2.1. Dielectric tensor and dispersion relation

We start from the relativistic Vlasov equation for the electronic distribution function f (r,p,t)

with v = p/(γm) and

The mass m will always be the electron mass in the sequel since we shall consider the ions form a fixed background. In the same way, q is the electron charge. One adds Maxwell's equations to the system and the following expression of the dielectric tensor elements is derived (Ichimaru, 1973)

where the integrals must be evaluated using the standard Landau contour for a proper kinetic treatment. The plasma frequency reads ω_p² = 4πn_p q²/m.

The basic form of the dispersion relation is

where ε(k,ω) is given by (3). If one makes the electrostatic approximation k × E_k = 0, the dielectric tensor takes the much simpler form

Without any assumption upon the nature of the waves, we have k × (k × E_k) = (k·E_k)k − k²E_k in Eq. (4) and get

Setting

non-trivial (E ≠ 0) solutions are obtained provided that det(T) = 0, that is,

This forms the most general expression of the dispersion relation. We can now start to detail the geometry of our problem. The velocity distribution anisotropy is set along the z-axis (see Fig. 1 for clarity). Without any restriction of generality, the cylindrical symmetry of the problem allows us to set k = (k_x,0,k_z). As far as the equilibrium function is concerned, we will use in the sequel electronic distribution functions f₀ such as

with ∫f₀(p) d³p = n_e. These distribution functions are isotropic in the (x,y) plane. We can notice that Eq. (9) implies a vanishing average momentum in the (x,y) plane. Under these assumptions, tensors elements T_α,y and T_y,α vanish for α = x,z and Eq. (8) reduces to (with η ≡ ω/c)

which displays two branches

and

These equations are valid for any orientation of the wave vector and any orientation of the electromagnetic field with respect to the wave vector. The first one is a typical dispersion equation for transverse waves. As for the second one, it can be checked, it reduces to an equation for transverse or longitudinal waves when k_x or k_z vanishes. For an arbitrarily oriented wave vector, the angle

equals the angle between k and the eigenvector of tensor T corresponding to the mode investigated.

Geometry of the problem. The angle φ between the electric field E and the wave vector k may take all values between 0 and π/2.

2.2. Presentation of the three models investigated

In order to detail plasma temperature and relativistic effects on oblique instabilities, we shall now focus on the following three models:

Model 1. The fluid, or cold, approximation, with distribution functions for the beam and plasma given by

This is the simplest approximation which displays the most basic trend of the phenomenon.

Model 2. The beam is still taken as cold whereas the plasma has a temperature in the direction normal to the beam with distribution function

This kind of waterbag distributions was used to derive analytical results in similar situations (Silva et al., 2002; Yoon & Davidson, 1987).

Model 3. Unlike the first two models where we restrict to the non-relativistic regime, here we shall consider relativistic effects together with the distributions functions used in model 2.

As one can see from the functions, we consider no finite size effects. Every model includes a return current which neutralizes the total current (charge neutralization comes from the fixed ions background). One therefore has in every cases n_pV_p = n_bV_b with P_p,b = γ_p,b mV_p,b (γ_p,b = 1 for models 1 and 2). We shall make use in the sequel of the dimensionless variables

and

.

Although lengthy, calculations for models 1 and 2 are straightforward. As far as model 3 is concerned, we set γ = 1 in the integrals involving the plasma distribution function because the two main velocities at stake regarding it are non-relativistic. The first one, the thermal velocity, is about 10 keV in a fusion plasma and remains much smaller than the 0.5 MeV required to tilt relativistic effects. The second velocity, V_p, is the one of the return current induced by the relativistic electron beam with V_p /c < n_b /n_p = α. It turns out that within the limits of the fast ignition scenario, α varies from 10⁻¹ (plasma edge) to 10⁻³ (plasma core) (Tabak et al., 1994). This shows that the return current velocity is non-relativistic so it is perfectly relevant to restrict relativistic effects to the beam.

4.1. Basic results in the normal and parallel directions

For wave vectors normal to the beam (θ_k = π/2, filamentation configuration), one finds a dispersion equation yielding unstable modes for (α,α_t << 1)

This is just the non-relativistic result times a factor γ_b so that relativistic effects are destabilizing the system at higher Z's. One also finds an infinite threshold for zero temperature (α_t = 0). This is consistent with the known fact that in the fluid model, the growth rate just saturates at high Z. The maximum growth rate in this direction

is found for

. Since δ_mπ/2 ∝ β(1 − β²)^1/4, the filamentation growth rate reaches a absolute maximum

with

For wave vectors parallel to the beam (θ_k = 0, two stream configuration), one finds a dispersion equation yielding unstable modes for (Godfrey et al., 1975)

in the limit α << 1. The maximum growth rate

is found at Z ∼ 1. It worth noticing that the same dispersion Eq. (12) yields these results for wave vectors normal as well as parallel to the beam, describing waves which are transverse on one side and longitudinal on the other (Fig. 3).

Numerical evaluation of the two-stream/filamentation growth rate in ωp units, in terms of Z = kVb /ωp. (a): model 1 with α = 0.1 and β = 0.2. (b): model 2 with . (c): model 3 with .

4.2. Arbitrary orientation for second branch

We now consider the dispersion Eq. (12) at any θ_k

Full expression of dispersion Eq. (25) after inserting the tensor elements (3) is obviously involved. We shall see a number of analytical conclusions can be reached however. The analysis detailed in the sequel refers to model 2, namely non-relativistic beam-hot plasma interaction. This simplifies exposition and relativistic corrections appear straightforward.

We start noticing that at large x, the asymptotic form of Q must be the dispersion relation at high frequency (ω² − ω_p² − ω_b²)(ω² − ω_p² − ω_b² − k²c²) = 0 so that the polynomial

can be considered as the base of the Q(x,θ_k) curve, which is just the P(x) curved disturbed by three singularities located at x₁, x₂, and x₃ with (see Fig. 4)

For θ_k = 0 one has x₁ = x₂. When the angle θ_k departs from 0, singularities x₁ and x₂ depart from each other. One has a typical curve such as the one depicted on Figure 4. We have circled the location where real roots may disappear, yielding the instability.

Plot of Q(x,θk) defined by Eq. (25) as a function of x. Parameters are chosen to display clearly the curve topology with . The circle indicates the place where real roots can appear or disappear. The dashed line is the curve of polynomial P(x) defined by Eq. (26).

As θ_k keeps increasing, we shall come across a very interesting transition coming from the order of the three singularities defined by Eq. (27). The first one is always negative, and always smaller than the second one since x₁ = x₂ − 2Zα_t sin θ_k. As for the second and the third one, a brief look shows that x₂ < x₃ for small angles but that x₃ is eventually greater than x₂ as the angles approaches π/2. The inversion occurs for a critical angle φ such as x₂ = x₃, that is

This expression is exact and independent of Z = kV_b /ω_p. Due to the singularity located at x₂ and x₃, the local minimum located in between increases when θ_k → φ. As a result, one has to increase Z higher and higher to recover two real roots between the singularities. Denoting Z_c(θ_k) the instability threshold, we see lim_θk→φ Z_c(θ_k) = ∞. This feature indicates that the boundary on the stability domain has the cone θ_k = φ as an asymptote in the k space.

The growth rate in the φ direction can be evaluated developing the dispersion equation around x = Z cos φ. In the limit α,α_t << 1, one finds at high Z (we include relativistic correction)

This result is exactly the growth rate calculated for θ_k = π/2 (see Eq. (21)). Indeed, it can also be derived from a continuity argument. Denoting δ_π/2^∞ and δ_φ^∞ the growth rates at high Z in the π/2 and φ directions, both quantities need to merge for α_t << 1 since lim_αt→0 φ = π/2.

Figure 3b displays a numerical evaluation of the growth rate for the two-stream/filamentation mode up to the point (50,3) in the (Z_x,Z_z) plane. This plot confirms the trend we have already anticipated. One identifies the two-stream instability growth rate along the Z_z axis and the filamentation one along the Z_x axis, but the most remarkable feature on this figure is the non-decreasing growth rate in the φ direction.

Figure 3a displays exactly the same plot but for the cold beam interacting with the cold plasma (model 1). There is no such thing as a critical angle here since Eq. (28) yields exactly π/2 for a cold plasma. Also noticeable is the filamentation saturation at high Z instead of a stabilization.

As mentioned previously, relativistic corrections are straightforward. The first point to emphasize is that the expression of the critical angle evidenced in the non-relativistic case remains unchanged. The origin of this critical angle lies in the dispersion relation singularities x_i (Eq. (27)). The critical angle φ is then defined by x₂ = x₃ and one easily check it bears no relativistic corrections. We now turn to numerical evaluation all over the wave vector space. We have used fast ignition scenario parameters: a relativistic 2 MeV (γ_b ∼ 5) electron beam with n_b = 10²⁰ cm⁻³ entering a 10 keV plasma with n_p = 10²¹ cm⁻³. This gives

. Actually, electronic plasma density ranges from 10²² to 10²⁶ cm⁻³ within this scenario (Tabak et al., 1994) so that α is always smaller than

. These parameters yield a critical angle φ = π/2.12 which is close to the normal direction. Figure 3c displays a numerical evaluation of the growth rate all over the plane (Z_x,Z_z). One notices the long unstable filamentation tail up to Z_x ∼ 40 in the normal direction, as well as the reduced two stream growth rate in the beam direction. The most striking features are indeed the steady growth rate in the φ direction and the maximum reached for Z_z ∼ 1 and Z_x ∼ 5. These results are detailed in Figure 5 which is a contour plot of Figure 3c. Note that the angle φ between the growth rate's “ridge” and the normal direction is amplified since the largest wave vector shown is Z = (50,2.5).

Contour plot of Figure 3c. Max. growth rate in ωp units is about 0.21 for Zz = 1 and Zx ∼ 5.3 (from Eq. (30)). One sees growth rates values of 0.16ωp up to Zx ∼ 25 in the φ direction.

4.3. Maximum growth rate all over the k space

It is obvious from Figure 3c that in the relativistic regime, the maximum growth rate all over the wave vector space is no longer on one axis but rather in between. We now turn to some analytical results concerning the maximum growth rate for any k, in the limit α,α_t << 1. Hopefully these limits are relevant for most experimental situations where the beam density is much lower than the target one, and the beam velocity much higher than the target thermal velocity.

The position of the maximum growth rate in the k, or Z space, can be derived from a continuity argument in the α_t << 1 limit. For a cold plasma the maximum growth rate dependance on Z_x is very weak (see Fig. 3a). The maximum growth rate in the beam direction being always near Z_z = 1, we can expect the same value for the Z_z maximum growth rate in the small temperature limit. As for the Z_x component of the maximum, its position coincides with the maximum filamentation growth rate. Having determined which wave vector Z_m leads to the maximum growth rate in the relativistic regime, we now make use of Figure 6 to find an analytical expression for the corresponding growth rate value δ_m(Z_m). An analysis of these plots shows δ_m behaves as α^1/3 for α << 1 and as 1/γ_b^1/3 in the relativistic regime. Figure 6c shows temperature has almost no effect in the small α_t limit so that we can guess δ_m(Z_m) ∝ (α/γ_b)^1/3. For this quantity to join the maximum two stream growth rate for γ_b ∼ 1, the coefficient must be

. As a conclusion, the highest growth rate is reached for

with

A result partially displayed in (

et al., 1970). Eq. (30) gives Z_xm = 5.3 with the parameters used to plot Figure 3c.

Comparison between maximum filamentation growth rate (δmπ/2, long dashed line), two stream growth rate (δm0, short dashed line) and the maximum at all k (δm, plain line) in terms of β, γb, αt and α. (a) . (b) . (c) . (d) are derived from Eqs. (21, 24) while δm is numerically evaluated. Black circles are evaluated from Eq. (31). All growth rates are in ωp units.

As for the transverse or longitudinal character of the waves, a plot of the electric field direction displayed on Figure 7 shows the transition domain between longitudinal two stream waves and filamentation transverse waves extends well bellow the φ ridge. A longitudinal wave has its field vector oriented towards the origin of the Z space, and one can see on this figure that waves are not longitudinal even for waves vector less oblique than φ.

Direction of the eigenvector electric field E(k,ωk) for Zz < 1 and Zx < 10 in the cold relativistic beam/hot plasma system. Same parameters as Figure 3c. The plain line indicates the φ direction. The arrow represents Zm, the wave vector yielding the maximum growth rate.