1. Magnetic Multipoles Jeffrey Eldred1 1
Magnetic Multipoles
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. Reminder on Del: Gradient: Divergence: Curl: Laplacian: 2 2 2 2 2
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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3. 3 3
Multipole Expansion 3
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4. Magnetic Fields in Source-free Regions
Insider the accelerator beampipe, there are no current sources (neglecting the beam itself).
Consequently, the magnetic field can be written as a scalar potential.
Importantly, the magnetic scalar potential is not the same as the electric potential:
Relatedly we can also think of As in the Coulomb gauge: 4
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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5. Magnetic Multipoles For transverse magnetic fields only we have:
Writing the scalar potential as a polynomial expansion we have:
And finding the Laplacian of this potential yields a recurrence relation:
Which means all the polynomial coefficients can be written as 5
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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6. List of Multipoles T. Zolkin Horizontal Dipoles: Regular Quadrupoles:
Skew Quadrupoles:
T. Zolkin 6
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7. List of Multipoles T. Zolkin Regular Sextupoles: Skew Sextupoles:
Regular Octupoles:
T. Zolkin 7
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8. Sector Multipoles T. Zolkin
In a bending orbit, the multipole decomposition is different.
These multipoles have only recently been rediscovered.
T. Zolkin 8
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9. Measuring Multipoles
Full 3D magnetic measurements are not strictly necessary to determine the magnetic multipole content.
Because of the multipole decomposition, it suffices to take measurements along the horizontal plane of the magnet.
Or, it suffices to take measurements along the boundaries in an enclosed volume within the magnet, for example a cylinder. 9
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10. Reference Point for Multipoles
Multipole describe how the magnetic field behaves as you move away from a specific reference point of the magnet.
Multipoles are a polynomial expansion around that reference point.
To say, for example “The dipole field drops off as you move away from the center” isn’t quite accurate. We should say the vertical component of the field drops off. The dipole field is the dipole field, the vertical field drops off from a sextupole or higher field.
All this to say that the “uniformity of the magnet” – the region in which the multipole expansion is roughly equivalent – is directly connected to the “purity of the multipole field”.
From symmetry odd multipoles tend to have odd-multipole errors, even multipoles tend to have even-multipole errors. 10
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11. 11 11
Fringe Field Effects 11
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12. Magnetic Length X. Huang
Often particle accelerators use a “hard-edge” model for accelerators in which the transverse magnetic fields drop instantly.
But that would actually violate of Maxwell’s equations!
It can be useful to define the “magnetic length” of the element:
X. Huang 12
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13. Fringe-Field Multipoles
If the field drops off with some function of Gn,0(s), then it will generate higher order transverse magnetic multipoles, as well as longitudinal magnetic fields:
As p increases, higher and higher multipoles are generated based on higher and higher derivatives of the original longitudinal profile.
At some point, the expansion has to be terminated.
X. Huang 13
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14. Fringe-Field Multipoles (cont.)
X. Huang 14
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15. Enge Function as a Fringe Field Model
X. Huang 15
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