introduction to accelerator physics - desy · 2014-07-31 · introduction to accelerator physics...

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 |

anglesolidareasource

flux spectral

anglesolidareasource

bandwidth 0.1% / s / photons Brilliance

×=

×=

Figure of merit: Brilliance


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)


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

��

�


vacuum chamberaccelerating devices

linear accelerator (linac)

vacuum chamber

magnet

accelerating device

injector

straight sections

circular accelerator: synchrotron


Motion in electric and magnetic fields

Equation of motion under Lorentz Force

( )BvEqFdt

pd rrrrr

×+==

charge velocity

of the particle

magnetic field

electric fieldmomentum


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

pdF

rrrrr

r⊥→×⋅==

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .electron

r

v

B (perpendicular)


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

pdF

rrr

r×⋅==

0cos)( 222

20

222

=×=×==

+=

φBvpqcBvpqcdt

pdpc

dt

dEE

EcpE

rrrrrrr

r

r

o90=φsince � � ⊥ � �


In general:

• Static magnetic fields � to guide (bend + focus) particle beams


acceleration with DC electric fields

E++++

++++

----

----

p+ -

+ - E++++

++++

----

----+ -

+ - E++++

++++

----

----+ -

+ -

∆V ∆V ∆V

3 ∆V

� � � �


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)


RF cavity basics: the pill box cavity

a quarterof a periodlater:

E++++

++++

----

----

p+ -

+ -

B. . .. . .

.

.. . .. . .

.

.

I

B


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:


� ∙ � � ��

� ∙ � � 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


� ∙ � � ��

� ∙ � � 0

� � � � ��

� � � ��


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 � �*��&


� ∙ � � ��

� ∙ � � 0

� � � � ��

� � � ��


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


+ boundary conditions

<

=

�

< : cavity radius

= : cavity length


� ∙ � � ��

� ∙ � � 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


boundary conditions

�& � ��6 �6>"< ,-./

�$ � 0

�% � 0�& � 0�$ � 0

�% � �?* <�6>�� ′6 �6>

"< ,-./


* � � �6><

<

=

�

< : cavity radius

= : cavity length

angular frequency :

�6> : n-th root of �6

�& " � < � ��6 �6> ,-./ � 0



�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


boundary conditions

�& � �� "< ,-./

�$ � 0

�% � 0�& � 0�$ � 0

�% � �?* <��

"< ,-./


* � � ��<

<

=

�

< : cavity radius

= : cavity length

�� 2.405

fundamental solution with �& � 0 (that is, � is transverse)

angular frequency :

�� : 1st root of ��

; � 0 and E �1


boundary conditions

�& � �� "< ,-./

�$ � 0

�% � 0�& � 0�$ � 0


<

=

�

< : cavity radius

= : cavity length


<

=

* � � ��< �� 2.405angular frequency :

<

�

�% � �?* <��

"< ,-./


boundary conditions

�& � �� "< ,-./

�$ � 0

�% � 0�& � 0�$ � 0


<

=

�

< : cavity radius

= : cavity length


<

* � � ��< �� 2.405angular frequency :

<

�% � �?* <��

"< ,-./


Pill box cavity: 3D visualisation of E and B

E B

beambeam


+ boundary conditions

set of solutions with �& � 0 (that is, � is transverse)

set of solutions with �& � 0 (that is, � is transverse)

<

=

�

< : cavity radius

= : cavity length


� ∙ � � ��

� ∙ � � 0

� � � � ��

� � � ��

TM modes(transverse magnetic modes)

TE modes(transverse electric modes)


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 : 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 :


http://en.wikipedia.org/wiki/Vibrating_string

Other examples of “standing waves”

http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum


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)


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


iris equator


Accelerating field map

beam

Simulation of the fundamental mode: electric field lines

beam

E

+++++

+ ----- -

--- +

++

+++++

+

+++

+++++

++++++

+ ---

-- -

---

EEEE

+VW � 1.3GHzmicrowaves: (L-band)

iris equator


Multipacting mitigation in superconducting cavities

beam

iris equator



beam

1 m


beam


pill box � called ‘cell’RF input portcalled ‘input coupler’

RF input portcalled ‘input coupler’

or ‘power coupler’

iris equator


Fundamental mode coupler (input coupler)

electric field

beam


beam

Fundamental mode coupler (input coupler)

electric field intensity



beam

1 m


beam


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


beam

� � \VW2 ↔ = � ⋯⋯ � 0.1154; � _`abb

homework !


Cavities inside a cryostat

beam


Number of cavities 8Cavity length 1.038 mOperating frequency 1.3 GHzOperating temperature 2 KAccelerating Gradient 23..35 MV/m

beam

beam


12.2 m


Cavities inside an accelerator module (cryostat)

beam

module installation in FLASH (2004)


Free-electron LASer in Hamburg (FLASH)

300 m, 1.2 GeVλ = 4 - 45 nm


Free-electron LASer in Hamburg (FLASH)

7 SC acceleration modules

6 undulator modules12 undulator modules

photon exp. halls


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


Superconducting cavities at HERA


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


Circular accelerators: the synchrotron

vacuum chamber

magnet

accelerating device

injector

straight sections


Low Energy Antiproton Ring (LEAR) at CERN



Dipole magnet

beam


. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

B (perpendicular)

R


� ⊥ � → d � e�

(circular motion)

d� ⊥ � → d � ; �<

e� � ;< → < � ;

e�

charge velocity

of the particle

magnetic field

momentum

d� � �� e� �


vacuum chamber

magnet

accelerating device

injector

straight sectionssynchrotron: R is constant,� increase B synchronously

with � � ; of particle


(circular motion)

� ⊥ � → d � e�d� ⊥ � → d � ; �

<e� � ;

< → < � ;e�


DESY (Deutsches Elektronen Synchrotron)

DESY: German electron synchrotron


DESY (Deutsches Elektronen Synchrotron)

DESY: German electron synchrotron, 1964, 7.4 GeV


Dipole magnet

beam


Electromagnet


Electromagnet

permeability of iron = 300…10000 larger than air

f


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


Dipole magnet cross section

increase B � increase current, but power dissipated P � < ∙ f�� large conductor cables



water cooling channels



increase B � increase current, but power dissipated P � < ∙ f�� large conductor cables� saturation effects


I

� ∶ permeability

ferromagnets

paramagnetsfree space

diamagnets

Saturation of iron: 1.6 – 2 T


I (Amps)

Saturation of iron: 1.6 – 2 T


Dipole magnet

beam

iron

currentloops



C magnet + C magnet = H magnet

beam


Dipole magnet cross section (another design)

beam

water cooling tubes

current leads

Power dissipated: 2IRP ⋅=

beam


Superconductivity

12.5 kAnormal conducting cables

12.5 kAsuperconducting cable


Superconductivity

resistance

critical temperature (Tc):

Tin

Copper


Superconducting dipole magnets

superconducting dipoles

LHC

HERA


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 =

�


Dipole field inside 2 conductors

densitycurrentuniform=J

J JB Br



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:



JJ

constant vertical field

B.

beam


From the principle … to the reality…

.B

56 mm

15 mm x 2 mm


LHC dipole coils in 3D

p beam

p beam

15 mm x 2 mm

Aluminium collar


LHC dipole coils in 3D

Bp beam

p beam

I


Computed magnetic field

Bferromagnetic iron

nonmagnetic collars

56 mm


LHC dipole magnet (cross-section)

beam tubes

superconducting coils

nonmagnetic collars

ferromagnetic iron

steel container for He

insulation vacuum

supports

vacuum tank

1 m


p

p


LHC dipole magnet interconnection:


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


Dipole magnets:

normal conducting dipoles

superconducting dipoles


beam

� � \VW2 ↔ = � ⋯⋯ � 0.1154; � _`abb

Homework

Demonstrate that the particle travels synchronous with the RF


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


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\


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)


[email protected]

Thank you for your attention

introduction to accelerator physics - desy · 2014-07-31 · introduction to accelerator physics...

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