COMSOL 6.4 - Quadrupole Lens in 2D

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. The model is set up in a cross section of the geometry.

Model Definition

The quadrupole consists of an assembly of four permanent magnets, as seen in Figure 1 below, 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 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 model is set up in a 2D cross section of any of the two identical quadrupoles Q1 and Q3. Neglecting fringe fields, the transverse magnetic field at a given point in a transverse plane in Q2 automatically has the same magnitude as the corresponding in Q1 and Q3, but point in the opposite direction. It is therefore sufficient to model the fields in one of the quadrupoles.

Domain Equations

The magnetic field is described using the Magnetostatics equation, solving for the z component of the magnetic potential A (Wb/m):

Here μ0 = 4π·10−7 H/m denotes the permeability of vacuum, Br is the remanent flux density (T), σ the conductivity (S/m), and v the velocity of the medium (m/s). In this example, the medium is not moving. The right-hand side of the equation holds an imposed current, specified in terms of an external current density Jze (A/m2). No currents are imposed. The iron domain uses a slightly different formulation of the same equation:

where μr = 4000 is the relative permeability. The magnetic potential is everywhere defined so that B = ∇ × A.

Boundary Conditions

The magnetic field is approximately parallel to the exterior boundary of the iron cylinder. To enforce this, use the magnetic insulation boundary condition, stating that Az = 0.

Results and Discussion

The magnetic field density and streamlines in a cross section of Q1 or Q3 appear in Figure 3 below.

Figure 3: The magnetic field density and flow lines in the center of one of the quadrupole magnets.

Each ion passing through the assembly experiences a Maxwell force equal to F = q v × B, where v (m/s) is the ion’s velocity. Next, assume that the z component of the velocity is constant and much larger than the x and y (transverse) components. Thus consider only the force contributions from the z component of the velocity. To find the transverse position as a function of time, you need to solve Newton’s second law for each ion, q v × B = m a, where m is the ion mass (kg), and a denotes its acceleration (m/s2). If the computed magnetic flux density in Q1 equals B', and the length of quadrupole i is Li (m), the flux density that the ion experiences is given by

where t (s) is the time of flight. This dependency of the magnetic flux density on the time of flight is fed to the particle-tracing algorithm as a logical expression. Figure 4 below shows how the ions travel in the transverse plane.

Figure 4: As the ions enter Q1, they start out evenly distributed around the larger circle, 3 cm from the z-axis. Q1 focuses along the x-axis and defocuses along the y-axis. The force on each ion is approximately proportional to its distance from the z-axis, so as the ions enter Q2, those that are far out on the x-axis rapidly turn around and move toward the center. Q3 stabilizes the motion and gets all ions on the right track. Finally the ions are left to drift toward a waist situated a little bit more than 1 m beyond Q3.

COMSOL_Multiphysics/Electromagnetics/quadrupole

Modeling Instructions

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Global Definitions

Parameters 1

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In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
M	11	11	Ion mass number
Z	5	5	Ion charge number
L1	1[m]	1 m	Length of first quadrupole
L2	2[m]	2 m	Length of second quadrupole
L3	1[m]	1 m	Length of third quadrupole
vz	0.01*c_const	2.9979E6 m/s	Ion velocity
m	M*mp_const	1.8399E-26 kg	Ion mass
q	Z*e_const	8.0109E-19 C	Ion charge
Br	8[mT]	0.008 T	Quadrupole remanent flux density

Here, c_const, mp_const and e_const are predefined constants for, respectively, the speed of light, the mass of the proton and the elementary charge.

Geometry 1

Rectangle 1 (r1)

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In the text field, type 0.177.

In the text field, type 0.07.

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In the text field, type 45.

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Circle 1 (c1)

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Intersection 1 (int1)

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Click in the window and then press Ctrl+A to select both objects.

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Rotate 2 (rot2)

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Now make two circles centered at the origin.

Circle 2 (c2)

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Circle 3 (c3)

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In the text field, type 0.12.

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Create a composite object subtracting the magnets from the union of the two circles. This complex operation can be done in one step by using the Compose geometry operation and specifying the formula.

Compose 1 (co1)

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In the text field, type c2+c3-(int1+rot2(1)+rot2(2)+rot2(3)).

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Make one more circle centered at the origin.

Circle 4 (c4)

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Add Material

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Materials

Iron (mat1)

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Magnets

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In the dialog, type 1, 3, 5-6 in the text field.

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In the window for , click to expand the section.

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Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Recoil permeability	murec_iso ; murecii = murec_iso, murecij = 0	1.05	1	Remanent flux density
Remanent flux density norm	normBr	Br	T	Remanent flux density