In general, the transfer matrix of a periodic cell can be expressed as M=
⎛
⎝M11 M12 M13
M21 M22 M23
0 0 1
⎞
⎠, (2.156)
whereM11, M12, M21andM22are given by Eq. (2.34). Using the closed-orbit condition of Eq. (2.151), we obtain
D=M13(1−M22) +M12M23
2−M11−M22 =M13(1−cos Φ +αsin Φ) +M23βsin Φ
2(1−cos Φ) ,
D�= M13M21+ (1−M11)M23
2−M11−M22 = −M13γsin Φ +M23(1−cos Φ−αsin Φ)
2(1−cos Φ) ,
where Φ is the horizontal betatron phase advance of the periodic cell, α, β and γ= (1+α2)/βare the Courant–Snyder parameters for the horizontal betatron motion at a periodic-cell locations, and DandD are the value of the dispersion function and its derivative at the same location. SolvingM13andM23 as functions ofDand D, the 3×3 transfer matrix is
M=
⎛
⎝
cos Φ+αsin Φ βsin Φ (1−cos Φ−αsin Φ)D−βDsin Φ
−γsin Φ cos Φ−αsin Φ γDsin Φ+(1−cos Φ+αsin Φ)D
0 0 1
⎞
⎠ (2.157) This representation of the transfer matrix is sometimes useful in studying the general properties of repetitive accelerator sections.
C. Effect of dipole or quadrupole field error on dispersion function In the presence of dipole and quadrupole field error, perturbation to the dispersion function ΔD(s) =D(s)−D0(s) obeys
(ΔD)+ [K0(s) +k(s)]ΔD(s) =
�1 ρ− 1
ρ0
�
−k(s)D0(s), (2.158) whereD0(s) is the unperturbed dispersion function, K0(s) andρ0(s) are the unper-turbed dipole and focusing functions, andk(s) is the quadrupole field error. The inho-mogeneous equation can be solved by employing Floquet transformation by neglecting the higher order termk(s)ΔD(s). The bottom plot of Fig. 2.20 shows ΔD(s)/√
βx induced by the gradient error of a single focusing quadrupole with 1% increase in strength. Outside the quadruple kick location, we find a pure sinusoidal betatron oscillation. Quadrupoles in dispersive locations can be used to produce a local dis-persion bump resembling that of closed orbit bump in Sec. III.3.
It appears that the factor 1ρ−ρ10 in Equation (2.158) can be large if length of all dipoles in a lattice is shortened or lengthened, e.g. ρcan be a factor of 2 larger or smaller thanρ0 in if the dipole length is shortened or lengthened by a factor 2. Can the change of dipole length causes a large perturbation to the dispersion function?
We have previously stated that the dispersion function of a FODO cell is essentially a function ofL1θ, nearly independent of the dipole length. HereL1andθare the length and bending angle in each half cell. This paradox is resolved by the fact that 1ρ−ρ10 has small stopband integrals at harmonics near the betatron tuneνx. On the other hand, if ρ(s)1 −ρ10 has a large stopband near [νx], the perturbation to the dispersion function will be substantial.
IV.2 H -Function, Action, and Integral Representation
The dispersionH-function is defined as
H(D, D) =γxD2+ 2αxDD+βxD2= 1 βx
[D2+ (βxD+αxD)2]. (2.159)
IV. OFF-MOMENTUM ORBIT 127 Since the dispersion function satisfies the homogeneous betatron equation of motion in regions with no dipole (1/ρ= 0), theH-function is invariant. In regions with dipoles, the H-function is not invariant. For a FODO cell, the dispersionH-function at the defocusing quadrupole is larger than that at the focusing quadrupole, i.e. HF≤ HD, where
HF=L1θ2sin Φ(1 +12sinΦ2)2
2(1 + sinΦ2) sin4 Φ2 , HD=L1θ2sin Φ(1−12sinΦ2)2
2(1−sinΦ2) sin4 Φ2 , (2.160) and the dispersionH-function is proportional to the inverse cubic power of the phase advance.
Now we define thenormalizeddispersion phase-space coordinates as
⎧⎪
⎨
⎪⎩
Xd= 1
√βxD=�
2Jdcos Φd, Pd=�
βxD+ αx
√βx
D=−�
2Jdsin Φd, (2.161) where the dispersion action isJd= 12H(D, D).In a straight section,Jd is invariant and Φd, aside from a constant, is identical to the betatron phase advance. In a region with dipoles, Jd is not constant. The change of the dispersion function across a thin dipole is ΔD = 0 and ΔD =θ, i.e. ΔXd = 0, ΔPd =√
βx ΔD = √ βxθ, whereθis the bending angle of the dipole. The change in dispersion action is ΔJd= (βxD+αxD)θ. For FODO-cell lattice shown in Sec. IV.1.A the normalized dispersion coordinateXdis nearly constant, i.e. D∼√
βx, andPdis small. Figure 2.31 shows the normalized dispersion phase-space coordinates in one superperiod of AGS lattice (see Fig. 2.5) that is approximately made of 5 FODO cells.
Figure 2.31: Left: Normalized dis-persion phase-space coordinates Xd andPd are plotted in a superperiod of the AGS lattice. Right: the coor-dinates are shown inXdvsPd. The scales for bothXd andPdare m1/2. Note thatXd is indeed nearly con-stant, and (Pd, Xd) propagate in a very small region of the dispersion phase-space (see also Fig. 2.38).
In contrast, the normalized dispersion phase-space coordinates for a double-bend achromat (DBA) lattice (see Sec. IV.5.A) shows different behavior. Figure 2.32 shows the normalized dispersion coordinates for the IUCF Cooler Ring, which is composed of 3 achromat straight-sections for electron cooling, rf cavities, etc., and 3 dispersive-sections for injection, momentum stacking, etc. The achromat dispersive-sections are described
by a single point at origin: Xd=Pd= 0. Inside dipoles, the normalized dispersion coordinates increase in magnitude. In dispersion matching sections, the normalized dispersion coordinates are located on invariant circles, that are nearly half-circles as shown in Fig. 2.32, i.e. the dispersion phase advance Φdis nearlyπin the dispersion matching section. Since the dispersion phase-advance is equal to the horizontal be-tatron phase-advance in a straight section, the horizontal bebe-tatron phase-advance is also nearlyπ.
Figure 2.32: Left: Normalized disper-sion phase-space coordinates Xd and Pdof the IUCF Cooler lattice are plot-ted. Right: The coordinates are shown inXdvsPdat the end of each lattice elements. The accelerator is made of six 60◦-bends forming a 3 double-bend achromat modules, where the disper-sion function is shown in Fig. 2.35. The scales for bothXdandPdare m1/2. The lattice function and the dispersion phase-space coordinates of the IUCF Cooler Ring differ substantially from the low emittance DBA lattice to be shown in Figs. 2.41. For an ion storage ring, the minimization of �H� plays no important role in beam dynamics. Instead, a minimumβzinside dipole will provide a criterion for the magnet gapg. Since the power required in the operation of a storage ring is proportional tog2, it is preferable to design a machine with a minimumβz inside dipoles, and the corresponding βx will be large in dipole. The resulting dispersion phase-space coordinates in dipoles are much larger than those of minimum emittance DBA lattices shown in Fig. 2.41.
The dispersion function can also be derived from the dipole field error resulting from the momentum deviation. The angular kick due to the off-momentum deviation is θ= Δpp
0 1
ρds, where ρis the bending radius and dsis the differential length of the dipole. The corresponding dipole field error is ΔBBρ = Δpp
0 1
ρ, Substituting the “dipole field error” in Eq. (2.92), we obtain the dispersion function of Eq. (2.101). The integral representation of the normalized dispersion functions is
⎧⎪
Since the synchronization of particle motion in a synchrotron depends critically on the total path length, it is important to evaluate the effect of the off-momentum
IV. OFF-MOMENTUM ORBIT 129 closed orbit on path length. Since the change in path length due to betatron motion is proportional to the square of the betatron amplitude [see Eq. (2.100)], the effect is small. The orbit deviation from a reference orbit of an off-momentum particle is linearly proportional to the product of the fractional off-momentum parameterδ and the dispersion functionD(s). The total path length will depend on the off-momentum parameter. The path difference and the “momentum compaction factor” are
ΔC=
where�D�iandθiare the average dispersion function and the bending angle of theith dipole, and the last approximate identity uses thin-lens approximation. Since D(s) is normally positive, the total path length for a higher momentum particle is longer.
For example, the momentum compaction factor for a FODO lattice is αc≈(DF+DD)θ
2L1 ≈ θ2
sin2(Φ/2)≈ 1 νx2,
whereL1andθare the length and the bending angle of one half-cell, Φ is the phase advance of a FODO cell, andνxis the betatron tune (see Exercise 2.4.2).
A. Transition energy and the phase-slip factor
The importance of the momentum compaction factor will be fully realized when we discuss synchrotron motion in Chap. 3. In the meantime, we discuss the phase stability of synchrotron motion discovered by McMillan and Veksler [21].
Particles with different momenta travel along different paths in an accelerator.
Since the revolution period isT = 1/f =C/v, whereCis the circumference, andvis the speed of the circulating particle, the fractional difference of the revolution periods between the off-momentum and on-momentum particles and the “phase-slip-factor”
are
whereT0= 1/f0 is the revolution period of a synchronous particle,δ= Δp/p0 is the fractional momentum deviation,γT ≡
1/αc is called the transition-γ, andγTmc2 or simplyγT is the transition energy. For FODO cell lattices,γT≈νx.
Below the transition energy, withγ < γTandη <0, a higher momentum particle will have a revolution period shorter than that of the synchronous particle. Because a high energy particle travels faster, its speed compensates its longer path length in the accelerator, so that a higher energy particle will arrive at a fixed location earlier
than a synchronous particle. Above the transition energy, withγ > γT, the converse is true. Without a longitudinal electric field, the time slippage between a higher or lower energy particle and a synchronous particle isT0ηδper revolution.
Atγ =γT the revolution period is independent of the particle momentum. All particles at different momenta travel rigidly around the accelerator with equal revo-lution frequencies. This is the isochronous condition, which is the operating principle of AVF isochronous cyclotrons.
B. Phase stability of the bunched beam acceleration
LetV(t) =V0sin(hω0t+φ) be the gap voltage of the rf cavity (see Fig. 2.33), where V0is the amplitude,φis an arbitrary phase angle,his an integer called the harmonic number, ω0 = 2πf0 is the angular revolution frequency, and f0 is the revolution frequency of a synchronous particle. A synchronous particle is defined as an ideal particle that arrives at the rf cavity at a constant phase angleφ=φs, whereφsis the synchronous phase angle. The acceleration voltage at the rf gap and the acceleration rate for a synchronous particle are respectively given by
Vs=V0sinφs, E˙0=f0eV0sinφs, (2.166) whereeis the charge,E0 is the energy of the synchronous particle, and the overdot indicates the derivative with respect to timet.
0 π/2 π 3π/2 2π
φ V0
−V0
0 φ s π/2
η<0 η>0 Acceleration
Deceleration φ s
Lower Energy Synchronous Energy
Higher Energy
Figure 2.33: Schematic drawing of an rf wave, and the rf phase angles for a synchronous, a higher, and a lower energy particles (Graph courtesy of D. Li, LBNL). For a stable synchrotron motion, the phase focusing principle requires 0< φs≤π/2 forη <0, andπ/2< φs≤π forη >0. Below the transition energy, with 0≤φs< π/2, a higher energy particle arrives at the rf gap earlier and receives less energy from the cavity. Thus the energy of the particle will becomes smaller than that of the synchronous particle. On the other hand, a lower energy par-ticle arrives later and gains more energy from the cavity. This process gives rise to the phase stability of synchrotron motion.
A non-synchronous particle arriving at the rf cavity gap has a phase angleφwith respect to the rf field. The phase φ varies with time, and the acceleration rate is E˙ =f eV0sinφ, wheref is the revolution frequency. Combining with Eq. (2.166), we find the rate of change of the energy deviation is (see also Chap. 3, Sec. I)
d dt
ΔE ω0
= 1
2πeV0(sinφ−sinφs), (2.167)
IV. OFF-MOMENTUM ORBIT 131 where ΔE = E −E0 is the energy difference between the non-synchronous and the synchronous particles. Similarly, the equation of motion for the rf phase angle φ=−hθ, whereθis the actual angular position of the particle in a synchrotron, is
d
dt(φ−φs) =−hΔω=hω0
ΔT T0
=hηω0
Δp p0
= ηhω02 β2E0
ΔE ω0
. (2.168)
Equations (2.167) and (2.168) form the basic synchrotron equation of motion for conjugate phase-space coordinates φ and ΔE/ω0. This is the equation of motion for a biased physical pendulum system, called synchrotron motion. The differential equation for the small amplitude phase oscillation is
d2(φ−φs)
dt2 =ηhω02eV0
2πβ2E0(sinφ−sinφs)≈ ηcosφshω20eV0
2πβ2E0 (φ−φs) =ωsyn2 (φ−φs);
ωsyn=ω0
heV0|ηcosφs| 2πβ2E0 .
0≤φs≤π/2 ifγ < γT orη <0,
π/2≤φs≤π ifγ > γT orη >0. (2.169) whereωsynis the small-amplitude angular synchrotron frequency. The phase stability condition isηcosφs <0. Below the transition energy, with 0≤φs < π/2, a higher energy particle arrives at the rf gap earlier and receives less energy from the rf cavity (see Fig. 2.33). Thus the energy of the particle will gradually becomes smaller than that of the synchronous particle. On the other hand, a lower energy particle arrives later and gains more energy from the cavity. This process gives rise to the phase stability of synchrotron motion. Similarly, the synchronous phase angle should be π/2< φs≤πatγ > γT.
Particles are accelerated through the transition energy in many medium energy synchrotrons such as the AGS, the Fermilab booster and main injector, the CERN PS, and the KEK PS. The synchronous angle has to be shifted fromφstoπ−φsacross the transition energy within 10 to 100μs. Fortunately, synchrotron motion around the transition energy region is very slow, i.e. ωsyn→0 atγ ∼γT. A sudden change in the synchronous phase angle of the rf wave will not cause much beam dilution.
However, when the beam is accelerated through the transition energy, beam loss and serious beam phase-space dilution can result from space-charge-induced mis-match, nonlinear synchrotron motion, microwave instability due to wakefields, etc.
An accelerator lattice with a negative momentum compaction factor, where the tran-sition γT is an imaginary number, offers an attractive solution to these problems.
Such a lattice is called an imaginaryγT lattice. Particle motion in an imaginary γT lattice is always below transition energy, thus the transition energy problems can be eliminated. Attaining an imaginaryγT lattice requires a negative horizontal disper-sion in most dipoles, i.e.
i�D�iθi<0. Methods of achieving a negative compaction lattice will be addressed in Sec. IV.8.
C: Effect of the dispersion function on orbit response matrix (ORM) A dipole-kickθj at positionsjchanges the closed orbit byG(s, sj)θj and the circum-ference by ΔC =D(sj)θj. The response matrix of the ORM experiment depends on the method of measurement:
1. Constant momentum: The change of the revolution period is ΔT = ΔC/βc= D(sj)θj/βcat a constant velocity. Similarly the rf frequencymustbe adjusted according to Δf /f =−ΔT /T in order to maintain a constant momentum, The beam motion at this new rf frequency is on-momentum, i.e. δ = 0 and the closed orbit is
xco(si) =G(si, sj)θj (2.170) or the response matrix is Ri,j = G(si, sj) of Eq. (2.110). Sometimes, the rf cavity is turned off during the ORM measurement in proton accelerators. The beam, at a constant injection momentum, is “on-momentum” and the response matrix isRi,j =G(si, sj).52
2. Constant path length: Some ORM experiments carry out at a constant rf fre-quency, i.e. the path length is constant. To maintain a constant pathlength, the beam has to orbit at an equivalent off-momentum “δ” = α1cΔCC0 to compensate path length change by the dipole bump. Thus the corresponding closed orbit is
xco(si) =G(si, sj)θj+D(si)δ=
G(si, sj) +D(si)D(sj) 2πRαc
θj, (2.171) whereαcis the momentum compaction factor,D(s) is the dispersion function, and R is the mean radius of the accelerator. The response matrix becomes Ri,j=G(si, sj) +D(s2πRαi)D(scj).
IV.4 Dispersion Suppression and Dispersion Matching
Since bending dipoles are needed for beam transport in arc sections, the dispersion function can not be zero there. If the arc is composed of modular cells, such as FODO cells, etc., the dispersion function is usually constrained by the periodicity condition, Eq. (2.151), which simplifies lattice design. In many applications, the dispersion function should be properly matched in straight sections for optimal accelerator op-eration.53 If the betatron and synchrotron motions are independent of each other,
52J. Kolski, Ph.D. Thesis (Indiana University, 2010) for ORM at PSR; Z. Liu, Ph.D. Thesis (Indiana University, 2011) for ORM at SNS.
53The curved transport line is usually called the arc, and the straight section that connects arcs is usually called the insertion, needed for injection, extraction, rf cavities, internal targets, insertion devices, and interaction regions for colliders.
IV. OFF-MOMENTUM ORBIT 133 the rms horizontal beam size isσx2(s) =βx(s)�x,rms+D2(s)�(Δp/p0)2�,where�x,rms
is the rms emittance. Thus the beam size of a collider at the interaction point can be minimized by designing a zero dispersion straight section. A zero dispersion function in the rf cavity region can be important to minimize the effect of synchro-betatron coupling resonances. We discuss here the general strategy for dispersion suppression.
First-order achromat theorem
The first-order achromat theorem states that a lattice ofn repetitive cells is achro-matic to first order if and only ifMn=I or each cell is achromatic.54 HereM is the 2×2 transfer matrix of each cell, andI is a 2×2 unit matrix. Let the 3×3 transfer matrix of a basic cell be
R=
whereM is the 2×2 transfer matrix for betatron motion, and ¯d is the dispersion vector. The transfer matrix ofncells is
Rn= function modules. A unit matrix achromat works like a transparent transport section for any dispersion functions.
Dispersion suppression
Applying the first-order achromat theorem, a strategy for dispersion function sup-pression can be derived. We consider a curved (dipole) achromatic section such that Mn=I. We note that one half of this achromatic section can generally be expressed as
Using the closed-orbit condition, Eq. (2.151), the dispersion function of the repetitive half achromat isD=d/2, D�=d�/2.If the dipole bending strength of the adjoining
−I section is halved, the transfer matrix and the dispersion function will be matched to zero value in the straight section, i.e.
R1/2=
54See K. Brown and R Servranckx, p. 121 in Ref. [16].
Thus the zero dispersion section is matched to the arc by the dispersion suppression section.
When edge focusing is included, a small modification in the quadrupole strengths is needed for dispersion suppression. This is usually called the missing dipole dispersion suppressor (see Exercise 2.4.3c). The reduced bending strength scheme for dispersion suppression is usually expensive because of the wasted space in the cells. A possible variant uses−Isections with full bending angles for dispersion suppression by varying the quadrupole strengths in the−I sections. With use of computer programs such as MAD and SYNCH, the fitting procedure is straightforward.
Is the dispersion function unique?
A trivial corollary of the first-order achromat theorem is that a dispersion function of arbitrary value can be transported through a unit achromat transfer matrix, i.e. a 3×3 unit matrix.
Now we consider the case of an accelerator or transport line with many repetitive modules, which however do not form a unit transfer matrix. Is the dispersion func-tion obtained unique? This quesfunc-tion is easily answered by the closed-orbit condifunc-tion Eq. (2.151) for the entire ring. The transport matrix ofnidentical modules is
Rn=
Mn (Mn−I)(M−I)−1d¯
0 1
, (2.175)
whereM is the transfer matrix of the basic module with dispersion vector ¯d. Using the closed-orbit condition, Eq. (2.151), we easily find that the dispersion function of the transport channel is uniquely determined by the basic module unless the transport matrix is a unit matrix, i.e. Mn =I. In the case of unit transport, any arbitrary value of dispersion function can be matched in the unit achromat. Since the machine tune can not be an integer because of the integer stopbands, the dispersion function of an accelerator lattice is uniquely determined.
IV.5 Achromat Transport Systems
If the dispersion function is not zero in a transport line, the beam closed orbit depends on particle momentum. However, it is possible to design a transport system such that the beam positions do not depend on beam momentum at both ends of the transport line. Such a beam transport system is called an achromat. The achromat theorem of Sec. IV.4 offers an example of an achromat.
A. The double-bend achromat
A double-bend achromat (DBA) or Chasman-Green lattice is a basic lattice cell fre-quently used in the design of low emittance synchrotron radiation storage rings. A
IV. OFF-MOMENTUM ORBIT 135 DBA cell consists of two dipoles and a dispersion-matching section such that the dispersion function outside the DBA cell is zero. It is represented schematically by
[OO] B {O QF O} B [OO],
where [OO] is the zero dispersion straight section and{O QF O} is the dispersion matching section. The top plot of Fig. 2.34 shows a basic DBA cell.
Figure 2.34: Schematic plots of DBA cells. Upper plot: standard DBA cell, where O and OO can contain doublets or triplets for optical match. Lower plot:
triplet DBA, where the quadrupole triplet is arranged to attain betatron and disper-sion function match of the entire module.
We consider a simple DBA cell with a single quadrupole in the middle. In thin-lens approximation, the dispersion matching condition is
We consider a simple DBA cell with a single quadrupole in the middle. In thin-lens approximation, the dispersion matching condition is