2.7 Equations of motion of 3D bunched beam
2.7.1 Space-charge field of uniform density beam and 3D bunched envelope equations 23
in the beam frame, which are given by [20, 42–44]
φ˜sc(x,y,ζ˜) =−πabcρ˜ beam Z ∞
0
ds
p(a2+s)(b2+s)(c˜2+s) x2
a2+s+ y2 b2+s+
ζ˜2
˜ c2+s
. (2.74) If we transform the space-charge potential to the laboratory frame as shown in Fig. 2.9 (b), it be- comes
φsc(x,y,ζ) =−πabγbcρlab Z ∞
0
ds q
(a2+s)(b2+s)(γb2c2+s) x2
a2+s+ y2
b2+s+ γb2ζ2 γb2c2+s
. (2.75)
Here,ρbeam=4πab˜3Qc is volume charge density andQ=qNbis total charge in a bunch in the beam frame, which is the increased value in the laboratory frame asρlab=γbρbeam.
Finally, the equations of motion for 3D bunched uniform density beam in laboratory frame is
x′′⊥(s) +κ⊥0(s)x⊥− q
γb3βb2mc2
∂ φsc(x,y,ζ)
∂x⊥ =0, ζ′′(s) +κ∥(s)ζ(s)− q
γb5βb2mc2
∂ φsc(x,y,ζ)
∂ ζ =0,
(2.76)
with the space-charge fields
∂ φsc(x,y,ζ)
∂x⊥ =3γ2bQx⊥R∞
0 ds (a2⊥+s)∆s,
∂ φsc(x,y,ζ)
∂ ζ =3γ
3 bQ
2 ζR0∞(γ2 ds bc2+s)∆s,
(2.77)
wherea⊥= (a,b),x⊥= (x,y), and∆s= q
(a2+s)(b2+s)(γb2c2+s).
The envelope equation in RMS size in the laboratory frame of 3D bunched beam is obtained as σrms,i′′ (s) +κ0(s)σrms,i(s)
−3 2
Nq2 γb2βb2mc2
1 5√
5σrms,i(s)Ai(σrms,x,σrms,y,γbσrms,ζ)− εrms,i2 σrms,i3 (s) =0,
(2.78)
whereAi(u1,u2,u3) =R0∞(u2ds
i+s)∆s,εrms,i2 = (εrms,⊥2 ,ε
2 rms,ζ˜
γb4 ), and∆s= q
(u21+s)(u22+s)(u23+s).
2.7.2 Space-charge field of Gaussian density beam
3D Gaussian density profile in the beam frame is ρbeam(x,y,ζ˜) = Q
(2π)3/2σrms,xσrms,yσ˜rms,ζ˜
exp
− x2
2σrms,x2 − y2
2σrms,y2 − ζ˜2 2 ˜σ2
rms,ζ˜
. (2.79)
Following the calculation of the space-charge potential of 2D Gaussian beam in Eqs. (2.63) and (2.65), the electrostatic potential of 3D bunched Gaussian beam in the laboratory frame becomes [40,41]
φsc(x,y,ζ) = 1 4π ε0
r2 π
Q σrms,xσrms,yσrms,ζ
Z ∞
0
e−
χ2x2 2(χ2σ2
rms,x+1)
q
χ2+σrms,x−2
e−
χ2y2 2(χ2σ2
rms,y+1)
q
χ2+σrms,y−2
e
− χ2ζ2
2(χ2σ2 rms,ζ+1)
q
χ2+σrms,ζ−2
dχ. (2.80)
Finally, the equations of motion for 3D axisymmetric bunched Gaussian beam under solenoid focus- ing in laboratory frame is
r′′(s) +κ⊥0(s)r(s)− q2Nb
4π ε0βb2γb3mc2r(s)A⊥(s) =0, ζ′′(s) +κ∥(s)ζ(s)− q2Nb
4π ε0βb2γb3mc2ζ(s)Az(s) =0,
(2.81)
with the space-charge fields
A⊥(s) =R0∞{σ2 dχ
rms,r(s)+χ}2{2γb2σrms,ζ2 (s)+χ}1/2 ×√2γb
πe−r2(s)/{σrms,r2 (s)+χ}−γb2ζ2(s)/{2γb2σrms,ζ2 (s)+χ}, Az(s) =R0∞{σ2 dχ
rms,r(s)+χ}{2γb2σrms,ζ2 (s)+χ}3/2 ×√2γb
πe−r2(s)/{σrms,r2 (s)+χ}−γb2ζ2(s)/{2γb2σrms,ζ2 (s)+χ}.
(2.82)
The phase advances of an entire beam distribution are defined by σ⊥ =R0S2εrms,⊥ ds
σrms,r2 (s) andσz= RS
0εrms,ζ ds
σrms,ζ2 (s), whereSis the length of lattice period.
For an infinitely long Gaussian beam withζ(s)→∞, Eq. (2.81) becomes r′′(s) +κ⊥0(s)r(s)−Kb
1−e−[r2(s)/σrms,r2 (s)]
r(s) =0, (2.83)
which is the same result with Eq. (2.69). Here,λ=√q
2π Nb
σrms,ζ is the line charge density [40].
Chapter 3
Space-charge driven coherent resonance
Section 3 is about the space-charge driven coherent resonance, which is also known as parametric instability. As mentioned in Sec. 2.5, for uniformly distributed K-V beam, the perturbed oscillations are excited from the accelerator structural errors and space-charge driven envelope mismatches. The mismatched conditions between design structures and beam parameters can cause perturbed motion of charged particles, which leads to the coherent resonance by means of the collective mode oscillations.
Figure 3.1: Periodic focusing lattices under (a) solenoid, (b) FODO quadrupole, and (c) quadrupole doublet with the matched envelope oscillations.Sis the length of a lattice period. For solenoid focusing channel, beam is axisymmetric [i.e.,a(s) =b(s) =rb(s)].
The matched envelope oscillationsa(s),b(s), andrb(s)by solving Eqs. (2.33) and (2.34) are plotted in Fig. 3.1. For solenoid focusing channel in Fig. 3.1 (a), beam is axisymmetric [i.e.,a(s) =b(s) =rb(s)].
The matched beams under the periodic focusing fields have the equilibrium oscillations along the lattice period S. Beam envelope radii are periodically oscillated, so that the values of the radii becomes the initial values after every lattice period;a(s0+S) =a(s0),b(s0+S) =b(s0), andrb(s0+S) =rb(s0).
Furthermore, the comparison between the matched beam and mismatched beam under a solenoid focusing field is shown in Fig. 3.2. Figure 3.2 (a) shows the matched beam, in which the envelope radius rb becomes the initial value at every lattice period, rb(s0+S) =rb(s0). On the other hand, Fig. 3.2 (b) shows the mismatched beam, in which the envelope radius has a perturbed termδr(s)that makes non-periodic oscillation at every lattice period.
Figure 3.2: Envelope oscillations along the solenoid focusing channel. (a) Matched beam in which the envelope radius is periodic at every lattice period, S. (b) Mismatched beam in which the envelope has a perturbed termδr(s)that makes non-periodic oscillation at every lattice period.
If we consider thesmooth focusing approximation[13], one may use a constant lattice coefficient κsf∼
"
κ⊥0(s)− Kb
rb(s)
#
, where the overline indicates an appropriate averaging over the lattice period.
Then, the matched envelope radius becomes constant function,rb(s) =rb=const. Under the periodic focusing, which is not constant along the lattice, the envelope does not stay constant but comes back to the initial value of radius at every lattice periods. The equilibrium keeps the beam stable along the propagation even under the strong space-charge forces.
However, if the beam is mismatched as shown in Fig. 3.2 (b), the mismatched beam generates perturbed terms in particle distribution functions and self potential functions. It affects the wave number termκ⊥(s) in single particle equation of motion, Eqs. (2.33) and (2.34). In general, the excitation of the parametric instability depends on the external focusing forces in which zero-current phase advance σ0 is decided. Also, the instability is manifested within a specific region of depressed phase advance σ, which depends on the beam current. The region of phase advances in which parametric instability occurs is calledstop band of the parametric instability.
The most well-known parametric instability is theenvelope instability, which is the second-order mode resonance of isotropic beam (i.e., same initial RMS emittances and phase advances inx and y directions). In Sec. 3.1, we will investigate the envelope instability stop bands by solving a transfer ma- trix of linearized perturbed equations and the resonance conditions of Breathing mode and Quadrupole mode.
In Sec. 3.2, we will discuss about the nth-order resonances in the envelope oscillations. The per- turbed envelope radius comes back to the starting point in every n-lattice periods of which the wave number is 2π/n. In this section, we carry out the particle-core model to observe the single particle motions under the condition of matched beam, mismatched beam, and the nth-order resonances.
Finally, in Sec. 3.3, we will introduce self-consistent collective mode instabilities by solving a lin-
earized Vlasov-Poisson equation. Finally, we can interpret theHofmann chartwhich is the fundamental map that implies the emittance exchange stop bands in terms of the initial phase advances and emittance ratios for both isotropic and anisotropic beams.
3.1 Envelope instability
Let
a(s) =rb(s) +δa(s), b(s) =rb(s) +δb(s),
(3.1) whererb(s)is the equilibrium envelope radius andδa(S)andδb(s)are the perturbed parts of radii inx andydirections. When we substitute the Eq. (3.1) into Eq. (2.41) by considering the smooth focusing approximation, we can get the perturbed equations
δa′′(s) +k2β0rb+k2β0δa(s)− 2Kb
2rb+δa(s) +δb(s)− ε⊥2
[rb+δa(s)]3 =0, δb′′(s) +k2β0rb+k2β0δb(s)− 2Kb
2rb+δa(s) +δb(s)− ε⊥2
[rb+δb(s)]3 =0.
(3.2)
By using Taylor expansion, 2Kb
2rb+δa(s) +δb(s)= 2Kb
2rb[1+δ2ra(s)
b +δb(s)2r
b ]
≃2Kb
2rb
1−δa(s) 2rb
−δb(s) 2rb
,
ε⊥2
[rb+δb(s)]3 = ε⊥2 r3b[1+δb(s)r
b ]3
≃ ε⊥2 r3b
1−3δb(s) rb
,
(3.3)
and the equilibrium equations under the continuous focusing, kβ02 rb−Kb
rb
−ε⊥2 r3b =0, k2β≡k2β0−Kb
r2b =ε⊥2
r4b =const,
(3.4)
Eq. (3.2) simplifies to
δa′′(s) +
kβ02 + Kb 2r2b+3ε⊥2
r4b
δa(s) + Kb
2r2bδb(s) =0, δb′′(s) +
kβ02 + Kb
2r2b+3ε⊥2 r4b
δb(s) + Kb
2r2bδa(s) =0
(3.5)
=⇒ δa′′(s) + 3
2kβ02 +5 2kβ2
δa(s) + 1
2k2β0−1 2k2β
δb(s) =0, δb′′(s) +
3
2kβ02 +5 2kβ2
δb(s) + 1
2k2β0−1 2k2β
δa(s) =0.
(3.6)
This is the linearized perturbed oscillations inxandydirections.
Now, let τ1 ≡δa−δb and τ2 ≡δa+δb. Then, by adding (subtracting) the two equations of Eq. (3.6), it becomes the simple harmonic oscillations ofτ1andτ2,
τ1′′(s) +k21τ1(s) =0, (3.7) where
k21 = 3
2kβ02 +5 2kβ2−1
2k2β0+1
2k2β (3.8)
= k2β0+3k2β (3.9)
= 4k2β0−3Kb
r2b =const. (3.10)
and
τ2′′(s) +k22τ2(s) =0, (3.11) where
k22 = 3
2kβ02 +5 2kβ2+1
2k2β0−1
2k2β (3.12)
= 2k2β0+2kβ2 (3.13)
= 4k2β0−2Kb
r2b =const. (3.14)
First, letδa(s) =δa0·eik1sandδb(s) =δb0·eik1swith Eq. (3.9) forKb̸=0. Then, Eq. (3.6) becomes 1
2(kβ02 −k2β)[δa0+δb0] =0
=⇒ δa0+δb0=0.
(3.15)
Second, let δa(s) =δa0·eik2sandδb(s) =δb0·eik2swith Eq. (3.13) forKb̸=0. Then, Eq. (3.6) becomes
1
2(k2β0−kβ2)[−δa0+δb0] =0
=⇒ δa0−δb0=0.
(3.16) Equation (3.16) says that the perturbed envelope radii inxandydirections have the same phase of perturbed oscillation with the wave frequency ofk2. It is calledbreathing mode (B) andk2 is called breathing mode frequency. On the other hand, Eq. (3.15) says that the perturbed envelope radii inxand ydirections have the opposite oscillation with the wave frequency ofk1. It is calledquadrupole mode (Q) andk1is called quadrupole mode frequency.
Figure 3.3 shows the breathing mode and the quadrupole mode of perturbed envelope oscillations.
Breathing mode is forδa0=δb0, which means that the initial perturbation inxandydirections are the same as shown in Fig. 3.3 (a). On the other hand, quadrupole mode is forδa0=−δb0, which means that the initial perturbation inxandydirections are the opposite as shown in Fig. 3.3 (b).
For the periodic focusing channel, let
a(s) =am(s) +δa(s), b(s) =bm(s) +δb(s),
(3.17)
Figure 3.3: Envelope perturbation of (a) Breathing mode and (b) Quadrupole mode.
wheream(si+S) =am(si)andbm(si+S) =bm(si)are the matched conditions of the transverse beam radii. The linearized perturbed oscillations of Eq. (3.6) can be written in the matrix form,
d ds
δa δb δa′ δb′
=
0 0 1 0
0 0 0 1
−κxm(s) −κ0m(s) 0 0
−κ0m(s) −κym(s) 0 0
δa δb δa′ δb′
, (3.18)
where
κxm(s)≡κx0(s) + 2Kb
[am(s) +bm(s)]2+ 3ε⊥2 a4m(s), κym(s)≡κy0(s) + 2Kb
[am(s) +bm(s)]2+ 3ε⊥2 b4m(s), κ0m(s)≡κx0(s) + 2Kb
[am(s) +bm(s)]2.
(3.19)
By solving the Eq. (3.18) with the 4×4 transfer matrixM(s|si)which satisfies(δa,δb,δa′,δb′)si+S= M(s|si)·(δa,δb,δa′,δb′)si, the solution matrix M(s|si) can be obtained based on the Floquet theo- rem [23]. As a result, there are 4 eigenvaluesλn(n=1,2,3,4) and 4 eigenvectorsEndefined by
M(S|0)·En=λn·En. (3.20)
The 4 eigenvalues can be expressed by agrowth factor (γn) of the perturbed mode per lattice period, and thephase advance of mode oscillation(σn) per lattice period.
λn≡γn·exp(iσn) (3.21)
The two eigenvalue pairs are conjugate relation, which corresponds toλ1=λ3∗andλ2=λ4∗. The sum of mode phase advances of the conjugate pairs isσ1+σ3=360◦andσ2+σ4=360◦.
From Eq. (3.21), if the absolute value of the growth factor|γn|is equal to 1, the perturbed oscillation has the stable motion. If|γn|>1 or|γn|<1, the perturbed oscillation becomes unstable and beam radius has the exponential growth or damping as the beam propagates through lattice periods. Beam stability
Figure 3.4: Plots of 4 eigenvaluesλn of the linearized perturbed oscillation in the complex plane (Re vs Im), and 4 types of the envelope resonance. The beam stability is represented by|λn|=1 for all n, and the instability is represented by|λn| ̸=1. Confluent resonance occurs only for quadrupole focusing channel between B and Q modes.
in 4D phase space (x,x’,y,y’) is represented byλnon the complex unit circle with|λn|=1 for all n, and the instability is represented by|λn| ̸=1.
There are 4 possible scenarios of the perturbed modes which are shown in Fig. 3.4.
1. Stable :|λn|=1 for all n.λnare on the complex unit circle.
2. Unstable - Confluent resonance : Resonance between the breathing and quadrupole modes. λ1 andλ2have the same phase angleσ1=σ2on the complex phase plane with|γn| ̸=1. This occurs only for quadrupole focusing channel.
3. Unstable - Lattice resonance : Resonance from one of the envelope modes with the focusing structure. One eigenvalue pair lies on the real axis having 180◦phase advance with|γn| ̸=1.
4. Unstable - Double lattice resonance : Resonance of two envelope modes with the focusing struc- ture simultaneously. Two eigenvalue pairs lie on the real axis having 180◦phase advances with|γn| ̸=1.
Envelope instability (B and Q modes) is manifested when zero-current phase advanceσ⊥0is larger than 90 degrees [21–23]. Also, the excitation of the envelope instability depends on the range of de- pressed phase advance σ⊥. Figure 3.5 shows the plots of envelope instability stop band in the tune depression space (σ⊥/σ⊥0). Note that the phase advancesσ⊥andσ⊥0in the tune depression space are calculated by Eq. (2.43) of the matched beamam(s)andbm(s). In the region of tune depression in which a growth factor is not 1, the envelope instabilities of parametric resonance or confluent resonance occur.
In this case, the amplitudes of the perturbed modes increase, because the non-zero |γn|is repeatably multiplied as the beam passes through the lattice periods. The zero-current phase advance in Fig. 3.5 is σ⊥0=115◦, andη is the focusing factor as shown in Fig. 2.2. Figure 3.5 (a) is for solenoid focusing lattice. Parametric instabilities of both B and Q modes are observed. Figure 3.5 (b) is for quadrupole FODO lattice. In this case, confluent resonance between B and Q modes is observed, as well as the
Figure 3.5: Envelope instability stop band in tune depression space (σ⊥/σ⊥0) under (a) solenoid and (b) FODO focusing channel. In the region of tune depression in which a growth factor is not 1, the envelope instabilities of B and Q modes are manifested. Here, zero-current phase advance isσ⊥0=115◦andηis the focusing factor as shown in Fig. 2.2.
parametric resonance of B mode.
When σ⊥0<90◦, the phase advance of stable B mode is σ+=2σ⊥02 +2σ⊥2, and that of stable Q mode isσ−=σ⊥02 +3σ⊥2, which correspond to the relations of Eqs. (3.9) and (3.13). Ifσ⊥0>90◦, the envelope instability can be manifested and the phase advances of B and Q modes are shifted because of the resonances. The plots of mode phase advanceσnand the envelope instability stop bands are shown in Fig. 3.6. In the region of parametric resonances, the mode phases have 180◦, which are shown in Fig. 3.6 (b) and (c). In the region of confluent resonances, the mode phases of B and Q modes are equal as shown in Fig. 3.6 (c).
The excitation of the envelope instability for the different zero-current phase advances under the solenoid focusing lattice are shown in Fig. 3.7. Forσ⊥0=89◦, there is no instability. Forσ⊥0>90◦in Fig. 3.7 (b), (c), and (d), the envelope instability is observed and the range of stop bands increase asσ⊥0
increases.