Ion Polarimetry

Abstract

The degree of polarization of the beams must be precisely measured, both to enable development and optimization of the beams, and to normalize the spin dependent effects observed in experiments. Ion beam polarimetry is particularly challenging since the physics processes available for polarimetry are themselves the subject of active physics research. This chapter describes ion polarimetry as implemented at the Relativistic Heavy Ion Collider (RHIC), the only high energy polarized proton collider.

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.

12.1 Polarimetry Requirements

Polarimetry is based on asymmetries measured in scattering of beam particles with target particles. For transverse polarization, spin effects are manifested as an azimuthal asymmetry of scattered particles. This results in an imbalance of particles scattered left and right in the plane perpendicular to the polarization vector. A diagram of such a scattering process is shown in Fig. 12.1.

Fig. 12.1

Diagram of scattering of a polarized beam particle scattering on an unpolarized target

Polarimetry of transversely polarized ion beams makes use of the Single Spin Asymmetry (SSA), where either the beam or target is polarized and the other unpolarized. If N_L and N_R are the numbers of particles scattered left and right, respectively, then for a beam or target with polarization P the left/right asymmetry 𝜖 is written as

$$\displaystyle \begin{aligned} \epsilon = \frac{N_R-N_L}{N_R+N_L} = P \cdot A_N . {} \end{aligned} $$

(12.1)

The proportionality constant A_N, referred to as the analyzing power, is the SSA for the process. It is a physics quantity depending on the particles involved, their energies and their scatting angles.

A polarimeter requires detectors sensitive to the azimuthal asymmetry. At minimum detectors must be placed left and right in the scattering plane perpendicular to the polarization vector as shown in Fig. 12.1. A more fine-grained azimuthal spacing of detectors will provide more information, such at the precise direction of the polarization vector.

12.1.1 Absolute Polarimetry

Unlike for electron polarimetry (Chap. 13), the analyzing power A_N for ion polarimetry is not known from established physics. In fact, it is the subject of spin physics studies. A procedure independent of A_N is required for measuring the absolute value of the polarization.

Such a procedure is possible if the target and beam are available in both spin states, up and down. By averaging over beam spin states the beam is effectively unpolarized, and the asymmetry with respect to target polarization may be measured. Similarly, averaging over target spin states allows measurement of the asymmetry with respect to the beam polarization. This principle is illustrated in Fig. 12.2 for a proton beam and target.

Fig. 12.2

Left: Polarized proton target with proton beam unpolarized by averaging over spin states. Right: Polarized proton beam with proton target unpolarized by averaging over spin states

In practice the ion beam typically has a mixture of bunches with both spin states; the data collected by the polarimeter can be sorted according to spin state. The target may also be operated in both spin states and the data similarly sorted. Then by averaging over beam spin states the asymmetry with respect to the target spin 𝜖_target is measured, and averaging over of target spin states the asymmetry with respect to the beam spin 𝜖_beam is measured. In terms of target and beam polarizations P_target and P_beam the asymmetries are

$$\displaystyle \begin{aligned} \epsilon_{\mathrm{target}} = P_{\mathrm{target}} \cdot A_N , \; \; \; \; \; \; \epsilon_{\mathrm{beam}} = P_{\mathrm{beam}} \cdot A_N . {} \end{aligned} $$

(12.2)

Furthermore, the target particles are at rest or almost at rest, and the target polarization P_target may be measured by conventional laboratory methods. Then from Eq. 12.2:

$$\displaystyle \begin{aligned} P_{\mathrm{beam}} = \frac{\epsilon_{\mathrm{beam}}}{\epsilon_{\mathrm{target}}} P_{\mathrm{target}} . {} \end{aligned} $$

(12.3)

The absolute beam polarization is thus determined in terms of measured quantities, independent of the analyzing power A_N.

12.1.2 Beam Details

Besides the absolute value other aspects of the beam polarization also need to be measured, both for beam optimization and use by experiments. An obvious example is the polarization lifetime, since the degree of beam polarization will inevitably decrease with time:

$$\displaystyle \begin{aligned} P(t) = P_{t=0} \; e^{-t/\tau} . {} \end{aligned} $$

(12.4)

The lifetime measurement requires several accurate polarimeter measurements throughout the life of a beam.

Polarization loss in stored beams occurs primarily through development of polarization profiles, with loss at the edges of the 6-dimensional beam bunch phase space. The bunch intensity and polarization may be parameterized by Gaussian distributions. For example, for the transverse position x intensity and polarization distributions are

$$\displaystyle \begin{aligned} I(x) = I_0 \; e^{-\frac{x^2}{2\sigma_I^2}} , \; \; \; \; \; \; P(x) = P_0 \; e^{-\frac{x^2}{2\sigma_P^2}} , {} \end{aligned} $$

(12.5)

as illustrated in Fig. 12.3.

Fig. 12.3

Example of Gaussian profiles for beam intensity I(x) and polarization P(x)

The polarimeters are sensitive to convolutions of the intensity and polarization profiles of single beams, and colliding beam experiments are sensitive to the profiles of both beams. The results of such convolutions are conveniently expressed in terms of the profile parameter R [1]:

$$\displaystyle \begin{aligned} R = \frac{\sigma_I^2}{\sigma_P^2} . {} \end{aligned} $$

(12.6)

For no depolarization the polarization profile is flat, σ_P →∞ and R = 0; R > 0 for a partially depolarized beam.

In general R may be different for the transverse dimensions x = (x, y); here for illustrative purposes we take R_x = R_y = R, with $P(\mathbf x) = P_0 \; e^{-\frac {x^2+y^2}{2\sigma _P^2}}$. A polarimeter averaging over x measures:

$$\displaystyle \begin{aligned} P_{\mathrm{avg}} = \frac{ \int d^2\mathbf{x} I(\mathbf x) P(\mathbf x) }{ \int d^2\mathbf{x} I(\mathbf x) } = \frac{P_0}{1+R} \; . {} \end{aligned} $$

(12.7)

For collisions consider two beams with the same profile parameter R and Gaussian normalizations P_0,1, P_0,2. Then the polarization for the Single Spin Asymmetry (SSA) with respect to beam 1 is:

$$\displaystyle \begin{aligned} P_{\mathrm{SSA}} = \frac{ \int d^2\mathbf{x} I_1(\mathbf x) I_2(\mathbf x) P_1(\mathbf x) }{ \int d^2\mathbf{x} I_1(\mathbf x) I_2(\mathbf x) } = \frac{P_{0,1}}{1+\frac{1}{2} R} \; , {} \end{aligned} $$

(12.8)

and similarly for the SSA with respect to beam 2. For Double Spin Asymmetry (DSA) measurements with both beams, the product of beam polarizations in collision is:

$$\displaystyle \begin{aligned} P^2_{\mathrm{DSA}} = \frac{ \int d^2\mathbf{x} I_1(\mathbf x) I_2(\mathbf x) P_1(\mathbf x) P_2(\mathbf x)}{ \int d^2\mathbf{x} I_1(\mathbf x) I_2(\mathbf x) } = \frac{P_{0,1}P_{0,2}}{1+R} \; . {} \end{aligned} $$

(12.9)

For R > 0 the polarimeter measurement P_avg needs to be corrected to the polarizations P_SSA and $P^2_{\mathrm {DSA}}$ required by collider experiments. The correction depends on R, and the polarimeter must be capable of measuring the relative widths of the intensity and polarization profiles as expressed in Eqs. 12.5.

12.2 Implementation at RHIC

The Relativistic Heavy Ion Collider (RHIC) is the only high energy polarized proton collider. It incorporates an ion polarimetry system meeting the needs of beam development and physics experiments. To meet the requirements outlined in Sect. 12.1 a two-pronged approach has been followed. Absolute polarimetry is provided by the hydrogen jet (Hjet) polarimeter. It is based on the process pp → pp, with a polarized atomic hydrogen jet target. The jet target is diffuse with a low data rate, requiring a long measurement period. Thus it is incapable of a statistically significant measurement of the time dependence of the polarization. Also, the target has transverse size large compared to beam bunches and can not resolve the transverse polarization structure. The fine grained time and spatial details are provided by the proton-carbon (pC) relative polarimeter. It is based on the process pC → pC, with an ultra-thin carbon ribbon target. The solid target produces a high data rate, allowing rapid measurements following the time dependence of polarization. The targets are also smaller than the beam bunch allowing resolution of transverse polarization structure. Ensembles of pC measurements are normalized to concurrent Hjet measurements, setting the absolute polarization scale.

12.2.1 Hjet Absolute Polarimeter

A side view of the Hjet is shown in the left of Fig. 12.4. The polarized atomic hydrogen target beam produced in the source at the top passes through the collider proton beams in the scattering chamber. The target beam polarization is measured in a Breit-Rabi polarimeter at the bottom, with typical values P_target ≈ 96%. In operation the polarization of the target is reversed every 5 minutes, providing both spin states as required for absolute polarimetry.

Fig. 12.4

Left: Side view of the Hjet polarimeter. The apparatus is approximately 3.5 m in height. Right: Diagram of the scattering chamber interior

The interior of the scattering chamber is shown in the right of Fig. 12.4. The collider beams cross but do not collide in the chamber, passing through the polarized target. Silicon strip detectors left and right of the target detect scattered protons, allowing an azimuthal asymmetry measurement. The detector segmentation into strips provides measurement of the polar angle of the scattered protons.

The silicon detector signals are read out in wave form digitizers (WFDs). The amplitude of the signal is a measure of E_kin, the kinetic energy of particles; the energy scale is calibrated with americium α particle sources. The time of the digitized signal provides the time of flight (TOF) of particles from the target to the detector. A two-dimensional histogram of TOF versus E_kin from the Hjet detectors is shown in Fig. 12.5. For non-relativistic protons in the MeV energy range $\mathrm {TOF} \propto 1/\sqrt { \mathrm {E}_{\mathrm {kin}}}$. This relation is shown by the curve in the figure. The accumulation of events near the curve are the signal protons and selected for event counting, rejecting the backgrounds apparent in the histogram. Protons in the energy range 1.6–6 MeV are used for the asymmetry measurement.

Fig. 12.5

TOF versus E_kin from the Hjet detectors

Absolute polarimetry requires elastic scattering, which for the Hjet polarimeter is the process pp → pp. The relation between kinetic energy and polar angle for an elastically scattered proton is unique, shown by the blue curve in the left of Fig. 12.6. For inelastic scattering pp → pX the energy-angle relation is different, with examples shown in the other curves in the figure. The strip number of the Hjet detectors is proportional to the scattering angle. The right of Fig. 12.6 shows a two-dimensional histogram of E_kin versus strip number for selected protons. The prominent accumulation of events similar to the blue curve in the left of the figure are elastic events. The events in each strip at lower energy are inelastic events and rejected for the asymmetry measurement.

Fig. 12.6

Left: E_kin versus polar scattering angle for elastic and inelastic pp scattering. Right: E_kin versus Hjet detector strip number for selected protons

The selected events are counted according to detector side, left L or right R, and spin state of the beam or target, up +  or down −. The asymmetry is measured with the relation [2]:

$$\displaystyle \begin{aligned} \epsilon = \frac{\sqrt{N_{R+} N_{L-} } - \sqrt{N_{L+} N_{R-}}} {\sqrt{N_{R+} N_{L-} } + \sqrt{N_{L+} N_{R-}}} \; . {} \end{aligned} $$

(12.10)

This relation is independent of the efficiencies of the left and right detectors and of the numbers of up and down spin state collisions. The asymmetry is measured separately for the beam and target spin states, averaged over the other, as described in Sect. 12.1.1, and the beam polarization is determined according to Eq. 12.3. The Hjet target is larger than the size of the beam and measures the transverse averaged polarization P_avg in Eq. 12.7. The low event rate of the Hjet, due to the diffuse target, allows only one statistically significant polarization measurement each RHIC fill of ≈ 4 − 8 hours.

12.2.2 pC Relative Polarimeter

A cross section of a pC polarimeter scattering chamber is shown in Fig. 12.7. A carbon ribbon target (vertical green line) is swept horizontally across the proton beam (red dot) at the center of the chamber. Six silicon strip detectors (red bars) are arranged azimuthally around the beam-target collision point, perpendicular to the beam direction. There are two such polarimeters in each RHIC beam, one with targets swept horizontally to measure horizontal polarization profiles, and one with targets swept vertically to measure vertical profiles.

Fig. 12.7

Cross section of a pC polarimeter scattering chamber. The beam direction is into the plane of the figure

The carbon ribbon targets are approximately 2.5 cm long, 10 μm wide, and 50 nm thick; the latter dimension is just a few hundred carbon atoms thick. The target width is significantly smaller than the transverse size of the beam, typically a few hundred μm, enabling the measurement of the transverse polarization profile. A photograph of a target is in the left of Fig. 12.8. The targets are mounted on a frame which rotates, sweeping the targets across the beam; a horizontally sweeping frame is shown in the center of Fig. 12.8. The frame holds six targets; when a target breaks, another can be positioned for beam sweeps, avoiding the need to break the chamber vacuum to replace individual targets. A photograph of a target in the beam is in the right of Fig. 12.8.

Fig. 12.8

Left: A carbon ribbon target. Center: Frame in scattering chamber holding six target. The frame rotates as shown by the arrow, sweeping the targets across the beam. Right:A target in the beam

The pC readout and particle selection are similar to the Hjet. The silicon detectors signals are read out in WFDs, and the energy scale is calibrated with americium and gadolinium α sources. The E_kin versus TOF relation is used to select scattered carbon nuclei; an example is shown in Fig. 12.9. Carbon nuclei in the energy range 400–900 keV are used for the asymmetry measurement.

Fig. 12.9

TOF versus E_kin from the pC detectors

A generalization of Eq. 12.10 is used to measure the asymmetry. For each pC detector an asymmetry with respect to the beam spin state is determined for a free parameter λ, the imbalance of beam spin up and down states. The asymmetries for each detector at azimuthal angle ϕ are fit to the form:

$$\displaystyle \begin{aligned} \epsilon(\phi) = \epsilon_0 \cdot \sin{}( \phi - \phi_0) \; , {} \end{aligned} $$

(12.11)

The sinusoidal form is motivated by fundamental spin physics. The parameter 𝜖₀ is the magnitude of the asymmetry, related to the polarization by the usual relation 𝜖₀ = P ⋅ A_N. The parameter ϕ₀ is the azimuthal tilt of the spin vector from vertical; for a 255 GeV proton beam in RHIC it is observed to deviate significantly from zero. An example fit is shown in Fig. 12.10.

Fig. 12.10

Fit to pC detector asymmetries. Results are asymmetry magnitude 𝜖₀, spin tilt ϕ₀, and beam state asymmetry λ

The targets are not rigid and their positions relative to the beam are not known from the position of the frame on which they are mounted. Thus, the transverse polarization profile can not be directly measured from the x dependence of $P(x) = P_0 \; e^{-\frac {x^2}{2\sigma _P^2}}$. Instead, the relations in Eqs. 12.5 may be rewritten in terms of the profile parameter R in Eq. 12.6:

$$\displaystyle \begin{aligned} P(x)/P_0 = \left( I(x)/I_0 \right)^R \; , {} \end{aligned} $$

(12.12)

which relates polarization P as a function of intensity of I, independent of x. In practice the asymmetry is measured in bins of intensity, or event rate, and the results fit to the power law Eq. 12.12, determining R. An example of such a fit is shown in Fig. 12.11.

Fig. 12.11

Fit to asymmetry versus intensity (event rate) determining the profile parameter R

The high event rate afforded by a solid target allows several statistically significant pC measurements during a RHIC fill, and the determination of the polarization lifetime. During a typical RHIC store of 6 hours, measurements are made at the beginning before and after beams are ramped to full energy, in the middle and at the end of the fill before beams are dumped. Results from a RHIC fill are shown in Fig. 12.12; they are fit to a linear time dependence:

$$\displaystyle \begin{aligned} P(t) = P_{t=0} \cdot (1 - t/\tau) , \; \; \; \; \; \; R(t) = R_{t=0} + R` \cdot t . {} \end{aligned} $$

(12.13)

Typical polarization lifetimes at RHIC are τ = 200 − 400 hours. The polarization is averaged over the target sweep across the beam and is the transverse averaged polarization P_avg in Eq. 12.7. The decline of P and growth of R with time demonstrates polarization loss through the development of profiles.

Fig. 12.12

pC polarization (top) and profile (bottom) measurements during a RHIC fill. The results are fit to linear time dependences. The first polarization measurement (marked by a cross) is at injection energy before ramping and not included in the fit

12.2.3 pC/Hjet Normalization

The pC analyzing power is not well known, and the measured asymmetries must be normalized to the Hjet results to determine the absolute polarization scale. The Hjet measures the average polarization over a RHIC fill weighted by the beam current I(t):

$$\displaystyle \begin{aligned} P_{\mathrm{H--jet}} = \frac{\int dt I(t) P(t)}{\int dt I(t)} \; . {} \end{aligned} $$

(12.14)

I(t) decreases with time as the beam decays, as shown for an example RHIC fill in Fig. 12.13. The Hjet measurement is clearly weighted toward the early part of the fill where polarization and beam current are highest.

Fig. 12.13

Beam current versus time for a RHIC fill

To compare to the Hjet, the pC results for each fill must be averaged weighted by the same I(t). In terms of the parameters from a fit to pC results in Eq. 12.13, the pC average is:

$$\displaystyle \begin{aligned} \overline{P_{\mathrm{pC}}} = P_{t=0} \cdot \left( 1 - \frac{1}{\tau} \cdot \frac{\int dt\,t I(t)}{\int dt I(t)} \right) \; . {} \end{aligned} $$

(12.15)

The scale of the pC polarization is then adjusted so that an average over fills of the pC to Hjet ratio is unity: $\left \langle \frac {\overline {P_{\mathrm {pC}}}} {\;\; P_{\mathrm {H--jet}}\;\; }\right \rangle _{\mathrm {fills}} = 1$. Typically the average is done for each of the four pC polarimeters over an entire year of RHIC fills. An example of these ratios after normalization is shown in Fig. 12.14.

Fig. 12.14

Ratio of current averaged pC and Hjet fill results after normalization versus fill number

12.3 Polarimetry Results

The polarimetry results are required by physics experiments to normalize observed spin dependent effects. Polarimeter measurements are also important for development and optimization of polarized beams, and to improve understanding of beam-spin physics. An example is the measurement of the spin tune.

12.3.1 Results for Experiments

The polarimeters measure the transverse averaged polarization, P_avg in Eq. 12.7. For Single Spin Asymmetry measurements with colliding beams P_SSA in Eq. 12.8 is required; it is related to P_avg by a function of the profile parameter R. From the fill results for P_avg(t) and R(t) in Eq. 12.13, P_SSA(t) is expressed to first order in the time t. Example results are shown in Fig. 12.15. Polarization is provided for the two RHIC beams BLUE and YELLOW. Included is a Unix time stamp for the start of the fill, t = 0, when the polarization is P0. Along with dP/dt, this allows determination of the polarization throughout the fill, for application when different data sets are collected by the experiments.

Fig. 12.15

Polarization results for spin physics experiments

The profile parameter typically has small values R = 0.1 − 0.2. From Eqs. 12.8,12.9, to lowest order in R:

$$\displaystyle \begin{aligned} P^2_{\mathrm{DSA}} = P_{\mathrm{SSA,1}} \cdot P_{\mathrm{SSA,2}} \; , {} \end{aligned} $$

(12.16)

provides the product of beam polarizations for Double Spin Asymmetry measurements.

12.3.2 Spin Tune Measurement

The RHIC spin flipper is a system of AC and DC dipole magnets allowing manipulation of the proton spin. A schematic is shown in Fig. 12.16. When operated in pulsed mode, it can flip the spin orientation. This feature has been demonstrated, observed as a sign change in the usual polarimeter measurements [3]. However, such operation often results in the loss of polarization, requiring time consuming refills of freshly polarized beams for detailed measurements.

Fig. 12.16

RHIC spin flipper. It consists of five AC dipoles and four DC dipoles

When the spin flipper is operated in continuous mode, it can induce a coherent spin precession about the stable spin direction [4]. Expressed as a fraction of the RHIC revolution frequency, the frequency of the precession is that of the spin flipper AC dipoles, ν_osc. The opening angle of the precession cone θ₀ is related to ν_osc and the spin tune ν_s by:

$$\displaystyle \begin{aligned} \tan{}(\theta_0) = \frac{|\epsilon|}{u_s - u_{\mathrm{osc}}} \; . {} \end{aligned} $$

(12.17)

𝜖 is the strength of the driven spin resonance and is known from the design of the spin flipper. Thus, if the flipper is driven with a frequency ν_osc, measurement of θ₀ provides a measurement of the spin tune ν_s.

The pC polarimeter measures the projection of the spin vector in the plane transverse to the beam direction (see Sect. 12.2.2). Figure 12.17 shows the projection of a spin precession cone in the transverse plane; the dotted red line is the path traced by the spin vector. θ₀ is the opening angle of the cone, as in Eq. 12.17. θ_tilt is a possible tilt from vertical of the un-driven stable spin direction.

Fig. 12.17

Parameters of spin precession projected into the plane transverse to the beam direction

The phase of the spin flipper AC dipoles was incorporated into the pC polarimeter readout, allowing measurements as a function of the dipole phase. The results of such measurements for one value of ν_osc are shown in Fig. 12.18, where the dipole phase is 2πν_osci. An oscillation of the measured tilt ϕ₀ as a function of the phase is clearly observed. The data are fit to the form:

$$\displaystyle \begin{aligned} \phi_0(2 \pi u_{\mathrm{osc}} {\mathrm{i}}) = \theta_{\mathrm{tilt}} \, + \, \tan^{-1} \left[ \tan \theta_0 \cdot \cos{}(2 \pi u_{\mathrm{osc}} {\mathrm{i}} -\Psi) \right] . {} \end{aligned} $$

(12.18)

Here Ψ is an arbitrary shift between the driver phase and pC readout introduced by cable delay. The resulting θ₀ then determines the tune shift via Eq. 12.17.

Fig. 12.18

Results of pC spin tilt measurements as a function of spin flipper driver phase for one value of ν_osc

A series of such measurements have been performed at both injection and full RHIC energies [4]. This has established the nondestructive measurement of the spin tune as a tool for further investigations of beam spin dynamics.