Foundation of Accelerator Beam Dynamics...Foundation of Accelerator Beam Dynamics - Part I and Part II - THz image KAERI-2019-092 Yujong Kim Future Accelerator R&D Team KAERI & UST,

Foundation of Accelerator Beam Dynamics- Part I and Part II -

THz image

KAERI-2019-092

Yujong Kim

Future Accelerator R&D Team

KAERI & UST, Korea

[email protected], [email protected]

October 19-20th, 2019, HIP, Japan

2

Outline

Acknowledgements & My Other Lecture Notes

Relativistic Particle Motion and Weak Focusing

Transfer Matrix, Betatron Oscillation, and Tune

Dispersion and Momentum Compaction

Magnets (dipole, quadrupole, sextupole, multipole expression, solenoid)

Strong Focusing

Beam Emittance

Chromaticity and Chromatic Effects

Short-Range Wakefields

Bunch Compressor

Accelerator Beamline Lattices (FODO, DBA, TBA, and MBA) and Brightness

Textbook:

An Introduction to the Physics of Particle Accelerators by Mario Conte

Other Reference:

Accelerator Physics by S. Y. Lee

Accelerator Physics Lecture Notes by Y. Kim

Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST

3

Acknowledgements


Y. Kim gives his sincere thanks to KAERI + UST colleagues and this school

chair, Prof. Masao Kuriki for their allowance of this lecture, and also to

following friends, references, and former & current supervisors:

KAERI, UST, and BMI: I. Jeong, S. Kim, Dr. J. Lee, and Dr. C. KimPAL & POSTECH: Prof. W. Namkung, Prof. I. S. Ko, and Prof. M. H. ChoSPring-8: Prof. T. Shintake (now at OIST), Prof. Kitamura, Dr. H. Tanaka,

Dr. T. Hara, Dr. T. Tanaka, and Dr. H. TomizawaKEK: Prof. K. Yokoya and Prof. H. MatsumotoPSI: Dr. S. Reiche, Dr. M. Pedrozzi, Dr. H. Braun, and Dr. T. Garvey,DESY: Dr. K. Floettmann, Dr. S. Schreiber, Dr. R. Brinkmann,

Prof. J. Rossbach, Dr. Y. ChaeAPS: Dr. J. Byrd, Dr. M. Borland, and Prof. Kwang-Je KimLANL: Dr. B. CarlstenIndiana University: Prof. S. Y. LeeINFN: Dr. M. FerrarioJefferson Lab: Dr. A. Hutton, Dr. H. Areti, and Dr. S. BensonDuke University: Prof. Y. WuIdaho State University & Idaho Accelerator Center: Prof. D. WellsHiroshima University: Prof. M. KurikiIBS RAON: Dr. D. Jeon, Dr. J. Kim, and Dr. M. Kwon

4

My Other Lecture Notes

Yujong Kim's Other Lecture Notes at Idaho State University, KAERI WCI,

POSTECH, KAIST, VITZRONEXTech, Korea-Japan Joint Summer School,

and ISBA.

Basic Accelerator PhysicsMagnets and Transverse Motion in Accelerators

RF System and Longitudinal Motion in Accelerators

Advanced Accelerator Physics Tutorial for XFEL Projects

Accelerator Beam Diagnostics

Linux Basic for Physicists

Laser Compton Scattering (LCS)

RF Technology and Electron Linear Accelerators

There are my lecture notes on beam dynamics for KoPAS2015.

There is also my lecture note on RF system for ISBA2018.

You can obtain them by sending an email to Yujong Kim:

[email protected], [email protected]


5

Short Review of Relativistic Particle Motion

Particle Accelerator Physics is a region of applied Special Relativity:

Law of physics may be expressed in equations having same form in all frames.

Speed of light in free space is the same value for all observers.

electronfor51099906.0/)MeV(/2

UmcU

please note that acceleration in linac gives

a growth of kinetic energy. At an electron gun exit,

energy gain = 500 keV → W = 0.5 MeV

U ~ 0.5 MeV + 0.511 MeV → γ ~ 1.97847358

U ~ pc for Ultra-Relativistic case.


c= 𝟐. 𝟗𝟗𝟕𝟗𝟐𝟒𝟓𝟖 × 𝟏𝟎𝟖𝐦/𝐬

= 𝑊 +𝑚𝑐2

6

Energy of Rest Mass - mc2 & Isotope

Unified Atomic Mass Unit (u or also known as amu)

1 u = 1.66053892173 ×10-27 kg = 931.49406121 MeV/c2 = 1822.88839 me ~ 1 mp or 1 mn

me = 9.1093897× 10-31 kg = 0.51099906 MeV/c2 for electron

mp = 1.6726231× 10-27 kg = 938.2723 MeV/c2 for proton

~ 1836.152725 me

mec2 = 0.51099906 MeV for electron

mpc2 = 938.2723 MeV = 1836.152725 mec

2 for proton

mnc2 = 939.5656 MeV for neutron

mdc2 = 1875.6134 MeV for deuterium (D or 2H) with one proton & one neutron

Mass and Rest Mass Energy

1 uc² = (1.66053892173 × 10−27 kg) × (2.99792458 × 108 m/s)²

≈ 1.49242 × 10−10 kg (m/s)² ≈ 1.49242 × 10−10 J × (1 MeV / 1.6021773 × 10−13 J) ~ 931.49 MeV


7

Short Review of Relativistic Particle Motion

protonfor2723.938/)MeV(

electronfor51099906.0/)MeV(

U

U

Ele

ctro

n K

inet

ic E

ner

gy

(M

eV)

Pro

ton

Kin

etic

En

erg

y (

MeV

)

MeV57139386651~ [email protected]~

[email protected]~

[email protected]~

2

c U - mW U

W

W

p


𝑈 = 𝛾𝑚𝑐2 = 𝑊 +𝑚𝑐2

𝛾 =𝑊 +𝑚𝑐2

𝑚𝑐2

8

β of IBS RAON Accelerator ?

IBS RAON Accelerator

600 MeV for proton

β ~ 0.8

(particle A)


HWR + QWR

SSR1 + SSR2

Site: 952,066㎡ (~ 290,000 pyeong)

9

Particle Motion in an Uniform Magnetic Fields

Under a constant magnet field in time, a positive charge q of a design particle

is performing a circular motion with a radius along a design orbit:

Lorentz force = centripetal force (for a reference particle on the design orbit)

betatron

B

&

xBRxBB yy ))(/(


B

10


magnet (or momentum) rigidity

[Measured Magnetic field of a dipole for Bunch Compressor of DESY FLASH Facility ]

B = B(I)


a relation to find beam energy or momentum in a dipole

for a singly charged particle (q = Ne = e with N =1)

11


magnet (or momentum) rigidity

a relation to find beam energy or momentum in a dipole

for a singly charged particle (q = Ne = e with N =1)

B = B(I)

[Measured Magnetic field of a dipole for Bunch Compressor of DESY FLASH Facility ]


12

Working Principle of Spectrometer

→

we can measure momentum or energy

from bending angle, effective length, magnetic field.DESY TN-04 02 by P. Castro

ρ

ρˑ


dl

B(l)

electrons dipole magnet

B field

13

Magnet Regidity for Heavy Ion Beams


N : charge number 𝑞 = 𝑁𝑒

𝑈nucleon GeV/u : energy per one proton or neutron

A: atomic mass number = number of proton

= v/c

𝐴

𝑁𝛽𝑈nucleon GeV/u ≅ 0.3𝐵 T 𝜌 m

R

q

Under a constant magnet field in time, a positive charge q of a design particle

is performing a circular motion with a radius along a design orbit.

Local Cartesian Coordinate System (x, y, z) : moving along a design particle

x : radial outward, x ≡ R - (continuous changing direction)

y : vertically up

z : direction of motion of the design particle

s : total traveling distance along the design orbit, s = s(θ ) = s(ωt)

negative charge : left-handed coordinates

Trajectory of general particles are slightly

different from the design orbit due to

energy spread and magnetic field gradient

, hence a change of slope of particle trajectories

after s moving (-oscillation).

See pages 60 & 66, An Introduction to the Physics of High Energy Accelerators by Edwards14

Equations of Motions for Weak Focusing

B

xBRxBB yy ))(/(


)1(

x

xR

15

Magnetic Fields and Unit Vectors

From Page 66, An Introduction to the Physics of High Energy Accelerators by Edwards

→

R


1616

Magnetic Fields and Unit Vectors

From An Introduction to the Physics of High Energy Accelerators by Edwards


1717

Now return to Page 20 in Conte's Book


r @ Edwards' book → R = + x @ Conte's book

ignore any energy change

from previous page and


18


define field index n

by applying the paraxial approximation, i. e.

18Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST

1919


similar to equations of simple harmonic oscillator!

Therefore, x & y motion can be stable and focused simultaneously only when

0 < n < 1 : weak focusing


2020

Weak Focusing - Lorentz Force

forces for coming out positive charge

center

horizontal defocusing for

radially opening poles

horizontal focusing for

radially closing poles

center

vertical focusing for

radially opening poles

vertical defocusing for

radially closing poles


2121

Weak Focusing - Resultant Force

Horizontal motion in Betatron is controlled by a resultant force of

centrifugal force (mv2/R) and centripetal Lorentz force (qvBy), Fres :

0

2

qvBv

m

betatron

B

here n: field index


2222

Weak Focusing

Horizontal motion in Betatron is controlled by a resultant force of

centrifugal force (mv2/R) and Lorentz force (qvBy), Fres , which acts to push

the particles towards the reference or equilibrium orbit under 0 < n < 1.

R

q

n = 0.5


𝒎𝒗𝟐

𝑹

qvBy

2323

Solutions of Motion Equations

with the condition 0 < n < 1 gives a possible solution:

If there are initial conditions at = 0; x(0) = x0, x'(0) = x0' = dx/ds = (1/)(dx/d) =0

or in transfer matrix form in the horizontal plane:


2424

Transfer Matrix, Betatron Oscillation, & Tune

From similarly, the transfer matrix in the vertical plane:

From solutions of equation of motion, we can find the fact that motion of particles

shows sinusoidal behaviors. This type of transverse oscillation is called the betatron

oscillation, which is induced due to the nonzero magnetic field gradient n.

horizontal and vertical betatron tunes QH and QV are defined as the numbers of

cycles of horizontal and vertical betatron oscillation which are made by a particle in

one turn circulation. QH & QV < 1 for weak focusing (0 < n < 1).

Define x & y betatron phase: &

for weak focusing

snn1


2525

Examples of Betatron Oscillation and Tunes

betatron oscillation & tune for strong focusing:

QH > 1

QV > 1

What is difficulty in weak focusing accelerator?

Can we observe the betatron oscillation in a region with quadrupoles?


2626

Transfer Matrix for Field Free Drift Space

x = x0 + Lx0'

x' = x0'

Transfer Matrix of a Drift Space with Length L

See also page 32 for other cases:

uniform magnetic field but no field gradient (n = 0)

90 degree, and opposite sign of n


old particle position

new particle position

particle angle

Lx0'

design orbit

then, the horizontal motion is described by:

2727

Momentum Dispersion - Beam Spreading

So far, we assumed that all particles have a same energy (mono-energy).

But in the real situation, there is a horizontal beam broadening in a dipole due to the

momentum spread. There is the beam broadening only in the horizontal or radial plane.

Why there is no beam broadening vertically?

0Bqp

ppmcp

ppmcp

)1(

x

xR


2828


by using definition of the field index

by ignoring nonlinear terms in x.

horizontal equation of motion under nonzero momentum spread


2929


by inserting

Similarly, the vertical motion is described by an equation at page 18 :

0Bqp

ignoring x2 term

)1(p

ppppmcp

)1(

x

xR


3030


By keeping linear terms, the equation of motion in the vertical plane is given by

vertical equation of motion under nonzero momentum spread

= same as previous case, without any momentum spread. Why?

Possible particular solutions of equations of motions in the x plane:

If there are initial conditions at = 0; x(0) = x0, x'(0) = x0' = dx/ds = (1/)(dx/d) =0


3131

Transfer Matrix under Momentum Spread

Therefore, solutions of motion equation in the horizontal plane under a momentum

spread can be given by

horizontal transfer matrix under momentum spread for weak focusing (0 < n < 1)

vertical transfer matrix is same as before without momentum spread (see page 24).


2p

p 22

p

p 22

3232

Examples - Electron Motion in Dipole Magnet

If a particle is in a uniform magnetic field such as in a wide dipole or cyclotron, n =0.

orbit of electrons under

entering uniform B field

If

If are initial conditions for x, x', p/p, then final x, x' and p/p of the particle:

beam broadening = p

p4


3333

Examples - Weak Focusing Synchrotron

Synchrocyclotron is a machine with a constant radius but with a varying magnetic field

in time according to the increasing momentum at an RF cavity.

cell or superperiod = a periodic block of elements

here cell number = 4 (one dipole + one drift space l0)

3 ways to make the transfer matrix for one cell in synchrotron above.

drift between dipoles = l0

drift = (l0/2)/=l0/2

synchrotron is a virtual circular machine:

= 2R


l0

3434


Find the transfer matrix for one cell in synchrotron!

First of all, let's ignore the momentum spread.

We will consider it when we study momentum compaction factor.

drift = (l0/2)/=l0/2

per cell

total tune for the synchrotron = 4 cells QH

see details in textbook

0

H

H

21,

lR

Q

R

sin(x+y) = sin(x).cos(y) + sin(y).cos(x) & cos(x+y) = cos(x).cos(y) - sin(x).sin(y)


a b

ab

3535


total vertical tune for the synchrotron = 4 cells QV

Similarly, for the vertical plane (try to drive these)

VQ

R


3636

Momentum Compaction Factor p

p is defined as the fractional difference in circumference of the reference orbit

with respect to the fractional difference in the particle momentum. That is,

in synchrotron, circumferences are different for particles with different momenta.

]m[]T[2998.0)GeV/c(,/

/ BpBqp

pdp

LdLp

From the weak focusing synchrotron as shown in page 33,

paths in straight section lo are same for all particles with different momenta.


3737

Momentum Compaction Factor p

For a weak focusing synchrotron, the momentum compaction factor p is larger than 1

because QH is smaller than 1. Therefore, the difference of circumference for particles

with a momentum spread is large. Hence the dimension of vacuum chamber becomes

large to build a high energy weak focusing synchrotron (problem).

But for the strong focusing machine, momentum compaction factor is small

enough because QH is much higher than 1.

Example of PLS storage ring: QH ~ 14.28, p << 1

→ no big circumference difference for particles with different energies.


(1 + 𝑥)−1≅ 1 − 𝑥 𝑓𝑜𝑟 𝑥 ≪ 1

3838

Liouville's Theorem

A particle density function f(x, y, z, px, py, pz, t) can be defined as the number of particle

per volume of six dimensional phase space (x, y, z, px, py, pz) at a given time t.

Here, x, y, z is the three spatial coordinates, and px, py, pz are their corresponding

momentum components.

Liouville's Theorem:

In the local region of a particle, the particle density in phase space is constant in time,

provided that the particles move in a general field consisting of magnetic fields and of

fields whose forces are independent of velocity (or non-dissipative systems

such as no radiation loss, space charge force, and wakefields).)(vFF

0dt

df


Typically, magnets in accelerator supply transverse or longitudinal magnetic fields.

From the Lorentz force, magnets are used to change of the direction of motion of a

particle; . Example, bending magnet.

Magnets with transverse fields (ex, dipole, quadrupole) are backbone of accelerator

and beam transport system, and magnets with longitudinal fields (ex, solenoid) can be

used to detect colliding beams in colliders or to focus beams in a low energy (ex, gun).

3939

Magnets in Accelerator

IL

NknIknIB

7

0 104]T[

http://hyperphysics.phy-astr.gsu.edu

magnetic field of Solenoid B [T], where

: permeability in material = k0

N : number of coil turns = nL

L : length of solenoid [m]

I : current [A]

k : relative permeability

~ 200 for magnetic iron

~ 20000 for -metal for magnetic shielding

(75% nickel, 2% chromium, 5% copper, 18% iron)


4040

Magnetic Field of Ferromagnetic Material

)()(000 materialindipolenetcurrent BBHMHMHB

where

H [A/m] is magnetic field strength driven by external driving current (I)

M [A/m] is magnetization, density of net magnetic dipole moments in a material

http://hyperphysics.phy-astr.gsu.edu H due to external driving current I


coil part iron core part

4141

Hysteresis Loop of Magnets with Iron Core

)(000 MHMHB

where






4242

Hysteresis Loop of Magnets with Iron Core

)(000 MHMHB


where



at a certain driving current I (or H), there are two different M depending on history or

direction of I.

We have to cycle all magnets with iron cores to reproduce the same magnetic fields B at

a certain current I.

M

H or I



4343

Cycling of Magnets with Iron Core

We have to cycle all magnets with iron cores to reproduce the same magnetic fields B at

a certain current I.

Generally, cycling should be done two times (at least) by changing current of power

supply gently to get a good reproducibility in magnetic field:

gently go Imax → waiting (until set value = reading value) → go Imin → waiting →

go Imax → waiting → go Imin → waiting → setting a current

cycling of a magnet with an unipolar (left) and a bipolar (right) power supply

I

M

9: setting a current3, 7

4, 8

1, 5

2, 6

I

M


4444

2D Magnet Model

In preliminary accelerator design, magnets can be considered with the 2D model

if the considering region is 1.5 times of a gap height d from the end of the magnet.

Specially, the 2D model works well if magnet length is long enough.

But in the real accelerator design, the end fringe field effects should be considered.

Until we learn the end fringe field effect, let's assume magnet with the 2D model.

pole gap height = dGood

2D Region

1.5d

d


4545

Maxwell Equations in 2D Magnet Model

For static field and if there is no current (ignoring beam current),

Then, magnetic field strength H can be expressed as the gradient of the magnetostatic

potential and it satisfies the Laplace equation.

0

0

0,0

H

B

jt

E

02

H


4646

Laplace Equation and Multipole Expansion

The general solution of the Laplace equation in the cylindrical coordinates (r, , z)

(see cylindrical harmonics in John R. Reitz's Foundations of Electromagnetic Theory)

can be found in 2D cylindrical coordinates;

Here a is the reference radius of the expansion ~ half of magnet gap.

While, we can also expand the field of bending magnets in a series of multipoles:

02 0

112

2

2

rrr

rr

y

x

a2

(r,)

HB

: normal strength of 2(n+1)th multipole

: skew strength of 2(n+1)th multipole


4747

Multipole Expansion

Normal Multipole Components:

n = 0 for normal dipole component

n = 1 for normal quadrupole component

n = 2 for normal sextupole component

n = 3 for normal octupole component

n = 4 for normal decapole component

Skew multipole Components :

poles are rotated by /2(n+1),

example,

skew dipole = 90 deg rotation for n =0

skew quadrupole = 45 deg rotation for n = 1

normal dipole & quadrupole

skew dipole & quadrupole

http://pbpl.physics.ucla.edu


4848

Equipotential Lines of → Pole Face

HB

equipotential lines of = pole face

Similarly to electric fields, magnetic field must be perpendicular to the equipotential

lines of , which gives the pole face of the magnet.

Equation of Optimum Pole Face for Normal Magnets:

Pole Face of Dipole :


http://pbpl.physics.ucla.edu

n =2 for QM

Example, Normal Dipole Magnet :

b0 = 1, all other bn & an = 0

note that

normal magnets have poles on y-axis for even n (dipole for n =0, sextupole n =2, ...),

normal magnets have no pole on any axis for odd n (quadrupole for n =1, octupole n =3, ...)

4949

Normal & Skew Magnets

for normal magnets for skew magnets

see also current relation in textbook!Yujong Kim for New Korean Synchrotron Light Source & nTOF, KAERI & UST

5050

Dipole Magnet

equipotential lines of = pole face

Apply Ampere's law to estimate the magnetic field in dipole gap

IldH

IldB

0

0


N turnsI

5151

Quadrupole Magnet

Apply Ampere's law to estimate the magnetic field in quadrupole

S

N

N

S

here note that

current direction is wrong.

it is reversed.

n = 1

r = a

= /4 for south

= -/4 for north

centerQMnear@00 HB


5252

2(n+1) Multipole Magnet

For sextupole, n = 2

centernear@00 HB


Here we can ignore Bz in the 2D model because length of QM is long enough (l > a).

By using

near QM center,

And from Ampere's law

5353

Quadrupole Lens

In the absence of electric field, which means no acceleration,

from page 6, motion equation becomes;

centerQMnear00 HB

0 B


5454

Quadrupole Lens

H

From QM magnetostatic potential at pages 48 & 52, rHB centeQMnear0

QMsperfectfor),(,2

),(

)GeV/c(

)A(2998.0

)GeV/c(

)T/m(2998.0)()(

0100

0

10

gxyxBgyya

Gy

aa

NIHyxB

p

IC

p

gsg

p

q

x

B

p

q

pa

qGsk

yxx

mag

x

y

see distributed paper on measurement g


5555

Quadrupole Lens

Here general solutions are used. From initial conditions (x = x0, x' = x0' at s =0):

In transfer matrix form:

After considering opposite sign k in y (k → -k), in vertical plane:

i


5656

Quadrupole Lens

Note that QM always makes opposite focusing in x & y planes:

horizontal focusing QM (QF) → vertical defocusing

vertical focusing QM (QD) → horizontal defocusing

If

Thin Lens Approximation for QM

here f is the focal length of the QM.

(x0, x'0 = 0)

(x = 0, x'= -x0/f)


5757

Fringe Field Effects in a Dipole

Up to now, we assumed that the design trajectory is orthogonal to the both ends of a

dipole magnet. But in real situation, there are many cases where the trajectory is not

orthogonal to the ends of dipoles.

→ there are additional focusing or defocusing fringe field effects in a dipole.

n

n

: normal vector with respect to the magnet yokes : rotation angle of the yoke-ends around vertical axis.

> 0 for the normal vector is at outside of the beam trajectory (entering - clockwise).

< 0 for the normal vector is at inside of the beam trajectory (exit - clockwise).

= 0 : sector dipole (no pole rotation).

beam trajectory

[top view of three different dipoles]

sector dipole

n


n

beam trajectory

5858

Fringe Field Effects in a Dipole

For > 0,

particles located at positive x take shorter paths in the dipole & to be bent weakly

particles located at negative x take longer paths in the dipole & to be bent strongly

→ horizontal defocusing & vertical focusing

For < 0,

particles located at positive x take longer paths in the dipole & to be bent strongly

particles located at negative x take shorter paths in the dipole & to be bent weakly

→ horizontal focusing & vertical defocusing

x > 0

x < 0

x > 0

x < 0


5959

Slanting Dipole = Normal Dipole + Two Wedges

Horizontal Transfer Matrix for a Slanting Dipole: Mslanting = MwedgeMsectorMwedge

Wedge is a kind of focusing or defocusing quadrupole magnet:

horizontal trajectory of positive charges in a wedge

we can consider this

wedge as a defocusing QM!

slanting dipole = wedge + sector dipole + wedge


From Conte's book page 84 or Reiser's book page 135-138,

Horizontal Transfer Matrix of a thin Wedge without any dispersion:

Horizontal Transfer Matrix of a thin Wedge with a dispersion p/p:

From Conte's book pages 52 & 56, Eq (3.84) and Eq (3.102), the 6 dimensional transfer

matrix (x, x' y, y', z, p/p) of a sector dipole without any wedge:

Note that a sector dipole can supply a focusing in the horizontal plane.6060

Transfer Matrix for Wedge

0

'

100

01tan

001

'

p

px

x

p

px

x

0'1

tan01

'

x

x

x

x


6161

Transfer Matrix for Wedge

Then, the total horizontal transfer matrix of a slanting dipole with wedges and a

dispersion can be obtained from a transfer matrix for wedges (Mw in page No. 60)

and the horizontal transfer matrix of a sector dipole without any wedge (= MH above):

00

H '

100

sincossin1

)cos1(sincos

')(M'

p

px

x

p

px

x

p

px

x

From the 6 dimensional matrix of a sector dipole, let's extract components of only

horizontal position, horizontal angle, and a dispersion (x, x', = p/p) →


6262

Transfer Matrix for a Rectangular Dipole

Similarly, from Reiser's book page 135-138 and the vertical components the transfer

matrix of a sector dipole magnet, the total vertical transfer matrix for a slating dipole

with wedges but without a dispersion (y, y') can be given by:

For a Rectangular Dipole Magnet with :

yw,Vyw,V M)(MMM θ

Note that a rectangular dipole can supply a focusing in the vertical plane.


6363

Angles of Rectangular Dipoles for Chicane

SCSS Bunch Compressor, details can be found from Y. Kim's NIMA 528 (2004) 421.

For the first dipole:bending angle = + 4 deg

edge angle1 for entrance = 0 deg

edge angle2 for exit = + 4 deg

For the second dipole:bending angle = - 4 deg

edge angle1 for entrance = - 4 deg

edge angle2 for exit = 0 deg

For the third dipole:bending angle = - 4 deg

edge angle1 for entrance = 0 deg

edge angle2 for exit = - 4 deg

For the fourth dipole:bending angle = + 4 deg

edge angle1 for entrance = + 4 deg

edge angle2 for exit = 0 deg

bending angle ~ 4 deg

dipole length ~ 0.2 m

beams go from left to right

note clockwise is positive

θedge1 + θedge2 = θbending


To keep beam size with a certain limits (< inner diameter of vacuum chamber), we need

high horizontal and vertical tunes (QH & QV > 1) : concepts of Strong Focusing.

QM gives the focusing only in one plane.

But continuous doublets can give the horizontal and vertical focusing:

From two thin lens combination, we can get a focal length of the two combined lens:

If fD = - fF,

f > 0, doublet gives a focusing system.

→ backbone of strong focusing!

6464

Strong Focusing

DFDF ff

l

fff

111

l


Separated function magnet has each own one function (bending or focusing)

Combined function magnet has more than one functions (bending and focusing)

[separated function dipole] [combined function dipole]

Generally, transfer matrix M(s) for a periodic cell can be made by multiplying transfer

matrices of machine components (drifts, quadrupoles, bending magnets, and so on).

For a generalized coordinate z

Note that the trace of matrix M(s) should be smaller than 2 for the stable periodic beam

motion in ring! (see details from Conte's book pages 97-100 & resonance conditions) .

6565

Twiss Parameters - x,y(s), βx,y(s), γx,y(s)


For a stable focusing beamline region satisfying , generally, the transfer matrix

can be parameterized with the Twiss parameters ((s), β(s), γ(s)) (see Conte's book pages

97-100) :

6666

Twiss Parameters - x,y(s), βx,y(s), γx,y(s)

Most transverse beam parameters can be expressed by the Twiss parameters, phase advance

emittance ε, and others. Ex, vertical rms beam size & rms divergence with no dispersion:

Design of a stable focusing machine lattice → optimization of the Twiss parameters!

)()()(,)()()( ' ssssss yyyyyy

2

1


67676767

Twiss Parameters for PSI 250 MeV injector

pickup

tuners

High Energy Transverse Deflecting Structure (TDS2)

developed with collaboration with INFN and PSI

resonance frequency : ~ 2997.912 MHz

deflecting mode : TM110

type : five cell SW cavity

physical length / average iris diameter : 441 mm / 36 mm

max available klystron power : 7.5 MW

max deflection voltage : 4.5 MV for about 4.1 MW

max slice number for 10 pC (200 pC) : 3 (12) slices

operation energy : ~ 250 MeV (gun region)

rms time resolution for 10 pC (200 pC) : 11 fs (16 fs) @ 4.5 MV TDS2 with five cells


6868

Diagnostic Section for PSI 250 MeV Injector5QMs TDS2 5QMs 3FODO 1QM DIPOLE

3FODO Cells

OTRs : 1 2 3 4 5 6 7 8

horizontal phase advance @ 3FODO = 55 deg per cell

vertical phase advance @ 3FODO = 25 deg per cell

cell length = 3.0 m

rms beam size @ 7FODO screens ~ 60 m for 200 pC

max dispersion by a dipole in DIAG1 ~ 0.2 m

With this special DIAG1,

we can measure followings

without change any optics:

- slice emittance

- slice energy spread

- longitudinal phase space

- bunch length

- arrival timing jitter

- projected emittance

- Twiss parameters

- optics matching\

only by turning on and/or

off TDS2 and a dipole in

DIAG1.


6969

Phase Advance along the Diagnostic Section

5QMs TDS2 5QMs 3FODO 1QM DIPOLE

3FODO Cells

OTRs : 1 2 3 4 5 6 7 8

Slice Emittance Measurement

βy should be high at TDS2 = 20 m

ψy = (nπ+π/2) at OTRs

=107.9 deg @ OTR6

ψx = 0 ~180 deg at OTRs

=189.2 deg @ OTR6

βy should be small at OTRs

= 6.5 m @ OTR6

βx should be small at OTRs

= 4.3 m @ OTR6

σx = 60 µm @ OTRs for 200 pC

Long. Phase Space Reconstruction

βy should be high at TDS2 = 20 m

ψy = (nπ+π/2) at OTR8

= 140.2 deg @ OTR8

high ηx @ OTR8

= 0.2 m @ OTR8

small β-beamsize σx at OTR8

= 26 µm without dipole

βx should be small at OTR8

= 0.96 m @ OTR8

εn, resolution ~ 0.05 µm for x,y = 20 µm

σδ,resolution ~ 1.3×10-4

σt,resolution ~ 16 fs

4.5 MV & 12 slices, 200 pC


7070

Beam Profiles along the Diagnostic Section

TDS2 Off for Projected Emittance Measurements

TDS2 = 4.5 MV, DM = 6 deg for Slice Emittance & Long. Phase Space Measurements

K1@Q18 = 4

X-band on

σδ,resolution ~ 1.3×10-4

σt,resolution ~ 16 fsQ = 200 pC

reconstruction @ OTR6


7171

Analytical Approach of Twiss Parameters

From Hamiltonian equation of charge particles in a sector dipole without any fringe field

(see Conte's book pages 42-54),

0p

p

''x

''y

consider linear terms only!


7272


After adding quadrupole focusing effects (see Conte's book page 82)

and bending effects of a sector dipole together, a generalized equation of motion of charge

particles in magnets supplying bending and focusing effects can be given by

here

and energy spread was considered.

0B


7373


From the generalized equation of motion of charge particles in magnets

which supply bending and focusing effects

Let's ignore the energy spread term (later we will consider it). Then we can get a

generalized Hill's equation:

here z represents x or y and k(s) represents kx(s) and ky(s).

Since k(s) is a periodic function with a period of L, the Hill's equation can have a quasi-

periodic solution, which is related to the Twiss parameters:

here , , w(s) is an amplitude function with a

period of L, w(s+L) = w(s), A & B are constants, (s) is a non-periodic phase.


Since zA(s) and zB(s) are solution of the Hill's equation, let's substitute zA(s) in the Hill's

equation:

To valid this equation for all values of , coefficients of sine and cosine functions should

be zero:

→

→

If we insert the last equation to the first equation:

7474


1''ln2'ln0'2

'

'' 22

wCwCw

w

w


From a general solution of the Hill's equation, we can get the slope z'(s):

☺

☺

here, we used .

By using initial conditions ( ), we can find two

constants A & B:

→

by inserting these A & B in z(s) and z'(s) above (see ☺ region), we can find transfer matrix

M(s) connecting two points z and z0 →7575



Transfer matrix M(s) connecting two points z and z0:

here

If we define following things (Twiss parameters)

7676



If we insert these relationships to a(s), b(s), c(s) , and d(s), then Transfer matrix M(s)

becomes:

From , , and → , then M(s) can be

written as:

and from

we can find that M(s) above is coincides with:

7777


Lss 0 00 ')(' wsw

)(/))(1()(

)()()(

)()()(

2

000

000

sss

sLss

sLss


From , the total phase advance per cell and total tune can be given by:

here, N is the number of cells, and QH and QV are the horizontal and vertical tunes.

Note the fact that if the transfer matrix of position and slope is given by

then, transfer matrix for the Twiss parameters is given by:

7878



From

→

If we re-arrange z(s):

Note that z(s) is a form of . Let's re-write z(s) as

we find 7979

Beam Phase Space and Emittance

from cos(A+B) = cos(A)cos(B) - sin(A)sin(B)


Similarly,

→

By squaring and adding following two equations, we can get constant W:

→

W is related to initial phase space coordinates (z0 and z'0).

Similarly, by squaring and adding z(s) and z'(s),

8080



→

W is also related to new phase space coordinates (z and z').

Therefore constant W is an invariant of motion, so called Courant-Snyder invariant, and it

is related to the area of a rotated ellipse equation of z and z' and beam emittance(see Conte's book page 105). Here W is similar to the total energy of a harmonic oscillator.

Since the area of a generalized rotated ellipse is given by

the total area of z and z' phase space ellipse is πWif we use .

At a certain accelerator location s1, initially, a charge particle is at a

point on a phase space ellipse (z, z'), the particle is continuously

located at some other point on the same ellipse in successive cycles

or turn of the periodic motion if tune is an irrational number.

8181


phase space (z, z') area of 100% particles = πW

z'(s1)

z(s1)

1st turn

2nd turn

3rd turn

4th turn


The phase space ellipse (z, z') in previous page corresponds to the phase space at a

point (s = s1) in the beamline. Therefore, the shape of the ellipse is not changed even

beams circulate many turns at s1. But its shape can be changeable at different locations

along the beamline as shown below. Even though its shape is changed along thebeamline, its area will be same (πW) at all locations if there is no any dissipative actions

such as space charge force, wakefields, or coherent synchrotron radiation.

8282


area @ s1= πW

z'(s1)

z(s1)

1st turn

2nd turn

3rd turn

4th turn

area @ s2= πW

z'(s2)

z(s2)

area @ s3= πW

z'(s3)

z(s3)

s = s1 s = s2

s = s3


Let's define a new smaller ellipse of the phase space (z and z'), which contains some fraction

of particles (example, 90%). If the area of the smaller ellipse is πε, we define ε as the beam

geometrical (or projected) core emittance of the fractional particles.

Geometrical (Projected) Core EmittanceIf the rms emittance is εrms, (see next page),

and the core emittance ε is nεrms, then, the

percentage of particles contained within the

ellipse of the core emittance is 100(1-e-n/2) (%).

8383


total area = πWcontained particles = 100%

z'

z

z'

z

total area = πε

contained particles

= some fraction, ex, 95%

see S.Y. Lee's book

square of 1D

z

z'

πεrms (= area in 39%)

πε = nπεrms

πW (100%)

n =

100(1-e-n/2)(%)


8484


The rms emittance rms for a normalized beam distribution ρ(y,y') with

is defined as:

here, <y> and <y'> are average of position and angle.

σy & σy' are the rms values of beam size and angle.

and r is the correlated coefficient.

rms is estimated for 100% particles but its value

is corresponding to area having about 39% particles.

see more details in S.Y. Lee's book

max amplitude of β-tron motion =

max divergence (angle) of β-tron motion =

example of (y,y'):

2

'

2'0

2

20

2

)'(

2

)(

)',(yy

yyyy

Aeyy

2

20

2

)(

22

1)( y

yy

y

ey

)(s

)(s


8585


In linac where beams are accelerated, generally, we use the projected normalized beam

emittance, which is an invariant and given by (see Conte's book pages 107-108).

Adiabatic damping of beamsize, angle, and emittance in linac:

Increased energy or γ → geometrical emittance ε becomes smaller, and beamsize and

divergence become smaller. But note that the normalized emittance is an invariant

without any dissipative actions such as space charge force,

wakefields, or coherent synchrotron radiation.

See Sadiq's & Chris's term projects

on QM scanning based emittance measurement method. http://www.physics.isu.edu/~yjkim/course/2010fall/2010fall_ap_term_project04.pdf

http://www.physics.isu.edu/~yjkim/course/2011spring/2011spring_ap_project07.pdf

http://www.physics.isu.edu/~yjkim/course/2011spring/2011spring_ap_project08.pdf

2

0/

1~for

cmU

c

v

γ

n

, σx,y, σ'x,y


The generalized equation of motion of charge particles in magnets supplying bending

and focusing effects is given by:

Since the generalized solution of the homogeneous equation with = 0 is given by

→ .

Then the generalized solution of the inhomogeneous equation with ≠ 0 can be written

as

→

Here D(s) is a particular solution with 0 = 1 (see Problem 5-8 in Conte's book).

8686

Dispersion Function (s)


8787


From the initial conditions (x0, x'0) at s = 0:

Since no change in energy spread is assumed, trajectory equations can be written in

matrix form for ≠ 0 :

Here, the trajectory x(s) has two parts: a part due to betatron oscillation, xβ(s) and

the other part due to dispersion or


If we insert the trajectory due to the periodic dispersion in the matrix

form:

For a periodic cell, x0 = (s0) 0=(s0+L) = (no energy spread change here).

If we solve equation above, we can find a periodic dispersion function (s) and '(s).

here we used tr(M)= C +S' = 2cos for .

8888


(see Problem 5-8 in Conte's book & S. Y. Lee's book)


8989

RMS Beamsize

In a horizontal dispersion region, two things contribute to the rms horizontal beamsize.

one is due to the betatron amplitude and the other is due to the dispersion function η

and the rms momentum spread σp:

Here we assumed that there is a horizontal bending.

Without any vertical dispersion, the rms vertical beamsize can be written:

222

Epp

Exx

nxp

xxnxp

xxxx

yyy

beamsizermsfor)()(,)()(p

ssxssxp

xp


9090

For Gaussian beam distribution,

probability for -1 < x < +1: 68.3%



Therefore, the beam full width is close to a region from -3 to +3 = 6 (99.7%).

x

beam full width = beam diameter ~ 6

full width at half maximum (FWHM)

35482.22ln22FWHM

Full Width & FWHM for Gaussian Distribution

courtesy of Wikipedia


If we focus beam very strongly to get a tiny beamspot at a undulator or radiator, can we

reduce beam emittance? How about beam divergence after the point?

If there is no any special action, the beam emittance is not reduced even though we focus

beam very strongly. Here, only -function (instead of emittance) is reduced or minimized

for the strongly focused beam;

for no dispersion case.

In this case, the beam divergence is dramatically increased, and there is a big beam loss

in a undulator gap with a tiny gap if we focus beam too strongly with a poor emittance;

To reduce the loss, we have to improve beam emittance or reduce strength of focusing.9191

Mis-concept with Strongly Focused Beam

)1

(

2

y,x

y,x

y,xy,xy,x'y,'x

y,xy,xy,x

undulator gap ~ 4-6 mm

THz-Radiator gap = 1.2 mm


9292

Dispersion Control by Quadrupole

By control phase advance or tune with QMs, we can control dispersion in the nono-zero

dispersion area.

See following pages on dispersion control in the HRRL beamline and Bunch Compressor.

See also dispersion suppressor at pages 122-125 in Conte's book.

0)(for0)( 0 ss


93

Dispersion Suppression @ HRRL Beamline

2.525 m

2.795 m

15 cm long QM (Quad2T) with 2 inch ID

24 cm long QM (Q1B) with 2 inch ID

1st and 2nd target

Kiwi dipole ~ 0.25 m (pole edges)

movable hole for transverse collimator

movable slit for energy collimation

beam pipe with ID > 36.1 mm

Faraday cup

movable screen

LINAC T1 TCOL1 1STTG T2 TCOL2

0.3 m

KIW

I T

3 2N

DT

G

dispersion controlling QM


94



dispersion controlling QMxxx

2

p

p

xxxx

After dipoles, horizontal dispersion (x) and energy spread

are the main contributions in the beam size!

Before dipoles, horizontal emittance (x) and beta function


x controlled


95



dispersion controlling QMxxx

2

p

p

xxxx

After dipoles, horizontal dispersion (x) and energy spread


Before dipoles, horizontal emittance (x) and beta function


x controlledx uncontrolled


96


initial assumed beam parameters

normalized emittance = 16 m

beam energy = 3 - 10 MeV

energy spread = 13% (FWHM) 5.52% (rms), 33.12% (FW)

Q = 50 pC

bunch length = 25 ps (FWHM)


KIWI KIWI dipole

controlled dispersion


97


initial assumed beam parameters

normalized emittance = 16 m

beam energy = 3 - 10 MeV

energy spread = 13% (FWHM) 5.52% (rms), 33.12% (FW)

Q = 50 pC

bunch length = 25 ps (FWHM)


KIWI KIWI dipole

controlled dispersion

uncontrolled dispersion

2

p

p

xxxx


9898

Dispersion Control @ Bunch Compressor

Two QMs in Bunch Compressor can control the residual dispersion after

Bunch Compressor.


Trajectories of a design particle and an off-momentum particle in an infinitesimal arc

is shown in figure below.

From the definition of the momentum compaction factor

(see Chapter 2 in Conte's book),

the circumference of a design particle can be given by

and the circumference of an off-momentum particle can be given by

The infinitesimal azimuthal angle d is common for both particles:

9999

Momentum Compaction Factor in Ring

d

p p+dp

xp

ds d


After using , then ∆L can be given

The momentum compaction factor can be a function of dispersion and bending radius.

100100

Momentum Compaction Factor in Ring

d

p p+dp

xp

ds d

→

→


101101

Momentum Compaction Factor R56 in BC

→

3

2

2

10

2

210

2

011566

0056

3

5666

2

56656

2

10

2

10

2

)('

)(

)(

)(

)(

here

p

p

p

p

p

p

p

px

p

p

p

p

dss

s

sLT

dss

sLR

p

pU

p

pT

p

pRdzdz

p

pL

p

pLL

p

p

p

p

L

L

p

if

Momentum Compaction Factors in Chicane Type Bunch Compressor

A. Nadji et al., NIMA 378 (1996) 376


102102

Momentum Compaction Factor R56 in BC

dz

Momentum Compaction Factors in Chicane Type Bunch Compressor

ELEGANT code supplies R56, T566, U5666 in BC


103103

Chromaticity

H. Wiedemann, Particle Accelerator Physics

Focal property of a lattice with QMs depends on the normalized strength k of QMs,

which, in turn, depend on the charge particle's momentum (chromatic effects).

→ beamsize growth hence beam emittance dilution due to energy spread.

0

0

00

0

0

0

11)(

)(

1

)()()(

p

pk

p

p

x

sB

p

e

x

sB

p

pp

e

x

sB

pp

e

x

sB

p

esk

y

yyy

Chromatic Effects becomes Stronger: larger β-function

larger energy spread = p/p

stronger QM normalized strength k

longer QM length LQM

βkLQM << 1 to ignore chromatic effects

see Sadiq's term project on chromatic effects!

1212

22

11


104104

Chromaticity Correction

H. Wiedemann, Particle Accelerator Physics

SF

SD

Natural Chromaticity ξN (ksi): the fractional difference in tune with respect to the fractional

difference in the particle momentum. The natural chromaticity can be corrected by putting

focusing and defocusing sextupoles as shown figures below.

before correction


105

Example of Chromatic Effects

SB03 SB04 XB 4QMs BC 5QMs 3FODO LOLA 2QMs DUMP

strong focusing optics against CSR in BC

x-function ~ 0, x-function ~ 6

Chromatic Effects becomes Stronger:

larger β-function

larger energy spread

stronger QM strength k


~ 2%

β ~ 60

K1@QM ~ 15

PSI 250 MeV Injector Version-1


106

SB03 SB04 XB 4QMs BC 5QMs 3FODO LOLA 2QMs DUMP

~ 2%

β ~ 60

K1@QM ~ 15

Example of Chromatic Effects

Chromatic Effects becomes Stronger:

larger β-function

larger energy spread

stronger QM strength k


PSI 250 MeV Injector Version-1


107

Interaction of charged beams & discontinuous surroundings

moving charged beam fields from the beam response from surroundings

fields from surroundings response of the beam changes in energy or emittance of

following bunches (or at the next turn of the same bunch) (long-range wakefields) /

changes in energy and angle (hence emittance) of following electrons in the same bunch

(short-range wakefields).

Wakefields - Definition


108

Short-Range Wakefields in Linac Accelerators

If an electron bunch moves in a periodic linac structure, there are interactions

between the electrons in a bunch and the linac structure, which induce

changes in beam energies and beam divergences (x' and y') of electrons in the

same bunch. We call these interactions between electrons in the same bunch

and the linac structure as the short-range wakefields, which change beam

energy spread and emittance of the bunch.

blue: an interaction between an electron at the head

region and a linac structure.

pink: short-range wakefield from the linac structure

to a following electron at the tail region.

2a

A. Chao's Handbook of Accelerator Physics & Engineering, p. 252

SLAC-AP-103 (LIAR manual)


Energy loss Ei of a test electron (or slice) i in a bunch due to the short-range

longitudinal wake function WL(s), which is induced by all other preceding

electrons j located at s = |i - j| distance from the test electron i is given by

Here qi and qj are charge of electron (or slice) i and j, and L is the length of the

linac structure. i or j = 1 means the head electron in the bunch, and the sum

term is only evaluated for i > 1.

The transverse trajectory deflection angle change xi' of a test electron i due to

the short-range transverse wake function WT(s), which is excited by all

preceding electrons j is given by

Here the sum term is only evaluated for i > 1.109

Short-Range Wakefields in Linac Accelerators

electron j moving with v ~ c

a test electron i with a distance s away

from preceding electron j and moving with v ~ c

.)(2

)0( 1

1

LqjiWqW

Ei

j

jLiL

i

.)(1

1

'

i

j

Tjji jiLWxqx



L

Longitudinal wake function WL (s) of the test particle in a bunch is the voltage

loss experienced by the test charged particle. The unit of WL (s) is [V/C] for a

single structure or [V/C/m] for a periodic unit length. The longitudinal wake

is zero if test particle is in front of the unit particle (s < 0). For a bunch of

longitudinal charge distribution z, the bunch wake (= voltage gain for

the test particle at position s) is given by

And the minus value of its average gives the loss factor and its rms

value gives energy spread increase:

where L is the length of one period cell, N is the number of electrons in the

bunch.

110

Longitudinal Short-Range Wakefields

a unit charged particle moving with v ~ c

a test charged particle with a distance s away

from the unit charged particle and moving with v ~ c


SLAC-PUB-11829

SLAC-PUB-9798

TESLA Report 2004-01



111


a unit charged particle moving with v ~ c

a test charged particle with a distance s away

from the unit charged particle and moving with v ~ c


SLAC-PUB-11829

SLAC-PUB-9798



red: without short-range wakefield

green: with short-range wakefield

increased nonlinearity in longitudinal

phase space


112


Longitudinal impedance is the Fourier transformation of the longitudinal

wake function:

Yokoya's wakefield model for periodic linac structure:

L


MHI 2π/3 Mode C-band Structure average inner radius a = 6.9535 mm

average outer radius b = 20.10075 mm

period p = 16.6667 mm

iris thickness t = 2.5 mm

cell number for 2 m structure = 119

attenuation constant τ = 0.452

average shunt impedance = 69.5 MΩ/m

filling time = 222 ns

RF pulse length = 0.5 µs

required RF power for 28 MV/m = 38 MW

one 50 MW klystron can drive 3 structures

This structure is used for linac Optimization-XIV

and Optimization-XV with RF Option-IV.

PSI 3π/4 Mode C-band Structure average inner radius a = 6.9545 mm

average outer radius b = 20.7555 mm

period p = 18.7501 mm

iris thickness t = 4.0 mm

cell number for 2 m structure = 106

attenuation constant τ = 0.630

average shunt impedance = 66.1 MΩ/m

filling time = 333 ns

RF pulse length = 0.5 µs

required RF power for 26 MV/m = 28.5 MW

required RF power for 28 MV/m = 33 MW

one 50 MW klystron can drive 4 structures

This structure is used for linac Optimization-XVII,

and Optimization-XVIII with RF Option-VII, VIII. 113113113

Wakefield of Two C-band Linac Structures

disk loaded type linac structure

2a

p

2b


MHI 2π/3 Mode C-band Structure (red lines in plots below)

This structure is used for SwissFEL linac Optimization-XIV and Optimization-XV with RF Option-IV.

PSI 3π/4 Mode C-band Structure (black lines in plots below)

This structure is used for SwissFEL linac Optimization-XVII, and Optimization-XVIII with RF Option-VII or

RF Option-VIII.

114114114

both structures have almost same short-range wakefields !

Short-Range Wakefields of Two C-band Structures


115

Why we need Bunch Compressor?

Optimization-XVIII with PSI C-band RF Structures for 2.7 kA

To reduce saturation length of XFEL, we need a high peak current (~ kA).

But normal guns can not supply such a high current due to the longitudinal

space charge force.

To generate femtosecond photon beams, the bunch length of electron beams

should also be femtosecond range.

To avoid the longitudinal space charge effect, we have to compress bunch

length at high energies. → We need the bunch compressor(s) at one or two

positions in linac.


116

Working Prinicple of Bunch Compressor (BC)

Bunch Compressor Layout for SCSS Project - Y. Kim et al, NIMA 528 (2004) 421

chicane. bendr rectangulathefor2

3where

))/(()/()/(

56566

32

56656

RT

EdEEdETEdERdzdz iiiif

)3

2(2

2

56 BB LLR

dt

dETail

Head

from precompressor linac from chicane

dz


117117117117

BC for PSI 250 MeV Injector Test Facility

ASTRA up to exit of INSB02 & ELEGANT from exit of INSB02 to consider space chare, CSR, ISR, and wakefields !


118118118118


10.5 m

0.75 m4.375 m

~ 0.5 m

0.25 m

QMs

length ~ 0.1 m

max gradient ~ 1.5 T/m

BPM

Collimator

OTR Screen

BM

BM BM

BC1 Dipoles

length = 0.25 m

max bending angle ~ 5 deg @ 250 MeV

0.4

04

m Radiation Port

E = 255.9 MeV

~ 1.673%

R56 = 46.8 mm

= 4.1 deg


119119119119

Knobs to minimize Emittance Growth @ BC

From field tolerance studies, we assumed following dipole field errors in BC

dipoles: Δb/b0 = 3.32×10-3 b1/b0 = 8.70×10-5

b2/b0 = 1.83×10-5 b3/b0 = 6.65×10-5 b4/b0 = 4.80×10-5

In this case, the minimum projected emittance can be obtained by

compensating residual dispersion with two small QMs in BC.

After BC1, zero residual dispersion

can be obtained when k1 of QMs

is -0.04 (1/m2). In this case,

horizontal projected

emittance after BC1 also

becomes its minimum.


Ipeak ~ 352 A

Ipeak ~ 22 A

E ~ 256 MeV, Q = 0.2 nC

~ 1.67%, z = 840 m

nx~ 0.35 m, ny~ 0.35 m

E ~ 256 MeV, Q = 0.2 nC

~ 1.67%, z = 58 m

nx~ 0.38 m, ny~ 0.35 m

120120120120



121

FODO Lattice

The most simplest lattice block of modern accelerators = a pair of QF and QD.

FODO cell = a simplest periodic lattice block with QF, O, QD, and O in a cell.

Here O can be a drift space, a bending magnet, undulator, or linac structure.

If the horizontal plane has a FODO cell, the vertical has a DOFO cell.

Betatron phase advances for x and y planes can be different (See SwissFEL DBC1).

Length of One FODO Cell with Two Linac Structures = l = 10.3 m

QF S-band Tube QD S-band Tube

4.3 m long S-band tube 4.3 m long S-band tube

0.7 m long diagnostic section

0.15 m long QM 0.15 m long QM

22 MV/m 22 MV/m


122

FODO Lattice

Length of a FODO Cell = l

QF Drift QD Drift

l/2 l/2

To find the maximum and minimum β-function and phase advance of a FODO cell,

let's consider QMs as thin lens ( )

Then,

Total transfer matrix for one FODO cell:

From generalized strong-focusing transfer matrix, we can put;

By taking trace of the matrix → →

1QMlk

1

101

1

01M DF,

fklQM

βmax

βmin


123

FODO Lattice

By comparing M12 component:

By using following relationships:

2sin1

sin41

sin

4sin

2

l

f

ll

f

ll

2sin1

sin

2sin1

sin

min

max

l

l

2sin1

2sin1

2sin2

2sin1

2sin2

2cos

2sin2sin

2

2sin4

fl


From , we can find a phase advance per FODO cell ().

→ total phase advance for Nc cell FODO lattice = Nc

→ betatron tunes for the Nc cell FODO lattice is given by

If we know length of one FODO cell l, phase advance per cell , then f and βmax and βmin

are automatically determined.

124

FODO Lattice


125

Example of FODO Lattice in a Linac

QF TUBE QD TUBE QF TUBE QD TUBE QF

length of one FODO cell in a LINAC

= two 4.3 m long PSI standard S-band tubes

+ two 0.7 m long PSI standard diagnostic sections

+ two 0.15 m long PSI standard QMs = 10.3 m

phase advance per FODO cell = 60 deg

max and min β-function ~ 17.3 m and 5.8 m

normalized QM strength K1 ~ 1.3 m-2

QM gradient @ 1.5 GeV ~ 6.5 T/m

FODO

ignorable chromatic effects at the LINAC

SwissFEL Linac1 Optimization-1


126126

5QMs TDS2 5QMs 3FODO 1QM DIPOLE

3FODO Cells

OTRs : 1 2 3 4 5 6 7 8

horizontal phase advance @ 3FODO = 55 deg per cell

vertical phase advance @ 3FODO = 25 deg per cell

cell length = 3.0 m

rms beam size @ 7FODO screens ~ 60 m for 200 pC

max dispersion by a dipole in DIAG1 ~ 0.2 m

With this special DIAG1,

we can measure followings

without change any optics:

- slice emittance

- slice energy spread

- longitudinal phase space

- bunch length

- arrival timing jitter

- projected emittance

- Twiss parameters

- optics matching\

only by turning on and/or

off TDS2 and a dipole in

DIAG1.

FODO Lattice - DIAG1 @ SwissFEL Injector


127127

SwissFEL OPT-III & VII

Optimization-III with a longer S-band RF Linacs for Chirp Compensation

Optimization-VII with Shortest C-band RF Linacs for Chirp Compensation


128128

SwissFEL - S-band based LINAC2 after BC2

length of one FODO cell in LINAC2



+ two 0.2 m long QMs = 10.4 m

pure active length per tube = 4.073032 m

number of cell per tube = 122 including two coupler cells

central cell length = 33.333 mm

iris diameter = 25.4 mm

total cells in LINAC2 = 34 FODO cells

No. of S-band tubes = SB23-SB90 for 34 FODO cells

total needed S-band tubes in LINAC2 = 68

total needed RF stations = 34 with two tubes per station

total needed QMs in LINAC2 = 2x34 = 68

total length of LINAC2 = 353.6 m

One FODO Cell for LINAC2 = 10.4 m

QF 4.3 m long S-band Tube QD 4.3 m long S-band Tube

2998 MHz S-band Tube 2998 MHz S-band Tube

0.7 m long diagnostic section

0.2 m long QM 0.2 m long QM

22 MV/m 22 MV/m

LINAC2 for Optimization-III

Optics for S-band Based LINAC2


129

Example of FODO Lattice in a Linac

QF TUBE QD TUBE QF TUBE QD TUBE QF

length of one FODO cell in a LINAC



+ two 0.15 m long PSI standard QMs = 10.3 m

phase advance per FODO cell = 60 deg

max and min β-function ~ 17.3 m and 5.8 m

normalized QM strength K1 ~ 1.3 m-2

QM gradient @ 1.5 GeV ~ 6.5 T/m from page 53

FODO

ignorable chromatic effects at the LINAC

SwissFEL Linac1 Version-1


130130

Cathode @ 0.0 m

Distance : 0.21 m 0.78 m 1.27 m 2.47 m 19.0 m 25.31 m 28.90 m 30.80 m 32.70 m 34.60 m 38.40 m

ACC2

~ 13 MV/m for first four cavities

~ 17 MV/m for last four cavities

0.0 degree for no compression

E = 127 MeV

R56= 181.3 mm

= 18.0355 deg

~ 0.2% (on crest)

~ 40 MV/m

38 degree

from zero

crossing

e- beams

Q ~ 1.0 nC

z ~ 4.4 ps rms

ACC1GUN BC2

3 FODO cells for emittance measurement

QMs QMs

for matching

Screen : 2GUN 3GUN 4DBC2 6DBC2 8DBC2 10DBC2

Solenoid Laser IDUMP dipole

TTF2 Injector = GUN + Booster (ACC1) + BC2 + 3 FODO Cells

1.3 GHz TESLA Module 1.3 GHz TESLA Module

By optimizing TTF2/FLASH injector properly, we could get an excellent emittance at

the injector. Without any bunch length compression, projected normalized emittance

is about 1.1 µm for 90% beam intensity in 1.0 nC and 4.4 ps (rms) long bunch.

FODO @ DESY FLASH Facility - Diagnostics


131

DESY FLASH Facility - 3 FODOs @ DBC2

Q4DBC2H Q5DBC2 Q6DBC2 Q7DBC2 Q8DBC2 Q9DBC2 Q10DBC2H

OTR4DBC2 OTR6DBC2 OTR8DBC2 OTR10DBC2

1.9 m

4 OTR in 3 FODO cells

-OTR4DBC2

-OTR6DBC2

-OTR8DBC2

-OTR10DBC2

Cell phase advance = 45 deg

One cell length = 1.9 m

7 QMs in three FODO Cells

- Q4DBC2 - defocusing

- Q5DBC2 - focusing


- Q7DBC2 - focusing


- Q9DBC2 - focusing


0.46 m 1.9 m 1.9 m

DBC2 section


To minimize emittance, light sources use various type dispersion

suppressed achromat lattice.

Lattice for Synchrotron Light Source


]cm[]T[934.0,2

12

uo

2

2

u

BK

K

Various Lattice for Synchrotron Light Sources


Double-Bend Achromat (DBA)

Triple-Bend Achromat (TBA),

Quadrupole-Bend Achromat (QBA)

Multi-Bend Achromat (MBA) Courtesy of Zhentang Zhao

DBA Vs. MBA based Storage Ring

ringcisomagnetifor108319.3

BMs

213

s

J

MBAfor

81

2

3

BM

2

n .xCN

Somewhat lower energy, longer bending radius, and many dipoles with focusing

QMs are better for us to get a small energy spread and a small emittance.

→ MBA based Diffraction Limited Storage Ring (DLSR),

(electron emittance photon emittance)

4xx,y


ESRF Hybrid MBA for Large Dynamic Aperture

135

The MBA lattice has both

strong focusing and small

dispersion small

dynamic apertures and

large chromaticities.

To solve this problem, the

Hybrid MBA (HMBA) has

two separate β-functions

and dispersion bumps

located between the ending

dipoles and the first and

last inner dipoles of the

MBA cell, where sextupoles

are placed to ease the

chromaticity correction.

Sextupoles for chromaticity correction


Courtesy of Zhentang Zhao

Average Spectral Brightness and Coherent Fraction

136

DBA Vs. MBA based Storage Ring


Emittance Vs. Synchrotron Generation

MBA based

4th Genertion

SynchrotronDBA/TBA based

3rd Generation

Synchrotron

137

Great Emittance with MBA!

~ 10 pm n ns ~ 0.12 m @ 6 GeV Storage Ring


MBA based 4th Generation Synchrotron

4GSLS Group with the Highest Perfomance

New Korean Ring will be close to PEP-X

PLS-II


3GSLS Vs. MBA based 4GSLS

3GSLS, ex) PLS-II

Structural Images of a Biological Sample with Different Wavelengths(0.5 - 5 Å )

139

4GSLS Group with the Highest Perfomance

New Korean Ring will be close to PEP-X


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