We consider an accelerator of circumferenceC =P LwithP identical superperiods.
The phase change per revolution isPΦ. The betatron tuneνy, orQy, defined as the number of betatron oscillations in one revolution, is
Qy=νy=PΦy
2π = 1 2π
s+C s
ds
βy(s). (2.44)
The betatron oscillation frequency isνyf0, where f0 is the revolution frequency. The general solution of Eq. (2.25) becomes
y(s) =a
βy(s) cos [ψy(s) +ξy] with ψy(s) = s
0
ds
βy(s), (2.45) wherea, ξy are constants to be determined from initial conditions. This is a pseudo-harmonic oscillation with varying amplitudeβy1/2(s). The local betatron wavelength isλ= 2πβy(s).
Introducing the coordinate η(φy), and ”time” coordinate to φy, Hill’s equation becomes
η= y
βy, φy(s) = 1 νy
s ds
βy(s), d2η
dφ2y+νy2η= 0. (2.46) The phase function (“time variable”)φyincreases by 2πin one revolution. The linear betatron motion is simple harmonic.
Example 1: FODO cell in thin-lens approximation
A FODO cell (Fig. 2.4) is made of a pair of focusing and defocusing quadrupoles with or without dipoles in between:
1
2QF O QD O 1
2QF
where O stands for either a dipole or a drift space: FODO cells are often used in beam transport in arcs and transport lines.9
9The accelerator lattice is usually divided into arcs and insertions. Arcs are curved sections that transport beams for a complete revolution. Insertions (or straight sections) are usually used for physics experiments, rf cavities, injection and extraction systems, etc.
II. LINEAR BETATRON MOTION 51 Figure 2.4: A schematic plot of a FODO cell, where the transfer matrices for dipoles (B) can be approx-imated by drift spaces, and QF and QD are the fo-cusing and defofo-cusing quadrupoles.
The transfer matrix in the thin-lens approximation, is10
M =
where the focusing and defocusing quadrupoles have focal lengths±frespectively and L1is the drift length between quadrupoles. Because of the repetitive nature of FODO cells, the transfer matrix can be identified with the Courant–Snyder parametrization of Eq. (2.34):
The parameter Φ is the phase advance per cell, and βF and αF are values of the betatron amplitude functions at the center of the focusing quadrupole. The betatron tune for a machine with N FODO cells is ν = NΦ/2π. The above procedure can be performed at any position of the FODO cell, and the corresponding Courant–
Snyder parameters are values of the betatron amplitude functions at that position.
For example, we find
at the center of the defocusing quadrupole, and at the midpoint between the QF and the QD respectively. We can also use the transfer matrix of Eq. (2.42) to find the betatron amplitude functions at other locations (see Exercise 2.2.8).
The solid and dashed lines in the upper plot of Fig. 2.5 show the betatron am-plitude functionsβx(s) andβz(s) for the AGS. The middle plot shows the dispersion
10The transfer matrices of dipoles are represented by those of drift spaces, where we neglect the effect of 1/ρ2focusing and edge focusing. The transfer matrix for vertical motion can be obtained by reversing focusing and defocusing elements.
Figure 2.5: The betatron amplitude func-tions for one superperiod of the AGS lat-tice, which made of 20 combined-function magnets. The upper plot shows βx (solid line) and βz (dashed line). The middle plot shows the dispersion functionDx. The lower plot shows schematically the placement of combined-function magnets. Note that the superperiod can be well approximated by five regular FODO cells. The phase advance of each FODO cell is about 52.8◦.
functionD(s), to be discussed in Sec. IV. The AGS lattice has 12 superperiods, each composed of 20 combined-function dipoles, shown schematically in the bottom plot of Fig. 2.5. The AGS lattice can be well approximated by 60 FODO cells with a phase advance of 52.8◦for a betatron tune of 8.8, and a half-cell length ofL1= 6.726 m for a complete circumference of 807.12 m.
Example 2: Doublet cells
The values of the horizontal and vertical betatron functions in FODO cells alternate in magnitude, i.e.
βx
βz ≈ 1 + sin Φ/2
1−sin Φ/2, and 1−sin Φ/2 1 + sin Φ/2,
at the focusing and defocusing quadrupoles respectively. The beam size variation increases with the phase advance of the FODO cell. In some applications, a paraxial beam transport system provides a simpler geometrical beam matching solution. Some examples of paraxial beam transport beam lines are the doublet, the triplet, and the solenoidal transport systems. In the following example, we consider a doublet beam line, shown schematically in Fig. 2.6.
Figure 2.6: A schematic plot of a doublet transport line, where two quadrupoles are sepa-rated by a distanceL1, and the long drift space L2between two quadrupoles can be filled with dipoles.
The phase advance of a doublet cell, in thin-lens approximation, and the maximum and minimum values of the betatron amplitude function are (see Exercise 2.2.13)
sinΦ 2 =
√L1L2
2f , βmax= L1+L2+L1L2/f
sin Φ , βmin= f2 L2 sin Φ.
where we have assumed equal focusing strength for the focusing and the defocusing quadrupoles,f is the focal length of the quadrupoles, andL1andL2are the lengths
II. LINEAR BETATRON MOTION 53 of the drift spaces shown in Fig. 2.6. IfL1�L2, the horizontal and vertical betatron amplitude functions are nearly identical along the transport line. Thus the doublet can be considered as an example of the paraxial transport system. Other paraxial transport systems are triplets and solenoidal focusing channels (see Exercise 2.2.12).
II.4 Action-Angle Variable and Floquet Transformation
The Hill equation,y+K(s)y= 0, can be derived from a pseudo-Hamiltonian H=1
2y2+1
2K(s)y2, (2.47)
where (y, y) are conjugate phase-space coordinates. We want to transform (y, y) to the action-angle coordinates, where ψ in Eq. (2.45) serves as the angle-coordinate.
There are two possible generating functions: eitherF1(y, ψ) orF3(y, ψ) (see Exercise 2.21). Using Eq. (2.45) we findF1, and obtain the action-coordinate as:
y=−y
β(tanψ−β
2), ⇐⇒ F1(y, ψ) = y
0
ydy=−y2
2β(tanψ−β
2), (2.48) J=−∂F1
∂ψ = y2
2βsec2ψ= 1
2β[y2+ (βy+αy)2].
where (ψ, J) are the angle (betatron phase) and action coordinates, andy=∂F1/∂y is verified easily from the generating functionF1(y, ψ). With the canonical transfor-mation, the new Hamiltonian becomes
H˜ =H+∂F1
∂s = J
β. (2.49)
Hamilton’s equation givesψ=∂H/∂J˜ = 1/β(s), which recovers Eq. (2.38). Since the new Hamiltonian is independent of the phase coordinateψ, the actionJis invariant, i.e. dJds =−∂∂ψH˜ = 0.Using Eq. (2.48), we obtain
y=
2βJ cosψ, y=−
2J
β [sinψ+αcosψ], (2.50) whereα = −β/2. The actionJ is the phase space area enclosed by the invariant torus:
J= 1 2π
torus
dydy= 1 2π
ydy. (2.51)
The Jacobian of the transformation from (y, y) to (J, ψ) is equal to 1. A word of convention: The area of the phase space ellipse is 2πJ, where we usually use (πJ) as action in unit of [π-mm-mrad] or [πμm], or [πnm], or [πpm]. Thus sometimes
Figure 2.7: The betatron phase space ellipses of a particle with actions J = 10π mm-mrad. The betatron parameters areβy= 10m, andαyshown by each curve. The scale for the ordinateyis mm, andyin mrad. The betatron parameters for each ellipse are marked on the graph. All ellipses has the maximumycoordinate at
2βyJ. The max-imum angular coordinate y is
2(1 +α2y)J/βy. All ellipses have the same phase space area of 2πJ.
Figure 2.8: The horizontal and vertical betatron ellipses for a particle with actionsJx=Jz= 0.5π mm-mrad at the end of the first dipole (left plots) and the end of the fourth dipole of the AGS lattice (see Fig. 2.5). The scale for the ordinatexorzis in mm, and that for the coordinate xorzis in the unit of πin action. Figure 2.7 shows the phase ellipses with identical action of J= 10πmm-mrad.
Figure 2.8 shows the phase-space ellipses (x, x) and (z, z) for a particle with actionsJx= Jz = 0.5π mm-mrad at the ends of the first and the fourth dipoles of the AGS lattice (see Fig. 2.5). Such a phase-space ellipse is also called the Poincar´e map, where the particle phase-space coordinates are plotted in each revolution. The consecutive phase-space points can be obtained by multiplying the transfer matrices,
i.e.
whereMxandMzare the transfer matrices of one complete revolution. The Poincar´e map of betatron motion at a fixed azimuth sis also called the Poincar´e surface of section. If the betatron tune is not a rational number, the consecutive phase-space points of the particle trajectory will trace out the entire ellipse. The areas enclosed by the horizontal and vertical ellipses are equal to 2πJx and 2πJz respectively. As the particle travels in an accelerator, the shape of the phase-space ellipse may vary but the area enclosed by the ellipse is invariant.
II. LINEAR BETATRON MOTION 55 A. Normalized phase space coordinates
We define the normalized conjugate phase-space coordinatePyas Py=βy�+αy=−
2βJsinψ. (2.52)
A particle trajectory in the normalized phase-space coordinates (y,Py) moves clock-wise on a circle with radius √
2βJ as phase advance ψ increases. In terms of the betatron amplitude matrix of Eq. (2.43), the normalized phase space coordinates are expressed as
B. Using the orbital angleθas the independent variable
The Hamiltonian ˜H of Eq. (2.49) depends on the independent variable s. Because β(s) is not a constant, the phase advance is modulated along the accelerator orbital trajectory. Sometimes it is useful to use the orbiting angle as “time” coordinate in order to obtain a global Fourier expansion of particle motion. We use the generating function for coordinate transformation: Hereθ=s/Ris the orbiting angle of the reference orbit andRis the mean radius of an accelerator. The transformation compensates the modulated phase-advance function with conjugate coordinates ( ¯ψ,J) and new Hamiltonian ˜¯ H = νJ /R. Scaling the¯
“time coordinate” fromstoθ, the re-scaled new Hamiltonian and the corresponding coordinate-transformation are
0 ds/β. The transformation is useful in expressing a general betatron Hamiltonian in action-angle coordinates for obtaining a global Fourier expansion in the nonlinear resonance analysis. Hereafter, the notation ( ¯ψ,J) is simplified to (ψ, J).¯
II.5 Courant–Snyder Invariant and Emittance
Using the general solutiony(s) of Eq. (2.45), we obtainβy�+αy=−aβ1/2(s) sin(νφ(s)+
δ).The Courant–Snyder invariant defined by C(y, y�) = 1
β
y2+ (αy+βy�)2
=γy2+ 2αyy�+βy�2 (2.56)
is equal totwice the action. The trajectory of particle motion with initial condition (y0, y0) follows an ellipse described by C(y, y) = �. The phase space enclosed by (y, y) of Eq. (2.56) is equal toπ� (see Fig. 2.9). The quantity “π�” is identify as emittance in the unit of [πmm-mrad] or [πμm] or [πnm] or simply [μ-meter] or [μm], [nano-meter] or [nm], [pico-meter] or [pm], etc.. In each unit, the factorπis implied, explicitly stated or not. The maximum betatron amplitude is√
β�, whereπis ignored in the calculation, or sometimes explicitly expressed as
β�/π to cancel theπis the unit of emittance.
Figure 2.9: The Courant-Snyder invari-ant ellipse. The area enclosed by the el-lipse is equal toπ�, where�is twice the be-tatron action;βis the betatron amplitude functions, andα=−12β,γ= (1 +α2)/β.
The maximum amplitude of betatron mo-tion is√
β�, and the maximum divergence (angle) of the betatron motion is√γ�(see the footnote in Eq. (2.50) for the conven-tion of the emittance and acconven-tion. The cen-ter of the ellipse is the reference orbit or closed orbit (c.o.)).
A. The emittance of a beam
A beam is composed of particles distributed in phase space. Depending on the initial beam preparation, we approximate a realistic beam distribution function by some simple analytic formula. Neglecting dissipation and diffusion processes, each particle in the distribution function has its invariant Courant–Snyder ellipse.
Given a normalized distribution function ρ(y, y) with
ρ(y, y)dydy = 1, the moments of the beam distribution are
�y�=
yρ(y, y)dydy, �y�=
yρ(y, y)dydy, σy2=
(y− �y�)2ρ(y, y)dydy, σy2=
(y− �y�)2ρ(y, y)dydy, σyy=
(y− �y�)(y− �y�)ρ(y, y)dydy=rσyσy,
where σy and σy are the rms beam widths, σyy is the correlation, and r is the correlation coefficient. The rms beam emittance is defined as
�rms≡
σ2yσy2−σ2yy=σyσy
√1−r2. (2.57)
II. LINEAR BETATRON MOTION 57 The rms emittance of a ring beam in phase space, i.e. particles uniformly dis-tributed in phase coordinateψ at a fixed actionJ of Eq. (2.50), is �rms=J. If the accelerator is composed of linear elements such as dipoles and quadrupoles, the emit-tance defined in Eq. (2.57) is invariant. The rms emitemit-tance is equal to the phase-space area enclosed by the Courant–Snyder ellipse of the rms particle (see Exercise 2.2.14).
Although incorrect, the term “emittance” is often loosely used as twice the action variable of betatron oscillations. The betatron oscillations of “a particle” with an
“emittance”�is
y(s) =
β� cos [νφ(s) +δ]. (2.58)
Figure 2.9 shows a Courant–Snyder invariant ellipse with phase space areaπ�,11 the rms beam width is
β(s)�, and the beam rms divergence y�is
γ(s)�. Since γ = (1 +α2)/β, the transverse beam divergence is smaller at a location with a largeβ(s) value, i.e. all particles travel in parallel paths. In accelerator design, a properβ(s) value is therefore important for achieving many desirable properties.
B. Theσ-matrix
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
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