Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 1

Lecture 14

ACCELERATOR PHYSICS

MT 2004

E. J. N. Wilson

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 2

Recap of previous lecture - Waves Buckets and Cavities

WAVES Two waves interfering (scissors)

(2waves.avi) Transverse magnetic mode in guide (wavegE.avi) MORE BUCKETS Small beta superconducting cavities (example RIA,

Argonne) RF frequency (scaling) Rf frequency (injection) RF bucket in collision Bunch rotation (LHCBunchRotationPresentation.AVI) CAVITIES Multicell travellilng wave (electrons) Fixed frequency electron cavity 50 MHz Ferrite tuned (FNAL) Low energy cavity tuned with ferrite for SSC Perspective view of ferrite loaded cavity for SSC PS 19 MHz cavity (prototype, photo: 1966) Ferrite cavity Gap of PS cavity (prototype) Examples of cavities CERN/PS 40 MHz cavity (for LHC) Iris loaded waveguide

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 3

Lecture 14 - Electrons I -contents

Synchrotron radiation Electrons in circular motion Retarded Potential Energy loss per turn Consequences of Radiation Loss Dipole radiation emission pattern Tangential observer’s view The spectrum Rate of emission of quanta Virtues of synchrotron radiation

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 4

Electromagnetic Radiation

Electrons accelerating by running up and down in a radio antenna emit radio waves

Radio waves are nothing more than Long Wavelength Light-

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 5

Particles radiate when v close to c and Happens at a few GeV for electrons but at a few TeV for protons because Really bremsstrahlung - power proportional to deceleration

In a synchrotron force is radial and there is an extra from the Lorentz transformation.

See below which explains this more rigorously and arrives at

Synchrotron radiation

1

m p 2000me

2

Force F

42

2

0

43

22

0 61

61

ce

cfeP

22 or forcepower Radiated z

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 6

Synchrotron radiation II From last page

For motion in a circleHence

Remember magnetic rigidity

Substitute for to obtain

(Equ. 14)

Remember too Finally a formula to remember

P

160

e2c2 4

where: re

e2

40m0c2

E / m0c2

P

160

e4

m4c5 B2E2

B p / e E / ce

P23

recmoc

2 3E4

2

3

22

061

czeP

// 22 cvz

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 7

Electrons in circular motion

Electrons in circular motion are also undergoing acceleration

When electrons are moving slowly the radiation comes out in all directions

When the electron speed gets close to the speed of light, the radiation comes out only in a narrow forward cone; a laser-like concentrated stream

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 8

Retarded Potential

Magnetic vector potential at point 1 from 2

But it was emitted at time t’ but observed when charge has moved at time t

A(1,t)

j 2, t r12 / c 40r12c

2 dv

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 9

Retarded Potential – Magnetic field

Assume blob is a point

Remember

Giving

and

Long range term

zqpdvjz

rp

ct

204

1

A

)/(11

41

41

20

20

crtpyrry

pcr

prc

Bx

)/( cvtpp

czq

rrzq

ccp

rrp

cBx

14

114

122

022

0

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 10

Retarded Potential – Magnetic field

Similarly we have for the electric field

Giving the power emitted over a sphere is

Using the rule for Lorenz transformation of transverse acceleration

czq

rEy

14

1

0

23

2

061 zceP

43

22

061

cfeP

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 11

Energy loss per turn

From last page we have a formula that tells us the power consumption (per particle)

We are very interested in how much energy is lost in the time it takes of a particle to circulate

Substitution gives this “energy loss per turn”:

2 / c

U0 43

rem0c

2 3E4

P23

recmoc

2 3E4

2

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 12

Consequences of Radiation Loss

The last line show how a future electron machine would lose half its energy each turn.

Table 1 - Synchrotron radiation from large synchrotrons.

Uo P

LEP at 50 GeV 220 MeV/turn 1.6 MW + 14 MW (ohmic)

LEP at 100 GeV 3.5 GeV/turn 16 MW + 224 MW (ohmic)

500 Gev in 250 km ring 220 GeV/turn 100 MW

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 13

Dipole radiation emission pattern

(a) circular particle motion(b) the trajectory and dipole radiation emission pattern as seen in a frame moving with the average electron speed c(c) the corresponding radiation pattern transformed into the lab. frame

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 14

Tangential observer’s view

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 15

The spectrum

Spectrum is broad and looks the same when normalised to

Every quantity is normalised to the frequency of a

characteristic quantum which is proportional to

uc h c

32

hc3

uc

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Rate of emission of quanta

If every quanta had the characteristic value

And the power emitted is

Then the rate of emission would be:

When the average over the spectrum is properly integrated:

uc h c

32

hc3

N

15 38Puc

Puc

P23

recmoc

2 3E4

2

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 17

Virtues of synchrotron radiation

• high intensity of photon flux;• continuous spectrum covering a broad range

from the far infra-red to hard X-rays;• small vertical angular divergence;• small source size, determined mainly by the

electron beam dimensions;• high "brightness" and hence high partial

coherence, resulting from the combination ofsmall source size and divergence;• polarization - linear in the orbit plane, with a

circular component above and below theorbit plane;• pulsed time structure, determined by that of

the electron beam;• calculable spectral intensity, allowing use as

a calibrated source.

Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 18

Electrons I – Summary

Synchrotron radiation Electrons in circular motion Retarded Potential Energy loss per turn Consequences of Radiation Loss Dipole radiation emission pattern Tangential observer’s view The spectrum Rate of emission of quanta Virtues of synchrotron radiation