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# Charged relativistic particles in a uniform magnetic field

Why are there boundaries?

### Physics

#### Charged relativistic particles in a uniform magnetic field

There are a number of reasons why one cannot accelerate all beams possible with a single machine, some of which become clear after we take a look at some equations describing motion of a particle moving with momentum p = m v and charge q in a homogeneous magnetic field B pointing in the z-direction. The resulting motion will be circular and confined to the xy plane. The radius r of the circle is given by the well known formula for the magnetic rigidity ("bee-rho value") B r = p / q Inserting p = mwr, expressing the mass m in MeV, the charge q in units of the electron charge, and equating w to the acceleration frequency 2p f/h, with h the harmonic number, we arrive at

• B = 2pi 1012 f/h u0 / (c2 Z/A)

with u0 the atomic mass unit expressed in MeV, c the speed of light, f/h the acceleration frequency in MHz, Z the charge state of the atom, and A the number of nucleons.

The kinetic energy (per nucleon) picked up by the particle is given (relativistically) by

• T / A = (γ-1) u 0

with γ the relativistic mass enhancement factor used in m = γm0. This factor can be linked back to the frequency through

• γ2 = (1 - β2)-1
• β = v / c = 2pi r/c 106  f/h

For a cyclotron, a reasonable approximation of the extraction energy is obtained by putting r equal to the extraction radius.

Summarizing, one can say that

1. The kinetic energy per nucleon (T / A) required fixes the acceleration frequency (f / h)
2. This frequency, together with the charge per nucleon (Z / A) of the particle, fixes the magnetic field (B).

Focusing Limit of a Cyclotron

Limitations arise from better models of a cyclotron. For example, the beam is stabilized in the z-direction by azimuthally modulating the magnetic field: as the particles spin around the centre of the cyclotron they experience a "wobble" in the field. This focuses the beam in the vertical direction, and the focusing limit Kf is the energy up to which particles can be accelerated without being defocused. This region is given by

• T / A  <  Kf  (Z / A)

### Bending Limit

Another parameter is the bending limit Kb, expressing the (magnetic field) strength of the cyclotron. Rewriting the formula for the kinetic energy one gets

• T = p2 / 2m = (qBR)2 / 2m

where R is the extraction radius, q (m) the charge (mass) of the particle and B the strength of the field. Because the magnet coils have a limiting current, there is an upper bound on the magnetic field, which means the kinetic energy of the accelerated particle is bounded by

• T / A  <  Kb  (Z / A) 2
• Kb  = (eBR)2 / 2m0

with e the electron charge and m0 the proton rest mass. The bending limit for AGOR is 600 MeV, which makes it a K600 cyclotron, as it is also called.

### Design Choices

Apart from physical limitations, there are also some choices made at the design stage which can place its limits on the operating diagramme:

• The RF system has a minimum and maximum frequency:. fmin < f / h < fmax
• At low fields, resonant instabilities start to occur, B > Bmin
• For some beams, the maximum dee voltage can be lower than optimal For a given harmonic mode,            Vdee ~ B0 2   and Vdee ~ Z/A.
• The maximum injection voltage produced by the ion source can limit operation. Vinj ~ B0 2 ,  Vinj ~ rinj , and Vinj ~ Z/A.

### Limits Specific to AGOR

#### Electromagnetic equipment

• The magnetic field can be chosen in the range 1.75 < B < 4.05 T
• The accelerating frequency f ranges from 24 to 62 MHz, harmonic modes are h = 2, 3, 4.
• The maximum dee voltage is 100 kV.
• Ion source specifications indicate a maximum injection voltage of 35 kV

#### Translation into (Z / A) and (T / A)

• At low Z / A and high T / A, the operating diagramme is bounded by the bending limit Kb = 600 MeV
• Above Z / A = 220 / 600,  the focusing limit Kf = 200 MeV becomes more restricting than the bending limit at high T / A.
• At high Z / A, the low T / A side is bounded by the magnetic field at which the strong nr + 2 nz = 3 resonance is occurring. In AGOR, this occurs at around 1.7 tesla.
• At low Z / A, the low T / A side is bounded by the low frequency limit of the RF system