Figure 6.1: Spinning flying objects in classical mechanics. Rotating beams are stabilized against small disturbances by means of the angular momentum conservation along the propagation axis [59].
Spinning flying objects such as American footballs, spinning rockets, and rifled bullets are stabilized against small disturbances by maintaining a large canonical angular momentum vector in a specific direction [58] as shown in Fig. 6.1. The spinning motion counteracts a misaligned thrust to the objects.
Motivated by this long-known stability principle of classical mechanics [59], we apply the spinning
effects on the charged particle beams to stabilized the halo formations along the beam propagation.
Note that the spinning addressed here is not about an atomic spin in quantum mechanics, but a rigid- rotor rotation around the center of a solid objects which has non-zero canonical angular momentum.
Charged particle beams are composed of many single charged particles, so we should calculate the average of canonical angular momentum of whole particles within a beam.
The evolution of symmetric transverse beam size rb of 2D coasting beam is given by envelope equation,
d2rb(s)
ds2 +κ⊥0(s)rb(s)− Kb
rb(s)− εT,⊥2
r3b(s)=0. (6.1)
Here,κ⊥is solenoid focusing function andεTindicates total emittance which is 4 times RMS emittance, εT,⊥=4εrms,⊥. For a non-spinning beam (of zero canonical angular momentum), the transverse RMS emittance is equal to the thermal emittanceεrms,⊥=εth,⊥=p
det(Σ), which is related to the determinant of the 4×4 beam matrixΣ=⟨zzT⟩, wherez= (x,y,x′,y′)Twith(· · ·)Tbeing the transpose operation. For a spinning beam with axisymmetry, the transverse RMS emittanceεrms,⊥ =q
εth,⊥2 +⟨Pˆθ⟩2/4 is com- posed of the transverse thermal emittanceεth,⊥and the normalized average canonical angular momentum
⟨Pˆθ⟩=⟨β γmcPθ ⟩[14,60,61]. Brackets⟨· · · ⟩denote the statistical average over the beam distribution. Also, we note that ˆPθ =r2θ′+ (qAθr/β γmc) corresponding to to a single particle is determined by the ini- tial conditions, where the azimuthal vector potential isAθ =Bz(s)r/2 to the leading order of solenoid magnetic fieldBz.
6.2.1 Equations of motion with non-zero canonical angular momentum
Basically, under the solenoid focusing, equations of motion of a single particle in real frame is coupled in x and y directions as shown in Eq. (2.13). In order to avoid the coupled terms, we transfer the laboratory frame into Larmor frame which becomes uncoupled between xandyas in Eq. (2.26). For the uniform density beam, equation of motion under quadrupole focusing in laboratory frame and that under solenoid focusing in Larmor frame are linear and uncoupled inxandyplanes [see Eqs. (2.31) and (2.32)]. However, for Gaussian density beam, the non-linear space-charge fields make coupling between xandywhen the initial canonical angular momentum is not zero even in Larmor frame. The equations of motion in Larmor frame of Gaussian space-charge fields [Eq. (2.71) with Eq. (2.72) and (2.73)] are written as
X′′(s) +κ⊥0(s)X(s)−Kb1−e−r2(s)/σr2(s) r2(s) X=0, Y′′(s) +κ⊥0(s)Y(s)−Kb
1−e−r2(s)/σr2(s) r2(s) Y =0,
(6.2)
whereκ⊥0(s)≡Ω2L(s)is solenoid focusing function expressed by Larmor frequency andr2(s) =X2(s) + Y2(s)is a radial coordinate which is invariant under Larmor transformation with RMS envelope radius σr(s) =rb(s)/√
2. The normalized canonical angular momentum can be calculated by ˆPθ =r2θ′ = xy′−yx′=XY′−Y X′ (for the drift space without vector potential), which is invariant under Larmor
transformation. If ˆPθ ̸=0, particle motions in X andY directions become coupled because ofr2(s) components in the last terms of Eq. (6.2).
The equation of motion of radial motionr(s)with a Gaussian space-charge is [14]
r′′(s) +κ⊥0(s)r(s)− Pˆθ2
r3(s)−Kb1−e−r2(s)/σr2(s)
r(s) =0. (6.3)
The third term of Eq. (6.3) adds a non-linear repulsive force when ˆPθ has a finite value. In this case, a particle never crosses the axis (r=0).
Figure 6.2: Particle angle on the real frame and the Larmor frame with canonical angular momentum.
A particle motion on real frame and Larmor frame can be expressed by r and angle relative to the x axis. The angle of a particle in real frame is
θ−θ0= Z z
z0
(− qBz
2γ βmc+Pˆθ2
r2)ds. (6.4)
The first term of the integration indicates the angle between real and Larmor frame,θL=−Rz0z 2γ βqBmcz ds.
A particle motion in the real frame (laboratory frame) is expressed by x=r·cosθ andy=r·sinθ. Also, we can express a particle motion in Larmor frame by X =r·cosθr andY =r·sinθr, where θr=θ−θL=Rz0z Prˆθ22ds+θ0is the rotating angle of a particle in Larmor frame. The angles and particle position on the real frame(x,y)and Larmor frame(X,Y)are shown in Fig. 6.2.
6.2.2 Generation of spinning beam in multiparticle simulation
For applying the spinning beams in multiparticle simulations, we can generate the spinning beams by loading initial particles at the center of a solenoid magnetic fieldBz without any average rotation [see Fig. 6.3]. Then the total canonical angular momentum is given by Pθ =qBz(s=0)r2/2. A beam goes out of the solenoid, whereBzvanishes, particles gain mechanical angular momentum with rotating term, Pθ =γ βmcr2θ′. This property has been experimentally adopted in research entailing electron beams [60, 62] and is generally referred to as Busch’s theorem [14, 62, 63]. For the simulation studies
Figure 6.3: Spinning beam generation by using multiparticle simulation. Beam is injected inside a solenoid magnetic fieldBzand gets rotating termθ′ as beam is going out of the solenoid by means of conservation of canonical angular momentum. The generated spinning beam passes through a periodic solenoid focusing lattice.
after generating the spinning beams, we apply the spinning beams to pass through a periodic solenoid focusing lattice as shown in Fig. 6.3.
A spinning beam has a non-zero average canonical angular momentum and exhibits rigid-rotor ro- tation around the beam propagation axis. This scheme is based on two notable achievements in beam physics : (i) a rigid-rotor beam equilibrium was obtained for an intense beam propagating through a periodic solenoid lattice [64]. (ii) a rotating beam was generated by stripping an ion beam inside a solenoid [65].
The technology of beam stripping for generating the spinning beam is based on several well-established experiments [65, 66]. Using a thin carbon foil (but without a solenoid field) in the MEBT section of the Spallation Neutron Source (SNS), a previous study reported the stripping of 2.5 MeV H− to a proton beam with an efficiency of 99.98% and emittance growth of only 10–20% [66]. If we strip a H−beam inside a solenoid field Bz as in the emittance transfer experiment (EMTEX) [65], ⟨Pˆθ⟩=2κ0σr2 with κ0={[(Bρ)in/(Bρ)out]−1}[Bz/2(Bρ)in], where(Bρ)in and(Bρ)outare the beam rigidities before and after the foil, respectively [67]. For a 2.5 MeV H−beam withσr=2 mm, we have|⟨Pˆθ⟩|≲35 mm-mrad for|Bz| ≤1 T. For a 2.5 MeV/u D−beam with similar conditions, we expect an outgoing deuteron beam with|⟨Pˆθ⟩|≲12 mm-mrad. Hence, the experimentally available ranges of⟨Pˆθ⟩would cover most of the simulation settings that will be presented in the next section.