2.1 Accelerator reviewed
2.1.1 RF acceleration and beam dynamics
Radio waves form specific propagation modes in the metallic cavity. In the linear accelerator, the interior of the cavity is separated by irises (disks) into many
2Synchrotron radiation is the EM wave emitted by a charged particle due to radial acceleration (the bending magnets).
.. coupled cells to reduce the phase velocity of the axial wave such that it meets the particle speed 𝛽𝑐. The cavity in the linac operates in 2𝜋/3-mode (the phase ad-vance per cell is 2𝜋/3). In other words, the synchronous particles “surf” on the crest of the traveling wave or standing wave and see an acceleration voltage at ev-ery cell3. In the storage ring, the ratio of the radio wave frequency (508.887 MHz) used in the RF cavity to the particle revolution frequency is an integer ℎ = 𝜔rf/𝜔rev, called the harmonic number. Namely, the radio wave in the cavity oscillates ℎ full cycles as a synchronous particle enters and exits the cavity, traveling along the beam pipe, and entering again at exactly the same phase 𝜑𝑠 = 𝜔𝑡𝑠 to gain an equal amount of energy 𝐸𝑠 = 𝑒𝑉accsin(𝜑𝑠) to that lost through synchrotron radi-ation over one revolution. In fact, there are many cavities along a storage ring, all separated by multiples of the RF wavelength.
For ultra-relativistic (𝛽 ∼ 1) electrons and positrons circulating in a mag-netic field, increased energy scales up the momentum linearly but almost doesn’t change the speed. Consequently, a non-synchronous particle with a higher en-ergy travels a longer circumference, and its revolution frequency is lowered. Thus, it enters the next cavity at a time 𝑡𝑛 > 𝑡𝑠later than the synchronous particle and gains a smaller energy 𝐸𝑛= 𝑒𝑉accsin(𝜑𝑛), Δ𝐸 ≡ 𝐸𝑛−𝐸𝑠 < 0. After the passage of several cavities, its energy falls below the synchronous particle, and it begins to enter the cavity earlier, picking up an energy larger than the synchronous particle.
Thus, all the particles in a bunch oscillate longitudinallly around the synchronous particle as they travel along the beam pipe. This is the synchrotron oscillation.
Similarly, they also oscillate in the transverse (horizontal and vertical) direction under the influence of the bending magnets. This is referred to as the betatron oscillation. The transition of the beam in the six-dimensional phase space defines the beam dynamics4.
We first discuss the betatron oscillation. In the absence of coupling between
3For a derivation of the propagation mode from Maxwell’s equations, see Ref. [82,83].
4For a formal treatment of the beam dynamics, see Ref. [84,83,85]. An introductory text is Ref. [81].
.. perpendicular directions or beam-beam interaction, the position 𝑥(𝑠) of a particle depending on the arc length 𝑠 is described by the Hill’s equation [81,31]
d2𝑥
d𝑠2 + 𝐾𝑥(𝑠) = 0,
where 𝐾𝑥(𝑠) includes the partial derivative of the bending magnetic field along the 𝑦-axis. The oscillating solution is
𝑥 = √𝜀𝛽(𝑠) cos(𝜓(𝑠) + 𝜑), 𝑥′ = √
𝜀 ( d
d𝑠√𝛽(𝑠)) cos(𝜓(𝑠) + 𝜑) − √ 𝜀
𝛽(𝑠)sin(𝜓(𝑠) + 𝜑),
where 𝜀 and 𝜑 are constants depending on initial conditions, 𝛽(𝑠) is the amplitude modulation of the oscillation due to the changing strength, that is, the focusing and defocusing quadrupole magnets over 𝑠. 𝑥′≡ d𝑥d𝑠 is the “tangential slope” of the transverse particle trajectory, but it can also be thought of as the momentum 𝑥′ ≡ d𝑥d𝑡, since 𝑡 proceeds along 𝑠. We have used 𝜓′(𝑠) = 1/𝛽(𝑠) to derive the second equation.
As a particle oscillates in the 𝑥 direction, it transform from one point in the phase space spanned by (𝑥, 𝑥′) to another. At a specific location 𝑠 of the ring, the oscillation amplitudes √𝜀𝛽(𝑠) and √𝜀/𝛽(𝑠) are the same, but the phase is differ-ent every time the particle passes 𝑠. Over many turns, its (𝑥, 𝑥′) form an ellipse in the phase space, as illustrated in Fig. 2.1a. Each particle in a bunch carries a different 𝜀, making an ellipse with a different oscillating amplitude. The horizon-tal rms emittance 𝜀𝑥is the phase space area5 containing a certain proportion, say 39%, of the particles in a bunch, divided by 𝜋. In a similar fashion, the vertical os-cillation gives rise to ellipses in the (𝑦, 𝑦′) phase space. The transverse bunch size 𝜎𝑖is characterized by the beta function 𝛽𝑖and the emittance 𝜎𝑖, 𝜎𝑖= √𝛽𝑖𝜀𝑖, where
5Although conceptually similar, there are various definitions of the emittance: Assuming that the bunch distribution in the phase space has a Gaussian profile, then the rms emittance covers 39% of the phase space area 𝜎𝑥= √⟨𝑥2⟩ = √𝛽𝜀rms. Another definition makes it cover 95% of the phase space area. Sometimes, the emittance is defined as area/𝜋; other times, it is just the area.
.. 𝑖 = 𝑥, 𝑦. At another point 𝑠′along the beam pipe, a different 𝛽(𝑠′) leads to a differ-ent ellipse and a differdiffer-ent bunch size, but the area of the ellipse 𝜀𝜋 as the particle moves along beam pipe remains the same6. On the other hand, the initial condi-tion prior to injeccondi-tion, acceleracondi-tion, particle loss, scattering and damping process can all change the emittance. SuperKEKB uses wiggler magnets and modifies the lattice design of focusing and defocusing quadrupole magnets7 along the arc to keep the emittance small, as shown in Fig. 2.2 [86]. Near the collision point, the beta functions are squeezed down by the final focusing magnets to achieve high luminosity. Figure 2.1: Phase space plots of the beam particles. Redrawn from [87]
Assuming that the energy of the synchronous particle varies very slowly com-pared to the energy difference between particles Δ𝐸, the time and energy differ-ence between cavities follow
d2
d𝑛2Δ𝑡𝑛+ (2𝜋𝑄𝑠)2Δ𝑡𝑛= 0,
where 𝑄𝑠 is the small amplitude synchrotron oscillation tune, and it is the
num-6Liouville’s theorem guarantees that the phase space density is conserved.
7A quadrupole magnet is focusing in the 𝑥-direction and defocusing in the 𝑦-direction at the same time. Rotated 90∘, it becomes focusing in the 𝑦-direction and defocusing in the 𝑥-direction.
..
Figure 2.2: Lattice design of the arc cell in (a) LER, (b) HER. The dispersion func-tion 𝜂 is kept small along the ring to suppress quantum excitafunc-tion, which enlarges emittance.
ber of synchrotron oscillations between RF cavities. The longitudinal phase space is spanned by Δ𝐸𝑛 and Δ𝑡𝑛, and the longitudinal emittance, analogous to the transverse ones, are defined as the area of the phase space as in Fig. 2.1b. The bunch length8 Δ𝑡max = √𝜀𝐿𝛽𝐿 and energy spread Δ𝐸max = √𝜀𝐿/𝛽𝐿 are sim-ilarly defined. The area around the synchronous particle, where the motion is bounded, is called the synchrotron bucket. The phase space trajectory on and outside the boundary is not an ellipse, since the small amplitude approximation does not apply any more. The circumference of the SuperKEKB storage ring con-tains 5120 different positions of synchronous particles, so there are totally 5120 buckets along the storage ring 9. When the energy difference Δ𝐸 of a particle grows larger than the bucket height (or the acceptance), the particle is no longer trapped and is lost in the beam pipe.
For stable beam operation, the bunch size must be kept smaller than both the longitudinal and the transverse momentum acceptance. Nevertheless, the EM in-teraction between particles in a bunch (the intra-bunch inin-teraction), or between beams of opposite charges (the beam-beam interaction), causes some particles to gain too much momentum and fall out of the bucket. When it happens near the interaction region in the detector, the outcasts give rise to the detector background
8The “bunch length” of 6 mm in LER and 5 mm in HER appearing in various talks and reports [86] is the value including intra-beam scattering and wake field from the ring impedance at the design beam current.
9Not all buckets have to be filled with bunches.
.. described in Sec. 2.2. Moreover, beam bunches can also kick back to the RF cav-ities through beam loading and couples to other consecutive bunches. Higher-order modes that satisfy the same Maxwell boundary conditions can also develop in the RF cavities. In fact, they are the main concern of bunch instability in Su-perKEKB [88].