# 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?)
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
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
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

• 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
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