# Physics:Multipole magnet

Multipole magnets are magnets built from multiple individual magnets, typically used to control beams of charged particles. Each type of magnet serves a particular purpose.

• Dipole magnets are used to bend the trajectory of particles
• Quadrupole magnets are used to focus particle beams
• Sextupole magnets are used to correct for chromaticity introduced by quadrupole magnets[1]

## Magnetic field equations

The magnetic field of an ideal multipole magnet in an accelerator is typically modeled as having no (or a constant) component parallel to the nominal beam direction ($\displaystyle{ z }$ direction) and the transverse components can be written as complex numbers:[2]

$\displaystyle{ B_y + i B_x = C_n \cdot ( x + iy )^{n-1} }$

where $\displaystyle{ x }$ and $\displaystyle{ y }$ are the coordinates in the plane transverse to the nominal beam direction. $\displaystyle{ C_n }$ is a complex number specifying the orientation and strength of the magnetic field. $\displaystyle{ B_x }$ and $\displaystyle{ B_y }$ are the components of the magnetic field in the corresponding directions. Fields with a real $\displaystyle{ C_n }$ are called 'normal' while fields with $\displaystyle{ C_n }$ purely imaginary are called 'skewed'.

First few multipole fields
n name magnetic field lines example device
1 dipole
3 sextupole

## Stored energy equation

Main page: Physics:Magnetic energy

For an electromagnet with a cylindrical bore, producing a pure multipole field of order $\displaystyle{ n }$, the stored magnetic energy is:

$\displaystyle{ U_n = \frac{n!^2}{2n} \pi \mu_0 \ell N^2 I^2 . }$

Here, $\displaystyle{ \mu_0 }$ is the permeability of free space, $\displaystyle{ \ell }$ is the effective length of the magnet (the length of the magnet, including the fringing fields), $\displaystyle{ N }$ is the number of turns in one of the coils (such that the entire device has $\displaystyle{ 2nN }$ turns), and $\displaystyle{ I }$ is the current flowing in the coils. Formulating the energy in terms of $\displaystyle{ NI }$ can be useful, since the magnitude of the field and the bore radius do not need to be measured.

Note that for a non-electromagnet, this equation still holds if the magnetic excitation can be expressed in units Amperes.

### Derivation

The equation for stored energy in an arbitrary magnetic field is:[3]

$\displaystyle{ U = \frac{1}{2}\int \left(\frac{B^2}{\mu_0} \right)\,d\tau. }$

Here, $\displaystyle{ \mu_0 }$ is the permeability of free space, $\displaystyle{ B }$ is the magnitude of the field, and $\displaystyle{ d\tau }$ is an infinitesimal element of volume. Now for an electromagnet with a cylindrical bore of radius $\displaystyle{ R }$, producing a pure multipole field of order $\displaystyle{ n }$, this integral becomes:

$\displaystyle{ U_{n} = \frac{1}{2\mu_0} \int^\ell\int^R_0\int^{2\pi}_0 B^2 \,d\tau. }$

Ampere's Law for multipole electromagnets gives the field within the bore as:[4]

$\displaystyle{ B(r) = \frac{n!\mu_0 NI}{R^n} r^{n-1}. }$

Here, $\displaystyle{ r }$ is the radial coordinate. It can be seen that along $\displaystyle{ r }$ the field of a dipole is constant, the field of a quadrupole magnet is linearly increasing (i.e. has a constant gradient), and the field of a sextupole magnet is parabolically increasing (i.e. has a constant second derivative). Substituting this equation into the previous equation for $\displaystyle{ U_{n} }$ gives:

$\displaystyle{ U_{n} = \frac{1}{2\mu_0} \int^\ell\int^R_0\int^{2\pi}_0 \left(\frac{n!\mu_0NI}{R^n}r^{n-1}\right)^2 \,d\tau, }$

$\displaystyle{ U_{n} = \frac{1}{2\mu_0} \int^R_0 \left(\frac{n!\mu_0NI}{R^n}r^{n-1}\right)^2 (2\pi\ell r\,dr), }$

$\displaystyle{ U_{n} = \frac{\pi\mu_0\ell n!^2 N^2 I^2}{R^{2n}} \int^R_0 r^{2n-1}\,dr, }$

$\displaystyle{ U_{n} = \frac{\pi\mu_0\ell n!^2 N^2 I^2}{R^{2n}} \left( \frac{R^{2n}}{2n} \right), }$

$\displaystyle{ U_{n} = \frac{n!^2}{2n} \pi\mu_0\ell N^2 I^2. }$

## References

1. Griffiths, David (2013). Introduction to Electromagnetism (4th ed.). Illinois: Pearson. p. 329.
2. Tanabe, Jack (2005). Iron Dominated Electromagnets - Design, Fabrication, Assembly and Measurements (4th ed.). Singapore: World Scientific.