Momentum Compaction Factor

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 α_c≈(D_F+D_D)θ

2L1 ≈ θ²

sin²(Φ/2)≈ 1 ν_x²,

whereL₁andθare the length and the bending angle of one half-cell, Φ is the phase advance of a FODO cell, andν_xis 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,γ_T ≡

1/α_c is called the transition-γ, andγ_Tmc² or simplyγ_T is the transition energy. For FODO cell lattices,γ_T≈ν_x.

Below the transition energy, withγ < γ_Tandη <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 > γ_T, 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 =γ_T 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=φ_s, whereφ_sis the synchronous phase angle. The acceleration voltage at the rf gap and the acceleration rate for a synchronous particle are respectively given by

V_s=V₀sinφ_s, E˙₀=f₀eV₀sinφ_s, (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< φ_s≤π/2 forη <0, andπ/2< φs≤π forη >0. Below the transition energy, with 0≤φ_s< 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−sinφ_s), (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−φ_s)

dt² =ηhω₀²eV₀

2πβ²E₀(sinφ−sinφ_s)≈ ηcosφ_shω²₀eV₀

2πβ²E₀ (φ−φ_s) =ω_syn² (φ−φ_s);

ω_syn=ω₀

heV₀|ηcosφ_s| 2πβ²E₀ .

0≤φ_s≤π/2 ifγ < γ_T orη <0,

π/2≤φs≤π ifγ > γ_T orη >0. (2.169) whereωsynis the small-amplitude angular synchrotron frequency. The phase stability condition isηcosφ_s <0. Below the transition energy, with 0≤φ_s < 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< φ_s≤πatγ > γ_T.

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 ∼γ_T. 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 γ_T is an imaginary number, offers an attractive solution to these problems.

Such a lattice is called an imaginaryγ_T lattice. Particle motion in an imaginary γ_T lattice is always below transition energy, thus the transition energy problems can be eliminated. Attaining an imaginaryγ_T lattice requires a negative horizontal disper-sion in most dipoles, i.e.

i�D�iθ_i<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-kickθ_j at positions_jchanges the closed orbit byG(s, s_j)θ_j and the circum-ference by ΔC =D(s_j)θ_j. 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

x_co(s_i) =G(s_i, s_j)θ_j (2.170) or the response matrix is R_i,j = G(s_i, s_j) 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 isR_i,j =G(s_i, s_j).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” = _α¹_c^ΔC_C0 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(s_i)D(s_j) 2πRαc

θj, (2.171) whereα_cis 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(s_2πRα^i)D(s_c^j).

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 isσ_x²(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.

R_1/2=

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

⎛

where f is the focal length of the quadrupole, θand Lare the bending angle and length of the dipole, andL₁is the distance from the end of the dipole to the center of the quadrupole. The zero dispersion value at the entrance to the dipole is matched to asymmetricconditionD_c= 0 at the center of the focusing quadrupole. The required focal length and the resulting dispersion function become

f =1

Note that the focal length needed in the dispersion function matching condition is independent of the dipole bending angle in thin-lens approximation, and it can easily be obtained from the geometric argument. The dispersion function at the symmetry point is proportional to the product of the effective length of the DBA cell and the bending angle.

Although this simple example shows that a single focusing quadrupole can attain dispersion matching, the betatron function depends on the magnet arrangement in the [OO] section, and possible other quadrupoles in the dispersion matching section.

The dispersion matching condition of Eq. (2.177) renders a horizontal betatron phase advance Φ_xlarger thanπ in the dispersion matching section (from the beginning of the dipole to the other end of the other dipole). The stability condition of betatron

motion (see Sec. II.6) indicates that betatron function matching section [OO] can not be made of a simple defocusing quadrupole. A quadrupole doublet, or a triplet, is usually used in the [OO] section. Such DBA lattice modules have been widely applied in the design of electron storage rings.

A simple DBA cell is the triplet DBA (lower plot of Fig. 2.34), where a quadrupole triplet is located symmetrically inside two dipoles. This compact lattice was used for the SOR ring in Tokyo. Some properties of the triplet DBA storage ring can be found in Exercise 4.3.6.

B. Other achromat modules

The beam transport system in a synchrotron or a storage ring requires proper dis-persion function matching. The design strategy is to use achromatic subsystems. An example of achromatic subsystem is the unit matrix module (see Sec. IV.4 on the first order achromat theorem). A unit matrix module can be made of FODO or other basic cells such that the total phase advance of the entire module is equal to an integer multiple of 2π. Achromatic modules can be optically matched with straight sections to form an accelerator lattice.

The achromatic transport modules are also important in the transport beamlines (see Exercises 2.4.12 to 2.4.15). The achromatic transport system find applications in high energy and nuclear physics experiments, medical radiation treatment, and other beam delivering systems.

IV.6 Transport Notation

In many applications, the particle coordinates in an accelerator can be characterized by a state vectorW�, where the transpose is

W� ^T = (W₁, W₂, W₃, W₄, W₅, W₆) = (x, x, z, z, βcΔt, δ), whereβcis the speed of the particle,βcΔtis the path length difference with respect to the reference orbit, and δ = Δp/p₀ is the fractional momentum deviation of a particle. The transport of the state vector in linear approximation is

W_i(s₂) = 6 j=1

R_ij(s₂|s₁)W_j(s₁), (i, j= 1,· · ·,6). (2.178) Note that the 2×2 diagonal matrices for the indices 1,2, and 3,4 are respectively the horizontal and verticalMmatrices. TheR₁₃, R₂₃R₁₄, R₂₄elements describe the linear betatron coupling. TheR₁₆, R₂₆ elements are the dispersion vectord�of Eq. (2.172).

Without synchrotron motion, we haveR₅₅ =R₆₆ = 1. All other elements of theR matrix are zero.

IV. OFF-MOMENTUM ORBIT 137 In general, the nonlinear dependence of the state vector can be expanded as

W_i(s₂) =

For example, particle transport through a thin quadrupole is Δx=− x Similarly, particle transport through a thin sextupole gives

Δx=− S

where S = −B2�/Bρis the integrated sextupole strength. Here we used the con-vention thatS > 0 corresponds to a focusing sextupole. Tracing the transport in one complete revolution, we get the momentum compaction factor asα_c=R₅₆. The program TRANSPORT⁵⁵ has often been used to calculate the transport coefficients in transport lines.

IV.7 Experimental Measurements of Dispersion Function

Digitized BPM turn by turn data can be used to measure the betatron motion. On the other hand, if the BPM signals are sampled at a longer time scale, the fast betatron oscillations are averaged to zero. The DC output provides the closed orbit of the beam. The dispersion function can be measured from the derivative of the closed orbit with respect to the off-momentum of the beam, i.e.

D= dx_co

d(Δp/p₀) =−η f₀dx_co

df₀ , (2.180)

Figure 2.35: The upper plot shows the closed or-bit at a BPM vs the rf frequency for the IUCF Cooler Ring. The slope of this measurement is used to obtain the “measured” dispersion func-tion. The lower plot compares the measured dis-persion function (rectangles) with that obtained from the MAD program (solid line).

wherex_cois the closed orbit,f₀is the revolution frequency,ηis the phase-slip factor, and the momentum of the beam is varied by changing the rf frequency.

The upper plot of Fig. 2.35 shows the closed orbit at a BPM location vs the rf frequency at the IUCF Cooler Ring. Using Eq. (2.180), we can deduce the dispersion function at the BPM location. In the lower plot of Fig. 2.35 the “measured” dispersion functions of the IUCF Cooler Ring is compared with that obtained from the MAD program [23].⁵⁶ The accuracy of the dispersion function measurement depends on the precision of the BPM system, and also on the effects of power supply ripple.

To improve the accuracy of the dispersion function measurement, we can induce frequency modulation to the rf frequency shift. The resulting closed orbit will have the characteristic modulation frequency. Fitting the resulting closed orbit with the known modulation frequency, we can determine the dispersion function more accurately.

IV.8 Transition Energy Manipulation

Medium energy accelerators often encounter problems during transition energy cross-ing, such as longitudinal microwave instability and nonlinear synchrotron motion.

These problems can be avoided by an accelerator having a negative momentum com-paction factor. The revolution period deviation ΔT for an off-momentum particle Δp = p−p₀ is given by Eq. (2.164). The accelerator becomes isochronous at the transition energy (γ=γ_T).

There are many unfavorable effects on the particle motion near the transition

There are many unfavorable effects on the particle motion near the transition

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