2 Beam Dynamics
21
We shall see that β (later to be called the envelope or betatron function) is a
property of the gutter, not the beam. In the synchrotron it varies around the ring
and is the envelope function we have plotted in Fig. 2.10 and again in Fig. 2.11.
By analogy, the “brim of the hat” which represents the alternating gradient focusing
system shown in this figure will vary its width and curvature around the crown and
β will follow this variation in some way.
2.1.6 Alternating Gradient Focusing
In Chap. 1 we described a major break-through in the design of synchrotrons: the
discovery of alternating gradient (AG) focusing (see [1] for an excellent summary
of the dynamics of AG focussing). This allowed designers to use much stronger
focusing systems with considerable savings in the space needed for the beam cross
section.
The principle is shown in Fig. 2.6 which depicts an optical system in which
each lens is concave in one plane while convex in the other and they alternate. It
is possible, even with lenses of equal strength, to find a ray which is always on
axis at the D lenses in the horizontal plane and therefore only sees the F lenses.
To appear like Fig. 2.6 the spacing of the lenses would have to be 2f. If the ray
is also central in the lenses which are vertically defocusing, the same condition
will apply simultaneously in the vertical plane. At least one particular particle
or trajectory corresponding to this ray will never be defocused and be contained
indefinitely.
The alternating gradient idea will work even when the rays in the D lenses do not
pass exactly at their centre and the lenses are not spaced by precisely 2f. In fact it is
sufficient for the lens strengths and spacing to be chosen to ensure that the particle
trajectories tend to be closer to the axis in D lenses than in F lenses as shown in Fig.
2.10.
Fig. 2.6 Optical analogy with an alternating pattern of lenses
21
We shall see that β (later to be called the envelope or betatron function) is a
property of the gutter, not the beam. In the synchrotron it varies around the ring
and is the envelope function we have plotted in Fig. 2.10 and again in Fig. 2.11.
By analogy, the “brim of the hat” which represents the alternating gradient focusing
system shown in this figure will vary its width and curvature around the crown and
β will follow this variation in some way.
2.1.6 Alternating Gradient Focusing
In Chap. 1 we described a major break-through in the design of synchrotrons: the
discovery of alternating gradient (AG) focusing (see [1] for an excellent summary
of the dynamics of AG focussing). This allowed designers to use much stronger
focusing systems with considerable savings in the space needed for the beam cross
section.
The principle is shown in Fig. 2.6 which depicts an optical system in which
each lens is concave in one plane while convex in the other and they alternate. It
is possible, even with lenses of equal strength, to find a ray which is always on
axis at the D lenses in the horizontal plane and therefore only sees the F lenses.
To appear like Fig. 2.6 the spacing of the lenses would have to be 2f. If the ray
is also central in the lenses which are vertically defocusing, the same condition
will apply simultaneously in the vertical plane. At least one particular particle
or trajectory corresponding to this ray will never be defocused and be contained
indefinitely.
The alternating gradient idea will work even when the rays in the D lenses do not
pass exactly at their centre and the lenses are not spaced by precisely 2f. In fact it is
sufficient for the lens strengths and spacing to be chosen to ensure that the particle
trajectories tend to be closer to the axis in D lenses than in F lenses as shown in Fig.
2.10.
Fig. 2.6 Optical analogy with an alternating pattern of lenses