Radiation Damping
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Radiation damping in
accelerator physics Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
is a phenomenon where
betatron oscillations Betatron oscillations are the fast transverse oscillations of a charged particle in various focusing systems: linear accelerators, storage rings, transfer channels. Oscillations are usually considered as a small deviations from the ideal reference ...
and longitudinal oscillations of the particle are damped due to energy loss by
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in some types ...
. It can be used to reduce the
beam emittance In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space. Each particle in a beam can be described by its position and momentum along each of t ...
of a high-velocity
charged particle beam A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The kinetic energies of the particles are much lar ...
. The two main ways of using radiation damping to reduce the emittance of a particle beam are the use of ''undulators'' and ''damping rings'' (often containing undulators), both relying on the same principle of inducing
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in some types ...
to reduce the particles' momentum, then replacing the momentum only in the desired direction of motion.


Damping rings

As particles are moving in a closed orbit, the lateral acceleration causes them to emit
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in some types ...
, thereby reducing the size of their momentum vectors (relative to the design orbit) without changing their orientation (ignoring the quantum fluctuations of the radiation for the moment). In longitudinal direction, the loss of particle impulse due to radiation is replaced by accelerating sections ( RF cavities) that are installed in the beam path so that an
equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...
is reached at the design energy of the accelerator. Since this is not happening in transverse direction, where the emittance of the beam is only increased by the quantization of radiation losses (quantum effects), the transverse equilibrium emittance of the particle beam will be smaller with large radiation losses, compared to small radiation losses. Because high orbit
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
s (low curvature radii) increase the emission of synchrotron radiation, damping rings are often small. If long beams with many particle bunches are needed to fill a larger
storage ring A storage ring is a type of circular particle accelerator in which a continuous or pulsed particle beam may be kept circulating, typically for many hours. Storage of a particular particle depends upon the mass, momentum, and usually the charge o ...
, the damping ring may be extended with long straight sections.


Undulators and wigglers

When faster damping is required than can be provided by the turns inherent in a damping ring, it is common to add
undulator An undulator is an insertion device from high-energy physics and usually part of a larger installation, a synchrotron storage ring, or it may be a component of a free electron laser. It consists of a periodic structure of dipole magnets. These ca ...
or wiggler magnets to induce more synchrotron radiation. These are devices with periodic magnetic fields that cause the particles to oscillate transversely, equivalent to many small tight turns. These operate using the same principle as damping rings and this oscillation causes the charged particles to emit synchrotron radiation. The many small turns in an undulator have the advantage that the cone of synchrotron radiation is all in one direction, forward. This is easier to shield than the broad fan produced by a large turn.


Energy loss

The power radiated by a charged particle is given by a generalization of the Larmor formula derived by LiƩnard in 1898 : P = \frac \gamma^6\left \dot)^2 - \frac\right/math>, where v=\beta c is the velocity of the particle, \dot v = \frac the acceleration, e the
elementary charge The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
, \epsilon_0 the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
,\gamma the
Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
and c the speed of light. Note: : p = \gamma m_0 v is the
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
and m_0 is the mass of the particle. : \frac=\gamma^3\frac : \frac = \gamma^3 \frac m_0 v + \gamma m_0 \dot v


Linac and RF Cavities

In case of an acceleration parallel to the longitudinal axis ( \times \dot = 0 ), the radiated power can be calculated as below : \frac = \gamma^3 m_0 \dot v Inserting in Larmor's formula gives : P_ = \frac\left(\frac\right)^2


Bending

In case of an acceleration perpendicular to the longitudinal axis ( . \dot = 0 ) : \frac = \gamma m_0 \dot v Inserting in Larmor's formula gives (''Hint: Factor 1/(\gamma m_0)^2 and use 1-v^2/c^2 = 1/\gamma^2'') : P_ = \frac\left( \frac \right)^2 Using magnetic field perpendicular to velocity : F_ = \frac = e v \times B : P_ = \frac\left(e \beta c B\right)^2 = \frac = \frac\beta^2 E^2 B^2 Using
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius ...
\dot v=\frac and inserting \gamma m_0 \dot v in P_ gives : P_=\frac\frac


Electron

Here are some useful formulas to calculate the power radiated by an electron accelerated by a magnetic field perpendicular to the velocity and \beta \approx 1. : P_=\fracE^2B^2 where E=\gamma m_e c^2 , B is the perpendicular magnetic field, m_e the electron mass. : P_=\frac\frac Using the
classical electron radius The classical electron radius is a combination of fundamental Physical quantity, physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic sel ...
r_e : P_=\frac\frac\frac=\frac\frac\frac=\frac r_e c \frac=\frac r_e m_e c^3 \frac where \rho is the
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius ...
,\rho=\frac \rho can also be derived from particle coordinates (using common 6D phase space coordinates system x,x',y,y',s,\Delta p/p_0): : \rho = \left, \frac \ \approx \frac Note: The transverse magnetic field is often normalized using the magnet rigidity: B\rho = \fracE_ \approx 3.3356 E_ m/math> Field expansion (using
Laurent_series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
): \frac = \sum_^ (ia_n + b_n) (x+iy)^n where (b_x,b_y) is the transverse field expressed in (a_n,b_n) the multipole field strengths (skew and normal) expressed in ^/math>, (x,y) the particle position and k the multipole order, k=0 for a dipole,k=1 for a quadrupole,k=2 for a sextupole, etc...


See also

* Particle beam cooling


References


External links


SLAC damping rings home page
including

of the damping rings at
SLAC SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center, is a federally funded research and development center in Menlo Park, California, United States. Founded in 1962, the laboratory is now sponsored ...
.
Studies Pertaining to a Small Damping Ring for the International Linear Collider
a report describing the constraints on minimum damping ring size. Accelerator physics Synchrotron radiation {{Accelerator-stub