Electron Polarization

Abstract

This chapter focuses on the introduction and discussion of electron polarization. In addition to the gyromagnetic ratio, the most different character of electrons compared to protons is that electrons radiate electromagnetic energy in a circular accelerator. A very small correction has to be applied to the electron spin flip to account for the synchrotron radiation. The different instantaneous spin flip probabilities, up to down and down to up, can build up the electron beam polarization state. However, mostly synchrotron radiation tends to disturb the electron orbital motion that is eventually balanced by the radiation damping along an equilibrium orbit. The electron spin motion is described by the modified Thomas-BMT equation with the radiative spin transition term included. Detail of the electron (de)polarization phenomena is described in this chapter. The lecture is extracted from various early theoretical papers, lectures, thesis and presentations (Lee, Accelerator Physics. World Scientific Publishing, 1999; Buon and Koutchouk, Polarization of Electron and Proton Beams. CERN-SL-94-80-AP, 1994; Montague, Phys. Rep. 113(1):1–96, 1984; Lee, Spin Dynamics and Snakes in Synchrotrons. World Scientific Publishing, 1997; Barber and Ripken, Handbook of Accelerator Physics and Engineering, 1st edn. World Scientific Publishing, 2006; Barber, An Introduction to Spin Polarisation in Accelerators and Storage Rings. Cockcroft Institute Academic Training Winter Term, 2014; Mane, Nucl. Instr. Methods Phys. Res. A 292:52–74, 1990; Berglund, Spin-Orbit Maps and Electron Spin Dynamics for the Luminosity Upgrade Project at HERA. DESY-THESIS-2001-044, 2001; Electron-Ion Collider Conceptual Design Report, 2020).

6.1 Synchrotron Radiation

One key aspect of electrons differing from ions in a circular accelerator is that electrons emit much more synchrotron radiation in dipoles. The energy loss of electrons due to the radiation is restored in the RF cavities. Synchrotron radiation spans a continuous spectrum. As shown in Fig. 6.1, the integrated spectral density up to the critical frequency ω_c contains half of the total radiated energy, with the peak occurring approximately at 0.3ω_c.

Fig. 6.1

Integrated spectral density from the synchrotron radiation of electrons

The critical photon energy is given by

$$\displaystyle \begin{aligned} u_c = \hbar \omega_c = \frac{3}{2}\frac{\hbar c}{\rho}\gamma^3. \end{aligned} $$

(6.1)

For electrons, the critical energy in practical units is

$$\displaystyle \begin{aligned} u_c[{\mathrm{keV}}] = 2.218 \frac{E^3[{\mathrm{GeV}}]}{\rho[m]} = 0.665 \cdot E^2[{\mathrm{GeV}}] \cdot B[T] . \end{aligned} $$

(6.2)

The radiation spectrum falls off exponentially beyond ω_c as $e^{-\omega / \omega _c}$ with the total integrated radiation power of

$$\displaystyle \begin{aligned} P_{classical} = \frac{cC_r E^4}{2\pi \rho^2}, \end{aligned} $$

(6.3)

where $C_r = \frac {4\pi r_e}{3(mc^2)^3} = 8.85\times 10^{-5} \frac {\text{m}}{(\text{GeV})^3}$. The total energy radiated in one revolution becomes [1]

$$\displaystyle \begin{aligned} U_0 = \frac{C_r E^4}{2\pi} \oint \frac{ds}{\rho^2}. \end{aligned} $$

(6.4)

The synchrotron radiation is basically a quantum mechanical process. When photons are emitted, the energy of the electron will decrease by the same discrete amount. The corresponding instantaneous radiation power will be reduced. To the first order in ħ, the quantum correction to the radiation power is given by Lee [2]

$$\displaystyle \begin{aligned} P_{qm} = P_{classical}(1 - \frac{55\hbar \omega_c}{8\sqrt{3}E}). {} \end{aligned} $$

(6.5)

The quantum correction factor is of the order of 10⁻⁵ and can not be easily measured. However, the quantum effect is observable in the phase space, where an equilibrium distribution is eventually reached in a balance between radiation damping (a classical phenomenon) and radiation excitation (a quantum mechanical effect) within a few damping times. The corresponding damping time constants in all three dimensions are [1]

$$\displaystyle \begin{aligned} \tau_x & = \frac{2E}{J_x U_0}T_0, \\ \tau_y & = \frac{2E}{J_y U_0}T_0, \\ \tau_E & = \frac{2E}{J_E U_0}T_0, \\ \end{aligned} $$

(6.6)

where E is the electron energy, U₀ is the energy loss in one revolution, T₀ is the revolution period, J_x = 1 − D, J_y = 1, J_E = 2 + D are the horizontal, vertical and longitudinal damping partition numbers respectively, and $D=\frac {1}{2\pi }\oint _{\mathrm {dipoles}} \frac {D_x(s)}{\rho ^2(s)}\, ds$ is the integral evaluated in dipoles where quadrupole focusing and/or defocusing effects are not considered.

The horizontal beam emittance and fractional energy spread in a planar accelerator are given by Lee [1]

$$\displaystyle \begin{aligned} \varepsilon_x & = C_q \gamma^2 \frac{\oint \frac{H}{|\rho|{}^3} ds}{J_x \oint \frac{1}{\rho^2} ds}, \\ (\frac{\sigma_E}{E})^2 & = C_q \gamma^2 \frac{\oint \frac{1}{|\rho|{}^3} ds}{J_E \oint \frac{1}{\rho^2} ds}. \end{aligned} $$

(6.7)

Here C_q = 3.84 × 10⁻¹³m. The dispersion H-function is defined as $H=\gamma _xD^2_x+2\alpha _xD_xD^{\prime }_x+\beta _xD_x^{'2}$. β_x, α_x, γ_x are the horizontal Twiss parameters and D_x and $D^{\prime }_x$ are the horizontal dispersion and its derivative, respectively. Note that the natural vertical emittance is orders of magnitude smaller than the horizontal one in a planar accelerator and ignored unless vertical emittance is excited on purpose.

6.2 Spin-Dependent Synchrotron Radiation

The electron has an intrinsic spin quantum number carrying the angular momentum. With the spin correction included, the radiation power is given by Eq. (6.8). Averaging overall spin orientations for an unpolarized beam, Eq. (6.8) reduces to Eq. (6.5). Spin dependent synchrotron radiation power is very small, however, the disparity of instantaneous spin flip transition rate is significant [2, 3].

$$\displaystyle \begin{aligned} P_{qm} = P_{classical}[1 - (\frac{55}{8\sqrt{3}} + \mathbf{S} \cdot \mathbf{z}) \frac{\hbar \omega_c}{E}]. {} \end{aligned} $$

(6.8)

Ternov et al. [4] discovered that the probability for an electron to emit a photon depends slightly on the initial spin state of the electron. In 1964, Sokolov and Ternov [5] completed the formula of the rate of photon emission for an electron with given initial s_i and final s_f spin states in the direction of the magnetic field

$$\displaystyle \begin{aligned} w(s_i,s_f)=\frac{5\sqrt{3}}{6}\frac{e^2c}{\hbar c\rho}\frac{E}{m_ec^2} \begin{cases} (\text{if } s_i = s_f) & 1-\frac{16\sqrt{3}}{45}\xi+\frac{25}{18}\xi^2-\frac{s_f}{5}(1-\frac{20\sqrt{3}}{9}\xi)\xi \\ (\text{if } s_i e s_f) & \frac{1}{6}(1-s_f\frac{8\sqrt{3}}{15})\xi^2 \end{cases} {} \end{aligned} $$

(6.9)

where ξ = ħω_c∕E. ξ is very small in general, for example 10⁻⁶ for the electron beam energy at 45 GeV. Three messages are delivered from Eq. (6.9):

• 1. the majority of photon emissions does not contribute to the spin flip, since w(s_i = s_f) ≫ w(s_i≠s_f).

• 2. a polarized beam radiates slightly less than an unpolarized beam, since $w(s_i = s_f)+w(s_i e s_f) \lesssim 1$.

• 3. there is an asymmetry in the spin flip probability, depending on s_f. This leads to the polarization build-up with w_↑↓≠w_↓↑.

Note that the probability for spin flipping (w_↑↓: spin flips from up ↑ to down ↓ and w_↓↑: spin flips from down ↓ to up ↑) versus non-spin flipping (w_↑↑: spin flips from up ↑ to up ↑, and w_↓↓: spin flips from down ↓ to down ↓) events is very small, in an order of 10⁻¹² given in Eq. (6.10). However, in the case of spin flip for electrons, the preference of A ≈ 92.4% for the spin state is antiparallel to the magnetic field. A stored positron beam can become polarized as well as an electron beam, with the direction of polarization parallel to the magnetic field.

$$\displaystyle \begin{aligned} \frac{w_{\uparrow \downarrow} + w_{\downarrow \uparrow}}{w_{\uparrow \uparrow} + w_{\downarrow \downarrow}} \sim \xi^2 {} \end{aligned} $$

(6.10)

$$\displaystyle \begin{aligned} A = \frac{w_{\uparrow \downarrow} - w_{\downarrow \uparrow}}{w_{\uparrow \downarrow} + w_{\downarrow \uparrow}} = \frac{8}{5\sqrt{3}} \approx 0.924 \end{aligned} $$

(6.11)

6.3 Sokolov-Ternov Effect

Simplify the conditions with the homogeneous magnetic field and absence of depolarizing effects, the dynamics of polarization can be calculated from Eq. (6.9). At a given time, the beam polarization and its time derivative are:

$$\displaystyle \begin{aligned} P = \frac{N_{\uparrow}-N_{\downarrow}}{N}, \end{aligned} $$

(6.12)

$$\displaystyle \begin{aligned} \frac{dP}{dt} = \frac{1}{N}(\frac{dN_{\uparrow}}{dt} - \frac{dN_{\downarrow}}{dt}). \end{aligned} $$

(6.13)

N_↑ and N_↓ are the numbers of electrons with spin up and spin down, respectively. Their rates of change may be deduced from the transition probabilities in Eq. (6.9):

$$\displaystyle \begin{aligned} \frac{dN_{\uparrow}}{dt} = N_{\downarrow}w_{\downarrow \uparrow} - N_{\uparrow}w_{\uparrow \downarrow} = Nw_{\downarrow \uparrow} - N_{\uparrow}(w_{\downarrow \uparrow} + w_{\uparrow \downarrow}) , \end{aligned} $$

(6.14)

$$\displaystyle \begin{aligned} \frac{dN_{\downarrow}}{dt} = Nw_{\uparrow \downarrow} - N_{\downarrow}(w_{\downarrow \uparrow} + w_{\uparrow \downarrow}) . \end{aligned} $$

(6.15)

Then, the growth of the polarization is

$$\displaystyle \begin{aligned} P(t) = A (1 - e^{-t/\tau_p}). \end{aligned} $$

(6.16)

The maximum degree of polarization is equal to the asymmetry $A=\frac {8}{5\sqrt {3}}\approx 0.924$. The characteristic time τ_p (noted τ_st to distinguish it from other characteristic times to be further introduced) of the polarization build-up is given by

$$\displaystyle \begin{aligned} \tau^{-1}_p = \tau^{-1}_{st} = (w_{\uparrow \downarrow} + w_{\downarrow \uparrow}) = \frac{5\sqrt{3}}{8} \frac{\lambda}{2\pi} \frac{cr_e\gamma^5}{|\rho|{}^3} = \frac{5\sqrt{3}}{8} \frac{\hbar r_e\gamma^5}{m_e|\rho|{}^3} \end{aligned} $$

(6.17)

The guiding field in a real accelerator is piece-wise constant, with changes of its strength and polarity. In the most common cases, the build-up polarization and time due to the Sokolov-Ternov effect [5] can be generalized as

$$\displaystyle \begin{aligned} P_{st} = - \frac{8}{5 \sqrt{3}} \frac{ \oint \frac{ds} {\rho^3(s)}}{\oint \frac{ds} {|\rho(s)|{}^3}}, \end{aligned} $$

(6.18)

$$\displaystyle \begin{aligned} \tau^{-1}_{st} = \frac{5\sqrt{3}}{8} \frac{\hbar r_e \gamma^5}{m_e C} \oint \frac{ds} {|\rho(s)|{}^3}. \end{aligned} $$

(6.19)

Here C is the accelerator circumference and r_e is the classical electron radius.

Table 6.1 lists several characteristic parameters of several electron storage rings. Note that the polarization build-up time is much longer than the phase space damping time in an electron storage ring. Typically, the polarization build-up time is of the order of minutes to hours, while the phase space damping time is of the order of milliseconds.

Table 6.1 Characteristic parameters of past and future polarized electron storage rings

6.4 Baier-Katkov-Strakhovenko Equation

In the late 1960s, Baier, Katkov and Strakhovenko (BKS) [6] generalized the spin-flip transition probability from the uniform magnetic fields to arbitrary magnetic field configurations. The general equation of evolution of electron polarization is derived as follows, in the presence of radiative polarization and absence of depolarizing effects due to the stochastic photon emission on the orbit [7]:

$$\displaystyle \begin{aligned} \frac{d \mathbf{P}}{dt} = \mathbf{\Omega}_{co} \times \mathbf{P} - \frac{1}{\tau_{st}(s)} [\mathbf{P} - \frac{2}{9} \mathbf{s} (\mathbf{P} \cdot \mathbf{s}) + \frac{8}{5\sqrt{3}} \mathbf{b}(s)]. \end{aligned} $$

(6.20)

The first part of the equation is the standard Thomas-BMT equation on the closed orbit, which can be solved in the form of $\mathbf {P}(s)=R^{co}_{3\times 3}(s,s_0)\mathbf {P}(s_0)$, described in the early chapters in this book. Here $R^{co}_{3\times 3}(s,s_0)$ is the 3 × 3 spin transfer matrix on the closed orbit. The second part of the equation describes the polarization motion with the spin flipping rate given by Baier and Katkov.

The BKS equation does not contain depolarizing terms, however, a derivation of its solution is a useful step towards incorporating the spin-orbit coupling depolarization effect. We replace t with θ, commonly used to describe the particle motions in an accelerator, as the independent variable in the Eq. (6.20), then the instantaneous polarization build-up rate is given by

$$\displaystyle \begin{aligned} \tau^{-1}_{ip} = \frac{5\sqrt{3}}{8} \frac{R \hbar r_e \gamma^5}{m_e c \rho^3}, \end{aligned} $$

(6.21)

where R is the average radius of the machine. Choosing the unit reference vectors (e₁, e₂, n₀) with respect to that the spin precesses at a constant rate ν₀, we have Ω_co = ν₀n₀ and the equation of motion for the electron polarization is written as

$$\displaystyle \begin{aligned} \frac{d\mathbf{P}}{d\theta} = u_0 {\mathbf{n}}_0 \times \mathbf{P} - \frac{1}{\tau_{ip}} [\mathbf{P} - \frac{2}{9} \mathbf{s} (\mathbf{P} \cdot \mathbf{s}) + \frac{8}{5\sqrt{3}} \mathbf{b}(s)]. \end{aligned} $$

(6.22)

Making the scalar product of n₀ yields the equation of evolution for the projection of P on the closed solution n₀, that is P₃ = n₀ ⋅P

$$\displaystyle \begin{aligned} \frac{dP_3}{d\theta} = - \frac{1}{\tau_{ip}} [P_3 \{1 - \frac{2}{9} ({\mathbf{n}}_0 \cdot \mathbf{s})^2 \} + \frac{8}{5\sqrt{3}}({\mathbf{n}}_0 \cdot \mathbf{b}) - \frac{2}{9}({\mathbf{n}}_0 \cdot \mathbf{s})\{P_1 + P_2\} ]. \end{aligned} $$

(6.23)

In most cases, $\tau _{ip}^{-1}$ is very small compared to ν₀. Hence, the fast-oscillating components P₁ and P₂ transverse to n₀ make a negligible contribution if Eq. (6.23) is averaged over one revolution and initial spin phases, then the third term can be dropped consequently

$$\displaystyle \begin{aligned} \frac{dP_3}{d\theta} = - \frac{1}{\tau_{ip}} [P_3 \{1 - \frac{2}{9} ({\mathbf{n}}_0 \cdot \mathbf{s})^2 \} + \frac{8}{5\sqrt{3}}({\mathbf{n}}_0 \cdot \mathbf{b}) ]. \end{aligned} $$

(6.24)

For the initial condition of P₃(0) = 0, Eq. (6.24) can be integrated to yield

$$\displaystyle \begin{aligned} P_3 = - \frac{B}{A} (1 - e^{-A\theta}). \end{aligned} $$

(6.25)

Since $\tau _{ip}^{-1}$ is small, the coefficients A and B can be expressed as averages over the machine circumference as

$$\displaystyle \begin{aligned} A & = \frac{5\sqrt{3}}{8} \frac{R \hbar r_e \gamma^5}{m_e c} \oint ds \frac{1 - \frac{2}{9} ({\mathbf{n}}_0(s) \cdot \mathbf{s})^2} {|\rho(s)|{}^3}, \\ B & = \frac{R \hbar r_e \gamma^5}{m_e c} \oint ds \frac{{\mathbf{n}}_0(s) \cdot \mathbf{b}(s)} {|\rho(s)|{}^3}. \end{aligned} $$

(6.26)

The asymptotic polarization for θ→∞ becomes [8]

$$\displaystyle \begin{aligned} P_{bks} = P_3(\infty) = - \frac{B}{A} = - \frac{8}{5 \sqrt{3}} \frac{ \oint ds \frac{{\mathbf{n}}_0(s) \cdot \mathbf{b}(s)} {|\rho|{}^3}}{\oint ds \frac{1 - \frac{2}{9} ({\mathbf{n}}_0(s) \cdot \mathbf{s})^2} {|\rho(s)|{}^3}}, \end{aligned} $$

(6.27)

with the BKS polarization build-up rate of [8]

$$\displaystyle \begin{aligned} \tau^{-1}_{bks} = \frac{5\sqrt{3}}{8} \frac{\hbar r_e \gamma^5}{m_e C} \oint ds \frac{1 - \frac{2}{9} ({\mathbf{n}}_0(s) \cdot \mathbf{s})^2} {|\rho(s)|{}^3}. \end{aligned} $$

(6.28)

Comments on the BKS equation:

• The asymptotic polarization of a beam is built up along the direction of the closed solution n₀, which does not in general lie along the direction of magnetic field b. In a perfectly aligned accelerator, where n₀ ×b = 0 and n₀ ⋅s = 0, the asymptotic polarization is $\frac {8}{5\sqrt {3}}\approx 92.4 \%$ with the direction anti-parallel (parallel) to the magnetic field for electrons (positrons).

• It is illusory that the negative sign of (n₀ ⋅s) would appear to increase P_bks, since (n₀ ⋅b) dominates over (n₀ ⋅s)² in the denominator.

• The (n₀ ⋅s)² term results in some reduction of the polarization rate. However, this is a rather small effect in practice because (n₀ ⋅s) cannot be permitted to differ substantially from zero over a large fraction of the machine circumference; otherwise it would imply a large average value of |d|² (the square of spin-orbit coupling function) and strong depolarization.

• The absolute value |ρ|³ appears inside the averaging brackets. This is because we assume implicitly that τ_ip is always positive, and the presence of regions where the magnetic field is not in the vertical direction is taken care of by the term (n₀ ⋅b) in the expression. For example, the machine may contain wigglers, composed of a sequence of dipole magnets with alternating polarities, which cause the sign of (n₀ ⋅b) to alternate.

6.5 Spin Diffusion

6.5.1 Spin Perturbation by Quantum Excitation

There are two distinct aspects related to the synchrotron radiation that modify depolarizing effects for electrons as compared with protons: sudden change of the energy due to the photon emission on the one hand and much slower energy recovery from the RF system on the other hand. As described in Eq. (6.9), the majority of photon emission is not associated with a spin flip. The abrupt energy jump causes the electron to initiate additional synchrotron and betatron oscillations since the closed orbit is in general energy dependent. During the subsequent evolution of the trajectory, the electron is subjected to a different sequence of magnetic fields from that corresponding to the original closed orbit, and the spin motion is consequently perturbed with respect to the initial state. The recovery of the lost energy from the RF system is accompanied by damping of orbit oscillations, and the electron damps to its initial orbit after several damping times. During the orbital damping time, the spin precesses many times and follows adiabatically the slow changes in magnetic field experienced by the electron. However, there is no damping of the spin motion: electron spin depends on the whole of its history following the photon emission and ends up with a different orientation from its initial one. The whole process can be explained by Fig. 6.2 Buon and Koutchouk [3].

Fig. 6.2

Evolution of the phase space and spin coordinates of a reference particle that emits a photon. The first row represents the initial state of the particle in the phase space just before the photon emission. The second row represents the particle state just after the photon emission. The third row represents the evolution of the coordinates in three phase space over several damping times

The first row of Fig. 6.2 represents the initial state of the particle in the phase space just before the photon emission: the particle is at the origin of the coordinates (i.e. the closed orbit) and the spin S is along an axis n to be vertical.

The second row of Fig. 6.2 presents the particle state just after the photon emission. The energy coordinate suddenly become negative due to the energy loss. The position of the particle does not change during the short time of photon emission. However, the closed orbit changes due to the change of particle’s energy. This results in the changes of dispersion orbit and betatron amplitude if the dispersion function does not vanish. Particle starts oscillating in the three phase spaces. Along with the new trajectories, the spin S is precessing about a new spin axis n, generally tilted with respect to the initial spin direction.

The third row of Fig. 6.2 shows the evolution of the coordinates in three phase spaces over several damping times. The spatial and momentum coordinates damp and reach to the equilibrium conditions. During this process, the spin axis n is gradually restored to its initial vertical position. However, the spin S precesses very rapidly and follows adiabatically the spin axis n, and finds itself tilted at an angle when the orbital coordinates are restored. Averaging over all particles in a beam, the horizontal component of polarization vanishes since the spin precesses in a stochastic way as photons are emitted. The remaining polarization is the projection of the initial polarization vector onto the spin axis after it has been tilted by the photon emission. This rather simple picture of depolarization due to the quantum excitation arises because of the very large difference among the time constants of three relevant phenomena: micro-second time scale for one spin precession, milli-second time scale for the damping of orbital motions and the gradual change of the n, minutes to hour time scale for the polarization build-up mechanism.

6.5.2 Spin-Orbit Coupling Function

To clarify the definition of d introduced by Derbenev and Kondratenko to describe the depolarizing influence of quantized synchrotron radiation in terms of the resulting random fluctuations of the precession axis, we use a simple model of the spin diffusion process: a single electron moving on the normal closed orbit with the design momentum and having its spin aligned along the nominal precession axis n₀ corresponding to this orbit, i.e., n₀ is the invariant spin field for a particle on the closed orbit. After the emission of photons, the orbit parameters change and so does the precession axis, n₀→n. Let us assume that the tilt of spin precession direction Δn = n −n₀ is proportional to the relative energy loss ΔE, then the spin-orbit coupling function d is defined as

$$\displaystyle \begin{aligned} \mathbf{d} = \frac{\partial{\mathbf{n}}}{\partial{\delta}}. \end{aligned} $$

(6.29)

Here $\delta =\frac {\Delta E}{E}$. d ≡d(u;s) is a vectorial quantity which depends on the azimuth s along the trajectory and summarizes the contributions from the betatron oscillations as well as the synchrotron oscillation u ≡ (x, p_x, y, p_y, z, δ).

The BKS equation describes the evolution of the polarization on the assumption that n₀ is constant at any specified azimuth, and is not subject to quantum fluctuations or oscillatory perturbations. In order to introduce the depolarizing effect of the spin-orbit coupling function, we supplement P₃ in Eq. (6.24) by an additional contribution ΔP₃ arising from the perturbation of n, with Δn is the change due to the emission of photons. Following the conclusion drawn from the solution of the BKS equation, this model can also constitute the generalization to the case of an ensemble of electrons with an average projection P₃ =< P ⋅n₀ > .

Since n is a unit vector and | Δn|≪ 1, the change ΔP₃ in the projection P₃ arising from Δn is

$$\displaystyle \begin{aligned} \Delta P_3 \approx -P_3\frac{|\Delta \mathbf{n}|{}^2}{2} = -\frac{P_3}{2}|\mathbf{d}|{}^2\delta^2. \end{aligned} $$

(6.30)

The rate of change of the energy fluctuation due to the quantum emission can be expressed in terms of the azimuthal variable θ from Eq. (6.21):

$$\displaystyle \begin{aligned} \frac{d\delta^2}{d\theta} = \frac{55}{24 \sqrt{3}}\frac{R \hbar r_e \gamma^5}{m_e c \rho^3} = \frac{11}{9}\frac{1}{\tau_{ip}}. \end{aligned} $$

(6.31)

Using Eq. (6.31) in Eq. (6.30), the fluctuation term can be obtained

$$\displaystyle \begin{aligned} \frac{d\Delta P_3}{d\theta} = -\frac{11}{18}\frac{|\mathbf{d}|{}^2}{\tau_{ip}}P_3, \end{aligned} $$

(6.32)

which can be introduced into the BKS equation Eq. (6.24) to obtain

$$\displaystyle \begin{aligned} \frac{dP_3}{d\theta} = - \frac{1}{\tau_{ip}} [P_3 \{1 - \frac{2}{9} (\mathbf{n} \cdot \mathbf{s})^2 + \frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2 \} + \frac{8}{5\sqrt{3}}(\mathbf{n} \cdot \mathbf{b}) ]. \end{aligned} $$

(6.33)

The solution is similar to the BKS equation but now with

$$\displaystyle \begin{aligned} A & = \frac{5\sqrt{3}}{8} \frac{R \hbar r_e \gamma^5}{m_e c} \oint ds \left\langle\frac{1 - \frac{2}{9} (\mathbf{n} \cdot \mathbf{s})^2 + \frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2 } {|\rho(s)|{}^3}\right\rangle_s, \\ B & = \frac{R \hbar r_e \gamma^5}{m_e c} \oint ds \left\langle\frac{\mathbf{n} \cdot \mathbf{b}} {|\rho(s)|{}^3}\right\rangle_s. \end{aligned} $$

(6.34)

A and B are expressed as averages over the machine circumference, as well as phase space at every azimuth s. The asymptotic polarization level is reduced to

$$\displaystyle \begin{aligned} P_{dk} = P_3(\infty) = - \frac{B}{A} = - \frac{8}{5 \sqrt{3}} \frac{ \oint ds \left\langle\frac{\mathbf{n} \cdot \mathbf{b}} {|\rho(s)|{}^3}\right\rangle_s} {\oint ds \left\langle\frac{1 - \frac{2}{9} (\mathbf{n} \cdot \mathbf{s})^2 + \frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2 } {|\rho(s)|{}^3}\right\rangle_s}. \end{aligned} $$

(6.35)

Note that, a value of |d| around unity results in a substantial reduction of the asymptotic polarization level.

The corresponding effective polarization rate becomes [8]

$$\displaystyle \begin{aligned} \tau^{-1}_{dk} = \frac{5\sqrt{3}}{8} \frac{\hbar r_e \gamma^5}{m_e C} \oint ds \left\langle\frac{1 - \frac{2}{9} (\mathbf{n} \cdot \mathbf{s})^2 + \frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2} {|\rho(s)|{}^3}\right\rangle_s. \end{aligned} $$

(6.36)

It can be written as

$$\displaystyle \begin{aligned} \tau^{-1}_{dk} = \tau^{-1}_{bks} + \tau^{-1}_{dep}, \end{aligned} $$

(6.37)

where $\tau ^{-1}_{bks}$ is given Eq. (6.28) and

$$\displaystyle \begin{aligned} \tau^{-1}_{dep} = \frac{5\sqrt{3}}{8} \frac{\hbar r_e \gamma^5}{m_e C} \oint ds \left\langle \frac{\frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2} {|\rho(s)|{}^3}\right\rangle_s. \end{aligned} $$

(6.38)

The time dependence of build-up of polarization from an initial polarization P₀ to equilibrium is [8]

$$\displaystyle \begin{aligned} P(t) = P_{dk}[1-e^{-t/\tau_{dk}}]+P_0e^{-t/\tau_{dk}} \end{aligned} $$

(6.39)

Figure 6.3 shows the time scales for an electron storage ring of 25 GeV: ranging from 10⁻¹⁰ s for the duration of the quantum emission process to over 10⁴ s for a desired depolarization time that exceeds the polarization build-up time by a factor of ten. Note that, the large separation between times for (de)polarization, radiation damping and orbital oscillation modes help to simplify calculations by the use of average methods. The separations between the oscillation-mode time scales, the interval between quanta emitted by a single electron and the duration of the quantum-emission permit the latter to be considered as an abrupt random process with no correction between successive photons.

Fig. 6.3

Characteristic time scales in a typical 25 GeV electron storage ring

6.6 Kinetic Polarization

In addition to the spin flip, Eq. (6.9) also describes the polarizing effect involving photon emission without the spin flip. This process combines two aspects, the dependence of synchrotron radiation intensity on the spin state and the energy dependence of the invariance spin field n. The principle can be explained using a simple model as shown in Fig. 6.4. We assume S₁ and S₂ initially have equal projections on n₀. After the photon emission, the precession axis abruptly changes by δn. The projection of spin vector S₁ on n is thereby increased, while the projection of spin vector S₂ is reduced evidently. If the probability of the photon emission, for the case of no spin flip, were the same for both two spin vectors, the net effect would be zero on average. However, The photon emission probability is higher for an electron with state S₁ than S₂, since S₁ has a larger projection on the magnetic field direction b than S₂. Averaging over all spin states, it results in an increase in the polarization along n.

Fig. 6.4

Kinetic polarization arising from the change of precession axis due to the photon emission

The polarizing contribution δP₃ arising from $\frac {\partial {\mathbf {n}}}{\partial {\delta }}$ can be introduced to Eqs. (6.27) and (6.35). The general equation for the asymptotic polarization is obtained by the Derbenev-Kondratenko formula: [8]

$$\displaystyle \begin{aligned} P_3(\infty) = - \frac{B}{A} = - \frac{8}{5 \sqrt{3}} \frac{ \oint ds \left\langle\frac{\mathbf{b} \cdot (\mathbf{n} - \frac{\partial{\mathbf{n}}}{\partial{delta}})} {|\rho(s)|{}^3}\right\rangle_s} {\oint ds \left\langle\frac{1 - \frac{2}{9} (\mathbf{n} \cdot \mathbf{s})^2 + \frac{11}{18}|\frac{\partial{\mathbf{n}}}{\partial{\delta}}|{}^2 } {|\rho(s)|{}^3}\right\rangle_s}. \end{aligned} $$

(6.40)

Comments on the kinetic polarization are given as follows.

• 1. The kinetic polarization effect is more favorable in rings where n₀ is horizontal, i.e. n₀ ⋅b = 0. In such rings, $\frac {\partial {\mathbf {n}}}{\partial {\delta }}$ has a vertical component in the dipole fields. This leads to a build-up of polarization, even though the pure Sokolov-Ternov effect vanishes. The rate is still $\tau ^{-1}_{dk}$.

• 2. The maximum kinetic polarization in an idealized model with certain constraints theoretically can reach to about 95%, comparing to 92.4% from the normal Sokolov-Ternov polarization. However, in practice, other constraints are likely to dominate the polarization.

• 3. In most typical electron storage rings, the n₀ is very closely parallel to b to avoid depolarizing effects arising from the $|\frac {\partial {\mathbf {n}}}{\partial {\delta }}|{ }^2$. Usually, the kinetic polarization effect is very small, unless there exist certain special conditions.

• 4. One needs pay attention to the kinetic polarizing mechanism in some special configurations of bending magnets where a substantial contribution to $\mathbf {b} \cdot \frac {\partial {\mathbf {n}}}{\partial {\delta }}$ occurs locally, such as spin rotators and Siberian snakes. Strong magnetic fields in these regions could usefully enhance or destructively reduce the asymptotic polarization, especially if b ⋅n contribution from other parts of the rings was insufficient.

6.7 Resonances

Analogous to the orbital motion, the behavior of the spin precession axis in the presence of an arbitrary perturbation can be expressed in terms of components with frequencies ν_i from the Fourier transform. This representation can be applied to the spin-orbit coupling function d by interpreting the perturbation as arising from the emission of a phonton. We start with the effective perturbation Δn of the precession axis immediately after the photon emission, i.e.,

$$\displaystyle \begin{aligned} \Delta \mathbf{n} = \sum_j \frac{\epsilon_j e^{-iu_j\theta_0}}{u_{spin} - u_j}{\mathbf{f}}_+ + h.c. \end{aligned} $$

(6.41)

where 𝜖_j is the magnitude of the perturbation, f₊ is the unit complex vector perpendicular to the third axis, and h.c. represents hermitian conjugate. Let us define 𝜖_j = c_jδ, then the spin-orbit coupling function becomes

$$\displaystyle \begin{aligned} \mathbf{d} = \sum_j \frac{c_j e^{-iu_j\theta_0}}{u_{spin} - u_j}{\mathbf{f}}_+ + h.c. \end{aligned} $$

(6.42)

In a case where the perturbations arise from the closed orbit errors, with the harmonics of amplitude z_k. Since the orbit harmonics are integer, we put ν_j = k and Eq. (6.41) becomes

$$\displaystyle \begin{aligned} \Delta \mathbf{n} = \sum_k \frac{\epsilon_k e^{-ik\theta_0}}{u_{spin} - k}{\mathbf{f}}_+ + h.c., \end{aligned} $$

(6.43)

and Eq. (6.42) becomes

$$\displaystyle \begin{aligned} \mathbf{d} = \frac{\partial \mathbf{n}}{\partial \delta} = \sum_k \frac{e^{-ik\theta_0}}{u_{spin} - k} \{ \frac{\partial c_k}{\partial \delta} - \frac{c_k}{u_{spin}-k}\frac{\partial u_{spin}}{\partial \delta} \} + h.c. \end{aligned} $$

(6.44)

The strength c_k of an integer spin resonance driven by an orbit harmonic of amplitude z_k is approximately given as

$$\displaystyle \begin{aligned} c_k = \frac{u_{spin} k^2 z_k}{R}, \end{aligned} $$

(6.45)

where R is the accelerator radius and z_k is related to the harmonic B_k of the magnetic field error, and ν_spin = γa. In a real machine with errors, the variation of c_k with energy comes from both ν_spin = γa of the spin tune and z from the vertical dispersion D_y. Therefore,

$$\displaystyle \begin{aligned} \frac{\partial c_k}{\partial \delta} = \frac{u_{spin} k^2}{R}(z_k+D_k), \end{aligned} $$

(6.46)

where $D_k=\frac {\partial z_k}{\partial \delta }$ is the kth harmonic of vertical dispersion, and $\gamma \frac {\partial {u _{spin}}}{\partial {\delta }}=u _{spin}$. Then, Eq. (6.44) becomes

$$\displaystyle \begin{aligned} \mathbf{d} & = \sum_k \frac{e^{-ik\theta_0}}{u_{spin} - k} \{ c_k[1 - \frac{u_{spin}}{u_{spin}-k}] + u_{spin} k^2 \frac{D_k}{R} \}{\mathbf{f}}_+ + h.c. \\ & = \sum_k \frac{u_{spin} k^2}{R}e^{-ik\theta_0} \{ \frac{D_k}{u_{spin}-k} - \frac{kz_k}{(u_{spin}-k)^2} \}{\mathbf{f}}_+ + h.c. \end{aligned} $$

(6.47)

The strongest contribution to d obviously comes from the harmonic k close to the spin tune ν_spin. Also, because of the different powers of the resonant dominator in Eq. (6.47), the influence of the dispersion harmonic D_k extends further than that of the orbit harmonics z_k. If we assume ν_spin to be mid-way between two integers, i.e. $u _{spin} - k = \frac {1}{2}$ and the relative importance of dispersion and orbit errors can be assessed by comparing D_k with 2kz_k. For example [7], in LEP at 50 GeV, k ≈ ν_spin ≈ 100 and z_k (before the orbit correction) might be typically around 5 × 10⁻² mm, then 2kz_k ≈ 10 mm. The rough estimates suggest that D_k should be approximately 5 mm, then the two terms are likely to be of similar orders of magnitude. With R ≈ 4 × 10³m, one obtains |d|≈ 7 in the worse case, which would give a large degree of depolarization. There is an obvious need for compensating the critical harmonics of both orbit and dispersion errors.

6.8 Synchrotron Sideband Resonances

The energy variation arising from synchrotron oscillations modulates the spin tune, leading to synchrotron sideband resonances in the vicinity of some orbit spin resonances. The synchrotron sideband resonances are more troublesome for high-energy electron storage rings than for proton rings. There are two main reasons:

• 1. the relatively large of synchrotron tune: The large value of synchrotron tune in electron rings results from a combination of the large RF voltage required to restore the energy loss from the synchrotron radiation and high frequency needed to minimize the overall cost of the RF system.

• 2. the large energy spread: The large energy spread of the electron beam is a consequence of the strong quantum excitation of orbit oscillations at high energies and the need to take account of particles far out in the tails of the Gaussian distribution. The spread of the spin tune is correspondingly large and tends to extend into the region where synchrotron sideband resonances of relatively low orders are present.

Figure 6.5 shows an example of measurements of the equilibrium polarization of the positron beam in the storage ring SPEAR, where P_max = 92.4%. Around the parent resonance ν − ν_x = 3, there are four synchrotron sideband resonances: ν − ν_x + ν_s = 3, ν − ν_x − ν_s = 3, ν − ν_x + 2ν_s = 3, ν − ν_x − 2ν_s = 3. These synchrotron sideband resonances narrow down the available working points in the machine operation and may be a troublesome to achieve a high polarization practically.

Fig. 6.5

Polarization measurements at SPEAR [9, 10]. The curve is a guide for the eye, not a theoretical calculation. The spin tune is ν, and the orbital tunes are ν_x,y,s

6.9 Maximization of the Polarization

6.9.1 Minimization of the Depolarization

The general mechanisms to increase the electron polarization are similar to the ones applied to increase the proton polarization, such as [3]

• 1. minimizing the machine imperfections. The mere correction of the orbit with respect to the beam monitors generally aligned on the near-by quadrupoles does not guarantee a good compensation of spin rotation. Figure 6.6 shows the calculated polarization in the LEP for given misalignments and residual orbits after the correction. The results are averaged over a sample of random imperfections. Aligning the LEP vertical orbit with the tightest tolerance has indeed increased the polarization.

  Fig. 6.6

  

  Dependence of the polarization on the machine alignment in LEP [11]. As it is shown that the correction of the orbit with respect to the quadrupoles is the key to compensate the spin rotation and improve the polarization

• 2. optimizing the correction. The correction of the vertical orbit is essential. In LEP, the adjustment of the lattice, together with improved beam monitors, reduced the rms residual orbit deviation from 1 mm to 0.3 mm. The correction of the orbit with a large number of orbit correctors reduced the vertical dispersion function significantly. This consequently reduced the excitation of the synchrotron spin resonances.

• 3. carefully choosing machine parameters. The natural polarization level can be maximized through a careful choice of beam tunes to avoid all or major strong systematic resonances.

6.9.2 Minimization of the Tilt of the n -Axis

In the high energy electron storage rings, the global minimization of depolarization sources is not sufficient to reach a high polarization. A mechanism, called spin matching procedure aiming at make the orbital motion transparent to the spin motion, is necessary to improve the asymptotic polarization.

The first step is to minimize the deviation of the invariant spin field n from the vertical in the dipoles that is mainly produced by vertical closed orbit distortions. This deviation is large on integer spin tune resonances. The depolarization effect can be associated with the tilt of n decreased with the 4^th power of the distance between the spin tune and the integer. To avoid any significant depolarization, n is normally at most a few milli-radians away from the vertical. This method is known as harmonic synchrobeta spin matching, which minimizes the strengths of depolarizing resonances by generating horizontal fields that are stationary in a (l₀, m₀, n₀) frame and adjusting their amplitudes and phases so as to compensate the driving term of each integer resonance. Here (l₀, m₀, n₀) are right hand orthonormal.

The generated harmonics usually do not affect the closed orbit significantly as they are far from the betatron tunes, and such compensation can be very empirical or computed from the measured orbit. However, it was successfully applied at PETRA, which increased the polarization from 40% to 80%. Generation of horizontal field harmonics could be achieved by a few short vertical closed orbit bumps rather than by an orbit perturbation all around the ring, as shown in Fig. 6.7. Figure 6.8 shows the rapid change of the build-up polarization in the LEP after the harmonic spin matching method was applied.

Fig. 6.7

In HERA, LEP and Tristan, a pattern of vertical π-bumps in the arcs is used to generate sine and cosine magnetic field harmonics to compensate the near-by integer spin resonances

Fig. 6.8

Improvement of the polarization build-up in LEP following a calculated harmonic correction of the vertical orbit [12]

6.9.3 Minimization of the Spin-Orbit Coupling Function

Harmonic spin matching can improve the electron polarization. However, the depolarization in the electron rings is dominated by the betatron and synchrotron spin resonances and not the integer resonances. At high electron energies, due to the large beam energy spread, the synchrotron spin resonances are overwhelming. The horizontal and vertical betatron resonances are also somewhat excited by the synchrotron-betatron coupling. Spin rotators that are introduced for spin manipulations may also significantly enhance the resonances related with horizontal betatron and synchrotron motions. The major transverse depolarization resonances are directly dependent on the tilt of the n axis. When the orbit is well corrected and controlled, the synchrotron spin resonances become dominant. It becomes more incentive to compensate more exactly the spin-coupling integrals at high energies. This method is known as strong synchrobeta spin matching that will be discussed in detail in the chapter of Spin Matching.

6.10 Polarized Electron Beams in Rings

Collisions of highly polarized electrons and protons at high energies enable a comprehensive study to understand the structure of the proton and neutron directly from the dynamics of their quarks and gluons. The electron serves as a probe to bear the unmatched precision of the electromagnetic interaction, while the proton determine the correlations of quark and gluon distributions. High values of polarization significantly reduce the uncertainties in determination of these correlations and provide the possibility of finding new physics with experimental evidences for the parity violation and weak interaction. Two accelerator facilities are utilized here to present the generation, manipulation and preservation of a polarized electron beam in a collider.

6.10.1 Polarized Electrons in HERA (1992–2007)

HERA, as shown in Fig. 6.9, was a 6.3 km long electron(positron)/proton collider located at Deutsches Elektronen Synchrotron, DESY, in Hamburg, Germany. The machine was routinely operated with collision energies of 27.5 GeV for electrons (positrons) and 920 GeV for protons at a center of mass energy of 318 GeV [13]. It was the only lepton-proton collider in the world while operating, and still the only one while this book is edited. Although the HERA experiments ended in 2007, the data analysis continues to study the proton inner structure and point the way for future particle physics experiments. The following is the major history of electron/positron polarization at HERA [14].

• 1981: first ideas on designing the HERA

  Fig. 6.9

  

  The HERA complex

• 1988–1991: transverse polarimeter designed and installed.

• 1991: first polarization measurement, ≈ 8% vertical polarization.

• 1992: better ring alignment and implemented harmonic bumps, >60% polarization.

• 1993–1994: first pair of spin rotators installed.

• 1994: successful operation with spin rotators.

• 1994–2000: routinely operation with polarization >50%

• 1997: longitudinal polarimeter installed.

• September 2000–Summer 2001: upgrade.

• March 2003–June 2007: 3 pairs of spin rotators.

• 2004–2007: “Fabry–Perot” cavity polarimeter for longitudinal polarization.

Figure 6.10 shows the invariant spin field n₀ on the closed orbit at HERA. As discussed in the early sections of this chapter, n₀ is designed to be vertical in arcs to provide an optimum asymptotic polarization and minimize the depolarization. The longitudinal polarization at the collision point is obtained by the spin rotators installed in the up- and downstream of the collision point. Figure 6.11 shows the electron orbital and spin motions in the HERA spin rotator. Such spin rotator is composed of interleaved horizontal and vertical dipole magnets, with a configuration of antisymmetric vertical magnetic field and symmetric radial field. With proper magnet placements and beam energy, the designed orbit is intact outside of the rotator. Spin matching was carried out in HERA successfully and it improved the polarization, as discussed in early sections. Figure 6.12 shows the electron polarization as a function of time at 26.7 GeV in HERA. The fifth power of the polarization build-up versus energy is observed when the beam energy is raised in the storage ring.

Fig. 6.10

The invariant spin field n₀ on the closed orbit in HERA

Fig. 6.11

HERA spin rotator [14]

Fig. 6.12

The electron polarization as a function of the time in the HERA [3]

6.10.2 Polarized Electrons in EIC

The Electron Ion Collider (EIC) [15] will be a 3.8 km long particle accelerator, shown in Fig. 6.13, built at the Brookhaven National Laboratory (BNL) in the United States of America. With the collisions of electrons and various ion species, the EIC will enable the physicists to understand the nature of matter at its most fundamental level, providing the clearest picture of how the elemental quarks and gluons interact to form the basic structure of atoms and nuclei. EIC has the following unique features:

• large center of mass energy range of 20–140 GeV, with ion beam energy from 20 to 275 GeV/u and electron beam energy from 5 to 18 GeV,

  Fig. 6.13

  

  EIC layout [15]. The ion and electron collider rings and electron injector, i.e. Rapid Cycling Synchrotron (RCS), share the same present RHIC tunnel. EIC can accommodate two collision points for experiments

• highly polarized ≥ 70% electron, proton and light ion beams, and

• high collision luminosity 10³³ to 10³⁴.

Both polarized electron and light ion beams are desired in the EIC. In particular, the electron polarization requirements are:

• high polarization ≥ 70% in the energy range of 5–18 GeV,

• longitudinal polarization orientation at the IP,

• opposite polarization helicities within the same store, and

• long polarization lifetime.

It is extremely challenging to reach all design goals. The experience on providing a highly polarized electron beam in the HERA is valuable, however, brainstorming on designing, preserving and manipulating the electron polarization are highly desired in order to meet all the stringent requirements in the EIC. In a summary, the strategies on the design of electron polarization in the EIC Electron Storage Ring (ESR) are:

• highly polarized electrons with two opposite polarization directions are injected in to the ESR,

• polarization is vertical in arcs to avoid spin diffusion and longitudinal at IP for physics experiments,

• spin rotators rotate the spin from the vertical in arcs to longitudinal at IP,

• spin matching is implemented to preserve high asymptotic (equilibrium) polarization and extend the polarization relaxation time, and

• electron bunches regular replacement down to a few minutes at highest beam energy 18 GeV is needed to obtain a high average polarization.

Figure 6.14 shows the Sokolov-Ternov times in the ESR at several EIC interested electron beam energies. Note that, it is not practical to build up the electron polarization in the EIC ESR due to its long Sokolov-Ternov time, up to more than 10 hours, in this relatively low electron beam energy range.

Fig. 6.14

Sokolov-Ternov time as a function of electron beam energy in the EIC ESR. The time has a strong dependence on the dipole bending radius (|ρ|³) and beam energy ($\frac {1}{\gamma ^{5}}$). The change of the slope in the low energies is due to the enhanced radiation by splitting the dipole structure at energies below 10 GeV. Overall, building up the electron polarization in the EIC ESR is not practical, especially in the low energies. EIC adopts a design of a full-energy injection of polarized electron bunches into the ESR with the desired spin direction from the polarized electron source

Figure 6.15 shows the electron spin rotator design in the EIC ESR. It is composed of interleaved solenoid and dipole fields. Such a spin rotator can rotate the electron spin from vertical to longitudinal, and vice versa, in the whole beam energy region from 5 to 18 GeV. Besides, the designed orbit is also constant in the whole energy region.

Fig. 6.15

Electron spin rotator in the EIC ESR to rotate the spin between the vertical and longitudinal directions. Spin rotation angles φ_1,2 from the solenoids are determined by the spin rotation angles ψ_1,2 from the dipole magnets. Note that dipole bending angles θ_1,2 are fixed in the whole energy range, while ψ_1,2 are scaled with the electron beam energy

Figures 6.16 and 6.17 show the asymptotic polarizations of the electron beam in the 10 and 18 GeV areas in the perfectly aligned EIC ESR, respectively. Note that P_bks ≈ 83%, less than 92.4%, is due to the (n₀ ⋅s)² term in Eq. (6.27) in the spin rotator regions. The spin matching is performed at 18 GeV electron beam energy to minimize the depolarization resulting in the asymptotic polarization of 68%, while the longitudinal spin matching can not be carried out perfectly at 10 GeV resulting in the asymptotic polarization of 50%. However, the depolarization in the 10 GeV area is ≈ 16 times slower than that in the 18 GeV area. With the proposed regular replacement of the electron beam, a high average polarization can be achieved in the whole interested electron energy region in the EIC.

Fig. 6.16

Asymptotic polarization for the 10 GeV polarized electron beams in the perfectly aligned EIC ESR

Fig. 6.17

Asymptotic polarization for the 18 GeV polarized electron beams in the perfectly aligned EIC ESR

Figure 6.18 shows that the asymptotic polarization decreases from 68% in the perfectly aligned machine shown in Fig. 6.17 down to 40% when quadrupole misalignments and orbital coupling introduced and corrected. Spin tracking simulations show many high-order spin resonances when the spin resonance condition meets.

Fig. 6.18

Asymptotic polarization from spin tracking simulations in the18 GeV area in the EIC ESR with misalignments and roll errors included and corrected [15]