Effect of Space-Charge Force on Betatron Motion

The betatron amplitude function w =

β_y of the Floquet transformation satisfies Eq. (2.38). Defining the envelope radius of a beam asRy =

βy�y,where�y is the emittance, the envelope equation becomes

R^��_y+KyRy− �²_y

R³_y = 0, (2.68)

where the prime corresponds to the derivative with respect tos. Based on the Floquet theorem, we can impose a periodic condition, Ry(s) = Ry(s+L) to the envelope equation, ifKyis a periodic function ofs, i.e. Ky(s) =Ky(s+L), withLas the length of a repetitive period. The periodic envelope solution, aside from a multiplicative constant, is equal to the betatron amplitude function. The envelope function of an emittance dominated beamisR_y=

β_y�_y. When the self-induced space-charge force is included in the betatron motion, what happens to the beam envelope?

12The transfer matrix M expressed in this form corresponds to the transfer matrix for phase space coordinates (q1, q2,· · ·, qn;p1, p2,· · ·, pn). If we choose the phase space coordinates as (q1, p1, q2, p2,· · ·, theJmatrix will be defined slightly differently.

II. LINEAR BETATRON MOTION 63 A. The Kapchinskij-Vladimirskij distribution

It is known that the Coulomb mean-field from an arbitrary beam distribution is likely to be nonlinear. For example, The Exercise 5.2.1 shows the Coulomb mean field of a Gaussian beam distribution. In 1959, Kapchinskij and Vladimirskij (KV) discovered an ellipsoid beam distribution that leads to a perfect linear space-charge force within the beam radius. This distribution function is called the KV distribution.¹³

Particles, in the KV distribution, are uniformly distributed on a constant total emittance surface of the 4-dimensional phase space, i.e.

ρ(x,Px, z,Pz) = Ne

whereN is the number of particles per unit length, eis the particle’s charge,a and bare envelope radii of the beam,xandzare the transverse phase-space coordinates, Px = R_x, and Pz = R_z are the corresponding normalized conjugate phase-space coordinates,�_x and�_z are the horizontal and vertical emittances, and the envelope radii area=√

β_x�_xandb=√

β_z�_z. Thus beam particles are uniformly distributed along an action line

Some properties of the KV distribution are as follows.

1. Integrating the conjugate momenta, the distribution function becomes ρ(x, z) = Ne KV particles are uniformly distributed in any two-dimensional projection of the four-dimensional phase space.

2. The rms emittances of the KV beam are

�_x,rms=�x²� β_x =�x

4, �_z,rms=�z²� β_z = �z

4. (2.73)

Thus the rms envelope radii are equal to half of the beam radii in the KV beam.

13I.M. Kapchinskij and V.V. Vladimirskij,Proc. Int. Conf. on High Energy Accelerators, p. 274 (CERN, Geneva, 1959).

B. The Coulomb mean-field due to all beam particles

The next task is to calculate the effect of the average space-charge force. Neglecting the longitudinal variations, beam particles can be viewed as a charge distribution in an infinite long wire with a line-charge density given by Eq. (2.72). The electric field at the spatial point (x, z) is

E(x, z) =� Ne

where�₀ is the vacuum permittivity. A noteworthy feature of the KV distribution function is that the resulting mean-field inside the beam envelope radii is linear!

If the external focusing force is also linear, the KV distribution is a self-consistent distribution function. Including the mean-magnetic-field, the force on the particle at (x, z) is

where γ is the relativistic energy factor. Hill’s equations of KV beams of motion become where the prime is a derivative with respect to the longitudinal coordinates, andKsc

is the “normalized” space-chargeperveanceparameter given by K_sc=2Nr₀

β²γ³, (2.77)

wherer₀ =e²/4π�₀mc² is the classical radius of the particle, and N is the number of particles per unit length. Performing Floquet transformation of the linear KV-Hill equationx=wxe^jψx andz=wze^jψz,we obtain

II. LINEAR BETATRON MOTION 65 Solving the KV envelope equation is equivalent to finding the betatron amplitude function in the presence of the space-charge force. The usefulness of the KV equation has been further extended to arbitrary ellipsoid distribution functions provided that the envelope functions a andb are equal to twice the rms envelope radii, and the emittances�_xand�_zare equal to four times the rms emittances.¹⁴

If the external force is periodic, i.e. K_x(s) =K_x(s+L), the KV equation can be solved by imposing the periodic boundary (closed orbit) condition (Floquet theorem) a(s) =a(s+L), b(s) =b(s+L). (2.81) A numerical integrator or differential equation solvers can be used to find the envelope function of the space-charge dominated beams. The matched beam envelope solution can be obtained by a proper closed orbit condition of Eq. (2.81).

For beams with an initial mismatched envelope, the envelope equation can be solved by using the initial value problem to find the behavior of the mismatched beams. For space-charge dominated beams, the envelope solution can vary widely depending on the external focusing function, the space-charge parameter, and the beam emittance. To understand the physics of the mismatched envelope, it is ad-vantageous to extend the envelope equation to Hamiltonian dynamics as discussed below.

C. Hamiltonian formalism of the envelope equation

Introducing the pseudo-envelope momenta asp_a=aandp_b =b, we can derive the KV equations (2.80) from the envelope Hamiltonian:

H_env=1 2

p²_a+p²_b

+V_env(a, b) Venv(a, b) = 1

2(Kxa²+Kzb²)−2Kscln(a+b) + �²_x 2a² + �²_z

2b², (2.82) whereV_env(a, b) is the envelope potential. The matched beam envelope is the equi-librium solution (the betatron amplitude function) of the envelope Hamiltonian. For example, if we start from the condition with envelope momenta pa = pb = 0, the matched envelope radii are located at the minimum potential energy location, i.e.

∂V_env

∂a (am, bm) = ∂V_env

∂b (am, bm) = 0,

wherea_mandb_mare the matched envelope radii. The envelope oscillations of a mis-matched beam can be determined by the perturbation around the mis-matched solution

V_env= 1 2

∂²Venv

∂a² (a−a_m)²+1 2

∂²Venv

∂b² (b−b_m)²+· · ·.

14P.M. Lapostolle,IEEE Trans. Nucl. Sci. NS-18, 1101 (1971); F.J. Sacherer,ibid. 1105 (1971);

J.D. Lawson, P.M. Lapostolle, and R.L. Gluckstern,Part. Accel. 5, 61 (1973); E.P. Lee and R.K.

Cooper,ibid. 7, 83 (1976).

Using the second-order derivatives, we can obtain the envelope tune, which is equal to twice betatron tune atKsc= 0.

D. An example of a uniform focusing paraxial system

First we consider a beam in a uniform paraxial focusing system, where the focusing function is

K_x= (2π/L)².

Here L is the betatron wavelength, and the betatron amplitude function is βx0 = L/2π. Witha=bin Eq. (2.80), the envelope Hamiltonian is

H_env= 1 When the space-charge force is negligible, we find that the matched envelope radius is am0=

�xL/2π=√

�xβx,and the second-order derivative at the matched envelope

radius

which is twice the betatron tune (see also Exercise 2.2.15) and is independent of the envelope-oscillation amplitude.

Now, we consider the effect of space charge on the envelope function. The matched envelope radius is obtained from the solution ofdVenv/da= 0, i.e.

a²_m=�_xβ_x=�_x whereκis the effective space-charge parameter, andL_totand Φ_totare the total length and total phase advance of a transport system.¹⁵ Equation (2.83) indicates that the betatron amplitude function increases by a factorκ+√

κ²+ 1 due to the space-charge force. The second-order derivative of the potential at the matched radius is

d²V_env

which is the phase advance per unit length of small amplitude envelope oscillation in the presence of the Coulomb potential. When the space-charge perveance parameter is zero, the phase advance of the envelope oscillation is twice of that of the betatron oscillation, and when the space-charge force is large, asκ→ ∞, the phase advance of the small-amplitude envelope oscillations can maximally be depressed to√

2 (2π/L).

15The Laslett (linear) space-charge tune shift is related to the space-charge perveance parameter byξsc≡Δνsc=KscLtot/4π�x=κν, whereνis the tune.

II. LINEAR BETATRON MOTION 67 There is a large envelope detuning from 2μ to √

2μ, where μ is the betatron phase advance. A nonlinear envelope resonance can be excited when perturbation exists and a resonance condition is satisfied.¹⁶

Figure 2.11: The phase advance of the enve-lope oscillations divided by the original betatron phase advance for a high space charge beam with K_sc = 10, μ = 2.28175. The matched radius is R₀ = a_m

2π/(μ�_xL) = 1.4199 in this example.

See Eq. (2.83) for the matched envelope radius.

When the envelope radius is mismatched fromR0, the envelope radius oscillates aroundR₀at an en-velope tune depending on its maximum radius os-cillation amplitude. The ordinateRis the normal-ized maximum envelope radius of the beam.

Figure 2.11 shows the envelope tune of a space charge dominated beam with K_sc = 10 and a phase advance of μ = 2.2817 radian (or ν = μ/2π for the un-perturbed betatron tune) as a function of the maximum amplitude of the envelope oscillation. At a large envelope amplitude, the envelope tune approaches twice the unperturbed betatron tune. Near the matched envelope radius (or small amplitude envelope oscillations), the envelope tune approaches√

2 times the unperturbed beta-tron tune.

The single particle betatron phase advance per unit length is obtained by sub-stituting Eq. (2.83) into Eq. (2.76), i.e. Φ_x = ^2π_L(√

κ²+ 1−κ). When the space charge parameterκis small, the incoherent space-charge (Laslett) tune shift is equal to Δν_sc= ξ_sc=κ. When the space charge parameter κis large, the betatron tune can be depressed to zero.

E. Space-charge force for Gaussian distribution

Since the emittance growth rate is usually much faster than a synchrotron period, this justifies the performance of only 2D simulation for a slice of the beam at the longitudinal bunch center. For a beam with linear particle densityN and bi-Gaussian charge distribution

ρ(x, z) = Ne

2πσ_xσ_ze^{−x2/2σ2x−z2/2σ2z}, (2.84)

16S.Y. Lee and A. Riabko,Phys. Rev. E51, 1609 (1995); A. Riabkoet al., Phys. Rev. E51, 3529 (1995); C. Chen and R.C. Davidson,Phys. Rev.E49, 5679 (1994);Phys. Rev. Lett.72, 2195 (1994). See also Ref. [8] for an exploration of the space-charge dynamics.

with σx,z being the rms horizontal and vertical beam radii including contribution coming from momentum dispersion, the transverse 2D space-charge potential is

V_sc(x, z) = K_sc whereK_scis the space-charge perveance of Eq. (2.77),r₀is the particle classical radius, and β andγ are the relativistic parameters. In the simulation, we set the bunch intensity with N_B particles and an rms bunch-lengthσ_s to obtainN =N_B/√

2πσ_s. The space-charge force on each particle is obtained by Hamilton’s equation. Thus each beam particle passing through a length Δsexperiences a space-charge kick

Δx^�

Δs =−∂V_sc

∂x , Δz^�

Δs =−∂V_sc

z . (2.86)

We expand the space-charge potential in Taylor series in order to study the sys-tematic space charge resonances: withr =σ_z/σ_x. The first term inside the curly brackets represents the linear force, which gives rise to linear space charge (Laslett) tune shift. The second and the third terms drive the 4th and 6th order resonances.

The linear space charge tune shift parameters become

ξsc,x/z≡ |Δνsc,x/z|=

Particles at the center of the beam has a betatron tune shift −ξ_sc,x/z, and large betatron amplitude particles have small betatron tune shift. Since particles at dif-ferent betatron amplitudes have different betatron tune shift, the space charge force produces anincoherenttune spreadξ_sc. The space charge parameter of the KV dis-tribution in Eq. (2.83) is

ξ_KV,sc= 2πRK_sc 16π�_rms.

The space charge parameter of Gaussian distribution is a factor of 2 larger than that of a beam with uniform distribution. The space charge tune shift of all particles in the KV beam is identicallyξ_KV,sc. It is stillincoherent.

EXERCISE 2.2 69

Exercise 2.2

1. The focusing functionK(s) for most accelerator magnets can be assumed to be piece-wise constant. Show that

�, in the thin-lens approximation, is M=

1 0

−¹_f 1

wheref = lim_�→0(K�)⁻¹,is the focal length of a quadrupole. For a focusing quad, f >0; and for a defocusing quad,f <0.

2. When a particle enters a dipole at an angleδwith respect to the normal edge of a dipole (see drawing below), there is a quadrupole effect. This phenomenon is usually referred to as edge focusing. Using edge focusing, the zero-gradient synchrotron (ZGS) was designed and constructed in the 1960’s at Argonne National Laboratory.

The ZGS was made of 8 dipoles with a circumference of 172 m attaining the energy of 12.5 GeV. Its first proton beam was commissioned on Sept. 18, 1963. See L.

Greenbaum, A Special Interest(Univ. of Michigan Press, Ann Arbor, 1971). We use the convention that δ > 0 if the particle trajectory is closer to the center of the bending radius. Show that the transfer matrices for the horizontal and vertical betatron motion due to the edge focusing are

Mx=

whereδ is the entrance or the exit angle of the particle with respect to the normal direction of the dipole edge. Thus the edge effect withδ >0 gives rise to horizontal defocusing and vertical focusing.

3. The particle orbit enters and exits asector dipolemagnet perpendicular to the dipole edges. Assuming that the gradient function of the dipole is zero, i.e. ∂Bz/∂x= 0, show that the transfer matrix is

M_x=

whereθis the bending angle,ρis the bending radius, and�is the length of the dipole.

Note that a sector magnet gives rise to horizontal focusing.

4. The entrance and exit edge angles of a rectangular dipole areδ₁=θ/2 andδ₂=θ/2, whereθis the bending angle. Find the horizontal and vertical transfer matrices for a rectangular dipole (Fig. 2.2b).

5. For a weak-focusing accelerator, Kz(s) = n/ρ² = constant and Kx = (1−n)/ρ², whereρis the radius of the accelerator. The focusing indexnis

n(s) = ρ(s)

where we have chosen the coordinate system shown in Fig. 2.1. Solve the following problems by using the uniform focusing approximation with constantn.

(a) Show that the horizontal and vertical transfer matrices are M_x=

(c) IfN equally spaced straight sections, withKx =Kz = 0, are introduced into the accelerator lattice adjacent to each combined-function dipole, calculate the mapping matrix for the basic period and discuss the stability condition.

6. The path length for a particle orbit in an accelerator is C= [1 + (x/ρ)]²+x²+z²ds.

Show that the average orbit length of the particle with a vertical betatron actionJz

is longer by

ΔC C = 1

2�1 +α²_z β_z �J_z,

whereα_z and β_z are betatron amplitude functions. In the smooth approximation, the betatron amplitude function is approximated by�β_z�=R/ν_z, and the betatron oscillations can be expressed as

whereR,νz and ˆz are the average radius, the vertical betatron tune, and the ver-tical betatron amplitude respectively, andχ_z is an arbitrary betatron phase angle of the particle. Show that the average orbit length of a particle executing betatron oscillations is longer by

ΔC C = ν_z²

4R²zˆ².

Thus the orbit length depends quadratically on the betatron amplitude.

7. In a strong-focusing synchrotron, the art (or science) of magnet arrangement is called lattice design. The basic building blocks of a lattice are usually FODO cells. A FODO cell is composed of QF OO QD OO, where QF is a focusing quadrupole, OO represents either a drift space or bending dipoles of length L₁, and QD is a defocusing quadrupole. The length of a FODO cell isL= 2L₁. Using the thin-lens approximation,

EXERCISE 2.2 71 (a) Find the mapping matrix and the phase advance of the FODO cell and discuss

the stability condition.

(b) Find the parametersβ, αat the quadrupoles and at the center of the drift space as a function ofL₁and Φ. Find the phase advance Φ that minimizes the betatron amplitude function at the focusing quadrupole location.

8. Using Eq. (2.41), show thatβ^���+ 4β^�K+ 2βK^�= 0. Solve this equation for a drift space and a quadrupole respectively, and show that the solution of this equation must be one of the following forms:

⎧⎨

|K|s+c, defocusing quadrupole.

(a) In a drift space, where there are no quadrupoles, Show also that the betatron amplitude function is given by

β(s) = β1−2α1s+γ1s² = β^∗+(s−s^∗)² β^∗ ,

where the parametersα₁, β₁andγ₁at betatron function values at the beginning of the element, ors= 0; andβ^∗is the betatron function at the symmetry point s=s^∗withβ^�= 0. This means thats^∗=β1/(α1+1/α1), andβ^∗=β1/(1+α²₁) = 1/γ^∗= 1/γ₁. Note thatγ₁= (1 +α²₁)/β₁= 1/β^∗, i.e. γ₁ is constant in a drift space.

(b) Using the similarity transformation Eq. (2.32), show that the Courant–Snyder parametersα2, β2, γ2ats2are related toα1, β1, γ1 ats1by

whereMijare the matrix elements ofM(s2|s1). Use these equations to verify your solution to part (a). Similarly, the betatron function inside the focusing and defocusing quadrupole are respectively given by

β_focusing(s) = 1 9. Use the transfer matrixM(s2|s1) of Eq. (2.42) to show that, when a particle is kicked

ats₁by an angleθ, the displacement at a downstream location is Δx₂=θ�

β₁β₂sinψ,

where β₁ and β₂ are values of betatron functions at s₁ and s₂ respectively, and ψ=ψ(s2)−ψ(s1) is the betatron phase advance betweens1and s2. The quantity

√β₁β₂sinψis usually called the kicker arm. To minimize the kicker magnet strength θ, the injection or extraction kickers are located at a high β locations with a 90^◦ phase advance.

10. Transforming the betatron phase-space coordinates onto the normalized coordinates show that the betatron transfer matrix in normalized coordinates becomes

M(s˜ 2|s1) =

cosψ sinψ

−sinψ cosψ

,

i.e. the betatron transfer matrix becomes coordinate rotation with rotation angle equal to the betatron phase advance. Show that the transfer matrix of Eq. (2.42) becomesM(s2|s1) =B2MB˜ ⁻¹₁ ,whereB2andB1are the betatron amplitude matrices ats=s₂ands₁respectively.

11. Show that the Floquet transformation of Eq. (2.55) transforms the Hamiltonian of Eq. (2.47) into Eq. (2.54).

12. Often a solenoidal field has been used to provide both the horizontal and the verti-cal beam focusing for the production of secondary beams from a target (see Exer-cise 2.1.4). The focusing channel can be considered as a focusing-focusing (FOFO) channel. We consider a FOFO focusing channel where the focusing elements are sep-arated by a distanceL. Use the thin-lens approximation to evaluate beam transport properties of a periodic FOFO channel.

(a) Show that the phase advance of a FOFO cell is sinΦ solenoid,g=B_�/2Bρis the effective solenoid strength,B_�is the solenoid field, and Θ =g�is the solenoid rotation angle.

(b) Show that the maximum and minimum values of the betatron amplitude func-tion are

βmax=L/sin Φ, βmin=fsin Φ.

13. The doublet configuration consists of a pair focusing and defocusing quadrupoles with equal focusing strength separated by a small distance L₁ as a beam focusing unit.

The doublet pairs are repeated at intervalsL₂�L₁for beam transport (Fig. 2.6).

These quadrupole doublets can be used to maintain round beam configuration during beam transport. Using the thin-lens approximation with equal focal length for the focusing and defocusing quadrupoles, describe the properties of betatron motion in a doublet transport line.

EXERCISE 2.2 73 (a) Show that the betatron phase advance in a doublet cell is

ψ=ψ_x,z= 2 arcsin

L₁L₂/2f , wheref is the focal length of the quadrupoles.

(b) Show that the maximum betatron amplitude function is approximately β_max= (L₁+L₂+L₁L₂/f)/sinψ.

(c) Show that the minimum betatron amplitude function is β^∗=

L₁(4f²−L₁L₂)/4L₂.

(d) Sketch the betatron amplitude functions and compare your results with that of the FODO cell transport line.

14. Statistical definition of beam emittance:¹⁷ We consider a statistical distribution of N non-interacting particles in phase space (x, x^�). Let ρ(x, x^�) be the distribution

function with

ρ(x, x^�)dxdx^�= 1.

The first and second moments of beam distribution are

�x�= 1 N

x_i=

xρ(x, x^�)dxdx^�, �x^��= 1 N

x^�_i=

x^�ρ(x, x^�)dxdx^�, σ_x²= 1

N

(x_i− �x�)², σ_x²= 1 N

(x^�_i− �x^��)², σ_xx= 1

N

(x_i− �x�)(x^�_i− �x^��) =rσ_xσ_x.

Hereσxandσx are rms beam widths, andris the correlation coefficient. The rms emittance is defined as

�_rms=σ_xσ_x

1−r².

(a) Assuming that particles are uniformly distributed in an ellipse x²/a²+x^�2/b²= 1,

show that the total phase-space area isA =πab = 4π�rms. The factor 4 has often been used in the definition of the full emittance, i.e. �= 4�_rms, to ensure that the phase-space area of such an ellipse isπ�.

(b) Show that the rms emittance defined above is invariant under a coordinate rotation

X=xcosθ+x^�sinθ, X^�=−xsinθ+x^�cosθ,

and show that the correlation coefficientR=σ_XX/σ_Xσ_X is zero if we choose the rotation angle to be

tan 2θ= 2σxσ_xr σ²_x−σ²_x

.

Show thatσ_X andσ_X reach extrema at this rotation angle.

17See P. Lapostolle,IEEE Trans. Nucl. Sci.NS-18, 1101 (1971), and J. Buon, CERN91-04, 30 (1991). The statistical definition of beam emittance is applicable to all phase space coordinates.

(c) In accelerators, particles are distributed in the Courant-Snyder ellipse:

I(x, x^�) =γx²+ 2αxx^�+βx^�2,

whereα, β, γare betatron amplitude functions. If the beam distribution function is a function ofI(x, x^�), show that

(d) Show that theσmatrix is transformed, in the linear betatron motion, according

to

where M^† is the transpose of the matrix M. Use this result to show that x^†σ⁻¹xof Eq. (2.60) is invariant under betatron motion and thus an invariant beam distribution function is a function ofx^†σ⁻¹x. The transport equation for theσ-matrix can be used to measure theσ-matrix elements and derive the rms beam emittance. For a thick quadrupole lens, show that Eq. (2.61) becomes

σ²_x(s₂) = σ²_x(s₁) quadrupole, andLis the length of the drift space between the quadrupole and the profile monitor.

(e) Particle motion in synchrotrons obeys Hamiltonian dynamics with x^�=dx

(e) Particle motion in synchrotrons obeys Hamiltonian dynamics with x^�=dx

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