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Ref. p. 3] 6.5 running head 1 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 𝛼𝑐 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 𝛼𝑐 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. Lando lt -Börnst ein New Ser ies I/21C 2 running head [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 Lando lt -Börnst ein New Ser ies I/21C Ref. p. 3] running head 3 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 Lando lt -Börnst ein New Ser ies I/21C