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12.1 Magnet Strength

There are many conventions for specifying magnetic fields in terms of a multipole, polynomial, or Taylor expansion, which leads to potential confusion. In elegant (as in MAD[2]), magnet strengths are specified in terms of Taylor series. For normal multipoles and y = 0, the expansion is

∑∞ Bnxn By(x,0) = -n!--, n=0
(19)

where B0 is the dipole, B1 is the quadrupole, etc. In general,

( n ) Bn = ∂-By- . ∂xn x=y=0
(20)

elegant follows MAD [2] in using a right-handed coordinate system (x,y,z) in which z is along the beam direction, x is to the left, and y is up.

This expansion for the normal multipole terms can be related to a multipole expansion that includes both normal and skew components. In this convention, positive normal multipole coefficients give positive By for x > 0 and y = 0. Rotating a positive normal multipole with N poles π∕N clockwise about the vector along the beam direction will convert it into a positive skew multipole. As a result, for a positive skew multipole, By will be non-negative and Bx will be negative for x > 0 along the line ϕ = π∕N.

We can satisfy these conventions if we write the scalar potential as

∑∞ iAn--1---Bn-1- n -inΔϕ V = n! (x + iy) e , n=1
(21)

where, as we’ll see, Am are skew components and Bm are normal components for a 2(m + 1)-pole. The coordinates (x,y) are in a right-handed system with the longitudinal coordinate z. Δϕ is the rotation angle of the magnet, where a clockwise rotation about the nominal trajectory corresponds to Δϕ > 0. The minus sign in e-inΔϕ is because we rotate the magnet while keeping the coordinate system fixed.

The magnetic fields are

∂V ∞∑ An + iBn n -i(n+1 )Δ ϕ By = - ℑ ∂y-= ℑ ---n!----(x+ iy) e , n=0
(22)

and

∂V ∞∑ - iAn + Bn Bx = - ℑ --- = ℑ -----------(x + iy)ne-i(n+1)Δϕ, ∂x n=0 n!
(23)

We can relate the coefficients to the Bm quantities used in MAD and elegant by noting that for Δϕ = 0

( ) ∂mBy-- Bm = ∂xm x=y=0
(24)

and

( ) ∂mBx-- Am = - ∂xm x=y=0
(25)

Note the minus sign in the last equation, which differs from commonly asserted conventions.

Multipole errors are typically specified as fractions of the main field harmonic at a reference radius R, e.g.,

n F = -KnR--∕n!-, n KmRm ∕m!
(26)

where m is the main harmonic and n is the error harmonic.

For electrons, the deflection from a thin element is

1 ∫ θ(x,y = 0) = -- B (x, y = 0)dl, H
(27)

where H = Bρ = -p∕e is the beam rigidity and p = mecβγ is the momentum. The geometric strengths Kn are defined as

Kn = Bn-. H
(28)

By convention in elegant, a positive Kn value deflects a particle at x > 0 toward x = 0. E.g., a positive K1 value indicates a horizontally focusing quadrupole.

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