Abstract
Full Text
Physics
G. I. Suramlishvili
On the Damping of a Langmuir Wave Near the Threshold for the Emission of Ion Sound
(Presented by Academician M. A. Leontovich, October 3, 1964)
As is known, in an isotropic homogeneous plasma there exist Langmuir and ion-sound oscillations ($l$- and $s$-plasmons), for which the dependences of the frequencies $\Omega_k$ and $\omega_q$ on the wave vectors $\mathbf{k}$ and $\mathbf{q}$ are given, respectively, by the expressions (we consider long-wavelength $l$- and $s$-plasmons):
\[ \Omega_k=\Omega_0(1+{}^3\!/_{2}k^2R_D^2),\qquad \omega_q=c_s q, \tag{1} \]
where $\Omega_0=\sqrt{4\pi e^2 n/m}$ is the Langmuir frequency; $R_D=\sqrt{T/4\pi e^2 n}$ is the Debye radius; $c_s$ is the sound velocity. With the dispersion law expressed by formula (1), damping of an $l$-plasmon is possible, caused by the emission of an $s$-plasmon with transition to an $l$-plasmon of lower frequency. In this case the conservation laws for the frequency and wave vector must be satisfied,
\[ \Omega_k=\Omega_{k-q}+\omega_q. \tag{2} \]
The indicated process is a threshold process, i.e., it is possible only for $k$ greater than a certain threshold value $k_{\mathrm{p}}$, since equation (2) has a solution with respect to $q$ for $k>k_{\mathrm{p}}$ and has no solution for $k<k_{\mathrm{p}}$, where
\[ k_{\mathrm{p}}=\frac{1}{3R_D}\left(\frac{m}{M}\right)^{1/2}. \]
Above the threshold, $c_s<\partial\Omega_k/\partial k$, i.e., the condition of Cherenkov radiation of $s$-plasmons by $l$-plasmons is fulfilled. Here the situation is in many respects analogous to that which occurs for elementary excitations in the theory of condensed media (${}^{1-5}$).
We take as the basis of the treatment the complete Hamiltonian of plasma oscillations with allowance for the indicated three-plasmon process:
\[ H=H_0+\varepsilon H' =\sum_k \Omega_k a_k^+a_k+\sum_q \omega_q c_q^+c_q +\varepsilon\sum_{kq}\Phi_{q;k\,k-q}a_{k-q}^+c_q^+a_k+\text{h.c.} \tag{3} \]
Here $a_k^+$, $a_k$ are, respectively, the creation and annihilation operators of an $l$-plasmon with energy $\Omega_k$ and momentum $\mathbf{k}$; $c_q^+$, $c_q$ have the same meaning for $s$-plasmons; $\varepsilon$ is a formally introduced expansion parameter.
$\Phi_{q;k\,k-q}$ is found by expanding the plasma Lagrangian in powers of the amplitudes of the collective plasma oscillations and has the form (${}^{6,7}$):
\[ \Phi_{q;k\,k-q} =\left(\frac{\pi}{2}\right)^{1/2}\frac{e c_T c_s^{1/2}}{T} \,\frac{k^2-\mathbf{k}\mathbf{q}}{|\mathbf{k}|\,|\mathbf{k}-\mathbf{q}|}\,q^{1/2}. \tag{4} \]
Let us introduce the one-particle two-time retarded Green function (${}^8$)
\[ G_k(t,t')=-i\theta(t-t')\langle [a_k(t);a_k^+(t')]\rangle, \tag{5} \]
where
\[ \langle\cdots\rangle=\operatorname{Sp}(e^{-H/T}\cdots)\{\operatorname{Sp}(e^{-H/T})\}^{-1}; \qquad \theta(t-t')= \begin{cases} 1, & t>t',\\ 0, & t<t', \end{cases} \]
\[ [a_k(t);a_k^+(t')]=a_k(t)a_k^+(t')-a_k^+(t')a_k(t). \]
Let us write the equation of motion for the functions (5):
\[ i\frac{dG_k(t,t')}{dt} = \delta(t-t') - i\theta(t-t') \left\langle \left[ i\frac{da_k(t)}{dt};\ a_k^+(t') \right] \right\rangle . \]
Expanding this equation with the aid of the Hamiltonian (3), we obtain:
\[ \begin{aligned} i\frac{dG_k(t,t')}{dt} &= \delta(t-t')+\Omega_kG_k(t,t') +\varepsilon\sum_q\Phi_{q;k\,k-q}D_{k-q,q,k} \\ &\quad +\varepsilon\sum_q\Phi_{q;k\,k+q}D^{(1)}_{k+q,q,k}, \end{aligned} \tag{6} \]
where
\[ \begin{aligned} D_{k-q,q,k} &= -i\theta(t-t') \left\langle \left[ a_{k-q}c_q;\ a_k^+(t') \right] \right\rangle, \\ D^{(1)}_{k+q,q,k} &= -i\theta(t-t') \left\langle \left[ a_{k+q}c_q^+;\ a_k^+(t') \right] \right\rangle . \end{aligned} \tag{7} \]
The equations of motion for the functions (7) will contain higher-order Green functions; for these Green functions, in turn, one can write equations of motion. Continuing this process, we obtain an infinite system of coupled equations. The decoupling of such a chain is, as a rule, justified each time by the physical aspect of the problem. Here we shall use the fact that the energy densities of the \(l\)- and \(s\)-plasmons greatly exceed the energy densities of their interaction; therefore, in the equations of motion for the functions (7), we single out and retain only those terms that contain the functions \(G_k\), \(D_{k-q,q,k}\), \(D^{(1)}_{k+q,q,k}\). This is achieved by means of the following approximations:
\[ \sum_{q_1}\Phi_{q_1;k-q_1,k-q+q_1} \left\{ -i\theta(t-t') \left\langle \left[ a_{k-q+q_1}c_{q_1}^+c_q;\ a_k^+(t') \right] \right\rangle \right\} \to \Phi_{q;k\,k-q}n_qG_k, \]
\[ \sum_{k_1}\Phi_{q;k_1\,k_1-q} \left\{ -i\theta(t-t') \left\langle \left[ a_{k-q}a_{k_1-q}^+a_{k_1};\ a_k^+(t') \right] \right\rangle \right\} \to \Phi_{q;k\,k-q}(1+N_{k-q})G_k, \]
\[ \sum_{q_1}\Phi_{q_1;k+q_1\,k+q-q_1} \left\{ -i\theta(t-t') \left\langle \left[ c_{q_1}a_{k+q-q_1}c_q^+;\ a_k^+(t') \right] \right\rangle \right\} \to \Phi_{q;k\,k+q}(1+n_q)G_k, \]
\[ \sum_{k_1}\Phi_{q;k_1\,k_1-q} \left\{ -i\theta(t-t') \left\langle \left[ a_{k+q}a_{k_1}^+a_{k_1-q};\ a_k^+(t') \right] \right\rangle \right\} \to \Phi_{q;k\,k+q}(1+N_{k+q})G_k, \]
\[ N_k=\langle a_k^+a_k\rangle,\qquad n_q=\langle c_q^+c_q\rangle . \]
Thus we have:
\[ i\frac{dD_{k-q,q,k}}{dt} = (\Omega_{k-q}+\omega_q)D_{k-q,q,k} + \varepsilon\Phi_{q;k\,k-q}(1+n_q+N_{k-q})G_k, \]
\[ i\frac{dD^{(1)}_{k+q,q,k}}{dt} = (\Omega_{k+q}-\omega_q)D^{(1)}_{k+q,q,k} + \varepsilon\Phi_{q;k\,k+q}(n_q-N_{k+q})G_k . \tag{8} \]
Passing to the Fourier representation \(A=\int A(\omega)e^{-i\omega t}\,d\omega\), from expressions (6) and (8) we obtain
\[ \left( \omega-\Omega_k -\varepsilon^2\sum_q\Phi_{q;k\,k-q} \frac{1+N_{k-q}+n_q}{\omega-\Omega_{k-q}-\omega_q} - \varepsilon^2\sum_q\Phi_{q;k\,k+q} \frac{n_q-N_{k+q}}{\omega+\omega_q-\Omega_{k+q}} \right)G_k(\omega) = \frac{1}{2\pi}. \]
Let us continue this function into the upper half-plane \(\omega \to \omega+i\alpha\) and use the symbolic identity:
\[ \frac{1}{\omega-\omega_0\pm i\alpha} = P\frac{1}{\omega-\omega_0}\mp i\pi\delta(\omega-\omega_0). \]
Here \(P\) denotes an integral in the sense of the principal value, and \((\omega-\omega_0)\) is regarded as a real quantity. We then obtain
\[ [\omega-\Omega_k-\varepsilon^2(Q_k^{(1)}(\omega)-iQ_k^{(2)}(\omega))]G_k(\omega)=\frac{1}{2\pi}, \tag{9} \]
where
\[ Q_k^{(1)}(\Omega_k) = P\sum_q \left( \Phi_{q;k\,k-q}^{2} \frac{1+n_q+N_{k-q}}{\Omega_k-\Omega_{k-q}-\omega_q} + \Phi_{q;k\,k+q}^{2} \frac{n_q-N_{k+q}}{\Omega_k+\omega_q-\Omega_{k+q}} \right); \]
\[ Q_k^{(2)}(\Omega_k) = \gamma_k^{(1)}(\Omega_k)+\gamma_k^{(2)}(\Omega_k) = \pi\sum_q \Phi_{q;k\,k-q}^{2} (1+n_q+N_{k-q}) \times \tag{10} \]
\[ \times\delta(\Omega_k-\Omega_{k-q}-\omega_q) + \pi\sum_q \Phi_{q;k\,k+q}^{2} (n_q-N_{k+q}) \delta(\Omega_k+\omega_q-\Omega_{k-q}). \]
The frequency shift and damping of the \(l\)-plasmon caused by interaction with the \(s\)-plasmon are expressed, respectively, through \(Q_k^{(1)}(\Omega_k)\) and \(Q_k^{(2)}(\Omega_k)\); here \(\gamma_k^{(1)}(\Omega_k)\) describes the damping due to decay, while \(\gamma_k^{(2)}(\Omega_k)\) describes the damping due to scattering accompanied by absorption of an \(s\)-plasmon. The latter is not a threshold effect, and therefore is of no interest to us.
Let us proceed to the concrete calculation of the damping quantity
\[ \gamma_k^{(1)}(\Omega_k) = \pi\sum_q \Phi_{q;k\,k-q}^{2} (1+n_q+N_{k-q}) \delta(\Omega_k-\Omega_{k-q}-\omega_q). \]
The argument of the \(\delta\)-function appearing in this expression has the roots:
\[ q_1=0,\qquad q_2=2k(t-z), \]
where
\[ t=\frac{(\mathbf{kq})}{kq},\qquad z=\frac{k_\Pi}{k}. \]
Replacing \(n_q\) and \(N_{k-q}\) by Rayleigh--Jeans distributions and retaining the leading terms, we obtain
\[ \gamma_k^{(1)}(\Omega_k) = \frac{1}{8\pi^2}\frac{T}{c_s} \int \frac{\Phi_{q;k\,k-q}^{2}}{q} \sum_i \frac{\delta(q-q_i)} {\left|\dfrac{\partial}{\partial q}(\Omega_k-\Omega_{k-q}-\omega)\right|_{q=q_i}} \,dq. \]
Using (4) and integrating the last expression over \(q\) with the aid of the \(\delta\)-function, we shall have:
\[ \gamma_k^{(1)}(\Omega_k) = \frac{1}{24\pi}\frac{\Omega_0}{N_D}(kR_D) \int_z^1 \frac{t^4-2t^3z-t^2(1-z^2)+zt+\tfrac14} {zt-z^2-\tfrac14} (z-1)\,dt, \tag{11} \]
where \(N_D=nR_D^3\). From formula (11) it follows that near the threshold \(z\simeq1\), \(l\)-plasmons damp due to emission of \(s\)-plasmons with decrement:
\[ \gamma_k^{(1)}(\Omega_k) \simeq 10^{-2}\frac{\Omega_0}{N_D}(kR_D)(1-z^2). \tag{12} \]
Thus, for \(z\ll1\),
\[ \gamma_k^{(1)}\simeq 10^{-2}\frac{\Omega_0}{N_D}(kR_D). \]
Note that \(\gamma_k^{(1)}\ll\gamma_{\mathrm{st}}\), where \(\gamma_{\mathrm{st}}\sim \dfrac{\Omega_0}{N_D}\ln N_D\) is the damping decrement of \(l\)-plasmons caused by collisions of electrons with ions.
\((\gamma_k^{(1)}/\gamma_{\mathrm{st}} = 10^{-2}(kR_D)/\ln N_D \ll 1)\). For the Landau damping decrement we have the expression: \(\gamma_L \sim \dfrac{\Omega_0}{(kR_D)^3} e^{-1/(kR_D)^2}\); therefore
\[ \frac{\gamma_k^{(1)}}{\gamma_L} = \frac{(kR_D)^4}{10^2 N_D} e^{1/(kR_D)^2}. \]
It follows from this that \(\gamma_k^{(1)} \gtrsim \gamma_L\) for waves whose wave vectors satisfy the condition \(k \lesssim k_0\), where \(k_0\) is found from the equation
\[ e^{1/(k_0R_D)^2} = 10^2 N_D \frac{1}{(k_0R_D)^4}. \]
I express my gratitude to A. A. Vedenov for his attention and comments.
Received
4 IX 1964
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