Basics of beam dynamics 25
radiation or coupling impedances. The energy of the synchronous particle on
successive turns satisfies
E s,n+1 = E s,n + eV n sin φ s − U 0,n ,
(1.88)
where subscripts n, n + 1 are turn numbers, V is the RF voltage, and U 0
contains all other energy losses. Similarly, the energy of a given particle over
the two turns satisfies
E n+1 = E n + eV n sin φ n − U 0,n ,
(1.89)
where φ n = φ s +ω rf τ n , with the arrival time error τ = T −T s and T s the arrival
time of the synchronous particle. Therefore, the energy errors, ∆E = E − E s ,
of this particle over two successive turns are related through
∆E n+1 = ∆E n + eV n [sin(φ s + ω rf,n τ n ) − sin φ s ].
(1.90)
On the other hand, the arrival time errors of the particle on the two successive turns are related through
τ n+1 = τ n + ∆T n+1 = τ n + ηδT 0,n+1 ,
where ∆T n+1 is the arrival time error accumulated over turn n + 1. Using
ω rf T 0 = 2πh and δ =
1
β 2
∆E
E , the above equation can be rewritten in terms of
the energy error coordinate,
τ n+1 = τ n +
2πhη(E s,n+1 )
ω rf,n+1 β 2
s,n+1 E s,n+1
∆E n+1 .
(1.91)
Eqs. (1.90) and (1.91) are the two mapping equations that describe the
longitudinal motion of particles in a synchrotron in (τ , ∆E) coordinates. Given
the ramping curves of the reference beam energy, the RF voltage, and the
RF frequency, the trajectory of a particle in the (τ , ∆E) phase space can
be determined by applying the equations over successive turns. Eq. (1.90)
describes the energy change at the RF cavity, and Eq. (1.91) describes the
arrival time change accumulated throughout the ring. They can be seen as
two maps that are applied sequentially. In each revolution one first calculates
the energy change at the cavity, followed by the calculation of the arrival time
change in the ring. Both maps are kick maps, i.e., maps in which an increment
is made only to one of the two conjugate coordinates. The determinant of the
Jacobian matrix of a kick map is unity, which means that the phase space
area of a beam distribution is preserved as it evolves with time.
The stability of the particle motion around the reference particle can be
analyzed after linearizing Eq. (1.90) with respect to the τ coordinate, which
becomes
∆E n+1 = ∆E n + τ n ω rf,n eV n cos φ s .
(1.92)
radiation or coupling impedances. The energy of the synchronous particle on
successive turns satisfies
E s,n+1 = E s,n + eV n sin φ s − U 0,n ,
(1.88)
where subscripts n, n + 1 are turn numbers, V is the RF voltage, and U 0
contains all other energy losses. Similarly, the energy of a given particle over
the two turns satisfies
E n+1 = E n + eV n sin φ n − U 0,n ,
(1.89)
where φ n = φ s +ω rf τ n , with the arrival time error τ = T −T s and T s the arrival
time of the synchronous particle. Therefore, the energy errors, ∆E = E − E s ,
of this particle over two successive turns are related through
∆E n+1 = ∆E n + eV n [sin(φ s + ω rf,n τ n ) − sin φ s ].
(1.90)
On the other hand, the arrival time errors of the particle on the two successive turns are related through
τ n+1 = τ n + ∆T n+1 = τ n + ηδT 0,n+1 ,
where ∆T n+1 is the arrival time error accumulated over turn n + 1. Using
ω rf T 0 = 2πh and δ =
1
β 2
∆E
E , the above equation can be rewritten in terms of
the energy error coordinate,
τ n+1 = τ n +
2πhη(E s,n+1 )
ω rf,n+1 β 2
s,n+1 E s,n+1
∆E n+1 .
(1.91)
Eqs. (1.90) and (1.91) are the two mapping equations that describe the
longitudinal motion of particles in a synchrotron in (τ , ∆E) coordinates. Given
the ramping curves of the reference beam energy, the RF voltage, and the
RF frequency, the trajectory of a particle in the (τ , ∆E) phase space can
be determined by applying the equations over successive turns. Eq. (1.90)
describes the energy change at the RF cavity, and Eq. (1.91) describes the
arrival time change accumulated throughout the ring. They can be seen as
two maps that are applied sequentially. In each revolution one first calculates
the energy change at the cavity, followed by the calculation of the arrival time
change in the ring. Both maps are kick maps, i.e., maps in which an increment
is made only to one of the two conjugate coordinates. The determinant of the
Jacobian matrix of a kick map is unity, which means that the phase space
area of a beam distribution is preserved as it evolves with time.
The stability of the particle motion around the reference particle can be
analyzed after linearizing Eq. (1.90) with respect to the τ coordinate, which
becomes
∆E n+1 = ∆E n + τ n ω rf,n eV n cos φ s .
(1.92)