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
Adams Accelerator Institute 10 - E. Wilson - 05/03/23 - Slide 16
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