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### Synchrotron Radiation

Eric Prebys, FNAL

Synchrotron Radiation

For a relativistic particle, the total radiated power (S&E 8.1) is

For a fixed energy and geometry, power goes as the inverse fourth power of the mass!

In a magnetic field

Lecture 13 - Synchrotron Radiation

Effects of Synchrotron Radiation

energy

damping time

energy lost per turn

period

Number of photons per period

Rate of photon emission

Average photon energy

Lecture 13 - Synchrotron Radiation

- Two competing effects
- Damping
- Quantum effects related to the statistics of the photons

The power spectrum of radiation is

“critical energy”

Calculate the photon rate per unit energy

Lecture 13 - Synchrotron Radiation

The mean photon energy is then

The mean square of the photon energy is

The energy lost per turn is

Lecture 13 - Synchrotron Radiation

It’s important to remember that ρ is not the curvature of the accelerator as a whole, but rather the curvature of individual magnets.

So if an accelerator is built using magnets of a fixed radius ρ0, then the energy lost per turn is

For CESR

For electrons

photons/turn

Lecture 13 - Synchrotron Radiation

Small Amplitude Longitudinal Motion

amplitude of energy oscillation

If we radiate a photon of energy u, then

Heating term

damping term

Lecture 13 - Synchrotron Radiation

Going way back to our original equation (p. 7)

heating

damping

The energy then decays in a time

Lecture 13 - Synchrotron Radiation

In a separated function lattice, there is no bend in the quads, so

Further assume uniform dipole field (ρ=ρ0)

probably the answer you would have guessed without doing any calculations.

Lecture 13 - Synchrotron Radiation

Equilibrium energy spread will be quads, so

- Effects of synchrotron radiation
- Damping in both planes
- Heating in bend plane

Lecture 13 - Synchrotron Radiation

Behavior of beams quads, so

We’re going to derive two important results

Robinson’s TheoremFor a separated function lattice

The equilibrium horizontal emittance

transverse damping times

photons emitted in a damping period

Mean dispersion

Lecture 13 - Synchrotron Radiation

Here we go… quads, so

Synchrotron radiation

Energy lost along trajectory, so radiated power will reduce momentum along flight path

If we assume that the RF system restores the energy lost each turn, then

Energy lost along the path

Energy restored along nominal path ”adiabatic damping”

Lecture 13 - Synchrotron Radiation

Recall quads, so

As we average this over many turns, we must average over all phase angles

Lecture 13 - Synchrotron Radiation

Calculating beam size quads, so

Note, in the absence of any heating terms or emittance exchange, this will damp to a very small value. This is why electron machines typically have flat beams. Allowing it to get too small can cause problems (discussed shortly)

Lecture 13 - Synchrotron Radiation

Horizontal Plane quads, so

Things in the horizontal plane are a bit more complicated because position depends on the energy

betatron motion

where

Now since the radiated photon changes the energy,but not the position or the angle, the betatron orbit must be modified; that is

Lecture 13 - Synchrotron Radiation

Going back to the motion in phase space quads, so

Lecture 13 - Synchrotron Radiation

Averaging over one turn quads, so

Average over all particle and phases

Lecture 13 - Synchrotron Radiation

Going back… quads, so

But remember, we still have the adiabatic damping term from re-acceleration, so

Robinson’s Theorem

As before…

Lecture 13 - Synchrotron Radiation

For a separated function, isomagnetic machine quads, so

Lecture 13 - Synchrotron Radiation

Approximate quads, so

Lecture 13 - Synchrotron Radiation

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