Accelerator Physics Homework #7
P470 (Problems: 1-4)
1. This exercise derives the linear transfer matrix for a skew quadrupole, where the magnetic eld is
Bz = B0 a1 z,
Bx = B0 a1 x,
Bs = 0;
with B0 a1 =
1
2
Bx Bz
x
z
,
x=z =0
where B0 is the main dipole

Accelerator Physics Homework #5
P470 (Problems: 1-5)
1. In a lossless transverse electromagnetic (TEM) wave transmission line, the equation
for the current and voltage is
V
I
= L ,
s
t
I
V
= C
,
s
t
where L and C are respectively the inductance and capaci

Solution:
1. In thin length approximation, 2R = 2N L, L = , =
c =
2
2N
1
2
DD
1
2N L2
DF
=
=
N
L+
L=
2R
2R sin2
sin2
2
2
2
2N sin
2
2
Similarly, we nd c = 2 /6R for DBA lattices.
2. The energy gain for a non-synchronous particle is
t
t
e
eVg
2
2
E =
V

Solution:
1. Floquet transformation:
(a) Dening a new coordinate = y/ and = (1/ )
and
s
0
ds/ , we nd ds/d = ,
d
1
ds d
1
1
=
= y 3/2 y = 1/2 y 1/2 y ,
d
d ds
2
2
d2
1
1
2
= 2 1/2 y 1/2 y 3/2 y .
2
d
2
4
Thus the equation of motion becomes
B
d2
+ 2 = 2

Solution:
1. Without loss of generality, we use the Frenet-Serret coordinate system of Fig. 2.1 and
derive equation of motion for positively charged ions in the accelerator. It is easy to
modify the equations of motion for electrons.
(a) The coordinate of

Solution:
1. Lorentz force provides the central force of circular motion:
F = qv B =
mv 2
B =
mv
p
=
,
Ze
Ze
where Z is the charge number of the particle. Using 1 [GeV/c]=109 [eV]/(2.9979
108 [m/s])=3.3357 e [kg m/s], we obtain
p
A
3.3357p [GeV/c]
= 3.33

Answer to the Midterm exam
1. In (, E ) space, the phase-space area A is proportional to a factor F = /| |. When
0
< T , F increases monotonously with , and when > T , the factor F has a
minimum at = 3T .
1
1
2
2
T
1
dF
=2
d
When =
2
3
3T , dF/d = 0 and