Collective Longitudinal Motion in Intense Ion Beams

Inertial Confinement Fusion (ICF) using high energy heavy ion beams requires tight focussing of the igniting ion beams in longitudinal, as well as transverse, space at the pellet target. This sets significant requirements for stability in longitudinal motion. It has been previously noted that this motion can contain significant resistive wall instability. Results of numerical simulations of this instability in perturbed bunched beams are presented and analyzed. It is found that reflection of perturbations off bunch ends is distorted and delayed by space charge forces and that "soliton" waves can appear after reflection.


Introduction
In earlier papersl'2ts we have explored some characteristics of longitudinal motion in intense ion beams. In references 1 and 2, longitudinal motion in bunched beams with "apace charge" forces is studied analytically and in reference 3 longitudinal motion in coasting and bunched beams with "apace charge" and "resistive wall" forces is explored analytically and numerically.
In reference 3, we find that the "apace charge" force causes wave propagation of disturbances in an equilibrium distribution and that a resistive force can cause unstable growth of these disturbances. In this paper we extend our discussion of this wave instability and note characteristics of the "wave reflectionI' which occurs when these disturbances reach the ends of a beam bunch.
"Soliton" formation is observed in reflection, as described below.

Equations of Motion
We choose the longitudinal position within the bunch z and the distance along the accelerator a as our dependent and independent variables.
We assume, as an approximation, completely decoupled and Then for the equation three forces: that transverse motion is the bunch is not accelerated. of motion we take the sum of 1) a "space charge" force where qe, M, and @c are the ion charge, mass and velocity, g a geometric factor, and h the ion charge density.
2) a "resistive wall" force where R1 is the resistive coupling per meter.
3) a bunching force F(z,a) provided by external fields.

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Our equation of motion is nonrelativistic: To demonstrate the coasting beam instability, we start with an initial distribution fo(z,z') = N' 6(z') where N' is the initial ion density and the velocity distribution is taken as a g-function to approximate the low velocity spread case of an ICF driver.
We take a perturbation: f = fl (2') e i(kz-ws) 1 and solve the linearized Vlaaov equation for w(k), obtaining flJ2 = N'(A k2-ikB) (5) When Ak >> B, as is true for HIF linaca, we find: Since Ret:) g ?v'G is independent of k, the wave velocity and group velocity are equal, which means disturbances propagate together as coherent wave packets along the beam. Also the motion is unstable, since Im(w) #O. A forward propagating wave decays (Im(w)<O) and a backward wave grows.

Reflection of Wave Packets at Bunch Ends
In bunched beams the wave packets of a perturbation will reach the bunch ends in a finite time.
A naive expectation is that a growing slow Wave propagation in a perturbed beam bunch with "resistive-wall" and "apace charge" forces. The perturbation splits into a decaying "fast" wave and a growing slow wave. The upper graphs show I(T) = AC-Z) ; the lower graphs show the phase apace distribution f(t,BE) 0~ f(-2,~').
wave will reach the end of the bunch, be immediately reflected to a decaying "faatl' wave by the external bunching field and produce no net instability. In HIF beams eL is small and will be ignored below. A disturbance reaching the end of the bunch perturbs 20 to z. +Azo(a). The equation for AZ0 is:  Figure  2-A-G. Reflection of perturbation off bunch ends with zero "resistive wall." Envelope oscillation is noticeable as well as particle oscillations to large Ap/p at the ends.

Velocity
Wave Formation In simulations with resistance, an interesting phenomenon can occur when a large perturbation is reflected off the bunch end. The return wave is changed in character from the initial wave and changes the average energy of the bunch.
In Figure 3A we show a phase apace distribution from a numerical simulation showing this characteristic wave form.
Similar behavior can be seen in a simple coasting beam model in which we collapse the velocity distribution to delta functions as in section 1.

<zO+Vwa
This has a "velocity" wave propagating with speed V w and with amplitude Vo, as shown in Figure 3B.
This anaatz must be self-consistent, which means: 1) The particle flux entering the wave front must equal the particle flux leaving the wave front: impulse -Vo from the space charge force as the wave front passes through them. The apace charge force is found from equation (1) dh (a -aI) F = -Ax Z-A ; where we have stretched the discontinuity of figure 3B over a distance 6. The impulse received as the wave passes is: