The origin of the magnetic field is the curl1 of the electric field,E, required to draw the return current and hence is a function of the plasma resistivity, η=η(x, y, z), [122]:
E=ηjr (3.9)
∂B
∂t =−∇ ×E=−∇ ×(ηjr) (3.10) ∇ ×B=µ0(jf +jr) (3.11)
The curl operator indicates that the B-field will form a rotational field and will grow if perfect current neutralisation does not occur: jf +jr 6= 0. The resulting net current
density will be∇×B/µ0, which is usually much smaller than bothjf andjr. The growth
rate ∂B/∂tcan be defined in terms of jf by combining the above equations:
jf +jr= ∇ ×B µ0 (3.12) ⇒ jr= ∇ ×B µ0 −jf (3.13) ∂B ∂t =−∇ × η µ0 ∇ ×B−ηjf (3.14) ⇒ ∂B ∂t =−∇ × η µ0 ∇ ×B +∇ ×(ηjf) (3.15) 1
Figure 3.1: (a) The resistivity curve for solid plastic material CH; Spatial gradients
within a plasma at high temperature tend to be uniform in terms of resistivity. This is reversed in cooler plasmas where the value of∇η is greatest. (b) A simulation of fast electron transport in a plastic target; The effects of both magnetic field components on the fast electron density is shown: (1) pinching at the injection region (∇η = 0) and (2) beam hollowing midway through the target where the temperature is relatively cool
(∇η= max).
The first term on the right-hand side of Equation 3.15 represents the resistive diffusion of magnetic field through the target, defined by the diffusion coefficientη/µ0. The second term is the source of magnetic field driven by the fast electron current. For a hot, highly ionised plasma the diffusion coefficient ≈1 and so the magnetic field equation reduces to:
∂B
∂t =∇ ×(ηjf) (3.16)
⇒ ∂B
∂t =η(∇ ×jf) +∇η×jf (3.17)
which generates a magnetic field that acts to push the electrons towards regions of higher current density and also towards regions of higher resistivity [123].
Consider a beam of fast electrons propagating through a plasma, the beam is denser on-axis and less dense towards the edges. The magnetic field can grow out of spatial variations in either the current density or the resistivity. The strongest gradients are associated along the radial direction across the beam. Consequently the magnetic fields are predominantly azimuthal. The first magnetic field component, η(∇ ×jf), is gen-
erated by spatial gradients in the current density. A radial force is therefore exerted,
F=−evf ×B, directed towards the beam axis which can effectivelypinch the beam
reducing its divergence (also referred to as Z-pinch). This force can also act on smaller variations within the beam producing a transverse breakup of the beam. This phenom- ena is known as the resistive filamentation instability, see Section 3.3.
The second magnetic field source, ∇η ×jf, arises from resistivity gradients in the
background plasma. As before, for the electron beam travelling through the plasma, the current density is greatest along axis. This gradient in current density induces a similar temperature gradient of the background plasma which is greatest on-axis. The plasma resistivity evolves with temperature and hence a resistivity gradient develops across the beam. The direction of this gradient can be positive or negative depending on the material specific resistivity curve. For the case where higher temperature plasma is less resistive, the resistivity is minimised on-axis and increases outwards, opposite in effect to the current density gradient. The resulting magnetic field pushes the fast electrons outwards from the beam axis, effectively hollowing the beam [123, 124]. There is clearly a competition between the two components of Equation 3.17, both effects have roles to play over the life time of the electron beam. Usually the current gradient component is stronger. However, at the edge of the beam, where there is an appreciable temperature gradient, the current density is falling to zero and hence the pinching effect is negligible. If η changes very little over a temperature gradient than there will be no resistive component. This usually occurs at high plasma temperatures where the resistivity curve is quiet flat. This is typical at the injection region where the plasma temperature is greatestkBTe≈1 keV. Deeper within the target, the plasma is relatively
cooler, and resistive gradients are present and beam hollowing may occur. An example of these competing effects are shown in Figure 3.1 for transport in a plastic target. In other materials, such as solid aluminium, the resistivity gradients are reversed at low temperature, which effectively provides another pinching component. Such increased levels of magnetic pinching can have significant effects relating to both beam divergence and beam filamentation, see results discussed in Chapters 5-6.
3.2.1 Magnetic pinching
Self induced collimation of the fast electron beam represents a tantalising possibility for shaping the constraints of Fast Ignition. Equation 3.17 provides in theory a self- generated magnetic field. This field is directed around the beam in a azimuthal fashion with the potential to indeed provide a pinching force to limit the beam’s divergence. The strength of the magnetic field can be estimated from Equation 3.10,
∂B ≈ ηjf∂t rf
(3.18)
and hence will exceed 1000 T at the injection zone where rf = rL ≈ 5µm and ∂t =
τL ≈ 1 ps . Obviously, if the beam spreads out, rf will increase and the value of jf
example, after propagating 100µm, a beam diverging with half-angle θ1/2 = 25◦ will extend torf ≈50µm and hence the magnetic field will drop by an order of magnitude.
This is apparent in Figure 3.2 which shows a simulation exemplifying the strength of the magnetic field and its effect on the fast electron beam.
The specifics of whether self-induced collimation can occur were considered by Bell and Kingham [94], who showed that the magnetic field must be sufficiently strong to deflect fast electrons through an angle θ1/2 in the distance rf/θ1/2 in which the beam radius doubles. For small θ1/2 , the theory requires that the ratio of the beam radius to the gyro-radius (rg), is:
rf
rg
> θ12/2 (3.19)
Using this condition acollimation factor(Γ) can be derived:
rg = mevf eB (3.20) ⇒ rf rg = rfeB γmevf > θ12/2 (3.21) Γ = rfeB γmevfθ21/2 (3.22)
Collimation occurs for Γ > 1. As an example, for an electron beam with kBTf =
5 MeV, with θ1/2 = 25◦, a magnetic field of>700T would be required for collimation. Assuming a resistivity of 2×10−6Ωm, the time needed to generate the field would be
rfB/ηjf = 20 fs. Indeed these conditions are easily attainable at the injection region
where the current density and hence magnetic field are greatest. The manner in which the magnetic field extends into the target interior is crucial if persistent collimation is to occur, otherwise the beam will diverge and begin spreading laterally once again. The general effect of such azimuthal magnetic fields acting upon the overall beam is termed
global pinching, and is the subject of an investigation presented in Chapter 5.
The effect of pinching on the fast beam should be accounted for self-consistently, as any collimation will increase the fast electron current density, which will increase the rate of magnetic field generation, i.e. once initiated, collimation occurs with positive feedback. Detailed study does require sophisticated numerical simulation. An example is shown in Figure 3.2 for transport in a relatively thick aluminium target using a hybrid code called LEDA which is described in Section 4.4.2.
Figure 3.2: Simulation of fast electron transport in a aluminium target using LEDA.
The fast electron density (nf) and the self-generated azimuthal magnetic field (Bz) are
shown at time 650 fs since laser incidence. The electron beam is injected from the left hand side and propagates along the x-direction. The magnetic field extends deep into
the target and acts to collimate the electron beam.