By Garcia J., De Lis C. S.

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2 Geometry of iron-dominated multipole magnets Normal-conducting magnets usually use iron cores to increase the flux density achieved by a given current. In such a magnet, the shape of the magnetic field depends mainly on the geometry of the iron. In this section, we shall determine the geometry required to generate a pure multipole of given order. To simplify the problem, we shall make some approximations: in particular, we shall assume that the iron core has uniform cross-section and infinite extent along z; that there are no limits to the iron in the x–y plane; and that the iron has infinite permeability.

The units of Cn depend on the order of the multipole. In SI units, for a dipole, the units of C1 are tesla (T); for a quadrupole, the units of C2 are December 13, 2013 16 8:45 World Scientific Book - 9in x 6in ws-book9x6 Beam Dynamics in High Energy Particle Accelerators T m−1 ; for a sextupole, the units of C3 are T m−2 , and so on. It is sometimes preferred to specify multipole components in dimensionless units. In that case, we introduce a reference field Bref and a reference radius Rref . The multipole expansion is then written, in a standard notation: ∞ By + iBx = Bref (bn + ian ) n=1 x + iy Rref n−1 .

62) where ∇2 is the Laplacian operator. 62) is Laplace’s equation: the scalar potential in a particular case is found by solving this equation with given boundary conditions. The geometry of iron required to generate a pure multipole field can be determined from the scalar potential for a pure multipole field, as follows. December 13, 2013 8:45 World Scientific Book - 9in x 6in Electromagnetic Fields in Accelerator Components ws-book9x6 23 Since the magnetic flux density B is obtained from the gradient of the scalar potential, the flux density at any point must be perpendicular to a surface of constant scalar potential.

### A bifurcation problem governed by the boundary condition I* by Garcia J., De Lis C. S.

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