34
An Introduction to Beam Physics
mines how widely the resulting derivations are compatible with general concepts in physics and mathematics. Below, the quantities with the subscript 0
are meant to indicate the reference particle.
Before launching the motion, we denote the energy deviation of a particular
particle of interest by δ by defining
δ =
K − K 0
K 0
,
where K is the initial kinetic energy of the particle under consideration while
K 0 is that of the reference particle. Finally, we introduce a space-like variable
l
l = κ (t − t 0 )
being the deviation of the time-of-flight t from that of the reference particle,
multiplied by a constant κ that has the dimension of velocity. Specifically,
κ = −v 0
γ 0
1 + γ 0
,
(2.1)
using the absolute value of the velocity of the reference particle v 0 and the
associated γ 0 , which can be expressed as
γ 0 =
1
1 − v 2
0 /c 2
=
( p 0 c) 2 + (mc 2 ) 2
mc 2
=
E 0
mc 2 ,
by referring to eq. (1.6). The specific form of κ, especially the fractional factor
involving γ 0 , is important for generating what turns out to be a canonical pair
of coordinates (l, δ); the details go beyond the scope of this book, and we refer
to [5] for details.
Then we form the vector
Z of particle optical coordinates as
Z =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x
a = p x /p 0
y
b = p y /p 0
l = κ (t − t 0 )
δ = (K − K 0 ) /K 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(2.2)
where p 0 is some previously chosen scaling momentum; a natural choice is
to select the momentum of the reference particle at the beginning. Likewise,
K 0 is a previously chosen scaling energy, for example the kinetic energy of
the reference particle, and similarly, κ is a scaling quantity introduced in eq.
(2.1).
Note that due to the definition of
Z, the reference particle itself corresponds
to
Z = 0, and hence the vector
Z does indeed describe the relative motion.
In a seemingly simple way, most of the problems of beam physics now revolve
around the question as to how
Z evolves as a function of s.
An Introduction to Beam Physics
mines how widely the resulting derivations are compatible with general concepts in physics and mathematics. Below, the quantities with the subscript 0
are meant to indicate the reference particle.
Before launching the motion, we denote the energy deviation of a particular
particle of interest by δ by defining
δ =
K − K 0
K 0
,
where K is the initial kinetic energy of the particle under consideration while
K 0 is that of the reference particle. Finally, we introduce a space-like variable
l
l = κ (t − t 0 )
being the deviation of the time-of-flight t from that of the reference particle,
multiplied by a constant κ that has the dimension of velocity. Specifically,
κ = −v 0
γ 0
1 + γ 0
,
(2.1)
using the absolute value of the velocity of the reference particle v 0 and the
associated γ 0 , which can be expressed as
γ 0 =
1
1 − v 2
0 /c 2
=
( p 0 c) 2 + (mc 2 ) 2
mc 2
=
E 0
mc 2 ,
by referring to eq. (1.6). The specific form of κ, especially the fractional factor
involving γ 0 , is important for generating what turns out to be a canonical pair
of coordinates (l, δ); the details go beyond the scope of this book, and we refer
to [5] for details.
Then we form the vector
Z of particle optical coordinates as
Z =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x
a = p x /p 0
y
b = p y /p 0
l = κ (t − t 0 )
δ = (K − K 0 ) /K 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(2.2)
where p 0 is some previously chosen scaling momentum; a natural choice is
to select the momentum of the reference particle at the beginning. Likewise,
K 0 is a previously chosen scaling energy, for example the kinetic energy of
the reference particle, and similarly, κ is a scaling quantity introduced in eq.
(2.1).
Note that due to the definition of
Z, the reference particle itself corresponds
to
Z = 0, and hence the vector
Z does indeed describe the relative motion.
In a seemingly simple way, most of the problems of beam physics now revolve
around the question as to how
Z evolves as a function of s.