BASIC PHYSICS OF HIGH BRIGHTNESS ELECTRON BEAMS

Wai-Keung LauWinter School on Free Electron Lasers 2016Jan. 18th 2016

OutlineWhat’s a beam?How to describe beam quality?Self fieldsRelativistic effectsSingle particle motion near the axis of axisymmetric fieldsEvolution of beam envelopThe Vlasov modelBeam injectors for XFELs

Application of CPBScientific:

High energy acceleratorsNovel light sourcesGeneration of high power microwavesElectron microscopy

Industrial or medical:e-beam welding, additive manufacturing (3D printing)e-beam lithographyIon implantation for semiconductor fabricationRadiosurgery, proton or heavy ion therapy

EnergyHeavy-ion fusionAccelerator-driven nuclear reactors

What is a Charged Particle Beam?an “ordered flow” of charged particles

all particles are movingalong the same trajectoryfor a perfect beam

a random distributionof charges

something in between(real world)

Single Particle Dynamics in EM Fields

Relativistic dynamicsApplied EM fields are usually expressed as expanded series called paraxial approximationComplicated EM field distributions can only be solved by numerical methods. Examples of simulation codes are:

POISSON/SUPERFISHHigh Frequency Structure Simulator (HFSS) -- http://www.ansoft.com/products/hf/hfss/CST Microwave Studio -- http://www.cst.com/Content/Products/MWS/Overview.aspx

( ) ( )BvEqvmdtd

dtpd rrrrr

×+== 0γ

The Lorentz factor γ is in general not a constant of time

Lorentz force law

6

Paraxial Approximation of Axisymmetric Electric and Magnetic Fields

General assumptions:Consider only static electric and magnetic fields at this point.That is, no time-varying fields and displacement currents are excluded.

Electric and magnetic fields for axisymmetric lenses in a beam focusing system do not usually have azimuthal components (i.e. ).Other field components have no azimuthal dependence

In paraxial approx., fields are calculated at small radii from the system axis with the assumption that the field vectors make small angles with the axis

0

0

=×∇

=⋅∇

B

Er

r

0=∂∂

=∂∂

θθrz EE

zr EE << zr BB <<

0

0

=×∇

=⋅∇

E

Br

r

0=∂∂

=∂∂

θθrz BB

0;0 == θθ BE

7

Define electrostatic and magnetic potentials such that:

For symmetric fields that satisfies the above assumptions, then

and therefore φe and φm are functions of r and z only. Therefore, φe and φmobeys Laplace equation:

The following form for electrostatic potential is useful to derive paraxial approximations for electric fields:

eE φ−∇=r

mB φ−∇=r

0=∂∂

=∂∂

θθrz EE 0=⎟

⎠⎞

⎜⎝⎛∂∂

∂∂

=⎟⎠⎞

⎜⎝⎛∂∂

∂∂

θφφ

θee

zz

0=⎟⎠⎞

⎜⎝⎛∂∂

∂∂

=⎟⎠⎞

⎜⎝⎛∂∂

∂∂

θφφ

θee

rr

( ) 01, 2

22 =

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∇zf

rfr

rrzrf

( ) ( ) L+++==∑∞

=

44

220

2

02, rfrffrzfzrf ν

νν

( ) 000

=∂∂

−===r

r rrE φ

( ) ( )xExE −=

f represents either φe or φm. They are not a function of θ

8

Substitute f into the Laplace equation of function f gives

From recursion formula for the coefficients f2ν

we have,

Therefore,

or

( )[ ] 01222 2

02

222

1=′′+−+ ∑∑

∞

=

−∞

=

ν

νν

νν

ν

ννν rfrf

( ) 022 2222 =′′++ + ννν ff

40

4

4

02

64141

zff

ff

∂∂

=

′′−=

( ) ( ) ( ) ( )L−

∂∂

+∂

∂−= 4

4

42

2

2 ,0641,0

41,0, r

zzfr

zzfzfzrf

( ) ( )( )

( ) ν

νν

νν

ν

2

02

2

2,0

!1, ⎟

⎠⎞

⎜⎝⎛

∂∂−

=∑∞

=

rz

zfzrf

9

Hence, the axial and radial fields in paraxial approximation are:

( )

( ) L

L

−∂∂

+∂∂

−=

−∂∂

+∂∂

−=

4

44

2

22

3

33

644,

162,

zEr

zErEzrE

zEr

zErzrE

z

r

( )

( ) L

L

−∂∂

+∂∂

−=

−∂∂

+∂∂

−=

4

44

2

22

3

33

644,

162,

zBr

zBrBzrB

zBr

zBrzrB

z

r

(1)

(2)

(3)

(4)

the total Coulomb force acting on q by a thin volume dVof charge Q is independent of r !!

Planar Diode with Space Charge – Child-Langmuir Law

02

22

ερφφ −==∇

dxd

constxJ x == &ρ

( ) 02

2 =− xexm φ&

( ) ( ) 2/12/10

2

2 1/2 φε

φmeJ

dxd

=⇒

( )C

meJ

dxd

+=⎟⎠⎞

⎜⎝⎛⇒ 2/1

2/10

2

/24 φ

εφ

xmeJ 4/12/1

0

4/3 2234 −

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

εφ ( )

3/4

0 ⎟⎠⎞

⎜⎝⎛=⇒dxVxφ 2

2/30

2/1

02

94

dV

meJ ⎟⎠⎞

⎜⎝⎛= ε

[ ]22

2/306 /1033.2 mAdVJ −×=

x

e-φ=0

φ=V0

cathode

C=0 under the boundary conditions: φ=0 and dφ/dx=0 at x=0; the condition dφ/dx=~x1/3=0 at x=0 implies that the special case electric field at cathode surface is null (a steady state solution).

Emission of electrons from cathodes:1.Thermionic emission2.Secondary emission3.Photo-emission

with

Child-Langmuir law

rzfocusing continuous beam

For a given focusing channel, the size of beam waist is in general determined by beam emittance, space charge forces etc..

beam expansion dueto space charge

propagation of a continuous beam and a bunched beam in drift space

change of particle distribution in phase space due to space charge (transverse beam size, bunch length, divergence, energy spread etc..)

Electrostatic repulsion forces between electrons tend to diverge the beamThe current density required in the electron beam is normally far greater than the emission density of the cathodeOptimum angle for parallel beam is referred to as Pierce electrodesConical diode is needed for convergent flowDefocusing effect of anode aperture has to be considereda simple diode gun 140 kV Electron Gun System for TLS

A Cylindrical Beam in a Strong Magnetic Field

Particles enter the drift tube with kinetic energy qφb. A new potential will be setup in the drift tube which will reduce the K.E. of the particles according to energy conservation law.As the potential is strong enough, K.E. of particles completely convert into potential energy. There exists a beam current limit!!

φb

φ0

φ(z)

( ) ( ) ( )rqrqTrT eb φφ ≡−=

E-BEAM

- +φb

DRIFT TUBE

STRONG MAGNETIC FIELD

a bCATHODE

( ) ⎟⎠⎞

⎜⎝⎛ +=

ab

cI

e ln214

00βπε

φ

electric fields of a uniformly moving charged particlev = 0 v < c v = c

1/γ

Er=2qδ(z-ct)/r ( ) 0ˆ =×+−≈ BzEeFrrr

v = c

Bθ=2qδ(z-ct)/r

!!

Liouville’s TheoremConsider a system of non-interacting particles in a 6-D phase space (qi, pi). The state of each particle at time t is represented by a point in the phase space. We can define a particle density n(x,y,z,px,py,pz,t) such that the number dN of particles in a small volume dV of phase space is given as

As the particles move under the action of some ‘external forces’, the whole volume they occupy in phase space also moves and changes its shape. However, total number of particles in the system does not change. Particles flowing into a unit volume dV must equal to the number increase in dV. That is, the motion of a group of particles in a volume dV must obey the continuity equation:

or( ) 0=

∂∂

+⋅∇tnvnr

zyx dpdpndxdydzdpndVdN ==

dV V

0=∂∂

+∇⋅+⋅∇tnnvvn rr

Liouville’s Theorem (cont’d)with

then we have

and since

Therefore, or

This implies the volume occupied by this system of particles in 6D-phase space remains constant throughout the motion.

03

1

223

1=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

≡⋅∇ ∑∑== i iiiii i

i

i

i

pqH

qpH

pp

qqv

&&r

0=∇⋅+∂∂ nvtn r

nvtnp

pnq

qn

tn

dtdn

i ii

ii

i

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

= ∑ ∑ r&&

.0 constnn ==0=dtdn

Liouville’s Theorem (cont’d)If particle motion in x-direction has no coupling to the other directions, the area in x-Px phase space defined by dxdPx remains constant. Liouville’s theorem is derived under the assumption of ‘non-interacting’ particles. However, it is still applicable in the presence of electric and magnetic self fields associated with the bulk spacecharge and current arising from the particles of the beam, as long as these fields can be represented by average scalar and vector potentials φ(x,y,z) and A(x,y,z).

Trace Space Area Definition of EmittanceTrace space area definition of emittance

The trace-space area Ax is related to the phase-space area in x-pxplane by

We can therefore useful to define normalized emittance that is independent of particle acceleration such that

However, beams with quite different distributions in trace space may have the same area!!

∫∫ ′== xdxdAxxε

many authors identify the emittance as trace-space area divided by π (unit: [m-rad])

[π m-rad]

∫∫∫∫ == xxz

x dxdpmc

dxdpp

Aγβ

11recall: under certain conditions, this integral can be a constant of t.

βγεε =n

The RMS Emittance DefinitionDefine rms emittance as

If x and x’ are not correlated (at beam waist where the beam is neither converging nor diverging)

RMS emittance provides a quantitative information on the quality of beamRMS emittance gives more weight to the particles in the outer region of the trace-space area. Therefore, remove some of the outer particles will significantly improve RMS emittance without too much degradation of beam intensity.

222~ xxxxx ′−′=ε

xxx xx ′=′= σσε ~~~

Effective EmittanceIn a system that all forces (space charge and external forces) acting on the particles are linear (i.e. proportional to particle displacement x from the beam axis), it is still useful to assume an elliptical shape for the area occupied by the beam in trace space such that

We are able to define an emittance as

The relation between and the corresponding RMS quantitiesare given by

( )mmx xxA ′= π

( )π

ε xmmx

Axx =′=

xthxx ε~,~,~ ′( ) xmm xx ε,, ′

( )xx

thm

m

xxxx

εε ~4

~2

~2

=

′=′=

x

x’

Thermal Emittance of a Beam from CathodeFor a beam from a thermionic cathode at temperature T, rms thermal velocity spread is related to rms beam divergence such that

Assume Maxwellian velocity distribution from a round cathode with radius rs,

where T is the temperature of the cathode, then

On the other hand, for a beam emitted from the cathode

Thermal emittance of such beam from the cathode is given by

0, /~~ vvx thx=′

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ ++−=

Tkvvvm

fvvvfB

zyxzyx 2

exp,,222

0

2/1~~ ⎟

⎠⎞

⎜⎝⎛==mTkvv B

yx

2/~~sryx ==

0

2/1

, 2~vmTk

r

B

syx

⎟⎠⎞

⎜⎝⎛

=ε

2/1

00

0

30

2

2

21

2

~~

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡==

∫∫drrn

drrnrx a

a

π

π

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y=e

xp(-X

.2 )

2/1

22~ ⎟⎠⎞

⎜⎝⎛=mcTkr B

snε

Beam BrightnessDefinition of beam brightness:

It is useful know that total beam current that can be confined within a 4-D trace space volume V4. We can define average brightness as

If any particle distribution whose boundary in 4-D trace space is defined by a hyperellipsoid

one finds and average brightness isRMS brightness is then defined as:

Ω=

Ω=

dsddI

dJB

4VIB = ∫∫ Ω= dsdV4

12

2

22

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ′++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′+

yx

ybbyxa

ax

εε

( )∫∫ =Ω yxdsd εεπ 2/2

yx

IBεεπ 2

2=

(note: 2/π2 is left out)yx

IBεε ~~

~ =

24

Particle motion near the axis of axisymmetric fields

Derive linear equations of particle motion in which only terms up to first order in r and r’=dr/dz are consideredAssumptions:

Cylindrical beam; self-fields are neglectedParticle trajectories remain close to the axis. That is, r << b. And b is the radii of electrodes, coils or iron pieces that produce the electric and magnetic fields. This also implies that the slopes of the particle trajectories remain small (i.e., r’ << 1 or ).Azimuthal velocity must remains very small as compared to the axial velocity (i.e., ). Thus, in this linear approximation, we have

zr && <<

θvzr && <<θ

( ) vrrvz ≈−−=2/12222 θ&&&

25

Let the electric potential on the axis be . i.e. .Paraxial approximation of the electric potential can be expressed as

From this, we obtained

Similarly, the first order magnetic field terms are

(note that the fields are axisymmetric)

( )zr,φ ( )zV ( ) ( )zVz =,0φ

( ) ( ) L−+′′−= 442

641

41, rVrVVzrφ

Vz

Ez ′−≈∂∂

−=φ

zErVr

rE zr ∂

∂−≈−′′=

∂∂

−=22

Lφ

rBBr ′−=21 BBz =

(5)

(7)

(6)

26

If we substitute the above relations into the equations of motion of a charged particle in EM fields, we have

In the paraxial approx. , we can neglect the termon the right hand side of Eq. (10).

Eq. 9 is a result of conservation of particle angular momentum in axis-symmetric field (Busch’s theorem).

( )

( ) BrqVqzdtdm

pBrqpqmr

BqrVqrrmrdtdm

′+′−=

+−=+−=

+′′=−

22

22

2

2

22

2

θγ

ψπ

θγ

θθγγ

θθ

&&

&

&&&

cvz β=≈& 2/2 Bqr ′θ&

(8)

(9)

(10)

27

And

or with ,

Thus, . And integration of this expression gives

This is just the energy conservation law T+U=constant. If V=0 when T=0, or γ=1, the constant is zero, then we get

3/ γγββ ′=′

( ) ( ) '2222 γγβγγ cczdtd

=+′= −&

( ) ( ) ( ) ( )ββγβγγγγ ′+′=== 22cvdzdvz

dzd

dtdzz

dtd

&&

Vqmc ′−=′γ2

( ) .1 2 constqVmc =+−γ

( ) ( ) ( )22 11

mczqV

mczqVz +=−=γ

–qV is always +ve.

(11)

28

From Eq. (9),

or,

with initial condition and

22 mrp

mqB

γγθ θ+−=&

22 rmcp

cmqB

c βγβγβθθ θ+−==′&

0θθ = 0zz =

dzrmc

pcm

qBz

z∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−

020 2 βγβγ

θθ θ

(12)

(13)

29

By substituting Eq. (12) into Eq. (8),

or,

Now,

Using the relation, . L.H.S. of Eq. (14) can be written as

And from Eq. (11),

( ) ( )qBmmr

mVqrr

dtd

++′′

= θγθγ &&

&2

⎟⎠⎞

⎜⎝⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

′′= 22 222 r

pqBmrp

mqB

mr

mVqr θθ

γγ

( ) 32

22 142 rm

pmqBr

mVqrr

dtd

γγγ θ+⎟

⎠⎞

⎜⎝⎛−

′′=&

( ) 222 crcrcrdzdcr

crzrrcz

βββββ

ββγγγ

′′+′′=′=

′=′=

′=′=

&&

&

&&

3/ γγββ ′=′

( ) ( )rrcrdtd ′′+′′= γγβγ 22&

γ ′′−=′′ 2mcVq

(14)

(15)

(16)

(17)

30

Substituting (15), (17) into (14)

or after dividing by gives the paraxial ray equation

In non-relativistic limit, we take

( ) ( ) 32

22222 1

42 rmp

mqBrmc

mrrrc

γγγγγβ θ+⎟

⎠⎞

⎜⎝⎛−′′−=′′+′′

22γβc

022 32222

22

22 =−⎟⎟⎠

⎞⎜⎜⎝

⎛+

′′+′

′+′′

rcmpr

mcqBrrr

βγβγγβγ

γβγ θ

VV

mcVq

VV

mcVqmcqV

cv

22

22

21

22

22

22

22

′′≈

′′⇒′′

−≈′′

′≈

′⇒′

−=′

−≈=

≈

βγγ

βγγ

β

γ

02842 3

222

=−+′′

+′′

+′′mqVrpr

mqVBqr

VVr

VVr θ

non-relativistic paraxial equation

(18)

(19)

Eq. 18 is still not a linear differential equation and it is not very useful in practice.

31

Nonlinear approx. for the angle θ is

the canonical angular momentum is recalled as .

Four cases are considered:

1. ,

2. ,

3.

4.

if we introduce the magnetic flux

then

( )dz

rmqVp

mqVBqz

z∫ ⎥⎦

⎤⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

022/1

22

0 28θθθ

0=θA

0=θA

0≠θA 00 =θ&00 =θ&

rqAmrp θθ θγ += &2

02

00 θγθ&mrp =

last term of paraxial ray equation vanishes !!

∫=Ψ 0

00 2rBrdrπ

πθ 20Ψ

=qp

(20)

(21)

0≠θA 00 ≠θ&

32

We can re-write the last term in the paraxial ray equation as

and

It is sometimes convenient to study the particle trajectories in Larmor frame. The anglebetween Larmor frame and the lab frame (θr) is given by

The angle θL of the particle in the Larmor frame is

When , particle motion in this frame is in a plane through the axis (meridionalPlane). In this case, the trajectory r(z) in the meridional plane may be found from Eq. (18)alone.

3

20

32222

2 12

1rmc

qrcm

p⎟⎟⎠

⎞⎜⎜⎝

⎛ Ψ=

βγπγβθ

( )32

20

3

2 18

12 rmqV

qrmqV

pπ

θ Ψ=

(relativistic)

(non-relativistic) (23)

(22)

dzmqBz

zr ∫−=0 2 βγγ

θ

020

θβγ

θθθ θ +=−= ∫ dzrmc

pz

zrL(24)

0=θp

33

The relativistic paraxial ray equation can be written as

by letting and .

General mathematical properties of Eq. (25) will be discussed in the case

Linear beam optics in an axisymmetric systemIn the case of magnetic fields, this description is in the rotating Larmor frame.

0)()( 21 =+′+′′ rzgrzgr

( )γβ

γ21′

=zg ( ) 22

2

22 2 czg L

βω

γβγ

+′′

=

(25)

Envelop evolution of a beam with finite emittance in drift space

34

12 211111

211 =′+′+ rcrrbra

12 222222

222 =′+′+ rcrrbra

( ) 2/12111

11bca

A−

==ππε ( ) 2/12

222

22bca

A−

==ππε

Suppose we have a distribution of particles at some initial position z1 such that the trace-space area is defined by an ellipse

The area occupied by this distribution at some other point z2 is then found by solving the transfer matrix relation for (r1,r1’) in terms of (r2,r2’) and substituting in Eq. (31).

Eq. (32) is still an ellipse since the coefficients are uniquely determined by the transfer matrix elements and initial coefficients. Emittance of the beam at z1 and z2 are given by

(31)

(32)

35

An ellipse in x’-y’ plane:

Ellipse in x-y plane:

Where

2

1

22

11

2

1

2

1

WW

AA

===γβγβ

εε

According to Liouville’s theorem, the emittance are related as

(33)

y’y

x

x’

θ12

2

2

2

=′′

+′′

by

ax baA ′′= π

12 22 =++ cybxyax

θθθθ

cossinsincosyxyyxx+−=′+=′

2

2

2

2

22

2

2

2

2

cossin

cossinsincos

sincos

bac

bab

baa

′+

′=

′−

′=

′+

′=

θθ

θθθθ

θθ

2bacbaA

−=′′=

ππ

36

At r = rmax, dr/dr’=0. Therefore, by differentiating the ellipse equation, we have

Substituting this result into the ellipse equation, one obtains

For a drift section, the transfer matrix is given by

RcbR −=′

cbaccR ε=−

= 2

cc

cbR

2′

=−=′εε

⎟⎟⎠

⎞⎜⎜⎝

⎛=

101 z

M

12 22 =′+′+ rcrbrar 1~~2~ 22 =′+′+ rcrrbra

22~

~~

azbzccazbb

aa

++=

+=

=

(49)

37

⎥⎦

⎤⎢⎣

⎡ ′−

′′=′′

ccc

ccR

42

2ε

22

43

42εε =⎥

⎦

⎤⎢⎣

⎡ ′−′′

=′′ cccRR

03

2

=−′′R

R ε

Beam envelope evolution in a drift space

( )2/1

2202

0

2

0020 2 ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′++′+= zR

RzRRRzR ε

(50)

(51)

Recall:

21

bac −=ε

38

Effects of a Lens on Trace Space Ellipse and Beam Envelope

A distribution of particle with finite emittance at any location is represented by an ellipse:

The coefficients a, b, c are function of z.

The envelop of the beam R can be defined as the maximum value of r (or the RMS value of r) of the beam.

12 22 =′+′+ rcrbrar

How do the trace space ellipse and beam envelope evolve?

(48)

01 2

34

x’

x

Uniform Beam ModelThe beam is assumed to have a sharp boundary

The uniformity of charge and current densities assures that the transverse electric and magnetic self-fields and the associated forces are linear functions of transverse coordinates (see below)This beam model allows us to extend the linear beam optics to include the space charge forces.

⎭⎬⎫

==constJconstρ

inside the boundary

0== Jρ everywhere outside the boundary

a b

beam pipe charged particle beam

evolution of beam envelop along the propagation direction is exaggerated!!

• A axisymmetric laminar beam• Particle trajectories obey

paraxial assumption that the angle with the z-axis is small.

• The variation of beam radius along z-axis is slow enough that Ez and Br can be neglected.

Based on the above assumptions, we have J, ρ and vz ≈ v are all constant values across the beam. Therefore, with denoting the charge density of the beam, we obtain

Since we assumed the electric field has only a radial component and by Gauss’ law,

vaI πρ 20 /=

vaI

aIJJ z

πρρ

π

20

2

==

==

for 0 ≤ r ≤ a

0== Jρ for r > a

(1)

vaIrrEr 200 22 πεε

ρ== for 0 ≤ r ≤ a (2a)

vrIEr

02πε= for r > a (2b)

the Lorentz force exerted on an electron in the beam has radial component only and is a linear function of r.

The magnetic field, on the other hand has only an azimuthal component, is obtained by applying Ampere’s circuital law:

20 2 aIrBπ

μθ =

rIBπ

μθ 20=

for 0 ≤ r ≤ a

for r > a

(4)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎠⎞

⎜⎝⎛+= 2

2

ln21ar

abVr sφ for 0 ≤ r ≤ a

( ) ⎟⎠⎞

⎜⎝⎛=rbVr s ln2φ for r > a

(5)

Integrate the electrostatic field along an arbitrary path from r = a to r = b,

ββγπεερ IIaVs

3044 00

20 ≈== (6)

by setting that φ = 0 at r = b, where

aIaVEa β/602≈=

( ) ( )[ ]abVV s ln210 0 +==φ

The motion of a beam particle in such field is described by the radial force equation

( ) ( )θγγ BvEqrmrmdtd

zr −== &&&

where we dropped the force term that is negligibly small and because there is no external acceleration.

zBqrθ& θ&r .const=γ

Substitution for Er from the first equation of (2), Bθ from the first equation of (3) and with , , we have2

00−= cμε cvz β==&

(8)

( )22

0

12

ββπε

γ −=ca

qIrrm && (9)

with rcdzrdvr z ′′== 222

22 β&& we have

(10)333202 γβπε mcaqIrr =′′

Define characteristic current (Alfve’n current) I0 as

qmc

qmcI

230

0 3014

≈=πε

which is approx. 17 kA for electrons and 31 (A/Z) MA for ions of mass number A and charge number Z.

(11)

Define “generalized perveance” K such that

330

2γβI

IK =

In terms of generalized perveance, the equation of motion can be expressed as

(12)

raKr 2=′′

Under the condition of laminar flow, the trajectories of all particles are similar and scale with the factor r/a. That is, the particle at r=a will always remain at the boundary of the beam. Thus, by setting r = a = rm, we obtain

Krr mm =′′

(13)

(14)

022 332

2

32222

22

22 =−−−⎟⎟⎠

⎞⎜⎜⎝

⎛+

′′+′

′+′′

mm

n

mmmmm r

Krrcm

prmcqBrrr

γβε

βγβγγβγ

γβγ θ

From the paraxial ray equation and considering the effects of finite emittance and linear space charge, we obtained the beam envelop equation:

(15)

Vlasov ModelDescribe the self-consistent beam motion under the action of electromagnetic fields

When the effect of the velocity spread is not negligible (compared with that of self-fields), the flow is then non-laminar.A system of identical charged particles is defined by a distribution function f(qi, pi, t) in 6D phase space.

Beam EM fields

Self fields Applied fields

Vlasov equation

Recall Liouville’s theorem:

The phase-space coordinates qi, pi follows Hamiltonian equations of motion:

where H is the relativistic Hamiltonian, that is

φ and A are the scalar and vector potentials of the EMfields, which is applied fields + self-fields. The self-fieldcontributions are determined by the charge and currentdensities via the wave equations:

03

1=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

= ∑=i

ii

ii

ppfq

qf

tf

dtdf

&&

,i

i pHq∂∂

=&i

i qHp∂∂

−=&

( ) ( )[ ] φqAepcmctpqH ii +−+=2/1222,,

rr

volume occupied by a number of particles Nin phase space is constant

∫∫ = .33 constpqdd

,0

2

2

002

ερφεμφ −=

∂∂

−∇t

JtAA

rr

r02

2

002 μεμ −=

∂∂

−∇

In terms of space coordinates and mechanical momentum Pi=pi-qAi,and if the distribution function f=f(qi,Pi,t)) in (qi,Pi) phase space satisfies Liouville’s theorem. That is:

∫∫ = .33 constPqdd

03

1=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

= ∑=i

ii

ii

PPfq

qf

tf

dtdf &&or

but ( )BvEqPrrr&r ×+= Lorentz force law

we have

( ) 03

1=⎥

⎦

⎤⎢⎣

⎡∂∂

×++∂∂

+∂∂ ∑

=i iii

i PfBvEqq

qf

tf rrr

&

relativistic Vlasov equation

∫∫∫∫ == .3333 constPqddpqdDd

( )( )ii

ii

PqpqD

,,

∂∂

=

In Vlasov beam model, one has to solve the following equations self-consistently (Maxwell-Vlasov equations):

( ) 03

1=⎥

⎦

⎤⎢⎣

⎡∂∂

×++∂∂

+∂∂ ∑

=i iii

i PfBvEqq

qf

tf rrr

&

( )

( )

0

,,

,,

3

0

003

0

=⋅∇

=⋅∇

∂∂

+=×∇

∂∂

−=×∇

∫

∫

B

PdtPqfqE

tEPdtPqfvqB

tBE

ii

ii

r

r

rrr

rr

ε

εμμ

2/1

22

2

1−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

cmP

mPvr

r

• Vlasov equation is, in general, nonlinear.

• given a initial beam distribution, integrate Vlasov numerically.

• determine an equilibrium state, linearize Vlasov equation w.r.t. this state and solve the linearized equation for small signals.

Equilibrium States of a Distribution of Particles

( ) 03

1

=⎥⎦

⎤⎢⎣

⎡∂∂

×++∂∂∑

=i iii

i PfBvEqq

qf rrr&

( )

( )

0

,,

,,

0

3

0

30

=⋅∇

=⋅∇

=×∇

=×∇

∫

∫

B

PdtPqfqE

PdtPqfvqB

E

ii

ii

r

r

rr

r

ε

μ

usual approach is to choose a distribution function that depends on the constants of motions. That is

with and Ijs are constants of motion

0=∂∂

=∑ dtdI

If

dtdf j

j j

( )jIff =

Linearization of Vlasov EquationLet the equilibrium distribution function be f0 and consider a small perturbation on f1 on the equilibrium state. i.e.

and now f1 satisfies

but now the EM fields are small associate with the perturbation f1.we can now assume that these quantities are small perturbationsfrom steady state. Solving the linearized Vlasov equation allows usto study system stability (instabilities).

( ) ( )tPqfftPqf iiii ,,,, 10 +=

( ) 03

1

111 =⎥⎦

⎤⎢⎣

⎡∂∂

×++∂∂

+∂∂ ∑

=i iii

i PfBvEqq

qf

tf rrr

&

Photo-injector for XFEL

The SACLA pulsed thermionic DC gun injector

The NSRRC Photo-injector System

The Accelerator Test Area @ NSRRC

SRF Gun System @ Helmholtz-Zentrum Rossendorf

APEX: the high repetition-rate photo-cathode rf gun at LBNL