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DESIGN NOTE TRIUMF TRI-DN-05-12 March 2005 Range and width, dwell time, acceleration eﬃciency, and normalized dispersion for serpentine channels of quadratic pendulum oscillator Shane Koscielniak Abstract This note deals with the choice of operating parameters for a nonscaling FFAG whose longi- tudinal dynamics is analgous to that of a pendulum oscillator with a cubic variation of speed on momentum. The choice depends upon a compromise between acceleration range, dwell time over that range, acceleration eﬃciency, and the dispersion of arrival time. Expressions are given for these four quantities for a variety of operating points and recommendations are made for a multi-GeV muon accelerator. Attention is drawn to the need to match the phase-space ellipse depending on the working point. Finally, these notions are explored and conﬁrmed by computer tracking of particle ensembles. The analysis described in this note was completed September 2004. TRIUMF 4004 WESBROOK MALL, VANCOUVER, B.C. CANADA V6T 2A3 Contents 1 Introduction 1 1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Where to operate? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Inﬂuence of scale constant y/E . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Adjusting ∆E to recover the nominal range . . . . . . . . . . . . . . 3 1.3 Dwell time, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Variation of dwell time . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Dispersion of ﬁnal coordinates . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Hamiltonian and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.2 Dispersion in period . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Minimizing distortion of phase-space ellipse . . . . . . . . . . . . . . . . . . 8 2 No slip reversal, case of b=0 8 2.1 Range and width of channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Half period, eﬃciency and dispersion . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Integration Range x = ±π/2, y = ±(3a/4)1/3 . . . . . . . . . . . . . . . . . . 10 2.3.1 On the central trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Oﬀ the central trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Summary for integration range x = ±π/2, y = ±(3a/4)1/3 . . . . . . 11 2.4 Integration Range y = ±1/2, x = ± arcsin(1/6a) . . . . . . . . . . . . . . . . 11 2.4.1 On the central trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.2 Oﬀ the central trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Two slip reversals, case b = 0 13 3.1 Explicit solutions y(x(z)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 End ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.2 Centre range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Range and width of channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Central trajectory spans y = ±1/2 . . . . . . . . . . . . . . . . . . . . . . . 16 . 3.4 Half period . . . . . . .√ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Integration Range y = ± b, x = ± arcsin(ac /a) . . . . . . . . . . . . . . . . 17 3.5.1 On the central trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 17 . 3.5.2 Oﬀ the central trajectory . . . . . . √ . . . . . . . . . . . . . . . . . . 18 3.5.3 Summary for integration range y = b, x = ± arcsin(ac /a) . . . . . . 19 3.6 Integration Range y = ±1/2, x = ± arcsin[(1 − 3b)/(6a)] . . . . . . . . . . . 20 3.6.1 Summary for integration range y = ±1/2 . . . . . . . . . . . . . . . . 21 3.7 Integration Range y = ±ˆ, x = ±π/2 . . . . . . . . . . . . . . . . y . . . . . . 21 3.7.1 Summary for integration range x = ±π/2 . . . . . . . . . . . . . . . . 23 4 Reprise 23 4.1 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1 1 Introduction The linear-ﬁeld, varying-tune Fixed-Field Alternating-Gradient (FFAG) accelerator is pro- posed for the neutrino factory and muon collider[1]. Though utilizing conventional linear magnetic elements, that are similar to those used in synchrotrons, these machines are in- tended to operate in a very novel way. Whereas the synchrotron increases the magnetic ﬁelds during acceleration, and typically tolerates a relative momentum spread 1 %, the FFAG operates at ﬁxed magnetic ﬁeld with a range of central momenta spanning up to ±50 % in δp/p. This has two consequences: (i) the transverse focusing strength falls with increasing momentum; and (ii) the particle beam moves across the radial aperture, during acceleration, leading to a signiﬁcant change in the orbit shape, which produces a quasi-parabolic time- of-ﬂight variation. The ﬁrst property leads to the crossing of many integer and half-integer betatron resonances. In a machine with ﬁxed radio-frequency, the second behaviour neces- sitates acceleration within a rotation manifold, a bundle of serpentine phase-space paths linking injection to extraction, rather than the customary libration manifold of the pendu- lum oscillator (a.k.a. rf bucket). This note deals with the novel longitudinal dynamics and the choice of operating parameters which depends up on a compromise between acceleration range, dwell time over that range, acceleration eﬃciency, and the dispersion of arrival time. Expressions will be given for these four quantities for a variety of operating points. 1.1 Equations of motion The linear-ﬁeld FFAGs posess magnet lattices that are almost isochronous across a large range of momenta. The variation of path length with momentum is almost parabolic. Let the time of ﬂight range per cell be ∆T over the energy range ∆E, and the peak energy increment per cell be δE. Let the index n denote iteration number, En be the particle energy and tn , Tn be the absolute and relative arrival times, respectively. Let τ0 be the cell traversal time at the reference energy Er . Then tn = Tn + nτ0 . The longitudinal motion in the variable-tune non-scaling FFAG accelerator may be modelled by the following simple diﬀerence equations: En+1 = En + δE cos(ωTn ) (1) Tn+1 = Tn + 4(En+1 − E)2 (∆T /∆E 2 ) − δT2 ¯ (2) ∆T = δT1 + δT2 , ˆ ˇ ∆E = E − E , ¯ ˆ ˇ E = (E + E)/2 . (3) The time slip δT2 represents the fact that the radio-frequency is synchronous with the orbital ¯ period at Er , which is not necessarily equal to the mean energy E. The reference energy is the solution of Tn+1 = Tn , namely Er = E ± (∆E/2) δT2 /∆T where the ratio δT2 /∆T may ¯ take any value between 0 and 1. Previously, in design note TRI-DN-03-13[2] and elsewhere, we had scaled the parame- ters such that all cases except δT2 = 0 were mapped into a single universal hamiltonian H = (y 3 /3 − y) − λ sin x depending on a single parameter λ with δE ∆T 1 a λ∝ = 3/2 . (4) ∆E δT2 ωδT2 b 1 To eﬀect this transformation, time was scaled as δT2 /∆T which eliminates any consideration of the case δT2 = 0. Moreover, the transformation obscures the fact that the range of motion depends on δT2 in addition to (δE/∆E). For these reasons, we shall now give a less sophisticated but more transparent formulation of the motion equations. We introduce dimensionless variables: y = (E − E)/∆E ¯ and x = ωT , (5) and approximate the motion by diﬀerential equations: dy/dn ≈ (δE/∆E) cos(x) (6) dx/dn ≈ (ω∆T )[4y 2 − (δT2 /∆T )] . (7) We introduce further dimensionless parameters: s ≡ nω∆T , a ≡ (δE/∆E)/(ω∆T ) , b ≡ (δT2 /∆T ) , (8) leading to the model equations: dy/ds = a cos x and dx/ds = (2y)2 − b . (9) The injection and extraction energies E, E correspond to y = ∓1/2, respectively. ˇ ˆ 1.2 Where to operate? For the 5-20 GeV muon application, cost and technical considerations limit a to approxi- mately 1/12 – but for an electron model values up to a = 1/4 are considered for a fuller investigation of the (a, b) parameter space. Previously, Koscielniak[3] suggested that the optimum value of b is one quarter. In this case, for a > 1/24 a serpentine channel extends between (and beyond) the full range y = ±1/2. Recently, Keil[4, 5] has proposed operation with b = 0 because it strongly reduces the nonlinear distortion of the occupied phase space - because the dispersion in arrival times is reduced. However, if operating with b = 0, then a must exceed 1/6 for the channel to extend over the full energy range. These developments, and the possibilities for exploration in the electron-model machine, prompt us to revisit this area. √ ¯ First we note that Er = E ± (∆E/2) b. This implies that when b alone changes so does the reference energy and the radio-frequency. But the magnetic lattice stays ﬁxed and so does its time-of-ﬂight parabola. When δT2 = 0, the range of acceleration scales as the cube root of voltage. As δT2 is progressively increased, so there is an opportunity for linear acceleration which reaches its maximum value when b = 1, that is δT2 = ∆T . As δT2 rises, so the overall range of acceleration increased; but the minimum voltage to open the serpentine channel also rises. For example, when δT2 = 0 the minimum acceleration to open the channel is a → 0, but the range is also zero. Contrastingly, for δT2 = ∆T the momentum y extends to twice the nominal acceleration range (i.e. y = ±1), but the channel does not open until a ≥ 1/3. 2 In general, the choice of operating point (a, b) will depend not only on acceleration range but also up on a compromise between dwell time (i.e. decay losses for muons), acceleration eﬃciency, and dispersion of arrival time (i.e. emittance distortion); and we must consider all four quantities. What may come as a surprise, is that for values of a > 1/6 it is not possible to place the extraction/injection momenta at precisely x = ±π/2 where the phase acceptance is symmetrical. 1.2.1 Inﬂuence of scale constant y/E It is instructive to mention one minor curiosity which stems from the arbitrary nature of the scale constant between y and E. The exact same magnetic lattice with the identical radio-frequency and voltage, but run over diﬀerent energy ranges will have diﬀerent values of a, b. For the same lattice, ∆T ∝ ∆E 2 . For example, if we assume there is no limitation of magnet aperture and propose to run the machine over twice the energy range, then ∆E ⇒ 2∆E , ∆T ⇒ 4∆E , a ⇒ a/8 , b ⇒ b/4 , ac ⇒ ac /8 . (10) The reference energy Er remains unchanged. With the new choices of a, b, the longitudinal phase space of the original energy range is now bounded by y = ±1/4. The fact that the ratio of a to b is changed when the scale changes gives some hint as to why the hamiltonian can be transformed into a single parameter (λ) system. These simple observations have far reaching implications. It is tantalizing to note that (a, b) = (1/3, 1) places extraction/injection at y = ±1, which is double the nominal range. However, this advantage is illusory. For example, it might be thought that selecting a 15-25 GeV machine with, say, (a, b) = (1/2, 1) is a clever way of designing a 10-30 GeV accelerator. But in fact it is simply a 10-30 GeV machine with (a, b) = (1/16, 1/4). This is not say that the energy range cannot be extended (a little) beyond the nominal by increasing b; it can, but the acceptance falls quickly. Incidentally, comparatively large values of a, such as 1/2, can be achieved by simply reducing the bend angle and increasing the number of cells; but the machine looks increasingly like a linac and eventually turns into a single pass accelerator. 1.2.2 Adjusting ∆E to recover the nominal range Later we shall encounter the fact that when (a, b) = (1/12, 1/4), the full energy range of the channel at x = ±π/2 is y = ±0.548956 which is larger than the range between the nominal extraction and injection energies y = ±0.5. This raises the question of whether it is possible to “reverse engineer” the range ∆E so that the full range becomes equal to the nominal range. Hence we now take ∆E to be a variable whose value will be found in terms of both the nominal range of the machine ∆En and previously chosen values of a, b. From equation (8), the nominal value of the ToF range is ∆Tn = (δE/∆En )/(a ω). The equations above are heavily constrained, but we may exploit the freedom to specify the exact relationship ∆T = (α∆E)2 which imposes a constraint on the magnet lattice. We 3 substitute this relationship into equation (8) and solve for δT2 and ∆E in terms of a, b. 2/3 1/3 α δE δE δT2 = b and ∆E = . (11) aω a α2 ω We have also the condition that the full range of y (for a ≥ 1/6) between x = ±π/2 is √ ∆y = 2 b cosh[(1/3)arccosh(3a/b3/2 )]. This quantity is a pure numerical factor because a, b are already chosen. From the deﬁnition of y, equation (5), it follows that ∆E = ∆En /∆y. Hence it follows that α2 = [δE/(aω)][∆y/∆En ]3 . This coeﬃcient may now be substituted to determine all other variables: ∆En ∆E = , δT2 = b∆T , ∆T = ∆y∆Tn , ∆T = ∆Tn ∆y 3 . (12) ∆y The meaning of these relations (for given a, b) is that if we design the machine for ∆E, ∆T , it 2 can be operated over ∆En , ∆T . Here ∆T = α2 ∆En is the actual permissable time interval over the nominal energy range. Given that ∆y ≥ 1, it should be apparent that the reverse engineered machine is less demanding than the nominal machine. For various values of b, ∆T and ∆T scales as follows. b ∆y ∆y 3 1/6 1.0 1.0 1/5 1.03967 1.1238 1/4 1.09791 1.32343 Thus for larger values of b, the permissable time of ﬂight variation ∆T is also larger. Of course, this advantage comes at the cost of reduced acceptance. In the case b = 1/6 the reverse-engineered machine reduces to the nominal machine. 1.3 Dwell time, etc Dwell time τ is the temporal interval to pass between two points. As the two points approach one another, one is led to the concept of the diﬀerential element dτ . We shall now obtain this quantity and its variation for a hamiltonian system. We adopt a Cartesian phase space with dimensionless coordinates (x, y) and associated orthogonal unit vectors i, j. Hence we may deﬁne a position vector x = ix + jy and a velocity ˙ ˙ vector v = ix + jy where the dot notation indicates time derivative. Given the hamiltonian ˙ H of conjugate variables x, y the components of the velocity vector are x = ∂H/∂y and y = √ ˙ −∂H/∂x. The vector element of length dl forms the hypoteneuse, of length dl = dx2 + dy 2, ˙ ˙ of a triangle having sides idx and jdy. The ratio of dx = dl(x/v) to dy = dl(y/v) depends on location on the path. From these deﬁnitions follows the relation between phase-space increments of path length dl = |dl|, dwell time dτ and speed v = |v|: dl dx2 + dy 2 dτ = = . (13) v x2 + y 2 ˙ ˙ 4 By using the relation ∂x/∂y = −x/y the increment may be formed holding, respectively, ˙ ˙ either y or x constant: dτ |y = dx/x ˙ or dτ |x = dy/y . ˙ (14) ˙ ˙ Whereas the second version (14) is prone to accidental singularities (where x or y are zero), ˙ the ﬁrst version (13) is only ill-deﬁned at essential singularities, i.e. stalling points, where x ˙ and y are both zero. From these variants follow directly the equations (15) below. To compute the dwell time, τ , a path in the two-dimensional x, y-plane must be spec- iﬁed, and the elements summed along that path. A particular value of the path is speciﬁed by the value h of the hamiltonian. Almost the same integral will give the expectation value f of some quantity f . ˆ x dx y dy ˆ dt = τ = = . (15) x x ˇ ˙ y y ˇ ˙ x dx ˆ y dy ˆ f τ = f = f . (16) x ˇ ˙ x) yˇ y˙ ˙ ˙ Here the reciprocals of speed x and acceleration y along the path gives a dwell-time weighting. From these integrals it follows that the averages of speed and acceleration are given by ˆ x ˆ y ˙ xτ= dx = x − x ˆ ˇ and y τ= ˙ dy = y − y . ˆ ˇ (17) ˇ x ˇ y Hence these averages may be computed trivially once the dwell time τ is known. 1.3.1 Variation of dwell time Of interest is how the dwell time changes when h is varied, that is dτ /dh. In performing ths derivative it must be recognised that the path and its end points change when h is varied. This is akin to the simple one-dimensional problem of ﬁnding the deriavtive of an integral with respect to a parameter. z2 ∂F z2 (h) ∂z z2 z2 (h) ∂f F = f dz , = f (z, h)dz = f + dz , (18) z1 ∂h z1 (h) ∂h z1 z1 (h) ∂h where terms are evaluated at the expansion point h = h0 . However, ours is a 2D problem; and moreover it is not a line integral with respect to two vectors as occurs in, say, the calculation of circulation in a vector ﬁeld. We are interested in: x2 (h+δh) dl(h + δh) x2 (h) dl(h) τ (h + δh) − τ (h) = − . (19) x1 (h+δh) v(h + δh) x1 (h v(h) First consider how the integration limits change, for example x1 (h + δh) = x1 (h) + δx1 . The increment δx1 is not a free variation, rather it is constrained to lie on a contour of constant hamiltonian with value h + δh. When H is varied we move in a direction given by the unit vector n ≡ ∇H/|∇H|, that is perpendicular to the line of constant hamiltonian. [∇H = −iy +jx.] We know that δH = ∇H ·δx, that δx = n|δx| and that (∇H)·(∇H) = v 2 . ˙ ˙ 5 Hence δH = v |δx| and δx = nδH/v where v = |v|. The unit vector tangential to the line of constant hamiltonian is m ≡ v/v. The local vector element of length along the phase-space path is dl = mdl. We must next ﬁnd the analogue of the term f (∂z/∂h)|z2 appearing in the 1D integral. z1 The relevant question is “how does the element of dwell time change as the endpoint of the path element is changed?” with answer δτ = ∇τ · δx. Now the Poisson bracket of conjugate variables such as time and energy (τ, H) is unity; from which it follows that ∇τ · v = 1. There are only two vectors to choose from which satisfy this condition for all v, and the natural choice is ∇τ = m/|v| = v/v 2 , which is perpendicular to ∇H. Thus, ﬁnally, v2 · δx2 v1 · δx1 ∂ x2 dl δh2 ∂ 2 x2 dl τ (h + δh) − τ (h) = 2 − 2 + δh + + ... (20) v2 v1 ∂h x1 v 2! ∂h2 x1 v Here the subscripts 1 and 2 denote the start and end points, respectively. 1.3.2 Dispersion of ﬁnal coordinates Equation (20) may be used in either of two ways: (i) to ﬁnd the diﬀerence in time elapsed for neighbour trajectories that are locally parallel at both their end points; and (ii) for equal dwell times what is the separation (to lowest order) of their end points. In the ﬁrst case, because v1 , δx1 are perpendicular, as are v2 , δx2 , it follows that these products are each zero. In the second case, we insist that τ (h + δh) − τ (h) = 0 and ask what δx2 satisﬁes this condition for a path that is initially locally parallel at x1 . To be precise, we ﬁnd the projection of δx2 on vector v2 , and so it is not completely deﬁned. The second case may be elaborated further. Suppose that we have independently computed the partial derivatives with respect to value of hamiltonian, and so know the dwell time in the form τ (h + δh) = τ (h)[1 + C1 δh + C2 δh2 + . . .]. It is known also that δh = |v1 ||δx1 |, and so v2 · δx2 = −v2 τ (h) C1 |δx1 |v1 + C2 |δx1 |2 |v1 |2 + . . . 2 (21) We shall pre-empt some later results. For paths within the rotation manifold bounded by the serpentine separatrix, and expanded about h0 = 0, the term C1 = 0 and C2 > 0 because h = 0 corresponds to the minimum dwell time. Further, we choose to measure the dwell time between diﬀerent points x1 , x2 for which v1 = v2 . Hence the displacement δx2 satisﬁes v1 · δx2 ≈ −v1 τ (h)|C2 ||δx1 |2 , 4 (22) and is negatively directed compared with v1 . Thus after the time interval τ (h = 0), particles with h = 0 lie behind (in either position or momentum) those on the central trajectory h = 0. 6 1.4 Hamiltonian and manifolds H(x, y, a, b) = (4/3)y 3 − yb − a sin x . (23) Let c be some particular value of the hamiltonian. The paths have the symmetry of inversion through the point x = y = c = 0. Thus if x(c), y(c) is path, then so will −x, −y be a path corresponding to −c. Further, there is always a central trajectory passing through (x, y) = (0, 0) for c = 0. Depending on (a, b, c) there are possibly two stable (libration) and three unstable (ro- tation) manifolds1 for each 2π range of x. In the former, motion is coperiodic in x and y; and in the latter motion is periodic only in y, and x is unbounded. We are interested in one of the three rotation manifolds. For large momentum oﬀset, there is no possibility of synchronism with the RF and the motion is uni-polar in the ordinate y. However, depending on (a, b, c) there is a serpentine channel in which the rotation is bi-polar in y; and this manifold may be used for acceleration. The value of the hamiltonian at the ﬁxed points of motion determines the separatrix for the serpentine channel; and the value is H = ±(a − b3/2 /3) for all settings of b. Hence for bi-polar motion |c| must be smaller than this amount. We shall use the prime notation to denote derivatives with respect to s; for example x = dx/ds. 1.4.1 Expectation values The half-period τ , the time to cross from low to high momentum is of signiﬁcance - par- ticularly in respect of decay losses. The average value of cos x along the path indicates the eﬃciency of acceleration and is also of interest. Both may be computed given the hamil- tonian (23), and the deﬁnition of expectation value above. In particular, the mean-square momentum and the acceleration eﬃciency satisfy 4 y 2 = b + (ˆ − x)/τ , x ˇ and aτ cos x = (ˆ − y ) . y ˇ (24) Though, in principle, the period may be formed by integration over x or y, the pres- ence/absence of singularities will lead to preferences. 1.4.2 Dispersion in period The dispersion in dwell times, for particles with diﬀering values (c) of the hamiltonian, is proportional to the normalized second derivative C2 ≡ (∂ 2 τ /∂c2 )/τ /2 evaluated at c = 0. Unfortunately, the derivatives w.r.t. c cannot be performed before the integration over x or y because the integrand becomes more strongly divergent. The signiﬁcance of C2 is that it leads, after acceleration, to a spread in x, y values. Let the half period and speed v = x be functions of the end points x and of the parameters a, c, ˙ ˇ ˆ etc. Consider the special case that y = 0 at the end points x, x. To ﬁnd the ﬁnal spread in x, we must see how far particles travel in a ﬁxed time, leading to the inverse problem ˆ x+∆x dx ˆ x dx = = τ (a, 0) , (25) ˇ x v(x, a, c) x ˇ v(x, a, 0) 1 Small angle motion of a pendulum is an example of the ﬁrst, whereas more than 360◦ rotation of the pendulum is an example of the second. It is worth noting that libration takes its root from the latin libra, e for ‘scales’, and not from the french libr´ for ‘free’. 7 to be solved for ∆x. Assuming the deviation ∆x is small, an approximate solution is ∆x ≈ [τ (a, 0) − τ (a, c)]v(ˆ, a, c) . x (26) Now τ (a, c) = τ (a, 0)[1 + C2 c2 + . . .], and so to lowest order the absolute spread is ∆x = −τ (a, 0)v(ˆ, a, 0)C2 c2 . The relative spread is ∆x/(ˆ − x) = −C2 c2 v(ˆ)/ v . Neither x x ˇ x τ × v nor v(ˆ)/ v varies strongly with b = [0, 1/4], and so it follows that C2 may be used x to rank the relative performance as function of (a, b). Ideally C2 is small. The variation of c with x, y depends on the values a, b. To facilitate comparison, we introduce the variation of c with δx, δy about the central trajectory for given x, y. The relevant quantity is 2 2 ∂c ∂c C 2 c2 = C 2 δx2 + δy 2 ≡ C2,x δx2 + C2,y δy 2 . (27) ∂x ∂y c=0 1.5 Minimizing distortion of phase-space ellipse The observation that the incremental dwell time is proportional to the square of the incre- ment c in hamiltonian, that is τ (a, c) = τ (a, 0)[1 + C2 c2 + . . .], is the key to minimizing the distortion of the occupied phase space. Basically, we must load the phase space about the reference particle on the central trajectory in such away that |c| takes (almost) the same value at all points on the perimeter of the ensemble. This implies that there is an optimum aspect ratio and orientation angle for the phase space ellipse, and the injected beam must match to these values. Given that c increases slowly/quickly in a direction parallel/perpendicular to the reference path, it is clear that ellipses wide in ∆x and narrow in ∆y will be favoured. However, in world coordinates (∆E, ∆T ) the aspect ratio may look very diﬀerent. 2 No slip reversal, case of b=0 In the case that b = 0, there is never any reversal of the x slip direction. The hamiltonian is cubic in y and (for given c) there are three roots for each value of x. However, one of the roots is complex and must be rejected. y(x) is a single-valued function, and so the remaining two roots are used for alternate ranges of x. Let x = − arcsin(c/a). The two real solutions ˜ of H = c and their ranges of applicability are: y1 = (3/4)1/3 (c + a sin x)1/3 x≤x≤π−x ˜ ˜ (28) 2/3 y3 = (−1) y1 −π−x≤x≤x. ˜ ˜ (29) These are, in fact, a simple particular case of the general equations (43-45). Depending on c, the motion is divided between three rotation manifolds. The central one is bounded by |c| ≤ a and is potentially useful for acceleration because the paths are bi-polar in y. The bounding condition is obtained by evaluating the hamiltonian at the ﬁxed points: H(x = ±π/2, y = 0) = ∓a. No path connects these points unless a = 0, that is δE = 0. 8 0.6 0.4 0.2 -3 -2 -1 1 2 3 -0.2 -0.4 -0.6 Figure 1: (a, b) = (1/6, 0). Central trajectory c = 0 (green). Upper and lower channel boundaries c = ±a (red and blue). Example paths c = ±a/2 (magenta and cyan). 2.1 Range and width of channel The range of the channel is given by that of the central trajectory, for which c = 0. The width of the channel depends on paths terminating on the ﬁxed points for which c = ±a. The acceleration range extends between y = ±ˆ and y = (3a/4)1/3 . At x = ±π/2, the y ˆ 1/3 momentum width of the channel is δy = (3a/2) – but the useful width is much less. Evidently, the serpentine channel opens for an inﬁnitessimal value of a, but its range on the central trajectory does not extend to injection or extraction (y = ±1/2) until a ≥ 1/6. However, even for a = 1/12 the width of the channel (from y = 0 to the separatrix) extends to (x, y) = ±(π, 1)/2 and one could imagine to inject below the central central trajectory and closer to the lower branch of the separatrix. This plan has some merit, but it still implies operation with a > 1/12 to achieve useful results. Moreover, in such a case the injection and extraction momenta are not symmetrically disposed about the mean energy ¯ E (i.e. y = 0). There is the further disadvantage that the dwell time is not a symmetric function of hamiltonian value when expanded about c = 0. But subject to the results of numerical investigation, the idea is not yet totally abandoned. 2.2 Half period, eﬃciency and dispersion To compute the half period, one may integrate over the position or momentum variable. The location and type of singularity of the integrand inﬂuences this choice. When integrating over y, the integrand is singular where y = 0. This occurs at (x, y) = ±(π/2, y ) where ˆ the singularity is of the form 1/ 1 − (y/ˆ) y 6 , which is divergent. Of course, for smaller, truncated ranges of y the integral is free of singularity. When integrating over x, the integrand is singular where x = 0. This occurs at (x, y) = (0, 0) and the singularity is of form 1/(a sin x)2/3 which permits convergence. 9 2.3 Integration Range x = ±π/2, y = ±(3a/4)1/3 The period and expectation values will vary with the range over which they are evaluated. Let us ﬁrst form the values on the ﬁxed range x = ±π/2, which implies a varying momentum range y = ±(3a/4)1/3 . 2.3.1 On the central trajectory For the central trajectory c = 0, the total dwell time is √ +π/2 dx π/2 dx π Γ(1/6) 2.20658 τu (a, 0) = =2 = ≈ . (30) −π/2 4y 2 0 (6a sin x)2/3 (6a)2/3 Γ(2/3) a2/3 On the same trajectory, the acceleration eﬃciency is found to be independent of a. 1 6 Γ(2/3) cos x = 2(3a/4)1/3 = √ ≈ 0.823503 (31) aτ π Γ(1/6) 2.3.2 Oﬀ the central trajectory In order to ﬁnd the time dispersion, we must compute the half period for motions oﬀ the central trajectory. The case that c = 0 is much more complicated than the previous working. However, with excellent tools such as Mathematica, progress can be made. We are compelled to subdivide the integration range x = ±π/2, and take the integral in the form: ˜ x dx +π/2 dx +π/2 dx +π/2 dx τ= 2 + 2 =+ + . (32) −π/2 4y3 ˜ x 4y1 x(+c) ˜ 4y1(+c)2 x(−c) ˜ 4y1 (−c)2 Here x = − arcsin(c/a). A “backdoor approach” that relies on the inversion symmetry of ˜ the paths reduces the work and avoids having the integrals become possibly undetermined at the limits. Note that we place few restrictions on y which may have extrema either less or greater than |y| = 1/2. The half period is τu (a, 0) a2/3 1 1 5 a+c 1 1 5 c−a τu (a, c) = √ √ (a + c)1/3 F21 , , , + (a − c)1/3 F21 , , , 3 a2 − c2 3 2 6 c−a 3 2 6 a+c (33) Here F21 (. . .) is the hypergeometric function. The half-period may be expanded ﬁrst in a Taylor series about c = 0. The lowest order terms in c are exactly 2 c 2 32 c 4 3136 c 6 τu (a, c) = τu (a, 0) 1 + + + ... . (34) 9 a 243 a 32805 a An alternative expansion retains the property that motion stalls on the separtrix at |c| = a: a 5 c 2 203 c 4 τu (a, c) ≈ √ τu (a, 0) 1 − − + ... (35) a2 − c2 18 a 1944 a The variation of c with y about x = π/2, y = (3a/4)1/3 is to lowest order ∂c/∂y = (6a)2/3 and ∂c/∂x = 0. The expectation value of cos x may also be obtained: a τ (a, c) cos x = 31/3 [(a + c)1/3 + (a − c)1/3 ]/22/3 ≈ (6a)1/3 [1 − (c/3a)2 + . . .] . (36) 10 2.3.3 Summary for integration range x = ±π/2, y = ±(3a/4)1/3 a 1/12 1/6 period 11.5657 7.28595 ±y range 0.39685 0.5 width 0.5 0.629961 C2,y 8×22/3 ≈ 12.7 8 2.4 Integration Range y = ±1/2, x = ± arcsin(1/6a) Notice in the working above, that we have not considered whether these paths cross the nominal acceleration range y = ±1/2. This ﬁrst occurs when a = 1/6 and the x-range extends between ±π/2. For larger values of a, the range of x for which |y| ≤ 1/2 is smaller and we must reduce the corresponding ranges of integration to x = ± arcsin[1/(6a)]. We shall refer to the dwell time between these points as the restricted period τr . Incidentally, τr (1/6, 0) = τu (1/6, 0) as may be observed in the ﬁgure 2, and likewise for cos x and C2 for a = 1/6. 2.4.1 On the central trajectory On the central trajectory c = 0, the restricted half period is 1 1 1 7 1 τr (a, 0) = F21 , , , . (37) a 6 2 6 (6a)2 Here F21 is the hypergeometric function. The period τr may also be written in a form that shows its relation to the unrestricted period τu . 1 1 1 1 τr = τu − 2/3 β 1− , , (38) (6a) (6a)2 2 6 where β[z, a, b] = 0z ta−1 (1 − t)b−1 dt is Euler’s incomplete beta function. We may also obtain the average value of cosine over the restricted range y − y = 1, ˆ ˇ 1 1 cos x r = = . (39) a τr F21 [. . .] This quickly approaches unity for a > 1/4. Approximations for the function F21 and its reciprocal are given in appendix A. 11 T <cosx> 11 0.975 10 0.95 9 0.925 8 7 a 0.25 0.3 0.35 0.4 0.45 0.5 6 0.875 5 0.85 a 0.125 0.15 0.175 0.2 0.225 0.25 0.825 Figure 2: Half period between x = ±π/2 Figure 3: cos x r for range y = ±1/2, (red), and between y = ±1/2 (blue). x = ± arcsin(1/6a). 2.4.2 Oﬀ the central trajectory To compute the time dispersion we must ﬁnd the dwell time for paths with c = 0. Again we restrict the x-range of the integrals2 to x = − arcsin[1/(6a)] to x = + arcsin[1/(6a)] with ˇ ˆ the condition that a ≥ 1/6. x ˜ dx x ˆ dx x ˆ dx x ˆ dx τr = 2 + 2 = 2 + 2 (40) x ˇ 4y3 x ˜ 4y1 x(−c) 4y1 (−c) ˜ x(+c) 4y1 (+c) ˜ 1 1 1 1 4 −1 + 6c 1 − 6c τr = √ (1 − 6c)1/3 A1 , , , , , 2 a 2 − c2 3 2 2 3 6(a + c) 6(a − c) 1 1 1 4 1 + 6c 1 + 6c + (1 + 6c)1/3 A1 , , , , , (41) 3 2 2 3 6(c − a) 6(a + c) Here A1 is the Appell hypergeometric function. τr may be Taylor expanded in powers of c, but in general the coeﬃcient of c2 , etc., are too lengthy to record here. However, we present a few special cases: a = 1/6, 1/5, 1/4. τr (1/6, c) = τr (1/6, 0) 1 + 8c2 + (512/3)c4 + (200704/45)c6 + . . . τr (1/5, c) = τr (1/5, 0)[1 + 3.49768c2 + 24.3792c4 + . . .] τr (1/5, 0) ≈ 5.37225 τr (1/4, c) = τr (1/4, 0)[1 + 0.687494c2 + . . .] τr (1/4, 0) ≈ 4.1581 √ τr (1/6, 0) = π Γ(1/6)/Γ(2/3) ≈ 7.28595 (42) In general the normalized second derivative C2 falls rapidly as a increases. The variation of c about x = arcsin(1/6a), y = 1/2 is to lowest order ∂c/∂y = 1 and ∂c/∂x = − a2 − 1/62. Hence C2,y = C2 and C2,x = (a2 − 1/62)C2 . The path-averaged value of cosine for c = 0 but with momentum restricted to |y| ≤ 1/2 is simply cos x aτr (a, c) = 1. 2 This implies the trajectories for c = 0 are not precisely restricted to −1/2 ≤ y ≤ +1/2. 12 3 Two slip reversals, case b = 0 √ When b = 0, the direction of x slip reverses at the values y = ±(1/2) b; and this leads to a system of four ﬁxed √ points, two each at x = ±π/2. Values of the hamiltonian at the ﬁxed √ points are H(+π/2, − b/2, a, b) = −a + (b3/2 /3) and H(−π/2, + b/2, a, b) = +a − (b3/2 /3). Paths emanating from these ﬁxed points constitute the separatrix of the serpentine channel, and this implies that bi-polar motion is bounded by |c| ≤ a − b3/2 /3. For larger values the of c the motion is uni-polar. The condition to link√ unstable ﬁxed points by a straight √ line segment with linear acceleration is H(−π/2, + b/2, a, b) = H(+π/2, − b/2, a, b) with solution a = (1/3)b3/2 ; this is the critical value of a for opening of the serpentine channel and it is denoted by the symbol ac . No further useful results can be obtained without solving for y(x). y 0.6 0.4 0.2 x -3 -2 -1 1 2 3 -0.2 -0.4 -0.6 Figure 4: (a, b) = (1/12, 1/5). Central trajectory passing through (0, 0) and separatrices. Colours red, green, blue denote solutions y1 , y2 , y3 respectively. Separate curves of the same colour have diﬀerent values of the hamiltonian c. 3.1 Explicit solutions y(x(z)) Let the three roots of r3 = −1 be: r1 = −1, r2 = e+iπ/3 and r3 = e−iπ/3 . A trajectory is a contour of constant hamiltonian H(x, y, a, b) = c, with solutions y(x) given by: y1 = −[r1 w + b/(r1 w)]/2 (43) y2 = −[r2 w + b/(r2 w)]/2 (44) y3 = −[r3 w + b/(r3 w)]/2 (45) where w = [3(c + a sin x) + −b3 + 9(c + a sin x)2 ]1/3 . (46) Here y1 is the upper, y2 is the middle, and y3 is the lower segment; see ﬁgure 4. How many of these solutions we shall need depends on the values of a, b, c and the ranges of x. When |c| ≥ a − b3/2 /3, the motion is uni-polar and y(x) is single valued, so one only from y1 or y3 is required. For paths within the serpentine channel, |c| ≤ a − b3/2 /3, the situation is 13 more complex: for a centre range of x, momentum y is triple valued (for a single value of c), and y1 , y2 , y3 are all needed; and for end ranges either y1 or y3 is suitable. The centre range corresponds to the case that the radicand within w is negative, and that w is complex; that is −b3 + 9(c + a sin x)2 < 0. The end ranges occur where there is no possibility of the radicand becoming complex; i.e. −b3 + 9(c + a sin x)2 > 0. This implies the ranges being divided by the turning points (ˇ, x) of the motion y2 , namely x ˆ x = − arcsin[(ac + c)/a] ˇ and x = arcsin[(ac − c)/a] , ˆ 3ac ≡ b3/2 . (47) We may write the solutions in the form y(x(z)) where the relation between x, z depends on the range x, c, as given in the table below. segment(s) x range transformation z range y1 x → +π/2 (c + a sin x) = +ac cosh z ˆ 0→z ˆ y1 , y2 , y3 x→x ˇ ˆ (c + a sin x) = ac sin z −π/2 → +π/2 y3 −π/2 → x (c + a sin x) = −ac cosh z ˇ z→0 ˇ where z = −arccosh[(a − c)/ac ] ˇ and z = +arccosh[(a + c)/ac ] , ˆ 3ac ≡ b3/2 . (48) In order to complete the explicit solutions, it is necessary to ﬁnd the relation between z and time-like s. This connection is the dwell time, introduced in (15) and elaborated in (60), etc. s x dx z dz dx ds = = (49) v v dz In general the integral cannot be found in closed form, nor can it be inverted for z(s). However, for the special case of a = ac = b3/2 /3 and c = 0, the relation may be constructed piecewise for y1 , y2 , y3. √ √ √ √ tanh[s1,3 b/ 3] = tan[(z ± π/2)/3]/ 3 , tanh[−s2 b/ 3] = tan[z/3]/ 3 . (50) The positive/negative sign is taken in s1 /s3 , respectively. 3.1.1 End ranges In√ coordinate z = ±arccosh[3(c + a sin x)/b3/2 ], the two solutions become y1 = −y3 = the + b cosh(z/3), and the speed is given by v1 = v3 = b[1 + 2 cosh(z/3)]. We take the posi- tive/negative signs in z for y1 , y3 respectively. We use y1 , y3 for positive/negative values of c, respectively. The same solutions in terms of the hyperbolic functions also deﬁne the motion for the case that motion is uni-polar and the paths are in the rotation manifolds above and below the serpentine channel. In this case, however, there are no turning points and the end range becomes the whole range. 14 3.1.2 Centre range In the new coordinate z = arcsin[3(c + a sin x)/b3/2 , the three solutions become √ y1 = + b cos[(z − π/2)/3] (51) √ y2 = − b sin[z/3] (52) √ y3 = − b cos[(z + π/2)/3] , (53) and the speed v ≡ (2y)2 − b is given by v1 /b = 1 + 2 cos[(2z − π)/3] (54) v2 /b = 1 − 2 cos[2z/3] (55) v3 /b = 1 + 2 cos[(2z + π)/3] . (56) 3.2 Range and width of channel The range of the serpentine channel is found by substituting x = ±π/2 in the central trajectory y(x, c = 0). The range extends between √ 1 a 1 b ±y = b cosh arccosh = +w , w = 31/3 (a + a2 − a2 )1/3 . c (57) 3 ac 2 w √ At the critical value ac (b) ≡ b3/2 /3, the range spans y(ac ) = ± b. The paths emanating from the ﬁxed points constitute the separatrix and deﬁne the momentum width at x = ±π/2. Substituting values of the hamiltonian at these points into √ y1 (x, c), we ﬁnd the lower and upper bounds to be respectively b and √ 1 2a 1 b b cosh arccosh −1 = +w , w = 31/3 [2a − ac + 2 a(a − ac )]1/3 . (58) 3 ac 2 w The two bounds become equal when a = ac (b). When a excedes the critical value by a small amount ε (that is a = ac + ε), the channel width becomes √ 1 6ε √ 2ε 8ε b cosh arccosh 1 + 3/2 − b≈ 1 − 3/2 . (59) 3 b 3b 9b The channel width increases as b diminishes and in the limit b → 0, the channel width tends to δy = (3a/2)1/3 at x = π/2. The channel collapses to zero in the case that b = (3a)2/3 , and so b must be smaller than this value. For the case of our muon FFAG case, a = 1/12 and a variety of b we ﬁnd the channel momentum range and width (at x = ±π/2) as follows: b 1/4 1/5 1/6 ac 0.0416667 0.0298142 0.0226805 ±y range 0.548956 0.519837 0.5 width 0.0888253 0.126333 0.154644 15 R,W 0.6 0.5 0.4 0.3 0.2 0.1 b 0.1 0.2 0.3 0.4 Figure 5: Range (red) and width (blue) of channel versus b at x = ±π/2 for a = 1/12 up to ˆ = (3a)2/3 . b 3.3 Central trajectory spans y = ±1/2 The intended injection/extraction momenta correspond to y = ∓1/2, respectively. One may ask over what range of x does the central path y(c = 0) span y ± 1/2. From y(w(x)) the answer is x = ± arcsin[(1 − 3b)/(6a)], with the implication that y = ±1/2 and x = ±π/2 when a = (1 − 3b)/6. For a > 1/6 there is no value of b which will move the extrema of motion to (x, y) = ±(π/2, 1/2). Of course, b is restricted to the range [0, 1]. The unique value of (a, b) which simultaneously satisﬁes the channel opening condition a = (1/3)b3/2 and the restricted range condition a = (1−3b)/6 occupies a special place in the parameter set; the values are (a, b) = (1/24, 1/4) and they form a starting point from which to set suitable acceleration parameters. In practise we need a channel of ﬁnite width, which implies a > 1/24; and the desire to keep the range equal to y = ±1/2 implies a lowering of b. For a ≤ 1/6 one can ﬁnd values of b ≥ 0 for which the x-range spans exactly ±π/2, but for larger values of a the x-range (at which y = ±1/2) falls; and more important the width of x accepted by the channel diminishes. In the proposed muon FFAGs, the acceleration rate is limited to a ≤ 1/12 for technical reasons - though this could be raised by reducing the bend per cell. We may ask the question: given a, for what value of b does the central trajectory exactly span (−x, y = −1/2) to (x, y = +1/2)? The answer is b = (1 − 6a sin x)/3. Substituting a = 1/12 and x = π/2 one ﬁnds the value b = 1/6. This is optimum in the sense that the momentum width of the channel (at x = ±π/2) is the largest possible, as is the width of x-accepted. The time dispersion is also minimized, but not the half-period. 3.4 Half period For the period, we may integrate over x or over y. The latter choice has the advantage that the integral is formed uni-directional. If the integral is formed over x, then the backward kink implies the integrand has to be broken into three pieces. However, there is also the issue of whether the integrands possess singularities, places where x = 0 or y = 0 for integrations over x or y, respectively. In both cases, the singularities are roughly of the form √ dq/ q − q which often converges; but this is not guaranteed. In light of these observations, ˜ integrations over x or y both look reasonable. However, to obtain analytic results one should 16 note that the singularities in the integral over x can be removed entirely by transforming to the variable z. Of course, the integral will still diverge for critical values of (a, b, c) for which the motion stalls at the ﬁxed points. √ 3.5 Integration Range y = ± b, x = ± arcsin(ac /a) Because it is mathematically the simplest case, we start by forming the half-period, eﬃciency and time dispersion for paths extending over the centre range of the serpentine channel. The speed v ≡ dx/ds = 4y 2 −b. We integrate 1/v over x and then transform to the new coordinate z given by (c + a sin x) = ac sin z. For hamiltonian value c, the dwell time is x(c) ˆ dx +π/2 dz dx b3/2 +π/2 dz cos z τ= = = . (60) x(c) ˇ v(x, c) −π/2 v(z, c) dz 3a −π/2 v(z) cos x In the centre range, y(z) is triple-valued and the path comprises three segments y1 , y2 , y3 and corresponding speeds v1 , v2 , v3 . We note that (1/v1 ) + (1/v3 ) = (−1/v2 ) from which it follows that the integral over (1/v1 ) + (1/v3 ) + (−1/v2 ) is equal to that over (−2/v2 ). That is the time to pass over segment y2 is equal to the total dwell time on segments y1 and y3 . This property holds for all values of c, because the range of z integration is always ±π/2. Hence √ xˆ dx b3/2 +π/2 dz cos z b +π/2 cos(z/3) τ = −2 = −2 = +2 dz . (61) x v2 [z(x)] ˇ 3a −π/2 cos x v2 (z) 3a −π/2 cos x(z) On the central trajectory c = 0, we have in addition that the integrals over 1/v1 and 1/v2 are equal and each is half that over −1/v2 . The integral is dominated by its singularities at the limits of integration. For that reason, cos(z/3) can be considered slowly varying, removed from the integrand and replaced by its average value cos(z/3) = 3/π. This leads to an excellent approximation for the dwell time: √ 2 b +π/2 dz τ (a, c) ≈ . (62) π −π/2 a2 − (c − ac sin z)2 3.5.1 On the central trajectory When c = 0 the integral is obtained exactly: √ √ 4 b a2 2 b 1 b3 1 b6 τ (a, 0) ≈ K c ≈ 1+ + + ... . (63) aπ a2 a 36 a2 576 a4 Clearly the series converges quickly for b3/2 < a. On the central trajectory, the acceleration √ eﬃciency cos x between the limits y = ± b is √ 2 b π/2 cos x = = , (64) aτ K[. . .] 17 which tends very quickly to unity; see ﬁgure 7.√ Corresponding to the momenta y = ± b is the position range x = ± arcsin(ac /a). Corresponding to the extraction/injection/ momenta y = ±1/2 is the position range x = ± arcsin[(1 − 3b)/(6a)]. These ranges become identical when b = 1/4, irrespective of the value of a. In this special case, the dwell time and eﬃciency are given by (63,64) for all a. T <cosx> 12 0.99 10 0.98 8 0.97 6 a a 0.125 0.15 0.175 0.2 0.225 0.25 0.125 0.15 0.175 0.2 0.225 0.25 Figure 6: The period τ (a, 0) between z = Figure 7: The eﬃciency cos x between ±π/2 versus a for b = 1/4, 1/5, 1/6 (red, z = ±π/2 versus a for b = 1/4, 1/5, 1/6 green, blue, respectively). (red, green, blue, respectively). 3.5.2 Oﬀ the central trajectory To ﬁnd the dispersion in arrival time, we must ﬁnd the dwell time for paths with c = 0. Essentially we desire to ﬁnd the coeﬃcient of c2 in a Taylor expansion about c = 0. √ 2 b c2 τ (a, c) ≈ τ (a, 0) + 2a2 E[. . .] + (a2 − a2 )K[. . .] (65) πa(a2 − a2 )2 c √ 4 c bc + 2 − a2 )4 8a2 (a2 + a2 )E[. . .] + (−5a4 + 2a2 a2 + 3a4 )K[. . .] + . . . c c c 2πa(a c The argument of the elliptic functions is (ac /a)2 = [b3 /(3a)2 ]. A signiﬁcant feature of the expansion is the resonant denominator terms (a2 − a2 ) = (a2 − b3 /9). These occur because, c for given a value, the larger is b so the motion is closer to a ﬁxed point; and so we should expect the dispersion in arrival times to increase with b. An alternative, more accurate expansion for τ (a, c) is given in appendix B. The normalized dispersion is the coeﬃcient of c2 divided by the period τ (a, 0): 1 ∂2τ 1 E C2 = = 2 − a2 )2 (a2 − a2 ) + 2a2 . (66) 2τ ∂c 2 2(a c c K To lowest order, the derivatives are ∂c/∂x = − a2 − a2 and ∂c/∂y = 3b. This implies c C2,x = (a2 − a2 )C2 and C2,y = (3b)2 C2 . For the muon FFAG case, a = 1/12 and a variety of c b, values are: b 1/4 1/5 1/6 C2 126.851 94.4815 83.921 C2,x 0.66068 0.57214 0.53962 C2,y 71.3534 34.0134 20.9802 18 √ When the momentum range is extended beyond ± b, the dwell time and time dispersion increases for b = 1/5, 1/6 - but not for b = 1/4 - and the picture changes somewhat. Hence, the advantages of operating with b close to 1/6 diminish. C2HyL 70 60 50 40 30 20 10 a 0.12 0.14 0.16 Figure 8: Analytic C2,y versus a for b = 1/4, 1/5, 1/6 (red, green, blue, respectively); z = ±π/2. The same function C2,y may also be obtained by numerical means, as indicated in ﬁgures 9,10. This method leads to very similar values, as given in the table below. This agreement gives us conﬁdence that numerical integrations for other ranges are also correct. THcL TH0L C2 1.12 140 1.1 120 100 1.08 80 1.06 60 1.04 40 1.02 20 dy dy -0.04 -0.02 0.02 0.04 -0.04 -0.02 0.02 0.04 Figure 9: shows τ (a, c)/τ (a, 0) versus δy Figure 10: C2,y versus δy for a = 1/12 and for a = 1/12 and b = 1/4, 1/5, 1/6 (red, b = 1/4, 1/5, 1/6 (red, green, blue, respec- green, blue, respectively); z = ±π/2. tively); z = ±π/2. √ 3.5.3 Summary for integration range y = b, x = ± arcsin(ac /a) Values for the muon FFAG with a = 1/12 and integration range z = ±π/2 for a variety of b: b 1/4 1/5 1/6 τ 12.8514 11.0928 9.98187 cos x 0.933751 0.967574 0.981576 ±x range π/6 = 30◦ 0.36588 = 20.9634◦ √ 0.275643 = 15.7932◦ √ ±y range 1/2 1/ 5 = 0.447214 1/ 6 = 0.408248 C2,y 70.3564 33.7677 20.892 19 3.6 Integration Range y = ±1/2, x = ± arcsin[(1 − 3b)/(6a)] The injection/extraction momenta are y = ±1/2. Hence this is the more relevant range over which to ﬁnd the period, eﬃciency and dispersion. The half-period integral is formed piecewise; as above for the centre range, and additional integrations over x extended into the end ranges. Closed forms for these additional integrals are not possible. Approximate forms based on the trapezium rule can be obtained. For c = 0, the sum of the end range integrals is approximately: (1 + 2b) √ arcsin[(ac /a) cosh[3arccosh(1/2 b)]] − arcsin(ac /a) (67) 3b(1 − b) However, we shall prefer to rely on more accurate numerical integrations, see ﬁgure 12, as performed by Mathematica. dT X 2 1.5 1.25 1.5 1 1 0.75 0.5 0.5 0.25 a 0.125 0.15 0.175 0.2 0.225 0.25 a 0.125 0.15 0.175 0.2 0.225 0.25 Figure 12: The incremental contribution Figure 11: The x-range between y = ±1/2 to the period between z = π/2 and y = 1/2 versus a for b = 1/4, 1/5, 1/6 (red, green, versus a for b = 1/4, 1/5, 1/6 (red, green, blue, respectively); c=0. blue, respectively); c=0. T <cosx> 14 0.975 0.95 12 0.925 10 a 0.12 0.14 0.16 0.875 8 0.85 a 0.825 0.12 0.14 0.16 Figure 13: The total period between y = Figure 14: The eﬃciency cos x between ±1/2 versus a for b = 1/4, 1/5, 1/6 (red, y = ±1/2 versus a for b = 1/4, 1/5, 1/6 green, blue, respectively); c=0. (red, green, blue, respectively); c=0. 20 THcL TH0L C2 80 1.1 70 60 1.08 50 1.06 40 1.04 30 20 1.02 10 dy dy -0.04 -0.02 0.02 0.04 -0.04 -0.02 0.02 0.04 Figure 15: shows τ (a, c)/τ (a, 0) versus δy Figure 16: C2,y versus δy for a = 1/12 and for a = 1/12 and b = 1/4, 1/5, 1/6 (red, b = 1/4, 1/5, 1/6 (red, green, blue, respec- green, blue, respectively); x-range = ±ˆ.x tively); x-range = ± arcsin[(1 − 3b)/(6a)]. The total half period τ (a, 0) and eﬃciency 1/(aτ ) on the central trajectory are given in ﬁgures 13,14. Note that the integration limits are x = ±ˆ independent of c, where x x = arcsin[(1 − 3b)/(6a)]. The relative variation of periods away from the central trajectory, ˆ τ (a, c)/τ (a, 0), is shown in ﬁgure 15. From this may be obtained C2,y , ﬁgure 16. The relationship between c and δy is c = (1 − b)δy to lowest order. 3.6.1 Summary for integration range y = ±1/2 For a = 1/12 and a variety of b, selected values are: b 1/4 1/5 1/6 period 12.8514 12.8862 14.691 cos x 0.933751 0.931227 0.816827 C2,x 0.4841 0.2042 0 C2,y 52.284 39.564 34.380 ±x range 30◦ 53.1301◦ 90◦ 3.7 Integration Range y = ±ˆ, x = ±π/2 y For c = 0 the sum of the end range integrals is approximately: 2 arccos[ac /a](2 + w) , w = cosh[(1/3)arccosh(a/ac )] . (68) 3b(1 + 2w) However, it is better to resort to numerical methods to obtain the half-period between x = ±π/2 and the acceleration√ y eﬃciency cos x = 2ˆ/(aτ ). These functions are given in ˆ ﬁgures 17,18. The range is y = b cosh[(1/3)arccosh(a/ac )]. The relative variation of dwell time for paths oﬀ the central trajectory τ (a, c)/τ (a, 0) and the normalized dispersion C2,y , the coeﬃcient of (δy)2 divided by τ , are shown in ﬁg- ures 20,21. The relation between c and y, to lowest order, is c = b(1+2 cosh[(2/3)arccosh(a/ac )])δy with ac = b3/2 /(3a). Of course, C2,x = 0 at x = ±π/2. 21 T <cosx> 0.85 15 14 0.8 13 0.75 12 11 a 0.125 0.15 0.175 0.2 0.225 0.25 a 0.12 0.14 0.16 0.65 Figure 17: The period between x = ±π/2 Figure 18: The eﬃciency cos x between versus a for b = 1/4, 1/5, 1/6 (red, green, x = ±π/2 versus a for b = 1/4, 1/5, 1/6 blue, respectively); c=0. (red, green, blue, respectively); c=0. Y 0.675 0.65 0.625 0.6 0.575 0.55 0.525 a 0.125 0.15 0.175 0.2 0.225 0.25 Figure 19: The y-range between x = ±π/2 versus a for b = 1/4, 1/5, 1/6 (red, green, blue, respectively); c=0. THcL TH0L C2 160 140 1.1 120 1.08 100 1.06 80 60 1.04 40 1.02 20 dy dy -0.04 -0.02 0.02 0.04 -0.04-0.03-0.02-0.01 0.01 0.02 0.03 0.04 Figure 20: shows τ (a, c)/τ (a, 0) versus δy Figure 21: C2,y versus δy for a = 1/12 and for a = 1/12 and b = 1/4, 1/5, 1/6 (red, b = 1/4, 1/5, 1/6 (red, green, blue, respec- green, blue, respectively); x = ±π/2. tively); x = ±π/2. 22 3.7.1 Summary for integration range x = ±π/2 For a = 1/12 and a variety of b, selected values are: b 1/4 1/5 1/6 period 15.5278 14.8391 14.691 cos x 0.848473 0.840757 0.816827 C2,y 84.8448 47.9727 34.3798 ±y range 0.548956 0.519837 0.5 4 Reprise We have found the ranges/limits of motion, and the widths of the rotation manifold for a variety of operating points (a, b) including the special case b = 0 where the two hyperbolic ﬁxed points become coincident. We have fould also the dwell time (τ ), the acceleration eﬃciency cos x , and the relative dispersion (∝ ∂ 2 τ /∂h2 ) over those varied ranges and over important sub-ranges such as y = ±1/2. The following short discussion explains the general trends in intuitive terms. y v 0.4 0.8 0.2 0.6 x -1.5 -1 -0.5 0.5 1 1.5 0.4 -0.2 0.2 -0.4 x -1.5 -1 -0.5 0.5 1 1.5 -0.2 Figure 22: Central trajectory for a = 1/12 and b = 1/4, 1/5, 1/6 (red, green, blue, Figure 23: Velocity v = 4y 2 − b on the respectively). Also lines of constant y = central trajectories for b = 1/4, 1/5, 1/6 ±1/2 (magenta and cyan). (red, green, blue, respectively). The central trajectory momentum y and speed v for each of the selected possible working points is shown in ﬁgures 22,23. These ﬁgures give some indication of why the acceleration eﬃciency is generally high; and also explains the variations depending on the chosen integration range - which has been the topic of most of this manuscript. The crest of the sinusoid is located at x = 0, and its zeros at x = ±π/2. The speed x = 4y 2 − b is largest at x = ±π/2 and so particles spend relatively little time where the acceleration is weak. However, where the speed is lower particles will cross the crest three times. √ Over the range y = ± b, the case b = 1/6 has the lowest velocity in the vicinity of crest, and so the maximum acceleration eﬃciency over this particular range. However, over the range x = ±π/2, the case b = 1/4 has the largest average speed and so the half period is least and the eﬃciency is highest. But it is clear also from ﬁgures 4,23 that the paths pass 23 √ closest to the ﬁxed points at ± b/2 when b = 1/4 and so the spread in dwell times oﬀ the central trajectory is largest. As has been mentioned the normalized dispersion C2 gives a ranking that predicts which cases will better preserve the macroscopic emittance; the smallest values of C2 leading to the least nonlinear distortion of the phase space ellipse. However, it cannot be used easily to determine what values of (a, b) will yield acceptable levels of emittance growth. In the absence of more sophisticated methods, we shall make this dermination based on tracking of test particles in a numerical schema that iterates the equations (1-2). 4.1 Particle Tracking Tracking of an ensemble of particles during acceleration with the various parameter values (a, b) discussed in the document has been completed, and is summarized in ﬁgures 24-37. For each case, two types of graph are presented: (i) a turn-by-turn summary of the acceleration; and (ii) a detailed comparison of the input and output phase spaces. The units adopted for the scales of the graphs are dimensionless and follow the deﬁnitions of x, y in equation (5). In all cases the phase space was loaded with an ellipse3 of area equal to 0.5 eV.s; and the optimum aspect ratio and orientation angle were respected. Notice that this area is a factor of π larger than the NFMC Study-2a[1] value4 of 0.16 eV.s. For almost every case, we took a 10–20 GeV FFAG with 90 cells and ω∆T = 0.02 and operating over roughly y = ±1/2. The survival rates are based purely on the machine acceptance and do not include the eﬀect of decay losses. In general the graphs conﬁrm the anticipated ranking of performance based on C2 ; and in particular they suggest that the range of operating points a = 1/12, b = [1/6, 1/5] will produce acceptable ﬁnal emittances. 5 Conclusion As was mentioned at the outset, the choice of working point (a, b) represents a compromise between acceleration range, dwell time, acceleration eﬃciency and the dispersion of dwell times about the central trajectory. These quantities have been found analytically for general (a, b), and the inﬂuence of the latter (dispersion) has been studied numerically via the computer tracking of particle ensembles. Given that a = 1/12 for the multi-GeV muon FFAG, the choice b = 0 is ruled out since the range is inadequate until a ≥ 1/6. A range of b values between 1/6 and 1/4 was considered. The latter is the “natural choice” and gives the greatest range and eﬃciency and smallest dwell time. However, the distance to the ﬁxed points (a2 − b3 /3) is also smallest leading to a strong dispersion of the period versus hamiltonian value. The former choice, b = 1/6 leads to the maximum acceptance and minimum dispersion; but the eﬃciency is lower and the decay losses will be increased by the longer dwell time. Between those limits, we have studied the case b = 1/5; this particular value being chosen arbitratily. This 3 Truly we mean the area, the product of the semi-axes would be 0.1592 eV.s. 4 This phase space area is usually given as π × 0.05 eV.s. 24 case exhibits performance in terms of range, period, eﬃciency and dispersion which appears to oﬀer a good compromise between the two previous extremes. Without a precise merit criterion to rank the cases, the best choice for (a, b) is ultimately subjective; but it is believed that it will be close to (a, b) = (1/12, 1/5). References [1] The Neutrino Factory and Muon Collider Collaboration: Neutrino Factory and Beta Beams Development Study 2a, BNL-72369-2004, FNAL-TM-2259. [2] S. Koscielniak: Phase-space trajectories and periods of motion for the quadratic pendu- lum; TRIUMF, Vancouver B.C. Canada, Design Note TRI-DN-03-13. [3] S. Koscielniak & C. Johnstone: Mechanisms for nonlinear acceleration in FFAGs withy ﬁxed RF; Nuc. Inst. Meths. in Phys. Res. A 523 (2004) 25-49. [4] E. Keil: New Muon Lattice - longitudinal emphasis, FFAG Workshop, TRIUMF, Van- couver Canada, 15-21 April 2004. [5] Keil, Berg, Sessler: Electron Model of an FFAG Muon Accelerator, Proc. European Par- ticle Accelerator Conf, Lucerne Switzerland, July 2004, p.587 25 Figure 24: Input (left) and output (right) phase space. (a, b) = (1/12, 1/6), tracking range x = ±π/2, 91.1% survival. Figure 25: Input (left) and output (right) phase space. (a, b) = (1/12, 1/5), tracking range x = ±π/2, 90.2% survival. Figure 26: Input (left) and output (right) phase space. (a, b) = (1/12, 1/4), tracking range x = ±π/2, 84.1% survival. 26 Figure 27: Input (left) and output (right) phase space. (a, b) = (1/12, 1/5), tracking range y = ±1/2, 87.4% survival. Figure 28: Input (left) and output (right) phase space. (a, b) = (1/12, 1/4), tracking range y = ±1/2, 76.0% survival. Figure 29: Input (left) and output (right) phase space. (a, b) = (1/6, 0), tracking range x = ±π/2, 85.7% survival. 27 Figure 30: Input (left) and output (right) phase space. (a, b) = (1/12, 0), tracking range x = ±π/2, 9.2% survival. Figure 31: (a, b) = (1/12, 0), tracking range x = ±π/2, 579 cells, τ = 11.580, y = 0.2683 28 Figure 32: (a, b) = (1/6, 0), tracking range x = ±π/2, 366 cells, τ = 7.320 Figure 33: (a, b) = (1/12, 1/6), tracking range x = ±π/2, 683 cells, τ = 13.660 29 Figure 34: (a, b) = (1/12, 1/5), tracking range x = ±π/2, 711 cells, τ = 14.220 Figure 35: (a, b) = (1/12, 1/4), tracking range x = ±π/2, 763 cells, τ = 15.260 30 Figure 36: (a, b) = (1/12, 1/5), tracking range y = ±1/2, 648 cells, τ = 12.960 Figure 37: (a, b) = (1/12, 1/4), tracking range y = ±1/2, 645 cells, τ = 12.90 31 A Hypergeometric function F21 In the region a ≥ 1/5 the function is well approximated by 1 1 7 1 1 1 1 1 F21 , , , ≈1 + × 2 + × 4 + ... (69) 6 2 6 (6a)2 504 a 44928 a In the region 1/6 ≤ a < 1/5 the function is well approximated by √ π Γ(7/6) 1 2 1 5 1 F21 ≈ 1 +2 a− − √ a− 1 + a− + ... (70) Γ(2/3) 6 3 6 6 6 A.1 Reciprocal Asymptotic expansion of the reciprocal of F21 : this function varies quickly for a < 1/5 and for larger values it quickly approaches unity. An approximation good for a < 1/4 is: 1 6ν 3 Γ(2/3) ≈ √ 1 + 4 ν a − 1/6 + [(48/π)ν 2 − 2][a − 1/6] + . . . ν= . F21 [. . .] π π Γ(1/6) (71) An approximation good for a ≥ 1/4 is: 1 1 1 121 1 ≈1 − × 2− × 4 + ... (72) F21 [. . .] 504 a 6604416 a √ B Approximate dwell time, b = 0 z = ±π/2, y = ± b An excellent approximation for the dwell time in the stated range is √ 2 b +π/2 dz τ (a, c) ≈ . (73) π −π/2 a2 − (c − ac sin z)2 We introduce α2 ≡ a2 − c2 and write the radicand as a2 − (c − b3/2 /3 sin z)2 = α2 − a2 (sin z)2 + 2 c ac sin z , c (74) and expand the integrand in powers of c. Only even powers remain after the integration: √ 2 b c2 (α2 + a2 ) τ (a, c) ≈ τ (α, 0) + c E[. . .] − K[. . .] (75) απ(α2 − a2 ) (α2 − a2 ) √ 4 c c bc (a4 + 9a2 α2 − 2α4 ) (a6 − 5a4 α2 − 5a2 α4 + α6 ) − c c K[. . .] + c c c E[. . .] + . . . πα3 (α2 − a2 )3 c 2 (α2 − a2 ) c The argument of the elliptic functions is a2 /α2 = [b3 /(3α)2 ]. A signiﬁcant feature of this c expansion is the resonant denominator terms (α2 − a2 ) = [(a2 − c2 ) − b3 /9]. Motion stalls on c the ﬁxed points when c = ± a2 − b3 /9. Hence, the larger is b the closer is the motion to a ﬁxed point; and so we should expect the dispersion in arrival times to increase with b. 32

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