3.1. Collective excitation in the electron gas described as a single fluid

Consider the propagation of plane longitudinal waves in an isotropic macroscopically motionless electron–ion plasma medium.

The equilibrium concentration of electrons $n_{0e}$ is equal to the equilibrium concentration of ions $n_{0i}$. The ions are assumed to be motionless for the consideration of the high-frequency excitations. The equilibrium velocity field are equal to zero $\boldsymbol {v}_{0e}=0$. The equilibrium pressure $p_{0e}^{\alpha \beta }$ is given by the isotropic Fermi pressure $p_{0e}^{\alpha \beta }=\delta ^{\alpha \beta }\cdot p_{\textrm {Fe}}$, where $p_{\textrm {Fe}}=(3{\rm \pi} ^{2})^{2/3}n_{0e}^{5/3}\hbar ^{2}/5m_{e}$, $p_{0e}^{\alpha \beta }=p_{0u}^{\alpha \beta }+p_{0d}^{\alpha \beta }$ is the superposition of partial pressures. The equilibrium third-rank tensor is equal to zero for the zero temperature isotropic fermions $Q_{0e}^{\alpha \beta \gamma }=0$ (see Supplementary material). The equilibrium fourth-rank tensor is expressed via the equilibrium concentration in accordance with the expression (2.8):

(3.1)\begin{equation} P_{0e}^{\alpha\beta\gamma\delta}=P_{0u}^{\alpha\beta\gamma\delta}+P_{0d}^{\alpha\beta\gamma\delta} =\frac{3{\rm \pi}^{2}}{35} I_{0}^{\alpha\beta\gamma\delta}(3{\rm \pi}^{2})^{{1}/{3}}\frac{\hbar^{4}n_{0e}^{{7}/{3}}}{m^{4}}. \end{equation}

The scalar potential of the electric field is equal to zero in the equilibrium state $\varPhi _{0}=0$.

Let us consider the small amplitude perturbations of all material fields involved in the presented model for perturbations propagating parallel to $x$-direction: $n_{e}=n_{0e}+ \tilde {n}_{e}$, $v_{e}^{x}=0+ \tilde {v}_{e}^{x}$, $\delta p_{e}^{\alpha \beta }=p_{0e}^{\alpha \beta } +\delta ^{x\alpha }\delta ^{x\beta } \tilde {p}_{e}^{xx}$, $Q_{e}^{\alpha \beta \gamma }=0+\delta ^{x\alpha }\delta ^{x\beta }\delta ^{x\gamma } \tilde {Q}_{e}^{xxx}$, $P_{e}^{\alpha \beta \gamma \delta }=P_{0e}^{\alpha \beta \gamma \delta }+\delta ^{x\alpha }\delta ^{x\beta }\delta ^{x\gamma }\delta ^{x\delta } \tilde {P}_{e}^{xxxx}$ and $\varPhi =0+\tilde {\varPhi }$, where

(3.2)\begin{equation} \tilde{P}_{e}^{xxxx}= \frac{(3{\rm \pi}^{2})^{4/3}}{5}\frac{\hbar^{4}n_{0e}^{4/3}}{m_{e}^{4}} \tilde{n}_{e} =\dfrac{1}{5}v_{\textrm{Fe}}^{4} \tilde{n}_{e}, \end{equation}

with the Fermi velocity $v_{\textrm {Fe}}=(3{\rm \pi} ^{2}n_{0e})^{1/3}\hbar /m_{e}$ and $I_{0}^{xxxx}=3$.

Perturbation of each function is presented in the form of a plane wave:

(3.3)\begin{equation} \left( \begin{array}{@{}c@{}} \tilde{n}_{e} \\ \tilde{v}_{e}^{x} \\ \tilde{p}_{e}^{xx} \\ \tilde{Q}_{e}^{xxx} \\ \tilde{\varPhi} \end{array} \right)=\left( \begin{array}{@{}c@{}} N_{e} \\ V_{e} \\ P_{e} \\ Q_{e} \\ \varPhi_{ampl} \end{array} \right) \exp({-\textrm{i}\omega t+\textrm{i} k x}). \end{equation}

The described procedure leads to the standard form of the linearized continuity equation:

(3.4)\begin{equation} \omega \tilde{n}_{e}=kn_{0e} \tilde{v}^{x}_{e}. \end{equation}

The linearized Euler equation has one modification. The single element of the pressure tensor $\tilde {p}_{e}^{xx}$ is presented as the independent function:

(3.5)\begin{equation} \omega m_{e}n_{0e} \tilde{v}^{x}_{e}-k \tilde{p}_{e}^{xx} -\frac{\hbar^{2}k^{3}}{4m_{e}} \tilde{n}_{e} =q_{e}n_{0e}k \tilde{\varPhi}. \end{equation}

The perturbation of the pressure $\tilde {p}_{e}^{xx}$ can be found from the linearized pressure evolution equation:

(3.6)\begin{equation} \omega \tilde{p}_{e}^{xx}=3kp_{0e}^{xx} \tilde{v}_{e}^{x}+k \tilde{Q}_{e}^{xxx}. \end{equation}

The first term on the right-hand side of the pressure evolution equation (3.6) provides the partial expression for the pressure perturbation in accordance with the kinetic model (Landau & Lifshitz Reference Landau and Lifshitz1980; Aleksandrov et al. Reference Aleksandrov, Bogdankevich and Rukhadze1984; Andreev Reference Andreev2016a):

(3.7)\begin{equation} \tilde{p}^{xx}_{partial}=3\frac{p_{0e}^{xx}}{n_{0e}} \tilde{n}_{e}=\tfrac{3}{5}v_{\textrm{Fe}}^{2} \tilde{n}_{e}. \end{equation}

Novel quantum effects enter the model via the second term on the right-hand side of (3.6).

The linearized equation for the third-rank tensor evolution has the following form:

(3.8)\begin{equation} \omega \tilde{Q}_{e}^{xxx}=k \tilde{P}_{e}^{xxxx} +\frac{\hbar^{2}}{4m^{2}} q_{e}n_{0e}k^{3} \tilde{\varPhi}. \end{equation}

The presented model contains the traditional form of the linearized Poisson equation:

(3.9)\begin{equation} k^{2}\tilde{\varPhi}=4{\rm \pi} q_{e} \tilde{n}_{e}. \end{equation}

Linearized equations (3.4)–(3.8) lead to the following dispersion equation, which is the quadratic equation relative to $\omega ^{2}$:

(3.10)\begin{gather} \omega^{4}-\omega^{2}\left(\omega_{\textrm{Le}}^{2}+\frac{3p_{0}^{xx}}{mn_{0}}k^{2}+\frac{\hbar^{2}k^{4}}{4m^{2}}\right)\nonumber\\ -\left(\omega_{\textrm{Le}}^{2}\frac{\hbar^{2}k^{4}}{4m^{2}}+k^{4}\frac{d P_{0}^{xxxx}}{d n_{0}}\right)=0, \end{gather}

where $\omega _{\textrm {Le}}^{2}=4{\rm \pi} e^{2}n_{0e}/m_{e}$ is the Langmuir frequency. The explicit form of the last term in (3.10) can be found from (3.2). However, it is kept in implicit form to track its source. The last group of terms is caused by the simultaneous contribution of the pressure evolution equation and the third-rank tensor evolution equations. This group consists of two terms. The first term is caused by the Coulomb interaction located on the right-hand side of the third-rank tensor evolution equation (2.7). The second term, which is proportional to $P_{0}^{xxxx}$, is caused by the last term on the left-hand side of (2.7).

It is useful to specify that the regime of electrostatic waves propagating parallel to the external magnetic field is considered. However, the quantum effects entering the model via the evolution of the third-rank tensor give an additional solution.

Here, the solution of (3.10) is presented as

(3.11)\begin{equation} \omega_{{\pm}}^{2}=\tfrac{1}{2}\left\{\omega_{\textrm{Le}}^{2}+\frac{3p_{0}^{xx}}{mn_{0}}k^{2}+\frac{\hbar^{2}k^{4}}{4m^{2}} \pm\left[\left(\omega_{\textrm{Le}}^{2}+\frac{3p_{0}^{xx}}{mn_{0}}k^{2}+\frac{\hbar^{2}k^{4}}{4m^{2}}\right)^{2} +4\omega_{\textrm{Le}}^{2}\frac{\hbar^{2}k^{4}}{4m^{2}}+k^{4}\frac{\delta P_{0}^{xxxx}}{\delta n_{0}}\right]^{1/2}\right\}. \end{equation}

Solution $\omega _{-}^{2}$ can be rewritten in the following form:

(3.12)\begin{equation} \omega_{-}^{2}= \frac{\omega_{-}^{2}\omega_{+}^{2}}{\omega_{+}^{2}} ={-}\frac{\omega_{\textrm{Le}}^{2}\frac{\hbar^{2}k^{4}}{4m^{2}}+\frac{1}{5}v_{\textrm{Fe}}^{4}k^{4}}{\omega_{+}^{2}}. \end{equation}

Equation (3.12) shows that $\omega _{-}^{2}<0$. This can be also seen from (3.11).

It is clear that the second solution exists in the quasi-classical limit. However, it is suppressed by the small thermal velocity $d P_{0}^{xxxx}/d n_{0}\sim v_{T}^{4}$ in the low-temperature limit. In the quantum regime with temperature $T$ below the Fermi temperature $T_{\textrm {Fe}}=mv_{\textrm {Fe}}^{2}/2$, the quasi-classical contribution is also suppressed because it is proportional to the high degree of the Planck constant $\hbar ^{4}$. However, the quantum-interaction part demonstrated in this paper is proportional to the $\hbar ^{2}$ and the large frequency value $\omega _{\textrm {Le}}^{2}$. Therefore, the existence of a solution is mainly caused by the first term on the right-hand side of (3.12) found from the right-hand side of (2.7).

The prefactor in front of the quantum Bohm potential can differ from the single-particle expression (Moldabekov, Bonitz & Ramazanov Reference Moldabekov, Bonitz and Ramazanov2018), while the linear part of the quantum Bohm potential can be strictly expressed via the concentration for the arbitrary many-particle wave function and arbitrary strength of interaction. However, the account of the evolution equations for the higher tensor rank like the pressure and the pressure flux leads to modification of the coefficient in front of the Fermi velocity. Moreover, further extension of the set of hydrodynamic equations can give correction of the prefactor in front of the quantum Bohm potential. However, it requires the derivation of the corresponding model beyond 20-moments approximation. Some phenomenological model for the coefficient in front of the quantum Bohm potential can be adopted. However, our comment is related to the systematic and consistent derivation of the coefficient in question in terms of the quantum hydrodynamic method.

Let us consider the small wave vector limit of the obtained solution:

(3.13)\begin{equation} \omega_{+}^{2}=\omega_{\textrm{Le}}^{2}, \end{equation}

and

(3.14)\begin{equation} \omega_{-}^{2}={-}\frac{\hbar^{2}k^{4}}{4m^{2}}-\tfrac{1}{5}k^{4}v_{\textrm{Fe}}^{2}r_{DF}^{2}, \end{equation}

where $r_{DF}=v_{\textrm {Fe}}/\omega _{\textrm {Le}}$ is the Debye radius for the degenerate electrons.

Let us present the dimensionless form of (3.10):

(3.15)\begin{equation} \xi^{4}-\xi^{2}(1+(3/5)\kappa^{2}+\varLambda\kappa^{4})-(\varLambda+0.2)\kappa^{4}=0, \end{equation}

where $\xi =\omega /\omega _{\textrm {Le}}$, $\kappa =kv_{\textrm {Fe}}/\omega _{\textrm {Le}}$ and $\varLambda =\hbar ^{2}\omega _{\textrm {Le}}^{2}/(4m^{2}v_{\textrm {Fe}}^{4})=[3{\rm \pi} (3{\rm \pi} ^{2})^{1/3}]^{-1}/(r_{B}n_{0}^{1/3})$, where $r_{B}=\hbar ^{2}/me^{2}$ is the Bohr radius. Figure 1 shows two wave solutions in accordance with (3.11)–(3.14). The short-wavelength decrease of the frequency of the Langmuir wave caused by the non-zero contribution of the third-rank tensor can be seen from a comparison of the two upper lines. The increment of the instability presented by the second wave solution (3.12) is also demonstrated in figure 1. It appears in the short-wavelength regime. Its stabilization can be expected under the influence of tensors of higher ranks.

The novel quantum effects presented by the third-rank tensor evolution manifest themselves at $\kappa \sim 1$, which corresponds to the wavelength $\lambda =2{\rm \pi} /k=\sqrt {{\rm \pi} }(3{\rm \pi} ^{2})^{1/3}\sqrt {ar_{D}}$, where the average interparticle distance is used $a=\sqrt [3]{n_{0}}$. As an estimation, we have $\lambda \approx 6\times 10^{-8}$ cm$\approx a\approx r_{D}$ for $n_{0}=10^{23}$ cm$^{-3}$, which corresponds to the limits of the quasi-classical motion.

3.2. Spin-electron acoustic waves

Consider the propagation of plane longitudinal waves in the direction of the external magnetic field. The external magnetic field is one of the mechanisms of the spin polarization formation. Magnetic conductive materials create a spontaneous spin polarization of the lattice and the electron gas. For these materials, spin polarization of electrons is non-zero even for the zero external magnetic field. The $z$-axis is directed parallel to the equilibrium spin polarization $S_{0z}=n_{0\uparrow }-n_{0\downarrow }$.

The structure of the equilibrium state and the form of the perturbations are similar to those described above for the single-fluid regime. Evolution of perturbations leads to the following dispersion equation:

(3.16)\begin{equation} 1=\sum_{s}\dfrac{\omega_{Ls}^{2}\left(1+\dfrac{\hbar^{2}k^{4}}{4m^{2}\omega^{2}}\right)}{ \omega^{2}-\dfrac{3p_{0s}^{xx}}{mn_{0s}}k^{2}-\dfrac{\hbar^{2}k^{4}}{4m^{2}}-\dfrac{k^{4}}{\omega^{2}}\dfrac{\textrm{d} P_{0s}^{xxxx}}{\textrm{d} n_{0s}}}, \end{equation}

where $\omega _{Ls}^{2}=4{\rm \pi} e^{2}n_{0s}/m$ is the partial Langmuir frequency.

The dispersion equation (3.16) is obtained for the regime, where two waves (the Langmuir wave and the spin-electron acoustic wave Andreev Reference Andreev2015) exist in the traditional separate-spin-evolution quantum hydrodynamics construct of two continuity equations and two Euler equations (Andreev Reference Andreev2015). Here, four waves are found. Hence, there are two new solutions. One of them is found in the single-fluid regime (3.14). Therefore, the regime of the separate spin evolution brings the novel solution.

If the contribution of the third-rank tensor is dropped, (3.16) simplifies to

(3.17)\begin{equation} 1=\sum_{s}\frac{\omega_{Ls}^{2}}{\omega^{2}-\dfrac{3p_{0s}^{xx}}{mn_{0s}}k^{2}-\dfrac{\hbar^{2}k^{4}}{4m^{2}}}. \end{equation}

Equation (3.17) includes the contribution of the pressure perturbations from the pressure evolution equation. Therefore, the speed of sound for the spin-electron acoustic waves corresponds to the kinetic model (Andreev Reference Andreev2016a) in contrast with hydrodynamics based on the continuity and Euler equations (Andreev Reference Andreev2015).

Let us present the dimensionless form of (3.16) as

(3.18)\begin{gather} \left[\xi^{4}-\left[\frac{3}{5}(1-\eta)^{{2}/{3}}\kappa^{2}+\varLambda\kappa^{4}\right]\xi^{2}-\frac{\kappa^{4}}{5}(1-\eta)^{{4}/{3}}\right]\nonumber\\ \times\left[\xi^{4}-\left[\frac{3}{5}(1+\eta)^{{2}/{3}}\kappa^{2}+\varLambda\kappa^{4}\right]\xi^{2}-\frac{\kappa^{4}}{5}(1+\eta)^{{4}/{3}}\right]\nonumber\\ -\frac{1}{2}(\xi^{2}+\varLambda\kappa^{4}) \left[(1-\eta) \left[\xi^{4}-\left[\frac{3}{5}(1-\eta)^{{4}/{3}}\kappa^{2}+\varLambda\kappa^{4}\right]\xi^{2}-\frac{\kappa^{4}}{5}(1-\eta)^{{4}/{3}}\right]\right.\nonumber\\ \left.+(1+\eta) \left[\xi^{4}-\left[\frac{3}{5}(1+\eta)^{{4}/{3}}\kappa^{2}+\varLambda\kappa^{4}\right]\xi^{2}-\frac{\kappa^{4}}{5}(1+\eta)^{{4}/{3}}\right]\right]=0, \end{gather}

where $\eta =\mid n_{\uparrow }-n_{\downarrow }\mid /(n_{\uparrow }+n_{\downarrow })$ is the spin polarization. It is used to plot the spectrum. The result is demonstrated in figures 2 and 3. The spin polarization $\eta$ is caused by the magnetic field $\eta =\tanh (\mu _{B}B_{0}/\varepsilon _{\textrm {Fe}})$, where $\mu _{B}$ is the Bohr magneton, $B_{0}$ is the external uniform magnetic field and $\varepsilon _{\textrm {Fe}}=mv_{\textrm {Fe}}^{2}/2$ is the Fermi energy. However, ferromagnetic metals demonstrate spin polarization. In the simplest way, it can be modelled as the result of action of the effective inner magnetic field $B_{\textrm {eff}}$. Hence, we have $\eta =\tanh (\mu _{B}(B_{0}+B_{\textrm {eff}})/\varepsilon _{\textrm {Fe}})$. Let us mention that the account of the internal field is a simple phenomenological assumption showing the influence of the ferromagnetic state of the sample on the conductivity electrons.

Figure 1. Numerical analysis of (3.11), (3.10) and (3.15) is presented. The three upper lines show the spectrum of the Langmuir wave in different regimes. The black continuous line corresponds to the hydrodynamic spectrum based on the continuity and Euler equations together with the Fermi pressure: $\omega ^{2}=\omega _{\textrm {Le}}^{2}+(v_{\textrm {Fe}}^{2}k^{2})/3+{\hbar ^{2}k^{4}}/{4m^{2}}$. The red dotted line corresponds to the application of the pressure evolution equation with the zero contribution of the third-rank tensor. The green dashed line presents the spectrum of the Langmuir wave under influence of the third-rank tensor evolution equation. The lowest line describes the module of the negative frequency square $\mid \omega ^{2}\mid$ of novel solution caused by the third-rank tensor evolution equation in accordance with (3.14). The value of parameter $\varLambda$ is presented in the figure. It corresponds to $n_{0}=10^{23}$ cm$^{-3}$.

Figure 2. Numerical analysis of (3.16) and (3.18). The blue continuous line shows the spectrum of the Langmuir wave. The green dashed line shows the spectrum of the spin-electron acoustic wave. The blue and black dotted lines correspond to the instabilities following from the application of the pressure evolution equation and the third-rank tensor evolution equation, respectively.

Figure 3. Numerical analysis of (3.16) and (3.18). The spin-electron acoustic waves are considered in different regimes. The green continuous line corresponds to the hydrodynamic spectrum based on the continuity and Euler equations together with the Fermi pressure, where coefficient $1/3$ is found in front of the square of the Fermi velocity $v_{\textrm {Fe}}^{2}$. The other three regimes correspond to the application of the pressure evolution equation. Hence, coefficient $3/5$ is found in front of the square of the Fermi velocity $v_{\textrm {Fe}}^{2}$. The red dotted line corresponds to the spectrum of spin-electron acoustic waves under influence of the third-rank tensor evolution equation at the zero contribution of the quantum Bohm potential. The black continuous line includes the contribution of the quantum Bohm potential with zero contribution of the third-rank tensor. The green dashed line presents the spectrum under the influence of all effects presented in (3.16).

The separate spin evolution leads to the appearance of the second electrostatic wave (for the motionless ions) (Andreev Reference Andreev2015). Moreover, (3.16) shows that the instability found in the single-fluid regime of electrons (3.11) splits into two instabilities. Frequency of these instabilities are shown in figure 2.

Modifications of the spectrum of the spin-electron acoustic waves under influence of different effects are demonstrated in figure 3.

3.3. Variation of the fourth-rank tensor under the evolution of higher-rank tensors

Further extension of the hydrodynamic model of electrons is considered for stabilization of the spectrum found above.

Here, we present the ‘main’ part of the fourth-rank tensor evolution equation

(3.19)\begin{equation} \partial_{t}\boldsymbol{\mathsf{Q}}_{4}+\partial \boldsymbol{\mathsf{Q}}_{5}=0, \end{equation}

where tensor $\boldsymbol{\mathsf{Q}}_{4}$ (tensor $\boldsymbol{\mathsf{Q}}_{5}$) is the fourth (fifth)-rank tensor in the comoving frame and $\partial$ presents the divergence of tensor $\boldsymbol{\mathsf{Q}}_{5}$. Tensors $\boldsymbol{\mathsf{Q}}_{4}$ and $\boldsymbol{\mathsf{Q}}_{5}$ are variations from the ‘equilibrium-like’ terms presented by (2.8) for the fourth-rank tensor. The equilibrium-like part of the fifth-rank tensor is equal to zero. Equation (3.19) is the ‘continuity’ equation for the fourth-rank tensor. Hence, it is assumed that the flux of tensor $\boldsymbol{\mathsf{Q}}_{4}$ (which is presented by tensor $\boldsymbol{\mathsf{Q}}_{5}$) is the major cause of the evolution of tensor $\boldsymbol{\mathsf{Q}}_{4}$. Interaction does not appear in (3.19), so it is not neglected there. We need to consider the contribution of the Coulomb interaction in the evolution of tensor $\boldsymbol{\mathsf{Q}}_{4}$. It can be found via the study of the evolution of tensor $\boldsymbol{\mathsf{Q}}_{5}$.

Let us present the simplified equation for the evolution of tensor $\boldsymbol{\mathsf{Q}}_{5}$:

(3.20)\begin{equation} \partial_{t}\boldsymbol{\mathsf{Q}}_{5}={-}\frac{\hbar^{4}}{16m^{5}}q_{e}n_{e}\partial^{(5)}\varPhi, \end{equation}

where $\partial ^{(5)}$ is the fifth-rank tensor composed of five space derivatives with different indexes. Here, the flux of tensor $\boldsymbol{\mathsf{Q}}_{5}$ is neglected. It is assumed that the major cause of evolution of tensor $\boldsymbol{\mathsf{Q}}_{5}$ is the interaction.

The application of additional equations (3.19) and (3.20) gives the extension of the model presented above (2.1)–(2.7).

The linear approximation gives the following dispersion equation for the single-fluid model of electrons:

(3.21)\begin{gather} \omega^{6}-\omega^{4}\left(\omega_{\textrm{Le}}^{2}+\tfrac{3}{5}v_{\textrm{Fe}}^{2}k^{2}+\omega_{q}^{2}\right)\nonumber\\ -\omega^{2}\left(\omega_{\textrm{Le}}^{2}\omega_{q}^{2}+\tfrac{1}{5}v_{\textrm{Fe}}^{4}k^{4}\right)-\omega_{\textrm{Le}}^{2}\omega_{q}^{4} =0, \end{gather}

where $\omega _{q}^{2}={\hbar ^{2}k^{4}}/{4m^{2}}$. Here the last term is caused by the evolution of the fourth-rank tensor and the fifth-rank tensor.

We do not present the numerical analysis of (3.21). We point out that no stabilization of the instabilities found above is obtained.

However, the instability found above is a rather fast instability which is not observed in plasmas. Therefore, we need to consider other mechanisms for the stabilization.

3.4. On exchange interaction

Two identical particles are indistinguishable in the quantum description. Hence, it is not enough to present the many-particle wave function as the product of the single-particle wave functions for the weakly interacting limit,

(3.22)\begin{equation} \varPsi(R,t)=\psi_{\kappa_{1}}(\boldsymbol{r}_{1},t)\boldsymbol{\cdot} \cdots\boldsymbol{\cdot}\psi_{\kappa_{N}}(\boldsymbol{r}_{N},t), \end{equation}

where $R=\{ \boldsymbol {r}_{1},\ldots, \boldsymbol {r}_{N}\}$, $\kappa _{j}$ is the full set of quantum numbers for $j$th particle. It is necessary to include the property of anisotropy of the wave function relative to the permutation of arguments:

(3.23)\begin{equation} \varPsi(\cdots, \boldsymbol{r}_{i},\ldots, \boldsymbol{r}_{j},\ldots, t) ={-}\varPsi(\cdots, \boldsymbol{r}_{j},\ldots, \boldsymbol{r}_{i},\ldots, t). \end{equation}

For the many-particle wave function, the weakly interacting particles can be presented in the form of the Slater determinant:

(3.24)\begin{equation} \varPsi(R,t)=\frac{1}{\sqrt{N!}}\left \| \begin{array}{cccc} \psi_{\kappa_{1}}(\boldsymbol{r}_{1},t) & \psi_{\kappa_{1}}(\boldsymbol{r}_{2},t) & \cdots & \psi_{\kappa_{1}}(\boldsymbol{r}_{N},t) \\ \psi_{\kappa_{2}}(\boldsymbol{r}_{1},t) & \psi_{\kappa_{2}}(\boldsymbol{r}_{2},t) & \cdots & \psi_{\kappa_{2}}(\boldsymbol{r}_{N},t) \\ \cdots & \cdots & \cdots & \cdots\\ \psi_{\kappa_{N}}(\boldsymbol{r}_{1},t) & \psi_{\kappa_{N}}(\boldsymbol{r}_{2},t) & \cdots & \psi_{\kappa_{N}}(\boldsymbol{r}_{N},t) \end{array} \right \|. \end{equation}

The average energy of interaction of each pair of particles appears in the form of two terms. One resembles the average energy which can also be obtained within the simple wave function (3.22). The other, called the exchange part, appears owing to the (anti)symmetrization of the wave function. This integral is non-zero if the wave functions of two interacting particles overlap each other in some area of space. The average collective exchange term is obtained for the hydrodynamic models (Lee & Jung Reference Lee and Jung2013; Jung & Akbari-Moghanjoughi Reference Jung and Akbari-Moghanjoughi2014; Andreev & Ivanov Reference Andreev and Ivanov2015; Andreev Reference Andreev2016b). It is found in the regime of small exchange interaction, so the Hartree part of the interaction in the Hartree–Fock approximation, for the mean-field interaction gives the major contribution in the collective effects.

From the practical point of view, the exchange interaction appears as an effective shift of pressure. Moreover, the exchange term is negative, so it decreases the contribution of the pressure. This is the decrease of the Fermi velocity from the constant value, where there is the mechanism giving a positive shift of the Fermi velocity depending on the wave vector. This mechanism is the another quantum effect, the quantum Bohm potential.

A systematic model of the exchange effects would require the calculation of the exchange interaction in the pressure evolution equation and in the third-rank tensor evolution equation.

3.5. Contribution of ions and stabilization of spectrum

Dispersion equations (3.10)–(3.18) are obtained in the high-frequency limit, where ions are assumed to be motionless. However, the analysis of the obtained solutions shows that the second solution $\omega _{-}^{2}$ approximately presented by (3.14) is a low-frequency solution. It shows that proper analysis of this phenomena requires an account of the ion motion.

Particularly, dimensionless frequency square $\xi _{-}^{2}$ at the dimensionless wave vector near the classical limit $\kappa =0.1$ gives the value $\xi _{-}^{2}\sim 10^{-5}$. We compare it with the contribution of ions. For the estimation of the contribution of ions, we choose the minimum of two parameters $min\{\omega ^{2}_{IS}/\omega ^{2}_{\textrm {Le}}, \omega ^{2}_{Li}/\omega ^{2}_{\textrm {Le}}\}$ $=\{(3/5)(m_{e}/m_{i})\kappa ^{2}, m_{e}/m_{i}\}$ $=(3/5)(m_{e}/m_{i})\kappa ^{2}$ $=6\times 10^{-3}\times (m_{e}/m_{i})$ $\approx 10^{-5}$ for the hydrogen ions. Here, $\omega _{IS}$ presents the spectrum of ion sound, $\omega ^{2}_{IS}=(3/5)(m_{e}/m_{i})v_{\textrm {Fe}}^{2}k^{2}$.

We substitute the contribution of all linearized hydrodynamic equations of electrons in the Euler equation:

(3.25)\begin{equation} \left(\omega^{2}-\omega^{2}_{q}-\tfrac{3}{5}v_{\textrm{Fe}}^{2}-\frac{v_{\textrm{Fe}}^{4}k^{4}}{5\omega^{2}}\right)\delta n_{e} =\frac{q_{e}n_{0}}{m_{e}}\left(1+\frac{\omega^{2}_{q}}{\omega^{2}}+\frac{\omega^{4}_{q}}{\omega^{4}}\right)k^{2}\delta\tilde{\varPhi}, \end{equation}

where $\omega ^{2}_{q}=\hbar ^{2}k^{4}/4m^{2}$ is the characteristic quantum frequency. Frequency $\omega ^{2}_{q}$ appears from different sources: the quantum Bohm potential and the quantum part of the Coulomb interaction in the third-rank tensor evolution equation. The term proportional to $\omega ^{4}_{q}$ presents the contribution of the fifth-rank tensor evolution. Similarly, the Euler equation for the ions appears in the following form:

(3.26)\begin{equation} \omega^{2}\delta n_{i}=\frac{q_{i}n_{0}}{m_{i}}k^{2}\delta\tilde{\varPhi}. \end{equation}

Here, we repeat the Poisson equation,

(3.27)\begin{equation} \triangle\tilde{\varPhi}={-}4{\rm \pi} q_{e}(\delta n_{e}-\delta n_{i}), \end{equation}

where the perturbations of ions are presented.

We combine (3.25), (3.26) and (3.27). As the result, we get the dispersion equation:

(3.28)\begin{equation} 1=\dfrac{\omega^{2}_{Li}}{\omega^{2}} +\dfrac{\omega^{2}_{\textrm{Le}}\left(1+\dfrac{\omega^{2}_{q}}{\omega^{2}}\right)}{\left(\omega^{2}-\omega^{2}_{q}-\tfrac{3}{5}v_{\textrm{Fe}}^{2}k^{2}-\dfrac{v_{\textrm{Fe}}^{4}k^{4}}{5\omega^{2}}\right)}, \end{equation}

where the term proportional to $\omega ^{4}_{q}$ is neglected.

Equation (3.28) can be simplified in the required limit. We consider $\omega ^{2}\sim \omega ^{2}_{q}$ at $\kappa \ll 1$. Hence, we find $\omega ^{2},\omega ^{2}_{q},{v_{\textrm {Fe}}^{4}k^{4}}/{5\omega ^{2}}\ll 3(v_{\textrm {Fe}}^{2}k^{2})/5$. Therefore, (3.28) reduces to

(3.29)\begin{equation} 1=\frac{\omega^{2}_{Li}}{\omega^{2}} -\frac{\omega^{2}_{\textrm{Le}}}{\tfrac{3}{5}v_{\textrm{Fe}}^{2}k^{2}} \left(1+\frac{\omega^{2}_{q}}{\omega^{2}}\right). \end{equation}

Equation (3.29) shows that the contribution of the third-rank tensor presented with $\omega ^{2}_{q}$ modifies the coefficient in the spectrum of the ion acoustic wave, and gives no new solution.

Let us show corresponding spectrum:

(3.30)\begin{equation} \omega^{2}=\frac{1}{1+\dfrac{1}{\kappa^{2}}} \left(\omega_{Li}^{2}-\frac{\omega_{q}^{2}}{\kappa^{2}}\right). \end{equation}

The quantum contribution appearing from the third-rank tensor evolution equation gives the decrease of the ion acoustic wave frequency. Let us mention that the quantum corrections, like the quantum Bohm potential for ions, are small in comparison with the described effects. This spectrum is illustrated in figure 4. It demonstrates the decrease of the frequency under the influence of the pressure evolution equation and the third-rank tensor evolution equation.

Figure 4. Numerical illustration of (3.30) is shown. The black solid line shows the spectrum of the ion acoustic wave obtained with no application of the pressure evolution equation. It corresponds to the absence of quantum term in (3.30). The blue dotted line shows the spectrum of the ion acoustic wave following from the application of the pressure evolution equation and the third-rank tensor evolution equation.