8 Accelerator Engineering and Technology: Accelerator Technology
339
is provided by the complex formalism [1] and its multipole expansion. Defining
the complex variable z = x + iy, where the plane (x,y) is that of the magnet cross
section, the function B y + iB x of the two non-zero components of the magnetic field
is expanded in series:
B y + iB x =
∞
n=1
(B n + iA n ) z
n−1 .
(8.5)
The coefficients B n and A n of the series expansion are the multipoles of the field,
and determine the shape of the field lines. As an example, a magnet in which only
the term B 1 is non-zero, corresponds to a magnetic field:
B x = 0; B y = B 1 ,
i.e. a perfect dipole field (constant in amplitude and direction) oriented in y
direction. If the y direction is taken perpendicular to the plane of the accelerator
(e.g. vertical), this is usually called a normal dipole, which provides bending in the
plane of the accelerator (e.g. horizontal). A magnet in which only A 1 is non-zero
results in a perfect skew dipole field:
B x = A 1 ; B y = 0,
which is in the plane of the accelerator (e.g. horizontal) and provides bending
perpendicular to it (e.g. vertical). Multipoles B 2 and A 2 correspond to magnets
generating a pure normal and skew quadrupole field. Higher order gradients
(sextupole, octupole, etc.) are obtained by simple analogy in the continuation of
the series.
The explicit expressions of the field components corresponding to the first four
multipoles, and a sketch of the corresponding field lines are reported in Table 8.1. It
is useful to remark that the coefficients B n and A n appearing in Table 8.1 have units
of [T/m n − 1 ] and are hence the generalized normal and skew gradient of order n.
Specifically, B 2 corresponds to the normal quadrupole gradient G discussed earlier.
To complete this short review of field configuration, we use the complex expansion
to evaluate the module of the field B in the case of a pure normal multipole (chosen
for convenience, the case of a pure skew multipole field yields identical results).
Simple algebra, writing that z = Re iθ where R is the module and θ is the argument
of z, gives the following result:
B = B n R
n−1 ,
(8.6)
which shows that the field strength in a pure multipole field of order n is proportional
to the generalized gradient and grows with the power n−1 of the distance from
the magnet centre. This extends the case of the dipole (n = 1, constant field), and
339
is provided by the complex formalism [1] and its multipole expansion. Defining
the complex variable z = x + iy, where the plane (x,y) is that of the magnet cross
section, the function B y + iB x of the two non-zero components of the magnetic field
is expanded in series:
B y + iB x =
∞
n=1
(B n + iA n ) z
n−1 .
(8.5)
The coefficients B n and A n of the series expansion are the multipoles of the field,
and determine the shape of the field lines. As an example, a magnet in which only
the term B 1 is non-zero, corresponds to a magnetic field:
B x = 0; B y = B 1 ,
i.e. a perfect dipole field (constant in amplitude and direction) oriented in y
direction. If the y direction is taken perpendicular to the plane of the accelerator
(e.g. vertical), this is usually called a normal dipole, which provides bending in the
plane of the accelerator (e.g. horizontal). A magnet in which only A 1 is non-zero
results in a perfect skew dipole field:
B x = A 1 ; B y = 0,
which is in the plane of the accelerator (e.g. horizontal) and provides bending
perpendicular to it (e.g. vertical). Multipoles B 2 and A 2 correspond to magnets
generating a pure normal and skew quadrupole field. Higher order gradients
(sextupole, octupole, etc.) are obtained by simple analogy in the continuation of
the series.
The explicit expressions of the field components corresponding to the first four
multipoles, and a sketch of the corresponding field lines are reported in Table 8.1. It
is useful to remark that the coefficients B n and A n appearing in Table 8.1 have units
of [T/m n − 1 ] and are hence the generalized normal and skew gradient of order n.
Specifically, B 2 corresponds to the normal quadrupole gradient G discussed earlier.
To complete this short review of field configuration, we use the complex expansion
to evaluate the module of the field B in the case of a pure normal multipole (chosen
for convenience, the case of a pure skew multipole field yields identical results).
Simple algebra, writing that z = Re iθ where R is the module and θ is the argument
of z, gives the following result:
B = B n R
n−1 ,
(8.6)
which shows that the field strength in a pure multipole field of order n is proportional
to the generalized gradient and grows with the power n−1 of the distance from
the magnet centre. This extends the case of the dipole (n = 1, constant field), and