lecture 3 - magnets and transverse dynamics i · lecture 3 - e. wilson –- slide 2 slides for...

Lecture 3 - E. Wilson –- Slide 1

Lecture 3 - Magnets and Transverse Dynamics I

ACCELERATOR PHYSICS

MT 2009

E. J. N. Wilson


Slides for study before the lecture

Please study the slides on relativity and cyclotron focusing before the lecture and ask questions to clarify any points not understood


Relativistic definitions

E m0c2

1 2

p mv m0v

1 2

m0c

1 2

1

1 v c 2

1

1 2

E

E0

E0 m0c2

Energy of a particle at rest

Total energy of a moving particle(definition of )

E E0 m0c2

Another relativistic variable is defined:

Alternative axioms you may have learned

You can prove:

momentum c

energy

pc

E

v

c

pc E m0c2 ( )


Newton & Einstein

Almost all modern accelerators accelerate particles to speeds very close to that of light.

In the classical Newton regime the velocity of the particle increases with the square root of the kinetic energy.

As v approaches c it is as if the velocity of the particle "saturates"

One can pour more and more energy into the particle, giving it a shorter De Broglie wavelength so that it probes deeper into the sub- atomic world

Velocity increases very slowly and asymptotically to that of light


dpdt

pddt

pdds

dsdt

p

dsdt

ev B edsdt

B

Magnetic rigidity

1

dds

B pe

pcec

Eec

E0

ec

m0ce

Fig.Brho 4.8

dθ dθ

GeV pc 3356.3

.T.m 1 smc

eVpcecpc

B


Equation of motion in a cyclotron

Non relativistic

Cartesian

Cylindrical

xyz

zxy

yzx

ByBxqdt

zmddtmvd

BxBzqdt

ymddt

mvd

BzByqdt

xmddtmvd

r

zr

z

BrrBqdt

zmd

BrBzqrmdtmrd

BzBrqmrdt

rmd

2

2

Bvv

qdtmd )(

Fv

dtmd


For non-relativistic particles (m = m0 ) and with an axial field Bz = -B0

The solution is a closed circular trajectory which has radius

and an angular frequency

Take into account special relativity by

And increase B with to stay synchronous!

Cyclotron orbit equation

20

0

0

2

0

z

z

m r r qr B

m r r qrB

m z

z

pR

qB

w q

m0

B0

0z

qB

m

000 E

Emmm


Cyclotron focusing – small deviations

See earlier equation of motion

If all particles have the same velocity:

Change independent variable and substitute for small deviations

Substitute

To give

20

0 0z

mvd dm ev B

dt dt

0v z

2 0z

d mrmr q r B zB

dt

000 - x, , BBBdsd

vdtd

zz

00 mvp

0 20 0 0 0

1 10zBd dx x

pmv ds ds B


From previous slide

Taylor expansion of field about orbit

Define field index (focusing gradient)

To give horizontal focusing

Cyclotron focusing – field gradient

0 20 0 0 0

1 10zBd dx x

pmv ds ds B

........!2

1 22

2

0 xxB

xx

BBB zz

z

xB

Bk z

00

1

011

200

xk

ds

dxp

ds

d

p


Cyclotron focusing – betatron oscillations

From previous slide - horizontal focusing:

Now Maxwell’s

Determines

hence In vertical plane

Simple harmonic motion with a number of oscillations per turn:

These are “betatron” frequencies

Note vertical plane is unstable if

011

200

xk

dsdx

pdsd

p

xB

z

B zx

0

0 B

kzBBx 00

01

00

kz

dsdz

pdsd

p

kQkQ zx ,1

2

yx QQ ,

2

1

k


Lecture 3 – Magnets - Contents

Magnet types

Multipole field expansion

Taylor series expansion

Dipole bending magnet

Magnetic rigidity

Diamond quadrupole

Fields and force in a quadrupole

Transverse coordinates

Weak focussing in a synchrotron

Gutter

Transverse ellipse

Alternating gradients

Equation of motion in transverse coordinates


Dipoles bend the beam

Quadrupoles focus it

Sextupoles correct chromaticity

Magnet types

x

By

x

B y

0

0



(r ,)Scalar potential obeys Laplace

whose solution is

Example of an octupole whose potentialoscillates like sin 4around the circle

2x2

2y2 0 or

1r2

2 2

1rr

rr

0

nrn sinn

n1



n

n1

r n sin n

Field in polar coordinates:

Br

r

, B 1

r

Br nnr n1 sinn , B nnr n1 cosn

Bz Br sin B cos

nnrn1 cos cosn sin sin n

nnr n1 cos n 1 nnxn1 (when y 0)

To get vertical field

Taylor series of multipoles

Fig. cas 1.2c

Octupole Sext Quad Dip.

...!3

12

!11

......4.3.2.

33

32

2

2

0

34

2320

xxB

xxB

xx

BB

xxxB

zzz

z


Diamond dipole


Bending Magnet

Effect of a uniform bending (dipole) field

If then

Sagitta

/ 2

BB

222sin

BB

2

1616

2cos12

2


dp

dt p

ddt

pdds

ds

dt

p

ds

dt

ev B edsdt

B

Magnetic rigidity

1 d

ds

B T.m pc eV

c m.s1 3.3356 pc GeV

B pe pc

ec E

ec E0

ec m0c

e

Fig.Brho 4.8

d


Diamond quadrupole



No field on the axis

Field strongest here

B x(hence is linear)

Force restores

Gradient

Normalised:

POWER OF LENS

Defocuses invertical plane

Fig. cas 10.8

1

. yBk

B x f

1

. yBk

B x

x

By



s



Vertical focussing comes from the curvature of the field lines when the field falls off with radius ( positive n-value)

Horizontal focussing from the curvature of the path

The negative field gradient defocuses horizontally and must not be so strong as to cancel the path curvature effect

The Cosmotron magnet


Gutter


Transverse ellipse

/

/. Area


Equation of motion in transverse coordinates

Hill’s equation (linear-periodic coefficients)

where at quadrupoles

like restoring constant in harmonic motion

Solution (e.g. Horizontal plane)

Condition

Property of machineProperty of the particle (beam) Physical meaning (H or V planes)

EnvelopeMaximum excursions

k 1B

dBz

dx

s ds

s

y s ˆ y / s

s

y s sin s 0

d 2yds2 + k s y 0


Summary

Magnet types



Dipole bending magnet

Magnetic rigidity

Diamond quadrupole




Gutter

Transverse ellipse


Equation of motion in transverse coordinates

lecture 3 - magnets and transverse dynamics i · lecture 3 - e. wilson –- slide 2 slides for...

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