6 Design and Principles of Synchrotrons and Circular Colliders
243
lost. The energy lost per turn is normally a small fraction of the total particle energy,
typically of the order of one part per thousand.
Transverse Oscillations
Since the radiation is emitted along the tangent to the trajectory, only the amplitude
of the momentum changes. As the RF cavities increase the longitudinal component
of the momentum only, the transverse component is damped exponentially with the
damping rate of the order of U 0 per revolution time. A typical transverse damping
time corresponds simply to the number of turns it would take to lose the amount of
energy equal to the particle energy. The damping times are very fast, in case of a
few GeV electron ring being on the order of a few milliseconds.
A ⊥ = A 0 e
−
t
τ , where
1
τ
=
U 0
2ET 0
.
(6.52)
In a given storage ring the damping time is inversely proportional to the cube of
the particle energy.
Longitudinal or Synchrotron Oscillations
Synchrotron oscillations are damped because the energy loss per turn is a quadratic
function of the particle’s energy. The damping rate is typically twice the rate for
transverse oscillations.
Damping Partition Numbers and Robinson Theorem
For particles that emit synchrotron radiation the dynamics is characterized by
the damping of particle oscillations in all three degrees of freedom. In fact, the
total amount of damping (Robinson theorem [49]), i.e. the sum of the damping
decrements depends only on the particle energy and the emitted synchrotron
radiation power:
1
τ x
+
1
τ y
+
1
τ ε
=
2U 0
ET 0
=
U 0
2ET 0
J x + J y + J ε
(6.53)
where we have introduced the usual notation of damping partition numbers that
show how the total amount of damping in the system is distributed among the three
degrees of freedom. A typical set of the damping partition numbers is (1,1,2) and
their sum is, according to the Robinson theorem, a constant.
J x + J y + J ε = 4.
(6.54)
Adjustment of Damping Rates
The partition numbers can differ from the above values, while their sum remains
a constant. In fact, under certain circumstances, the motion can become “antidamped”, i.e. the damping time can become negative, leading to an exponential
growth of the oscillations amplitudes. From a more detailed analysis of damping
243
lost. The energy lost per turn is normally a small fraction of the total particle energy,
typically of the order of one part per thousand.
Transverse Oscillations
Since the radiation is emitted along the tangent to the trajectory, only the amplitude
of the momentum changes. As the RF cavities increase the longitudinal component
of the momentum only, the transverse component is damped exponentially with the
damping rate of the order of U 0 per revolution time. A typical transverse damping
time corresponds simply to the number of turns it would take to lose the amount of
energy equal to the particle energy. The damping times are very fast, in case of a
few GeV electron ring being on the order of a few milliseconds.
A ⊥ = A 0 e
−
t
τ , where
1
τ
=
U 0
2ET 0
.
(6.52)
In a given storage ring the damping time is inversely proportional to the cube of
the particle energy.
Longitudinal or Synchrotron Oscillations
Synchrotron oscillations are damped because the energy loss per turn is a quadratic
function of the particle’s energy. The damping rate is typically twice the rate for
transverse oscillations.
Damping Partition Numbers and Robinson Theorem
For particles that emit synchrotron radiation the dynamics is characterized by
the damping of particle oscillations in all three degrees of freedom. In fact, the
total amount of damping (Robinson theorem [49]), i.e. the sum of the damping
decrements depends only on the particle energy and the emitted synchrotron
radiation power:
1
τ x
+
1
τ y
+
1
τ ε
=
2U 0
ET 0
=
U 0
2ET 0
J x + J y + J ε
(6.53)
where we have introduced the usual notation of damping partition numbers that
show how the total amount of damping in the system is distributed among the three
degrees of freedom. A typical set of the damping partition numbers is (1,1,2) and
their sum is, according to the Robinson theorem, a constant.
J x + J y + J ε = 4.
(6.54)
Adjustment of Damping Rates
The partition numbers can differ from the above values, while their sum remains
a constant. In fact, under certain circumstances, the motion can become “antidamped”, i.e. the damping time can become negative, leading to an exponential
growth of the oscillations amplitudes. From a more detailed analysis of damping