Storage Ring Lattice Calibration Using Resonant Spin Depolarization

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 16, 074001 (2013)

Storage ring lattice calibration using resonant spin depolarization

K. P. Wootton,1,* M. J. Boland,1,2 W. J. Corbett,3 X. Huang,3 G. S. LeBlanc,2 M. Lundin,4 H. P. Panopoulos,1,† J. A. Safranek,3 Y.-R. E. Tan,2 G. N. Taylor,1 K. Tian,3 and R. P. Rassool1 1School of Physics, The University of Melbourne, Melbourne, VIC, 3010, Australia 2Australian Synchrotron, 800 Blackburn Road, Clayton, VIC, 3168, Australia 3SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 4MAX IV Laboratory, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden (Received 22 February 2013; published 1 July 2013) This paper presents measurements of the GeV-scale electron beam energy for the storage rings at the synchrotron light source facilities Australian Synchrotron (AS) and SPEAR3 at SLAC. Resonant spin depolarization was employed in the beam energy measurement, since it is presently the highest precision technique and an uncertainty of order 106 was achieved at SPEAR3 and AS. Using the resonant depolarization technique, the beam energy was measured at various rf frequencies to measure the linear momentum compaction factor. This measured linear momentum compaction factor was used to evaluate models of the beam trajectory through combined-function bending magnets. The main bending magnets of both lattices are rectangular, horizontally defocusing gradient bending magnets. Four modeling approaches are compared for the beam trajectory through the bending magnet: a circular trajectory, linear and nonlinear hyperbolic cosine trajectories, and numerical evaluation of the trajectory through the measured magnetic ﬁeld map. Within the uncertainty of the measurement the momentum compaction factor is shown to agree with the numerical model of the trajectory within the bending magnet, and disagree with the hyperbolic cosine approximation.

DOI: 10.1103/PhysRevSTAB.16.074001 PACS numbers: 29.27.Hj, 29.40.Mc, 41.75.Ht

I. INTRODUCTION and momentum compaction factor. The technique is used because it is the highest precision energy measurement This paper presents measurements of the momentum presently available, and the typical measurement accuracy compaction factor using resonant spin depolarization, to is of order 105–106. This is one of very few methods for calibrate the model of horizontal defocusing rectangular measurement of the momentum compaction factor, which gradient bending magnets. We present experimental results can be calculated from measurements of the synchrotron in the storage rings of light sources SPEAR3 [1] and the frequency [9]. Pioneering work on polarization was per- Australian Synchrotron (AS) [2], which are modern light formed on eþ–e collider rings such as ACO [10,11], sources of intermediate energy with rectangular defocusing VEPP-2M [12,13], SPEAR [14], and LEP [15]. Our gradient dipoles in the double-bend achromat lattices [3]. method of polarizing and depolarizing the beam follows In literature, there are very few measurements of the mo- the technique used at BESSY I [5], BESSY II [6], ALS [4], mentum compaction factor with rectangular gradient SLS [7], and ANKA [8]. Independent recent measurements bending magnets. The momentum compaction factor was at Diamond [16] and SOLEIL [17,18] achieve the same measured at ALS, with only a small departure from the high precision. However, of the above storage rings, the model [4], however no details of the lattice model used in ALS alone employs rectangular gradient bending magnets. that study are available. At the time of that measurement, Many existing [1,19–22], upgrading [23] and planned the bending magnets of the ALS lattice were all rectangular rings [24–28] incorporate defocusing gradients into the defocusing gradient magnets. main bending magnets, as part of a strategy to reduce the Electron beam energy measurements using resonant horizontal equilibrium emittance. The trajectory through spin depolarization have been performed at storage rings rectangular gradient magnets does not follow a circular arc for calibrating many aspects of the machine [4–8], most as in the case of pure dipole magnets, and the modeling of notably the absolute beam energy, beam energy stability, gradient bending magnets is challenging. In this work, the electron trajectory through the gradient dipoles is modeled using trajectories that are circular, linear hyperbolic cosine, *[email protected] † nonlinear hyperbolic cosine, or numerical integration of Present address: Centre for P.E.T., Austin Hospital, 145 the measured magnetic ﬁeld [21]. Studley Road, Heidelberg, VIC 3084, Australia.

Published by the American Physical Society under the terms of II. POLARIZATION THEORY the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and A familiar description of storage ring accelerators is that the published article’s title, journal citation, and DOI. particles are deﬂected according to their electrical charge,

1098-4402=13=16(7)=074001(13) 074001-1 Published by the American Physical Society K. P. WOOTTON et al. Phys. Rev. ST Accel. Beams 16, 074001 (2013) mass, and energy using electric and magnetic fields. These of this relativistic electron beam. If the bending radius ðsÞ fields are arranged such that beams perform stable, oscil- varies around the ring circumference, rather than use the latory motion over many thousands of turns, which can be average value we make the substitution of the third measured to great precision as a frequency spectrum. synchrotron radiation integral [30,32], Measurement and control of resonances at the revolution 1 1 I 1 and rf frequencies, betatron and synchrotron tunes, informs ! ¼ 3 I3 2 j ð Þj3 ds; (4) the global properties of the linear lattice [29]. Here, we R s exploit another property of the electron—its spin—to in- where R denotes the mean ring radius. form and calibrate our model of the dipole lattice of storage rings. B. Resonant spin depolarization A thorough review of theory and experiments with polarized beams of protons, electrons, and muons was Spin transport is described by the Thomas–Bargmann- undertaken by Mane [30]. We will revisit the main theories Michel-Telegdi equation [33]. The electron spin pre- of radiative polarization of electron beams, adiabatic reso- cesses about the polarization axis at the spin precession nant spin depolarization, and Møller scattering cross- frequency [15] section polarimetry in the following sections. ~ qe BMT ¼ ð1 þ aeÞB? þð1 þ aeÞBk me A. Radiative polarization ~ E~ A beam of electrons in a storage ring with an initial ae þ ; (5) 1 þ c random distribution of spin orientations (unpolarized) develops polarization over time, by the Sokolov-Ternov where qe is the electric charge and ae ¼ðge 2Þ=2 effect [31]. Under the action of emission of spin-flip pho- the anomalous magnetic moment of the electron, and c tons, the population of beam electron spins aligns parallel the speed of light in vacuum. As defined for Eq. (2), the or antiparallel with the main guide field of the bending direction of electric fields E~ are considered with respect to magnets. The population of spin-up and spin-down parti- the relativistic velocity of the electron ~ ¼ v=c~ . Normally cles is biased by the asymmetry of transition probabilities the storage rings of light sources do not include any sig- ð Þ of spin-flip radiation [31], and the polarization P t of the nificant solenoid magnetic fields nor transverse electric beam develops by [30] fields, that is Bk ¼ 0, ~ E~ ¼ 0. Hence, we can make t=ST PðtÞP0ð1 e Þ; (1) the simplifying assumption that the electron spin precesses about a polarization axis which is antiparallel to B? (for where time is denoted by t. The polarization PðtÞ bending magnets of a ring, a vertical magnetic field), at a approaches an equilibrium [6] frequency given by Eq. (5) which can be simplified to the H 3 spin tune spin [15]: 8 B ds ¼ pffiffiffi H ? P0 3 : (2) 2 5 3 jB?jds ge E spin ¼ a ; (6) e 2 m c2 Magnetic fields are considered in the directions perpen- e dicular (B?) and parallel (Bk) to the curvilinear trajectory where E is the beam energy. If the beam is excited by a of the beam, s. In a storage ring, both vertical and radial radial magnetic field fkick resonant at any harmonic to the magnetic fields are denoted here by B?. To accommodate spin tune, the polarization axis of the beam can be coher- reverse bends and wiggler insertion devices, we integrate ently rotated away from its equilibrium vertical orienta- both B? and its absolute value jB?j around the trajectory s. tion. The beam is hence resonantly depolarized at the ¼ By inspection, P0 approaches a maximum for a storage frequency fkick fdep. ring without reverse bends or wiggler insertion devices, and for a beam of electron species with gyromagnetic C. Current fundamental and experimental 2 factor ge , the theoretical maximum of polarization is uncertainties P0 ¼ 0:9238 in an isomagnetic, planar ring [31]. The The gyromagnetic factor ge for electrons has been mea- characteristic Sokolov-Ternov polarization time ST is sured to precision within the 12th significant figure [34]. given by [31] ¼ pffiffiffi We use the NIST CODATA values [35] for ae 1 5 3 @ 5 1 0:00115965218076ð27Þ and m ¼0:510998928ð11ÞMeV. 1 ¼ re e ST 3 ; (3) As a point of interest, with a relative uncertainty of 40 8 me 10 8 ae=ae ¼ 2:3 10 , me=me ¼ 2:2 10 , the un- where is the local bending radius, me, re the classical certainty in the electron mass has improved by almost an electron mass and radius, 0 the permittivity of free space, order of magnitude since 1994 [15], and hence the theo- @ the reduced Planck’s constant, and is the Lorentz factor retical fundamental limit of uncertainty in the resonant

074001-2 STORAGE RING LATTICE CALIBRATION USING ... Phys. Rev. ST Accel. Beams 16, 074001 (2013)

TABLE I. Relative experimental uncertainties for AS and Touschek scattered electrons [38–40]: the intrabunch cross SPEAR3 experiments. section resulting from betatron oscillations. The particle Parameter Relative uncertainty loss rate dN=dt is described in terms of the polarization ð Þ 10 P t by [12,30,37] ae 2:3 10 2 2 108 NðtÞ2c me : dN ¼pffiffiffi ½ þ ð Þ2 1 1010 2 f1 f2P t ; (8) frf 2 0 0 7 dt xx yy z fkick 1 10 6 fdep 1 10 where the number of electrons per bunch is NðtÞ.Fora stored beam of current IðtÞ with equal current in several bunches, the bunch population NðtÞ/IðtÞ. The horizontal, depolarization technique is reduced to approximately vertical, and longitudinal beam dimensions are denoted by 8 E=E ¼ 2:2 10 . In practice, this limit remains orders x, y, z, and divergences denoted x0 , y0 in the hori- of magnitude lower than other experimental uncertainties, zontal and vertical directions, respectively. The functions as detailed for these experiments in Table I. f1 and f2 can be treated for a given measurement as As outlined in Table I, the rf frequency frf is calibrated constants. Importantly, because NðtÞ/IðtÞ, an instanta- to high precision and the excitation frequency fkick can be neous normalized loss rate Rnorm can be defined as calibrated against a reference clock. Hence from Eq. (6), 1 dN 2 measurement of the spin tune spin gives a direct measure- Rnorm ¼ / f1 þ f2PðtÞ : (9) IðtÞ2 dt ment of the beam energy, with experimental uncertainty dominated by uncertainty in fitting the depolarizing The normalized loss rate is the figure of merit used to frequency fdep. evaluate changes in the level of beam polarization.

D. Depolarizing effects III. DEPOLARIZATION EXPERIMENTS The effective polarization time eff is given by [17,36] A. Lattice and beam parameters 1 1 1 ¼ þ ; (7) Depolarization experiments were conducted at both eff ST dep the AS and SPEAR3 storage rings [41,42]. The pertinent design parameters of the two rings are summarized in where ST represents the Sokolov-Ternov polarization time Table II. [Eq. (3)], and dep a depolarization time governed by radial Using Eq. (6), the spin tune of a 3 GeV electron beam magnetic ﬁeld errors [17]. Storage rings of several GeV was calculated as spin ¼ 6:8081, as presented in Table II. achieve Sokolov-Ternov polarization times on the order of As outlined in Sec. II D, betatron tunes are depolarizing if 15–20 minutes, with depolarization times exceeding sev- overlapping with the spin tune. For the initial measurement eral hours [37]. Because the depolarization effects have a of both rings, quadrupole strengths were changed to reduce much longer characteristic time, the effective polarization the fractional vertical tune to approximately 0.1. time is dominated by the Sokolov-Ternov polarization time. A strong depolarizing resonance to be avoided is the B. Polarization time choice of stored beam energy corresponding to integer spin For both the AS and SPEAR3 storage rings, an unpolar- tune [Eq. (6)] [11]. Also depolarizing is the overlap of the ized beam of electrons was injected into the storage ring, spin tune with betatron or synchrotron tunes. The vertical which was observed to polarize over time. The measured betatron tune has been usefully employed for resonant spin depolarization [14]. In that experiment, the vertical TABLE II. Storage ring design parameters. betatron tune was swept as the depolarizer. The width of the vertical betatron tune resonance limited the uncertainty Parameter AS SPEAR3 in the beam energy measurement to approximately Beam energy E 3.00 3.00 GeV 4 E=E ¼ 10 . In this experiment at the AS and Lattice periodicity 14 18 SPEAR3, a feedback kicker is excited with a sinusoidal Lorentz factor 5871 5871 oscillation, because it can have a narrower frequency Spin tune spin 6.8081 6.8081 spread than the betatron tune. Betatron tunes x 13.290 14.130 y 5.216 6.194 E. Møller scattering polarimetry Bending radius 7.69 7.86 m Circumference C 216.000 234.144 m Møller scattering is electron-electron scattering and rf frequency frf 499.671 476.300 MHz occurs within a bunch in the storage ring. The polarimetry Polarization time ST 807 1003 s observable is the Møller scattering cross section of

074001-3 K. P. WOOTTON et al. Phys. Rev. ST Accel. Beams 16, 074001 (2013)

0.56 TABLE III. Polarization time in AS and SPEAR3.

Machine Measured eff (s) ST (s) 0.55 AS 806 21 807 0.54 SPEAR3 840 17 1003

0.53 kicker magnet and the oscillating frequency was scanned 0.52 close to the predicted spin tune. Additional detail concern- ing the depolarizing kicker is given in Appendix B. As the 0.51 magnetic field frequency crosses the spin tune the normal- 0.5 ized loss rate increases indicating the beam is depolarized. Figure 3 shows a depolarizing event at AS where the 1000 1500 2000 2500 3000 3500 4000 change in the rate is clearly observable over and above the fluctuations due to counting statistics. The uncertainty in the central frequency of the spin tune FIG. 1. Measurement of polarization time (AS). Fit to is not the same as the width of the spin tune resonance. The normalized count rate gives eff ¼ 806 21 s. 6 width of the spin tune resonance f=fdep 5 10 was observed to be much larger than the uncertainty in the normalized loss rate Rnorm was fitted by Eqs. (1) and (9)to central frequency of the spin tune resonance fdep=fdep determine the polarization time. This is shown for the AS 2 106. In addition, the width of the spin tune resonance in Fig. 1, and for SPEAR3 in Fig. 2. is much narrower than the energy spread of the beam, 3 Using Eq. (3) and design parameters in Table II the which for this lattice is E=E 1 10 [2]. Sokolov-Ternov polarization time was calculated. The The beam energy for AS and SPEAR3 were extracted measured eff and theoretical ST are compared in from spin tune frequency measurements, and the results are Table III. presented in Table IV. The high precision which was Importantly, for both rings a change in Rnorm is observed, achieved with this energy measurement technique is ex- which can be sensibly attributed to a change in polarization ploited to measure the momentum compaction factor of the PðtÞ by the Sokolov-Ternov effect. Hence, with this appa- storage rings. ratus we will be able to observe the depolarization of the beam at the spin tune. Additional detail about the detector D. Momentum compaction factor apparatus is given in Appendix A. The momentum compaction factor is the change in circumference with change in energy [29,43]. For a set of C. Spin tune and beam energy measurement rf frequencies, the beam energy was measured using the With the stored electron beam polarized, a time-varying radial magnetic field was applied to the beam through a fast 1.09

1.08 2.2 1.07 2.15 1.06 2.1 1.05 2.05 1.04 2 1.03

1.95 1.02 17.818 17.819 17.82 17.821 17.822 1.9 1000 2000 3000 4000 FIG. 3. Resonant spin depolarization (AS). A central frequency of fdep ¼ 17819790 30 Hz is ﬁtted with a width of FIG. 2. Measurement of polarization time (SPEAR3). Fit to f ¼ 110 40 Hz. An error function is ﬁtted, allowing for the normalized count rate gives eff ¼ 840 12 s. subsequent repolarization of the beam.

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TABLE IV. Measured spin tune and beam energy. TABLE V. Momentum compaction factor measured using resonant spin depolarization. Machine fdep (Hz) spin ¼ aeE(eV) Machine AS 17819790 30 6:83859 0:00002 3013416 9 c SPEAR3 253620 20 6:80192 0:00002 2997251 7 AS 0:00211 0:00005 SPEAR3 0:00164 0:00001 resonant spin depolarization technique. The rf frequency provides an accurate constraint on the circumference of the were used to describe the trajectory of an electron beam closed orbit, and resonant spin depolarization an accurate through a rectangular gradient bending magnet [21], as measurement of stored beam energy. In order to keep the used in the present experiments. Without a rigorous deri- beam energy stable, rf frequency feedback and fast-orbit vation, we quote the results of trajectories described by feedback were turned off. The rf frequency was varied by circular, analytical linear, analytical nonlinear, and numeri- small changes of 500 and 1000 Hz from the nominal cal models. The model lattice optical functions and frequency given in Table II, resulting in small changes to momentum compaction factor are analyzed and compared the stored beam energy. The corresponding change in beam to measured values. energy was measured, as illustrated in Fig. 4 for AS. A similar measurement was implemented for SPEAR3. The A. Coordinates and parameters measured momentum compaction factors of each ring are In this section of the paper, the coordinates of the beam summarized in Table V. These values are later compared trajectory will be given with reference to the straight, with calculated momentum compaction factor for the dif- rectangular gradient bending magnet as illustrated in ferent models of the gradient dipoles in the rings as de- Fig. 5 for the AS storage ring. scribed in Sec. IV F. This Cartesian coordinate system is right handed in ðx; y; zÞ, with x pointing radially outward from the storage IV. GRADIENT BENDING MAGNET MODELING ring center, y vertically upwards, and z parallel to the center line of the gradient magnet. The trajectory through As demonstrated in Sec. III D, we can exploit resonant a gradient bending magnet can be modeled as a beam depolarization to measure the momentum compaction fac- traveling off center through a large quadrupole. We adopt tor of a storage ring. The momentum compaction factor c a coordinate system [21], and define ðx; y; zÞ¼ð0; 0;zÞ as of a lattice can be calculated by integrating around the the straight line along the center of this quadrupole, with curvilinear trajectory s as [44] z ¼ 0 at the longitudinal center of the quadrupole as shown 1 I C ð Þ in Fig. 5. The bending magnet will be considered to be ¼ x s c ds; (10) centered at ðx; y; zÞ¼ðxQ; 0; 0Þ, where xQ ¼ B0=B1 and C 0 ðsÞ @nBðx; zÞ where C is the circumference, ðsÞ the horizontal disper- Bnðx; 0;zÞ¼ ; (11) x @xn sion, and ðsÞ the local bending radius. In previous work, several analytical models as well as a numerical model and n is the order of the transverse derivative of the magnetic field. This is further illustrated in Fig. 6. 1 The numerically evaluated trajectory will be evaluated in terms of a magnetic field map measured in the ðx; 0;zÞ plane, but components of the numerically integrated mag- 0.5 netic field will subsequently be specified in a curvilinear coordinate system ðu; v; sÞ. The curvilinear system is right handed with s tangential to the trajectory of the beam, u perpendicular and radially outwards, and v perpendicular 0 and vertically upwards. The curvilinear system is selected for the numerical trajectory because it is easy to implement curvilinear magnetic elements in existing accelerator −0.5 tracking codes. Parameters of the bending magnet are summarized in Table VI. −1 −3 −2 −1 0 1 2 3 B. Circular arc trajectory We make the simplifying assumption that the trajectory FIG. 4. Momentum compaction factor measurement (AS). is approximated by a circular arc, as a base against which

074001-5 K. P. WOOTTON et al. Phys. Rev. ST Accel. Beams 16, 074001 (2013)

FIG. 5. Half sector of the AS storage ring [2]. The rectangular gradient bending magnet is shown in yellow, quadrupoles in red, and sextupoles in green. to make comparisons. Ignoring the horizontal defocusing approximation x0ðzÞ¼0. We quote the trajectory evaluated gradient, we approximate the pole profile as a pure dipole. using this linear approximation, and direct the interested For the beam rigidity B, bending angle , and effective reader to its derivation [21]. The beam trajectory xLðzÞ from length Leff given in Table VI, the mean dipole field and magnet center to exit (and symmetrically, entrance) is bending radius are presented in Table VII. Unless other- expressed by wise specified only the vertical component of the magnetic pffiffiffi field on the midplane will be used in the following analysis xLðzÞ¼xLð0Þ coshð kzÞ; (14) and discussion, i.e., Bðx; zÞ¼Byðx; 0;zÞ. In the coordinate system given, the circular trajectory tanð=2Þ xcircðzÞ within the magnet is expressed by xLð0Þ¼pffiffiffi pffiffiffi : (15) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k sinhð kLeff=2Þ ð Þ¼ 1 ð Þ2 xcirc z z= xQ: (12) The deflection of the beam is chosen to be symmetric longitudinally about the magnet center z ¼ 0 so that the Outside the effective length of the dipole Leff, the trajec- 2 tory is straight. magnet deflects the beam through a bending angle = within the length Leff=2, with maximum orbit amplitude at the magnet center. C. Linear hyperbolic cosine trajectory The field profile of a gradient bending magnet can be D. Nonlinear hyperbolic cosine trajectory considered to be the field of a quadrupole of very large bore, laterally offset from the center line of the bending magnet. A nonlinear analytic solution describing the horizontal This quadrupole is considered to have a strength, k, which is trajectory of the electron through the bending magnet can be obtained on substitution of the linear solution given by defined in terms of the defocusing gradient B1 and beam Eq. (14) into the equation of motion, Eq. (13). This non- rigidity (B)byk ¼ B1=ðBÞ. The equation of motion of the beam through this quadrupole is given by [21] linear analytic solution is given by [21] Bðx; zÞ x00ðzÞ¼ ½1 þ x0ðzÞ23=2; (13) B TABLE VI. Storage ring bending magnet parameters. Parameter Symbol AS SPEAR3 units where the prime denotes the derivative along z.A linear solution to Eq. (13) can be obtained by making the Beam rigidity B 10 10 T m Bending angle 2=28 2=34 rad Defocusing gradient @B=@x 3.35 3.63 Tm1 Iron length Liron 1.700 1.450 m Effective length Leff 1.726 1.505 m

TABLE VII. Circular radius approximation. Parameter Symbol AS SPEAR3 FIG. 6. Description of gradient bending magnet field by the Effective field B0 1.300 1.228 T rQ field of a single quadrupole of radius , laterally offset from the Bending radius 7.692 8.144 m nominal center line of the bending magnet by a distance xQ [21].

074001-6 STORAGE RING LATTICE CALIBRATION USING ... Phys. Rev. ST Accel. Beams 16, 074001 (2013)

−0.35 −0.2 −0.4 −0.6 −0.4 −0.8 −1 −1.2 −0.45 −1.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

FIG. 7. Vertical component of magnetic ﬁeld of AS horizontally defocusing gradient bending magnet, measured with a Hall probe. The x-z axes corresponds to the coordinate system of Fig. 5. The numerically evaluated trajectory is represented by a solid blue line. The color scale shows the vertical component of the magnetic ﬁeld Byðx; 0;zÞ in units of Tesla.

pffiffiffi The n ¼ 0; 1; 2; 3 order coefficients are correspondingly 1 1 coshð kzÞ xNLðzÞ¼xNLð0Þpffiffiffi sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 referred to as the dipole, quadrupole, sextupole, and k 1 þ 1=½kxNLð0Þ octupole components of the magnetic field. The field 1 1 components for the AS bending magnet are shown in sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (16) 2 1 þ 1=½kxNLð0Þ Fig. 8. The horizontal trajectory of the particle through the where measured magnetic field (blue curve in Fig. 7) is calculated by numerically solving Eq. (13) with a fourth order Runge- tanð=2Þ xNLð0Þ¼pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (17) Kutta integrator (with no variable step size). The trajectory 2 k sinhð kLeff=2Þ 1 þ tan ð=2Þ is constrained to longitudinal symmetry, i.e. xðLeff=2Þ¼ 0 0 RxðLeff=2Þ, x ðLeff=2Þ¼x ðLeff=2Þ, and a deflection of B0ðsÞds ¼ðBÞ. E. Numerically integrated trajectory using The numerical trajectory is modeled with field compo- measured magnetic field data nents given by Eq. (18). The bending magnet model is A model of the AS bending magnet can be created by constructed by performing piecewise integration of the using a group of sector dipole elements with higher order field map along the trajectory s at points si. For the ith multipole field components (quadrupole, sextupole, and slice, the integrated multipole component is given by octupole). This method has also been implemented for modeling the SPEAR3 ring [45], and has the benefit of ensuring that particle tracking is symplectic. 0 6 The magnetic field map of the defocusing gradient bending magnet was measured by a three-axis Hall probe on the −0.5 4 horizontal midplane to give Bx;y;zðx; 0;zÞ. The vertical component of the magnetic field Byðx; 0;zÞ is presented −1 2 in Fig. 7. Superimposed on the map is the trajectory of a 3 GeV electron passing through the magnet as calculated −1.5 0 using the numerical integration method. −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 The vertical magnetic fields experienced by the electron while traversing the gradient magnet can be approximated by calculating the coefficients of the Taylor expansion of 1 10 the local magnetic field relative to the trajectory. As out- 0.5 lined in Sec. IVA, this curvilinear system is selected for the 0 0 numerical trajectory because it is easy to implement a −10 sequence of curvilinear magnetic elements in existing −0.5 accelerator tracking codes. −20 The first step in the calculation is a coordinate trans- −30 −1 formation from the Cartesian ðx; y; zÞ coordinates to a −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 curvilinear coordinate system ðu; y; sÞ, where s is tangential and u perpendicular to the reference trajectory. FIG. 8. Field components Bnðu; sÞ, along the curvilinear tra- Therefore, all field components are evaluated as jectory plotted in Fig. 7. (a) Dipole component. (b) Quadrupole component. The peaks at the ends of the bending magnet are due @nBðu; sÞ B ðu; sÞ¼ : (18) to edge focusing. (c) Sextupole component. (d) Octupole n @un component.

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Z n 1 siþ1 @ Bð0;sÞ both the bending radius ðsÞ and the horizontal dispersion i ¼ mn n @s; (19) ðsÞ. In this section, we will determine and evaluate both siþ1 si s @u x i of these. evaluated about the center of the trajectory u ¼ 0. This The trajectories of each of the four models are plotted sequence of sector elements describing the bending magnet in Fig. 9. As illustrated in Figs. 9(b) and 9(c), we compare model can readily be incorporated in an accelerator lattice the four trajectories relative to the initial position model. Because the edge effects are encapsulated by the xðz ¼1:234Þ. The circular trajectory approximation modeling approach, the individual segments are modeled gives the greatest deviation from the numerically evaluated i with higher order multipole components mn. trajectory. It is easier to see the local changes of the trajectory by F. Evaluation of modeling approaches inspecting the local bending radius of these four modeling ð Þ In the preceding subsections, different analytical ap- approaches. The bending field B0 z of the gradient bend- proaches were presented for the modeling of a straight, ing magnet is described by a virtual quadrupole of trans- rectangular gradient bending magnet. We propose to com- verse quadrupole gradient B1, and hence the dipole field varies in the longitudinal coordinate z by B0ðzÞ¼B1xðzÞ. pare the measured and modeled momentum compaction ð Þ factor as an evaluation of these models. As defined in As a result, the local bending radius z is given by Eq. (10), the momentum compaction factor depends upon ðzÞ¼p=ðqeB0Þ¼p=½qeB1xðzÞ; (20)

where p is the momentum of the beam electron. The local bending radii of each of the four models are plotted in −0.38 Fig. 10. There is longitudinal variation in the bending radius −0.4 ðzÞ, with the highest bending ﬁelds corresponding to lower bending radii at large z, as highlighted in Fig. 10(b). This −0.42 Circular results in a trajectory with greatest bending near the ex- Linear trema of the bending magnet, and less bending in the center. −0.44 Nonlinear Numerical We can also compare the lattice parameters [29,43] with −0.46 bending magnets modeled by the linear hyperbolic and −1 −0.5 0 0.5 1

0.1 60 Circular 0.08 Linear Nonlinear 40 0.06 Numerical

0.04 20

0.02 0 0 0.2 0.4 0.6 0.8 1 0 −1 −0.5 0 0.5 1

0.0905 8

0.09 7.8 7.6 0.0895 7.4

0.089 7.2

0 0.2 0.4 0.6 0.8 1 0.0885 −0.1 −0.05 0 0.05 0.1

FIG. 10. Bending radius as a function of the longitudinal FIG. 9. (a) Electron beam trajectories in coordinates ðx; zÞ for coordinate z, for each of the four models described in Sec. IV. each of the four models described in Sec. IV, with equal entrance (a) Bending radius ðzÞ with longitudinal position z, highlighting and exit angles. (b) Translation in x of trajectories in (a) by the longitudinal extent of the fringe field in the numerical model. xðzÞxð1:234Þ, to give equal entrance and exit positions and (b) Magnification of (a), highlighting variation of bending radius angles for each model. (c) Magnification of trajectories in (b). within the iron length of the magnet.

074001-8 STORAGE RING LATTICE CALIBRATION USING ... Phys. Rev. ST Accel. Beams 16, 074001 (2013) numerical trajectories. The storage ring is simulated using dispersion function xMEAS is compared to numerical and the ACCELERATOR TOOLBOX (AT) code [46] for each of the linear models in Fig. 12. analytical linear model given by Eq. (14), and numerically For the horizontal dispersion shown in Fig. 12, the evaluated bending magnet fields given by Eq. (19). The mean and standard deviation of the difference between equation for the path length of the trajectory using these the measured and model dispersion of the BPMs in the models is given explicitly in Ref. [21]. Linear and numeri- center of the arcs is for the linear model xMEAS xL ¼ cal models of the trajectory are specifically compared, as 5:8 1:4mm, and for the numerical model xMEAS the linear method is commonly implemented in accelerator xNUM ¼0:6 1:4mm. Hence, the measured disper- tracking codes, and elements commonly included in accel- sion function agrees with the numerical model to within erator tracking codes can be used to yield the numerically the limits of this lattice calibration. The difference between evaluated trajectory. For each model, the three families of the measured dispersion and linear model is attributable to storage ring quadrupoles are matched to give the same the difference between the linear and numerical models. betatron tunes and horizontal dispersion in the straights. As presented in Sec. III D, to evaluate which is a more Figure 11 shows the difference between the betatron and accurate model of the bending magnet, we have measured dispersion functions for the AS lattice. the momentum compaction factor to high precision using We observe in Fig. 11(b) that the solution of quadrupole resonant spin depolarization. Measured and modeled val- strengths for matched tunes and dispersion in the center of ues of the momentum compaction factors of both lattices the straights yields a significant difference in the dispersion are compared in Table VIII. and betatron functions. As an input to the momentum Within the uncertainty of the measurement the momen- compaction factor, we focus on the change in the differ- tum compaction factor is shown to agree with the numeri- ence between the dispersion functions, which at the beam cal model of the trajectory within the bending magnet, and position monitor (BPM) in the center of the arc is a peak disagree with the linear hyperbolic cosine approximation. difference of xNUM xL ¼ 4:8mm. The measured The accuracy of the numerical model comes from using the

0.7 40 0.6

30 0.5

0.4 20 0.3

10 0.2

0.1 0 0 5 10 15 0 0 50 100 150 200

10 0.8 8 0.6 6

0.4 4

2 0.2 0 0 −2 0 5 10 15 −4 0 50 100 150 200

FIG. 11. (a) Lattice functions evaluated using numerical model for trajectory, ﬁtting quadrupoles for the 0.1 m dispersion lattice FIG. 12. (a) Measured and model dispersion functions for the of the AS [2]. (b) Difference between lattice functions of AS. (b) Difference between measured dispersion, and dispersion numerical and linear hyperbolic cosine trajectory. evaluated using linear hyperbolic and numerical trajectory.

074001-9 K. P. WOOTTON et al. Phys. Rev. ST Accel. Beams 16, 074001 (2013)

TABLE VIII. Momentum compaction factor. proaches of the electron beam trajectory. The trajectory AS SPEAR3 through the gradient dipoles is modeled using circular, linear hyperbolic cosine, nonlinear hyperbolic cosine Linear hyperbolic cosine 0.00205 0.00162 approximations, and numerical integration of the measured Numerical 0.00211 0.00165 magnetic field. Within the uncertainty of the measurement Measured 0:00211 0:00005 0:00164 0:00001 the momentum compaction factor is shown to agree with the numerical model of the trajectory within the bending magnet, and disagree with the hyperbolic cosine approxi- correct distribution of the dipole field component as illus- mation. Linear and numerical models of the trajectory are trated in Figs. 8 and 10, subsequently giving a better model specifically compared, as the linear method is commonly for the horizontal dispersion as illustrated in Fig. 12. implemented in accelerator tracking codes, and elements commonly included in accelerator tracking codes can be V. DISCUSSION used to yield the numerically evaluated trajectory. Resonant spin depolarization has been usefully em- This is an extended article of measurements presented at ployed previously at rings to confirm effects principally recent International Particle Accelerator Conferences related to beam energy stability. In the present work, we [41,42]. benefit from the precision of the technique in the calibration of lattice models of gradient dipole magnets. We have ACKNOWLEDGMENTS compared the momentum compaction factor for analytic Portions of this research were carried out at the Stanford linear and numerically evaluated models, and measure- Synchrotron Radiation Lightsource, a Directorate of SLAC ments of storage ring lattices incorporating rectangular National Accelerator Laboratory and an Office of Science gradient bending magnets. User Facility operated for the U.S. Department of Energy As a modeling technique, the numerical evaluation of Office of Science by Stanford University. Parts of this trajectory could find potential application with various research were undertaken on the storage ring at the proposed accelerators. The design of present and future Australian Synchrotron, Victoria, Australia. K. P.W. third-generation storage ring light sources already consider thanks P. Kuske (BESSY), J. Zhang (LAL, Orsay), and this approach [22,47]. Ultimate storage ring light sources I. P.S. Martin (Diamond) for useful discussions at and [24–28] plan to employ gradient dipole magnets as part of following IPAC’10 and IPAC’11. a strategy to reduce the equilibrium horizontal emittance. Fixed-field, alternating gradient accelerators are enjoy- APPENDIX A: DETECTOR CHOICE ing a recent resurgence in interest [48]. In particular, scal- ing lattices employing gradient dipoles [49,50] could find The change in beam polarization is observed in the benefits to trajectory and focusing from numerical model- normalized loss rate Rnorm, given by Eq. (9). Two main ing, as well as nonscaling lattices with real quadrupoles at approaches are considered in literature: evaluation of large transverse offsets [51]. Also, with beams of antipro- the Touschek lifetime from the storage ring DC current ton species, this modeling approach could prove useful to the Recycler ring at Fermilab [52]. For proton accelerators, 80 this numerical evaluation of trajectory has been demonstrated to natively account for modeling of magnet fringe 0.34 fields [53]. 75 0.32 CONCLUSION 0.3 The beam energy was measured for two electron storage 70 rings AS and SPEAR 3 with defocusing gradient bending 0.28 magnets. Resonant spin depolarization was employed, achieving an uncertainty of order 106 in the beam energy. 65 0.26 To measure the momentum compaction factor, the rf frequency provided an accurate constraint on the circumference of the closed orbit, and resonant spin depolarization an accu- 0.24 60 rate measurement of stored beam energy. Measurements and 253.4 253.6 253.8 254 254.2 models of the momentum compaction factor were compared to evaluate analytical and numerical models of the beam trajectory through a defocusing gradient dipole magnet. FIG. 13. Resonant depolarization measured for SPEAR3 using Armed with precision measurements, we made a critical NaI scintillator and DCCT, increasing the excitation frequency evaluation of the suitability of different modeling ap- in time. The lifetime is calculated from 30 samples of the DCCT.

074001-10 STORAGE RING LATTICE CALIBRATION USING ... Phys. Rev. ST Accel. Beams 16, 074001 (2013) transformer (DCCT), and detection of the electromagnetic 1.1 shower from Touschek scattered beam particles striking the vacuum chamber when lost [5,54]. Figure 13 presents 1 measurements of a depolarization using each of these 0.9 techniques. Figure 13 shows that both approaches could work in the 0.8 identification of a depolarization. The measured NaI loss 0.7 monitor count rate responds to the resonant depolarization 0.6 within 1 s, while calculation of the lifetime with small uncertainty requires approximately 30 s of measurements 0.5 of the stored beam current and the drop in lifetime is 0.4 observed approximately 30 s after the depolarization. 0.3 The beam lifetime is calculated from the time derivative of the stored beam current, while the loss monitor mea- 17.76 17.78 17.8 17.82 sures the absolute loss rate, which is the derivative of the beam current with respect to time. For the technique of resonant spin depolarization it is FIG. 14. Measured normalized loss rate resulting from excita- best to have a beam lifetime dominated by Touschek tion of the beam with a horizontal magnetic field, oscillating scattering. One of the simplest ways to control this is to sinusoidally in time at the excitation frequency, using the AS alter the fill pattern in the storage ring to maximize the storage ring. Large drops in counts correspond to exciting the single bunch current. beam at synchrotron sidebands of the vertical betatron tune. The spin tune was measured as a step increase in normalized loss rate The beam loss monitor was a 75 mm diameter NaI at 17.82 MHz. The vertical axis Rnorm is normalized according to scintillator and photomultiplier tube at AS, and 50 mm Eq. (9): the value Rnorm ¼ 1 has no special significance. diameter at SPEAR3. For both experiments, the scintillator was installed in the orbit plane of the ring, on the inner side For both rings, the excitation frequency was swept at a of the vacuum chamber. At SPEAR3, the detector was rate of 10 Hz s1. Depolarization is a resonant effect—it is installed adjacent to the scraper defining the minimum particularly important to scan slowly [6]. energy aperture of the SPEAR3 storage ring, to maximize the count rate. This is immediately downstream of the central focusing quadrupole, which is the point of maximum horizontal dispersion in one of the double-bend achromat arc cells. At AS, the detector was installed at [1] J. Corbett, C. Limborg, Y. Nosochov, J. Safranek, and A. the upstream end of an insertion straight, where the hori- Garren, in Proceedings of European Particle Accelerator zontal dispersion is 0.1 m. Conference 1998 (JACoW, Stockholm, Sweden, 1998), In summary, the change in polarization needs to be quite pp. 574–576. large to observe the depolarization using the beam lifetime. [2] J. W. Boldeman and D. Einfeld, Nucl. Instrum. Methods From a beam physics perspective, large changes of polar- Phys. Res., Sect. A 521, 306 (2004). ization are not necessary to measure the beam energy: one [3] R. Chasman, G. K. Green, and E. M. Rowe, IEEE Trans. is interested in the precession frequency at which depolar- Nucl. Sci. 22, 1765 (1975). [4] C. Steier, J. Byrd, and P. Kuske, in Proceedings of the ization occurs. The time delay of 20–30 s observed in the European Particle Accelerator Conference 2000 (JACoW, beam lifetime measurement compromises this precision Vienna, Austria, 2000), p. MOP5B03. measurement of depolarization frequency. [5] R. Thornagel, G. Ulm, P. Kuske, T. Mayer, and K. Ott, in Proceedings of the European Particle Accelerator APPENDIX B: DEPOLARIZATION KICKER Conference 1994 (JACoW, London, England, 1994), pp. 1719–1721. Resonant depolarization of the beam is achieved with a [6] P. Kuske, R. Goergen, J. Kuszynski, and P. Schmid, in magnetic field which is radial in orientation (perpendicular Proceedings of the International Particle Accelerator to both the beam trajectory and main bending field), and Conference 2010 (JACoW, Kyoto, Japan, 2010), oscillating sinusoidally in time. Presented in Fig. 14 is the p. MOPD083. measured loss rate while scanning the excitation frequency. [7] S. C. Leemann, M. Bo¨ge, M. Dehler, V. Schlott, and A. Streun, in Proceedings of the European Particle Exciting the beam at a betatron resonance results in a Accelerator Conference 2002 (JACoW, Paris, France, decreased loss rate (increased lifetime) since the bunch 2002), p. TUPRI011. vertical size increases, hence the scattering rate decreases. [8] A.-S. Mu¨ller, I. Birkel, E. Huttel, F. Pe´rez, M. Pont, and R. Crossing a spin resonance the count rate increases since the Rossmanith, in Proceedings of the European Particle Møller scattering cross section increases when the polar- Accelerator Conference 2004 (JACoW, Lucerne, ization is reduced. Switzerland, 2004), p. THPKF022.

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