24
E. Wilson and B. J. Holzer
applies to the motion in the vertical plane while in the horizontal plane the effect of
the dipole magnets has to be included:
x
+
1
ρ 2 (s)
− k(s)
x = 0.
(2.16)
Here the sign in front of k(s) is reversed so that the quadrupole focuses. The extra
focusing term 1/ρ 2 due to the curvature of the orbit can be significant in small rings.
In the old constant gradient synchrotrons, this weak focusing term was the only form
of horizontal focusing.
We see in Fig. 2.10, the pattern of one cell of a simple synchrotron lattice—a
pattern which is repeated many times around the circumference as may be seen in
Fig. 2.11 which shows—in addition to the focusing and defocusing lenses also the
bending magnets—bending magnets. Within this pattern of dipole and quadrupole
focusing and defocusing (F and D), particles make betatron oscillations within the
envelopes described by β x and β z , or more precisely, the square roots of these
quantities (here we use the variable y to represent either the horizontal or the vertical
coordinate, x or z)
y =
εβ(s) sin (φ(s) + φ 0 ) .
(2.17)
If one tries to verify that this is the solution of Hills Equation an important and
necessary condition emerges:
φ
= 1/β
(2.18)
From which we see that 2πβ is the local wavelength of the transverse oscillations.
2.1.9 Matrix Description
Usually in alternating gradient (AG) machines, the ring is a repetitive pattern of
focusing fields that we call the “lattice”. Each lattice element may be expressed
by a matrix and whole sections of the ring which transport the beam from place
to place may be represented as the product matrix of the single element matrices
involved, which makes the description of particle trajectories very simple and very
elegant at the same time. Any linear differential equation, like Hill’s Equation, has
solutions which can be traced from one point, s 1 , to another, s 2 , by a 2 × 2 matrix,
the transport matrix:
y (s 2 )
y (s 2 )
=
a b
c d
y (s 1 )
y (s 1 )
= M 21
y (s 1 )
y (s 1 )
.
(2.19)
E. Wilson and B. J. Holzer
applies to the motion in the vertical plane while in the horizontal plane the effect of
the dipole magnets has to be included:
x
+
1
ρ 2 (s)
− k(s)
x = 0.
(2.16)
Here the sign in front of k(s) is reversed so that the quadrupole focuses. The extra
focusing term 1/ρ 2 due to the curvature of the orbit can be significant in small rings.
In the old constant gradient synchrotrons, this weak focusing term was the only form
of horizontal focusing.
We see in Fig. 2.10, the pattern of one cell of a simple synchrotron lattice—a
pattern which is repeated many times around the circumference as may be seen in
Fig. 2.11 which shows—in addition to the focusing and defocusing lenses also the
bending magnets—bending magnets. Within this pattern of dipole and quadrupole
focusing and defocusing (F and D), particles make betatron oscillations within the
envelopes described by β x and β z , or more precisely, the square roots of these
quantities (here we use the variable y to represent either the horizontal or the vertical
coordinate, x or z)
y =
εβ(s) sin (φ(s) + φ 0 ) .
(2.17)
If one tries to verify that this is the solution of Hills Equation an important and
necessary condition emerges:
φ
= 1/β
(2.18)
From which we see that 2πβ is the local wavelength of the transverse oscillations.
2.1.9 Matrix Description
Usually in alternating gradient (AG) machines, the ring is a repetitive pattern of
focusing fields that we call the “lattice”. Each lattice element may be expressed
by a matrix and whole sections of the ring which transport the beam from place
to place may be represented as the product matrix of the single element matrices
involved, which makes the description of particle trajectories very simple and very
elegant at the same time. Any linear differential equation, like Hill’s Equation, has
solutions which can be traced from one point, s 1 , to another, s 2 , by a 2 × 2 matrix,
the transport matrix:
y (s 2 )
y (s 2 )
=
a b
c d
y (s 1 )
y (s 1 )
= M 21
y (s 1 )
y (s 1 )
.
(2.19)