Using a Cartesian coordinate system, a particle given some velocity in thezxplane will remain in the same plane and follow a circular path if there is a magnetic field directed along the y axis1. The radius of this path is given by,
ρ= p
qBy
. (2.2)
Figure 2.1 illustrates this case. A disc within the zx plane has been taken, within the disc there is a uniform magnetic field directed along the y axis. A charged particle given the appropriate starting conditions (position and velocity) will orbit around the centre of the disc (the path of such a particle is marked red in Fig. 2.1). Taking two more particles of the same energy, and giving one a slightly different starting velocity (Fig. 2.1a) and the second a different starting location (Fig. 2.1b), it is seen that the orbits of these two extra particles can be said to oscillate around the orbit of the original particle, but that in all cases each particle has the same location and velocity at the end of one orbit as it had at the start. The number of oscillations of a particle around the original path per orbit is called the betatron tune, which for pure dipole field is 1.
æ æ æ æ z x
(a) Momentum offset from equi- librium orbit. æ æ æ æ z x
(b) Spatial offset from equilib- rium orbit.
Figure 2.1: A particle with velocity in the zx plane travels on a disc that has a magnetic field directed along the y axis, the force experienced by the particle causes it to orbit around the centre of the plot (red path). If the initial velocity (blue path) or position (green path) of the particle is changed, then these new paths are seen to
oscillate around the original path.
The disc is now divided into quarters, and the quarters of the disc are moved outwards (as illustrated in Fig. 2.2) so that they are separated by regions with no magnetic field: these field free regions are referred to as drift spaces. Figure 2.2a shows a path for which a particle will have the same location and velocity at the end of an orbit as at the start; this path is referred to as the closed (or equilibrium) orbit. Particles following any other path are seen to oscillate around the closed orbit. However, unlike in Fig. 2.1, the particles do not return to their starting conditions after one orbit, but instead do so after three. The betatron tune in the case shown in Fig. 2.2 is 4/3.
These two examples are crude representations of the arrangement of magnets within a cyclotron (Fig. 2.1) and synchrotron (Fig. 2.2). For both arrangements, the return of the particles to their starting conditions after a given number of orbits is evidence of the focusing properties of a uniform magnetic field, there is clearly a range of initial conditions within the zxplane for which a particle will orbit indefinitely. The particle sources used for accelerators emit particles which have a spread of initial locations, ve- locities and energies; focusing methods are important in ensuring that as many particles as possible survive to the end of the acceleration cycle without being lost to the walls of the accelerator.
Equation 2.2 tells us that the radius of an arc subtended by a charged particle that is travelling through a magnetic field increases with momentum, the result of which is a closed orbit that is dependent on momentum. The change in closed orbit with the fractional offset in momentum, δ, is referred to as dispersion. The momentum of the particle tracked through the split dipole is now increased, and the effects of dispersion can be seen in Fig. 2.3a. The betatron tune of the high momentum particle is not 4/3, but is instead either an irrational number or expressed by a fraction with a denominator much
æ æ
z
x
(a) Equilibrium orbit.
æ æ
z
x
(b) Momentum offset from equi- librium orbit.
æ æ
z
x
(c) Spatial offset from equilib- rium orbit.
Figure 2.2: Separating the quadrants of the disc shown in Fig. 2.1 (with regions that have no fields) changes the particle dynamics. Figure (a) shows the equilibrium orbit for the new dipole configuration (this path is marked by the red dashed line in Figs. (b) and (c)); if particles have a momentum (Fig. (b)) or spatial (Fig. (c)) offset from the equilibrium orbit, then these particles will oscillate around the equilibrium orbit. In
this example the betatron tune is 4/3.
larger than 3 (Fig. 2.3b). The change in tune with the fractional offset in momentum is called the chromaticity. We now select an entrance plane to one of the dipole quarters as an observation point, and track a particle through 200 orbits in the disc. Each time the particle passes an observation point, its position and momentum along the x axis are recorded. Figure 2.3 shows that a plot of the recorded momentum vs. position forms an ellipse, at the centre of which is the closed orbit. The orientation of the ellipse in a plot of the transverse dynamical variables relative to the closed orbit is dependent upon where along the closed orbit the measurement is made; however, the area enclosed by the ellipse will remain constant (a matrix describing the propagation of the motion is symplectic).
æ æ æ æ z x
(a) Change in closed orbit with momentum.
æ æ
z
x
(b) Spatial offset from equilib- rium orbit. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ x px (c) Poincar´e plot.
Figure 2.3: The momentum of the particle in these plots is greater than that in Fig. 2.2. Dispersion leads to a closed orbit that is found at a larger radius: the closed orbit for the new and previous momenta are shown by the solid and dashed red lines in Fig. (a) respectively. Figure (b) shows that the path of a particle that is offset from the equilibrium orbit no longer closes every three turns; the change in betatron tune with a fractional change in momentum is called the chromaticity. A particle is tracked through 200 turns, and the phase space variables, position and momentum, of the particle in the direction of thexaxis are recorded at a boundary of the lower right quadrant (marked orange in Fig. (c)); the plot of px vs.x at the boundary traces out an ellipse. If the
values of the phase space variables are recorded at successive points in the remainder of the quadrant (along an axis transverse to the closed orbit), then the ellipse will be seen to change shape, whilst the area enclosed by the ellipse remains constant. Given the periodicity of the system, the evolution of the ellipse is the same for each quadrant.
x
s
z
y
machine centreFigure 2.4: Coordinate system local to the equilibrium path of a particle beam. x
is the longitudinal coordinate (tangential to the closed orbit). y andz are transverse horizontal and vertical respectively.
accelerated the increase in velocity is compensated by the increase in orbital circumfer- ence, so the orbital period remains constant. However, in a synchrotron, the fields are ramped up during acceleration so as to overcome the effect of dispersion and maintain a fixed closed orbit. In FFAGs, the magnetic fields increase with radius, which results in a reduction of the dispersion. For both synchrotrons and FFAGs the motion of particles can be described in terms of small oscillations (compared to the orbital radius) around closed orbits: for this reason it is convenient to use a coordinate system that has an origin at or close to the position of the closed orbit of a beam at any point around the circumference of the accelerator (Fig. 2.4).
The focusing provided by a uniform field directed along theyaxis acts only in thezx
plane. Eventually, any particle with some component of velocity in the y direction will be lost. A number of non-uniform field profiles (introduced in section 2.3) are commonly applied in accelerators to provide both horizontal and vertical focusing and to ensure that the beam dynamics within an accelerator are as required for any given application.