9.1 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=0  n!

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

     (  n   )
B  =   ∂-By-       .
 n     ∂xn   x=y=0

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
V =     -------------(x + iy)ne-inΔϕ,
    n=1      n!

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

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


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

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

      (∂mBy  )
Bm =   ---m--
        ∂x    x=y=0


       (  m    )
Am = -   ∂--Bx-
          ∂xm   x=y=0

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.,

      K  Rn∕n!
Fn = --n--m----,
     KmR   ∕m!

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

For electrons, the deflection from a thin element is

θ(x,y = 0) = -1   B (x,y = 0)dl,

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

Kn =  H  .

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.