Theσ-matrix of a beam distribution is defined as σ=
σ11 σ12 σ12 σ22
=
σ2y σyy
σyy σ2y
=�(y− �y�)(y− �y�)†�,
σ(s2) =M(s2|s1)σ(s1)M(s2|s1)†. (2.59) where yis the betatron state-vector of Eq. (2.27), y† = (y, y�) is the transpose of y, and �y� is the first moment. The rms emittance defined by Eq. (2.57) is the determinant of theσ-matrix, i.e. �rms=√
detσ (see also Exercise 2.2.14). It is easy to verify thaty†σ−1yis invariant under linear betatron motion. An invariant beam distribution is
ρ(y, y�) =ρ(y†σ−1y). (2.60) C. Emittance measurement
The emittance can be obtained by measuring the σ-matrix. The beam profile of protons and ions is usually measured by using wire scanners or ionization profile monitors. Synchrotron light monitors are commonly used in electron storage rings.
More recently, laser light has been used to measure electron beam size in the sub-micron range. Using the rms beam width and Courant–Snyder parameters, we can
11The accelerator scientists commonly use π-mm-mrad as the unit of emittance. However, the factorπis also often omitted. In beam width calculation, we getσy=
π�yβy/π. The synchrotron light source community also uses nano-meter (nm) as the unit for emittance. In fact, the factorπis implied and omitted in the literature.
deduce the emittance of the beam. Two methods commonly used to measure the rms emittance are discussed below.
C1. Quadrupole tuning method Using Eq. (2.59), we find
σ11(s2) =σ11(s1)
M11+σ12(s1) σ11(s1)M12
2
+ �2rms
σ11(s1)M122, (2.61) whereσij(s1)’s are elements of the σmatrix at the entrance of the quadrupole with
�2rms = σ11σ22 −σ122, and σ11(s2) is the 11-element of the σ-matrix at the profile monitor locations2. For a setup of a quadrupole and a drift space, we find M12 = (1/√
K) sin(√
K�q) +Lcos(√
K�q) andM11= cos(√
K�q)−√
KLsin(√
K�q), where K = B1/Bρ is the focusing function, �q is the length of the quadrupole, andL is the distance between the quadrupole and the beam profile monitor. In thin lens approximation, we findM12→(L+�2q)M11→1−(L+�2q)g, and
σ11(s2)≈σ11(s1)
1−(L+�q
2)g+σ12(s1) σ11(s1)(L+�q
2) 2
+ �2rms
σ11(s1)(L+�q 2)2, whereg=Klqis the effective quadrupole strength.
Theσ11(s2) data by varying quadrupole strengthgcan be used to fit a parabola.
The rms emittance�rmscan be obtained from the fitted parameters. This method is commonly used at the end of a transport line, where a fluorescence screen or a wire detector (harp) is used to measure the rms beam size.
C2. Moving screen method
Using a movable fluorescence screen, the beam size at three spots can be used to determine the emittance. Employing the transfer matrix of drift space, the rms beam widths at the second and third positions are
R22=σ11+ 2L1σ12+L21σ22,
R32=σ11+ 2(L1+L2)σ12+ (L1+L2)2σ22, (2.62) whereσ11=R12,σ12andσ22are elements of theσmatrix at the first screen location, andL1andL2 are respectively drift distances between screens 1 and 2 and between screens 2 and 3. The solutionσ12andσ22of Eq. (2.62) can be used to obtain the rms beam emittance: �rms=
σ11σ22−σ212.
If screen 2 is located at the waist, i.e. dR22/dL1 = 0, then the emittance can be determined from rms beam size measurements of screens 1 and 2 alone. The resulting emittance is
�2=
R21R22−R42 /L21.
This method is commonly used to measure the electron emittance in a transport line.
II. LINEAR BETATRON MOTION 59 D. The Gaussian distribution function
The equilibrium beam distribution in the linearized betatron phase space may be any function of the invariant action. However, the Gaussian distribution function
ρ(y, y�) =Nexp −1
2 detσ(σ22y2−2σ12yy�+σ11y�2)
(2.63) is commonly used to evaluate the beam properties. Expressing the normalized Gaus-sian distribution in the normalized phase space, we obtain
ρ(y,Py) = 1
2πσy2e−(y2+P2y)/2σ2y, (2.64) where�y2�=�p2y�=σ2y =βy�rmswith an rms emittance�rms. Transforming (y,Py) into the action-angle variables (J, ψ) with
y=
2βyJcosψ, Py=−
2βyJsinψ.
The Jacobian of the transformation is ∂(y,P∂(ψ,J)y) = βy, and the distribution function becomes
ρ(J) = 1
�rmse−J/�rms, ρ(�) = 1
2�rmse−�/2�rms, (2.65) where�= 2J. The percentage of particles contained within�=n�rms is 1−e−n/2, shown in Table 2.1.
Table 2.1: Percentage of particles in the confined phase-space volume
�/�rms 2 4 6 8
Percentage in 1D [%] 63 86 95 98 Percentage in 2D [%] 40 74 90 96
The maximum phase-space area that particles can survive in an accelerator is called theadmittance, or thedynamic aperture. The admittance is determined by the vacuum chamber size, the kicker aperture, and nonlinear magnetic fields. To achieve good performance of an accelerator, the emittance should be kept much smaller than the admittance. Note that some publications assume 95% emittance, i.e. the phase-space area contains 95% of the beam particles,�95%≈6�rmsfor a Gaussian distribu-tion. For superconducting accelerators, a dynamic aperture of 6σor more is normally assumed for magnet quench protection. For electron storage rings, quantum fluctu-ations due to synchrotron radiation are important; the machine acceptance usually requires about 10σfor good quantum lifetime.
Accelerator scientists in Europe use�= 4�rmsto define the beam emittance. This convention arises from the fact that the rms beam emittance of a KV beam is equal to 1/4 of the full KV beam emittance [see Eq. (2.73)]. A uniform phase space distribution in an ellipsey2/a2+y2/b2= 1 has an rms emittance equal toπab/4.
E. Adiabatic damping and the normalized emittance: �n=βγ�
The Courant–Snyder invariant of Eq. (2.56), derived from the phase-space coordinate y, y, is not invariant when the energy is changed. To obtain the Liouville invariant phase-space area, we should use the conjugate phase-space coordinates (y, py) of the Hamiltonian in Eq. (2.14). Sincepy=py=mcβγy,wheremis the particle’s mass, pis its momentum, andβγis the Lorentz relativistic factor, thenormalized emittance defined by�n=βγ�is invariant. The beam emittance decreases with increasing beam momentum, i.e. �=�n/βγ. This is calledadiabatic damping. The adiabatic phase-space damping of the beam can be visualized as follows. The transverse velocity of a particle does not change during acceleration, while the transverse angley=py/p becomes smaller as the particle momentum increases, and thus the beam emittance
�=�n/βγbecomes smaller. The adiabatic damping also applies to beam emittance in proton or electron linacs.
On the other hand, the beam emittance in electron storage rings increases with en-ergy (∼γ2) resulting from the quantum fluctuation (see Chap. 4). The corresponding normalized emittance is proportional toγ3, whereγis the relativistic Lorentz factor.
II.6 Stability of Betatron Motion: A FODO Cell Example
In this section, we illustrate the stability of betatron motion using a FODO cell example. We consider a FODO cell with quadrupole focal lengthf1and−f2, where the ± signs designates the focusing and defocusing quadrupoles respectively. The transfer matrix of{12QF1O QD2 O 12QF1}is
where L1 is the drift length between quadrupoles. Identifying the transfer matrix with the Courant–Snyder parametrization, we obtain
cos Φx= 1 +L1
II. LINEAR BETATRON MOTION 61 The stability condition, Eq. (2.33), of the betatron motion is equivalent to the fol-lowing conditions:
|1 + 2X2−2X1−2X1X2| ≤1 and |1−2X2+ 2X1−2X1X2| ≤1, (2.66) whereX1=L1/2f1andX2=L1/2f2. The solution of Eq. (2.66) is shown in Fig. 2.10, which is usually called the necktie diagram. The lower and the upper boundaries of the shaded area correspond to Φx,z = 0 orπ respectively. Since the stable region is limited byX1,2≤1, the focal length should be larger than one-fourth of thefull cell length.
Figure 2.10: The “necktie diagram” for the sta-ble region of a FODO cell lattice shown in the shaded area of focusing strengthsX1=L1/2f1vs X2=L1/2f2. whereL1is the half cell length, f’s are focal lengths. The lower and upper boundaries correspond to Φx,z= 0 or 180◦respectively. When X1 is at the lower part of stability boundary, the phase advance of the FODO cell is Φx = 0. At the boundary of the stability X1 = 1, the phase advance Φx=π.
The phase advances Φx and Φz of repetitive FODO cells should be less than π. The phase advances of a complex repetitive lattice-module with more than 2 quadrupoles can be larger than π. For example, the phase advance of a flexible momentum compaction (FMC) module is about 3π/2 (see Sec. IV.8 and Exercise 2.4.17) and the phase advance of a minimum emittance double-bend achromat module is about 2.4π(see Sec. III.1; Chap. 4). In general, the stability of betatron motion is described by |cos Φx| ≤ 1 and |cos Φz| ≤ 1 for any type of accelerator lattice or repetitive transport line.
II.7 Symplectic Condition
The 2×2 transfer matrix M with detM = 1 satisfies ˜M JM = J, where ˜M is the transpose of the matrixM, andJ=
0 1
−1 0
. In general, the transfer matrix of a Hamiltonian flow ofndegrees of freedom satisfies
M JM˜ =J, J=
0 I
−I 0
, (2.67)
where ˜Mis the transpose of the transfer matrixM, andJ2=−I, J˜=−J, J−1=−J withI as then×nunit matrix. A 2n×2nmatrix,M, is said to be Symplectic if it satisfies Eq. (2.67).12 The matricesI andJ are symplectic.
If the matrix M is symplectic, then M−1 is also symplectic and detM = 1. If M andN are symplectic, then MN is also symplectic. Since the set of symplectic matrices satisfies the properties that (1) the unit matrixI is symplectic, (2) ifM is symplectic thenM−1is symplectic, and (3) ifMandN are symplectic, thenMN is also symplectic, the set of symplectic matrices form a group denoted bySp(2n). The properties of real symplectic matrices are described below.
• The eigenvalues of symplectic matrixMmust be real or must occur in complex conjugate pairs, i.e. λandλ∗. The eigenvalues of a real matrixM or the roots of the characteristic polynomialP(λ) =|M−λI|= 0 have real coefficients.
• Since|M|= 1, zero can not be an eigenvalue of a symplectic matrix.
• Ifλ is an eigenvalue of a real symplectic matrixM, then 1/λmust also be an eigenvalue. They should occur at the same multiplicity. Thus eigenvalues of a symplectic matrix are pairs of reciprocal numbers. For a symplectic matrix, we haveK−1( ˜M−λI)K=M−1−λI =−λM−1(M−λ−1I) or P(λ) =λ2nP(1λ).
If we defineQ(λ) =λ−nP(λ), thenQ(λ) =Q(1λ).
II.8 Effect of Space-Charge Force on Betatron Motion
The betatron amplitude function w =
βy 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− �2y
R3y = 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 beamisRy=
βy�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 = Rx, and Pz = Rz are the corresponding normalized conjugate phase-space coordinates,�x and�z are the horizontal and vertical emittances, and the envelope radii area=√
βx�xandb=√
βz�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� βx =�x
4, �z,rms=�z2� βz = �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 Ksc=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=wxejψx andz=wzejψz,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. Kx(s) =Kx(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 aspa=aandpb =b, we can derive the KV equations (2.80) from the envelope Hamiltonian:
Henv=1 2
p2a+p2b
+Venv(a, b) Venv(a, b) = 1
2(Kxa2+Kzb2)−2Kscln(a+b) + �2x 2a2 + �2z
2b2, (2.82) whereVenv(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.
∂Venv
∂a (am, bm) = ∂Venv
∂b (am, bm) = 0,
whereamandbmare the matched envelope radii. The envelope oscillations of a mis-matched beam can be determined by the perturbation around the mis-matched solution
Venv= 1 2
∂2Venv
∂a2 (a−am)2+1 2
∂2Venv
∂b2 (b−bm)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
Kx= (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
Henv= 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.
a2m=�xβx=�x whereκis the effective space-charge parameter, andLtotand Φtotare 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
d2Venv
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 Ksc = 10, μ = 2.28175. The matched radius is R0 = am
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 Ksc = 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. Φx = 2πL(√
κ2+ 1−κ). When the space charge parameterκis small, the incoherent space-charge (Laslett) tune shift is equal to Δνsc= ξsc=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
2πσxσze−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
Vsc(x, z) = Ksc whereKscis 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 NB particles and an rms bunch-lengthσs to obtainN =NB/√
2πσs. 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 =−∂Vsc
∂x , Δz�
Δs =−∂Vsc
z . (2.86)
We expand the space-charge potential in Taylor series in order to study the sys-tematic space charge resonances: withr =σz/σx. 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 −ξsc,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 spreadξsc. The space charge parameter of the KV dis-tribution in Eq. (2.83) is
Particles at the center of the beam has a betatron tune shift −ξsc,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 spreadξsc. The space charge parameter of the KV dis-tribution in Eq. (2.83) is