2 Beam Dynamics
31
Fig. 2.11 One cell of the
CERN SPS representing
1/108 of the circumference.
The pattern of dipole (B)
magnets and quadrupole (F
and D) lenses is shown above
size. The reader should be persuaded that particles do not follow the β(s) curves but
oscillate within them in a form of modified sinusoidal motion whose phase advance
is described by φ(s). The phase change per cell in the example shown is close to π/2
but the rate of phase advance is modulated throughout the cell.
2.2 Coupling
Until now we have considered the motion in the vertical and horizontal direction to
be orthogonal and independent. This is the ideal case. Now we look at what happens
when there is a skew quadrupole or solenoidal field in the machine which couples
horizontal motion into vertical and vice versa. This is rather a special case affecting
mainly electron synchrotrons and the reader may choose to skip to Sect. 2.3 and
leave coupling to a second reading.
In a fully coupled machine the betatron oscillations in the two transverse
directions are like two harmonic oscillators which transfer energy from one to the
other with a frequency which is just the difference between their Q’s. They act like
coupled pendula. In this way all the horizontal “emittance” can add to the vertical
emittance and the beam exceeds the available vertical aperture.
The phenomenon is particularly important in electron rings. The electron beam
would damp to zero emittance were it not for quantum emission in the horizontal
plane exciting betatron oscillations. There is no comparable excitation in the vertical
plane and only coupling of the horizontal oscillations into the vertical plane gives
31
Fig. 2.11 One cell of the
CERN SPS representing
1/108 of the circumference.
The pattern of dipole (B)
magnets and quadrupole (F
and D) lenses is shown above
size. The reader should be persuaded that particles do not follow the β(s) curves but
oscillate within them in a form of modified sinusoidal motion whose phase advance
is described by φ(s). The phase change per cell in the example shown is close to π/2
but the rate of phase advance is modulated throughout the cell.
2.2 Coupling
Until now we have considered the motion in the vertical and horizontal direction to
be orthogonal and independent. This is the ideal case. Now we look at what happens
when there is a skew quadrupole or solenoidal field in the machine which couples
horizontal motion into vertical and vice versa. This is rather a special case affecting
mainly electron synchrotrons and the reader may choose to skip to Sect. 2.3 and
leave coupling to a second reading.
In a fully coupled machine the betatron oscillations in the two transverse
directions are like two harmonic oscillators which transfer energy from one to the
other with a frequency which is just the difference between their Q’s. They act like
coupled pendula. In this way all the horizontal “emittance” can add to the vertical
emittance and the beam exceeds the available vertical aperture.
The phenomenon is particularly important in electron rings. The electron beam
would damp to zero emittance were it not for quantum emission in the horizontal
plane exciting betatron oscillations. There is no comparable excitation in the vertical
plane and only coupling of the horizontal oscillations into the vertical plane gives