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A method to remove synchrotron frequency from the spectrum of momentum-forced radial oscillations
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Nuclear Instruments and Methods in Physics Research A 587 (2008) 1–6 www.elsevier.com/locate/nima
A method to remove synchrotron frequency from the spectrum of momentum-forced radial oscillations Yuri F. Orlov Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853, USA Received 1 October 2007; received in revised form 5 December 2007; accepted 11 December 2007 Available online 9 January 2008
Abstract Using the case of a storage ring for the resonance electric dipole moment (EDM) measurement, we demonstrate how to transform the frequency of forced radial oscillations from the frequency induced by the synchrotron oscillations of momentum, into a fraction of that frequency. This transformation, achieved through exploiting non-linearity, an oscillating external field, and destructive interference, cancels many parasitic spin resonances imitating the EDM. r 2008 Elsevier B.V. All rights reserved. PACS: 29.27.Hj; 29.27.Bd; 21.10.Ky; 14.20.Dh Keywords: Storage ring; Non-linearity; Electric dipole moment; Deuteron
1. Introduction In a storage ring, synchrotron oscillations of a particle’s momentum induce radial oscillations of the particle with the same frequency and phase. Here, we propose a method of removing synchrotron frequency from the spectrum of radial oscillations without removing the momentum’s synchrotron oscillations. This approach is possible only for one single phase of synchrotron oscillations. For this reason, we consider here only synchrotron oscillations that possess a mode whose frequency and phase are the same for all particles in the ring. Then, we can remove this mode from the radial oscillations of all particles. The basic idea is rather simple. Consider an oscillator whose free oscillations have frequency o, and an external force oscillating with frequency O. If the oscillator is linear, then d2 x=dt2 þ O2 x ¼ f cosðot þ fÞ; the right-side term describes the external force. The oscillations are a sum of the free and the forced oscillations and, if o6¼O, their Fourier spectrum contains two frequencies, O and o. Free oscillations are defined by individual initial conditions; Tel.: +1 607 255 3502; fax: +1 607 254 4552.
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forced oscillations are defined by f, o, and O. We can remove frequency o from the spectrum of x, while keeping untouched the external force possessing this frequency, by (i) applying the additional external force, F cos(ot/k+j), oscillating with frequency o/k, where k is an integer; (ii) introducing the non-linearity of the k order, bxk , into the oscillator design; and (iii) adjusting non-linearity strength b, amplitude F, and phase j of the additional force in such a way that frequency o disappears from the spectrum of x. Note that without non-linearity we would have only three frequencies in the spectrum—O, o, o/k— the last two being the frequencies of the x-oscillations induced by the two external forces. However, non-linearity xk produces many more components of the forced oscillations, among them a component with frequency k(o/k) ¼ o. The two components of the forced oscillations with the same frequency o (one induced by the original external force and the other by our additional force, plus non-linearity) can be adjusted to cancel each other. Below, we apply this method to the proposed measurement [1] of the deuteron electric dipole moment (EDM) ~ Any EDM oriented along the deuteron spin, d~ / S. oriented along spin violates the fundamental T and P symmetries. T-symmetry violation has never been observed
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Y.F. Orlov / Nuclear Instruments and Methods in Physics Research A 587 (2008) 1–6
directly; moreover, the observation of a non-zero deuteron EDM at the level 1029 e cm promised in Ref. [1] will lead to physics beyond the current Standard Model. In the experiment, the deuterons of some 1–1.5 GeV/c momentum, polarized mostly in the horizontal plane, rotate in a small storage ring, R2 m at the magnetic field B2 T. In its restframe, the deuteron spin precesses around the orbit (the so-called g-2 precession) due to the interaction ~ between the magnetic dipole moment, oriented along S, ~ and the vertically oriented B-field. The basic idea of the experiment is to use a restframe electric field, ~ and oscillating in resonance with E, perpendicular to B ~ the g-2 precession. If a non-zero EDM oriented along S exists, then we will observe a resonance spin-flip due to the ~ interaction between this EDM, precessing together with S, ~ and the oscillating E-field. Statistically, the spin-flip will be seen by polarimeter(s) as a small spin rotation in the vertical plane. In order to get as big an E as possible, the authors of Ref. [1] propose to use specially designed synchrotron oscillations of the deuteron momentum, the same for all deuterons. As a result, the restframe radial ~ will oscillate due to oscillations electric field g½~ v B of the deuteron velocity, ~ vðtÞ ¼ ~ v0 þ ðD~ vÞ0 cosðot þ fÞ, where o ¼ oa, the g-2 precession frequency. Subscript a refers to the anomalous gyromagnetic factor, a ¼ (g2)/2, a ¼ 0.143, for deuterons. In terms of the abstract oscillator described above, the momentum synchrotron oscillations are the oscillations of the external force, while the radial betatron oscillations induced by the momentum oscillations are the forced oscillations of the oscillator. Variable x of that oscillator is simply the deuteron radial deviation from the designed equilibrium orbit. Thus, o ¼ oa is the frequency of the forced radial oscillations. Of course, there exist also the free radial oscillations around the forced ones. A problem arising from the existence of the forced radial oscillations with the g-2 frequency and phase is that the presence of any ring imperfection coupling the horizontal, x, and the vertical, y, betatron oscillations leads to forced y-oscillations with the same g-2 frequency and phase. These forced y-oscillations in turn lead to parasitic spin resonance imitating the designed EDM spin resonance. This results in systematic errors. However, if we remove the g-2 frequency from the x-spectrum, this frequency will be automatically removed from the y-oscillations. Although we cannot cancel all false EDM resonances by doing so, we can considerably reduce their number and strength. In the following analysis, s ¼ s(t) is the particle coordinate along the designed reference orbit as a function of time. Correspondingly, t ¼ t(s). Using a ‘‘smooth’’ approximation, synchrotron oscillations of particle momentum in a usual storage ring, DpðsðtÞÞ ¼ ðDpÞ0 cos ot þ higher modes
(1)
induce radial oscillations, xðsðtÞÞ ¼ DðsÞðDp=pÞ0 cos ot þ higher modes
(2)
where o is the frequency of synchrotron oscillations and D(s) ¼ D(s(t)), the momentum dispersion function, is defined by the ring lattice. This approximation works well if o is small compared with the revolution frequency, so Dp is almost constant during one turn. Dp in fact changes during a turn, but only when the particle passes an accelerating RF cavity; it remains constant between cavities. Application of our method to a concrete EDM experiment is partly based on this detail. The main difference between a usual storage ring and a resonance EDM ring [1] is that the Dp of every particle in the latter includes not only free synchrotron oscillations with some individual eigenfrequency o, but also specially designed oscillations with the same frequency oa and phase f for all particles. In a preferable version of Ref. [1], synchrotron oscillations are highly non-linear and little resemble the usual combination of free and forced oscillations. For this reason, and to distinguish them from other forced oscillations, we will sometimes call the synchrotron oscillations common to all particles in the EDM ring ‘‘coherent’’ oscillations. In this and the next sections, we consider only coherent oscillations. In an EDM ring without non-linearities, the equation for x ¼ x(s) in the smooth and linear approximation would be 1 nðsÞ 1 Dp x þ cosðoa tðsÞÞ x¼ 2 RðsÞ p 0 R ðsÞ RqB=qx s tðsÞ ¼ . n¼ B v 00
(3)
Since the equation is linear, the forced x-oscillations in Eq. (2) resulting from Eq. (3) have the same frequency as the external force, o ¼ oa; the external force in Eq. (3) is proportional to Dp/p. These x-oscillations would be harmless for the spin in an ideal ring, but when combined with ring imperfections they create perturbative fields oscillating with the same frequency, o ¼ oa. As we have already noted, the obvious result is a number of parasitic spin resonances imitating EDM. We need to remove frequency oa from the x-oscillations without removing the external force, because in this experiment we need Dp to oscillate with the g-2 frequency. We propose the following three-pronged solution: (a) include a non-linear term, xk, in Eq. (3), that is, introduce non-linear fields into the ring lattice; (b) also include in Eq. (3) some ‘‘active’’ lattice element(s) that oscillate with frequency oa/k; and (c) use destructive interference to eliminate the terms oscillating with frequency oa. The active element(s) will induce oscillations of the xk-term with the oa-frequency due to multiplication, k (oa/k) ¼ oa. In the case analyzed below, the active elements oscillate initial and final conditions for x at the entrances and exits of the big magnet sections. The destructive interference will occur between these oscillations and those of the external force with frequency oa. This interference will cancel the dangerous oa-mode of the x-oscillations without touching the oa-frequency of Dp/p.
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2. Non-linear forced oscillations The EDM ring under discussion contains two big magnet sections and two big straight sections. (Fig. 1 shows 1/4 of the ring.) Each big magnet section consists of several small magnets with relatively small straight sections between them. The magnets are designed to have different non-linear magnetic fields and different x-deviations. As described below, in the small straight sections we place lenses that ‘‘sew together’’ different x-values in different magnets. The big straight sections contain the usual strong focusing gradient lenses, RF cavities, etc. Between the big straight sections and the big magnet sections lie transition sections, our ‘‘active elements’’. A transition section may occupy some 2–3 m. Each section contains a deflecting RF cavity, the TM110 mode of which sews together the forced x-deviations in the adjacent straight and magnet sections. The deflecting RF cavities with the needed properties have recently been investigated [2]. Let us use the notations x and y for the forced vertical and horizontal deviations, and r and z for the free horizontal and vertical betatron oscillations around them. y(s)0, and x ¼ 0 in the two big straight sections. Inside any magnet, we have by design (instead of Eq. (2)), o a x ¼ a þ b cos ðt0 þ nTÞ 3
that we use a cubic non-linearity, x3, k ¼ 3, to cancel frequency oa of the radial oscillations. The phase in Eq. (4) is such that every maximum of x coincides with every third maximum of (Dp/p). In any small straight section between two magnets, x(s) changes its values along s from a value in one magnet to a different value in another. In Fig. 1, a ¼ 0 and b changes its sign when the particle goes from one magnet to another. With y ¼ 0, the proposed equation for x inside a magnet is x00 þ n2 x þ ax2 þ bx3 þ d ¼ d cos f f oa ðt0 þ nTÞ
Fig. 1. One quarter of the EDM ring. The line with the arrows shows a forced orbit of the particles. 0–00 , set of lenses making (1) matrices for both x and y; 00 –1, thin focusing lens; 1/F ¼ (qB/qx)(l/BR)E0.8 m1, lE0.2 m; 1–2, free interval 0.7 m; 2–3, defocusing thin lens, 1/FE1.75 m1, lE0.2 m; 4–5, RF deflector, lE0.7 m, (Ev)max15 MV/m; 6–7, thick defocusing lens, lE1 m, 1/FE1.75 m1; 8–9, thick focusing lens, lE1.22 m, 1/FE0.67 m1; 10–11, magnet, l ¼ pR/4, R ¼ 2 m, B ¼ 1.7 T, b ¼ +50 m4, a ¼ 0; 12–13, half of the (pR/2)-magnet, the same R, B, and a, but b ¼ 50 m4; 11–12, set of lenses making (1)-matrices for both x and y. All unlisted intervals are lE0.3 m. Lenses placed in the big straight sections to correct the CS-beta-functions inside the magnets are not shown.
(5)
where n2 ¼
1n R2
(6)
a¼
q2 B=qx2 2BR
sextupole non-linearity
(7)
b¼
q3 B=qx3 6BR
octupole non-linearity
(8)
d¼
DB BR
(4)
where a and b differ in different magnets. p and (t0+nT) are constants between accelerating cavities; T is the period and n the number of revolutions. Thus, our x-deviation inside a magnet is constant at a given n, but oscillates as a function of n with the frequency oa/3. The factor 1/3 means
3
an additional homogeneous B field
(9)
ðDp=pÞ0 . (10) R We include the quadratic non-linearity here only for the sake of generality. Though the sextupole field is needed to prolong spin coherence time [3] (a separate problem of the EDM experiment), it is neither necessary nor desirable to combine it with the magnets. We want only to show that our interference technique does not forbid having a set of different as along our ring. The inclusion of a homogeneous field (9) is a consequence of our including sextupole field (7). Quadratic non-linearity shifts the equilibrium relative to the RF deflector, and d corrects this. For simplicity, Eqs. (5)–(10) omit relatively small ‘‘inertial’’ sextupole terms proportional to n/R3 and similar octupole terms proportional to a/R. But we have kept the inertial quadrupole term 1/R2 in Eq. (6) because it may turn out not to be small compared with n/R2. The omitted terms can only slightly change Eqs. (11)–(14), below. From assumption Eq. (4) and Eq. (5), with x00 ¼ 0, we get four equations for the four different modes of x as a function of the number of turns, n 1=3 bb3 4d ¼ d; b ¼ for cos f (11) b 4
d¼
a ; a¼ 3b
2f for cos 3
b n2 þ 2aa þ 3ba2 þ 3d ¼ 0;
(12) f for cos 3
(13)
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4
b2 3ab2 2 3 þb a þ þd ¼0 n aþa a þ 2 2 2
for constant. (14)
2
Eq. (13) connects n with a and b, n2 ¼
a2 3b1=3 d2=3 3b 41=3
and Eq. (14) defines d, ! a 2a2 a2 d2=3 2 n d¼ . ¼a 3b 9b 27b2 41=3 b2=3
(15)
(16)
We see that d ¼ a ¼ 0 if a ¼ 0. To get some feeling for the numerical values of b and b, assume p ¼ 1 GeV/c, B ¼ 1.7 T, and (Dp/p)0 ¼ 0.5 102. Then R ¼ 2 m, d ¼ 2.5 103 m3. To estimate the biggest possible b, consider some reasonable limit on the octupole field, say, ðDBÞoct =B ðq3 B=qx3 Þx3 =6B ¼ bRx3 ¼ 20% at x ¼ +0.05 m. Note that we are free to choose the sign of b, so b ¼ jbj bb40.
(17) 4
Obviously, d40. Thus, we may choose any |b|o800 m . This gives the amplitude |b|42.3 cm, with sign b ¼ sign b. In Fig. 1, bE750 m4, bE70.058 m (the scale is arbitrary). An important requirement for the set of bs along the ring is that our non-linear oscillations of the forced orbit, Eq. (4), not influence the synchrotron oscillations. This requirement can be met if the length of the orbit remains constant when x oscillates coherently with the frequency oa/3. That is, if X bk l k ¼ 0. (18) Rk Here, bk is the positive or negative amplitude of the orbit oscillations inside magnet #k; lk is this magnet’s length; and Rk its radius. If Eq. (18) is satisfied, then the orbit length does not depend on the coherent oscillations of the momentum. But it depends, of course, on the free momentum oscillations and the radial oscillations corresponding to them. These will be discussed in Section 3. We see from Eq. (18) that we need bs, and hence bs, of different signs along the orbit, as in Fig. 1. If we decide to use sextupole fields not only in the big straight sections, but also in the magnet sections, then the choice of a depends on whether we want sign(a) ¼ sign(b) ¼ sign b and therefore ao0, or want sign(a) ¼ sign(b) ¼ sign b and therefore a40; see Eq. (12). As shown below, once we choose a certain sign of ab, and hence the sign of a ¼ 3ab, we cannot change it along a given big magnet section. For, if we do, we will not be able to sew the orbit inside that section. But it is permitted to have a set of as of different signs if they are at different big magnet sections, since the big magnet sections are separated from one another by the big straight sections, where x ¼ 0. Because x ¼ 0, the two transition sections at
the ends of the same big straight section are mutually independent. We may estimate the numerical values of as from Eq. (12). If, say, we want a ¼ 1.5 cm or a ¼ 1.5 cm, we get a ¼ 836 m3 when b ¼ 800 m4. Then from Eq. (15), with d ¼ 2.5 103 m1, we get n2 ¼ 0.165 m2; and if R ¼ 2 m, then n ¼ 0.34. But when b ¼ 800, we need to reduce a if we want a positive n2 inside the given magnet. For example, if a ¼ 71 cm, then a ¼ 724 and we get n2 ¼ 0.135 m2, n ¼ 0.46. If b ¼ 800 and a ¼ +24, then x (which is constant inside a given magnet) oscillates as a function of the turn number, n, as t0 þ nT xðnÞ ¼ 1 2:3 cos oa cm ðexampleÞ. (19) 3 Let k be the magnet number inside a given big magnet section. We can sew xk(n) to xk+1(n) only if xkþ1 ðnÞ ¼ mk;kþ1 xk ðnÞ
(20)
where mk;kþ1 is constant for every n. Transformation (20) means akþ1 ¼ mk;kþ1 ak and bkþ1 ¼ mk;kþ1 bk . So, for the sewable deviations, the product ab (and hence a) does not change its sign. Since dx/ds ¼ 0 inside the magnets, transformation (20) can be written as ! mk;kþ1 l k;kþ1 xk xkþ1 ¼ (21) 1 0 0 0 mk;kþ1 where matrix element l k;kþ1 is arbitrary and mk;kþ1 ¼ xkþ1 =xk . There are no problems in building such matrices by using gradient lenses. But they cannot transform a+b cos (f/3) into ab cos (f/3) when these cosines oscillate. In the Fig. 1 version of the ring, we choose to have (1)matrices transforming both x-x and y-y. Such an arrangement decouples operations on lenses from operations on magnets. Its disadvantage is that a straight section for a (1)-matrix cannot be very small. In a transition section, sewing the x ¼ 0 of a big straight section to the x6¼0 of a big magnet section requires an RF cavity producing the properly phased deflecting radial electric field and vertical magnetic field. These fields oscillate with either frequency o ¼ oRF+oa/3 or frequency oRFoa/3, where oRF is the frequency of the usual accelerating cavity of the ring. Also required are defocusing and focusing lenses (as in Fig. 1) and, when a6¼0, a constant BV-deflector. Note that the strong focusing lenses and the accelerating cavities placed in the big straight sections do not influence the forced x-oscillations inside the big magnet sections. That independence considerably simplifies the problem of stabilizing the free betatron oscillations around the forced oscillations. This is the picture for the horizontal plane. For the vertical plane, the matrices that sew xk+1s generally look
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different from Eq. (21). They will be used for focusing the free z-oscillations around y ¼ 0. The Fig. 1 parameters of the main ring elements are not optimized in any sense. AA and BB are two reflection symmetry axes of the ring. 3. Stability of the free oscillations The alternating cubic non-linearity, 7|b|, permits us to cancel more than the influence of our x-oscillations on the synchrotron oscillations; see Eq. (18). We can also cancel their influence on the designed betatron tunes, that is, on the stability of the free betatron oscillations around x(s, n) and y(s, n)0. We first consider the stability of the free radial oscillations. The equation for small deviations, r, from the forced orbit x(s, n) is 1 qB=qx t0 þ nT r00 þ 2 þ r þ 2aðsÞ aðsÞ þ bðsÞ cos oa r BR 3 R t0 þ nT 2 r ¼ 0. (22) þ 3bðsÞ aðsÞ þ bðsÞ cos oa 3 Here, a, b, a, b vary from magnet to magnet but remain constant inside every magnet. In all non-magnet areas, a ¼ b/R ¼ 0, so only the gradient focusing is present. Taking a and b from Eqs. (11)–(15) into Eq. (22), we get " # qB=qx b 1=3 2=3 00 r þ rþ 3 d r BR S 4 M 1=3 b t þ nT 0 2=3 þ6 d cos 2oa r ¼ 0. (23) 4 M 3 Note that parameters a and n2 of the magnets disappear from this linear equation. b ¼ b(s) is present only inside magnets; hence, subindex M. In this equation (not in the lattice!), [1/R2+(qB/qx)/BR] ¼ (qB/qR)/BR is non-zero only outside magnets; hence, subindex S. In what follows, we will sometimes omit these subindices. If the gradient lenses in the big straight sections, in the transition sections, and between magnets collectively provide sufficiently strong focusing (which can be achieved rather easily), then we can treat the cosine term in Eq. (23) as a perturbation. So r00 þ Kr þ DKr ¼ 0 1=3 t þ nT b 2=3 DKðsÞ ¼ 6d cos 2oa 3 4
(24)
where the term in the square brackets is constant for a given turn. The betatron tune remains unchanged in the first approximation and the r-oscillations therefore remain stable if [4] X 1 cos 2pnx cos 2pnx0 ¼ sin 2pnx0 ðb¯ CS Þi DK i l i ¼ 0 2 i (25)
or simply if X 1=3 ðb¯ xCS Þi bi l i ¼ 0
5
(26)
i
where the bar means averaging along magnet #i. bxCS(s) is the Courant–Snyder (CS) beta-function [4] for the horizontal oscillations corresponding to the non-perturbed equation with DK ¼ 0; nx and nx0 are the perturbed and non-perturbed horizontal tunes. The structure of Eq. (23) defines the so-called momentum compaction factor, ap ¼ (DL/L)/(Dp/p). We need to take into account that (Dp/p) contains two parts: a coherent part, d cos f in Eq. (5), which is the same for all particles, and an incoherent part, (Dp/p)inc. ðDp=pÞinc ¼ ðDp=pÞi cosðOt þ wÞ, where Oaocoh ¼ oa and w is a phase of a particle’s free synchrotron oscillations around the coherent forced oscillations. We should put R1 ðDp=pÞinc into the right side of Eq. (23). Due to the non-linearity of synchrotron oscillations, all particles have different Os and hence different full phases, which makes these oscillations incoherent. In the first approximation, we can neglect DK in Eq. (24). If we then solve the new equation for r as a linear function of (Dp/p)inc, we will get our momentum distribution function, D, and ap ¼ /D/RS, the average along the ring—which is essential for the free synchrotron oscillations. These oscillations are stable if ap6¼1/g2, where g is the relativistic factor. Let us now consider the stability of the vertical free oscillations, of course taking into account the Maxwell equations connecting different components of the fields. The equation for small vertical deviations z from y ¼ 0 is ðqB=qxÞ t0 þ nT 00 z 2aðsÞ aðsÞ þ bðsÞ cos oa z z BR 3 2 t0 þ nT z¼0 (27) 3b aðsÞ þ bðsÞ cos oa 3 which gives " # 1=3 qB=qx 2a2 b 2=3 00 3 z þ d z BR S 4 3b M 1=3 b t0 þ nT 6 d2=3 cos 2oa z ¼ 0. 4 M 3
(28)
As in Eq. (23), term (qB/qx)/BR in this equation (not in the lattice!)6¼0 only outside the magnets, while a and b6¼0 only inside them. Let, m and byCS be the tune and the Courant–Snyder beta-function of the non-perturbed vertical free oscillations. Then, m is not changed by the oscillating part of the gradient in Eq. (28)—that is, by our x-oscillations—if X 1=3 ðb¯ yCS Þi ðbÞi l i ¼ 0 (29) i
where the bar means averaging along magnet #i. It is easy to maintain the stability of the oscillations without the cosine term in Eq. (28), and the presence of that term does not affect the stability if condition (29) is satisfied.
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Y.F. Orlov / Nuclear Instruments and Methods in Physics Research A 587 (2008) 1–6
4. The side effects The above non-linear method applied to the EDM experiment cancels with some accuracy all parasitic spin resonances connected with radial oscillations induced by momentum oscillations—though only for field multipoles proportional to x0, x1, x2, and skew multipoles corresponding to them. (A perturbative skew multipole appears when the normal multipole is rotated around the longitudinal axis.) The cancellation follows from the fact that when x oscillates with the frequency oa/3, none of these multipoles can produce the spin resonance frequency oa. However, the octupole non-linearity, which is proportional to x3 and is used for these cancellations, cannot cancel a spin resonance produced by its own perturbative skew octupole. This is the main side effect of our method. The advantage of the method is that this perturbation appears only in the octupole mode of the field, and in this respect is much better defined than other perturbations. The normal octupole acts as ½ðq3 B=qx3 Þ=6BRðx3 3xy2 Þ in the horizontal plane and as ½ðq3 B=qx3 Þ=6BRðy3 3yx2 Þ in the vertical plane. The skew octupole acts as ½ðq3 B= qx2 qyÞ=6BRð3x2 y y3 Þ in the horizontal plane and as ½ðq3 B=qx2 qyÞ=6BRð3xy2 þ x3 Þ in the vertical plane. x oscillates with the frequency oa/3 in our ring. So obviously the skew x3-term now oscillates in the vertical plane with the spin resonance frequency oa. This is a small and rather well-defined perturbation, jq3 B=qx2 qyj ¼ yjq3 B=qx3 j, y51, where y is a rotation angle around the longitudinal axis. But it is a perturbation that needs to be canceled. One way
to reveal and measure it is to excite coherent vertical oscillations with the frequency oz ¼ (or)/3, where or is the frequency of the free radial oscillations, and observe the resonance build-up in the horizontal oscillations. Description of other, more sophisticated ways of doing so is beyond the scope of this paper. A possible additional side effect of the non-linear method is the appearance of the frequency oa/3 in the Fourier spectrum of the g-2 precessions, giving us a nonlinear version of the very situation we are trying to avoid. We estimate that, in the worst case, these new resonances will fortunately be some three to four orders of magnitude weaker than the resonances we have canceled. In the best case, we can choose a lattice in which the frequency oa/3 appears only beyond the accuracy level defined in our method by errors in field measurements. Acknowledgments The author thanks M. Tigner, Y.K. Semertzidis, R. Talman, and S. Orlov for valuable discussions. References [1] Y.F. Orlov, W.B. Morse, Y.K. Semertzidis, Phys. Rev. Lett. 96 (2006) 214802. [2] V. Shemelin, S. Belomestnykh, RF design of the deflecting cavity for beam diagnostics in ERL injector, Cornell LNS Report ERL 07–2, /http://www.lns.cornell.edu/public/ERL/2007/S. [3] B. Vasserman, et al., Phys. Rev. B 198 (1987) 302. [4] A. Wu Chao, M. Tigner (Eds.), Handbook of Accelerator Physics and Engineering, World Scientific, Singapore, 1998.
Nuclear Instruments and Methods in Physics Research A 587 (2008) 1–6 www.elsevier.com/locate/nima
A method to remove synchrotron frequency from the spectrum of momentum-forced radial oscillations Yuri F. Orlov Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853, USA Received 1 October 2007; received in revised form 5 December 2007; accepted 11 December 2007 Available online 9 January 2008
Abstract Using the case of a storage ring for the resonance electric dipole moment (EDM) measurement, we demonstrate how to transform the frequency of forced radial oscillations from the frequency induced by the synchrotron oscillations of momentum, into a fraction of that frequency. This transformation, achieved through exploiting non-linearity, an oscillating external field, and destructive interference, cancels many parasitic spin resonances imitating the EDM. r 2008 Elsevier B.V. All rights reserved. PACS: 29.27.Hj; 29.27.Bd; 21.10.Ky; 14.20.Dh Keywords: Storage ring; Non-linearity; Electric dipole moment; Deuteron
1. Introduction In a storage ring, synchrotron oscillations of a particle’s momentum induce radial oscillations of the particle with the same frequency and phase. Here, we propose a method of removing synchrotron frequency from the spectrum of radial oscillations without removing the momentum’s synchrotron oscillations. This approach is possible only for one single phase of synchrotron oscillations. For this reason, we consider here only synchrotron oscillations that possess a mode whose frequency and phase are the same for all particles in the ring. Then, we can remove this mode from the radial oscillations of all particles. The basic idea is rather simple. Consider an oscillator whose free oscillations have frequency o, and an external force oscillating with frequency O. If the oscillator is linear, then d2 x=dt2 þ O2 x ¼ f cosðot þ fÞ; the right-side term describes the external force. The oscillations are a sum of the free and the forced oscillations and, if o6¼O, their Fourier spectrum contains two frequencies, O and o. Free oscillations are defined by individual initial conditions; Tel.: +1 607 255 3502; fax: +1 607 254 4552.
E-mail address: [email protected] 0168-9002/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.12.042
forced oscillations are defined by f, o, and O. We can remove frequency o from the spectrum of x, while keeping untouched the external force possessing this frequency, by (i) applying the additional external force, F cos(ot/k+j), oscillating with frequency o/k, where k is an integer; (ii) introducing the non-linearity of the k order, bxk , into the oscillator design; and (iii) adjusting non-linearity strength b, amplitude F, and phase j of the additional force in such a way that frequency o disappears from the spectrum of x. Note that without non-linearity we would have only three frequencies in the spectrum—O, o, o/k— the last two being the frequencies of the x-oscillations induced by the two external forces. However, non-linearity xk produces many more components of the forced oscillations, among them a component with frequency k(o/k) ¼ o. The two components of the forced oscillations with the same frequency o (one induced by the original external force and the other by our additional force, plus non-linearity) can be adjusted to cancel each other. Below, we apply this method to the proposed measurement [1] of the deuteron electric dipole moment (EDM) ~ Any EDM oriented along the deuteron spin, d~ / S. oriented along spin violates the fundamental T and P symmetries. T-symmetry violation has never been observed
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Y.F. Orlov / Nuclear Instruments and Methods in Physics Research A 587 (2008) 1–6
directly; moreover, the observation of a non-zero deuteron EDM at the level 1029 e cm promised in Ref. [1] will lead to physics beyond the current Standard Model. In the experiment, the deuterons of some 1–1.5 GeV/c momentum, polarized mostly in the horizontal plane, rotate in a small storage ring, R2 m at the magnetic field B2 T. In its restframe, the deuteron spin precesses around the orbit (the so-called g-2 precession) due to the interaction ~ between the magnetic dipole moment, oriented along S, ~ and the vertically oriented B-field. The basic idea of the experiment is to use a restframe electric field, ~ and oscillating in resonance with E, perpendicular to B ~ the g-2 precession. If a non-zero EDM oriented along S exists, then we will observe a resonance spin-flip due to the ~ interaction between this EDM, precessing together with S, ~ and the oscillating E-field. Statistically, the spin-flip will be seen by polarimeter(s) as a small spin rotation in the vertical plane. In order to get as big an E as possible, the authors of Ref. [1] propose to use specially designed synchrotron oscillations of the deuteron momentum, the same for all deuterons. As a result, the restframe radial ~ will oscillate due to oscillations electric field g½~ v B of the deuteron velocity, ~ vðtÞ ¼ ~ v0 þ ðD~ vÞ0 cosðot þ fÞ, where o ¼ oa, the g-2 precession frequency. Subscript a refers to the anomalous gyromagnetic factor, a ¼ (g2)/2, a ¼ 0.143, for deuterons. In terms of the abstract oscillator described above, the momentum synchrotron oscillations are the oscillations of the external force, while the radial betatron oscillations induced by the momentum oscillations are the forced oscillations of the oscillator. Variable x of that oscillator is simply the deuteron radial deviation from the designed equilibrium orbit. Thus, o ¼ oa is the frequency of the forced radial oscillations. Of course, there exist also the free radial oscillations around the forced ones. A problem arising from the existence of the forced radial oscillations with the g-2 frequency and phase is that the presence of any ring imperfection coupling the horizontal, x, and the vertical, y, betatron oscillations leads to forced y-oscillations with the same g-2 frequency and phase. These forced y-oscillations in turn lead to parasitic spin resonance imitating the designed EDM spin resonance. This results in systematic errors. However, if we remove the g-2 frequency from the x-spectrum, this frequency will be automatically removed from the y-oscillations. Although we cannot cancel all false EDM resonances by doing so, we can considerably reduce their number and strength. In the following analysis, s ¼ s(t) is the particle coordinate along the designed reference orbit as a function of time. Correspondingly, t ¼ t(s). Using a ‘‘smooth’’ approximation, synchrotron oscillations of particle momentum in a usual storage ring, DpðsðtÞÞ ¼ ðDpÞ0 cos ot þ higher modes
(1)
induce radial oscillations, xðsðtÞÞ ¼ DðsÞðDp=pÞ0 cos ot þ higher modes
(2)
where o is the frequency of synchrotron oscillations and D(s) ¼ D(s(t)), the momentum dispersion function, is defined by the ring lattice. This approximation works well if o is small compared with the revolution frequency, so Dp is almost constant during one turn. Dp in fact changes during a turn, but only when the particle passes an accelerating RF cavity; it remains constant between cavities. Application of our method to a concrete EDM experiment is partly based on this detail. The main difference between a usual storage ring and a resonance EDM ring [1] is that the Dp of every particle in the latter includes not only free synchrotron oscillations with some individual eigenfrequency o, but also specially designed oscillations with the same frequency oa and phase f for all particles. In a preferable version of Ref. [1], synchrotron oscillations are highly non-linear and little resemble the usual combination of free and forced oscillations. For this reason, and to distinguish them from other forced oscillations, we will sometimes call the synchrotron oscillations common to all particles in the EDM ring ‘‘coherent’’ oscillations. In this and the next sections, we consider only coherent oscillations. In an EDM ring without non-linearities, the equation for x ¼ x(s) in the smooth and linear approximation would be 1 nðsÞ 1 Dp x þ cosðoa tðsÞÞ x¼ 2 RðsÞ p 0 R ðsÞ RqB=qx s tðsÞ ¼ . n¼ B v 00
(3)
Since the equation is linear, the forced x-oscillations in Eq. (2) resulting from Eq. (3) have the same frequency as the external force, o ¼ oa; the external force in Eq. (3) is proportional to Dp/p. These x-oscillations would be harmless for the spin in an ideal ring, but when combined with ring imperfections they create perturbative fields oscillating with the same frequency, o ¼ oa. As we have already noted, the obvious result is a number of parasitic spin resonances imitating EDM. We need to remove frequency oa from the x-oscillations without removing the external force, because in this experiment we need Dp to oscillate with the g-2 frequency. We propose the following three-pronged solution: (a) include a non-linear term, xk, in Eq. (3), that is, introduce non-linear fields into the ring lattice; (b) also include in Eq. (3) some ‘‘active’’ lattice element(s) that oscillate with frequency oa/k; and (c) use destructive interference to eliminate the terms oscillating with frequency oa. The active element(s) will induce oscillations of the xk-term with the oa-frequency due to multiplication, k (oa/k) ¼ oa. In the case analyzed below, the active elements oscillate initial and final conditions for x at the entrances and exits of the big magnet sections. The destructive interference will occur between these oscillations and those of the external force with frequency oa. This interference will cancel the dangerous oa-mode of the x-oscillations without touching the oa-frequency of Dp/p.
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2. Non-linear forced oscillations The EDM ring under discussion contains two big magnet sections and two big straight sections. (Fig. 1 shows 1/4 of the ring.) Each big magnet section consists of several small magnets with relatively small straight sections between them. The magnets are designed to have different non-linear magnetic fields and different x-deviations. As described below, in the small straight sections we place lenses that ‘‘sew together’’ different x-values in different magnets. The big straight sections contain the usual strong focusing gradient lenses, RF cavities, etc. Between the big straight sections and the big magnet sections lie transition sections, our ‘‘active elements’’. A transition section may occupy some 2–3 m. Each section contains a deflecting RF cavity, the TM110 mode of which sews together the forced x-deviations in the adjacent straight and magnet sections. The deflecting RF cavities with the needed properties have recently been investigated [2]. Let us use the notations x and y for the forced vertical and horizontal deviations, and r and z for the free horizontal and vertical betatron oscillations around them. y(s)0, and x ¼ 0 in the two big straight sections. Inside any magnet, we have by design (instead of Eq. (2)), o a x ¼ a þ b cos ðt0 þ nTÞ 3
that we use a cubic non-linearity, x3, k ¼ 3, to cancel frequency oa of the radial oscillations. The phase in Eq. (4) is such that every maximum of x coincides with every third maximum of (Dp/p). In any small straight section between two magnets, x(s) changes its values along s from a value in one magnet to a different value in another. In Fig. 1, a ¼ 0 and b changes its sign when the particle goes from one magnet to another. With y ¼ 0, the proposed equation for x inside a magnet is x00 þ n2 x þ ax2 þ bx3 þ d ¼ d cos f f oa ðt0 þ nTÞ
Fig. 1. One quarter of the EDM ring. The line with the arrows shows a forced orbit of the particles. 0–00 , set of lenses making (1) matrices for both x and y; 00 –1, thin focusing lens; 1/F ¼ (qB/qx)(l/BR)E0.8 m1, lE0.2 m; 1–2, free interval 0.7 m; 2–3, defocusing thin lens, 1/FE1.75 m1, lE0.2 m; 4–5, RF deflector, lE0.7 m, (Ev)max15 MV/m; 6–7, thick defocusing lens, lE1 m, 1/FE1.75 m1; 8–9, thick focusing lens, lE1.22 m, 1/FE0.67 m1; 10–11, magnet, l ¼ pR/4, R ¼ 2 m, B ¼ 1.7 T, b ¼ +50 m4, a ¼ 0; 12–13, half of the (pR/2)-magnet, the same R, B, and a, but b ¼ 50 m4; 11–12, set of lenses making (1)-matrices for both x and y. All unlisted intervals are lE0.3 m. Lenses placed in the big straight sections to correct the CS-beta-functions inside the magnets are not shown.
(5)
where n2 ¼
1n R2
(6)
a¼
q2 B=qx2 2BR
sextupole non-linearity
(7)
b¼
q3 B=qx3 6BR
octupole non-linearity
(8)
d¼
DB BR
(4)
where a and b differ in different magnets. p and (t0+nT) are constants between accelerating cavities; T is the period and n the number of revolutions. Thus, our x-deviation inside a magnet is constant at a given n, but oscillates as a function of n with the frequency oa/3. The factor 1/3 means
3
an additional homogeneous B field
(9)
ðDp=pÞ0 . (10) R We include the quadratic non-linearity here only for the sake of generality. Though the sextupole field is needed to prolong spin coherence time [3] (a separate problem of the EDM experiment), it is neither necessary nor desirable to combine it with the magnets. We want only to show that our interference technique does not forbid having a set of different as along our ring. The inclusion of a homogeneous field (9) is a consequence of our including sextupole field (7). Quadratic non-linearity shifts the equilibrium relative to the RF deflector, and d corrects this. For simplicity, Eqs. (5)–(10) omit relatively small ‘‘inertial’’ sextupole terms proportional to n/R3 and similar octupole terms proportional to a/R. But we have kept the inertial quadrupole term 1/R2 in Eq. (6) because it may turn out not to be small compared with n/R2. The omitted terms can only slightly change Eqs. (11)–(14), below. From assumption Eq. (4) and Eq. (5), with x00 ¼ 0, we get four equations for the four different modes of x as a function of the number of turns, n 1=3 bb3 4d ¼ d; b ¼ for cos f (11) b 4
d¼
a ; a¼ 3b
2f for cos 3
b n2 þ 2aa þ 3ba2 þ 3d ¼ 0;
(12) f for cos 3
(13)
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b2 3ab2 2 3 þb a þ þd ¼0 n aþa a þ 2 2 2
for constant. (14)
2
Eq. (13) connects n with a and b, n2 ¼
a2 3b1=3 d2=3 3b 41=3
and Eq. (14) defines d, ! a 2a2 a2 d2=3 2 n d¼ . ¼a 3b 9b 27b2 41=3 b2=3
(15)
(16)
We see that d ¼ a ¼ 0 if a ¼ 0. To get some feeling for the numerical values of b and b, assume p ¼ 1 GeV/c, B ¼ 1.7 T, and (Dp/p)0 ¼ 0.5 102. Then R ¼ 2 m, d ¼ 2.5 103 m3. To estimate the biggest possible b, consider some reasonable limit on the octupole field, say, ðDBÞoct =B ðq3 B=qx3 Þx3 =6B ¼ bRx3 ¼ 20% at x ¼ +0.05 m. Note that we are free to choose the sign of b, so b ¼ jbj bb40.
(17) 4
Obviously, d40. Thus, we may choose any |b|o800 m . This gives the amplitude |b|42.3 cm, with sign b ¼ sign b. In Fig. 1, bE750 m4, bE70.058 m (the scale is arbitrary). An important requirement for the set of bs along the ring is that our non-linear oscillations of the forced orbit, Eq. (4), not influence the synchrotron oscillations. This requirement can be met if the length of the orbit remains constant when x oscillates coherently with the frequency oa/3. That is, if X bk l k ¼ 0. (18) Rk Here, bk is the positive or negative amplitude of the orbit oscillations inside magnet #k; lk is this magnet’s length; and Rk its radius. If Eq. (18) is satisfied, then the orbit length does not depend on the coherent oscillations of the momentum. But it depends, of course, on the free momentum oscillations and the radial oscillations corresponding to them. These will be discussed in Section 3. We see from Eq. (18) that we need bs, and hence bs, of different signs along the orbit, as in Fig. 1. If we decide to use sextupole fields not only in the big straight sections, but also in the magnet sections, then the choice of a depends on whether we want sign(a) ¼ sign(b) ¼ sign b and therefore ao0, or want sign(a) ¼ sign(b) ¼ sign b and therefore a40; see Eq. (12). As shown below, once we choose a certain sign of ab, and hence the sign of a ¼ 3ab, we cannot change it along a given big magnet section. For, if we do, we will not be able to sew the orbit inside that section. But it is permitted to have a set of as of different signs if they are at different big magnet sections, since the big magnet sections are separated from one another by the big straight sections, where x ¼ 0. Because x ¼ 0, the two transition sections at
the ends of the same big straight section are mutually independent. We may estimate the numerical values of as from Eq. (12). If, say, we want a ¼ 1.5 cm or a ¼ 1.5 cm, we get a ¼ 836 m3 when b ¼ 800 m4. Then from Eq. (15), with d ¼ 2.5 103 m1, we get n2 ¼ 0.165 m2; and if R ¼ 2 m, then n ¼ 0.34. But when b ¼ 800, we need to reduce a if we want a positive n2 inside the given magnet. For example, if a ¼ 71 cm, then a ¼ 724 and we get n2 ¼ 0.135 m2, n ¼ 0.46. If b ¼ 800 and a ¼ +24, then x (which is constant inside a given magnet) oscillates as a function of the turn number, n, as t0 þ nT xðnÞ ¼ 1 2:3 cos oa cm ðexampleÞ. (19) 3 Let k be the magnet number inside a given big magnet section. We can sew xk(n) to xk+1(n) only if xkþ1 ðnÞ ¼ mk;kþ1 xk ðnÞ
(20)
where mk;kþ1 is constant for every n. Transformation (20) means akþ1 ¼ mk;kþ1 ak and bkþ1 ¼ mk;kþ1 bk . So, for the sewable deviations, the product ab (and hence a) does not change its sign. Since dx/ds ¼ 0 inside the magnets, transformation (20) can be written as ! mk;kþ1 l k;kþ1 xk xkþ1 ¼ (21) 1 0 0 0 mk;kþ1 where matrix element l k;kþ1 is arbitrary and mk;kþ1 ¼ xkþ1 =xk . There are no problems in building such matrices by using gradient lenses. But they cannot transform a+b cos (f/3) into ab cos (f/3) when these cosines oscillate. In the Fig. 1 version of the ring, we choose to have (1)matrices transforming both x-x and y-y. Such an arrangement decouples operations on lenses from operations on magnets. Its disadvantage is that a straight section for a (1)-matrix cannot be very small. In a transition section, sewing the x ¼ 0 of a big straight section to the x6¼0 of a big magnet section requires an RF cavity producing the properly phased deflecting radial electric field and vertical magnetic field. These fields oscillate with either frequency o ¼ oRF+oa/3 or frequency oRFoa/3, where oRF is the frequency of the usual accelerating cavity of the ring. Also required are defocusing and focusing lenses (as in Fig. 1) and, when a6¼0, a constant BV-deflector. Note that the strong focusing lenses and the accelerating cavities placed in the big straight sections do not influence the forced x-oscillations inside the big magnet sections. That independence considerably simplifies the problem of stabilizing the free betatron oscillations around the forced oscillations. This is the picture for the horizontal plane. For the vertical plane, the matrices that sew xk+1s generally look
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different from Eq. (21). They will be used for focusing the free z-oscillations around y ¼ 0. The Fig. 1 parameters of the main ring elements are not optimized in any sense. AA and BB are two reflection symmetry axes of the ring. 3. Stability of the free oscillations The alternating cubic non-linearity, 7|b|, permits us to cancel more than the influence of our x-oscillations on the synchrotron oscillations; see Eq. (18). We can also cancel their influence on the designed betatron tunes, that is, on the stability of the free betatron oscillations around x(s, n) and y(s, n)0. We first consider the stability of the free radial oscillations. The equation for small deviations, r, from the forced orbit x(s, n) is 1 qB=qx t0 þ nT r00 þ 2 þ r þ 2aðsÞ aðsÞ þ bðsÞ cos oa r BR 3 R t0 þ nT 2 r ¼ 0. (22) þ 3bðsÞ aðsÞ þ bðsÞ cos oa 3 Here, a, b, a, b vary from magnet to magnet but remain constant inside every magnet. In all non-magnet areas, a ¼ b/R ¼ 0, so only the gradient focusing is present. Taking a and b from Eqs. (11)–(15) into Eq. (22), we get " # qB=qx b 1=3 2=3 00 r þ rþ 3 d r BR S 4 M 1=3 b t þ nT 0 2=3 þ6 d cos 2oa r ¼ 0. (23) 4 M 3 Note that parameters a and n2 of the magnets disappear from this linear equation. b ¼ b(s) is present only inside magnets; hence, subindex M. In this equation (not in the lattice!), [1/R2+(qB/qx)/BR] ¼ (qB/qR)/BR is non-zero only outside magnets; hence, subindex S. In what follows, we will sometimes omit these subindices. If the gradient lenses in the big straight sections, in the transition sections, and between magnets collectively provide sufficiently strong focusing (which can be achieved rather easily), then we can treat the cosine term in Eq. (23) as a perturbation. So r00 þ Kr þ DKr ¼ 0 1=3 t þ nT b 2=3 DKðsÞ ¼ 6d cos 2oa 3 4
(24)
where the term in the square brackets is constant for a given turn. The betatron tune remains unchanged in the first approximation and the r-oscillations therefore remain stable if [4] X 1 cos 2pnx cos 2pnx0 ¼ sin 2pnx0 ðb¯ CS Þi DK i l i ¼ 0 2 i (25)
or simply if X 1=3 ðb¯ xCS Þi bi l i ¼ 0
5
(26)
i
where the bar means averaging along magnet #i. bxCS(s) is the Courant–Snyder (CS) beta-function [4] for the horizontal oscillations corresponding to the non-perturbed equation with DK ¼ 0; nx and nx0 are the perturbed and non-perturbed horizontal tunes. The structure of Eq. (23) defines the so-called momentum compaction factor, ap ¼ (DL/L)/(Dp/p). We need to take into account that (Dp/p) contains two parts: a coherent part, d cos f in Eq. (5), which is the same for all particles, and an incoherent part, (Dp/p)inc. ðDp=pÞinc ¼ ðDp=pÞi cosðOt þ wÞ, where Oaocoh ¼ oa and w is a phase of a particle’s free synchrotron oscillations around the coherent forced oscillations. We should put R1 ðDp=pÞinc into the right side of Eq. (23). Due to the non-linearity of synchrotron oscillations, all particles have different Os and hence different full phases, which makes these oscillations incoherent. In the first approximation, we can neglect DK in Eq. (24). If we then solve the new equation for r as a linear function of (Dp/p)inc, we will get our momentum distribution function, D, and ap ¼ /D/RS, the average along the ring—which is essential for the free synchrotron oscillations. These oscillations are stable if ap6¼1/g2, where g is the relativistic factor. Let us now consider the stability of the vertical free oscillations, of course taking into account the Maxwell equations connecting different components of the fields. The equation for small vertical deviations z from y ¼ 0 is ðqB=qxÞ t0 þ nT 00 z 2aðsÞ aðsÞ þ bðsÞ cos oa z z BR 3 2 t0 þ nT z¼0 (27) 3b aðsÞ þ bðsÞ cos oa 3 which gives " # 1=3 qB=qx 2a2 b 2=3 00 3 z þ d z BR S 4 3b M 1=3 b t0 þ nT 6 d2=3 cos 2oa z ¼ 0. 4 M 3
(28)
As in Eq. (23), term (qB/qx)/BR in this equation (not in the lattice!)6¼0 only outside the magnets, while a and b6¼0 only inside them. Let, m and byCS be the tune and the Courant–Snyder beta-function of the non-perturbed vertical free oscillations. Then, m is not changed by the oscillating part of the gradient in Eq. (28)—that is, by our x-oscillations—if X 1=3 ðb¯ yCS Þi ðbÞi l i ¼ 0 (29) i
where the bar means averaging along magnet #i. It is easy to maintain the stability of the oscillations without the cosine term in Eq. (28), and the presence of that term does not affect the stability if condition (29) is satisfied.
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4. The side effects The above non-linear method applied to the EDM experiment cancels with some accuracy all parasitic spin resonances connected with radial oscillations induced by momentum oscillations—though only for field multipoles proportional to x0, x1, x2, and skew multipoles corresponding to them. (A perturbative skew multipole appears when the normal multipole is rotated around the longitudinal axis.) The cancellation follows from the fact that when x oscillates with the frequency oa/3, none of these multipoles can produce the spin resonance frequency oa. However, the octupole non-linearity, which is proportional to x3 and is used for these cancellations, cannot cancel a spin resonance produced by its own perturbative skew octupole. This is the main side effect of our method. The advantage of the method is that this perturbation appears only in the octupole mode of the field, and in this respect is much better defined than other perturbations. The normal octupole acts as ½ðq3 B=qx3 Þ=6BRðx3 3xy2 Þ in the horizontal plane and as ½ðq3 B=qx3 Þ=6BRðy3 3yx2 Þ in the vertical plane. The skew octupole acts as ½ðq3 B= qx2 qyÞ=6BRð3x2 y y3 Þ in the horizontal plane and as ½ðq3 B=qx2 qyÞ=6BRð3xy2 þ x3 Þ in the vertical plane. x oscillates with the frequency oa/3 in our ring. So obviously the skew x3-term now oscillates in the vertical plane with the spin resonance frequency oa. This is a small and rather well-defined perturbation, jq3 B=qx2 qyj ¼ yjq3 B=qx3 j, y51, where y is a rotation angle around the longitudinal axis. But it is a perturbation that needs to be canceled. One way
to reveal and measure it is to excite coherent vertical oscillations with the frequency oz ¼ (or)/3, where or is the frequency of the free radial oscillations, and observe the resonance build-up in the horizontal oscillations. Description of other, more sophisticated ways of doing so is beyond the scope of this paper. A possible additional side effect of the non-linear method is the appearance of the frequency oa/3 in the Fourier spectrum of the g-2 precessions, giving us a nonlinear version of the very situation we are trying to avoid. We estimate that, in the worst case, these new resonances will fortunately be some three to four orders of magnitude weaker than the resonances we have canceled. In the best case, we can choose a lattice in which the frequency oa/3 appears only beyond the accuracy level defined in our method by errors in field measurements. Acknowledgments The author thanks M. Tigner, Y.K. Semertzidis, R. Talman, and S. Orlov for valuable discussions. References [1] Y.F. Orlov, W.B. Morse, Y.K. Semertzidis, Phys. Rev. Lett. 96 (2006) 214802. [2] V. Shemelin, S. Belomestnykh, RF design of the deflecting cavity for beam diagnostics in ERL injector, Cornell LNS Report ERL 07–2, /http://www.lns.cornell.edu/public/ERL/2007/S. [3] B. Vasserman, et al., Phys. Rev. B 198 (1987) 302. [4] A. Wu Chao, M. Tigner (Eds.), Handbook of Accelerator Physics and Engineering, World Scientific, Singapore, 1998.