In the low-current limit, the beam does not take away any energy from the accelerating structures and the beam is placed on the crest of the rf wave to achieve both maximum acceleration and minimum energy spread within the bunch. At higher beam currents while invoking BNS damping, the klystrons in the first part of the linac are phased to impart relatively higher energy to the head of the bunch while in the latter part of the linac the klystrons are phased to restore the energy spread to less than the energy acceptance of the downstream target area or beam-delivery system.

At even higher bunch currents, one must carefully balance the two terms in square brackets in (8.21), that is cancel the energy variation along the bunch arising from the slope of the rf and that from the longitudinal wake field. Shown in Fig. 8.15 are sketches illustrating such cancellation. The effective rf gain representing the vector sum of all accelerating stations is plotted versus time (left) together with the projection of the charge distribution which shows the resultant energy spread of the bunch (right). At low current, a bunch placed on crest has minimum energy spread. Off crest, a position-energy correlation is introduced and the energy spread is increased. At high current, due to longitudinal wake field, or beam loading, a bunch placed on crest has a large energy spread. For short, high intensity bunches, the energy spread may

Fig. 8.15. Effective energy gain (left) and energy spread (right) for low (a) and high (b) current bunches illustrating optimum phasing of the rf structures for minimum energy spread

be minimized by placing the beam off-crest as shown. In this case the beaminduced wake field (dotted line) exactly cancels the slope of the rf across the bunch.

The energy spread can also be modified by changing the longitudinal beam distribution in the linac. A clever technique of adjusting the bunch shape in the SLC linac consisted in raising or lowering the compressor voltage, relative to the ideal case of maximum compression (see Sect. 8.1). These two modes of operation are called overcompression or undercompression, respectively. Although the same rms bunch length can be obtained with either mode, the shape of the bunch is quite different, which is partly due to an initial distortion caused by impedance effects in the upstream damping ring. Operation with overcompression produced the desired forward-peaked distribution [42] and was used to avoid undesirable energy tails at the end of the linac [43] (see also Fig. 8.20).

Simulation results using (8.3) are shown in Fig. 8.16. While perhaps nonintuitive, simply by 'overcompressing' the bunch, the tails in the energy spread distribution could be eliminated without diluting the longitudinal beam emittance. Measurements of the beam at the end of the SLC linac in a dispersive region are shown [43] in Fig. 8.17 with undercompression (left) and with overcompression (right). The absence of the low energy tails in the latter case justified the routine use of over-compressed beams at the SLC.

In linear accelerators with high current bunches the longitudinal density profile can be further adjusted (so-called bunch shaping) to minimize transverse emittance dilutions arising from short-range wake fields and/or dispersion. This is particularly useful since the outgoing energy spread, and especially energy 'tails', often cause even further emittance dilutions downstream due to chromatic aberrations in the final focus systems.

Minimization of the energy spread depends critically on the single-bunch charge. In the single particle approximation, the energy gained through an accelerating structure is

Fig. 8.16. Longitudinal phase space (top) and projections onto time axis (bottom) with undercompression (left) and with overcompression (right) (Courtesy F.-J. Decker, 1999)

Fig. 8.17. Measurements of the beam profile at a dispersive location at low compressor voltage (left) and with bunch overcompression (right) (Courtesy F.J. Decker, 2000)

$$eV = E\_k L\_k \cos \phi\_k \ , \tag{8.24}$$

where Ek is the accelerating gradient in units of MeV/m, Lk is the length of the accelerating region, and φk is the time-like variable representing the phase of the particle relative to the crest of the rf. The magnitude of Ek

and the definition of the phase φk include possible effects of klystron phase errors.

For a bunch of particles, the energy seen by a given particle is reduced due to loading of the accelerating structure by leading particles within the bunch. Letting φ0 represent the phase at the head of the bunch with respect to the crest of the rf, the total energy gain at the end of the linac becomes

$$eV(\phi\_k) = E\_k L\_k \cos \phi\_k + N\_b e \int\_0^{\phi\_0 - \phi\_k} \rho(\phi') W\_{\parallel}(\phi\_0 - \phi\_k - \phi') \, \mathrm{d}\phi' \,, \tag{8.25}$$

where ρ(φ ) represents the bunch charge distribution, and W is the wake function for the entire accelerator and is given by the product of the singlebunch wake field times the number of accelerating structures.

Minimum energy spread within the bunch requires that V (φk) is independent of φk; that is

$$\frac{\partial V(\phi\_k)}{\partial \phi\_k} = 0 \,. \tag{8.26}$$

It has been shown [42] that there exists a solution for the bunch charge distribution ρ(φ) which satisfies this criterion, and which can be obtained by numerically solving

$$\rho(x) = \frac{E\_k L\_k}{W\_{\parallel}(0)} \sin \left(\phi\_0 - \tilde{\phi}\right) + N\_b e \int\_0^{\tilde{\phi}} \frac{\frac{\partial W\_{\parallel}}{\partial \phi} \left(\tilde{\phi} - \phi'\right) \rho(\phi')}{W\_{\parallel}(0)} d\phi',\tag{8.27}$$

Fig. 8.18. Optimum bunch shape for beam loading compensation in the SLC linac. The bunch head is on the left (Courtesy G. Loew, 1999)

where φ˜ ≡ (φ0 − φk). The interpretation of (8.27) is quite analogous to that shown in Fig. 8.10. Only in this case, the voltage provided by the higher harmonic cavity is replaced by the decelerating voltage induced by beam loading of the leading particles within the bunch.

The solution given in (8.27) is shown in Fig. 8.18 for the SLC linac. The horizontal axis shows the phase angle of particles within the bunch measured with respect to φ0 with the leading particles located at zero phase angle. The different curves labelled by φ0 designate the position of the head of the bunch with respect to the crest of the rf in units of degrees, and the points marked 'T' indicate where the integrated bunch charge reaches the design single bunch population of Nb = 5 × 1010 particles per bunch. As can be seen, for minimum energy spread, the preferred charge distributions tend in general to have a steeply rising edge.

Fig. 8.19. Conceptual illustration for optimizing the relative phase of the beam for the case of very long bunches. The accelerating field Erf and the longitudinal wake field W are shown in (a). The longitudinal profiles and their projections onto the energy axes are shown for different relative phases (between the beam and the rf wave) in (b)–(d) (Courtesy J. Seeman, 1999)

#### 196 8 Longitudinal Phase Space Manipulation

The tradeoff between bunch length and energy spread in a linear accelerator is illustrated in Fig. 8.19, which shows the effect of the combined voltages from the power source ERF and the longitudinal wake field W. On the left is depicted the longitudinal phase space of a long bunch while the projection onto the energy axis is given on the right. In the limit of long bunches one can see that a 'double-horned' distribution produces the minimum rms energy spread.

Measured energy spread profiles taken at the end of the SLAC linac are shown in Fig. 8.20. A wire scanner located in a dispersive region of a downstream transport line was used to measure the profile σw. The energy spread σδ was inferred by subtracting out, in quadrature, the contribution from the betatron beam size σβ = √ xβx:

$$
\sigma\_{\delta} = \sqrt{\sigma\_{w}^{2} - \sigma\_{\beta}^{2}} \,. \tag{8.28}
$$

The angle φ denoted in the figure shows the BNS phase angle at the time of the measurement. These data show clearly the effects of not only misphasing the linac, but the additive contributions of the short-range longitudinal wake field and have been used together with simulation [44, 45] to determine the longitudinal bunch distribution at the SLC.

Fig. 8.20. Energy spread measurements taken at the end of the SLAC linac for different BNS phases (φ) (Courtesy K. Bane, 1999)

# 8.7 Energy Compression

The energy spread of a single bunch can be made smaller (at the expense of increased bunch length) using reverse application of a bunch length compressor. In this case, the beam first passes through a region with an energy-dependent path length, in which the particles of different energy are separated in time. Following this is an accelerating section which decelerates and accelerates the high and low energy particles, respectively. Energy compression for single bunches was designed and implemented [46] for the positron transport line into the SLAC positron damping ring. The net increase in particle yield was observed to be about 10%.

# 8.8 Beam Loading and Long-Range Wake Fields

In the quest for obtaining ever-increasing total beam currents, both newly constructed and future accelerators have in common bunch trains consisting of multiple, closely spaced, high-current bunches. In both linear and circular accelerators this may lead to a relative phase shift between the bunches of a bunch train (an example was shown in Fig. 8.13).

When a beam passes through an accelerating cavity, it induces a voltage Vb,m in each mode m of the cavity. The induced voltage is always retarding; that is, the beam-induced voltage always acts to decelerate the beam. Expressed another way, the beam always takes energy away from the cavity. This is refered to as beam loading.

The fundamental theorem of beam loading [47] is relevant on short time scales (i.e., for a single pass through an accelerating cavity). The theorem states that the voltage that a test particle would experience when trailing a (point) source particle at time t0 > 0 is exactly twice its beam induced voltage at t0 = 0. More generally, the induced voltage, or wake potential, Vδ is given by

$$V\_\delta(t\_0) = 0 \quad \text{for} \quad t\_0 < 0$$

$$= -kq \quad \text{for} \quad t\_0 = 0$$

$$= -2kq e^{-\frac{t\_0}{\tau\_f}} \cos \omega\_{\text{rf}} t\_0 \quad \text{for} \quad t\_0 > 0 \; , \tag{8.29}$$

where τf = 2Q/ωrf is the fill time of the cavity and Q is the cavity quality factor. Here, the variable

$$k = \frac{\omega}{2} \left(\frac{R}{Q}\right) \tag{8.30}$$

is called the loss parameter which tends to be determined by the structure geometry close to the beam. In practice, k is often calculated for each cavity mode using numerical programs, e.g., MAFIA. The wake potential describes the electromagnetic field that the point-like beam generates as it interacts with its surroundings and how this field acts back on the beam, thus perturbing its motion.

We consider next the effect of multi-bunch or multi-turn beam loading, which is caused by the interaction of the beam with the fundamental accelerating mode of the rf cavity on successive bunch passages. In addition, there is also a short-range effect, which consists of the superposition of many higher-order cavity modes. These higher-order modes are usually damped, so that the short-range wake field quickly decays. The fundamental mode on the other hand is used for the acceleration, and, hence, it necessarily exhibits a weak damping, or a high Q value.

In a circular accelerator containing a single accelerating cavity and a single particle bunch, the steady-state (t τf ) beam-induced voltage Vb is given by summing over the contribution from all previous turns. For short bunches, using (8.29) with Trev denoting the revolution frequency,

$$\begin{split} V\_{b} &= -kq - 2kq \sum\_{n=1}^{\infty} e^{-\frac{t\_0}{\tau\_f}} \cos \omega t\_o \delta(t\_0 - nT\_{\text{rev}}) \\ &= -2kq \left[ \sum\_{n=0}^{\infty} e^{-\frac{t\_0}{\tau\_f}} \cos \omega t\_o + \frac{1}{2} \right] \delta(t\_0 - nT\_{\text{rev}}) \\ &= -2kq \left[ \sum\_{n=0}^{\infty} e^{-\frac{nT\_{\text{rev}}}{\tau\_f}} \cos n\omega T\_{\text{rev}} + \frac{1}{2} \right] . \end{split} \tag{8.31}$$

Driving the cavity near its resonance frequency, i.e., taking ω = ωrf, and noting that cos(nωTrev) = 1 for all n, we can simplify this expression and find

$$V\_b = -2kq \left[ \sum\_{n=0}^{\infty} e^{-\frac{n\omega\_{\rm rf}}{2Q}T\_{\rm rev}} + \frac{1}{2} \right]$$

$$= -2kq \left[ \frac{1}{1 - e^{-\frac{n\omega\_{\rm rf}}{2Q}T\_{\rm rev}}} + \frac{1}{2} \right] \,. \tag{8.32}$$

Neglecting the small self-loading term (factor 1/2), and applying

$$\frac{1}{1 - e^{-x}} \approx \frac{1}{x} \text{ for } \ x = \frac{\omega\_{\text{rf}} T\_{\text{rev}}}{2Q} \ll 1\,\,\,\tag{8.33}$$

we then obtain

$$\begin{split} V\_b &= -2kq \left( \frac{2Q}{\omega\_{\rm rf} T\_{\rm rev}} \right), \quad k = \frac{\omega\_0}{2} \left( \frac{R}{Q} \right) \\ &= -2qf\_{\rm rev} R \\ &= -I\_b R \,. \end{split} \tag{8.34}$$

This simple result shows that with the cavity tuned to resonance, the beaminduced voltage is simply given by the beam current at the resonance frequency (Ib = 2Idc) times the cavity impedance. (Note that this is the loaded impedance, i.e., the combined impedances of the cavities, the connecting wave guides and rf power source, and the possible influence of rf feedback loops. The minus sign, again, indicates that the beam takes energy away from the accelerating cavity.

Application of (8.29 and 8.34) to the transport of high current particle beams is a subject of great interest in modern accelerators. In the extreme short-range limit (t<σzω/c), the variable t0 may represent the time interval between particles within a single bunch in which case, by causality, the charge q represents the charge of all preceding particles within the bunch. As the beam current is increased, eventually, as many experimental and theoretical studies have shown, the ensuing motion can become unstable. BNS damping in a linac is used, for example, to avoid the beam breakup instability associated with short-range wake fields.

At the SLC high beam currents were achieved using BNS damping and single, widely spaced bunches each of high charge (about 4 × 1010 particles per bunch). At such high bunch charges, the effects of long-range wake fields (t ∼ τbb, where τbb is the spacing between bunches) were also observed. Shown in Fig. 8.21 is a measurement [48] showing the orbit oscillation induced after changing τbb by 1 (of about 170) rf buckets. Using similar measurements with variable bunch spacings, the frequency of the driving mode could be roughly determined and compared with numerical models. Interestingly, the simplicity of the SLC (FODO) lattice and the different sign of accelerated particle species was such that the deflections due to the higher-order wake fields added coherently along the SLC linac. To minimize this resonant build-up, the symmetry of the focussing was changed using a so-called 'split-tune' lattice [48].

Fig. 8.21. Measured vertical difference orbit of the trailing electron bunch (solid line) after displacement of the leading positron bunch by 1 rf bucket (dashed line) in the SLC linac

Here the horizontal and vertical phase advances were changed so as not to be equal. In this way, resonant excitation of the leading positron bunch had less influence on the trajectory of the trailing electron bunch (the horizontal phase advance of the electrons equals the vertical phase advance of the positrons). The pulse-by-pulse jitter of the electron bunch was reduced [48] by about 15% horizontally (from 0.4σx to about 0.35σx) and 30% vertically (from 0.75σy to about 0.50σy).

The preferred way nowadays to achieve even higher beam currents, while avoiding intrabunch beam instabilities arising from increased single-bunch beam currents, uses multiple bunches (often called a bunch train) each with lower single-bunch beam currents. In this case, t0 in (8.29) refers to the spacing between appropriate bunches. Now the beam-induced voltage experienced by a particular bunch is given by the sum of the voltages induced by all preceding bunches each obtained by evaluating (8.29) at the appropriate time t0 = Nτbb, where N denotes the number of bunch spacings to the preceeding bunch of interest. In future linear collider designs with multi-bunch beams, research and development in the design of the accelerating structures aims towards minimizing the strength of the offending modes. An example,

Fig. 8.22. Horizontal (crosses) and vertical (diamonds) wake field measurements and predictions (solid lines) in the X-band structure test at ASSET at SLAC (courtesy C. Adolphsen, 2002)

from the Accelerator Structure Setup (ASSET) facility at SLAC, is shown in Fig. 8.22 [49]. Here again, the spacing between the drive and trailing bunches was varied and the particle orbits were recorded and analyzed to give deflection seen by the trailing bunch and so determine the strength of the transverse wake field.

In existing circular accelerators operating already with high current, multi-bunch beams, much effort has been devoted to carefully developing and testing new cavity and feedback designs. At KEKB novel ARES [50] energy-storage cavities, super conducting cavities, and multi-bunch feedback are used to minimize the effect of wake fields. At PEP2 higher-order mode dampers and both multi-bunch and rf feedback are used. While bunch-to-

Fig. 8.23. Measured beam currents and beam phases for the PEP-II electron and positron beams (Courtesy P. Corredoura, 2000)

Fig. 8.24. Measured phase difference between the PEP-II electron (HER) and positron (LER) beams (Courtesy P. Corredoura, 2000)

bunch stability at both accelerators is excellent, even at total currents in the range of Amperes, beam loading is still an issue.

As an example, measured phase changes across the bunch fill patterns are shown in Fig. 8.23 from the PEP2 B-factory. The bunch distribution was uniform with a 5% gap in the fill pattern. This nonuniformity in the fill results in a change in cavity voltage along the bunch train and, from (7.5), a change in synchronous phase. This phase variation leads to a reduction in gain of the feedback loops. In addition, the phase difference between the two trains must be minimized to ensure the desired (longitudinal) collision point and hence the highest possible luminosity. The difference in the phase transients of the two beams is shown in Fig. 8.24.

# 8.9 Multi-Bunch Energy Compensation

Two methods, known as Δf and Δt compensation have been proposed to combat multibunch phase transients in linear accelerators. Shown in Fig. 8.25 is the principle of Δt compensation [51]. Here the voltage Vk represents the voltage response of a finite bandwidth accelerating structure to a step function input pulse. The lower curve represents the beam-induced voltage Vb of the entire bunch train. By injecting the beam prior to the time the linac structure is at peak voltage, the vector sum Vk+Vb is observed to be flat over the duration of the bunch train. The projected energy spread is therefore minimized and the phase relationship between the bunches is constant.

The principle of the Δf compensation is illustrated [52] in Fig. 8.26. In this design from the ATF in Japan, some fraction of the many accelerating structures are slightly detuned by ±δf. The different bunches therefore obtain a different energy gain which depends on their location within the train.

Fig. 8.25. Conceptual diagram illustrating multi-bunch, Δt beam loading and energy compensation

Fig. 8.26. Conceptual diagram illustrating multi-bunch, Δf beam loading and energy compensation [53] (Courtesy J. Urakawa, 1999)

With some structures detuned by +δf and some by −δf the position-energy correlations introduced by the slope of the rf cancels. In this way, not only is the projected energy spread along the bunch train minimized, but the energy spread of each bunch is also preserved.

Beam loading compensaton using the Δt method may be advantageous since the correction may be applied locally at each accelerating section. On the other hand it is anticipated [51] that about 10% more power is required relative to the Δf compensation scheme.

# 8.10 Damping Partition Number Change via RF Frequency Shift

The generation of small emittance beams is a key issue for synchrotron light sources, collider rings, and for future linear colliders. Dedicated accelerators have been designed to produce such beams, but techniques to further reduce the design emittances would yield immediate improvements. At injection into such accelerators, the transverse beam emittances are large and often fill a large fraction of the dynamic aperture. For lepton beams, at later times after the beam has radiation damped, the horizontal damping time and equilibrium emittance may be reduced by shifting the rf frequency, such that the particle orbit moves inwards. By passing off-center through the quadrupoles in regions of nonzero dispersion, the horizontal partition number Jx is changed. This reduces both the horizontal damping time and equilibrium emittance. In addition, to the extent that the vertical emittance is determined by betatron coupling, the reduction in horizontal emittance may be accompanied by a corresponding reduction in vertical emittance.

The horizontal damping time and beam emittance are both inversely proportional to the horizontal partition number Jx = 1 − D, where [54]

204 8 Longitudinal Phase Space Manipulation

$$\mathcal{D} = \frac{\int D\_x G(G^2 + 2k) \,\mathrm{d}s}{\int G^2 \,\mathrm{d}s} \,\mathrm{.}\tag{8.35}$$

Here Dx is the horizontal dispersion, G = 1/ρ and k describe the inverse bending radius in m−1 and the quadrupole focusing gradient m−2, respectively, and the integrals are evaluated around the ring circumference. For the non-combined function magnets in the SLC damping ring, Gk ≈ 0.

For a beam orbit offset Δx in the quadrupoles, the change in D is given approximately by

$$
\Delta \mathcal{D} \approx \frac{2k^2 D\_{x,q} L\_q N\_q}{2\pi/\rho} \Delta x \,, \tag{8.36}
$$

where k = ec E ( ∂B ∂x ) with e = 1.6 × 10−19 C, c = 3 × 1010 m/s, E = 1.19 GeV, ρ is the local bending radius, Dx,q is the dispersion at the quadrupoles, and Lq and Nq are respectively the quadrupole length and number of quadrupoles.

The orbit may be offset in the quadrupoles by either changing the accelerating frequency or by physically displacing the magnet support girders. Emittance optimization using the accelerating frequency has been used in e+e− storage rings previously [55] and is used routinely at LEP [56]. The effect of changing the geometric ring circumference was already dicussed in Sect. 4.3.1. The circumference adjustment is applicable provided that the transverse acceptance is not limited and that the injected beam energy spread is small compared to the energy acceptance. At the SLC, the electron damping ring was 'stretched' [57] in 1992 by 9 mm for a 15% increase in Jx. In doing so, the energy aperture at injection was reduced yet without any loss in transmitted beam current. For the case of the positron damping ring, the incoming beam filled the entire aperture so stretching the accelerator was not an option.

Shown in Fig. 8.27 is a calculation of the horizontal emittance γ x as a function of time for 4 different frequency offsets for the case of the SLC damping rings. It is assumed that the beam is injected at the nominal rf frequency of 714 MHz with an initial emittance of 20 × 10−5 m-r. The accelerating frequency is increased after 1 ms (dashed line) for which the longitudinal emittance has damped by about a factor of 2. The simulations (using SAD [58]) with a trapezoidal approximation for the bending magnet fringe fields show a half unit reduction (i.e., 15–20%) in normalized emittance with a 100 kHz frequency change, while the damping time reduces from 3.4 ms to 3.0 ms.

At storage rings and colliders, there is no tight tolerance on maintaining the desired rf frequency. In a damping ring, the time required to reset the frequency and relock the beam phase to the desired extraction phase is critical since the frequency must be ramped back to nominal just before extraction in order not to introduce any energy and/or phase errors in downstream subsystems (in our example, the SLC bunch compressor and the SLC linac). Minimizing this time [59] is critical since reverting to the nominal rf frequency is associated with corresponding antidamping of the beam.

Fig. 8.27. Normalized horizontal beam emittance as a function of store time for different frequency shifts calculated for the SLC damping rings. The full store time is shown with an expanded view near beam extraction at 8.33 ms shown in the insert. The curves, when viewed from top to bottom, correspond to frequency shifts of 0, 50, 100, and 150 kHz, respectively

As a practical point, with a change in accelerating frequency, the accelerating cavities are detuned by an amount characterized by the tuning angle φz which is given by

$$\phi\_z = \tan^{-1} \left[ 2Q \left( \frac{f\_0 - f\_{\rm rf}}{f\_0} \right) \right], \tag{8.37}$$

where Q is the loaded cavity quality factor, f0 is the resonant frequency of the cavity, and frf is the frequency of the applied rf. With the cavity tuners fixed, the new tuning angle φz corresponding to the new applied rf frequency frf = frf + δfrf is given by

$$\tan \phi\_z \,' = 2Q \left[ 1 - (1 - \frac{1}{2Q} \tan \phi\_z) \frac{f\_{\rm rf} \,'}{f\_{\rm rf}} \right] \,. \tag{8.38}$$

An example is shown in Fig. 8.28(a). Typically, the tuning angle is set for minimum reflected power:

$$\left.\phi\_z\right|\_{\phi l=0} = -\frac{I\_b R}{\hat{V}} \sin\phi\_b \,, \tag{8.39}$$

where Ib is twice the dc beam current, R is the total loaded impedance, Vˆ is the total cavity voltage in units of V, and φb is the synchronous beam phase

Fig. 8.28. Cavity tuning considerations. The initial and final tuning angles at frf and frf = frf + 100 kHz, respectively, are shown in (a). The stability boundary for cavity voltage regulation at 600 kV is given by the two solid curves in (b). The dashed curves show the loading angle φl which is measured and regulated by the tuner feedback loops. At zero beam current the loading angle is equal to the tuning angle

measured with respect to the crest of the accelerating rf. The loading angle φl is related to the tuning angle φz by

$$\tan \phi\_z = \left( 1 + \frac{I\_b R}{V\_c} \cos \phi\_b \right) \tan \phi\_l - \frac{I\_b R}{V\_c} \sin \phi\_b \ . \tag{8.40}$$

In the case of the SLC, at zero beam current the cavity detuning exceeded the power capabilities of the power source as indicated by the lower solid curve in Fig. 8.28(b).

Assuming sufficient frequency aperture, which may be restricted, e.g., by transverse betatron resonances that can be encountered during the frequency ramp with nonzero chromaticity or by a physical aperture in a region with nonzero dispersion, the maximum frequency change may be limited by either available rf power as discussed above or by the damping poles [60] at which the damping rate becomes zero.

Fig. 8.29. Nominal emittance at injection to the SLC linac without an rf frequency shift in the upstream damping ring. The measured normalized emittance was 3.30 ± 0.07 m

Fig. 8.30. Normalized emittance measurement at injection to the SLC linac with a 62.5 kHz frequency shift in the upstream damping ring. The normalized emittance was reduced from 3.30 ± 0.07 m to 2.66 ± 0.06 m

#### 208 8 Longitudinal Phase Space Manipulation

Measurements [61] showing the effect of a frequency ramp in the SLC positron ring are given in Figs. 8.29 and 8.30. In this experiment, the downstream compressor was turned off in order to more cleanly detect the effect of the frequency shift on the beam emittance. From Fig. 8.30, with a 62.5 kHz shift, the reduction in normalized emittance was from 3.30±0.07 m to 2.66 ± 0.06 m. For the electron damping ring, the frequency shift reduced the horizontal normalized emittance from 3.22 ± 0.08 m to 2.93 ± 0.07 m. We expected that the emittances at the entrance of the final focus would be reduced by a similar amount to γ x = 5.2 × 10−5 m and γ y = 1.1 × 10−5 m, and, using the 1997 SLC interaction point parameters with rms angular divergences of θx = 450 μrad and θy = 250 μrad, the corresponding rms beam sizes at the collision point were estimated to be σx = 1.3 μm and σy = 0.5 μm. With a 1 mm bunch length and 4 × 1010 particles per bunch, the luminosity was computed by the code Guinea-Pig [62] to be 4.3 × 1032 m−2 per collision. This corresponded to an estimated increase of over 40% in luminosity by application of the frequency shift in the damping rings. Unfortunately, no rigorous study was undertaken to experimentally quantify the effect of the rf frequency shift on the SLC luminosity.

Decreasing the beam emittance by changing the damping partition numbers is also part of the HERA luminosity upgrade [63]. To maintain matched beam sizes, with a reduced proton beam size resulting from the modified optics at the interaction points of HERA, the electron beam size must be reduced. The approach that was taken is twofold: stronger focussing in the arcs, which reduces the horizontal equilibrium emittance, and a +200 Hz rf frequency shift, which changes the damping partition numbers.

# Exercises

### 8.1 Phase Tolerances in a Bunch Compressor

Consider a 1.2 GeV electron beam passing through a 2856 MHz compressor cavity and a transport line with R56 = −0.6 m. Sketch the error in the final phase as a function of injection phase error for different accelerating voltages. At which compressor voltage is the final beam phase, behind the compressor region, minimially sensitive to initial phase errors?

### 8.2 Bunch Precompression

a) Describe (or sketch) the motion of the beam centroid in longitudinal phase space for the following process:

$$\begin{aligned} \hat{V} &= V\_0 \quad \text{for} \quad t < t\_0\\ &= 0.75V\_0 \quad \text{for} \quad t\_0 < t < t\_1 = \frac{\tau\_{s,l}}{8} \\ &= V\_0 \quad \text{for} \quad t\_1 < t < t\_2 = \frac{\tau\_{s,h}}{4} \end{aligned}$$

$$t = 0.75V\_0 \quad \text{for} \quad t\_2 < t < t\_3 = \frac{\tau\_{s,l}}{8}$$

$$t = V\_0 \quad \text{for} \quad t\_3 < t < t\_1,\tag{8.41}$$

where τs,l is the synchrotron period at the lower voltage and τs,h is the synchrotron period at the higher voltage.

b) Taking into consideration a multiparticle beam, describe the particle distribution in phase space during the process given by comparing initial and final states. Show that the the bunch length is compressed (at the expense of increased energy spread).

### 8.3 Harmonic Cavities

a) Verify the equations given in 8.20.

b) Sketch the relative phase between the two rf systems of Fig. 8.10 as the ratio of radiative losses to primary rf voltage varies from zero (proton beam limit) to slightly less than one.

### 8.4 Minimum Voltage Required for Beam Storage

The power Pγ radiated due to synchrotron radiation per turn by an electron or positron may be expressed [54] as

$$P\_{\gamma} = \frac{cC\_{\gamma}}{2\pi} \frac{E^4}{\rho^2} \,, \tag{8.42}$$

where c is the speed of light, Cγ = 8.85 × 10−5 m-GeV−3, E is the beam energy in GeV, and ρ is the local radius of curvature of the bending magnets.

a) For an accelerator (without insertion devices) with ρ = 2 m and E = 1 GeV, what is the total radiated power for 1011 particles?

b) With a 100 ns particle revolution period, at what voltage could the beam no longer be captured? What is the synchronous phase at this voltage? Assume that there are no other energy loss sources.

c) For low current beams, the electron bunch length scales with the total accelerating voltage V as V − 1 2 . What is the disadvantage of lowering the cavity voltage for increased bunch length compared with the use of harmonic cavities?

### 8.5 Phase Shift along a Bunch Train

The cavity fill time τf describes the time evolution of the cavity voltage in response to a step function. For example, if a cavity initially at amplitude V0 has suddenly its power source turned off, then the cavity voltage decays as

$$
\hat{V}(t) = V\_0 e^{-\frac{t}{\tau\_f}}, \quad \text{where} \quad \tau\_f = \frac{2Q}{\omega\_{\text{rf}}}.\tag{8.43}
$$

For the case of a storage ring with fast feedback, estimate the change in synchronous phase across a 500 ns long bunch train of 100 mA average beam current along the train, and cavities with a loaded Q of Q = 3000, a total impedance of 5 MΩ, an rf frequency of 476 MHz, and an external rf voltage of 10 MV.

# 9 Injection and Extraction

In transferring the beam from one accelerator to another, preservation of the beam properties is essential. Injection should be accomplished with minimum beam loss and often minimal emittance dilution. Single-turn injection, in which a single bunch of particles is injected into a single empty rf bucket, is usually straightforward. In many cases, however, to attain higher bunch currents, one may also wish to accumulate beam in a storage ring by reinjecting different beam pulses into the same rf bucket. This is called multi-turn injection. In addition to conventional schemes, there are several new or more exotic injection techniques, devised to control and improve the properties of the stored beam.

Extraction refers to the removal of beam from an accelerator. It is roughly the reverse process of injection. One difference is that usually at extraction the beam energy is higher. Thus space charge effects are less important, but the hardware requirements for the septa and kicker magnets are more challenging. A high extraction efficiency is necessary to avoid activation of accelerator components and also to make optimum use of the accelerated beam, e.g., to achieve the maximum luminosity. Which extraction procedure is chosen depends on the specific application. Fast one-turn extraction is used for transferring bunches between different circular machines in an accelerator chain. For fixed-target experiments, slow extraction by the controlled excitation of nonlinear betatron resonances is a common technique, which provides a slow uniform depletion of particles in the ring, i.e., 'spill'. Again, several novel extraction schemes are being studied, for example, extraction using a bent crystal.

A good overview of conventional beam injection and extraction can be found in [1] and [2].

# 9.1 Transverse Single-Turn Injection

For single-turn injection, the beam is brought onto the central orbit using a septum magnet and a fast kicker element, as illustrated in Fig. 9.1. In the following, we assume that the injection is performed horizontally. The expressions derived can be extended easily to the vertical case, or to a combined horizontal and vertical injection.

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3\_13

Fig. 9.1. Schematic of single-turn injection with septum and kicker: (top) magnet arrangement, (bottom) phase-space diagram. The septum strength is adjusted to place the beam onto the line describing an angle μ (the phase advance from septum to kicker) with respect to the vertical [1]

At the exit of the septum, the injected beam must be at a horizontal distance xsep from the center of the machine aperture:

$$x\_{\rm sep} \ge n\_{xi}\sigma\_{xi} + n\_{xs}\sigma\_{xs} + D\_x\left(\frac{\Delta p}{p}\right)\_i + x\_{\rm co,rms} + x\_{i,\rm rms} + d\_{\rm sep} \,, \tag{9.1}$$

where σxi, σxs are the rms beam sizes of the incoming beam and of the stored beam, respectively, nxs is the beam stay-clear in the ring required for sufficient beam lifetime or negligible injection losses in units of the rms beam size of the stored beam (e.g., reasonable values may be nx ≥ 8 for electron rings, and nx ≥ 4 for proton rings), nxi is the beam stay-clear for the injected beam also in units of its rms beam size, (Δp/p)i the relative momentum deviation of the injected beam with respect to the ring energy, Dx the dispersion function, xco,rms the rms closed-orbit offset at the location of the septum, xi,rms the rms orbit variation of the injected beam, and dsep the thickness of the septum. For simplicity in (9.1) we have assumed that the injected and the stored beam have the same beam stay-clear, in units of their rms beam size.

The injected beam must be positioned at the center of the aperture when it reaches the kicker. Let R denote the 2 × 2 transport matrix between the septum and the kicker. Requiring that the beam be on-axis at the kicker, the condition xkic = R11xsep + R12x sep = 0, where xkic denotes the position of the beam with respect to the center of the kicker magnet, determines the correlation of angle x sep and offset xsep of the injected beam at the exit of the septum:

$$x'\_{\rm sep} = -\frac{R\_{11}}{R\_{12}} \ x\_{\rm sep} = -\frac{\alpha\_{\rm sep} + \cot \mu}{\beta\_{\rm sep}} \ x\_{\rm sep} \ , \tag{9.2}$$

where μ denotes the phase advance from septum to kicker, and αsep and βsep are the beta and alpha function at the septum. The angle x sep can be adjusted by changing the strength of the septum magnet, in order to meet condition (9.2).

Using (9.2) the slope of the injected beam at the kicker is

$$x\_{\rm kic}^{\prime} = R\_{21}x\_{\rm sep} + R\_{22}x\_{\rm sep}^{\prime} = \left(R\_{21} - \frac{R\_{11}}{R\_{12}}R\_{22}\right)x\_{\rm sep} = -\frac{1}{R\_{12}}x\_{\rm sep}.\tag{9.3}$$

To position the injected beam on the design orbit in the ring, the kicker must apply the opposite angular deflection, which is

$$\theta\_{\rm kic} = -x\_{\rm kic}' = \frac{x\_{\rm sep}}{\sqrt{\beta\_{\rm sep} \beta\_{\rm kic}} \sin \mu} \,\,\,\,\tag{9.4}$$

A large value of the beta function at the kicker, βkic, reduces the kicker strength, and a large value of βsep reduces the relative contribution to θkic which arises from the septum thickness dsep (see (9.1) and recall that σxi, σxs ∝ βsep).

In case of a FODO lattice, the septum and kicker are best placed downstream of a focusing quadrupole, where the beta functions are close to maximum. In the particular case that the phase advance μ is π/2, the above formulae simplify to x sep = −αsepxsep/βsep, θkic = xsep/ βsepβkic.

The septum may consist of either dc septum magnets or dc electrostatic wires. In either case, the stray or leakage fields of the septum are a concern. These nonlinear fields can affect the quality of the stored beam. The stray fields may be reduced by a magnetic shielding, which extends beyond the septum ends.

The kicker magnets must be fast, since their rise and fall times often determine the minimum size of the gaps between bunches or between bunch trains in the ring, which in turn may depend on the repetition rate of the accelerator. Typical time constants are tens of nanoseconds, and voltage and current levels of 80 kV and 5000 A, with fields of 500 Gauss, are not uncommon. Frequently ferrites are used for field containment, and sometimes a ceramic vacuum chamber is inserted between ferrite and the beam, with a conducting layer deposited on the inside of the ceramic. The coated ceramic reduces the impedance seen by the beam, as well as the associated heating of the ferrite. It is remarkable that the conducting layer can be much smaller than the skin depth and still provide adequate shielding of the beam fields, since shielding occurs already when the thickness of the metal coating is larger than the square of the skin depth divided by the thickness of the ceramic [3]. The coupling impedance experienced by the beam should be measured prior to installation of the kicker chamber. As an example, measurements for a prototype LHC kicker chamber are documented in [4]. A non-flat shape of the kicker pulse results in unequal deflections for different bunches in the beam. If necessary, a double-kicker system can relax the tolerance on the pulse shape (see the discussion of the KEK/ATF extraction scheme in Sect. 9.7).

Other important injection issues include transient beam loading and phase-space matching. Reference [5] gives a thorough review of beam loading compensation in storage rings, including a discussion of direct rf feedback and of problems that can arise from klystron power limititions. Horizontal and vertical dispersion and beta functions, as well as the ratio of bunch length and energy spread must be matched to the ring (or linac) optics. The transverse optics is matched by varying the strength of quadrupole magnets in the injection line, and, in case of dispersion, possibly also the strength of steering magnets. The longitudinal matching can be achieved by optimizing the amplitude of the storage-ring rf voltage, or, in certain cases, via bunch rotation, bunch compression, or energy compression prior to injection (cf. Sect. 8).

# 9.2 Multi-Turn Injection

For many applications, the beam is first accumulated in a ring, to increase its intensity, before it is further accelerated or sent towards its final destination. Typically a current-limited cw beam from the injector is thereby converted into a pulsed beam at higher-intensity and energy. The accumulation requires multi-turn injection.

### 9.2.1 Transverse Multi-Turn Injection

Multi-turn injection usually employs a ramped orbit bump in the vicinity of the septum, i.e., a slow change in the position of the ring closed orbit from turn to turn, instead of a fast kicker. The injection scheme is different for electrons and for protons or heavy ions.

In case of electron rings, radiation damping is utilized. First, a single bunch is injected. Then the orbit bump is reduced over a few revolution periods. After a few damping times, when the beam size has shrunk to its small equilibrium value, the orbit bump is reintroduced, and another bunch is injected into the same bucket. Similar schemes, though usually in the synchrotron phase space (see below), are employed for accumulation of electron cooled proton beams and stochastically cooled antiproton beams.

For proton or heavy ions beams, the orbit bump is reduced slowly in time, and bunches are injected into different regions of the ring acceptance, so that the early bunches occupy the central region, and the later ones the outer parts of the acceptance. Some emittance dilution is inherent to this scheme. If beam is injected over Nt turns ('Nt-turn injection'), the final emittance can be estimated from the rough formula [1]:

$$
\epsilon\_f > 1.5 N\_t \epsilon\_i \, . \tag{9.5}
$$

Much larger emittance dilutions arise at low beam energy or high intensity, when space charge effects are important.

Fig. 9.2. Injection process for PEP-II [6] (Courtesy M. Donald, 2002)

### 216 9 Injection and Extraction

It is also possible and indeed more elegant, to install two kickers in the ring, powered in parallel by the same pulser. They are arranged such that only the second kicker of the pair deflects the injected beam, while both kickers act on the stored beam. If the phase advance between the two kickers is π or 2π, and the sign of the kick appropriately chosen, the kicker deflections generate a closed bump for the circulating beam. The advantage is that in this case the rise and fall times of the kicker do not have to be smaller than the bunch spacing, but can be on the order of the revolution time. The requirements on the kickers can be further alleviated by a dc orbit bump, which brings the stored beam closer to the septum prior to the injection. Figure 9.2 illustrates such a scheme, which is used at PEP-II [6].

### 9.2.2 Longitudinal and Transverse Multi-Turn Injection

The accumulation efficiency can be increased by combining transverse and longitudinal injection. This option was studied for LEAR [7], where the injected bunches come from a linac, which allows for an easy variation of their

Fig. 9.3. Plots of the simulated horizontal phase space for multiturn injection into LEAR [7]: (top) 10 turns after start of purely transverse injection and (bottom) 20 turns after start of combined longitudinal transverse injection. Each bunch is represented by three ellipses with slightly different momentum deviations. The two vertical lines on the right represent the thickness of the septum (Courtesy Ch. Carli, 2002)

energy. As in the purely transverse multi-turn injection scheme, a local orbit bump is created and then decreased during the injection of successive bunches. At the same time the linac energy is ramped such that, at the injection septum, the closed orbit corresponding to the instantaneous linac energy remains constant. In other words, the change in the bump amplitude xbump(t) and the simultaneous variation of the momentum δ(t) are related by

$$D\_x \delta(t) = -x\_{\text{bump}}(t) + x\_0 \,, \tag{9.6}$$

where x0 is a constant and Dx the horizontal dispersion at the injection septum. In this scheme, the final transverse emittance is smaller than for the conventional multi-turn injection at the expense of an increased momentum spread. Figure 9.3 compares simulated phase space distributions [7] for transverse and combined injection into LEAR. Figure 9.4 shows the predicted improvement in the accumulation efficiency [7].

Fig. 9.4. Effective number of turns neff stored in LEAR as a function of the number of injected turns ninj [7]. The solid lines represent the combined transverse longitudinal injection scheme, for three different LEAR optics (crosses: standard optics; triangles: improved optics with large beta functions at the electron cooler; squares: improved optics with intermediate beta functions at the electron cooler). For comparison, the dashed line is for a purely transverse injection and the standard optics (Courtesy Ch. Carli, 2002)

### 9.2.3 Longitudinal Multiturn Injection

If for an electron ring the time between subsequent injections is short compared with the radiation damping time, multiturn transverse injection becomes difficult. In such a case, longitudinal injection offers a solution. Here the circulating beam is brought close to the septum with an ac bump and the incoming beam is injected with a negative energy offset such that the product of this offset in energy and the dispersion is equal to the distance between

#### 218 9 Injection and Extraction

the newly injected and the stored beam. The injected bunches execute slow synchrotron oscillations.

Consider as an example the injection scheme employed at LEP [8]. Half a synchrotron period after the first bunch is injected, the next injection occurs. At this time, the first bunch is at its maximum distance from the septum. The situation is illustrated in Fig. 9.5. Similarily one could conceive injecting every 1/4 oscillation period, thus accumulating 4 injected bunches in one rf bucket. An advantage of longitudinal injection is a factor two faster radiation damping of the injection oscillations, since the longitudinal damping partition number is twice the transverse (Jz ≈ 2Jx; compare Sect. 4.3). A possible disadvantage is that the acceptable time separation of successive injections is constrained by the synchrotron frequency.

Fig. 9.5. Double injection into the same rf bucket; bunches are injected off-energy at a point with dispersion [8]. Time between the two injections is half a synchrotron oscillation period modulo a full period (Courtesy P. Baudrenghien and P. Collier, 2001)

### 9.2.4 Phase-Space Painting

For proton and ions beams, the multi-turn injection is often described as "phase-space painting" [9, 10]. This term refers to the injection of many small (linac) bunches into different spots of a 2- or 6-dimensional storage-ring phase space, so as to generate a desired beam distribution, e.g., an approximately uniform distribution with reduced space-charge effects. The combined longitudinal-transverse injection into LEAR discussed in Sect. 9.2.2 can be considered as an example of phase-space painting. In the simplest case the beam is injected at a fixed position in the longitudinal (or transverse) phase space, and the painting is done automatically by the synchrotron oscillations.

During the injection process, the position of the injected beam in phase space can also be moved adiabatically, i.e., at a speed which is slow compared with the synchrotron oscillations. The injected phase-space density P(r) and its projection p(x) are related via

$$p(x) = 2\int\_{x}^{R} \frac{rP(r) \, \mathrm{d}r}{\sqrt{r^2 - x^2}} \,, \tag{9.7}$$

where R denotes the maximum radius in phase space at which bunches are injected. From a desired function p(x), the corresponding radial density P(r) can be computed using (9.7). The radial increment in the injected beam position between two successive bunches is [9]

$$
\Delta r \approx \frac{1}{2\pi r P(r) N\_{\rm inj}}\,,\tag{9.8}
$$

with Ninj the total number of injected bunches.

More complicated schemes are frequently used. Similar to the above horizontal-longitudinal injection for LEAR, one can also combine horizontal and vertical painting. For example, at Rutherford Appleton Laboratory (RAL) a vertical steering magnet in the injection line is ramped, while the guide field in the ring is decreased [10]. Initially, there are small horizontal and large vertical oscillations, while at the end of the injection the situation is reversed. Instead of the vertical steering in the injection line, a programmable vertical orbit bump in the ring could be employed alternatively.

# 9.3 H Charge Exchange Injection

The principle of injection using H− exchange originated in Novosibirsk [11]. It is now the preferred injection scheme for proton machines [1]. In this scheme H− ions are accelerated by a linac and are stripped to protons, when they traverse a thin foil during injection into the ring [1, 12], as illustrated in Fig. 9.6.

The stripping of the H− ions to protons occurs within the ring acceptance. Since during the stripping the particles change their charge, Liouville's theorem on the conservation of the beam density in phase space does not apply, and, thus, in principle, a high proton density could be attained by injecting successive bunches into the same region of phase space. In most practical applications, however, vertical steering in the injection line is combined with a

Fig. 9.6. Schematic of H− stripping injection

ramped horizontal orbit bump in the ring in order to provide a very uniform filling of the phase space and to minimize space-charge effects.

The heating of the stripping foil and stripper scattering effects limit the foil thickness. Typical thicknesses range between 50 and 200 μg cm−2 (less than 1 μm), with stripping efficiencies of 98% for 50 MeV protons [1]. As foil materials, polyparaxylene, carbon and aluminum oxide have been used. The rms scattering angle for a single foil traversal is typically on the order 0.2 mrad. The total scattering angle increases as the square root of the average number of passages through the foil. The stripping foils are supported at three edges, so that vertical beam motion cannot does not reduce the number of foil traversals.

Stacking simultaneously in both betatron and synchrotron phase space reduces the number of foil traversals, however. This can be achieved by changing the magnitude of an orbit bump, while also ramping the frequency and phase of the rf system during the injection cycle. The following lattice parameters at the location of the foil are considered advantageous [1]: D x = 0, αx = 0, and D(Δp/p) > √Aβ, where Δp/p is the momentum acceptance and A is the transverse acceptance (in emittance units). A location between two symmetric defocusing quadrupoles is suitable for meeting these conditions. The second quadrupole assists in deflecting the unstripped H− ions.

Stripping foils are used for heavy ions as well. The final charge distribution of the ions depends on the foil thickness and on the particle energy [13]. In the extreme case, the ions can be fully stripped.

Foils are also used for extracting H0 atoms or protons from H− storage rings, where the neutral atoms are generated by capturing electrons from the foil. For these applications, also a gas jet or a laser beam [14] can facilitate the extraction in a similar way.

# 9.4 Resonant Injection

Another injection scheme worth mentioning is a proposal for resonant injection [15]. Here 'bumper' magnets with dipole, quadrupole and octupole fields are excited to produce a separatrix with two stable regions in phase space; the stored beam is in one region, and a bunch is injected into the other region. Afterwards, the fields are adjusted to merge the two parts of the beam. Then the injection condition is reestablished.

# 9.5 Continuous Injection

Continuous injection has been proposed as a means for maximizing the luminosity of a circular collider [16]. The motivation is obvious; if the stored beam could be continually replenished so that the current per bunch stays constant, then the average luminosity would roughly equal the peak luminosity. Continuous injection also reduces fill-to-fill variations and avoids transient phenomena, e.g., transient beam loading, thereby establishing quasi-static conditions, which is of interest not only for colliders, but also for light sources. Finally, in a colliding-beam storage ring the beam lifetime τ decreases inversely with the luminosity L, 1

$$\frac{1}{\tau} = -\frac{1}{N} \frac{\mathrm{d}N}{\mathrm{d}t} \propto \frac{L}{N} \,. \tag{9.9}$$

Thus, continuous injection supporting a much reduced lifetime could provide a substantial gain in average luminosity.

As an example, taking all these effects together, continuous injection is estimated [16] to potentially increase the average luminosity of the PEP-II B factory by about a factor of 5, assuming that each bunch in both rings can be replenished every 2.1 s. In this example, a 67-ns long orbit bump would move the injected bunches transversely to about 4σ from the stored beam core. This is done so that the injected bunches have an unobstructed passage through the physics detector. The effective minimum beam lifetime which can be supported is given by [16]

$$
\tau = \frac{N\_b}{\Delta N\_b} \,\,\Delta t\,\,\,\tag{9.10}
$$

where Nb is the nominal number of particles per bunch, ΔNb is the number of particles added into a single bunch per injection, and Δt the time period between successive injections into the same bunch. Extreme parameters for PEP-II are [16] Nb ≈ 1.2 × 1011, ΔNb ≈ 109, and Δt ≈ 2.1 s, yielding a minimum supportable beam lifetime of τ ≈ 4.2 minutes.

# 9.6 Injection Envelope Matching

At injection into a storage ring, if the incoming beam distribution is not properly matched to the ring optics, the beam envelope in phase space will rotate around the matched design envelope. This oscillation will result in turn-to-turn beam-size variations, which can be measured using a synchrotron light monitor and a fast-gated camera.

An injection-mismatch measurement from the SLC damping ring [17, 18] is shown in Fig. 9.7. The different pictures correspond to successive turns after injection, at the indicated turn number. Each picture is an average over 8 individual images. Clearly visible is a variation of the bunch shape from turn to turn.

Fig. 9.7. Beam images of the first twelve turns after injection into the SLC damping ring, illustrating the effect of an injection mismatch [17, 18]. These are pictures from a synchrotron light monitor taken with a gated camera. Each image is an average over 8 beam pulses. The beam-size variation from turn to turn is an indication of injection mismatch

At the SLC damping ring, the matching of the injected beam distribution consists of minimizing the measured beam size after 1250 turns, by varying several quadrupoles at the end of the injection beam transport line. A number of 1250 turns was chosen, because at this time the initial beta and dispersion mismatch has completely filamented. Since, on the other hand, the time scale is much shorter than the radiation damping time, the emittance is given directly by (Bmag · ), where is the emittance of the injected beam, and Bmag the mismatch factor defined in (4.59).

The beam size variation can be analyzed in the frequency domain by a Fourier transform to determine the amplitude of the mismatch factor and ultimately, for a well calibrated monitor, the beam emittance at injection. A beta mismatch will appear as a frequency line at twice the betatron tune, while a horizontal dispersion mismatch will be evident as a line at the betatron tune itself [18]. If only a beta mismatch is present, the ratio ρ of the dc Fourier component and the component at 2Qx is equal to (Bmag/ B2 mag − 1). From this, Bmag = 1/ -1 − ρ−2 can be determined [18, 19].

Figure 9.8 shows the beam size squared for the first 100 turns after injection, as well as the FFT (multiplied with γ/βx where γ is the relativistic Lorentz factor and βx the beta function). Clearly visible are peaks at 2Qx in the horizontal signal and at (1−2Qy) in the vertical one. The final emittance after filamentation, (Bmag · ), is given by the dc component of the FFT.

Fig. 9.8. Horizontal (top) and vertical (bottom) beam sizes for the first 100 turns after injection into the SLC damping ring (left) and their FFT (right) [18]. Clearly visible in the frequency spectra are lines at 2Qx (top) and at (1 − 2Qy) (bottom), whose amplitude is a measure of the amount of beta mismatch

Fig. 9.9. Same as Fig. 9.8, after beta matching [18]. The peaks at twice the betatron tune have disappeared

The standard matching procedure reduces the FFT signals at 2Qx, (1 − 2Qy) and Qx, as illustrated in Fig. 9.9.

# 9.7 Fast Extraction

Fast extraction is similar to single-turn injection. Orbit bumps are generated which move the stored beam close to a septum magnet. Then a fast kicker is fired, which deflects the next bunch, or group of bunches, into the extraction channel. If only one kicker is used, the kicker rise time must normally be smaller than the separation between two circulating bunches (or groups of bunches). The pulse length and fall time of the kicker are determined by the number of bunches to be extracted, and by the ring fill pattern. From (9.4), the minimum deflection angle required is

$$\theta\_{\rm kic} = \frac{x\_{\rm sep}}{\sqrt{\beta\_{\rm sep} \beta\_{\rm kic}} \sin \mu} \,, \tag{9.11}$$

where βsep and βkic are the beta functions at septum and kicker, μ is the phase advance between these two elements, and xsep is the minimum displacement at the septum required for clean extraction. The initial orbit bump reduces the value of xsep. In a FODO lattice with a phase advance per cell of, e.g., 90◦, the kicker can be positioned just upstream of the focusing quadrupole, and the septum at the identical position one cell downstream. In this way, the beta functions are maximized, and so the deflection angle required for the kicker is minimized.

For extraction from the damping ring of a linear collider, it is extremely important that the deflection imparted by the kicker has a very small pulseto-pulse fluctuation ('jitter') and is sufficiently flat over the length of a bunch train. In order to confine the orbit variation at the interaction point (IP) to 0.1σ∗ (σ∗ is the IP spot size), the orbit variation at the septum should be smaller than 0.1σsep. The tolerance on the relative deflection error then is Δθkic/θkic < 0.1σsep/xsep, where σsep is the rms beam size at the septum, and xsep the transverse displacement of the kicked beam. Without fast orbit bumps, this can also be rewritten as [20]

$$\frac{\Delta\theta\_{\rm kic}}{\theta\_{\rm kic}} \le \frac{1}{10} \frac{\sqrt{\epsilon\_{\rm ext} \beta\_{\rm sep}}}{d\_{\rm sep} + n\_s \sqrt{\epsilon\_{\rm inj} \beta\_{\rm sep}}},\tag{9.12}$$

where inj and ext are the injected and extracted beam emittances, βsep the beta function at the septum, and ns the distance between the closed orbit and the septum plate in units of the injected rms beam size, when the beam is largest (i.e., the injected beam size enters, because the aperture at the septum must be large enough to accommodate the injected beam). For electron rings, one must have ns ≥ 7. Using typical parameters for a linear-collider damping ring, the relative jitter tolerance for the kicker, Δθkic/θkic, is on the order of a few 10−4, and it is mainly determined by the ratio of the extracted beam emittance to the injected emittance [20]. A possible solution is the use of a double kicker system, separated by a betatron phase advance of π, to cancel the jitter [20, 21]. This compensation scheme is illustrated in Fig. 9.10. One kicker would be placed before the septum and the other in the extraction line. If a pulser feeds both kickers in parallel, with appropriate cable delays, kicker pulse errors in the first magnet are canceled by those in the second

Fig. 9.10. Damping ring extraction with double kicker system, for reducing downstream beam orbit fluctuations. A change in the pulse shape alters the deflection from both kickers equally. The R matrix between the two kickers and cable delay times are chosen such that the effects of the two deflection errors cancel exactly, and the final beam trajectory is unchanged

magnet. A double-kicker system of this type has been built and installed at the KEK ATF damping ring [22].

Similar techniques can be applied to compensate for drifts of the septum field. For example, the NLC design contemplates the use of a compensating bending magnet in the extraction line, which is powered in series with the septum and placed such that field fluctuations will cancel [21].

# 9.8 Kickers

There are several different types of kicker magnets [6, 23] such as: (1) a current loop inside the vacuum, (2) a terminated transmission line inside the vacuum, (3) a ferrite magnet outside the vacuum, and (4) a multi-cell transmission line with ferrite flux returns [21]. As an example, Fig. 9.11 shows the ferrite kicker and the kicker pulser circuit adopted for PEP-II [6].

Fig. 9.11. Schematic of PEP-II kickers [6]: (top) kicker magnet cross section; (bottom) pulsing circuit with FET switch (Courtesy M. Zisman, 2002)

Typical kicker rise and fall times are 50–150 ns (SLC and NLC design: 60 ns, PEP-II: 120 ns). For a fast horizontal kicker with ferrite yoke, the characteristic time constant of the kicker scales as lw/g, where l is the length, w the width, and g the vertical gap of the kicker. This time constant can be reduced by dividing the kicker into several segments of shorter length. The kicker magnets are powered by kicker pulsers, usually based on thyratron cable discharges. The pulse shape can be modified by adding filters and capacitors in parallel with the charge line. Spark gaps and solid-state FETs, such as in Fig. 9.11, are thyratron alternatives with potentially shorter rise and fall times [21].

For many future applications with closely spaced bunch trains, shorter kicker time constants are desired. A very fast counter-travelling wave kicker was designed and built for the TESLA project [24]. This kicker scheme uses two parallel conducting plates or electrodes. These are excited by short pulses from a generator, generating an electromagnetic wave which travels in a direction opposite to the beam, and produces a horizontal kick. At the end of

Fig. 9.12. Very fast kicker prototype [24]: (top) layout; (bottom) measured output rf pulses (Courtesy V. Shiltsev, 1998)

the kicker plates, the wave passes into two ceramic outputs, in which ideally it is fully absorbed without reflection. For a beam travelling opposite to the wave the effects of the magnetic and electric forces add, while they cancel each other for a beam moving in the same direction. The maximum integrated kicker strength in units of voltage is given by

$$S\_0 \approx \frac{2U\_ml}{a} \,, \tag{9.13}$$

where Um is the maximum pulse voltage at each plate, a is the half aperture, and l the length. A kicker was tested with Um = 2 kV, a = 25 mm, and total length l of 0.5 m. Figure 9.12 shows output pulses measured on this kicker prototype, demonstrating a zero-to-zero pulse length of about 6 ns. The maximum pulse height corresponds to the predicted kick strength of

Fig. 9.13. Beam-beam kicker: 'head-on' (top) and 'cross' scheme (bottom) [25] (Courtesy V. Shiltsev, 1998)

80 kV, or 2.76 Gm. For electron or positron beams with E = 3.3 GeV, this would amount to a deflection angle of 24 μrad.

An ultrafast beam-beam kicker was proposed [25], in order to provide even faster kicker pulses. Here, a wide high-charge low-energy bunch traverses the beam pipe either parallel to the beam direction or perpendicular to it. The electro-magnetic or electric field of this bunch is used to deflect (and extract) a bunch circulating in the ring. The pulse length of the beam-beam kicker is determined by the length of the low-energy bunch and can be on the order of 2 ns. Figure 9.13 illustrates two possible geometries.

# 9.9 Septa

As we have seen in (9.1) and (9.4), a small septum thickness dsep reduces the requirements on the kicker and increases the extraction efficiency. For this reason, electrostatic wire septa have been employed since many years, for example, during fast extraction at the Fermilab Tevatron [26]. The Tevatron electrostatic septum consists of two 354 cm long sections with 86 cm space in between, made from 75% tungsten and 25% rhenium wires of 0.002 inch diameter and 0.1 inch spacing with an angle of 25 μrad between sections. The voltage of 93 kV results in an electric field of 83 kV/cm [26]. Very similar electrostatic deflectors have been proposed for the muon collider [27]. At high energies, the integrated strength of a wire septum often cannot provide a deflection angle large enough for clean extraction, and, in such cases, an additional thin septum magnet is positioned immediately downstream.

In general, two types of septum magnets are widely used [28]: Lambertson iron septum dipoles and current-carrying septum dipoles. A Lambertson magnet is illustrated in Fig. 9.14. The triangular cut-out in the window frame leaves space for the circulating beam. As shown, a kicker deflects the beam horizontally into the septum, by which it is then bent vertically.

Figure 9.15 depicts a current sheet septum. A current carrying septum with thickness d and current density J generates a field B = μ0Jd. For d of the order of a millimeter, the septum is used in a pulsed mode to provide

Fig. 9.14. Schematic of Lambertson septum iron magnet [2]; the symbol c represents the circulating beam, the symbol e the extracted beam

Fig. 9.15. Cross section of current sheet septum [6] (Courtesy M. Zisman, 2002)

enough field strength. For larger thicknesses, dc operation is common. Septum leakage fields which affect the circulating beam are a concern. In addition to normal dipole and higher order fields, the septum stray field may contain a skew quadrupole component.

# 9.10 Slow Extraction

The beam can be slowly extracted by exciting a third order nonlinear resonance, using sextupoles. Also a second order linear resonance can be used, in combination with octupoles. The extraction efficiency depends on the ratio of the betatron amplitude growth per turn and the septum thickness. It can be improved with a high-beta insertion at the septum.

Figure 9.16 depicts the phase space near the 3rd order resonance, excited by sextupole magnets. Particles inside the inner triangle are stable. Outside the triangle the oscillation amplitude of a particle grows exponentially. Therefore, particles in this region are rapidly lost, along a particular direction in phase space (in this example, towards the right). The size of the triangle depends on the strength of the sextupoles and on the betatron tune.

Near the third-integer resonance, (3Q−q) ≈ 0, with integer q, the particle motion can be described by a Hamiltonian of the form

$$H(I, \psi, \vartheta) = (Q - q/3)I + \frac{1}{24}(2I)^{3/2}|\tilde{K}\_s|\sin(3\psi + \theta\_0)\,,\tag{9.14}$$

where ϑ is the azimuthal position around the ring, which acts as the time-like variable, and I and ψ are the action-angle variables, which are related to the

Fig. 9.16. Phase space schematic for slow extraction near the 3rd integer resonance (axes in arbitrary units). The sextupole excitation pattern around the ring is chosen such that the maximum excursion in the horizontal coordinate x occurs at the location of the septum. The position of the septum wire is indicated as a vertical line close to the unstable fixed point

transverse particle coordinates at the septum via xsep = -2βsepI cos ψ and x sep = − -2I/βsep sin ψ−α -2I/β cos ψ. The term |K˜s| is the absolute value and θ0 the complex phase of the resonant Fourier harmonic of the sextupole distribution around the ring appropriately weighted by the beta function:

$$|\tilde{K}\_s|e^{-iq\theta\_0} = \frac{1}{2\pi} \int\_0^{2\pi} k\_s(\theta) \beta^{3/2}(\theta) e^{-iq\theta} \text{ d}\theta \,. \tag{9.15}$$

Here θ is the azimuthal angle around the ring, and the sextupole strength (in units of m−3) is given by ks(θ) = ∂2Bz(θ)/∂x2/(Bρ), with Bρ the magnetic rigidity.

Suppose the tune is slightly below the 3rd integer resonance, (3Q−q) < 0. Then a corner point of the separatrix coincides with the horizontal position coordinate xsep at the septum, if θ0 = π/2. Above the resonance, (3Q−q) > 0, the optimum choice would be θ0 = −π/2. The value of θ0 can be adjusted by changes to the sextupole configuration, or by changes to the ring optics. The particles arrive at the septum with a large amplitude on every 3rd turn. The amplitude growth over three turns, for a particle near the unstable fixed point (at the asymptotic angle ψ = π/6), is approximately

$$
\Delta x\_{\rm sep} \approx \frac{3\pi x\_{\rm sep}^2 |\tilde{K}\_s|}{4\beta\_{\rm sep}^{1/2}} \,\, . \tag{9.16}
$$

This shows that large sextupole strengths and a large beta function at the septum (since xsep ∼ βsep) are advantageous.

A slow spill can be controlled by adjusting either the strength of the sextupoles or the betatron tune. Extraction may also involve beam steering. Also making use of chromaticity, particles of different momenta can progressively be brought onto the resonance. Extraction starts when the beam particles at one end of the momentum distribution fill the triangular stable area in phase space. The stable area then shrinks to zero for these particles, and subsequently particles of different momenta are extracted.

To achieve a slow extraction efficiency greater than 98%, the thickness of the septum must typically be of the order of 100 μm.

# 9.11 Extraction via Resonance Islands

A novel method for multi-turn extraction from a circular particle accelerator was explored at the CERN PS [29, 30]. Here, in addition to fast or slow extraction an intermediate extraction mode is needed, which is called multiturn extraction. The PS serves as injector to the SPS. The latter has an 11 times larger circumference. In order to fill the SPS ring with only two 'shots' from the PS, each PS beam is extracted over 5 turns.

The conventional technique used for this extraction is called the 'continuous transfer' (CT). The principle is illustrated in Fig. 9.17. The tune is moved closed to the quarter integer resonance. Then the beam is deflected so that a fraction of it is shaved off at the electrostatic septum blade. Three other slices are transferred on subsequent turns. The central part is extracted last during the fifth turn, by applying a larger deflection. Since during extraction the beam is cut into 5 transverse pieces, the slices transferred ideally have a five times lower transverse emittance than the original beam.

However, there are various problems with this approach: (1) beam losses at the septum are unavoidable; (2) the extracted slices do not match the natural circular shape of the phase-space trajectories which implies emittance growth

Fig. 9.17. Principle of the conventional 'continuous-transfer' extraction from the CERN PS ring. The beam is shaved by means of an electrostatic septum [29, 30]. The horizontal tune of the PS is set to 6.25 (Courtesy M. Giovannozzi, 2001)

in the downstream SPS; (3) unequal emittances of the extracted slices. For these reasons the CT extraction is not particularly suitable for the CERN neutrino to Gran Sasso (CNGS) proton beam [31].

The alternative novel scheme [29] makes use of stable nonlinear resonance islands. By exciting sextupole and octupole magnets, islands are created in phase space. The position and width of the islands are controlled by moving the betatron tune across the quarter-integer resonance. Initially, the islands are introduced adiabatically near the origin. The beam is thereby split into 5 components, all round in shape, and well matched to the circular phase space structure. Then the tune is shifted away from the resonance, so that the islands separate and approach larger amplitudes. Now the beam can be deflected as in the conventional CT scheme described above.

In this case, however, beam losses can be avoided by deflecting an empty region of phase space (between the resonance islands) onto the septum blade. Each slice is well matched, and, hence, the emittance growth is negligible. Finally, by properly adjusting the island parameters, the slices can be equally populated and be produced such that their emittances are equal. Figure 9.18 shows the proposed tune evolution for this extraction scheme, and Fig. 9.19 the simulated beam distribution at various times of the trapping process. At the end of the process, the islands are well separated. In the simulation, no particles are lost, neither during the island creation ('capture process') nor when shifting the island positions.

Open questions concern the quantitative relation between slice emittance and island parameters, the optimization of the tune change, and the robustness against perturbing effects such as tune modulation, e.g., caused by power-supply ripple.

Fig. 9.18. Variation of the small-amplitude tune Q as a function of turn number n during resonant multiturn extraction [29]. The solid squares refer to tune values for which phase-space portraits are shown in Fig. 9.19 (Courtesy M. Giovannozzi, 2001)

Fig. 9.19. Simulated evolution of the beam distribution during the trapping process of resonant multiturn extraction (axes in arbitrary units) [29]. The different plots correspond to the tune values which are represented by solid squares in Fig. 9.18. Each plot represents 2.25 × 104 points (Courtesy M. Giovannozzi, 2001)

# 9.12 Beam Separation

A problem similar to injection and extraction is the beam separation near the collision point of a collider, or the beam (re-)combination before and after the arcs of a recirculating linac.

To illustrate the concept and a possible approach, we describe a design example for horizontal beam separation at a Very Large Lepton Collider (VLLC) [32]. The purpose of the beam separation is to feed the two beams into the two separate magnet channels of the collider arcs. The separation is launched in the straight section close to the collision point. The optical lattice in the straight is assumed to be the same FODO lattice as in the arcs, with a cell length Lp and quadrupole focal length f.

An electrostatic separator is placed next to a horizontally focusing quadrupole. Its field is chosen such that the beam are offset by ±Nσ rms beam sizes at the next quadrupole downstream, which is horizontally defocusing. The separation NσσD (where σD denotes the rms beam size at the quadrupole), the deflection angle φe and the integrated strength of the septum are related via [32]

$$\phi\_e = \frac{2N\_\sigma \sigma\_D}{L\_p} \,, \tag{9.17}$$

where

$$E\_x l\_e = \phi\_e E \; , \tag{9.18}$$

in which Ex denotes the (horizontal) electric field and E the beam energy. A LEP separator consists of 4-m long electrodes, and provides a nominal field of 2.5 MV over a gap of 0.11 m [33]. About two of these separators would be needed for the VLLC application (beam energy 184 GeV) [32]. The defocusing quadrupole enhances the slope between the two beams. Downstream are dc magnetic septum magnets with opposite vertical fields on either side of a current sheet. They add enough slope to the beam that they may be brought into separate channels at the next focusing quadrupole. We denote the half separation of the two channels at that (focusing) quadrupole by dm, the deflection angle of the magnetic septum by φm, its integrated magnetic field by Blm, and the electrostatic separation at the intermediate defocusing quadrupole with focal length f by xs. The required deflection by the magnetic septum is then given by

$$
\phi\_m = \frac{2(d\_m - x\_s)}{L\_p} - \left(\phi\_e + x\_s/f\right). \tag{9.19}
$$

The deflection φm is generated as

$$
\phi\_m = Bl\_m c / E\_b \, , \tag{9.20}
$$

where B is the septum field and lm the septum length. For a length of 10 m, a septum field of about 0.25 T is required, which could be produced by a septum of thickness 7 mm, assuming a current density of 60 A/mm2 in the septum sheet, which is the operation value for the d.c. septum at the SPS. The entire beam separation is illustrated schematically in Fig. 9.20. Dispersion generated by the separation is compensated in the arc dispersion suppressors.

Fig. 9.20. Schematic of beam separation at the VLLC as described in [32]

# 9.13 Crystal Extraction

Crystal extraction is conceptually quite different from the extraction methods. It was first studied at Dubna and Protvino [34], and later tested extensively at the CERN SPS [35, 36] and at the Fermilab Tevatron [37]. Here, particles in the transverse beam halo, entering a crystal placed close to the beam, are trapped between the crystalline planes [38]. If the crystal is slightly bent, the particles can be deflected outwards, and subsequently be transported to a fixed-target experiment. Figure 9.21 shows a schematic view of crystal extraction.

Crystal extraction is foreseen as an option for the LHC. It would be parasitic to the normal collider operation, and re-utilize the halo particles which do no longer contribute to the collider luminosity.

Fig. 9.21. Extraction from the transverse halo of a circulating proton beam by means of a bent crystal. Particles incident with a large impact parameter are channeled and deflected outwards. Particles hitting the inefficient crystal surface layer experience multiple scattering, and may be channeled on a later turn; this is called multi-pass extraction

Channeling occurs if the incident angle of the particles is smaller than the Lindhard critical angle [38]. The critical angle depends on the orientation of the crystal and on the material. The crystal orientation can be defined with respect to an axial direction [ijk] or with respect to a plane (ijk)1. For the (110) planar direction in silicon the critical angle is [38]

$$
\Psi\_{\rm crit} \approx 5 \text{ } \mu \text{rad} \, \frac{\sqrt{Z}}{\sqrt{p[\text{TeV}/c]}} \,, \tag{9.21}
$$

where p is the momentum of the incident particle and Z is its charge in units of the electron charge.

1 In a crystal with cubic symmetry, a vector [ijk] with components along the three orthogonal symmetry axes has a perpendicular atomic plane, which is denoted (ijk).

Thermal vibrations, the discreteness of the crystal lattice, and the presence of the electrons in the target all increase the transverse energy of a channeled particle, and can ultimately lead to dechanneling. This is approximated by an exponential depletion of the number n of channeled particles with the traversed distance z:

$$n = n\_0 \exp(-z/L\_0) \,. \tag{9.22}$$

The empirical parameter L0 is called the dechanneling length, and it increases linearly with momentum. For silicon, we have [38] L0 ≈ 0.9 m p[TeV/c]. Since scattering on nuclei is an important dechanneling process, the channeling can be improved by cooling of the target.

Another concern are imperfections on the crystal surface. These give rise to an inefficient surface layer, typically a few micrometers thick, in which no channeling takes place. To be extracted in a single pass, a particle must enter the crystal with an impact parameter larger than the thickness of the surface layer. On the other hand, particle passing through the surface area experience multiple scattering, and can re-enter the crystal on subsequent revolutions, this time at a larger impact parameter and under the right conditions to be channeled and extracted.

Channeling is possible only for bending radii larger than a minimum 'critical' radius, whose value depends on the crystal, its orientation, and the beam energy. For a proton beam incident parallel to the (110) plane2 of a silicon crystal, this critical radius is [38]

$$R\_c \approx 0.4 \text{ m } p \text{ [TeV/}c] \text{ .} \tag{9.23}$$

The efficiency of cystal extraction is defined as the number of particles extracted divided by the number of particles lost. Proton extraction efficiencies up to 18% have been obtained [36]. Using a crystal coated with a 30 μm amorphous SiO layer, pure multi-pass extraction with an efficiency of 4–7% was demonstrated [36]. The importance of multi-pass extraction implies that not only the initial impact parameter, but also machine parameters such as the beta function at the crystal and the betatron tune play an important role for the overall efficiency.

Finally, in addition to protons also heavy ions can be extracted by a bent crystal. For fully stripped Pb ions (Z = 82) at 22 TeV, an extraction efficiency of 10% was achieved at the SPS [36]. This value was slightly lower than for protons of equivalent energy per nucleon.

2 In a crystal with cubic symmetry, every vector [ijk] defines a perpendicular atomic plane (ijk).

# Exercises

### 9.1 Septum Fields for Injection and Extraction

Suppose that the minimum beam separation at the septum is xsep > nsσx. Derive an expression for the integrated kick strength BkicLkic, with Bkic the kicker magnetic field and Lkic its length, as a function of normalized emittance x,N and energy for a proton beam.

Assume ns = 10, β = 100 m, μ = π/2, a kicker length of Lkic = 5 m, and a normalized emittance x,N = 4 μm. Compute the magnetic field Bkic required at a beam energy of 10 GeV and at 10 TeV.

### 9.2 Emittance Dilutions due to Injection Errors

Consider injection into a storage ring with a 1 mm orbit error at β = 100 m, in both betatron phases.

a) Estimate the corresponding (growth in) normalized emittance after complete filamentation for proton, muon and electron beams at 10 GeV and at 1 TeV. Compare this with the design normalized emittances of the LHC (3.75 μm), the multi-TeV Muon Collider (50 μm), and the NLC (3 nm vertically). Note that mec2 = 511 keV, mpc2 = 938 MeV, and mμc2 = 105.7 MeV.

b) In general the orbit error results in emittance growth comparable to the design emittance when it is of the same order as the rms beam size. Calculate the rms beam sizes for a 7-TeV proton beam (LHC), a 500-GeV electron beam (NLC) and a 2-TeV muon beam (MC) at β = 100 m.

### 9.3 Filamentation

Consider a point bunch which is injected somewhere in phase space at a radius r. Compute the projected beam density p(x), normalized to unity, after filamentation.

### 9.4 Particle Impact for Slow Extraction

Derive (9.16).

### 9.5 Crystal Channeling

What is the maximum bending angle over a length of 3 cm, for the LHC beam energy of 7 TeV?

# 10 Polarization Issues

The study of spin dynamics in synchrotrons has evolved over the years as has the desire for achieving polarized particle beams of the highest possible beam energies. A selection of reviews of the dynamics of polarized beams may be found in [1]–[9]. In this chapter, we focus on experimental data and describe spin transport in circular accelerators and transport lines. Except where explicitly mentioned, radiative effects in electron accelerators or very high energy proton accelerators are not treated here. We begin with a review of the Thomas-BMT equation for spin motion. This will be given in terms of the SU(2) spinor representation. Spinor algebra will be introduced and applied in the description of techniques used for preserving the beam polarization during acceleration through depolarizing resonances at moderate beam energies.

# 10.1 Equation of Spin Motion

The concept of particle spin was first introduced by Uhlenbeck and Goudsmit in 1926 to explain certain features of atomic spectra. They presupposed that the (in this case) electron of mass m and charge e, possessed both a magnetic moment μ and the spin angular momentum s, related to one another by

$$
\mu = \frac{ge}{2m} \mathbf{s} \,, \tag{10.1}
$$

where g is the gyromagnetic ratio whose value was empirically taken to be 2 for electrons to explain certain experimental observations. In 1927 Thomas [10] showed that once a relativistic kinematic effect was taken into account, the value of g = 2 was consistent with the atomic spectra measurements.

The equation of motion for the spin angular momentum in an external magnetic field [5] is given, in the particle rest frame, by

$$\begin{split} \frac{d\mathbf{s}}{dt} &= \boldsymbol{\mu} \times \mathbf{B} \\ &= \boldsymbol{\Omega} \times \mathbf{s} \end{split} \tag{10.2}$$

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3\_13

where the angular velocity of the spin precession is

$$
\mathcal{Q} = -\frac{ge}{2m} \mathcal{B}.\tag{10.3}
$$

In the above equations, the spin angular momentum s of a single particle takes on discrete values of magnitude |s| = ¯h/2 for spin- 1 2 particles (i.e. electrons and protons). It is convenient to normalize s and work with the spin vector S, with |S| = 1, defined as the normalized spin expectation value in the rest frame.

# 10.2 Thomas-BMT Equation

In the laboratory frame, the spin precession for a relativistic particle in external electromagnetic fields is given by the Thomas-BMT equation [10]–[13]:

$$\frac{d\mathbf{S}}{dt} = -\frac{e}{\gamma m} \left[ (1 + a\gamma) \cdot \mathbf{B}\_{\perp} + (1 + a) \cdot \mathbf{B}\_{\parallel} + \left( a\gamma + \frac{\gamma}{\gamma + 1} \right) \frac{\mathbf{E} \times \mathbf{v}}{c^2} \right] \times \mathbf{S} \,, \tag{10.4}$$

where B⊥ and B represent the magnetic fields perpendicular and parallel to the particle velocity respectively, β = v/c with v the particle velocity, and γ = 1/ -1 − β2 the Lorentz factor or ratio of the particle energy to mass.

The factor a in (10.4) is the gyromagnetic anomaly of the electron. It is

$$a = \frac{g - 2}{2} = 0.00115966\ ,\tag{10.5}$$

and deviates from zero due to radiative corrections. For protons, which are composite, we replace a with the symbol G, where

$$G = \frac{g - 2}{2} = 1.792846\ . \tag{10.6}$$

For many practical applications there are no significant electric fields1, and the Thomas-BMT equation is simply

$$\frac{d\mathbf{S}}{dt} = -\frac{e}{\gamma m} \left[ (1 + a\gamma) \mathbf{B}\_{\perp} + (1 + a) \mathbf{B}\_{\parallel} \right] \times \mathbf{S} \,. \tag{10.7}$$

From this equation applied to protons, the spin precession due to transverse magnetic fields depends on the particle energy through the factor 1/γ + G while the amount of precession due to a longitudinal magnetic field scales as (1+G)/γ. We will see later that this has implications for spin rotator design.

1 More precisely, the term β × E is nearly zero since the electric fields in an accelerator are usually parallel to the particle velocity.

# 10.3 Beam Polarization

The polarization P of a bunch is defined as the ensemble average over the spin vectors S of the individual particles:

$$\mathbf{P} = \langle \mathbf{S} \rangle = \left| \frac{1}{N} \sum\_{p=1}^{N} \mathbf{S}\_p \right|, \tag{10.8}$$

where N denotes the number of particles in the bunch.

As an illustration of the formulae presented so far, we calculate the beam depolarization due to spin precession and energy spread in a transport line neglecting radiation effects. We suppose a transverse magnetic field bends an electron orbit by the angle θ. Then, according to (10.4) one finds that the electron spin direction precesses by φ = (aγ)θ relative to the orbit in the laboratory frame. Considering a relativistic electron beam of many particles with a finite energy spread, the spin vectors of different electrons in the beam precess by different angles, since φ depends on the particle energy. If the initial beam polarization is longitudinal with magnitude P0 and the beam is bent horizontally by an angle θ, not only is the final polarization vector Pf rotated by an angle aγ0θ, but also its magnitude is reduced as

$$P\_f = P\_0 e^{-\left(a\gamma\_0 \sigma\_\delta \theta\right)^2/2},\tag{10.9}$$

where γ0 is the Lorentz factor corresponding to the mean energy of the beam, and σδ the rms relative momentum spread assuming a Gaussian momentum distribution. Equation (10.9) is strictly valid, if the orbit is bent in one plane. The same formula was also used to model the spin transport and the depolarization in the SLC North Collider Arc [14, 15], which constituted a nonplanar transport line with horizontal and vertical bends used to follow the terrain of the SLAC site. In this application, an 'effective' bending angle θ, entering in (10.9), was determined by measuring the final direction of the polarization vector as a function of the beam energy [15].

# 10.4 Spinor Algebra Using SU(2)

We can use methods of quantum mechanics [5] to describe spin transport. It is mathematically advantageous to do so since transporting 2 component spinors (Ψ) is simpler than transporting the 3-dimensional spin vector S. The relationship between the two is given by

$$S\_i = \Psi^\dagger \sigma\_i \Psi \,, \tag{10.10}$$

with the Pauli matrices defined2 as

2 Caution. Different authors adopt different conventions; here we adopt the convention (specified in the introduction; c.f. Fig. 1.1) that x is radial, s is longitudinal, and y is vertical. A cyclic permutation may be used to transform between conventions.

242 10 Polarization Issues

$$
\sigma\_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma\_s = \begin{pmatrix} 0 & -\mathbf{i} \\ \mathbf{i} & 0 \end{pmatrix}, \quad \sigma\_y = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \tag{10.11}
$$

Together with the 2 × 2 identity matrix I2, these 4 matrices generate an irreducible representation of the SU(2) group.

# 10.5 Equation of Spin Motion

For generality we can reexpress [1] the equation of spin motion (10.2) in terms of a time-like variable θ defined as

$$\theta = \int\_0^s \frac{\mathrm{d}s'}{\rho(s')} \, , \tag{10.12}$$

which is equal to the accumulated bending angle or so-called orbital angle. Then, as we will show below, the equation of spin motion (10.2) is equivalent to

$$\frac{\mathrm{d}\Psi}{\mathrm{d}\theta} = \frac{i}{2}H\Psi^{\prime},\tag{10.13}$$

where H = Ω˜ ·σ denotes the effective spin Hamiltonian, which is represented here as a matrix [5] with

$$
\tilde{\Omega} = \Omega / \frac{\mathrm{d}\theta}{\mathrm{d}t} \,. \tag{10.14}
$$

In the absence of depolarizing resonances (see Sect. 10.7), for a particle circulating in the horizontal plane under the influence of vertical magnetic fields only, H may be expressed as

$$H = \begin{pmatrix} -\kappa & 0\\ 0 & \kappa \end{pmatrix} \,, \tag{10.15}$$

with κ = Gγ for protons and κ = aγ for electrons.

Equation (10.13) may be expressed as

$$\frac{d\Psi}{d\theta} = -\mathbf{i}\frac{\lambda}{2}(\sigma \cdot \mathbf{\hat{b}})\Psi \; , \tag{10.16}$$

where λ = |Ω˜| denotes the amplitude of the precession frequency and depends in general on the particle coordinates (to concentrate on the underlying principles, we defer the explicit expressions until needed in Sect. 10.12) and ˆb = Ωˆ is a unit vector aligned with the precession axis. The vector ˆb and λ depend on the longitudinal coordinate and on the position of the particle in the sixdimensional phase space [7]. Therefore ˆb may have different orientations at fixed orbital angle θ on subsequent turns.

The solution of (10.16), which defines the spinor transformation matrix M, is

$$
\Psi(\theta) = M\Psi(0) \,. \tag{10.17}
$$

If λˆb is constant along a section of the orbit, then

$$M = e^{-\mathrm{i}(\frac{\lambda}{2})(\sigma \cdot \mathbf{\hat{b}})\theta} \,. \tag{10.18}$$

Using the algebra of the σ matrices, after expanding the exponential, the solution for the spinor is

$$\Psi(\theta) = \left[ \cos \left( \frac{\lambda \theta}{2} \right) - i (\sigma \cdot \hat{\mathbf{b}}) \sin \left( \frac{\lambda \theta}{2} \right) \right] \Psi(0) \,. \tag{10.19}$$

To complete this section we use the Pauli algebra to confirm that the equation of motion for the spinor Ψ (10.13) indeed leads to the equation of motion for the spin vector S. Since the components of the spin are given by the expectation value of the Pauli spin matrices (S = Ψ†σΨ), the equation of spin motion for the polarization vector is obtained by taking the derivative of the latter expression

$$\frac{d\mathbf{S}}{d\theta} = \frac{d\Psi^\dagger}{d\theta} \sigma \Psi + \Psi^\dagger \sigma \frac{d\Psi}{d\theta} \,. \tag{10.20}$$

Using (10.16) and its Hermitian conjugate, dΨ† dθ = iλ 2 Ψ†(σ · ˆb), one finds

$$i\frac{d\mathbf{S}}{d\theta} = i\frac{\lambda}{2}\Psi^\dagger[(\sigma\cdot\hat{\mathbf{b}})\sigma - \sigma(\sigma\cdot\hat{\mathbf{b}}))|\Psi\ . \tag{10.21}$$

Making use of the Pauli algebra σiσj = 1 for i = j and σiσj = i ijkσk for i = j, one can show that

$$\mathbf{i}\left[\boldsymbol{\sigma}\cdot\boldsymbol{\hat{b}},\boldsymbol{\sigma}\right] = 2\mathbf{i}(\mathbf{\hat{b}}\times\boldsymbol{\sigma})\,.\tag{10.22}$$

Therefore, we may write

$$\frac{d\mathbf{S}}{d\theta} = -\lambda \Psi^\dagger (\hat{\mathbf{b}} \times \boldsymbol{\sigma}) \Psi = -\lambda \hat{\mathbf{b}} \times (\Psi^\dagger \boldsymbol{\sigma} \Psi) = -\lambda (\hat{\mathbf{b}} \times \mathbf{S})\,,\tag{10.23}$$

which may be compared with (10.4).

# 10.6 Periodic Solution to the Equation of Spin Motion

The first step for studying spin motion in a circular accelerator is to find the periodic solution to the equation of spin motion on the closed orbit. Here we write a spin transfer matrix M as a product of n precession matrices, each of which characterizes a section of constant magnetic field causing spin precession; i.e., M = M1M2 ··· Mn.

The transfer matrix corresponding to one turn around the accelerator is refered to as the one turn spin transfer map denoted by M0. For the closed orbit M0 is periodic: M0(θ + 2π) = M0(θ). Because the norm of the vector in the precession equation is an invariant, M0 is unitary so that it may be expressed as

$$M\_0 = e^{-i\pi\nu\_0(\sigma \cdot \hat{\mathbf{n}}\_0)} = I\_2 \cos\pi\nu\_0 - i(\sigma \cdot \hat{\mathbf{n}}\_0)\sin\pi\nu\_0 \,, \tag{10.24}$$

or

$$M\_0 = \begin{pmatrix} \cos \pi \nu\_0 - \mathrm{i} \sin \pi \nu\_0 \cos \alpha\_y & -\sin \pi \nu\_0 \cos \alpha\_s - \mathrm{i} \sin \pi \nu\_0 \cos \alpha\_x \\ \sin \pi \nu\_0 \cos \alpha\_s - \mathrm{i} \sin \pi \nu\_0 \cos \alpha\_x & \cos \pi \nu\_0 + \mathrm{i} \sin \pi \nu\_0 \cos \alpha\_y \end{pmatrix},\tag{10.25}$$

The unit vector nˆ0 is the precession axis for the one turn map M0. It is periodic and fulfills the Thomas-BMT equation [5]; a spin set parallel to ˆn0 at orbital location θ will, after one turn (θ + 2π), also be parallel to ˆn0. Thus nˆ0 is refered to as the 'stable spin direction' on the closed orbit and may be described by direction cosines:

$$
\hat{m}\_{\mathbf{0}} = \left(\cos\alpha\_x, \cos\alpha\_s, \cos\alpha\_y\right) \tag{10.26}
$$

with normalization cos2 αx + cos2 αs + cos2 αy = 1.

The parameter ν0, called the spin tune, represents the number of times the spin of a particle on the closed orbit precesses about the stable spin direction in one turn around the ring. The fractional part of the spin tune may be obtained from the trace of the (periodic) spin precession matrix:

$$\begin{aligned} \text{Tr } M\_0 &= 2 \cos \pi \nu\_0 \quad \text{or} \\ \nu\_0 &= \frac{1}{\pi} \cos^{-1} \left( \frac{\text{Tr } M\_0}{2} \right) \,. \end{aligned} \tag{10.27}$$

For the spin motion of particles not on the closed orbit, M is in practice not periodic since accelerators are not typically operated on resonance (for which the particle returns to the same point in phase space after an integer number of turns). The unit vector ˆb in the equation of spin motion (10.16) may therefore have different orientations at fixed orbital coordinate θ on subsequent turns (θ, θ + 2π, θ + 4π,...).

# 10.7 Depolarizing Resonances

The spin of a particle executing synchro-betatron motion around the closed orbit is perturbed by magnetic fields sampled at the betatron and synchrotron frequencies which are characteristic of the particle motion. Since the particles within a bunch have generally different amplitudes and phases, the perturbation to the spin is different for different particles resulting in a spread of the particle spins and thus a lower polarization.

Depolarizing resonances occur whenever the spin tune beats with any of the natural oscillation frequencies of the orbital motion; that is when the spin tune, ν0, equals a resonance tune, νres, by satisfying

$$\nu\_0 = \nu\_{\rm res} \equiv m + qQ\_x + rQ\_y + sQ\_s \, \,, \tag{10.28}$$

where Qx and Qy are the horizontal and vertical betatron tunes, Qs is the synchrotron tune, while m, q, r,and s are integers3. Here m is the product of an integer times the periodicity of the lattice. The quantity |q| + |r| + |s| is called the order of the resonance.

The general resonance condition specifies the criteria for many different types of resonances. Imperfection depolarizing resonances, for which

$$
\nu\_0 = \nu\_{\rm res} = m = \text{integer}\,,\tag{10.29}
$$

arise, for example, from horizontal magnetic fields experienced by the orbiting particle due to magnet imperfections, dipole magnet rotations about the beam direction, and to vertical quadrupole magnet misalignments. Lowest order intrinsic resonances, which result from the horizontal fields of quadrupoles, occur if

$$
\nu\_0 = \nu\_{\rm res} = m + rQ\_y \,. \tag{10.30}
$$

In practice, the above two types of resonances have proven the most significant in the energy regimes of existing accelerators with polarized beams.

Other higher order spin depolarizing resonances may occur for any combination of integers which satisfy (10.28). Studies have shown that higher-order intrinsic resonances of the form

$$
\nu\_0 = \nu\_{\rm res} = m + rQ\_y + sQ\_s \,. \tag{10.31}
$$

become increasingly important at higher beam energies. Due to the interaction with the particle longitudinal momentum, such resonances are also refered to as synchrotron sideband depolarizing resonances.

Resonant spin motion has been observed in many accelerators. Interestingly, it was observed in the SLC collider arcs [14], through which bunches pass only once. The 1 mile arcs were used to transport 45.6 GeV polarized electrons from the linac to the interaction point where they collided head-on with positrons. The arc consists of 23 achromatic sections with a 108◦ phase advance per cell. The vertical beam trajectory and the components of a spin vector are shown in Fig. 10.1 assuming an initial vertical offset of 0.5 mm and random quadrupole misalignments. For the nominal SLC operating energy, the phase advances of (betatron) orbit and spin in the SLC arcs were almost identical. In Fig. 10.1 this equality is seen to contribute to a net tilt of the spin vector as evidenced by the increase in the vertical component of a spin along the arc. In practice, vertical bumps were used to optimally align the polarization to be longitudinal at the interaction point.

3 For particles in the bunch with large orbital amplitudes, ν0 in (10.28) should be replaced by spin tune of the individual particle ν [7]–[9].

Fig. 10.1. Beam orbit (solid line) and spin transport in the SLC collider arc (Courtesy P. Emma, 1999)

# 10.8 Polarization Preservation in Storage Rings

For proton and deuteron accelerators, for which the polarization of the beam is produced at the particle source, the first requirement of maintaining a beam's polarization is preserving this polarization at injection into downstream accelerators. Ignoring synchro-betatron motion, this is easily achieved by orienting (using upstream spin rotators) the beam polarization so that it is aligned with the stable spin direction of the downstream transport line or storage ring.

A mismatch at injection results in a cosine-like reduction of the timeaveraged beam polarization. Letting (cos αx, cos αs, cos αy) denote the orientation of the injected polarization Pinj and (cos βx, cos βs, cos βy) the orientation of the stable spin direction nˆ0 in the laboratory reference frame, then the projection of the injected polarization vector Pinj onto nˆ0 is

$$\begin{array}{l} \|\mathbf{P}\| = \mathbf{P}\_{\text{inj}} \cdot \hat{\mathbf{n}}\_{\mathbf{0}} \\ = P\_{\text{inj}} (\cos \beta\_x \cos \alpha\_x + \cos \beta\_s \cos \alpha\_s + \cos \beta\_y \cos \alpha\_y) \, . \end{array} \tag{10.32}$$

The components of the time-average polarization one would measure at the injection point are then obtained by projecting the polarization onto the three coordinate axes:

$$P\_y = \|\mathbf{P}\|\cos\beta\_y, \quad P\_x = \|\mathbf{P}\|\cos\beta\_x, \quad P\_s = \|\mathbf{P}\|\cos\beta\_s. \tag{10.33}$$

A conceptual illustration is given in Fig. 10.2. At other locations in the ring, the measurable polarization components are obtained by performing a second projection using Pnˆ0(s) where nˆ0(s) is the stable spin direction at the point of interest.

Fig. 10.2. Graphical representation showing the projection of the injected polarization Pinj onto the stable spin direction ˆn0 at the injection point. The time-averaged vertical polarization one would measure at that point in denoted by Py

Once the beam is successfully injected without loss of polarization, it must be ramped to high energy thereby encountering numerous depolarizing resonances along the energy ramp. With considerable effort, polarized proton beams were accelerated through many intrinsic and imperfection depolarizing resonances to GeV energies at the ZGS [16], Saturne [17], the AGS [18], and KEK [19]. The methods employed were based on overcoming each depolarizing resonance individually. In this section we review techniques used to overcome these resonances. In the next section we describe 'siberian snakes' which are used for preserving the beam polarization during the energy ramp. Proof of principle experiments with snakes were initially carried out at the Indiana University Cyclotron Facility (IUCF) [20]–[22]. Siberian snakes are now used in routine operation during acceleration of polarized protons at RHIC [23] and constitute the preferred method of polarization preservation for future high energy hadron accelerators.

### 10.8.1 Harmonic Correction

Harmonic correction of imperfection depolarizing resonances was used at the AGS [18] to ramp vertically polarized proton beams to about 22 GeV. There 96 correction dipoles were employed whose currents were programmed during the acceleration process such that the Fourier harmonics of the radially outward and longitudinal fields in the measured particle trajectories for the most nearby resonances were minimized. The Fourier harmonics are given by an cos nθ + bn sin nθ, where n denotes the resonance harmonic of interest. As will be shown later, the resonance strength depends on the vertical beam displacement (in quadrupoles, for example, the nominally vertical polarization experiences a radial precession field with an off-axis beam). To eliminate depolarization, the coefficients an and bn were experimentally adjusted to minimize the horizontal magnetic fields causing each imperfection resonance. Shown in Fig. 10.3 are traces on an oscilloscope from the AGS [18] showing

Fig. 10.3. Oscilloscope traces showing the currents in pulsed magnets during the energy ramp to 16.5 GeV/c at the AGS. Shown are the pulsed dipole currents (top), the pulsed quadrupoles currents (middle), and the main dipole current (bottom) (Courtesy A. Krisch, 1999)

the corrector dipole currents (top trace) and the main dipole current (bottom trace), which is proportional to the beam energy.

Harmonic correction methods [24]–[26] were also applied in the case of high energy electron beams at both HERA [27]–[29] and, deterministically, at LEP [30]. In electron accelerators, an initially unpolarized beam may after some time become polarized due to the emission of synchrotron radiation. This is known as the Sokolov–Ternov effect [31] which also predicts a maximum possible beam polarization of 92.4% for electrons. In practice however this level of polarization is not reached due to spin-orbit coupling caused by the trajectory oscillations which result from photon emission [3]. Minimizing the strength of the harmonics nearest the beam energy thus minimizes the

Fig. 10.4. Measured transverse beam polarization at HERA showing improvements gained using harmonic correction (Courtesy of the HERMES experiment, 2002)

influence of the depolarizing effects due to synchrotron radiation. At HERA and LEP, instead of empirically varying the whole closed orbit, closed orbit bumps were used to minimize the strength of the nearest imperfection resonances. A recent result from HERA is shown [29] in Fig. 10.4 obtained after implementing a new optic as intended for the HERA upgrade.

### 10.8.2 Adiabatic Spin Flip

The method of adiabatic spin flip, which was also used at the AGS, is based on the results of Froissart and Stora [32]. The Froissart–Stora formula, which describes the spin transport through a single, isolated imperfection or lowestorder intrinsic resonance, is

$$P\_y(\infty) = \left(2e^{-\frac{\pi|\epsilon|^2}{2\alpha}} - 1\right)P\_y(-\infty)\,,\tag{10.34}$$

where is the resonance strength (see Sect. 10.12 and [1]), α = dν0/dθ, and Py(−∞) or Py(∞) refer to the initial and final polarizations, respectively.

Fig. 10.5. Spin flip of a vertically polarized beam at the IUCF with kinetic energy 105.4 MeV and a frequency sweep range of 2 kHz. The estimated resonance strength at 15 kV applied voltage is 1.9×10−4 (Courtesy A. Krisch, 1999)

The Froissart–Stora formula mandates that the spin of the orbiting particle will flip if the passage is slow and the resonance is strong (the argument in the exponent of (10.34) is large). This behavior has been verified by experiment [17, 33, 34]. An example taken from the IUCF cooler ring [34] is shown in Fig. 10.5. Here a solenoid was used to produce a sinusoidally varying longitudinal magnetic field of frequency frf and peak amplitude Vsol. This served to create a depolarizing resonance such that

250 10 Polarization Issues

$$\nu\_0 \approx c\_1 \pm \frac{f\_{\rm rf}}{f\_{\rm rev}} = c\_2 \pm \left(\frac{f\_{\rm rev} - f\_{\rm rf}}{f\_{\rm rev}}\right) \,, \tag{10.35}$$

where c1 and c2 are integers. In these measurements a frequency sweep of span 2 kHz centered about the revolution frequency was used. After each data point, the beam was dumped and a new beam was injected. The curves in Fig. 10.5 show the prediction by the Froissart–Stora equation computed for the measured resonance strength for two different ramp rates.

### 10.8.3 Tune Jump

Intrinsic depolarizing resonances were overcome at the AGS using the method of tune jump [18]. From the Froissart–Stora relation, if the resonance is crossed sufficiently quickly (the exponent approaches zero), then the polarization will be preserved. Therefore, as the spin tune ν0 = Gγ increases during acceleration, the resonance can be traversed without loss of polarization by rapidly shifting the vertical betatron tune, νy. A classic example [18] from the AGS was shown in Fig. 10.3 in which the current of the pulsed quadrupoles (middle trace) is depicted. To achieve this, strong pulsed quadrupoles and special ceramic beam pipes (to allow passage of the field) were required.

# 10.9 Siberian Snakes

The above mentioned correction schemes were anticipated to be of limited applicability when accelerating polarized beams to very high energies. The harmonic correction employed at the AGS was complicated and time consuming; the empirically found corrections also depended on the closed orbit of the accelerator, which was observed to drift with time and change between running periods. At very high energies, where the resonances will be overlapping due to an increase in the resonance strength with increasing energy (see (10.49) below), the method of adiabatic spin flip fails [35]. The method of tune jump is stopband limited since, for a very strong intrinsic resonance, the vertical betatron tune shift required to overcome the resonance may exceed the separation between the machine betatron resonances. Finally, the number of resonances to be crossed increases with energy. At the SSC, where there would have been more than 104 imperfection and first-order intrinsic resonances, overcoming each resonance individually clearly would have been impractical.

An ingenious arrangement of magnets was proposed [36, 37] by Derbenev and Kondratenko in 1976 (see also [13]). Use of this technique4 would simultaneously overcome all imperfection and lowest-order intrinsic resonances by making the spin tune energy independent. A so-called type-1 snake rotates

4 This scheme was dubbed 'siberian snake' by E. Courant.

the spin of each proton by 180 degrees about the longitudinal axis on each turn around the ring without changing the closed orbit outside of the snake. This forces the spin tune to be 1/2 and the stable spin direction nˆ0 to lie in the horizontal plane. The resonance condition of (10.29) and (10.30) can therefore never be satisfied regardless of the beam energy: the condition for imperfection resonances with integer spin tune (10.29) is never satisfied and, for betatron tunes not equal to 1/2 (corresponding to half integer orbit resonances), intrinisic resonances (10.30) are also avoided. A type-2 siberian snake precesses the spin about the radial direction. A type-3 siberian snake precesses the spin about the vertical direction.

The most cost-effective construction of a siberian snake depends upon the energy range of interest. A siberian snake consisting of a solenoidal field (and skew quadrupoles for coupling correction) requires a field strength (given here for protons) of

$$\int B\_{\parallel} \text{ dl} = \frac{mc\beta\gamma}{(1+G)e}\psi \,, \tag{10.36}$$

in which ψ (= π for a full siberian snake) is the angle through which the spin is precessed. In this case, the required field integral depends linearly on γ. Due to technical constraints the field strength and magnet length cannot be increased indefinitely. Therefore siberian snakes made with solenoids are better suited for low energy operation.

Alternatively a 'conventional' siberian snake consisting of eight transverse field dipoles each of which precesses the spin by π 2 requires a field strength of

$$\int B\_{\perp} \text{ dl} = \frac{mc\beta}{Ge} \psi \quad \text{(transverse scale)}, \tag{10.37}$$

which is independent of γ. This type of siberian snake therefore has the advantage that a single set of dipole operating currents suffices for all beam energies once the relativistic factor β is close to 1. However, in a dipole magnet, the orbit deflection angle is ψ/Gγ and so depends on the energy. For low beam energies (γ < 10), a siberian snake consisting of dipoles would thus require the construction of large and costly dipoles. Siberian snakes consisting of dipole magnets are therefore more suitable for operation at high beam energies.

With a single type-1 siberian snake, the one-turn spin transfer matrix evaluated at the orbital angle θ is

$$M = e^{-\mathbf{i}\pi\nu\_0(\hat{\mathbf{n}}\_0 \cdot \boldsymbol{\sigma})} = \left[ e^{-\mathbf{i}\frac{G\gamma}{2}(\pi-\theta)\sigma\_\mathbf{y}} e^{-\mathbf{i}\frac{\boldsymbol{\sigma}}{2}\sigma\_\mathbf{s}} \right] \left[ e^{-\mathbf{i}\frac{G\gamma}{2}(\pi+\theta)\sigma\_\mathbf{y}} \right],\tag{10.38}$$

where η gives the spin precession about the longitudinal in radians. Taking the trace of (10.38), the particle spin tune is

$$
\cos \pi \nu\_0 = \cos \left( \pi G \gamma \right) \cos \left( \frac{\eta}{2} \right) \,\, . \tag{10.39}
$$

If the siberian snake is off (η = 0), then ν0 = Gγ as expected. With the snake fully turned on (η = π), then cos πν0 = 0 and ν0 = 1/2 (modulo 2π).

For a siberian snake design optimum for high beam energies, the use of transverse magnetic fields for spin precession has the unfortunate consequence of also deflecting the particle orbit. The design of a siberian snake therefore uses closed horizontal and vertical bumps so that the orbit outside of the snake region is unchanged (optical transparency). However this is achieved only at the expense of increased snake length which may become costly. Many different snake designs have been proposed. Some of the earlier designs by Steffen are given in [38]–[40] and the design currently used at RHIC is described in [41, 42] have been proposed. Shown in Fig. 10.6 is one such design consisting of alternating horizontal and vertical dipoles. This design is conveniently expressed as

$$\mathrm{V}\left(\frac{\pi}{2}\right)\mathrm{V}\left(-\frac{\pi}{2}\right)\mathrm{H}\left(\frac{\pi}{2}\right)\mathrm{V}\left(-\frac{\pi}{2}\right)\mathrm{H}\left(-\pi\right)\mathrm{V}\left(\frac{\pi}{2}\right)\mathrm{H}\left(\frac{\pi}{2}\right),\tag{10.40}$$

where H and V represent horizontal and vertical dipoles rotating the spin through the angle of the argument.

Fig. 10.6. A design of a type-1 siberian snake showing the geometry (top) and the design orbit in the vertical (middle) and horizontal (bottom) planes. The beam moves from the left to the right (Courtesy A. Chao, 1999)

In addition to preserving the polarization, it is often desirable to rotate the spin from vertical to longitudinal [4] at one of possibly many interaction points in a storage ring as is done at HERA [43], for example. An optically transparant spin rotator which does that was proposed by Montague [5, 44]. The spin rotation matrix R is given by

$$R = \operatorname{V}\left(\alpha\right)\operatorname{V}\left(-\alpha\right)\operatorname{V}\left(-\frac{\pi}{4}\right)\operatorname{H}\left(-\pi\right)\operatorname{V}\left(\frac{\pi}{4}\right)X\operatorname{V}\left(-\frac{\pi}{4}\right)\operatorname{H}\left(\pi\right)\operatorname{V}\left(\frac{\pi}{4}\right),\tag{10.41}$$

where α is an arbitrary precession angle and 'X' indicates the location of the interaction point. For transverse magnetic fields, the orbital bending angles are obtained by the precession angle divided by Gγ. In contrast to spin transformations for which the transfer matrices of each magnet do not commute, the small orbital deflections do essentially commute and are seen to sum to zero.

Radiation in strong vertical bending magnets, used in spin rotators for example, in an electron storage ring can cause excitation of vertical betatron motion which may then lead to depolarization due to the radial component of quadrupole fields [5]. In addition, when rotators are used to precess the polarization into the longitudinal direction, the horizontal motion excited by radiation in the arcs can cause depolarization in the vertical fields of the quadrupoles in the interaction regions. These depolarization mechanisms may be avoided by invoking spin matching conditions to obtain spin transparency [4, 26, 45].

Fig. 10.7. Measurements of time-averaged radial polarization at the IUCF near a Gγ = 2 imperfection resonance (at about 106.4 GeV) at 5 different beam kinetic energies with a 100% siberian snake on (left) and off (right)

Fig. 10.8. Measurements of time-averaged vertical polarization at the IUCF near a Gγ = 2 imperfection resonance (at about 106.4 GeV) at 5 different beam kinetic energies with a 100% siberian snake on (left) and off (right)

Interestingly, it was not until 1989 that the siberian snake concept was tested experimentally [20, 21]. Shown in Figs. 10.7 and 10.8 are measurements which demonstrated the use of siberian snakes for control of the beam polarization. Plotted are the time-averaged vertical and radial polarization measured at different beam kinetic energies in the vicinity5 of the Gγ = 2 imperfection resonance. The horizontal axis shows the strength of an error field introduced diametrically opposite to the siberian snake. In these measurements, the error field was set, the beam was injected, the polarization was measured, and then the beam was dumped. The polarization of the injected beam was always oriented to match the stable spin direction at the injection point; i.e. in the horizontal plane with the snake turned on or vertically with

5 From the figures, the resonance was found to lie between 105.9 MeV and 107.8 MeV rather than at kinetic energy (2 − G)/mG = 108.4 MeV one might expect using Gγ = 2. This was later explained by the presence of an unintentional type-3 snake in the accelerator.

no snake. From Fig. 10.7, with the snake turned on, the radial polarization was not affected by the nearby resonance. From Fig. 10.8, with the snake turned off, the measured time-averaged vertical polarization was observed to be less near the resonance. The curves in Figs. 10.7 and 10.8 were obtained using the periodic solution to the equatio of spin motion taking into account the presence of the type-3 snake.

It is worth mentioning that the presence of a siberian snake in an accelerator may introduce a new kind of resonance dubbed a snake resonance [46]. The snake resonance condition is given by uνres = ν0 + n in which ν0 (=1 2 ) is the spin tune determined by the snake and u and n are integers. These are relevant when the fractional betatron tune Q is 1/integer, which is not a condition under which an accelerator is typically operated. Nonetheless, the presence of snake resonances has been both predicted and verified experimentally [47].

# 10.10 Partial Siberian Snakes

In 1989 Roser [48] proposed an elegant and intuitive variant of the siberian snake called a partial siberian snake. These are magnetic devices which enable the polarization to be maintained when accelerating through imperfection resonances.  At low beam energies, a partial snake consisting of dipoles does not induce too large an orbit excursion, while a solenoidal partial snake requires significantly less magnetic field than a full snake. As mentioned above the disturbance to the polarization at imperfection resonances can be minimized by minimizing certain harmonics in the fields on the closed orbit. Partial snakes imply the opposite approach, namely the snake induces a large predetermined imperfection onto the spin motion which dominates all other imperfections. Then during normal acceleration with the large resonance strength, the Froissart– Stora formula guarantees a full spin flip without loss of polarization. The polarization will flip again after passage through the next imperfection resonance.

The expression for the dependence of the spin tune on energy and the precession per turn was given in (10.39) and is shown in Fig. 10.9 for different percentages of applied longitudinal field (η = π denotes a full snake which is designated in the figure by 100%). The diagonal line shows the spin tune with no snake. As can be seen, even a relatively weak (∼ few %) snake can cause a significant deviation of the spin tune from Gγ during acceleration and a large spin tune gap thus avoiding the imperfection resonance at ν0 = n = integer.

An example showing the first demonstration of the ability of partial siberian snakes to avoid imperfection depolarizing resonances, in this case at fixed beam energy, is shown in Fig. 10.10. With a 10% snake full polarization was maintained despite the applied error field. At the AGS a 5%

Fig. 10.9. The dependence of the spin tune on Gγ for various snake strengths (indicated by percentage of full 180 degree spin precession)

Fig. 10.10. Demonstration from the IUCF of polarization preservation in the vicinity of an imperfection resonance using a 10% partial siberian snake (left) and loss of polarization without the partial snake (right). In these measurements the kinetic energy of the beam was fixed at 105.9 MeV

partial siberian snake has been installed and is used routinely to ramp polarized protrons through numerous imperfection resonances to the required transfer energy (∼ 25 GeV) for injection into RHIC [49, 50]. Note that for fixed beam energy, once the betatron tunes are known, the strength of the partial siberian snake could be set such that a nearby intrinsic resonance is also avoided.

# 10.11 RF Dipole

At the AGS, the many intrinisic resonances not avoidable the energy ramp are now overcome using an rf dipole magnet [51] (see also Sect. 3.10). From the Froissart–Stora equation (10.34), full spin flip may be expected if the resonance strength is large. The resonance strength may be artificially increased (see (10.48) below) by exciting a coherent vertical betatron oscillation thus inducing a complete spin flip [52]. If the location of externally induced resonance is placed close to an intrinsic resonance, then depending on the relative tune separation, phase, and strength, the induced resonance can be made to dominate the spin motion and full spin flip can be achieved [51]. At the AGS, the rf dipole is adiabatically turned on and off to avoid the emittance growth that would be observed with a pure impulse excitation [51].

# 10.12 Single Resonance Model

Until now we have avoided the use of complicated formulas and have presented basic concepts useful for practical applications of spin transport and preservation. In this section we define the resonance strength parameter used previously and show explicitly, for the case of an isolated resonance in the absence of siberian snakes, its effect on the beam polarization.

We rely on the work of Courant and Ruth [1], who expressed the magnetic fields in the Thomas-BMT equation in terms of the particle coordinates. They found that the equation of spin motion reduces to

$$\frac{\mathrm{d}S\_x}{\mathrm{d}\theta} = -\kappa S\_s - rS\_y; \quad \frac{\mathrm{d}S\_s}{\mathrm{d}\theta} = +\kappa S\_x - tS\_y; \quad \frac{\mathrm{d}S\_y}{\mathrm{d}\theta} = +rS\_x + tS\_s \,, \tag{10.42}$$

where κ, r, and t are functions of the transverse coordinates of the particle orbit. In the cartesian coordinate system (with ˆx radially outward, ˆs along the beam direction, and ˆy vertical) and ρ the local radius of curvature of the reference orbit, these variables are given by [1]

$$\begin{aligned} \kappa &= G\gamma - (1 + G\gamma)\rho x''' \approx G\gamma\\ r &= (1 + G\gamma)y' - \rho(1 + G)\left(\frac{y}{\rho}\right)'\\ t &= (1 + G\gamma)\rho y''' \end{aligned} \tag{10.43}$$

where the derivatives are with respect to the longitudinal coordinate s.

Equation (10.42) can then be transformed [1] into an equivalent spinor representation, for which

$$\frac{\mathrm{d}\Psi}{\mathrm{d}\theta} = \frac{\mathrm{i}}{2}H\Psi\,. \tag{10.44}$$

Here H is the spinor precession matrix given by

$$H = \begin{pmatrix} -\kappa & -t - \mathrm{i}r \\ -t + \mathrm{i}r & \kappa \end{pmatrix} \,, \tag{10.45}$$

and Ψ is a two component complex spinor. In first approximation the function H is uniquely determined by the properties of the synchrotron. In the preceding analyses we have assumed that the function H is piecewise constant. As before the spin components are obtained by taking the expectation value of the Pauli matrix vector, σ, i.e.,

$$S\_i = \Psi^\dagger \sigma\_i \Psi \,. \tag{10.46}$$

Due to the periodic nature of a synchrotron, the coupling term of (10.45) may be expanded in terms of the Fourier components, i.e.,

$$\mathbf{i}\,t + \mathbf{i}r = \sum\_{k} \epsilon\_{k} e^{-\mathbf{i}\nu^{\pm}\_{\text{ rus},k}\theta} \,, \tag{10.47}$$

in which θ is the particle's orbital angle, ν±res,k = k for imperfection resonances, ν±res,k = k ± Qy for the first order intrinsic resonances, and k is the resonance strength which is given by the Fourier amplitude

$$\epsilon\_k = \frac{1}{2\pi} \int (t + \mathrm{i}r) e^{\mathrm{i}\nu^{\pm}\_{\mathrm{rms},k}\theta} \mathrm{d}\theta \;. \tag{10.48}$$

For the case of an imperfection resonance, the resonance strength is given approximately by summing over the radial error fields encountered by a particle on the closed orbit in one turn around the ring:

$$\epsilon\_k \approx \frac{1+G\gamma}{2\pi} \sum\_l \Delta s\_l \, \frac{\partial B\_x / \partial y}{B\rho} y e^{\mathbf{i}\nu\_{rm}\lambda} \,\,,\tag{10.49}$$