In the inertial confinement fusion (ICF) study,^[1–3] electron transport is an important but difficult problem. For example, for fast ignition (FI) the laser energy is first deposited into fast electrons in the interaction region, which is usually 50 μm–100 μm away from the dense core, and then the energy flux transports into the dense core. Studies^[4] showed that electrons with an energy of 1 MeV–3 MeV are optimal for FI. The transport coefficients are usually required for both target design and magnetohydrodynamic codes. Because of its significance, many theoretical and numerical models have been established in search of electron transport features, such as the Spitzer–Härm model,^[5,6] the Braginskii model,^[7] and the nonlocal electron transport model,^[8] and so on.

In laser–plasma interaction physics, in order to obtain accurate electron transport coefficients one must consider two effects, one is associated with the self-generated magnetic field and the other is the plasma collision.^[9–13] Certainly the electron distribution^[14–16] has also played a key role. The self-generated magnetic field has been studied in laser interacting with plasma over the period of time.^[9,17–22] Recently, experimental measurements by Tatarakis et al.^[23] showed that the self-generated magnetic field, from a few of MGs (1 Gs = 10⁻⁴ T) to several hundred of MG, can be generated. The corresponding electron cyclotron frequency ω_Be lies within ∼ 1.8 × 10¹⁵ Hz–1.8 × 10¹⁷ Hz. Such a strong magnetic field is believed to have an important impact on electron transport.

Besides, the collision effect on the transport of electrons has been considered and studied in the high density region.^[24–26] For example, Kemp et al.^[27] showed that the collision effect cannot be neglected when the number density is beyond ∼ 10²³ cm⁻³, due to the collision frequency being close to the plasma frequency. Usually, the number density can be 10²² cm⁻³–10²⁶ cm⁻³ in ICF, and the corresponding electron–ion collision frequency ν_ei is ∼ 1.5 × 10¹⁵ Hz–3.8 × 10¹⁷ Hz, which is obviously comparable to ω_Be. The most previous studies about the electronic transverse transport coefficients were focused on the calculations of the weak collision, i.e. ω_Be ≫ ν_ei, with a Maxwellian distribution function.^[28] However, the combination effect of collision and strong magnetic field for different distributions have not been considered, in particular when ω_Be is comparable to ν_ei. Therefore, the electronic transverse transport needs to be studied further in these cases.

Plasma consists of electrons and ions. With the condition of assuming the electron temperature equals the ion temperature and the mass of the electrons is much smaller than that of the ions, the speed of electrons is much faster than the speed of the ions. Therefore, it is feasible assuming that electrons are moving while ions are stationary, so that ions are not affected by the collision and electromagnetic field. Besides, for simplicity we just consider the involving problem for the Lorentz plasma,^[28–30] in which the electron–electron collision ν_ee ∼ e⁴N_e can be ignored in comparison with the electron–ion collision ν_ei ∼ e⁴z_iN_e, i.e., ν_ee ≪ ν_ei, where N_e denotes the electron density and z_i is the atomic number. In order to control the divergence of electrons, Robinson et al.^[31] showed that an electron beam could be collimated well by the self-generated magnetic field in the high-Z material area. Without loss of generality, we can assume that the atomic number is z_i ≫ 1.^[30,32,33]

In this paper, the electronic transverse transport coefficients, such as electric and thermal conductivity, are studied by solving the Boltzmann transport equation for Lorentz plasma with a collision effect in the presence of a magnetic field. The different electrons distribution functions are also considered. The analytical formula is given and the corresponding numerical results are illustrated for the transport coefficients. It is found that the strong magnetic field and quasi-monoenergetic distribution are both beneficial to reduce the electronic transverse transport.

The paper is organized as follows. Section 2 gives the theoretical frame and analytical formula. Section 3 exhibits the results in the presence of both magnetic field and collision for different distribution functions. The results are also analyzed physically. A brief summary and discussion are given in Section 4.

Assuming that f(r,υ,t) is the electrons distribution function at position r, velocity υ, and time t, then the stationary transport equation of Lorentz plasma in a magnetic field reads^[28]

In order to calculate the electronic transport coefficients such as the electric conductivity and thermal conductivity, the current density and the generalized Ohm law as well the heat flux and the generalized heat flow law are need. They are respectively given as (refer to Ref. [28])

Before giving the detailed physical analysis and numerical results, we first see the concrete expressions for the current density and the heat flux, respectively. Substituting Eq. (5) into Eq. (6), one can obtain

For simplicity the following normalized variables are introduced as follows:

In Section 2, we have given the theoretical frame and analytical formula. Let us now turn to calculate the transverse transport coefficients for the different distribution functions.

3.1. Results for the Maxwellian distribution function

In the case of a Maxwellian distribution function, f₀ can be written as

By substituting Eqs. (10) and (13) into Eq. (7), one can obtain the electric conductivity

where

is defined, and σ_⊥ is normalized by in the weak collision.^[28,29] Similarly, by substituting Eqs. (10) and (13) into Eq. (9), one can obtain the thermal conductivity

where κ_⊥ is normalized by .

The σ_⊥ and κ_⊥ against Ω are plotted in Fig. 1. First from Fig. 1(a) when Ω → 0, one can see that the collision leads to the enhanced electronic mobility and heat conduction due to the large transverse divergency by the electron–ion collision scattering. With the increase of Ω, σ_⊥ decreases and recovers to the value σ_⊥0 when Ω → ∞ due to the strong magnetic field decreasing the impact of the collision. This is not surprising because the magnetic field reduces the mobility of electrons as the Larmor radius becomes smaller than the collision mean free paths,^[32,34] which results in that the mobility of an electron is limited. Similarly, since the magnetic field can suppress heat flow by decreasing the heat-carrying electronic mobility, the κ_⊥ decreases also with the increase of Ω and recovers to the value κ_⊥0 of ultra-strong magnetic field when Ω → ∞, as seen in Fig. 1(b). As the reduction of σ_⊥ and κ_⊥, electronic transverse mobility and heat flow are greatly suppressed.

Fig. 1.

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	Fig. 1. Transverse electric conductivity σ⊥ (a) and thermal conductivity κ⊥ (b) against Ω for the case of Maxwellian distribution. The black dashed line (case 1) and red solid line (case 2) represent the condition of weak collision (ωBe ≫ νT, see Refs. [28] and [29]) and the full condition without any limitation to the magnetic field and collision.

It is interesting to see the decreasing degree when the magnetic field increases, for example, in the typical case of ICF (ω_Be ∼ ν_ei), compared to that without magnetic field, the σ_⊥ reduces about by 55.27%, however, κ_⊥ reduces dramatically at about by 99.21%. This means that the spontaneous magnetic field can reduce the influence of collision on the electronic transport, in particular the heat flux. This would lead to the electrons being highly collimated by the magnetic field in the transverse direction.

3.2. Results for the quasi-monoenergetic distribution function

It is worth pointing out that in many cases, for example in some regions of ICF or/and laser interacting with a solid target, the distribution of electrons may be non-Maxwellian, therefore, the electronic transverse transport may exhibit different behavior. Now we consider a typical case of the quasi-monoenergetic distribution. For convenience as well as comparison with the previous Maxwellian case, the electrons distribution function is written as

where ɛ is the full width at half maximum (FWHM) or the quasi-monoenergetic degree, and υ_T is the virtual “electron thermal speed” which is associated to the electrons peak energy E_m through and the corresponding virtual “temperature” T_m = 2E_m. Here we assume that the directional speed υ_T is approximately longitudinal, i.e. υ_T ≈ (0,0,v_T), so that the electric field is still ignored. If the collision is not significant, i.e., when Ω → ∞, one has

where

and ɛ̄ = ɛ/v_T are defined and introduced, respectively. However, when the electron–ion collision is included, the transverse electric conductivity would become

where

is defined. It is ready to check that in the limit of Ω → ∞, equation (18) recovers to Eq. (17). Thus, it is convenient to get the following normalized σ_⊥ by σ_⊥0 as

Similarly, for the thermal conductivity we can get respectively

and

where both κ_⊥0 and κ_⊥ are normalized by N_eT_m/mν_T.

The σ_⊥ and κ_⊥ dependent of Ω are shown in Fig. 2 for a quasi-monoenergetic distribution with different energy spread width ɛ. From Fig. 2 one can see that there exist the similar effect of magnetic field on lowering the transverse transport coefficients for a certain ɛ̄ as in the case of Maxwellian distribution. Moreover, it is also found that there exist some delicate discrepancies for different distributions. First the transverse transport coefficients decrease rapidly as ɛ̄ decreases, which means that an electron’s motion and heat flow can be suppressed greatly. Second, for the given ɛ̄, as the magnetic field increases, the reduction effect of the magnetic field on the electric transport in the regime of a strong magnetic field (see the lower right curves of Fig. 2(a)), as well heat transport in the regime of a weak magnetic field (see the upper left curves of Fig. 2(b)), would become weaker and weaker. These results may be attributed to the fact that the distribution is far away from the Maxwellian distribution. It is not surprising to understand the lowering transport dependence of decreasing width. Because the narrower width would lead to the decrease of high-energy electrons, for example, in Ref. [33] it is demonstrated that the heat-flow is dominated by a small number of high energy electrons within 3v_T–4v_T, which would limit the heat flow greatly. Therefore, it concludes that the smaller the ɛ is, the less amount the transverse transport exhibits.

Fig. 2.

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	Fig. 2. Transverse electric conductivity σ⊥ (a) and thermal conductivity κ⊥ (b) against Ω with different width ɛ. The dashed line represents the condition of ɛ̄ → 0.

On the other hand, it is also noted that when ɛ̄ varies from 0.5–0.9, the conductivity can still be affected by Ω at 10⁴–10⁶. This can be explained by the relation between Larmor radius r_b and magnetic field B:

where E is the electron energy. We can see that r_b is proportional to E while inversely proportional to B. For ɛ̄ varies from 0.5–0.9, which means that there are more electrons in the higher energy region, so that a stronger magnetic field is required to collimate the transverse transport of electrons.

It is worth noting that for our special interested cases of ICF when ω_Be ∼ ν_ei, σ_⊥ and κ_⊥ are also reduced as the ɛ̄ decreases while this influence becomes weaker and weaker. A more interesting aspect is that the transport is hardly affected by the magnetic field when ɛ̄ → 0 for σ_⊥ (as seen in the dashed lines in Fig. 2(a)) as well for κ_⊥ when Ω < 1 (as seen in the dashed lines in Fig. 2(a)). This will be studied further in the following subsection and discussed in the final section.

3.3. Results for the Delta distribution function

In the limit of ɛ → 0, quasi-monoenergetic distribution will recover to the Delta one with almost perfect monoenergetic electron density. Thus it is necessary to discuss this special condition in this subsection. First the distribution function is now written as

As mentioned above, in the limit of Ω → ∞, σ_⊥ and κ_⊥ are given as

When electron–ion collision is included, one obtains

where all of the coefficients are normalized by e²N_e/mν_T and N_eT_m/mν_T, respectively.

In the comparison with each other in two cases (the weak collision, i.e., ω_Be ≫ ν_ei), σ_⊥ does not change over all the parameter change region of Ω, and κ_⊥ also hardly changes at the region of Ω < 1 (as seen in the left red line of Fig. 3(b)). The reason for the conductivity insensitivity to the magnetic field is due to the combinational effects of the Hall term and the thermal pressure term. And it is also seen as the result of υ_T ∥ B. Certainly if the higher order is included or there exists the perpendicular directional velocity, the conductivity would not keep constant to the magnetic field. In particular, for example in our interested case of ICF where ω_Be ∼ ν_ei, the κ_⊥ is reduced about one half. On the other hand the region hardly affected on the transverse transport κ_⊥ is now only limited to the parameter changed region of Ω < 1. When Ω > 1, the κ_⊥ is dramatically reduced similar to the case of weak collision (ω_Be ≫ ν_ei),^[28] due to the κ_⊥ being always proportional to 1/Ω² or 1/(Ω + 1)².

Fig. 3.

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	Fig. 3. Same as in Fig. 1 except for the case of Delta distribution.

In summary, we have investigated the electronic transverse transport with the collision effect in the presence of a magnetic field for different electrons density distributions. According to our analyses, the transverse transport is not only affected by magnetic field and collision, but also modified by distribution functions.

First, for the Maxwellian one, when Ω < 1, the transverse transport is dominated by a collision effect. But with the competition of magnetic field and collision, σ_⊥ and κ_⊥ are obviously reduced due to the Larmor radius becoming smaller than the collision mean free paths and magnetic field effect will be dominant at Ω > 1. Therefore, the magnetic field leads to great suppression of the electronic mobility and heat flux.

Second when the distribution function is far away from the Maxwellian, the transport is different from the classical transport theory. In this case, the transverse transport coefficients are significantly modified by a magnetic field as well as the distribution. On the one hand, similar to the Maxwellian one, all of the transverse transport coefficients decrease with the increase of Ω. On the other hand, the electronic mobility and heat flow are suppressed greatly as ɛ̄ decreases. As a result, the smaller the ɛ is, the less amount the transverse transport exhibits. On the other hand, however, with the decrease of ɛ̄, the magnetic field and collision balances with each other in the region of Ω < 1 so that σ_⊥ and κ_⊥ change weaker and weaker. Until to the region of Ω > 1, in which κ_⊥ is rapidly reduced due to the fact that the magnetic field effect is dominated over the collision effect, which destroys the weak balance.

Finally, in the limit of ɛ → 0, the distribution will tend to be the Delta function with almost perfect monoenergetic density distribution. Under this circumstance, the thermal pressure of affecting σ_⊥ and the Hall effect of affecting σ_⊥ are canceled with each other, which causes a perfect balance so that the σ_⊥ keeps a constant for the whole region of Ω. But this behavior is approximately valid to κ_⊥ only when Ω < 1 and again the magnetic field effect is dominated over the collision effect when Ω > 1 in which the κ_⊥ decrease rapidly.

In a word, the results studied in this paper indicate that the magnetic field and proper quasi-monoenergetic distribution can effectively reduce the transverse transport and suppress the transverse divergence of electrons. It would be helpful to understand the transport problem of Lorenz plasma as well the fast electrons transport in ICF background. Besides, we consider the first-order approximation in the process of calculation. High-order small quantities may also slightly affect the results, which we will investigate in the future.