Introduction to Accelerator PhysicsPart 2
Pedro Castro / Accelerator Physics Group (MPY)Introduction to Accelerator PhysicsDESY, 28th July 2014
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 2
anglesolidareasource
flux spectral
anglesolidareasource
bandwidth 0.1% / s / photons Brilliance
×=
×=
Figure of merit: Brilliance
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 3
e+ e-
production rate of a given event (for example, Z particle production):
Ldt
dNR z
zZ ⋅Σ==
cross section of Z production
luminosity (independent of the event type)
number of events
**4 yx
eeb NNfL
σσπ−+=
luminositytransverse bunch sizes (at the collision point *)
number of colliding bunches per second number of positrons per bunch
number of electrons per bunch
Figure of merit: Luminosity definition
(simplified expression)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 4
Figure of merit: Emittance
xy
z
���bunch of particles
phase space diagram
area/π = emittance (units: mm.mrad)
222 xxxxx ′−′=εemittance definition:
high emittance beamlow emittance beam
�� � ����
�
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 5
vacuum chamberaccelerating devices
linear accelerator (linac)
vacuum chamber
magnet
accelerating device
injector
straight sections
circular accelerator: synchrotron
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 6
Motion in electric and magnetic fields
Equation of motion under Lorentz Force
( )BvEqFdt
pd rrrrr
×+==
charge velocity
of the particle
magnetic field
electric fieldmomentum
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 7
Magnetic fields do not change the particles energy, only electric fields do !
Motion in magnetic fields
if the electric field is zero (E=0), then
vFBvqdt
rrrrr
r⊥→×⋅==
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .electron
r
v
B (perpendicular)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 8
Magnetic fields do not change the particles energy, only electric fields do !
Motion in magnetic fields
if the electric field is zero (E=0), then
Bvqdt
rrr
r×⋅==
0cos)( 222
20
222
=×=×==
+=
φBvpqcBvpqcdt
pdpc
dt
dEE
EcpE
rrrrrrr
r
r
o90=φsince � � ⊥ � �
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 9
In general:
• Static magnetic fields � to guide (bend + focus) particle beams
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 10
acceleration with DC electric fields
E++++
++++
----
----
p+ -
+ - E++++
++++
----
----+ -
+ - E++++
++++
----
----+ -
+ -
∆V ∆V ∆V
3 ∆V
� � � �
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 11
In general:
• Static magnetic fields � to guide (bend + focus) particle beams
• Static electric fields � accelerate particle beams (low energy)
• Radio-frequency EM fields � accelerate particle beams (high E)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 12
RF cavity basics: the pill box cavity
a quarterof a periodlater:
E++++
++++
----
----
p+ -
+ -
B. . .. . .
.
.. . .. . .
.
.
I
B
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 13
RF cavity basics: the pill box cavity
a quarterof a periodlater:
L
C
+ -
L
I
C
E++++
++++
----
----
p+ -
+ -
B. . .. . .
.
.. . .. . .
.
.
I
B
half a periodlater:
E++++
+++++
+----
-----
-
L
C
+-
LC circuit (or resonant circuit) analogy:
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 14
� ∙ � � ���
� ∙ � � 0
� � � � ����
� � � ���� � ��������
Maxwell's equations(differential formulation in SI units)
0
0
�� � 1��
����� � � 0
�� � 1��
����� � � 0
Wave equations for �and for � ∶
��� � 1"
��" " ��
�" � 1"�
����#� � ���
���
��� � 1"
��" " ��
�" � 1"�
����#� � ���
���z
#
"
cylindrical coordinates
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 15
� ∙ � � ���
� ∙ � � 0
� � � � ����
� � � ���� � ��������
Maxwell's equations(differential formulation in SI units)
0
0
��
� � � ��������
�$ � 0�% � 0
�& ' 0
� ∙ � � 0
��&�� � 0
�& � 0�$ � 0�% ' 0
�� � 1��
����� � � 0
1"
��" " ��&
�" � 1"�
���&�#� � ���&
��� � 1��
���&��� � 0
0
1"
�("�$)�" � 1
"��%�# � ��&
�� � 0
0
we choose rotational symmetry
1"
��&�" � ���&
�"� � 1��
���&��� � 0
1"
��&�" � ���&
�"� � *��� �& � 0
�& � +("),-./�
01230/1 � �*��&
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 16
� ∙ � � ���
� ∙ � � 0
� � � � ����
� � � ���� � ��������
Maxwell's equations(differential formulation in SI units)
0
0
�� ��+(�)��� � � �+(�)
�� � �� � 4� +(�) � 0Bessel’s differential equation
�´6 : derivative of the Bessel´s functions
�´6 � � �67� � � �68� �2 +:"; ' 0
�´� � � ��� �
�6 : Bessel´s functions
"� ∙ 1"
��&�" � ���&
�"� � *��� �& � 0
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 17
+ boundary conditions
<
=
�
< : cavity radius
= : cavity length
Maxwell's equations(differential formulation in SI units)
� ∙ � � ���
� ∙ � � 0
� � � � ����
� � � ���� � ��������
0
0
conductor wall�
conductor wall�
no component of the E vector may be parallel to a metallic surface
no component of the B vector may be perpendicular to a metallic surface
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 18
boundary conditions
�& � ���6 �6>"< ,-./
�$ � 0
�% � 0�& � 0�$ � 0
�% � �?* <�6>�� ���′6 �6>
"< ,-./
�6 : Bessel´s functions
* � � �6><
<
=
�
< : cavity radius
= : cavity length
angular frequency :
�6> : n-th root of �6
�& " � < � ���6 �6> ,-./ � 0
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 19
�6 : Bessel´s functions
�6> : n-th root of �6 (that is, �6 �6> � 0)
��� ��� ���
��
����
��� ���
A �6� �6� �6�0 2.405 5.520 8.654
1 3.832 7.016 10.173
2 5.136 8.417 11.620
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 20
boundary conditions
�& � ���� ���"< ,-./
�$ � 0
�% � 0�& � 0�$ � 0
�% � �?* <����� ���� ���
"< ,-./
�6 : Bessel´s functions
* � � ���<
<
=
�
< : cavity radius
= : cavity length
��� � 2.405
fundamental solution with �& � 0 (that is, � is transverse)
angular frequency :
��� : 1st root of ��
; � 0 and E �1
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 21
boundary conditions
�& � ���� ���"< ,-./
�$ � 0
�% � 0�& � 0�$ � 0
�6 : Bessel´s functions
<
=
�
< : cavity radius
= : cavity length
fundamental solution with �& � 0 (that is, � is transverse)
<
=
* � � ���< ��� � 2.405angular frequency :
<
�
�% � �?* <����� ���� ���
"< ,-./
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 22
boundary conditions
�& � ���� ���"< ,-./
�$ � 0
�% � 0�& � 0�$ � 0
�6 : Bessel´s functions
<
=
�
< : cavity radius
= : cavity length
fundamental solution with �& � 0 (that is, � is transverse)
<
* � � ���< ��� � 2.405angular frequency :
<
�% � �?* <����� ���� ���
"< ,-./
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 23
Pill box cavity: 3D visualisation of E and B
E B
beambeam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 24
+ boundary conditions
set of solutions with �& � 0 (that is, � is transverse)
set of solutions with �& � 0 (that is, � is transverse)
<
=
�
< : cavity radius
= : cavity length
Maxwell's equations(differential formulation in SI units)
� ∙ � � ���
� ∙ � � 0
� � � � ����
� � � ���� � ��������
TM modes(transverse magnetic modes)
TE modes(transverse electric modes)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 25
boundary conditions set of solutions with �& � 0 (that is, � is transverse)
�& � 0
indices:
A � F, H, I, …: number of full period variations in K of the fields
L � H, I, … : number of zeros of the axial field component in MN � F, H, I, … : number of half period variations in O of the fields
�6 : Bessel´s functions
�´6 : derivative of the Bessel´s functions
�6> : n-th root of �6 (that is, �6 �6> � 0)
* � � �6><
� � �P=
�
<
=
�
< : cavity radius
= : cavity length
�& � ���6 �6>"< cos ;# cos �P
= � ,-./
�$ � � �P= <
�6>���´6 �6>
"< cos ;#sin �P
= � ,-./
�% � � �P= ;<�
�6>�"���6 �6>"< sin ;#sin �P
= � ,-./
�$ � �?* ;<��6>�"�� ���6 �6>
"< sin ;#cos �P
= � ,-./
�% � �?* <�6>�� ���´6 �6>
"< cos ;# cos �P
= � ,-./
angular frequency :
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 26
http://en.wikipedia.org/wiki/Vibrating_string
Other examples of “standing waves”
http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 27
Superconducting cavities for acceleration
• Free-electron LASer in Hamburg (FLASH)
• European X-ray Free-Electron Laser (XFEL)
• International Linear Collider (ILC)
(future project, 30 km, 250 GeV)
(in construction, 3 km, 17.5 GeV)
(in operation, 300 m, 1.2 GeV)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 28
Superconducting cavity used in FLASH (0.3 km) and in XFEL (3 km)
beam
1 m
pill box � called ‘cell’
beam
Superconducting cavity used in FLASH and in XFEL
pill box � called ‘cell’
iris equator
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 29
Accelerating field map
beam
Simulation of the fundamental mode: electric field lines
beam
E
+++++
+ ----- -
--- +
++
+++++
+
+++
+++++
++++++
+ ---
-- -
---
EEEE
+VW � 1.3GHzmicrowaves: (L-band)
iris equator
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 30
Multipacting mitigation in superconducting cavities
beam
iris equator
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 31
Superconducting cavity used in FLASH (0.3 km) and in XFEL (3 km)
beam
1 m
pill box � called ‘cell’
beam
Superconducting cavity used in FLASH and in XFEL
pill box � called ‘cell’RF input portcalled ‘input coupler’
RF input portcalled ‘input coupler’
or ‘power coupler’
iris equator
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 32
Fundamental mode coupler (input coupler)
electric field
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 33
beam
Fundamental mode coupler (input coupler)
electric field intensity
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 34
Superconducting cavity used in FLASH (0.3 km) and in XFEL (3 km)
beam
1 m
pill box � called ‘cell’
beam
Superconducting cavity used in FLASH and in XFEL
pill box � called ‘cell’RF input portcalled ‘input coupler’or ‘power coupler’
RF input portcalled ‘input coupler’
or ‘power coupler’
HOM coupler
pick up antenna HOM coupler Higher Order Modes port(unwanted modes)
HOM coupler
iris equator
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 35
beam
� � \VW2 ↔ = � ⋯⋯ � 0.1154; � _`abb
homework !
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 36
Cavities inside a cryostat
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 37
Number of cavities 8Cavity length 1.038 mOperating frequency 1.3 GHzOperating temperature 2 KAccelerating Gradient 23..35 MV/m
beam
beam
Cavities inside a cryostat
12.2 m
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 38
Cavities inside a cryostat
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 39
Cavities inside an accelerator module (cryostat)
beam
module installation in FLASH (2004)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 40
Free-electron LASer in Hamburg (FLASH)
300 m, 1.2 GeVλ = 4 - 45 nm
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 41
Free-electron LASer in Hamburg (FLASH)
7 SC acceleration modules
6 undulator modules12 undulator modules
photon exp. halls
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 42
100 accelerator modules (cryostats) in XFEL
European X-ray Free-Electron Laser (XFEL) (in construction, 3 km, 17.5 GeV)
Hamburg
Schleswig-Holstein
λ = 0.05 - 6 nm
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 43
Superconducting cavities at HERA
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 44
Other accelerators using superconducting cavities
• 5 de-commissioned• 11 in operation• 4 in construction• 9 in design phase
Total = 29
full list: http://tesla-new.desy.de/sites/site_tesla/content/e163749/e163751/infoboxContent163765/SRFAccelerators.pdf
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 45
Circular accelerators: the synchrotron
vacuum chamber
magnet
accelerating device
injector
straight sections
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 46
Low Energy Antiproton Ring (LEAR) at CERN
Circular accelerators: the synchrotron
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 47
Dipole magnet
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 48
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
B (perpendicular)
R
Circular accelerators: the synchrotron
� ⊥ � → d � e�
(circular motion)
d� ⊥ � → d � ; �<
e� � ;< → < � ;
e�
charge velocity
of the particle
magnetic field
momentum
d� � ����� � e� �
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 49
vacuum chamber
magnet
accelerating device
injector
straight sectionssynchrotron: R is constant,� increase B synchronously
with � � ; of particle
Circular accelerators: the synchrotron
(circular motion)
� ⊥ � → d � e�d� ⊥ � → d � ; �
<e� � ;
< → < � ;e�
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 50
DESY (Deutsches Elektronen Synchrotron)
DESY: German electron synchrotron
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 51
DESY (Deutsches Elektronen Synchrotron)
DESY: German electron synchrotron, 1964, 7.4 GeV
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 52
Dipole magnet
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 53
Electromagnet
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 54
Electromagnet
permeability of iron = 300…10000 larger than air
f
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 55
Dipole magnet
beam
air gap
flux lines
beam
Ampere’s law:
f
g h�i � j h�i�$k>
� j h�ilmn
� of
j ���$k>
�i�$k>
� j ���
�ilmn
� of
j ���
�ilmn
� �p��
� of
� � ��ofp
gap height
N
S
g h�i � fa>`bkqar � of
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 56
Dipole magnet cross section
increase B � increase current, but power dissipated P � < ∙ f�� large conductor cables
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 57
Dipole magnet cross section
water cooling channels
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 58
Dipole magnet cross section
increase B � increase current, but power dissipated P � < ∙ f�� large conductor cables� saturation effects
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 59
I
� ∶ permeability
ferromagnets
paramagnetsfree space
diamagnets
Saturation of iron: 1.6 – 2 T
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 60
I (Amps)
Saturation of iron: 1.6 – 2 T
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 61
Dipole magnet cross section
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 62
Dipole magnet
beam
iron
currentloops
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 63
Dipole magnet cross section
C magnet + C magnet = H magnet
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 64
Dipole magnet cross section (another design)
beam
water cooling tubes
current leads
Power dissipated: 2IRP ⋅=
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 65
Superconductivity
12.5 kAnormal conducting cables
12.5 kAsuperconducting cable
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 66
Superconductivity
resistance
critical temperature (Tc):
Tin
Copper
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 67
Superconducting dipole magnets
superconducting dipoles
LHC
HERA
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 68
Superconducting dipole magnets
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 69
Dipole field inside 1 conductor
B
Ampere’s law:
r
g � ∙ �i� � g ��i � 2P"� � ��P"��
g � ∙ �i� � ��f�: uniform current density
�
� � ���2 "
θr
θµsin
20 rJ
Bx −=
θµcos
20 rJ
By =
�
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 70
Dipole field inside 2 conductors
densitycurrentuniform=J
J JB Br
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 71
Dipole field inside 2 conductors
JJ
0=J
densitycurrentuniform=J
θµsin
20 rJ
Bx −=
θµcos
20 rJ
By =
1θ1r
2θ2r
)cos(cos 2211 θθ rrd −+=
2211 sinsin θθ rrh ==
0)sinsin(2 22110 =+−= θθµ
rrJ
Bx
dJ
rrJ
By 2)coscos(
20
22110 µθθµ =−=
.
one conductor:
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 72
Dipole field inside 2 conductors
JJ
constant vertical field
B.
beam
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 73
From the principle … to the reality…
.B
56 mm
15 mm x 2 mm
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 74
LHC dipole coils in 3D
p beam
p beam
15 mm x 2 mm
Aluminium collar
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 75
LHC dipole coils in 3D
Bp beam
p beam
I
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 76
Computed magnetic field
Bferromagnetic iron
nonmagnetic collars
56 mm
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 77
LHC dipole magnet (cross-section)
beam tubes
superconducting coils
nonmagnetic collars
ferromagnetic iron
steel container for He
insulation vacuum
supports
vacuum tank
1 m
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 78
p
p
Superconducting dipole magnets
LHC dipole magnet interconnection:
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 79
Summing-up of part 1,2,3 (today)
Applications:
• HEP (example: LHC)• light source (example: PETRA)• medicine (example: PET)• industry (example: electron beam welding)• cathode ray tubes (example: TV)
RF cavities:
pill-box cavity
superconducting cavities
Circular accelerators: the synchrotron
Dipole magnets:
normal conducting dipoles
superconducting dipoles
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 80
beam
� � \VW2 ↔ = � ⋯⋯ � 0.1154; � _`abb
Homework
Demonstrate that the particle travels synchronous with the RF
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 81
Homework
Calculate the resonant frequency of the fundamental mode in a ‘coca-cola’ tin
assume a cylindrical shapewith a diameter of 6.4 cm and a height of 12.1 cm
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 82
Homework
1) Obtain an expression for the number of turns that this particle will travel aroundthe synchrotron during the particle’s mean lifetime at the lab reference system t∗ � vtas a function of the dipole magnetic field B
Assume a non-stable charged particle with mean lifetime of t circulating in a synchrotron whose dipoles have a magnetic field B and occupy half its circumference (dipole fill factor of 0.5)
vacuum chamber
dipole magnet
accelerating device
injector
straight sectionshint: synchrotron circumference: L � 2 ∙ (2P<)
where< is the bending radius inside the dipoles
2) Apply the expression obtained in (1) for the muon with:
mean lifetime t � 2.2�icharge q � 1.6 ∙ 107�{|mass at rest ;� � 1.88 ∙ 107�~�p
and � � 7\
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 83
Hollywood? Artistic view?
“Electromagnetic fields accelerate the electrons in a superconducting resonator “
https://media.desy.de/DESYmediabank/?l=en&c=3980&r=4199&p=1&f2165=1
DESY�Press�Media database�XFEL (with filter: media type=movies)
Pedro Castro / MPY | Accelerator Physics | 28th July 2014 | Page 84
Thank you for your attention