Low Energy Beam Transport Design Optimization for RIBs

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					Low Energy Beam Transport Design
      Optimization for RIBs


           R. Baartman
            TRIUMF


            May, 2002




                              1
Optics Scaling
One of the characteristic features of heavy ion facilities such as
ISAC is that a wide variety of particles can be provided for the
experimenters: particle mass, charge and even energy can be
customized to the user’s request. How is the optics adjusted
when one of these parameters is changed? We can answer this
by looking at the scaling of the non-relativistic Hamiltonian, which
in Cartesian coordinates is
                   1                 2
               H=    P − qA              + qV (x, y, z).             (1)
                  2m
Here, P is not the kinetic momentum p, but the canonical mo-
mentum p + q A as appropriate for cases with magnetic fields
given by the vector potential A.
All dynamics can be derived from Hamilton’s variational principle:

                      δ    (Pidqi − Hdt) = 0                         (2)

((P1 , P2 , P3 ) = (Px , Py , Pz ), (q1 , q2 , q3 ) = (x, y, z), and re-
peated indices follow summation convention.) We can rescale
any variable as long as the variational principle remains true.
We want all particles to always follow the same trajectories, so
x, y, z are inviolate. To keep track of what we’re doing, let’s do
                                                           √
it in 2 steps. First scale the momenta by a factor mqV0 . I.e.
define
                                     P
                           P∗ = √                                    (3)
                                     mqV0

                                                                 2
Optics Scaling–cont’d
According to the variational principle, we must scale either t or
H the same amount. Let’s scale t. Then we get
                      qV0 ∗        2
                                 ∗
                H=        P − A + qV,                             (4)
                       2
where we’ve defined a new vector potential
                                   q
                         A∗ =         A.                          (5)
                                  mV0

Now rescale H by qV0 without changing the momenta. This can
be done by rescaling t by the reciprocal. We then find
                        1 ∗         2
                     ∗
                  H =             ∗
                           P − A + V ∗,                           (6)
                        2
where we’ve introduced the scaled potential
                          V ∗ = V /V0 .                           (7)
Taken together, the 2 steps have scaled time t by the total
amount
                                  qV0
                          t∗ =        t.                          (8)
                                   m



                                                              3
Optics Scaling–cont’d
To recap: we’ve transformed the Hamiltonian to one which
contains neither charge nor mass, without rescaling spatial
coordinates. This proves that trajectories are preserved through
the transport system as long as the scaling relations are
obeyed. In particular:

  • Electric fields are unchanged when changing either mass
    or charge: in other words, in an entirely electrostatic
    beamline from source to experiment, all particles have the
    same trajectory, irrespective of mass or charge.

  • By eqn. 7, when changing energy, all potentials scale with
    the energy change.

  • Magnetic fields scale as eqn. 5, in particular, the mass
    separator dipole can go from one mass to the next by
    scaling the magnetic field as the square root of the mass
    over charge.

  • Time scales as eqn. 8. If the time scale must remain fixed,
    for example as in a fixed-frequency linac, the potentials, in
    particular the voltage on the linac, scale ∝ m/q.




                                                          4
RIB LEBTs

At beam voltages (energy ÷ charge state) <∼ 1 MV,
electrostatic optics is more economical than
magnetic.

As well, Radioactive Low Energy Ion Beams are of a
wide variety of species (masses). Electrostatic optics
has the nice scaling feature that element strengths
depend only on beam voltage and not on mass (or
charge).

For these reasons, we concentrate on electrostatic
optics.




                                                 5
What’s better, a whole lot of little bitty quads,
or a few large ones?


Size Scale – Economics
One would like to transport a beam of given emittance. Say the
aperture radius is a, and the distance between focusing
elements is Lc. Then maximum angles are a/Lc and
acceptance is on the order of ˆ ∼ a2 /Lc. (More precisely,
ˆ = a2 /βT , where βT is the Twiss β-function, but βT ∼ Lc).

For example, say we want ˆ = 200 mm-mrad. Then the
following combinations will work:

          Length Scale    aperture    Looks like ...
              5m           50 mm     CERN-ISOLDE
              1m           22 mm     TRIUMF-ISAC
             2 cm          3 mm         an RFQ

Also, cost is roughly ∝ a2 (cross-sectional area of vacuum
chamber), and ∝ 1/Lc (i.e. the number of optical elements
assuming a fixed distance from source to destination).

Therefore, cost ∝ a2 /Lc ∼ ˆ, so whether Lc is 5 or 1 metre, the
cost is roughly the same. In other words, we have plenty of
freedom to choose a scale for the optics.

                                                          6
Large scale (like ISOLDE)...

             4
                                            x-envelope (cm)
             3                              y-envelope (cm)

             2
             1
             0
            -1
            -2
            -3
            -4
                 0   50   100   150     200     250   300     350   400
                                   distance (in)



Small scale (like ISAC)...

             4
                                            x-envelope (cm)
             3                              y-envelope (cm)

             2
             1
             0
            -1
            -2
            -3
            -4
                 0   50   100   150     200     250   300     350   400
                                   distance (in)


                                                                          7
Size Scale – Stray Fields
Are there other considerations? Yes! The equation of motion for
regularly-focused particles is
                              2
                       x + x/βT = 1/ρ                          (9)
where ρ = (Bρ)/Bstray is the radius of curvature of the beam n
the stray field. So on average, the displacement ∆x due to the
               2
stray field is βT /ρ ∼ L2 /ρ. This effect is nasty because it is
                        c
mass-dependent through Bρ. To avoid light masses being
steered off-axis compared with heavy masses at the same
voltage, we would like ∆x to be small compared with the beam
     √         √
size    βT ∼      LC. In other words,

                         LC     (ρ2 )1/3                      (10)
This clearly favours small optics.

Example:

In ISAC, the smallest Bρ is 400 gauss-metres. This is for the
lightest mass (A = 6) at the energy 2 keV/u required for the
RFQ. In the earth’s magnetic field, we find ρ=1 km. For
acceptance =100 mm-mrad, we would like Lc        5 m. In fact,
we have Lc = 1 m. This makes us fairly tolerant, but
CERN-ISOLDE at Lc ∼ 3 m is much more susceptible.



                                                          8
Size Scale – Space Charge
RIBs are generally of such low intensity that space charge is not
an issue. However, in setting up stable beams for
commissioning purposes, it is useful to know how much current
is allowed before space charge becomes a non-negligible effect.
To find this, we can compare the space charge and emittance
terms in the Kapchinsky-Vladimirsky envelope equation.
                 a + k2 (z)a +    2
                                    /a3 + 2A/(a + b)   =   0             (11)
                  b + k2 (z)b +    2 3
                                    /b + 2A/(a + b)    =   0,            (12)
where a and b are x- and y-envelopes, A is the space charge parameter.
                   space charge potential well depth   I × 30 Ω
           A=                                        =                   (13)
                            beam voltage                 βVB

 This gives the following condition for the negligibility of space
charge.
             2
     A           /a2 ,   2
                             /b2 ∼ (beam divergence)2 ∼ /LC            (14)
 This means that in spite of smaller beam sizes and stronger
space charge forces, stronger focusing allows higher current.

Example:

In ISAC, LC ∼ 1 m, and a good beam has emittance
10 mm-mrad. This leads to A     10−5 . As VB ∼ 30 kV, and
β = 0.002, we find that to avoid space charge effects, we need
I    20 µA.


                                                                   9
One often wonders: Are large quad apertures better because
they make the nonlinear fields weaker, or worse because they
make the nonlinear fields longer? In fact the two effects cancel.



Size Scale – Aperture (What about nonlinearity?)
The quadrupole potential field
                                  k 2
                         V (x, y) = (x − y 2 )                 (15)
                                  2
is a solution to Laplace’s equation, but only if the quadrupole is
infinitely long (k=constant). For finite quads, we use the
expansion
                   k 2            k                k
   V (x, y, z) =     (x − y 2 ) −    (x4 − y 4 ) +     (x6 − y 6 ) − ...        (16)
                   2              24               720
(Note that this assumes no quad ‘shape’ errors.)
The quartic term gives a cubic force term which leads to the
following focusing error,
                           −1      7 3 1 2
                    ∆x =            x − xy ,                   (17)
                         f 2 LQ 6       2
 where LQ is the quad length and f the focal length. It is
important to note that this is independent of aperture size or
fringe field hardness: indeed, the aberration is not affected by
changing the fringe field shape. In a given context of fixed optics
and quad locations, the only way to reduce quad aberration
is to lenghten the quads.
BTW, similar formulas can be found for other types of optical
elements.
                                                                           10
Nonlinearity – cont’d




It is clear that for the higher order errors to have a
negligible effect on the beam quality, the distortion in
the emittance ellipse, ∆x , should be small
compared with the local divergence /x.


                                                   11
Nonlinearity – cont’d
Roughly speaking,
                     ∆     ∆x     x4
                         =     ∼ 2                          (18)
                            /x   f LQ
In a fairly smooth-focusing transport channel, x2 = βT ∼ LC
and f ∼ LC, so somewhat surprisingly the optics length scale
cancels and
                          ∆
                              =                             (19)
                                  LQ
Even if the emittance is as large as 100 mm-mrad, this gives the
easily-satisfied constraint LQ     0.1 mm.

However, this is for a regular FODO transport case. Matching
sections often have much tighter constraints. An example is
when matching to an RFQ. Since the RFQ requires a very small
beam, the matching system must transform a relatively large
beam with small divergence to a small beam with large
divergence. The magnification in the case of the ISAC RFQ is
xf /xi = 1/5. The focal length of the last doublet is 1/5 of the
typical FODO focal length of 1 m. Apply eqn. 18, with same x,
but 5 times smaller f . This makes the aberration 25 times
worse, now requiring LQ       2.5 mm.



                                                          12
Catch-22 situation: Making the quad longer helps, but increases
the beam size, which hinders.



Nonlinearity – cont’d
The situation is actually worse than this because of using
quadrupoles rather than stigmatic lenses. The beam in the
next-to-last quadrupole is much larger in the defocusing plane
of the last quadrupole than in the focusing plane. This forces
the last two quads close together, and requires a very short last
lens.
        2
                           50 pi mm-mrad
                          x-envelope (cm)
      1.5                 y-envelope (cm)
                      x-focal power (arb.)
                      y-focal power (arb.)
        1

      0.5

        0

      -0.5

       -1

      -1.5
             0   10         20       30           40       50   60   70        80
                                             distance (in)


                                                                          13
Optimization technique
There are many optimization techniques for designing and
tuning beam transport lines. Some are built into the transport
codes themselves. Almost all of these work on the basis of
reducing an error to zero by finding local derivatives of the error
with respect to the parameters. This finds only local minima and
is not useful for more than about 3 variable parameters. A
technique I use is simulated annealing. The book Numerical
Recipes by Press, Flannery, et al. gives the following
description.
     Offered a succession of options, a simulated thermodynamic
     system was assumed to change its configuration from energy
     E1 to energy E2 with probability p = exp[(E2 − E1 )/(kT )].
     Notice that if E2 < E1 , this probability is greater than unity; in
     such cases the change is arbitrarily assigned a probability
     p = 1, i.e., the system always took such an option. This general
     scheme, of always taking a downhill step while sometimes
     taking an uphill step, has come to be known as the Metropolis
     algorithm. To make use of the Metropolis algorithm for other
     than thermodynamic systems, one must provide the following
     elements:
      1. A description of possible system configurations.
      2. A generator of random changes in the configuration; these
         changes are the options presented to the system.
      3. An objective function E (analog of energy) whose
         minimization is the goal of the procedure.
      4. A control parameter T (analog of temperature) and an
         annealing schedule which tells how it is lowered from high
         to low values, e.g., after how many random changes in
         configuration is each downward step in T taken, and how
         large is that step.

                                                                       14
Simulated Annealing applied to Beam Optics Design
Applied to the design of a transport channel, we can identify
these 4 points as

  1. A set of parameters such as quad strengths, locations, and their
     allowed ranges.

  2. We use the vertices of a “simplex” in the space of all possible
     parameters, and change these randomly, independently according to
     T . For T = 0, the technique reduces to the standard “simplex”
     method of minimization.

  3. E = the sum of all possible effects to be minimized, such as mismatch,
     emittance growth, etc. with their appropriate weights.

  4. Use T such that T = 1 implies that parameters are varied through
     their entire ranges, specify N the number of random changes at each
     temperature, and α(< 1), the factor with which T is multiplied at each
     step.

For as few as 3 parameters, α can be 0.8, with N = 5, resulting
in only about 100 evaluations of the beamline to achieve 10−4
accuracy. But this number of evaluations increases
exponentially with number of parameters. For 9 parameters,
need α = 0.98, N = 50, and this results in roughly 104 − 105
evaluations before convergence is obtained.



                                                                    15
Conclusions

 • Optics of any scale can be made to work:
   elements can be many metres apart, down to cm
   apart.


 • Smaller scale means more focusing per unit
   length. This makes the beam less sensitive to
   perturbations (stray fields, misalignments, space
   charge, ...)


 • Smaller scale also means more customizablility
   for adding new experiments.


 • Beam transport design can be accomplished
   with simple tools like first order matrix codes,
   augmented by “back of the envelope” formulas
   for higher orders. (This is not true however of
   aberration-limited designs like spectrometers.)

                                               16
17

				
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posted:8/19/2011
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