instability differs from the classical calculation. Specifically, for both cases the author considers in Sections 2 and 3, the regime of the instability predicted by the Lindhard approach is considerably larger than the classical one as illustrated in Figs. 2 and 4. The discussion on the cutoff in Figs. 2 and 4 suggests that the kinetic approach on the quantum plasma is superior to the quantum fluid theory. The author also has shown that the unstable Langmuir wave due to the two-stream instability can have the frequency as large as 1.5ω pe , which is impossible in the classical plasmas. Theses findings in this paper can have major implications on the Backward Raman scattering [20–26] and the beam stopping by the dense background plasmas. In the backward Raman scattering, the momentum conservation and energy conservation between the photons and the plasmon will be modified especially when the plasmon wave vector is high. The degeneracy will probably enhance or weaken the backward Raman scattering depending on the physical regime; some quantum aspects of the backward Raman scattering have been studied by the author [27]. On the beam stopping, the two-stream instability has larger regime of instability from the quantum theory than the classical theory; the probability of the electron beam or the ion beam exciting the Langmuir waves via the two-stream instability is higher so that the quantum theory of the two-stream instability implies the stronger electron or ion stopping than the classical theory. On the other hand, the electron degeneracy normally reduces the stopping power from the electron
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collisions. The above consideration suggests that there could be a physical regime where the beam stopping by the electrons could be dominated by the two-stream instability rather than the electron collisions. Especially, this is possible when the drift velocity is comparable to the Fermi energy. Those exciting questions should be addressed in the future researches. One important limitation of this paper is the fact that the electron degeneracy is short lived; the consideration in this paper would be only relevant to the initial phase of the plasma evolution because the heating from the two-stream instability will reduce the electron degeneracy eventually. References [1] M. Tabak, J. Hammer, M.E. Glinsky, W.L. Kruerand, S.C. Wilks, J. Woodworth, E.M. Campbell, M.J. Perry, R.J. Mason, Phys. Plasmas 1 (1994) 1626–1634. [2] S. Son, N.J. Fisch, Phys. Rev. Lett. 95 (2005) 225002. [3] S. Son, N.J. Fisch, Phys. Lett. A 329 (2004) 76–82. [4] S.P.D. Mangles, C.D. Murphy, Z. Najmudin, A.G.R. Thomas, J.L. Collier, A.E. Dangor, E.J. Divall, P.S. Foster, J.G. Gallacher, C.J. Hooker, et al., Nature 431 (2004) 535–538.
[5] M. Tatarakis, F.N. Beg, E.L. Clark, A.E. Dangor, R.D. Edwards, R.G. Evans, T.J. Goldsack, K.W.D. Ledingham, P.A. Norreys, M.A. Sinclair, M.S. Wei, M. Zepf, K. Krushelnick, Phys. Rev. Lett. 90 (2003) 175001. [6] T.P. Fleming, J.M. Stone, J.F. Hawley, Astrophys. J. 530 (2000) 464. [7] D.E. Innes, B. Inhester, W.I. Axford, K. Wilhelm, Nature 386 (1997) 811–813. [8] E. Nakar, Phys. Rep. 442 (2007) 166–236. [9] D. Biskamp, Phys. Fluids 29 (1986) 1520–1531. [10] F. Haas, G. Manfredi, M. Feix, Phys. Rev. E 62 (2000) 2763–2772. [11] G. Manfredi, F. Hass, Phys. Rev. B 64 (2001) 075316. [12] P.K. Shukla, B. Eliasson, Phys. Usp. 53 (2010) 51–76. [13] I. Zeba, W.M. Moslem, P.K. Shukla, Astrophys. J. 570 (2012) 71. [14] S.H. Glenzer, L.M. Divol, R.L. Berger, C. Geddes, R.K. Kirkwood, J.D. Moody, E.A. Williams, P.E. Young, Phys. Rev. Lett. 86 (2001) 2565–2568. [15] R.P. Drake, S.H. Batha, Phys. Fluids B 3 (1991) 2936–2938. [16] D.A. Horner, J.D. Kress, L.A. Collins, Phys. Rev. B 81 (2010) 214301. [17] K.P. Driver, B. Militzer, Phys. Rev. Lett. 108 (2012) 115502. [18] S. Son, N.J. Fisch, Phys. Lett. A 356 (2006) 65–71. [19] J. Lindhard, K. Dan, Vidensk. Sels. Mat. Fys. Medd. 28 (1954) 8. [20] S. Son, S. Ku, S.J. Moon, Phys. Plasmas 17 (2010) 114506. [21] S. Son, S. Ku, Phys. Plasmas 17 (2010) 010703. [22] S. Son, Sung Joon Moon, Appl. Phys. Lett. 98 (2011) 081501. [23] C.J. McKinstrie, A. Simon, Phys. Fluids 29 (1986) 1959–1970. [24] V.M. Malkin, N.J. Fisch, Phys. Rev. Lett. 99 (2007) 205001. [25] S. Son, Sung Joon Moon, Phys. Plasmas 18 (2011) 084504. [26] S. Son, S. Ku, Sung Joon Moon, Phys. Plasmas 17 (2010) 112709. [27] S. Son, Phys. Plasmas 21 (2014) 034502.