Quadrupole Lens

Just like optical lenses focus light, electric and magnetic lenses can focus beams of charged particles. Systems of magnetic quadrupole lenses find a common use in focusing both ion and particle beams in accelerators at nuclear and particle physics centers such as CERN, SLAC, and ISIS. This COMSOL Multiphysics model shows the path of B5+ ions going through three consecutive magnetic quadrupole lenses. This 3D model takes fringing fields into account, and the calculation of the forces on the ions uses all components of their velocities.

Model Definition

The quadrupole consists of an assembly of four permanent magnets, as seen in Figure 1, where the magnets work together to give a good approximation of a quadrupole field. To strengthen the field and keep it contained within the system, the magnets are set in an iron cylinder.

Figure 1: Cross-sectional view of one of the magnetic quadrupoles used in the lens. The arrows show the direction of the magnetization.

The ions are sent through a system of three consecutive quadrupole assemblies. The middle one is twice as long as the other ones, and is rotated by 90 degrees around the central axis. This means the polarity of its magnets is reversed. Figure 2 gives a full view of the magnetic quadrupole lens.

Figure 2: Cutout of the quadrupole lens. The second quadrupole (Q2) has its polarities reversed compared to Q1 and Q3. After traveling through the lens, the ions are left to drift 1 m.

An accelerator feeds the system with ions traveling with the velocity 0.01c along the central axis. To study the focusing effect of the quadrupoles, track a number of ions starting out from a distance of 3 cm from the central axis, evenly distributed along the circumference of a circle in the transverse plane. They are all assumed to have a zero initial transverse velocity. Each quadrupole focuses the ion beam along one of the transverse axes and defocuses it along the other one. The net effect after traveling through the system of the three quadrupoles and the drift length is focusing in all directions. As the ions exit the system, they are all contained within a 1 cm radius in the transverse plane.

The geometry of the quadrupole lens is composed of three quadrupoles in a row, followed by 1 m of empty space, where the ions are left to drift. The AC/DC Module features a physics interface for magnetostatics in absence of currents. The formulation used in this physics interface reduces the memory usage considerably compared to the formulation including currents.

Domain Equations

The magnetic field is described using a static magnetic equation solving for the magnetic scalar potential Vm (Wb/m):

where μ0 = 4π·10−7 H/m denotes the permeability of vacuum and M is the magnetization (A/m). In the iron domain

where μr = 4000 is the relative permeability. The magnetic scalar potential is everywhere defined so that H = ∇Vm.

Boundary Conditions

The magnetic insulation boundary condition, reading n · B = 0, is used all around the iron cylinder, and at the lateral surfaces of the air domain that encloses the drift length.

Results

The x-component of the magnetic field density and arrows showing its local direction are shown in the figure below.

Figure 3: Arrows of the magnetic field and slices of its x-component in the quadrupole lens.

Each ion passing through the assembly experiences Maxwell forces equal to F = q v × B, where v (SI unit: m/s) is the velocity of the ion. To find the transverse position as a function of time, solve Newton’s second law for each ion: qv × B = ma. This particle tracing operation can be performed in postprocessing (in a Plot Group) and does not require the Particle Tracing Module. Figure 4 shows the traces of the ions as they fly through the quadrupole lens.