**9.1 The U3DS design**

This magnet comprises a single U-shaped iron yoke as shown in **Figures 13** and **14**, providing two rectangular zones of uniform dipole field extending across the large dimension of the ribbon beam. Three optional iron bars labeled P1b, P2b, and P3 are shown placed on the opposite side of the beam. They are not mandatory, but serve the auxiliary functions of homogenizing the field, lowering the ampere-turn requirement, and thus the power requirement, and enabling some reduction in aberrations. Two simple identical rectangular coils provide the magnetic induction.

The U3DS magnet deflects the beam in an S-shaped path, bending in its major (breadth) x-direction and back, as illustrated in **Figure 15**, causing an offset in its path, and this deflection increases with the field strength. The net angular deflection in the breadth dimension is zero and is achromatic. However, unexpectedly, the beam is also deflected away from the exciting coils through an angle which is proportional to

**Figure 14.** *Sectional view of the U3DS analyzer.*

the square of the field strength, and is simultaneously strongly focused in this direction. The trajectories have 3D S-shaped paths, and this y-direction deflection allows the device to be used as a spectrometer in the conventional manner. The dispersion is surprisingly high, and therefore so is the resolving power. The ribbon beam is bent through a modest angle of 15–25 degrees in its thin direction, and strongly focused, while the peak deflection in the S-shaped path in the breadth direction is about twice this amount. The dispersion achieved is twice that of a simple dipole with the same bend angle, so the performance is as good as a magnet bending 50 degrees. The focal length is substantially shorter.

The new magnet provides unperturbed uniformity along the beam breadth direction because there is no intrinsic variation of any field component along the breadth, and this avoids several of the worst aberrations of conventional dipole magnets. Nevertheless, there is a first-order aberration limiting resolving power, proportional to the intrinsic random divergence spread in the beam breadth direction. This limits the resolving power in practical situations to about 30.

Manufacture of IG6 flat panel displays requires a beam broader than 1.5 m, able to analyze an 80 keV P<sup>+</sup> beam. A suitable U3DS magnet would weigh about 7 tons. I estimate that a suitable dipole magnet with a 1600 mm pole gap would weight about 100 tons. The beam current carrying capacity is many times greater, as discussed in detail below.

In **Figure 15**, three zones of magnetic field, q, r, and s, are labeled. Zone q is where the trajectory first passes between two poles (labeled pole P1a and P1b in **Figure 14**), where the field will be substantially uniform and orthogonal to the trajectory. The field will fall at the edges; the effective length of uniform field is denoted by dimension a in **Figure 14**, and the gap between poles by g. The magnetic field direction is reversed in the right-hand half. Magnetic field zone s in **Figure 15** corresponds to this reversed reflection of zone q, adjacent to pole P2a. Central zone r is characterized by a predominant component of field from left to right in both views, but this field is nonuniform, and the field lines have some curvature. Poles P1b, P2b, and P3 are

**Figure 15.** *Plan view: S-shaped path of reference trajectory.*

supplementary pole pieces which concentrate the H-field where it is required, thereby greatly reducing the required coil current and power. They can be used to control the field shape to minimize aberrations.

The deflection in the y-direction arises because in the central zone r the ions have a large component of motion in the x-direction, and the magnetic field is predominantly in the z-direction—hence the deflection is normal to both—and is in the y-direction. Being proportional both to the z-field and to the amount of motion in the x-direction, which is itself a function of B, the amount of y-deflection scales with B<sup>2</sup> .

Optical elements are defined as illustrated in **Figure 16**. The focal length can be estimated from the geometry, given that the total deflection 2β = �2 h/f and 2h= �3 g, and assume g = d, using symbols defined in **Figure 16** This works well in practice. Greater accuracy requires full 3-dimensional trajectory modeling.

The beam plasma instability in magnetic fields discussed above applies to any magnet design. Experimentally, as discussed above, the background ion plasma density seems to be the most important factor. Ion sound waves in this plasma can carry instabilities through the magnet and cause them to be amplified, if the threshold for this instability is significantly exceeded. For a rectangular beam of high aspect ratio A > > 1 whose width (narrow dimension) within the magnetic field B is tb, and in an ambient pressure P, the dimensionless figure of demerit for instability in residual gas is.

$$\Delta\Omega\_{pi}/\Omega\_{\rm c} \cong \frac{\mathcal{M}\_{i}}{B} \sqrt{\frac{J\_{b}t\_{b}P\sigma}{\varepsilon\_{0}qkT\_{0}\sqrt{\mathcal{M}\_{i}kT\_{\rm c}}}} \tag{8}$$

Mi is the ion mass, B is the magnetic field in the beam, Jb the current density, tb is the beam thickness in the magnet in the narrow direction, and thus the quantity Jbtb is the one-dimensional current density along the beam breadth direction, a quantity of direct interest. σ is the cross-section for slow ion production by charge exchange with beam ions, T0 is the residual gas temperature, and Te is the plasma electron temperature. This equation follows from Eq. (7), assuming a very broad beam whose thickness can be neglected in this context. As an example, for the magnet used as a worked example above, the figure of demerit for the background gas instability for a beam of

15 keV P<sup>+</sup> at 0.43 T Tesla, in a pressure of 2 <sup>10</sup><sup>5</sup> mbar, at a current of 500 mA per meter, is 1. This is a current well in excess of existing commercial requirements; the instability usually becomes problematic when this ratio is 2 or more; so for practical purposes we do not expect to see beam instability with this device.
