256
An Introduction to Beam Physics
first order phase slippage factor defined in eq. (10.3). The quantity μ t is
called the synchrotron tune which is proportional to the square root of the
harmonic number, the accelerating voltage and the slippage factor. Usually
the slippage factor is expressed in terms of Δp/p 0 , as defined in eq. (10.4),
denoted here as η
p
1 . Hence the relation between the two is
η
ph
1 =
γ 0
1 + γ 0
η
p
1 .
Taking into account that μ t is the phase advance per turn, the synchrotron
tune in terms of revolution per second can be written as
ω t =
1
t 0
2πh (qE 0 LT ) η
p
1 sin (φ 0 )
β 2
0 γ 0 mc 2
= ω 0
h (qE 0 LT ) η
p
1 sin (φ 0 )
2πβ 2
0 γ 0 mc 2
,
which is the usual form that appears in most textbooks. For a circular accelerator with GeV level energy, the synchrotron tune ω t is usually between
0.1% and 1% of the revolution frequency ω 0 . The betatron tunes, on the other
hand, are usually between a few to a few hundred times of ω 0 . As a result,
the coupling between the betatron and the synchrotron motions is usually
weak. As an example, let us take a look at the storage ring of the Advanced
Light Source (ALS) at Lawrence Berkeley National Laboratory (LBNL, LBL),
California, USA, which is an electron machine that operates at the energy of
1.9 GeV. The main purpose of the RF cavity is to restore the energy loss due
to synchrotron radiation from bending magnets and insertion devices, which
is on the order of 0.5 MeV per turn per electron. The harmonic number is
328 and the slippage factor is roughly 1.4 × 10
−3 . As a result, we have
ω t
ω 0
=
328 × 1.4 × 10 −3 × 0.5
2π × 3718 × 0.511
= 4.4 × 10
−3 .
Using eqs. (8.1), we obtain
α t = −
qE 0 LT
2K 0 sin μ t
ω
κ
(l|δ) sin (φ 0 ) = 1
1
2
μ t ,
β t =
(l|δ)
sin μ t
= 1
(l|δ)
μ t
,
γ t = −
qE 0 LT
K 0 sin μ t
ω
κ
sin (φ 0 ) = 1
μ t
(l|δ)
.
Since μ t 1, we have α t 1. Consequently, the invariant ellipse is basically
upright. For a given longitudinal emittance t , the maximum bunch length is
l max = −2
v 0
κ
β t t = 1 −2
v 0
κ
(l|δ) t
μ t
,
An Introduction to Beam Physics
first order phase slippage factor defined in eq. (10.3). The quantity μ t is
called the synchrotron tune which is proportional to the square root of the
harmonic number, the accelerating voltage and the slippage factor. Usually
the slippage factor is expressed in terms of Δp/p 0 , as defined in eq. (10.4),
denoted here as η
p
1 . Hence the relation between the two is
η
ph
1 =
γ 0
1 + γ 0
η
p
1 .
Taking into account that μ t is the phase advance per turn, the synchrotron
tune in terms of revolution per second can be written as
ω t =
1
t 0
2πh (qE 0 LT ) η
p
1 sin (φ 0 )
β 2
0 γ 0 mc 2
= ω 0
h (qE 0 LT ) η
p
1 sin (φ 0 )
2πβ 2
0 γ 0 mc 2
,
which is the usual form that appears in most textbooks. For a circular accelerator with GeV level energy, the synchrotron tune ω t is usually between
0.1% and 1% of the revolution frequency ω 0 . The betatron tunes, on the other
hand, are usually between a few to a few hundred times of ω 0 . As a result,
the coupling between the betatron and the synchrotron motions is usually
weak. As an example, let us take a look at the storage ring of the Advanced
Light Source (ALS) at Lawrence Berkeley National Laboratory (LBNL, LBL),
California, USA, which is an electron machine that operates at the energy of
1.9 GeV. The main purpose of the RF cavity is to restore the energy loss due
to synchrotron radiation from bending magnets and insertion devices, which
is on the order of 0.5 MeV per turn per electron. The harmonic number is
328 and the slippage factor is roughly 1.4 × 10
−3 . As a result, we have
ω t
ω 0
=
328 × 1.4 × 10 −3 × 0.5
2π × 3718 × 0.511
= 4.4 × 10
−3 .
Using eqs. (8.1), we obtain
α t = −
qE 0 LT
2K 0 sin μ t
ω
κ
(l|δ) sin (φ 0 ) = 1
1
2
μ t ,
β t =
(l|δ)
sin μ t
= 1
(l|δ)
μ t
,
γ t = −
qE 0 LT
K 0 sin μ t
ω
κ
sin (φ 0 ) = 1
μ t
(l|δ)
.
Since μ t 1, we have α t 1. Consequently, the invariant ellipse is basically
upright. For a given longitudinal emittance t , the maximum bunch length is
l max = −2
v 0
κ
β t t = 1 −2
v 0
κ
(l|δ) t
μ t
,