Past, Present, and Future of Polarized Hadron Beams

Abstract

The acceleration and storage of high energy polarized proton beams has made tremendous progress over the last 40 years challenging along the way the technologies, precision and the understanding of the beam dynamics of accelerators. This progress is most evident in that one can now contemplate high energy colliders with polarized beams and high luminosity at the same time. After a brief summary of the development and history of polarized proton beam acceleration this chapter will focus on the acceleration of polarized proton beams from MeV to the 100s of GeVs and the possibility of accelerating polarized beams to even higher energies in the future. Elements of the history of polarized electron beams, subject to the effects of synchrotron radiation, will be found in the electron beam polarization dedicated chapters in these lectures.

This manuscript has been authored by Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy. The United States Government and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

1.1 Spin Dynamics, Resonances and Siberian Snakes

Accelerating polarized beams requires an understanding of both the orbital motion and spin motion. Whereas the effect of the spin on the orbit is negligible the effect of the orbit on the spin is usually very strong. The evolution of the spin direction of a beam of polarized protons in external magnetic fields such as exist in a circular accelerator is governed by the Thomas-BMT equation [1],

$$\displaystyle \begin{aligned} \frac{d\mathbf{P}}{dt}=-\left( \frac e{\gamma m}\right) \left[ G\gamma {\mathbf{B}}_{\mathbf{\bot}}+\left( 1+G\right) {\mathbf{B}}_{\mathbf{{\Vert}}}\right] \times \mathbf{P} \end{aligned}$$

where the polarization vector P is expressed in the frame that moves with the particle and the magnetic field is expressed in the laboratory frame. This simple precession equation is very similar to the Lorentz force equation, which governs the evolution of the orbital motion in an external magnetic field:

$$\displaystyle \begin{aligned} \frac{d\mathbf{v}}{dt}=-\left( \frac e{\gamma m}\right) \left[ {\mathbf{B}}_{\mathbf{{\bot}}}\right] \times \mathbf{v}. \end{aligned}$$

From comparing these two equations it can readily be seen that, in a pure vertical field, the spin rotates Gγ times faster than the orbital motion. Here G = 1.7928 is the anomalous magnetic moment of the proton and γ = E∕m. In this case the factor Gγ then gives the number of full spin precessions for every full revolution, a number that is also called the spin tune ν_sp. At top energies of the Brookhaven Relativistic Heavy Ion Collider (RHIC) this number reaches about 400. The Thomas-BMT equation also shows that at low energies $\left ( \gamma \approx 1\right )$ longitudinal fields B_∥ can be quite effective in manipulating the spin motion, but at high energies transverse fields B_⊥ need to be used to have any effect beyond the always present vertical holding field.

The acceleration of polarized beams in circular accelerators is complicated by the presence of numerous depolarizing spin resonances. During acceleration, a spin resonance is crossed whenever the spin precession frequency equals the frequency with which spin-perturbing magnetic fields are encountered. There are two main types of spin resonances corresponding to the possible sources of such fields: imperfection resonances, which are driven by magnet errors and misalignments, and intrinsic resonances, driven by the focusing fields.

The resonance conditions are usually expressed in terms of the spin tune ν_sp. For an ideal planar accelerator, where orbiting particles experience only the vertical guide field, the spin tune is equal to Gγ, as stated earlier. The resonance condition for imperfection depolarizing resonances arise when ν_sp = Gγ = n, where n is an integer. For proton beams, imperfection resonances are therefore separated by only 523 MeV energy steps. The condition for intrinsic resonances is ν_sp = Gγ = kP ± ν_y, where k is an integer, ν_y is the vertical betatron tune and P is the superperiodicity. For example at the Brookhaven Alternating Gradient Synchrotron (AGS), P = 12 and ν_y ≈ 8.8.

Close to a spin resonance the spin tune deviates away from its value of Gγ of the ideal flat machine. For a resonance with strength 𝜖, which is the total spin rotation per radian of orbit deflection due to the resonance driving fields, the new spin tune is given by the equation

$$\displaystyle \begin{aligned} \cos \left(\pi u_{sp}\right)= \cos \left(\pi G \gamma \right) \cos \left( \pi\epsilon \right) . \end{aligned}$$

Figure 1.1 shows the solutions of this equation with and without a resonance. A similar calculation can be done for the effective precession direction or, as it is often called, the stable spin direction or n₀. The stable spin direction describes those polarization components that are repeated every turn. Note that both the stable spin direction and the spin tune are completely determined by the magnetic structure of the accelerator and the beam energy. The magnitude and sign of the beam polarization, however, depend on the beam polarization at injection and the history of the acceleration process.

Fig. 1.1

The evolution of the spin tune during the crossing of a resonance at Gγ = K with strength 𝜖

The spin tune and stable spin direction calculations apply only to a time-independent static situation or if parameters are changed adiabatically. Far from the resonance the stable spin direction coincides with the main vertical magnetic field. Close to the resonance at Gγ = K, the stable spin direction is perturbed away from the vertical direction by the resonance driving fields. When a polarized beam is accelerated through an isolated resonance at arbitrary speed, the final polarization can be calculated analytically [2] and is given by

$$\displaystyle \begin{aligned} \frac{P_f}{P_i}=2e^{\textstyle -\frac{\pi \left| \epsilon \right|{}^2}{2\alpha }}-1, \end{aligned}$$

where P_i and P_f are the polarizations before and after the resonance crossing, respectively, and α is the change of (K − ν_sp) per radian of the orbit deflection. When the beam is slowly ($\alpha \ll \left | \epsilon \right |{ }^2$) accelerated through the resonance, the spin vector will adiabatically follow the stable spin direction resulting in a spin flip as is indicated in Fig. 1.1. However, for a faster acceleration rate partial depolarization or partial spin flip will occur.

1.2 Polarized Proton Acceleration from ZGS to RHIC

In the first efforts to accelerate polarized proton beams to high energy, the intrinsic resonances were overcome by using a betatron tune jump, which makes the resonance crossing speed α large, and the imperfection resonances were overcome with harmonic corrections of the vertical orbit to reduce the resonance strength 𝜖. Both of these methods aim at making the resonance crossing non-adiabatic. They were used very successfully for the acceleration of polarized proton beams to high energy at the Argonne Zero Gradient Synchrotron (ZGS) and at the AGS. At the ZGS polarized protons were accelerated to 12 GeV/c in 1973, a maximum polarization of 70% was reached using tune jump quadrupoles [3]. Later, polarized protons were accelerated at the strong-focusing AGS to 22 GeV/c. The intrinsic resonance strengths at the strong-focusing AGS were much stronger than at the ZGS requiring stronger and faster pulsed quadrupoles [4]. Figure 1.2 shows the layout used at the AGS as well as tuning curves used to adjust the timing of the pulsed quadrupoles and the strengths of the correction dipoles. These resonance correction techniques require very accurate adjustments at every one of the many imperfection resonance crossings, which becomes very difficult and time consuming.

Fig. 1.2

The first effort to accelerate polarized protons at the AGS used pulsed quadrupoles and correction dipoles, shown in the layout on the right to overcome depolarizing resonances. Typical tuning graphs for the pulsed quadrupole timing (top) and correction dipole strengths (bottom) are shown on the right [4]

Several new techniques to cross both imperfection and intrinsic resonances adiabatically have been developed at the AGS. The correction dipoles used to correct the imperfection resonance strength to zero were replaced by a localized spin rotator or ‘partial Siberian snake’, which makes all the imperfection resonance strengths large and causes complete adiabatic spin flip at every imperfection resonance [5]. The tune jump quadrupoles were initially replaced by a single rf dipole magnet, which increased the strength of the intrinsic resonances by driving large coherent betatron oscillations [6].

Later two strong partial Siberian snakes were installed in the AGS that can overcome both imperfection and intrinsic resonances. With strong enough partial snakes a gap between the spin tune and an integer opens up that becomes large enough to place the fractional part of the betatron tune and therefore the intrinsic resonance inside this gap. Figure 1.3 shows the measured asymmetry during acceleration in the AGS showing adiabatic spin flip at every integer value of Gγ.

Fig. 1.3

Left-right asymmetry measured by the AGS polarimeter during the acceleration cycle. Note that the measured asymmetry flips sign at every integer value of Gγ. The drop of the magnitude of the asymmetry during acceleration is mainly due to the decreasing analyzing power of the analyzing reaction of small-angle proton-carbon scattering

At higher energies a ‘full Siberian snake’ [7], which is a 180^∘ spin rotator of the spin about a horizontal axis, will keep the stable spin direction unperturbed at all times as long as the spin rotation due to the Siberian snake is much larger than the spin rotation due to the resonance driving fields. Therefore the beam polarization is preserved during acceleration. An alternative way to describe the effect of the Siberian snake comes from the observation that the spin tune with the snake is a half-integer and energy independent. Therefore, neither imperfection nor intrinsic resonance conditions can ever be met as long as the betatron tune is different from a half-integer.

Figure 1.4 shows the lay-out of the Brookhaven accelerator complex highlighting the components required for polarized beam acceleration in the AGS and RHIC. The ‘Optically Pumped Polarized Ion Source’ [8] is producing 10¹² polarized protons per pulse. A single source pulse is captured into a single bunch, which is ample beam intensity to reach a RHIC bunch intensity of 2 × 10¹¹ polarized protons.

Fig. 1.4

The RHIC accelerator complex with the elements required for the acceleration and collision of polarized protons highlighted

In the AGS two partial Siberian snakes are installed. One of them is an iron-based helical dipole [9] that rotates the spin by 11^∘. The other is a superconducting helical dipole that can reach a 3 Tesla field and a spin rotation of up to 45^∘ [10]. Both helical dipoles have the same design with a variable pitch along the length of the magnet to minimize orbit excursions and also to fit into the 3 m available straight sections in the AGS. With the two partial snakes strategically placed with one third of the AGS ring between them all vertical spin resonances can be avoided up to the required transfer energy to RHIC of about 25 GeV as long as the vertical betatron tune is placed at 8.98, very close to an integer [11]. This was achieved reliably over the whole acceleration cycle. With an 80% polarization from the source 65% polarization was reached at AGS extraction energy. The majority of the remaining polarization loss in the AGS comes from weak spin resonances driven by the horizontal motion of the beam.

The full Siberian snakes (two for each ring) and the spin rotators (four for each collider experiment) for RHIC each consist of four 2.4 m long, 4 T helical dipole magnet modules each having a full 360^∘ helical twist. The two full snakes in RHIC both rotate the spin by 180^∘ around an axis in the horizontal plane and are placed such that the beam deflection between the two snakes is exactly 180^∘, which guarantees that the spin tune is energy independent. The spin tune is then given by

$$\displaystyle \begin{aligned} u_{sp}=\left(\alpha_b - \alpha_a\right)/\pi. \end{aligned}$$

Here α_a and α_b are the angles of the rotation axes of the two snakes relative to the beam direction. At RHIC these angles are ± 45^∘ to achieve a half-integer spin tune. A spin tune of 1/2 reduces the number of snake resonances to a minimum giving more room for placing the betatron tune. With such orthogonal snake rotation axes the spin tune is also independent of the betatron amplitude for a single spin resonance [12], which could also minimize the amplitude dependent spin tune shift in the presence of multiple resonances.

Figure 1.5 shows the circulating beam intensity in the blue and yellow ring, the measured circulating beam polarizations and luminosities of a RHIC store with a 255 GeV beam energy. A peak luminosity of about 2.5 × 10³² cm⁻² s⁻¹ was reached. The beam polarization of about 55% was calibrated with an absolute polarimeter. This beam polarization is averaged over both beams and over the full store length of about 10 h [13]. To preserve beam polarization in RHIC during acceleration and storage the vertical betatron tune had to be controlled to better than 0.005 and the orbit had to be corrected to better than 1 mm rms to avoid depolarizing snake resonances [14].

Fig. 1.5

Circulating beam intensity in the blue and yellow ring and the measured circulating beam polarization in the blue and yellow RHIC ring (blue(dark) and yellow(light) symbols), respectively are shown in the top plot. The lower plot shows the luminosity at PHENIX (black) and STAR (red(grey)) collider experiments for one typical store

More than 20 years after Y. Derbenev and A. Kodratenko [7] made their proposal to use local spin rotators to stabilize polarized beams in high energy rings, it was demonstrated at RHIC that their concept is working flawlessly even in the presence of strong spin resonances at high energy.

Table 1.1 gives a concise overview of important milestones towards high energy polarized beam accelerator facilities.

Table 1.1 A list of polarized ion beam milestones (col. 1); col. 2: first polarized beam; col 3: polarized species accelerated; col. 4: beam momentum (proton); col. 4: beam polarization achieved (proton); col. 6: milestone. And the future: the electron ion collider, under design, using RHIC rings for polarized p, D and helion beam acceleration

1.3 Polarimetry, Spin Manipulation and Spin Flipping

In addition to maintaining polarization, the accurate measurement of the beam polarization is of great importance. Very small angle elastic scattering in the Coulomb-Nuclear interference region offers the possibility for an analyzing reaction with a high figure-of-merit, which is not expected to be strongly energy dependent [15]. For polarized beam commissioning in the AGS and RHIC an ultra-thin carbon ribbon is used as an internal target, and the recoil carbon nuclei are detected to measure both vertical and radial polarization components. The detection of the recoil carbon with silicon detectors using both energy and time-of-flight information shows excellent particle identification. It was demonstrated that this polarimeter can be used to monitor polarization of high energy proton beams in an almost non-destructive manner and that the carbon fiber target could be scanned through the circulating beam to measure polarization profiles. A polarized atomic hydrogen jet was also installed in RHIC as an internal target for small angle proton-proton scattering, which allows the absolute calibration of the beam polarization to better than 3%.

Artificially driven spin resonances can be used to control the polarization of the stored proton beam. Typically a horizontal magnetic dipole field that is modulated at or close to the spin tune frequency is used. These devices are called either rf or ac dipoles depending on the frequency of modulation. As described above an rf dipole was used in the AGS to overcome intrinsic resonances. By ramping the drive frequency through the resonance condition a full spin flip can be achieved [16]. For RHIC with a spin tune of 0.5 the two resonances that a simple rf dipole produces interfere during the resonance crossing and prevent a full spin flip. A new device that consists of two ac dipoles with a spin rotator in between can avoid this problem by producing only a single resonance by properly adjusting the relative amplitude and phase of the ac dipole excitations [17].

This device can also be used to measure the spin tune. By adiabatically exciting the spin flipper at a drive tune close to the spin tune the beam polarization will be tilted away from the otherwise stable vertical direction. If the strength of the spin flipper is larger than the spin tune spread the whole beam polarization will tilt in the same way and can be measured with the polarimeter. The ratio of the vertical and horizontal polarization component is then proportional to the difference between the drive tune and the spin tune [20].

1.4 Accelerating Polarized Protons to Even Higher Energy

The strength of imperfection resonances approximately increases linearly with energy and the strength of intrinsic resonances increases with the square root of the beam energy. To first order the total spin rotation due to all full or partial Siberian snakes will have to be at least as large as the total spin rotation due to the resonance driving term. More realistically the snakes should at least provide twice the spin rotation of the resonances. This simple rule fits the experience of the AGS, where the partial snakes can overcome the typical AGS resonance strengths of 0.07 and at RHIC, where the two snakes can overcome resonance strengths up to about 0.5. It follows that at higher energy than RHIC multiple snake pairs will be needed. For example, acceleration of polarized protons in the 7 TeV LHC would require at least 16 full snakes or 8 snake pairs to cope with the expected resonance strengths of about 4.

For a ring with N snakes pairs the spin tune is energy independent and equal to 1/2 if

$$\displaystyle \begin{aligned} \sum_{i=1}^N\left(\theta_b^i - \theta_a^i\right)=\pi\ \mathrm{and}\ \sum_{i=1}^N\left(\alpha_b^i - \alpha_a^i\right)/\pi=1/2. \end{aligned}$$

Here $\theta _a^i$, $\theta _b^i$, $\alpha _a^i$, and $\alpha _b^i$ are the azimuthal locations and snake rotation angles of the i-th snake pair, respectively. After satisfying the above conditions these parameters can then be chosen to maximize the beam polarization of particles with large betatron amplitude. This was first examined by K. Steffen [21]. S. Mane [22] showed that snake configurations with rotation axis angle that increase by equal amounts around the ring preserve the property of the two snake configuration of RHIC that the spin tune is independent of the beam emittance for an isolated single spin resonance and should therefore be a promising configuration with minimized amplitude dependent spin tune shift in the presence of multiple spin resonances. Figure 1.6 shows such configurations up to 16 snakes, which would, for example, be applicable to LHC with one snake pair per arc.

Fig. 1.6

Multiple snake configurations of equally spaced Siberian snakes with rotation axis with a steadily increasing angle [22]. Such configurations have been shown to have an emittance independent spin tune in a single resonance model. The question marks mean that these colliders have stopped operation and have never been used for polarized protons

The increasing strength of intrinsic spin resonances can be overcome with an increasing number of properly configured Siberian snakes. However, the random residual orbit distortions, after orbit corrections have been performed, drive imperfection resonances that are unaffected by Siberian snakes. S.Y. Lee and E.D. Courant [23] have shown that the maximum tolerable strength for such a resonance is about 0.05. For RHIC this corresponds to about 250 μm rms residual error of the vertical orbit, which is achievable. However, for LHC this translates to a very challenging 10 μm rms residual orbit error.