Landau damping of electron-acoustic waves due to multi-plasmon resonances

In many equilibrium plasmas, electrons can be divided into two groups with different temperatures Goswami and Buti (1976). Such a division may be possible as certain electrons are preferentially heated by some external sources like beams or waves until a final equilibrium is reached. The existence of two-temperature electrons was also experimentally observed in a sputtering magnetron plasma Sheridan et al. (1991). However, in $1977$, it was investigated by Watanabe and Taniuti Watanabe and Taniuti (1977) that if electrons in a plasma with background ions have two different temperatures, there can exist an electron-acoustic wave (EAW). They also reported that if the hot electron number density is much higher than that of cold ones, the phase velocity of such waves lies in between the thermal velocities of cold and hot species, and the EAWs do not experience strong Landau damping. Later, in $1991$, Holloway and Dorning Holloway and Dorning (1991) noted that a small-amplitude undamped EAW can exist in plasmas with the dispersion relation of the form $\omega\approx 1.31kv_{\text{th}}$, where $\omega~{}(k)$ is the wave frequency (number) and $v_{th}$ is the electron thermal speed. Recently, based on the observations from the Van Allen Probes, Vasko et al. Vasko et al. (2017) reported the existence of EAWs in the inner magnetosphere with the similar properties as above but a different linear dispersion law (as per their notations): $\omega=kv_{0}\left(1-k^{2}d^{2}/2\right)$, where $v_{0}$ and $d$ are, respectively, the electron-acoustic velocity and dispersive scale. They also elucidated the observed asymmetric solitary waves in terms of electron-acoustic solitons and shocks. In an another work by Dillard et al. Dillard et al. (2018) it has been shown by means of theoretical and numerical analysis of hydrodynamic equations that the asymmetric solitary waves can appear due to the steepening of initially quasi-monochromatic EAWs. The beam-generated EAWs due to trapping of electrons by whistler waves has also been reported with the phase velocity slightly large than that of the whistler waves An et al. (2019). Although, such EAWs have similar properties with those stated before, their phase velocity is predicted a bit higher, i.e., $1.8\lesssim\omega/kv_{\text{th}}\lesssim 2.2$.

The EAWs are typically electrostatic plasma oscillations which propagate with a frequency below that of the Langmuir wave. The characteristics of EAWs have been extensively studied theoretically by means of Vlasov-Poisson simulation Holloway and Dorning (1991); Valentini et al. (2012) and PIC simulation Valentini et al. (2006), as well as experimentally Anderegg et al. (2009); Chowdhury et al. (2017). However, all these investigations were limited to the classical regimes. Recently, a liner theory of EAWs in degenerate plasmas with two groups of electrons has been studied by Fahad et al. Fahad et al. (2019). They considered a fully degenerate plasma with two different Fermi temperatures of electrons. However, such a model may not be physically sound in the context of quantum mechanics as there should not be more than one species of electrons in a fully degenerate plasma or a zero-temperature Fermi gas.

In plasmas with two-temperature electrons, the nonlinear theory of EAWs has been investigated by many authors, however, using the fluid theory approach, i.e., without the Landau damping effects Lakhina et al. (2008); Akter et al. (2016). On the other hand, after the work of Ott and Sudan Ott and Sudan (1969) on nonlinear theory of ion-acoustic waves with linear Landau damping, many authors have studied the Landau damping of low-frequency electrostatic waves W. Vandam and Taniuti (1973); Misra and Barman (2015); Barman and Misra (2017) as well as high-frequency wave envelopes Chatterjee and Misra (2015); Misra et al. (2017), however, no nonlinear theory of EAWs with Landau damping has been developed in the context of quantum plasmas. While the linear plasmon resonance is known in the literature, the multi-plasmon resonance Brodin et al. (2017) in the nonlinear regime has not yet been fully explored, especially in the context of Korteweg-de Vries (KdV) equation.

In this work, we will generalize the previous works on EAWs to include short-scale quantum effects. This is of interest for dense plasmas where the characteristic de Broglie wavelength of electrons is larger than the nearest neighboring distance. Such dense plasmas include those appear in various astrophysical environments and certain laboratory systems, e.g., cases where high power lasers compress solid density targets. We consider weakly coupled systems only such that a background distribution, deviating from the thermodynamic equilibrium, can be sustained. More specifically, we study the linear and nonlinear theories of EAWs in a partially degenerate plasma consisting of two groups of electrons with their thermodynamic temperatures $(T_{j})$ exceeding the Fermi temperature $(T_{Fj})$ somewhat (i.e., $T_{j}>T_{Fj}$, where $j=l,h$ stand for low and high temperature electrons) and stationary positive ions. The purpose of this study is to find the quantum generalization of known classical results of EAWs including wave-particle interaction. As a consequence, we need to consider the Wigner theory rather than the classical Vlasov or semi-classical Vlasov theories to take into account the effects of quantum dispersion and the wave-particle resonance. A specific and novel feature introduced by the Wigner theory, is a nonlinear wave-particle interaction referred to as multi-plasmon interaction (see e.g. Refs. (Misra et al., 2017; Brodin et al., 2017)) where we can have simultaneous absorption (or emission) of multiple plasmon quanta. Including this mechanism into the calculation scheme, an additional wave damping due to the two-plasmon resonance is seen to occur in the evolution of EAWs along with the resonance of linear theory. For these types of effect to be significant, we have to focus on the regime of short-wavelengths, i.e., the thermal de Broglie wavelength can not be much smaller than that of EAWs $(\hbar k/m\lesssim v_{tl})$ which still accounts for some quantum effects to enter the picture. Here, $\hbar=h/2\pi$ is the reduced Planck’s constant, $k$ is the wave number, $m$ is the electron mass, and $v_{tl}$ is the thermal velocity of low-temperature electrons.

An important aspect for the details of EAWs is the properties of the electron background distribution function. For dense plasmas, as can be found, e.g., in the interiors of giant stars like white dwarfs and neutron stars Horn (1991); Chabrier et al. (2002), as well as in gas giants (e.g., Jupiter) Coppari , typically the density is high enough for the electron distribution function to be degenerate or partially degenerate. In our study, the case of partial degeneracy is in the focus. Moreover, we are concerned with the background distributions deviating from the thermodynamic equilibrium. The high density (partially degenerate) plasmas where the electron background deviates from the thermodynamic equilibrium can appear in the context of plasmas produced by lasers or ion beams hau ; Gibbon (2005). Importantly, the system energy flows mostly into the electrons, and, as a result, there may appear partially degenerate electrons with high temperature tails, allowing the electron background to be modeled by two different temperatures. In such systems, EAWs are likely to be excited and play a role in the nonlinear dynamics.

The paper is organized as follows: In Sec. II we present the basic physical model and the underlying assumptions. The linear theory of EAWs comprising the wave dispersion and Landau damping are studied in Sec. III. Section IV presents the derivation of the modified KdV equation and the occurrence of multi-plasmon resonances. We have verified the conservation laws for the KdV equation as demonstrated in Sec. V. In Sec. VI, an approximate soliton solution of the KdV equation is obtained and the effect of Landau resonance is discussed. Finally, Sec. VII is left to summarize and conclude the results.

We consider the nonlinear wave-particle interaction and evolution of small-amplitude EAWs in a partially degenerate plasma with two-temperature electrons and stationary ions. Our basic assumptions for the study are the following:

• •

  Under certain conditions electrons have a relatively high-temperature tail such that electrons can be grouped into two distinct components with different thermodynamic temperatures $T_{l}$ and $T_{h}$ [Hereafter we call low- $(\alpha=l)$ and high-temperature $(\alpha=h)$ electrons with $T_{h}>T_{l}$ and $T_{\alpha}\gtrsim T_{F\alpha}$, where $T_{F\alpha}$ denotes the Fermi temperature of $\alpha$-species electrons].

• •

  Although, the theory is independent of the background distribution, however, we consider the initial equilibrium of electrons as given by the Fermi-Dirac distribution at finite temperature $(T_{\alpha}\neq 0)$. The case of full degeneracy of electrons at zero temperature restricts only one species of electrons and thereby inadmissible to the present study.

• •

  There exists a low-frequency mode with the frequency $(\omega)$ proportional to the wave number $(k)$ such that the linear wave damping is small and the nonlinear evolution of EAWs can be described by the Korteweg de Vries (KdV) equation.

• •

  Similar to classical plasmas, the equilibrium number density of high-temperature electrons is higher than that of low-temperature species, i.e., $n_{h0}>n_{l0}$.

• •

  The EAWs are weakly dispersive such that in the regime $\hbar k/m\lesssim v_{tl}$, the Wigner-Moyal equation is still valid and some quantum effects come into play in the process of wave-particle interactions.

We start with the Wigner-Moyal equation, given by,

and the Poisson equation

where $f_{\alpha}$ is the Wigner distribution function for $\alpha$-species electrons; $e,m,v$, respectively, denote the charge, mass, and velocity of electrons, $\phi$ is the self-consistent electrostatic potential, and $n_{0}$ is the number density of background ions. Moreover, we consider the propagation of EAWs along the $x$-axis and the background distribution function for $\alpha$-species electrons in one-dimension as (writing $v_{x}$ as $v$) Manfredi (2005)

where $E=(m/2)(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})$ is the kinetic energy and $\mu_{\alpha}$ the chemical potential which satisfies the following charge neutrality condition.

In the nondegenerate limit $(T_{\alpha}\gg T_{F\alpha})$, the parameter $\xi_{\alpha}=\mu_{\alpha}/k_{B}T_{\alpha}$ is large and negative, while it is large and positive in the fully degenerate limit $(T_{\alpha}\ll T_{F\alpha})$. We, however, consider the case of $T_{\alpha}\gtrsim T_{F\alpha}$ such that $\xi_{\alpha}~{}(<0)$ is of moderate value.

We stress here that there are certain parameter restrictions imposed by the Pauli-principle, since we cannot have a phase space density of the background Wigner-function exceeding $2(m/2\pi\hbar)^{3}$. As a result, the parameters of the high- and low-temperature distributions cannot be chosen independently. The strictest criterion appears for $E=0$ and reads

For a partially degenerate low-temperature distribution (i.e., $k_{B}T_{l}\sim E_{F}$, $\mu_{l}\sim 2E_{F}$), this condition is typically fulfilled if the high-temperature distribution is not too far from the classical Maxwell-Boltzmann regime such that $-\mu_{h}/k_{B}T_{h}\gtrsim 3$.

Before we proceed to the nonlinear theory of wave-particle interactions, it is imperative to study the linear theory of EAWs and the associated wave damping. To this end, we linearize Eqs. (1) and (2) by splitting up the functions $f$ and $\phi$ into their equilibrium and perturbation parts, i.e., $f_{\alpha}(x,v,t)=f_{\alpha}^{(0)}+f_{\alpha}^{(1)}$ and $\phi(x,t)=\phi^{(1)}$, and assume the perturbations to be of the form $\sim\exp(ikx-i\omega t)$, i.e., plane waves with frequency $\omega$ and wave number $k$. Thus, we obtain the following dispersion relation.

where $\omega_{p\alpha}=\sqrt{4\pi n_{\alpha 0}e^{2}/m}$ is the plasma frequency for $\alpha$-species electrons and $v_{q}=\hbar k/2m$ is the velocity associated with the plasmon quanta.

For weak wave damping of EAWs, the time asymptotic solution for the complex frequency $(\omega=\omega_{r}+i\gamma_{L})$ can be obtained by finding the roots of the dispersion equation $D(\omega,k)\equiv D_{r}(\omega_{r},k)+iD_{i}(\omega_{r},k)+i\gamma_{L}[\partial D_{r}(\omega_{r},k)/\partial\omega_{r}]=0$. Thus, separating the real and imaginary parts we obtain

and the Landau damping rate, given by,

where

in which $v^{\text{res}}_{\pm}=\omega_{r}/k\pm v_{q}$ denotes the plasmon resonance velocity.

Next, substituting the distribution function (3) into Eq. (7) and evaluating the integrals in two different regimes $|v|<\omega_{r}/k\pm v_{q}$ and $|v|>\omega_{r}/k\pm v_{q}$, i.e.,

and noting that the exponential function in the distribution function [Eq. (3)] is small in the partially degenerate plasma regime, we obtain

Equation (11) describes both the high-frequency and relatively low-frequency wave eigenmodes of EAWs. In the limit of $\omega_{r}/k\pm v_{q}\gg v_{th}$, the high-frequency Langmuir wave (LW) mode can be obtained from Eq. (11) by considering the first and the second terms as

where $v_{t\alpha}=\sqrt{2k_{B}T_{\alpha}/m}$ is the thermal velocity and $v_{F\alpha}=\sqrt{2k_{B}T_{F\alpha}/m}$ the Fermi velocity of $\alpha$-species electrons.

On the other hand, for $\omega_{r}/k\pm v_{q}\ll v_{tl}$, the first and the third terms of Eq. (11) can be combined to yield the following dispersion relation for the EAW mode.

From Eq. (13) we note that, similar to the ion-acoustic waves in an electron-ion plasma, the EAW in two-temperature-electron plasmas has also the cubic order of dispersion and for small wave numbers $\omega/k$ approaches a constant value, i.e., the EAW becomes dispersionless. These are the prerequisites for the description of small-amplitude EAWs by a KdV-like equation. Further inspecting on Eq. (13) we find that for typical plasma parameters $n_{l0}=2\times 10^{24}$ cm^-3, $n_{h0}=10-12n_{l0}$, $T_{l}=10^{6}$ K and $T_{h}=2.5-3T_{l}$, and with $kv_{th}/\omega_{ph}\equiv k\lambda_{h}\ll 1$, the ratio of the two terms (without the factor $k$ and the term involving $k^{2}$) on the right-hand side of Eq. (13) roughly varies in between $2.2$ to $2.5$ times the thermal velocity of high-temperature species ($v_{th}$). In this situation, Eq. (13) reduces to $\omega_{r}\approx 2.5kv_{th}$. Thus, it follows that the phase velocity of EAWs in a plasma with finite temperature degeneracy is a bit higher than predicted in the classical theory Holloway and Dorning (1991) $(\omega_{r}\approx 1.31kv_{th})$. Figure 1 displays the dispersion curves for high-frequency LWs and low-frequency EAWs [panels (a) and (c)], as well as the Landau damping rates for EAWs [panels (b) and (d)]. It is seen that with lower values of the temperature ratio $T=T_{l}/T_{h}$ or with higher values of the density ratio $N=n_{l0}/n_{h0}$, a critical value $K_{c}$ of $K\equiv k\lambda_{h}$ exists, beyond which the EAW frequency can turn over, going to zero, and then assume negative values. Such a nature of the wave frequency, which is unusual in classical plasmas where the high- and low-frequency branches join to form a thumb-like curve Holloway and Dorning (1991), may be due to finite temperature degeneracy of background electrons. As a result, the linear damping rate $|\gamma_{L}/\omega_{ph}|$, which was initially low $(\sim 0.01)$, starts increasing with certain values of $K<K_{c}$ and then decreases at relatively higher values of $K>K_{c}$. This enables us to consider the regime $0\lesssim K\lesssim 0.5$ or $\hbar k/mv_{th}\lesssim 0.2$ where the linear Landau damping rate of EAWs remains weak.

It is to be noted that the approximate dispersion Eqs. (12) and (13) are obtained by assuming the smallness of the exponential function in the distribution function (3) as applicable in the partially degenerate regime. However, the Fermi-Dirac distribution approaches the Maxwell-Boltzmann distribution in the limit of low density or high temperature, i.e., when the integrand in Eq. (3) is much smaller than the unity. So, the classical results of LWs and EAWs cannot be recovered directly from Eqs. (12) and (13).

In what follows, we calculate the equilibrium chemical potentials from the following relation.

which gives

On the basis of the results of the linear theory of EAWs (Sec. III), i.e., EAW has a cubic order dispersion and the Landau damping rate is small, we proceed to the nonlinear theory of small-amplitude EAWs. Here, we derive an evolution equation for the weakly nonlinear EAWs in a degenerate plasma using the multiple-scale perturbation technique. Some special attention must be devoted to the higher order (in the amplitude) resonances that occur in the Wigner theory. In particular, due to the nonlinearities, we will have Landau resonances with resonant velocities that are shifted an amount $\pm n\delta v_{q}$ in momentum space (See e.g., Ref. (Brodin et al., 2017)) compared to the resonant velocity of classical theory (i.e., the phase velocity). Here, $n=1$ gives the velocity shift already appeared in the linear theory as described in the previous section III. Close to the resonant velocities, the Wigner equation must be analyzed in more detail. As a starting point, we will divide the velocity space into resonant and non-resonant regions. While a rigorous derivation of the nonlinear higher order resonances, based on the Wigner equation, has been given in Ref. (Brodin et al., 2017), we will here give a heuristic derivation based on elementary quantum mechanics, following the steps of Ref. (Brodin et al., 2016) Our starting point is the quantum mechanical modifications of the classical linear resonance, where the resonant velocity is adjusted according to

Let us study the physical meaning of this modification. When a particle absorbs or emits a wave quantum it can increase or decrease the momentum according to

and at the same time the energy changes according to

Next we identify $\hbar k_{1}/m$ (or equally well $\hbar k_{2}/m$) with the resonant velocity $v^{\text{res}}$ and note that for small amplitude waves, the particle frequencies and wavenumbers $(\omega_{1,2},k_{1,2})$ obey the free particle dispersion relation $\omega_{1,2}=\hbar k_{1,2}^{2}/2m$. Using these relations, we immediately see that the energy momentum relations Eqs. (17) and (18) imply the modification of the resonant velocity seen in Eq. (16). An interesting possibility, which was studied in Ref. (Brodin et al., 2017), is the simultaneous absorption (or emission) of multiple wave quanta, rather than a single wave quantum at a time. In that case Eqs. (17) and (18) are replaced by

and

where $n=1,2,3,\ldots$ is an integer. Accordingly, performing the same algebra as for the linear case, the resonant velocities now becomes

When we pick the minus sign in Eq. (21), the resonant velocity for absorbing multiple wave quanta can be considerably lower, provided the wavelengths are short. As a consequence, in the case of Langmuir waves, the damping rate due to absorption of multiple wave quanta can be much larger than the standard linear damping rate. This is due to the larger number of resonant particles in the former case.

For the non-resonance region of velocity space we have $|v-v^{\text{res}}_{\pm n}|\gg o(\epsilon)$ with $\epsilon~{}(>0)$ denoting the scaling parameter and $v^{\text{res}}_{\pm n}=\omega/k\pm n\hbar k/2m$, $n=0,1,2,3,...$ the resonant velocity. In the non-resonance regime, we consider the following expansions for $f_{\alpha}$ and $\phi$.

However, in the resonance region where $|v-v^{\text{res}}_{\pm n}|\sim o(\epsilon)$, a slightly different ordering for both $f_{\alpha}$ and $\phi$ needs to be considered. This will be verified shortly. The idea of considering different orderings for the distribution function and the electrostatic potential in the resonance and non-resonance regions was proposed and rigorously discussed by Vandam and Tanuity W. Vandam and Taniuti (1973) on the consideration that the Landau damping of KdV solitons is a far-field approximation of the Vlasov equation. Although, the present nonlinear theory considers a similar approach, the mathematical treatment is a bit different from the above due to the Wigner function. A particular ordering for each of $f_{\alpha}$ and $\phi$ cannot give rise the resonance and nonresonance contributions properly rather with different orderings, the terms involving the resonance and nonresonance parts can be combined with the closure equation, i.e., the Poisson equation which involves both the resonance and nonresonance contributions of the distribution function. Also, the Gardner-Morikawa transformation, $\xi=\epsilon^{1/2}(x-\lambda t),~{}\tau=\epsilon^{3/2}t$, which is applicable for the fluid theory of a KdV equation, may not be directly applied to the Vlasov-Poisson or Wigner-Poisson systems because of the singularities in the integrals due to resonant particles. Thus, we modify the Gardner-Morikawa transformation as W. Vandam and Taniuti (1973); Barman and Misra (2017)

and assume the following multi-scale Fourier-Laplace transforms for the perturbations.

where $W$ is the usual Laplace transform contour which excludes the poles of the integrals, and $\tilde{f}_{\alpha}^{(n)}$ and $\tilde{\phi}^{(n)}$ are analytic for all $\omega$. Equation (24) represents a plane wave propagating along the $\xi$-direction with the phase speed $\lambda$.

Substituting the stretched coordinates (23) and the expansions (22) and (24) into Eqs. (1) and (2) we obtain in one-dimensional geometry

and

where the symbol $\doteq$ is used to denote the equality in the weak sense. In the following subsections IV.0.1-IV, we obtain different expressions for the first and second order perturbations to obtain the modified KdV equation, as well as the conditions for multi-plasmon resonances.