30
E. Wilson and B. J. Holzer
Unlike early lattice designers we have computers to help when we come to
multiply these elements together to form the matrix for a ring or a period of the
lattice [3–5]. A lattice program such as MAD [6] does all the matrix multiplication
to obtain (a, b, c, d) from each specified point, s, and back again. It prints out β
and ϕ and other lattice variables in each plane, and we can plot the result to find
the beam envelope around the machine. This is the way machines are designed.
Lengths, gradients, and numbers of FODO normal periods are varied to match the
desired beam sizes and Q values.
In Fig. 2.8 we saw the trajectory of a particle, oscillating in a pattern of alternating
focussing and defocusing quadrupoles (FODO). The trajectories in general all lie
within an envelope which has the general features of the optical model in Fig. 2.6. If
we were to repeat the observation of the displacement and divergence of a particle
on successive turns we would find the elliptical locus of its motion (Fig. 2.5). The
aspect ratio of this ellipse would depend upon where in the ring we choose to make
the observation. The ellipse would be squat near D lenses and elongated near F’s.
The figure would appear just the same if we were to plot it between what are F
quadrupoles in the vertical plane. Of course, the whole pattern of quadruples and
the envelope is shifted by the distance between adjacent quadrupoles because Fquadrupoles in one plane are D in the other (et vice versa).
2.1.11 The Betatron Envelopes
To recapitulate, a modern synchrotron consists of pure bending magnets and
quadrupole magnets or lenses which provide focusing. These are interspersed
among the bending magnets of the ring in a pattern called the lattice. By suitable
choice of strength and spacing of the lenses the envelope function β(s) can be made
periodic in such a way that it is large at all F quadrupoles and small at all D’s.
Symmetry will ensure this is true also in the vertical plane. Particles oscillating
within this envelope will always tend to be further off axis in F quadrupoles than
in D quadrupoles and there will therefore be a net focusing action. We have already
seen that β is the aspect ratio of the phase space ellipse (see also [7, 8]).
In Fig. 2.11 we see an example of such a magnet pattern which is one cell, or
about 1% of the circumference, of the 400 GeV SPS at CERN. Although the SPS is
now considered a rather old fashioned machine its simplicity leads us to use it as an
example. The focusing structure is FODO and in this pattern half of the quadrupoles
(F) focus, while the other half, defocus (D) the beam. Bending magnets, which
in a first approximation do no focussing are represented together with other nonfocussing elements by the letter “O”. The envelope of these oscillations follows a
function β(s) which has waists near each defocusing magnet and has a maximum
at the centres of F quadrupoles. Since F quadrupoles in the horizontal plane are
D quadrupoles vertically, and vice versa, the two functions β h (s) and β v (s) are
one half-cell out of register in the two transverse planes. The function β has the
dimensions of length but the units bear no relation at this stage to physical beam
E. Wilson and B. J. Holzer
Unlike early lattice designers we have computers to help when we come to
multiply these elements together to form the matrix for a ring or a period of the
lattice [3–5]. A lattice program such as MAD [6] does all the matrix multiplication
to obtain (a, b, c, d) from each specified point, s, and back again. It prints out β
and ϕ and other lattice variables in each plane, and we can plot the result to find
the beam envelope around the machine. This is the way machines are designed.
Lengths, gradients, and numbers of FODO normal periods are varied to match the
desired beam sizes and Q values.
In Fig. 2.8 we saw the trajectory of a particle, oscillating in a pattern of alternating
focussing and defocusing quadrupoles (FODO). The trajectories in general all lie
within an envelope which has the general features of the optical model in Fig. 2.6. If
we were to repeat the observation of the displacement and divergence of a particle
on successive turns we would find the elliptical locus of its motion (Fig. 2.5). The
aspect ratio of this ellipse would depend upon where in the ring we choose to make
the observation. The ellipse would be squat near D lenses and elongated near F’s.
The figure would appear just the same if we were to plot it between what are F
quadrupoles in the vertical plane. Of course, the whole pattern of quadruples and
the envelope is shifted by the distance between adjacent quadrupoles because Fquadrupoles in one plane are D in the other (et vice versa).
2.1.11 The Betatron Envelopes
To recapitulate, a modern synchrotron consists of pure bending magnets and
quadrupole magnets or lenses which provide focusing. These are interspersed
among the bending magnets of the ring in a pattern called the lattice. By suitable
choice of strength and spacing of the lenses the envelope function β(s) can be made
periodic in such a way that it is large at all F quadrupoles and small at all D’s.
Symmetry will ensure this is true also in the vertical plane. Particles oscillating
within this envelope will always tend to be further off axis in F quadrupoles than
in D quadrupoles and there will therefore be a net focusing action. We have already
seen that β is the aspect ratio of the phase space ellipse (see also [7, 8]).
In Fig. 2.11 we see an example of such a magnet pattern which is one cell, or
about 1% of the circumference, of the 400 GeV SPS at CERN. Although the SPS is
now considered a rather old fashioned machine its simplicity leads us to use it as an
example. The focusing structure is FODO and in this pattern half of the quadrupoles
(F) focus, while the other half, defocus (D) the beam. Bending magnets, which
in a first approximation do no focussing are represented together with other nonfocussing elements by the letter “O”. The envelope of these oscillations follows a
function β(s) which has waists near each defocusing magnet and has a maximum
at the centres of F quadrupoles. Since F quadrupoles in the horizontal plane are
D quadrupoles vertically, and vice versa, the two functions β h (s) and β v (s) are
one half-cell out of register in the two transverse planes. The function β has the
dimensions of length but the units bear no relation at this stage to physical beam