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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.

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