Download 6.5 Synchrotron radiation and damping

Ref. p. 3]
6.5
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Synchrotron radiation and damping
L. Rivkin
EPFL, Lausanne and PSI, Villigen
6.5.1 Basic properties of synchrotron radiation
Charged particles radiate when they are deflected in the magnetic field [1] (transverse acceleration). In
the ultra-relativistic case, when the particle speed is very close to the speed of light,  ≈ c, most of the
radiation is emitted in the forward direction [2] into a cone centred on the tangent to the trajectory and
with an opening angle of 1/, where  is the Lorentz factor (since for a few GeV electron or a few TeV
proton,  ≈ 1000, the photon emission angles are within a milliradian of the tangent to the trajectory).
The power emitted by a particle is proportional to the square of its energy E and to the square of the
deflecting magnetic field B:
𝑃𝑆𝑅 ∝ 𝐸 2 𝐵2
and in terms of Lorentz factor  and the local bending radius  can be written as follows:
2
𝛾4
𝛼𝑐 2 2
3
𝜌
𝑃𝑆𝑅 =
where  is the fine-structure constant and the Plank’s constant is given in a convenient conversion
constant:
1
𝛼=
and 𝑐 = 197 ∙ 10−15 Mev ∙ fm
137
The emitted power is a very steep function of both the particle energy and particle mass, being
proportional to the fourth power of .
Integrating the above expression around the machine we obtain the amount of energy lost per turn:
𝑈0 =
4𝜋
𝛾4
𝛼𝑐
3
𝜌
The emitted radiation spectrum consists of harmonics of the revolution frequency and peaks near the socalled critical frequency or critical photon energy. It is defined such that exactly half of the radiated
power is emitted below it
𝜀𝑐 =
2
𝛾3
𝑐
3
𝜌
On the average a particle then emits 𝑛𝑐 ≈ 2𝜋𝛼𝛾 photons per turn.
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[Ref. p. 3
6.5.2 Radiation damping
In a storage ring the steady loss of energy to synchrotron radiation is compensated in the RF cavities,
where the particle receives each turn the average amount of energy lost. The energy lost per turn is a
small fraction of the total particle energy, typically on 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 on the order of
𝑈0 per revolution time. 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.
𝑡
𝐴 = 𝐴0 𝑒 −𝜏
where
1
𝜏
=
𝑈0
2𝐸𝑇0
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 due to the fact that 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 [3]), i.e. the sum of the damping decrements depends
only on the particle energy and the emitted synchrotron radiation power:
1
1
1 2𝑈0
𝑈0
+ + =
=
(𝐽 + 𝐽𝑦 + 𝐽𝜀 )
𝜏𝑥 𝜏𝑦 𝜏𝜀 𝐸𝑇0 2𝐸𝑇0 𝑥
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. Typical set of the
damping partition numbers is (1,1,2) and their sum is, according to the Robinson theorem, a constant.
𝐽𝑥 + 𝐽𝑦 + 𝐽𝜀 = 4
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 “anti-damped”, i.e. the
damping time can become negative, leading to an exponential growth of the oscillations amplitudes. From
a more detailed analysis of damping rates [4] the damping time can be written as
𝐷
1
∮ 𝜌 (2𝑘 + 2 ) 𝑑𝑠
1
𝑈0
1
𝑈0
𝜌
(2 + ) and
(1 − ) 𝑤ℎ𝑒𝑟𝑒  ≡
=
=
𝑑𝑠
𝜏𝜀 2𝐸𝑇0
𝜏𝑥
2𝐸𝑇0
∮ 2
𝜌
The constant introduced above is an integral of the dispersion function D and the magnetic guide field
functions, i.e. bending radius and gradient around the ring and is independent of the particle energy. It
deviates substantially from zero only when a particle encounters combined function elements, i.e. where
the product of the field gradient and the curvature is non-zero. The damping partition numbers then are:
𝐽𝑥 = 1 −  ,
𝐽𝜀 = 2 + ,
𝐽𝑥 + 𝐽𝜀 = 3
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The vertical damping partition number is usually unchanged as the vertical dispersion is zero in storage
rings that are built in one (horizontal) plane.
The amount of damping can be repartitioned between the horizontal and energy-time oscillations by
altering the value of the D constant [4]. This can be achieved by either using combined function magnetic
elements in the lattice, or by introducing a special combined function wiggler magnet (so-called Robinson
wiggler). Values of horizontal partition number as high as 2.5 have been obtained that way. Values of D
> 1 lead to anti-damping of horizontal betatron oscillations, while for D < -2 the synchrotron oscillations
become unstable.
References
[1]
J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, 1998
[2]
A. Hofmann, The Physics of Synchrotron Radiation, Cambridge, 2004
[3]
K. W. Robinson, Phys. Rev. 111 (1958) p. 373
[4]
H. Wiedemann, Particle Accelerator Physics, Springer, 2007
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