1

lecture 4

• multiparticle effects i.e. the effect of the self field of the beam

– 1) ion linac : space charge

– 2) electron linac : wake field

• virtual tour of the CERN hadron LINACS

2

space charge

• Coulomb force between particles in a bunch

• Forces that pushes particles away from each other, equivalent to a defocusing force (in both planes)

• Space charge force is zero on the beam centre (for symmetry)

3

space charge

• we have to keep into account the space charge force when we determine the transverse and longitudinal focusing in the accelerator

• Part of the focusing needed in LINAC goes to counteract the space charge forces

4

estimate of the space charge

• assume the beam as a uniformly charged ellipsoid

effect iszero on the centre of the beam : the contribution of the two red particles on the green one canceles out.

zrrr

f

c

IE

yrrrr

f

c

IE

xrrrr

f

c

IE

zyxz

zyxyy

zyxxx

3

4

1

)(

13

4

1

)(

13

4

1

0

20

20

I=beam current, rx,y,z are the semiaxis of the ellypsoid, f is a form factor

LINEAR!!!!

5

transverse phase advance with space charge

• transverse force balance

• • quadrupole focusing (1)

• • RF defocusing (2)

• • space charge defocusing (3)

zyx

t rrrmc

fqIZ

mc

TqE32

30

320

2

40

8

13sin

8

(1) (2) (3)

6

transverse phase advance with space charge

zyx

t rrrmc

fqIZ

mc

TqE32

30

320

2

40

8

13sin

8

Z0 is the free-space impedance (376.73 Ohm), I is the beam current, f is a geometrical factor, function of the ratio of the transvese beam dimensions, rx,y is the average transverse beam dimension, rz the longitudinal

beam volume

beam current

when we bunch the beam we

increase the effect of space charge !!!!

7

non linear effects of space charge

• the more the beam is compressed in real space, the more the space charge effect is non linear

• non linear space charge effect generates emittance growth

• at low energy space charge is the limiting factor for the minimum emittance that can be produced out of an accelerator

8

EMITTANCE OUT VS EMITTANCE IN

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

emittance in ( rms, norm, mm mrad)

em

itta

nc

e o

ut

ein=eout (CHASMAN PLOT) (CHASMAN PLOT)

FOR CERN RFQ2 FOR CERN RFQ2 (200MHz., 200 mA (200MHz., 200 mA PROTONS)PROTONS)

9

emittance growth due to filamentation

• velocity of rotation in the transverse phase space with no space charge doesn’t depend on the amplitude.

• with linear space charge it is lowered but it still doesn’t depend on the amplitude

• with non linear space charge it does depend on amplitude and therefore there are areas of the phase space that move at different velocity. This generates emittance growth.

10

Non-linear forces : Filamentation

11

Filamentation : emittance increase

Linear force Non linear force

Evolution of the emittance along an accelerator under the influence of linear forces only (blue line) or non-linear forces (red line)

12

DTL output longitudinal emittance

(1) (2)

(1) el_in=0.21

(2) el_in=0.18

(3) el_in=0.12

(4) el_in=0.03

emittance r.m.s. in deg MeV

(3) (4)

13

14

• • Halo can produce beam loss and activation that makes maintenance difficult and time-consuming.

• • Control of beam halo and beam loss is necessary for high beam availability in high-power proton linacs.

• Calculation of the behaviour of the halo is at the limit of the capability of modern computers

• Rule for hand-on-maintenance is to control the losses to 1Watt/m. For a 2 GeV 40 mA proton beam this correspond to an accuracy of the calculation of the order of 10-8

Why beam halo is important

15

instabilities in e-linac

• Phenomenon typical of high energy electrons traveling in very high frequency structures (GHz).

• Electromagnetic waves caused by the charged beam traveling through the structure can heavily interact with the particles that follows.

• The fields left behind the particle are called wake fields.

16

wake field

a (source) charge Q1 traveling with a (small) offset x1 respect to the center of the RF structure perturbs the accelerating field configuration and leaves a wake field behind. A following (test) particle will experience a transverse field proportional to the displacement and to the charge of the source particle:

L=period of the structure

W= wake function, depends on the delay between particles and on the RF frequency (very strongly like f3)

17

wake field effect

• this force is a dipole kick which can be expressed like :

decreases with energy

121" xLmc

weQx

18

wake field effects

• Effect of the head of the bunch on the tail of the bunch (head-tail instabilities)

• In the particular situation of resonance between the lattice (FODO) oscillation of the head and the FODO+wake oscillation of the tail we have BBU (Beam breakUp) causing emittance growth (limit to the luminosity in linear colliders)

• Effect of one bunch on the following.

19

summary of lecture 4

• two multiparticle phenomena : space charge (low energy, low frequncy-ion linac) and wake field (high energy,high frequency- electron linac)

• Self field generated by the beam are the main limiting factor to LINAC performances

20

credits

• much of the material is taken directly from Thomas Wangler USPAS course (http://uspas.fnal.gov/materials/SNS_Front-End.ppt.pdf) and Mario Weiss and Pierre Lapostolle report (Formulae and procedures useful for the design of linear accelerators, from CERN doc server)

• from previous linac courses at CAS and JUAS by Erk Jensen, Nicolas Pichoff, Andrea Pisent and Maurizio Vretenar , (http://cas.web.cern.ch/cas)

21

Further reading

Thomas Wangler “RF Linear Accelerators”, Wiley Series in Beam physics and accelerator technology, ISBN 0-471-16814-9

22

QUESTIONS ????

23

beam dynamics tour of LINAC3

24

•SOURCE : produces 0.8 mA of lead ions composed of 10 charge states around 25+

•LEBT : selects one charge states (nominal : 25+)

•RFQ (100MHz) + 3 IH tanks (100 and 200MHz) : increase the energy from 0.0025 to 4.2 MeV/u

•STRIPPER : converts lead 25+ in lead 54+ (and 4 adjacent charge states)

•FILTER LINE : selects one charge states and delivers 25 μA of lead 54+

25

Low Energy Beam Transport

• continous beam• 10 charge states• generate a dispersion to select the wanted

charge• resimmetrize the

beam by means of a triplet

• match to the RFQ by means of a solenoid

26

Radio Frequency Quadrupole

increase of energy of 100 times,

increase of velocity of 10 times : the last cell is 10 times longer than the first one

27

Radio Frequency Quadrupole

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80 100 120 140 160 180

z (cm)

mod

ulat

ion

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

phi (

deg)

modulation

synchronous phase

fast bunching

boosting

max value = -23

28

IH tanks

208

Design ionEnergy range (MeV/u)Effective voltage gain (MV)Total length (mm)No. of 0 deg synchr. particle sectionsNo. of tanksFrequency (MHz):Tank 1Tank 2 and Tank 3No. of quadrupole tripletsNo. of accelerating gaps

Pb25+0.25 – 4.2

32.98129

53

101.28202.56

499

check the difference in transverse size for the two frequencies!!!Which is the bigger?

29

stripping

• why stripping at 4.2 MeV/u ???

• why not start directly wit high charge state?

30

•SOURCE : produces 0.8 mA of lead ions composed of 10 charge states around 25+

•LEBT : selects one charge states (nominal : 25+)

•RFQ (100MHz) + 3 IH tanks (100 and 200MHz) : increase the energy from 0.0025 to 4.2 MeV/u

•STRIPPER : converts lead 25+ in lead 54+ (and 4 adjacent charge states)

•FILTER LINE : selects one charge states and delivers 25 μA of lead 54+

31

stripping

• why stripping at 4.2 MeV/u ???

• why not start directly wit high charge state?

SPACE CHARGE LIMITATIONS !!!!