Synchrotron Radiation

Eric Prebys, FNAL

Synchrotron Radiation

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 2

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

In a magnetic field

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

Effects of Synchrotron Radiation Two competing effects

Damping

Quantum effects related to the statistics of the photons

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 3

damping time

period

energy

energy lost per turn

Number of photons per

periodRate of photon emission

Average photon energy

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 4

The power spectrum of radiation is

“critical energy”

Calculate the photon rate per unit energy

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 5

The total rate is:

The mean photon energy is then

The mean square of the photon energy is

The energy lost per turn is

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 6

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 electrons For CESR

photons/turn

Small Amplitude Longitudinal Motion

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 7

amplitude of energy oscillation

If we radiate a photon of energy u, then

damping term Heating term

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 8

Evaluate integral in damping term

Recall Dependence of field

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 9

Putting it all together…

use

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 10

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

damping

heating

The energy then decays in a time

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 11

In a separated function lattice, there is no bend in the quads, soFurther assume uniform dipole field (ρ=ρ0)

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 12

Equilibrium energy spread will be

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

Behavior of beams

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 13

We’re going to derive two important results1. Robinson’s Theorem

For a separated function lattice

2. The equilibrium horizontal emittance

transverse damping times

photons emitted in a damping period

Mean dispersion

Here we go…

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 14

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, thenEnergy lost along the path

Energy restored along nominal path ”adiabatic damping”

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 15

Recall

A

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 16

Calculating beam size

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)

Horizontal Plane

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 17

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

n 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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 18

Going back to the motion in phase space

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 19

Averaging over one turn

Average over all

particle and phases

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 20

As before

Same procedure as p. 9

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 21

Going back…

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

As before…

Robinson’s Theorem

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 22

Equilibrium emittance use

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 23

For a separated function, isomagnetic machine

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 13 - Synchrotron Radiation 24

Approximate