2 Beam Dynamics
23
The angular deflection given to a particle passing through a short quadrupole of
length, and strength k, at a displacement x is therefore:
Δx
= θ = B/ (Bρ) = B
x/ (Bρ) = kx.
(2.11)
The use of x to indicate the divergence angle of a trajectory is defined in Fig.
2.5. Compare this with a converging lens in optics:
Δx
= −x/f
(2.12)
and we see that the focal length of a horizontally focusing quadrupole is
f = −1/ (kk)
(2.13)
The particular quadrupole shown in Fig. 2.7 would focus positive particles
coming out of the paper or negative particles going into the paper in the horizontal
plane. A closer examination reveals that such a quadrupole deflects particles with
a vertical displacement away from the axis—vertical displacements are defocused.
This can be seen if Fig. 2.7 is rotated through 90 ◦ .
2.1.8 The Equation of Motion
Earlier we derived an expression for the change in divergence of a particle passing
through the quadrupole. A horizontally focusing quadrupole (which is at the same
time vertically defocusing) has a negative k.
We first look at the vertical plane. The angular deflection given to a particle
passing through a short quadrupole of length ds and strength k at a displacement
z is therefore:
dz
= −kzds.
(2.14)
From this we can deduce a differential equation for the motion
z
+ k(s)z = 0.
(2.15)
Here we would like to make a clear statement: While inside a lattice element,
say a quadrupole lens, the normalised gradient k is constant and we get a equation
that we know from Hook’s law in classical mechanics, (see Eq. 2.12), the situation
now is more general. We allow k(s) to change, while our particles are running
through the accelerator. The corresponding equation (2.15) is called Hill’s Equation,
a second order linear equation with a periodic coefficient, k(s) which describes the
distribution of focusing strength around the ring. The above form of Hill’s equation
23
The angular deflection given to a particle passing through a short quadrupole of
length, and strength k, at a displacement x is therefore:
Δx
= θ = B/ (Bρ) = B
x/ (Bρ) = kx.
(2.11)
The use of x to indicate the divergence angle of a trajectory is defined in Fig.
2.5. Compare this with a converging lens in optics:
Δx
= −x/f
(2.12)
and we see that the focal length of a horizontally focusing quadrupole is
f = −1/ (kk)
(2.13)
The particular quadrupole shown in Fig. 2.7 would focus positive particles
coming out of the paper or negative particles going into the paper in the horizontal
plane. A closer examination reveals that such a quadrupole deflects particles with
a vertical displacement away from the axis—vertical displacements are defocused.
This can be seen if Fig. 2.7 is rotated through 90 ◦ .
2.1.8 The Equation of Motion
Earlier we derived an expression for the change in divergence of a particle passing
through the quadrupole. A horizontally focusing quadrupole (which is at the same
time vertically defocusing) has a negative k.
We first look at the vertical plane. The angular deflection given to a particle
passing through a short quadrupole of length ds and strength k at a displacement
z is therefore:
dz
= −kzds.
(2.14)
From this we can deduce a differential equation for the motion
z
+ k(s)z = 0.
(2.15)
Here we would like to make a clear statement: While inside a lattice element,
say a quadrupole lens, the normalised gradient k is constant and we get a equation
that we know from Hook’s law in classical mechanics, (see Eq. 2.12), the situation
now is more general. We allow k(s) to change, while our particles are running
through the accelerator. The corresponding equation (2.15) is called Hill’s Equation,
a second order linear equation with a periodic coefficient, k(s) which describes the
distribution of focusing strength around the ring. The above form of Hill’s equation