Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro /...
Transcript of Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro /...
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Introduction to Accelerator PhysicsPart 1
Pedro Castro / Accelerator Physics Group (MPY)Zeuthen, 26th July 2018
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DESY CERN
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Applications of Accelerators (1)
Particle colliders for High Energy Physics (HEP) experiments
Fixed target experiments
Two beams collider experiments
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Applications of Accelerators (1)
Particle colliders for High Energy Physics experiments
Example: the Large Hadron Collider (LHC) at CERN
superconducting magnets(inside a cryostat)
built between 2001 and 2009Higgs discovery: July 2012
8.6 km
Mont BlancLake Geneva Geneva
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Applications of Accelerators (2)
B
Light sources for biology, physics, chemistry… experiments
Electromagnet
• structural analysis of crystalline materials• X-ray crystallography (of proteins)• X-ray microscopy• X-ray absorption (or emission) spectroscopy• …
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7 GeV
Example: Positron-Elektron-Tandem-Ring-Anlage (PETRA)‘positron-electron tandem ring accelerator’ at DESY
built between 1976 and 1978HEP experiments: 1978-1986gluon discovery: 1979
40 GeVprotons
12 GeVe-/e+
HERA
protons
pre-accelerator for HERA 1987-2007synchrotron radiation since 1987
0.5 GeVe-/e+2.3 km long
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Example: Positron-Elektron-Tandem-Ring-Anlage (PETRA)‘positron-electron tandem ring accelerator’ at DESY
e-/e+
built between 1976 and 1978HEP experiments: 1978-1986gluon discovery: 1979
pre-accelerator for HERA 1987-2007synchrotron radiation since 1987PETRA III since 2009
Max von Laue hall300 m
2.3 km long
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e-
e-
photons
14 beamlines30 exp. stations
Max von Laue hall
http://photon-science.desy.de/facilities/petra_iii/beamlines/index_eng.html
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Example: Positron-Elektron-Tandem-Ring-Anlage (PETRA)‘positron-electron tandem ring accelerator’ at DESY
e-/e+2.3 km long
hall nord
hall east
built between 1976 and 1978HEP experiments: 1978-1986gluon discovery: 1979
pre-accelerator for HERA 1987-2007synchrotron radiation since 1987PETRA III since 2009
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For radioisotope production
Applications of Accelerators (3)
For radiotherapy and radiosurgery:
• x-rays and gamma-rays
• ions (from protons to atoms with atomic number up to 18, Argon)
• neutrons
proton beam + stable isotopetransmutation
radioactive isotope
Accelerators in medicine
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For example: p
Oxygen-18 + positron
97% of decays
Applications of Accelerators (3)
18 MeV proton accelerator Oxygen-18 (stable)
target
Fluorine-18 (half-life time = 110 min.)
(transmutation)
Accelerators in medicine
For radioisotope production
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For example: p
Fluorine-18 (half-life time = 110 min.)
Applications of Accelerators (3)
18 MeV proton accelerator Oxygen-18
target
Fludeoxyglucose (18F)
(transmutation)
Accelerators in medicine
For radioisotope production
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Applications of Accelerators (3)
Fluorine-18
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Positron EmissionTomography (PET)
Applications of Accelerators (3)
detectors
�
�
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Positron EmissionTomography (PET)
Applications of Accelerators (3)
detectors
�
�
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For industrial applications:
Applications of Accelerators (4)
approx. numbers from 2007 (worldwide)
Application
Ion implantation ~ 9500
Electron cutting and welding ~ 4500
Electron beam and x-ray irradiators ~ 2000
Ion beam analysis (including AMS) ~ 200
Radioisotope production (including PET) ~ 900
Nondestructive testing (including security) ~ 650
Neutron generators (including sealed tubes) ~ 1000
with energies up to 15 MeV
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For industrial applications:
an example: electron beam welding
‘deep welding effect’
up to 15 cm
acceleration up to 60-200 keV
Applications of Accelerators (4)
magnets as‘focusing lenses’as well as‘deflectors’
electron beam
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> About 120 accelerators for research in “nuclear and particle physics”
> About 70 electron storage rings and electron linear accelerators used as light sources (so-called ‘synchrotron radiation sources’)
> More than 7,000 accelerators for medicineradiotherapy (>7,500), radioisotope production (200)
ion implantation (>9,000) , electron cutting and welding (>4,000) …
> More than 18,000 industrial accelerators
Worldwide …
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> About 120 accelerators for research in “nuclear and particle physics”
> About 70 electron storage rings and electron linear accelerators used as light sources (so-called ‘synchrotron radiation sources’)
> More than 7,000 accelerators for medicineradiotherapy (>7,500), radioisotope production (200)
ion implantation (>9,000) , electron cutting and welding (>4,000) …
> More than 18,000 industrial accelerators
< 1%
Worldwide …
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Many millions of television sets, oscilloscopes using CRTs (Cathode Ray Tube)
Applications of Accelerators (5)
TV
oscilloscope
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Many millions of television sets, oscilloscopes using CRTs (Cathode Ray Tube)
Applications of Accelerators (5)
acceleration
magnets as ‘focusing lenses’as well as ‘deflectors’
25 frames / s
625lines
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Applications of Accelerators (6)
X-ray tubes
DC high voltage (20-150 kV)
VACUUM
braking radiationor bremsstrahlung
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Working with accelerators in the control room …
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The case begins…
Accelerator Control RoomHamburg, DESYSat. 12th June 20102 o’clock a.m.PETRA runs with a beam
current of 75 mA
02:24 a.m.: beam lost
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Circular accelerators: the synchrotron
vacuum chamberbending magnet
accelerating device
injector
straight sections
beam
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Dipole magnet
beam
� dipole magnets: tomorrow
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Circular accelerators: the synchrotron
vacuum chamberbending magnet
accelerating device
injector
straight sections
� radiofrequency acceleration: tomorrow
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Circular accelerators: the synchrotron
vacuum chamber
accelerating device
bending magnet
Low Energy Antiproton Ring (LEAR) at CERN (built in 1982)
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. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
B (perpendicular)
R
Circular accelerators: the synchrotron
charge velocity
of the particle
magnetic field
momentum
�� =���
��= �� × �
� ⊥ � → � = ��
(circular motion)
�� ⊥ � → � = ��
�
�� =�
�→ � =
�
��
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vacuum chambermagnet
accelerating device
injector
straight sections
Circular accelerators: the synchrotron
(circular motion)
� ⊥ � → � = ��
�� ⊥ � → � = ��
�
�� =�
�→ � =
�
��= ��������
� increase B synchronouslywith � = � of particle
�
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DESY (Deutsches Elektronen Synchrotron)
DESY: German electron synchrotron
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DESY: German electron synchrotron, 1964, 7.4 GeV
DESY (Deutsches Elektronen Synchrotron)
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back to the case…
Hamburg, DESYSat. 12th June 20102 o’clock a.m.PETRA runs with 75 mA
02:24 a.m.: beam lost
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The Main Accelerator Control Room
beam current [mA]
time
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost
02:24 a.m.: beam lost
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Run number 4: 60 Bunches; 23 rd – 30th March, 2011
magnetpower supply
power off incrate at DORIS
RF??
top-uptiming problems
RF??beam losswithout dump
RF problemsand longitudinalinstabilities
vertical beam blow up
RF circulatorwater cooling
RF??
RF circulatorwater cooling
One example of PETRA run over 7 days
Source: K. Balewski (MAC report 2011)
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http://ttfinfo.desy.de/petra/show.jsp?dir=/2010/23/11.06_n&pos=2010-06-12T02:26:30
The link to the electronic logbook:
Beam lost at 02:24 a.m.
What to do?
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Alarm overview: the Machine Protection System
MPS
control system
status andalarm signals
stop beam
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The Machine Protection System status from 12th June 2010 at 02:26
12th June 2010 02:26
Alarm overview: the Machine Protection System
• vacuum system ok• magnet system ok• radio-frequency ok• water cooling ok…“all systems up and running”
• vacuum system ok• magnet system ok• radio-frequency ok• water cooling ok…“all systems up and running”
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Electrons can be injected but cannot be stored !
beam current at injection
beam current after a few turns
injection problem?
beamcurrent
time
500 µs ≈ 65 turns
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Next suspect: injection
stored beam reference trajectory
vacuum chamber
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Next suspect: injection
injected beam
stored beam
bending magnet
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Next suspect: injection
dipoledipoledipole
stored beam
septumhomogeneous field
free field region
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Next suspect: injection
dipoledipoledipole
stored beam
septumhomogeneous field
free field region
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Next suspect: injection + accumulation
dipoledipoledipole
stored beam
septum
injected beam
PETRAseptum
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septum at the Proton Synchrotron Booster (PSB) at CERN
homogeneous fieldfree field region
Next suspect: injection + accumulation
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septum at the Proton Synchrotron Booster (PSB) at CERN
homogeneous fieldfree field region
beaminjected beamstored beam
Next suspect: injection + accumulation
(1)
(2)
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septum at the Proton Synchrotron Booster (PSB) at CERN
homogeneous fieldfree field region
beaminjected beamstored beam
Next suspect: injection + accumulation
(1)
(2)
(3)
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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stored beam
septum
kicker(very fast dipole)
kicker(very fast dipole)
kicker(very fast dipole)
injected beam
time
kickerfield
~20 µs
Next suspect: injection + accumulation
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Page 49
Electrons can be injected but cannot be stored !
beam current at injection
beam current after a few turns
beamcurrent
time
500 µs ≈ 65 turns
injection problem?
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Next suspect: a problem with vacuum chamber
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in accelerator
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Next suspect: the new octant in ‘Max von Laue hall’
e-
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e-
Next suspect: the new octant in ‘Max von Laue hall’
e-beam
22.5 m
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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e-
undulators
beam
22.5 m
Next suspect: the new octant in ‘Max von Laue hall’
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 54
Undulator PU 10
undulator field lines
Next suspect: the new octant in ‘Max von Laue hall’
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant
�NS
NS
permanentmagnets
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Undulator PU 10
Next suspect: the new octant in ‘Max von Laue hall’
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant
electrons
photons
undulator
undulator field lines
beam�N
S
NS
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Undulator PU 10
very flat undulator vacuum chambers
undulator field lines
7 mm
9 mm
Next suspect: the new octant in ‘Max von Laue hall’
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant
beam�N
S
NS
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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a couple of months earlier…
vacuum chamber
Next suspect: the new octant in ‘Max von Laue hall’
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 58
citation from the logbook: “Visual inspection of new octant: no findings”
No findings in visual inspection
The electronic logbook:
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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citation from the logbook: “What we have tried so far: …”
citation from the logbook: “Visual inspection of new octant: no findings”
timeof entries
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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…when you have eliminated the impossible,whatever remains, however improbable, must be the truth
Sherlock Holmes, The Sign of the FourSir Arthur Conan Doyle
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Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan
Next suspect: an aperture problem
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Need of focusing
beam / bunch
vacuum chambermagnet
accelerating device
injector
straight sections
beam
we need to focus the beam !
��
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Page 63
Quadrupole magnetsquadrupole magnet:
B
beam
beam
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Quadrupole magnetsquadrupole magnet:four iron pole shoes of hyperbolic contour
hyperbola
x
y
)gradient quadrupole ( 2
0
R
Ig
µ=
RB
�� = �� ∙ �
�� = �� ∙ �
![Page 65: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/65.jpg)
Page 65
Quadrupole magnets
x
y
B
�� ��
�� = �� × � (Lorentz force)
�� = �� ∙ � → �� = �� ∙ �
� �
��
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Classical mechanics: harmonic oscillator
restoring force:
� = � �
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Page 67
Quadrupole magnets
focusing !
defocusing
x
y
B �!
�!
�� = �� ∙ � → �� = �� ∙ �
�� = �� ∙ � → �� = � ∙ �
�
���
![Page 68: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/68.jpg)
Page 68
In light optics…
defocusinglens
focusinglens
light rays
1
#∗=
1
#%+
1
#'�
�
#%#'
#: focal length
(light optics)#∗: systemfocal length
1
#∗=
�
#�> 0if #% = �#' = #
![Page 69: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/69.jpg)
Page 69
Quadrupole magnets
QD + QF = net focusing effect:
charged particle
defocusingquadrupole(rotated 90°)
focusingquadrupole
center of quadrupoles
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Page 70
Quadrupole magnets
B B
N
N
S
S
I
I
II
vertical defocusinghorizontal focusing
vertical focusinghorizontal defocusing
rotated 90°I = -I
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Page 71
Quadrupole magnets
QD + QF = net focusing effect:
focusingquadrupole
defocusingquadrupole
beam
y-plane:
defocusingquadrupole
focusingquadrupole
beam
x-plane:
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Quadrupole magnets
QD + QF = net focusing effect:
defocusingquadrupole
focusingquadrupole
beam
x-plane:
defocusingquadrupole
focusingquadrupole
QF QD QF QD QF QD QF QD QF QD
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Circular accelerator
cell
dipolemagnet
dipolemagnet
defocusingquadrupole
focusingquadrupole
focusingquadrupole
beam
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Page 74
Circular accelerator
cell
dipolemagnet
dipolemagnet
defocusingquadrupole
focusingquadrupole
focusingquadrupole
beam
PETRA
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Page 75
Circular accelerator
cell
dipolemagnet
dipolemagnet
defocusingquadrupole
focusingquadrupole
focusingquadrupole
beam
corrector dipole
corrector dipole
corrector dipole
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Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan
defocusingquadrupole
focusingquadrupole
QF QD QF QD QF QD QF QD QF QD
beam
Next suspect: an aperture problem
corrector dipole184 hor. corrector dipoles194 ver. corrector dipoles
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan
defocusingquadrupole
focusingquadrupole
QF QD QF QD QF QD QF QD QF QD
beam
Next suspect: an aperture problem
corrector dipole184 hor. corrector dipoles194 ver. corrector dipoles
?
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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First useful hint: aperture problem
Beam position monitor
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan13:20 a.m.: beam stored
244 beam position monitors
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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First useful hint: horizontal aperture problem
horizontalbeam pos.[mm]
verticalbeam pos.[mm]
N
S
EW
NENW
SESW
W N E S
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan13:20 a.m.: beam stored
244 monitors
radiofrequencysystems
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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First useful hint: horizontal aperture problem
N
S
EW
NENW
SESW
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan13:20 a.m.: beam stored
defocusingquadrupole
focusingquadrupole
QF QD QF QD QF QD QF QD QF QD
beam
corrector dipole
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 81
First useful hint: horizontal aperture problem
horizontalbeam pos.[mm]
N
S
EW
NENW
SESW
W N E S
new octant
Hamburg, DESYSat. 12th June 2010
02:24 a.m.: beam lost07:00 a.m.: visual inspection
in new octant11:52 a.m.: start aperture scan13:20 a.m.: beam stored
after‘flattening’the orbit
244 monitors
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 82
citation from the logbook: “the problem is at the end of the new octant”
horizontal aperture problem in the new octant
second hint: vacuum pressure raise in the new octant
beam current [mA] vacuum pressure [mb]
beam current
vacuum pressure
vacuum pressure
02:24 a.m.beam lost
aperture scan+ trajectory corrections
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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horizontal aperture problem in the new octant
second hint: vacuum pressure raise in the new octant
beam current [mA] vacuum pressure [mb]
beam current
vacuum pressure
vacuum pressure
02:24 a.m.
beam
60 m
PU13 PU14
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 84
visual inspection inside the vacuum chamber…
beam
PU13 PU14
endoscope
beam
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 85
visual inspection inside the vacuum chamber…
beam
PU13 PU14
endoscope
beam
vacuumbellows
beam
undulatorvacuum chamber
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 86
visual inspection inside the vacuum chamber…
an example of vacuum bellows
undulatorvacuum chamber
vacuumbellows
beam
dipole magnetvacuum chamber
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 87
the problem was found: RF fingers
beam
beam
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 88
the problem was found: RF fingers
beam
beam
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 89
RF fingers and wakefields
vacuum chamber wall
beam
+ : electric field ?
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 90
electric field of a charged particle
e
+
= 0
+?
e
= -�
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 91
electric field of a relativistic particle
e
� = 1- = 0
+
+ =�
4/01
(1 � -�)
1 � -� sin� 7 8/�:�:�
:
� =1
1 � -�
+; 7 = 0 =�
4/011
��:�:�
:
+< 7 =/
2=
�
4/01�
:�:�
:
+ =�
4/01
1
:�:�
:
= -�
+?
e z7
cylindrical coordinates
= 0- = 0
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 92
electric field of a relativistic particle
e
+
+ =�
4/01
(1 � -�)
1 � -� sin� 7 8/�:�:�
:
� =1
1 � -�
� ≫ 1- ≅ 1
+
e
+; 7 = 0 =�
4/011
��:�:�
:
+< 7 =/
2=
�
4/01�
:�:�
:
� → ∞0
� → ∞∞
z
� = 1- = 0
+ =�
4/01
1
:�:�
:
= -�- ≅ 1
= 0- = 0
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
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Page 93
RF fingers and wakefields
vacuum chamber wall
beam
+ +++ +
+ +++ +
mirror currents
+
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
![Page 94: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/94.jpg)
Page 94
RF fingers and wakefields
vacuum chamber wall
beam
+ +++ +
+ +++ +
mirror currents
+
simulation
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
![Page 95: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/95.jpg)
Page 95
RF fingers and wakefields
vacuum chamber wall
beam
+ +++ +
+ +++ +
mirror currents
RF fingers
beam
RF fingers
+
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
![Page 96: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/96.jpg)
Page 96
RF fingers: improvements done
old RF fingers were tilted outwards by 2 degrees
new RF fingers have stronger tilt, more tension
new design withRF fingers outside
beam
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
![Page 97: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/97.jpg)
Page 97
RF fingers: improvements done
old RF fingers were tilted outwards by 2 degrees
new RF fingers have stronger tilt, more tension
new design withRF fingers outside
beam
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018
![Page 98: Introduction to Accelerator Physics · Introduction to Accelerator Physics Part 1 Pedro Castro / Accelerator Physics Group (MPY) Zeuthen, 26th July 2018. Page 2 ... Applications of](https://reader030.fdocuments.net/reader030/viewer/2022040205/5f0491637e708231d40e9cc2/html5/thumbnails/98.jpg)
Page 98
Summing-up of this part
advancedaccelerator physics
RF shielding (RF fingers)
Circular accelerators: the synchrotron
basic components
dipole, quadrupole, undulator magnets, corrector dipoles
injection system (kickers and septum)
vacuum pumps, vacuum pressure monitors
beam position monitors
basic conceptsin operation aperture scans
trajectory (orbit) corrections
vacuum chambers, bellows
mirror currents and wakefields
electric field of a relativistic particle
| Introduction to Accelerator Physics | Pedro Castro, 26th July 2018