Studies of Magnetic Dipole Edge Shaping
J.A. Maloney
TRIUMF
1. Abstract
This note explores a study related to the use of shaping the entrance and exit edges of a
magnetic dipole to correct nonlinear effects in the beam line.
2. Introduction
a. Motivation
Design of the CANREB HRS requires correction of nonlinear effects in the beam line to
maintain its target resolution. Several methods to correct these aberrations are known.
Current design of the HRS utilizes a basic curvature of the entrance and exit edges of the
dipole magnets to correct most of the 2nd order aberrations effecting horizontal beam
width. Also, an electrostatic multipole is used to correct the residual 2nd order aberrations
to horizontal beam width as well as other aberrations through 5th order. This will require
these multipole fields to be tuned during operation.
An alternative methodology is to incorporate the corrections for 3rd through 5th order into
the shaping of the entrance and exit edges of the dipole magnets. This would correct the
basic nonlinearities of the beam line design, but still allow the electrostatic corrector to be
used as a “fine tune” to resolve any unintended nonlinearities in the beam line that may
arise from misalignments, manufacturing tolerances, etc.
b. Simulations of the Magnetic Dipole in COSY Infinity
In the current version of beam physics package for COSY Infinity (COSY.fox ver. 9.1 –
January 2013), the modelling magnetic bending elements can be effected in several
manners. Initial modeling of the CANREB HRS beam line utilized COSY’s internal
coding for a homogenous magnetic dipole with angled and curvature on the edges. This
procedure (known as “DI”) specifies a homogeneous magnetic dipole based on the
following parameters:
DI
R Theta D E1 H1 E2 H2
Where R is the dipole radius of curvature (in 1/meters), Theta is the dipole bending angle
(in degrees), E1 and E2 are the angles made by the magnets’ entrance and exit edges
from normal (perpendicular to the beam’s reference trajectory), and H1 and H2 represent
the radius of curvature for the dipole entrance and exit edges as measured from a point
along the beam’s reference trajectory.
Using this procedure, the entrance and exit angles were adjusted to correct for the vertical
focusing effects caused by the dipole fringe fields. The curvatures were used to correct
the 2nd order aberration in the beam line as much as possible, and a correcting
electrostatic multipole has been design to correct the residual 2nd order aberrations and
aberrations from 3rd through 5th order affecting the horizontal beam width.
For a more detailed description of the dipole, the general magnetic bending element
procedure (known as “MC”) can be used instead. This element specifies the dipole based
on the following parameters:
MC
R Theta D N S1 S2 ni
Where R, Theta and D are the same parameters specified in the procedure “DI”. N is an
array that can be used to express the radial dependency of the magnetic field in the case
of an inhomogeneous dipole. In the case of a homogeneous dipole, all coefficients in this
array are zero. S1 and S2 are arrays that specify the curvature of the entrance and exit
edges as a function of the horizontal offset from the reference trajectory. The form of
this function is:
S(x) = S(1)*x + S(2)*x2 + S(3)*x3 + … + S(n)*xn
Lastly, ni is a parameter reflecting the highest order of the terms in the arrays N, S1 and
S2.
The correlation between the two procedures is not a simple one, but is it geometric in
nature. In terms of the “DI” parameters E1, H1, E2 and H2, the first 3 terms of the “MC”
polynomials S1 and S2 are given by:
𝑆1(1) = tan (𝐸1 ∗
2𝜋
)
360
𝑆2(1) = tan (𝐸2 ∗
2𝜋
)
360
1
2𝜋 3
∗ 𝐻1 ∗ (1/cos⁡(𝐸1 ∗
) )
2
360
1
2𝜋 3
𝑆2(2) = ∗ 𝐻2 ∗ (1/cos⁡(𝐸2 ∗
) )
2
360
1
2𝜋
2𝜋 5
𝑆1(3) = ∗ 𝐻12 ∗ (sin(𝐸1 ∗
)/cos⁡(𝐸1 ∗
) )
2
360
360
1
2𝜋
2𝜋 5
𝑆2(3) = ∗ 𝐻22 ∗ (sin(𝐸2 ∗
)/cos⁡(𝐸2 ∗
) )
2
360
360
𝑆1(2) =
There are additional terms in the geometric expansion beyond the 3rd order terms, but
they represent and extension of the same geometric calculations. These correlations have
been verified using COSY in both the hard edge case and with fringe fields.
3. Correction of Aberrations using Magnetic Dipole Edge
Shaping
a. Study of Corrections of Second Order Aberration.
In the CANREB HRS design, the 3 largest 2nd order aberrations affecting the final
horizontal beam width have been identified. These are referred to, using COSY’s
convention, as (x|aa), (x|yy) and (x|bb). For example, (x|aa) represents the coefficient in
the Taylor map showing beam evolution in the final horizontal beam half width (x) based
on the square of the initial half spread in horizontal beam angle (px/p0). Correcting these
aberrations to the extent possible is essential to maintaining high resolution in the
separator.
In the current HRS design, a curvature was introduced and fit for a homogenous magnetic
dipole using the “DI” procedure that minimized these aberration coefficients to the
greatest extent possible. As additional constraints, symmetry was maintained for the
dipole entrance and exit edges, and both magnetic dipoles in the HRS are identical.
Given the number of terms we are seeking to minimize, it was not possible to achieve this
through the curvature alone. For this reason, the sextupole component of the HRS
electrostatic multipole corrector was used to eliminate the residual aberrations.
Using the 2nd order term for the edge shaping in the “MC” dipole, and imposing the same
constraints, an identical result was obtained from trying to minimize the identified 2nd
order aberration terms. Accordingly, for the remaining simulation studies, the original
edge curvature values corresponding to the “DI” model were used along with the
sextupole component of the HRS electrostatic multipole corrector.
b. Study of Corrections for 3rd through 5th Order Aberrations
Using the base HRS layout, with fringe fields, fits were done using the edge shaping
coefficients to attempts to correct the identified aberrations terms at 3rd, 4th and 5ht order
effecting final horizontal beam width. These terms are (x|aaa), (x|aaaa) and (x|aaaaa).
All these aberrations related to the initial horizontal angular spread in the beam.
Using the “MC” procedure, fits were found that minimized all the identified terms
without using the octupole or higher order multipole fields from the HRS electrostatic
multipole corrector. For these fits, symmetry between the entrance and exit edge shapes
was also maintained. The coefficients for the shaping function are given in Table 1 along
with the corresponding poletip voltage that corresponded to the same correction.
Table 1: MC edge shaping fit results from 3rd through 5th order
Multipole Order
Shaping Polynomial Value used
in “MC”
3rd (octupole)
4th (decapole)
th
5 (duodecapole)
0.04576578
-0.06252355
0.14451963
Corresponding Electrostatic
Corrector Poletip voltage (in
kV)
.00214308
.00017982
-.00001787
c. Conclusions
The ability to shape dipole edges looks promising from this basic simulation. More
particularly, it will be important to examine (1) what effects this shaping has on the
effective field length and flatness of the dipoles, and (2) whether this shaping can be
accomplished through shaping of the magnetic field clamps versed shaping the entire
magnet surface.
4. References