syllabus and slides

Syllabus and slidesSyllabus and slides

• Lecture 1: Overview and history of Particle accelerators (EW)• Lecture 2: Beam optics I (transverse) (EW)• Lecture 3: Beam optics II (longitudinal) (EW)• Lecture 4: Liouville's theorem and Emittance (RB)• Lecture 5: Beam Optics and Imperfections (RB)• Lecture 6: Beam Optics in linac (Compression) (RB)• Lecture 7: Synchrotron radiation (RB)• Lecture 8: Beam instabilities (RB)• Lecture 9: Space charge (RB)• Lecture 10: RF (ET)• Lecture 11: Beam diagnostics (ET)• Lecture 12: Accelerator Applications (Particle Physics) (ET)• Visit of Diamond Light Source/ ISIS / (some hospital if possible)

The slides of the lectures are available at

http://www.adams-institute.ac.uk/training/undergraduate

Dr. Riccardo Bartolini (DWB room 622) [email protected]

R. Bartolini, John Adams Institute, 8 May 2013 2/30

Lecture 10

Synchrotron radiation

properties of synchrotron radiationsynchrotron light sourcesLienard-Wiechert potentialsAngular distribution of power radiated by accelerated particlesAngular and frequency distribution of energy radiated: Radiation from undulators and wigglers

3/30

What is synchrotron radiationElectromagnetic radiation is emitted by charged particles when accelerated

The electromagnetic radiation emitted when the charged particles are accelerated radially (v a) is called synchrotron radiation

It is produced in the synchrotron radiation sources using bending magnets undulators and wigglers

R. Bartolini, John Adams Institute, 8 May 2013

High Flux: high intensity photon beam, allows rapid experiments or use of weakly scattering crystals;

High Brilliance (Spectral Brightness): highly collimated photon beam generated by a small divergence and small size source (partial coherence);

High Stability: submicron source stability

Polarisation: both linear and circular (with IDs)

Pulsed Time Structure: pulsed length down to tens of picoseconds allows the resolution of process on the same time scale 4/30

Synchrotron radiation sources properties

Broad Spectrum which covers from microwaves to hard X-rays: the user can select the wavelength required for experiment;

synchrotron light

Flux = Photons / ( s BW)

Brilliance = Photons / ( s mm2 mrad2 BW )


5/30

A brief history of storage ring synchrotron radiation sources

• First observation:

1947, General Electric, 70 MeV synchrotron

• First user experiments:

1956, Cornell, 320 MeV synchrotron

• 1st generation light sources: machine built for High Energy Physics or other purposes used parasitically for synchrotron radiation

• 2nd generation light sources: purpose built synchrotron light sources, SRS at Daresbury was the first dedicated machine (1981 – 2008)

• 3rd generation light sources: optimised for high brilliance with low emittance and Insertion Devices; ESRF, Diamond, …

• 4th generation light sources: photoinjectors LINAC based Free Electron Laser sources; FLASH (DESY), LCLS (SLAC), …


6/30

Peak Brilliance

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Bri

gh

tnes

s (P

ho

ton

s/se

c/m

m2 /m

rad

2 /0.1

%)

Candle

X-ray tube

60W bulb

X-rays from Diamond will be 1012 times

brighter than from an X-ray tube,

105 times brighter than the SRS !

diamond


7/30

Layout of a synchrotron radiation source (I)

Electrons are generated and accelerated in a linac, further

accelerated to the required energy in a booster and injected and stored in

the storage ring

The circulating electrons emit an intense beam of synchrotron radiation

which is sent down the beamline


Layout of a synchrotron radiation source

9/30

Main components of a storage ring

Insertion devices (undulators) to generate high brilliance radiation

Insertion devices (wiggler) to reach high photon energies


10/30

Many ways to use x-rays

scattering SAXS& imaging

diffractioncrystallography& imaging

photo-emission(electrons)

electronic structure& imaging

from the synchrotron

fluorescenceEXAFSXRF imaging

absorption

SpectroscopyEXAFSXANES & imaging

to the detector


1992 ESRF, France (EU) 6 GeVALS, US 1.5-1.9 GeV

1993 TLS, Taiwan 1.5 GeV1994 ELETTRA, Italy 2.4 GeV

PLS, Korea 2 GeVMAX II, Sweden 1.5 GeV

1996 APS, US 7 GeVLNLS, Brazil 1.35 GeV

1997 Spring-8, Japan 8 GeV1998 BESSY II, Germany 1.9 GeV2000 ANKA, Germany 2.5 GeV

SLS, Switzerland 2.4 GeV2004 SPEAR3, US 3 GeV

CLS, Canada 2.9 GeV2006: SOLEIL, France 2.8 GeV

DIAMOND, UK 3 GeV ASP, Australia 3 GeVMAX III, Sweden 700 MeVIndus-II, India 2.5 GeV

2008 SSRF, China 3.4 GeV2009 PETRA-III, D 6 GeV 2011 ALBA, E 3 GeV

33rdrd generation storage ring light sources generation storage ring light sources

ESRF

SSRF

11/30

12/30

Diamond Aerial views

June 2003

Oct 2006

13/30

Heuristic approach to synchrotron radiation

Synchrotron radiation is emitted in an arc of circumference with radius , Angle of emission of radiation is 1/ (relativistic argument), therefore

c

T transit time in the arc of dipole

During this time the electron travels a distance

Tcs

The time duration of the radiation pulse seen by the observer is the difference between the time of emission of the photons and the time travelled by the electron in the arc

3

2

c2)1(

1

c1

1

cc

sT

The width of the Fourier transform of the pulse is

3c2


14/30

Lienard-Wiechert Potentials (I)

The equations for vector potential and scalar potential

have as solution the Lienard-Wiechert potentials

with the current and charge densities of a single charged particle, i.e.

02

2

22

tc

1

0

22

2

22 1

cJ

t

A

cA

))((),( )3( trxetx ))(()(),( )3( trxtvetxJ

ret0 R)n1(

e

4

1)t,x(

ret0 R)n1(

e

c4

1)t,x(A

c

tRtt

)'('

[ ]ret means computed at time t’


15/30

The electric and magnetic fields generated by the moving charge are computed from the potentials

Lineard-Wiechert Potentials (II)

velocity field

retretRn

nn

c

e

Rn

netxE

3

0232

0 )1(

)(

4)1(4),(

ritEn

c

1)t,x(B

and are called Lineard-Wiechert fields

t

AVE

AB

acceleration field R

1 nBE ˆ

Power radiated by a particle on a surface is the flux of the Poynting vector

BE1

S0

dntxStS ),())((

Angular distribution of radiated power [see Jackson]

22

)1)(( RnnSd

Pd

radiation emitted by the particle


16/30

velocity acceleration: synchrotron radiation

Assuming and substituting the acceleration field [Jackson]

22

22

30

2

22

5

2

20

2

22

)cos1(

cossin1

)cos1(

1

4)1(

)(

4

c

e

n

nn

c

e

d

Pd

When the electron velocity approaches the speed of light the emission pattern is sharply collimated forward

cone aperture

1/


17/30

Total radiated power via synchrotron radiation

2254

0

4

2

4

0

22

2

320

24

4

2

0

24

2

0

2

66666BE

cm

ece

dt

pd

cm

e

E

E

c

e

c

eP

o

Integrating over the whole solid angle we obtain the total instantaneous power radiated by one electron (Larmor formula and its relativistic generalisation)

• Strong dependence 1/m4 on the rest mass

• proportional to 1/2 ( is the bending radius)

• proportional to B2 (B is the magnetic field of the bending dipole)

The radiation power emitted by an electron beam in a storage ring is very high.

The surface of the vacuum chamber hit by synchrotron radiation must be cooled.


18/30

Energy loss via synchrotron radiation emission in a storage ring

i.e. Energy Loss per turn (per electron) )(

)(46.88

3)(

4

0

42

0 m

GeVEekeVU

4

0

2

0 3

2 e

cPPTPdtU b

Power radiated by a beam of average current Ib: this power loss has to be compensated by the RF system

Power radiated by a beam of average current Ib in a dipole of length L

(energy loss per second)

In the time Tb spent in the bendings the particle loses the energy U0

e

TIN revb

tot

)(

)()(46.88

3)(

4

0

4

m

AIGeVEI

ekWP b

2

4

20

4

)(

)()()(08.14

6)(

m

GeVEAImLLI

ekWP b


19/30

The radiation integral (I)

Angular and frequency distribution of the energy received by an observer

2

)/)((22

0

23

)1(

)(

44

dten

nn

c

e

dd

Wd ctrnti

Neglecting the velocity fields and assuming the observer in the far field: n constant, R constant

22

0

3

)(ˆ2

EcRdd

Wd

Radiation Integral

dttERcdtd

Pd

d

Wd 20

22

|)(|

The energy received by an observer (per unit solid angle at the source) is

0

20

2

|)(|2 dERcd

Wd

Using the Fourier Transform we move to the frequency space


20/30

The radiation integral (II)

2

)/)((2

0

223

)(44

dtennc

e

dd

Wd ctrnti

determine the particle motion

compute the cross products and the phase factor

integrate each component and take the vector square modulus

)();();( tttr

The radiation integral can be simplified to [see Jackson]

How to solve it?

Calculations are generally quite lengthy: even for simple cases as for the radiation emitted by an electron in a bending magnet they require Airy integrals or the modified Bessel functions (available in MATLAB)


21/30

Radiation integral for synchrotron radiation

Trajectory of the arc of circumference [see Jackson]

sincossin)( ||

ctctnn

cossin

)( ct

ct

c

trnt

Substituting into the radiation integral and introducing 2/32231

c3

In the limit of small angles we compute

0,t

csin,t

ccos1)t(r

)(

1)(1

3

2

162

3/122

222

3/2

222

2

20

3

23

KK

cc

e

dd

Wd


22/30

Critical frequency and critical angle

Using the properties of the modified Bessel function we observe that the radiation intensity is negligible for >> 1

11c3

2/3223

Critical frequency3

2

3

cc

Higher frequencies have smaller critical

angle

Critical angle

3/11

c

c

)(K

1)(K1

c3

2

c16

e

dd

Wd 23/122

222

3/2222

2

20

3

23

For frequencies much larger than the critical frequency and angles much larger than the critical angle the synchrotron radiation emission is negligible


23/30

Frequency distribution of radiated energyIt is possible to verify that the integral over the frequencies agrees with the previous expression for the total power radiated [Hubner]

2

4

0

2

0

3/50

2

0

0

6)(

9

211

c

cedxxKd

c

e

Td

d

dW

TT

UP c

bbb

The frequency integral extended up to the critical frequency contains half of the total energy radiated, the peak occurs approximately at 0.3c

50% 50%

3

2

3

ccc

For electrons, the critical energy in practical units reads

][][665.0][

][218.2][ 2

3

TBGeVEm

GeVEkeVc

It is also convenient to define the critical photon energy as


24/30

Synchrotron radiation emission as a function of beam the energy

Dependence of the frequency distribution of the energy radiated via synchrotron emission on the electron beam energy

No dependence on the energy at longer

wavelengths3

2

3

ccc

Critical frequency

3

2

3

cc

Critical angle

3/11

c

c

Critical energy

][][665.0][

][218.2][ 2

3

TBGeVEm

GeVEkeVc

for electrons! How does this

change for protons?


Brilliance with IDs (medium energy light sources)Brilliance with IDs (medium energy light sources)

Medium energy storage rings with in-vacuum undulators operated at low gaps (e.g. 5-7 mm) can reach 10 keV with a brilliance of 1020

ph/s/0.1%BW/mm2/mrad2

25/30

Brilliance dependence

with current

with energy

with emittance


26/30

Polarisation of synchrotron radiation

In the orbit plane = 0, the polarisation is purely horizontal

Polarisation in the orbit plane

Polarisation orthogonal to the orbit plane

Integrating on all the angles we get a polarization on the plan of the orbit 7 times larger than on the plan perpendicular to the orbit

22

22

2/5220

52

0

32

17

51

)1(

1

64

7

e

ddd

Id

d

Wd

Integrating on all frequencies we get the angular distribution of the energy radiated

)(

1)(1

3

2

162

3/122

222

3/2

222

2

20

3

23

KK

cc

e

dd

Wd


27/30

Undulators and wigglers

Insertion devices (undulators) to generate high brilliance radiation

Periodic array of magnetic poles providing a sinusoidal magnetic field on axis:

),0),sin(,0( 0 zkBB u

Constructive interference of radiation emitted by different electrons at different poles


28/30

Synchrotron radiation from undulators and wigglers

Continuous spectrum characterized by c = critical energy

c(keV) = 0.665 B(T)E2(GeV)

eg: for B = 1.4T E = 3GeV c = 8.4 keV

(bending magnet fields are usually lower ~ 1 – 1.5T)

Quasi-monochromatic spectrum with peaks at lower energy than a wiggler

undulator - coherent interference K < 1

wiggler - incoherent superposition K > 1

bending magnet - a “sweeping searchlight”

2

2 21

2 2u u

n

K

n n

2

2

[ ]( ) 9.496

[ ] 12

n

u

nE GeVeV

Km


29/30

Summary

Accelerated charged particles emit electromagnetic radiation

Synchrotron radiation is stronger for light particles and is emitted by bending magnets in a narrow cone within a critical frequency

Undulators and wigglers enhance the synchrotron radiation emission

Synchrotron radiation has unique characteristics and many applications


30/30

Bibliography

J. D. Jackson, Classical Electrodynamics, John Wiley & sons.

E. Wilson, An Introduction to Particle Accelerators, OUP, (2001)

M. Sands, SLAC-121, (1970)

R. P. Walker, CAS CERN 94-01 and CAS CERN 98-04

K. Hubner, CAS CERN 90-03

J. Schwinger, Phys. Rev. 75, pg. 1912, (1949)

B. M. Kincaid, Jour. Appl. Phys., 48, pp. 2684, (1977).


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