Post on 01-Jun-2020
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