USPAS course on
Recirculated and Energy Recovered Linacs
Ivan Bazarov, Cornell University
Geoff Krafft, JLAB
Electron Sources: Single Particle Dynamics,
Space Charge Limited Emission
January 26, 2008 USPAS’08 R & ER Linacs 2
Contents
• Child-Langmuir limit
• Space charge limit with short pulses
• Busch’s theorem
• Paraxial ray equation
• Electrostatic and magnetostatic focusing
• RF effects on emittance
• Drift bunching
January 26, 2008 USPAS’08 R & ER Linacs 3
Child-Langmuir limit
So far we have discussed current density available from a cathode.
Child-Langmuir law specifies maximum current density for a space-
charge limited, nonrelativistic, 1-D beam regardless of available current
density from the cathode. The law has a limited applicability to
R&ERLs guns (applies to continuous flow, few 100s kV DC guns), but
provides an interesting insight.
-V
1D problem.d
2
2/3
0 2
9
4
d
V
m
eJ
ε=
]cm[/]MV/m[33.2]A/cm[ 2/32dEJ =
January 26, 2008 USPAS’08 R & ER Linacs 4
Bunch charge limit in guns
Let’s estimate bunch charge limit of a short pulse in a gun.
Assume ‘beer-can’ with rms σx,y σt
also that Ecath does not change much over the
bunch duration (usually true for photoguns)
If or
motion during emission stays nonrelativistic.
Aspect ratio of emitted electrons near the cathode
after the laser pulse has expired:
xσ4
xσ12
1)(
2<<
×
mc
ceE tcath σ1
511
]mm)[(]MV/m[<<
× tcath cE σ
2
2
])mm[(
]mm[
]MV/m[
341
)()(3
2
|| t
x
cathtcatht
x
cEceE
mc
cA
σ
σ
σσ
σ==
⊥=
January 26, 2008 USPAS’08 R & ER Linacs 5
Bunch charge limit in guns
(contd.)
0
..ε
σ=csE →
nC ]mm[]MV/m[11.0
4
2
2
0
xcath
xcath
E
Eq
σ
σπε
×=
=
]MV/m[/]eV[]nC[4]mrad-mm[ caththn EEq≥ε
More often than not A >> 1 in photoinjectors, i.e. the bunch looks
like a pancake near the cathode (!).
From PHYS101 (note a factor of 2 due to image charge)
if emittance is dominated by thermal energy of emitted electrons,
the following scaling applies (min possible emit.)
January 26, 2008 USPAS’08 R & ER Linacs 6
Beam dynamics
Beam dynamics without collective forces is simple.
Calculating orbits in known fields is a single particle problem.
0=⋅∇ Brr
ρ=⋅∇ Drr
Jt
DH
rr
rr+
∂
∂=×∇
t
BE
∂
∂−=×∇
rrr
ED
HBrr
rr
ε
µ
=
=
0=∂
∂+⋅∇
tJ
ρrr
)()( BvEecmdt
d rrrr×+=βγ
January 26, 2008 USPAS’08 R & ER Linacs 7
Effects on the phase space
bunch phase space
time varying fields:
RF focusing
coupler kicks
aberrations:
geometric
chromatic
collective space
charge forces
Single particle
solution integrated
over finite bunch
dimensions / energy
(this lecture)
Trickier space charge
forces (next lecture)
January 26, 2008 USPAS’08 R & ER Linacs 8
Emittance
xxxxn xxxxxppxmc
βγεβγε =′−′=−=222222
,
1
Since emittance is such a central concept / parameter in the accelerator physics, it
warrants few comments.
For Hamiltonian systems, the phase space density is conserved (a.k.a. Liouville’s
theorem). Rms (normalized) emittance most often quoted in accelerators’ field is
based on the same concept and defined as following [and similarly for (y, py) or (E, t)]
Strictly speaking, this quantity is not what Lioville’s theorem refers to, i.e. it does not
have to be conserved in
Hamiltonian systems (e.g.
geometric aberrations ‘twist’
phase space, increasing effective
area, while actual phase space
area remains constant). Rms
emittance is conserved for linear
optics (and no coupling) only.
January 26, 2008 USPAS’08 R & ER Linacs 9
Emittance measurement
The combination of two slits give
position and divergence → direct
emittance measurement. Applicable
for space charge dominated beams (if
slits are small enough).
We’ll see later that envelope equation
in drift is
Vary lens strength and measure size to
fit in eqn with 3 unknowns to find ε.
OK to use if
3
2
3
0
1
2 σ
ε
σγσ +≈′′
I
I
IIn /2 0γεσ <<
Usefulness of rms emittance: it enters envelope equations &
can be readily measured, but it provides limited info about the
beam.
slits / pinholes lens / quadrupole scan
January 26, 2008 USPAS’08 R & ER Linacs 10
Busch’s theorem
)(1
)( 2θγθ&&& mr
dt
d
rBzBreF rz =−−=
∫=Ψr
zdrrB0
2π
0=⋅∇ Brr
⇒−=Ψ
)(2 rz BzBrrdt
d&&π )(
202
Ψ−Ψ−
=mr
e
πγθ&
formula) sBusch'get /2 ,2( 0
2 →→Ψ→Ψ+= ePrAmrerAP θθθθ ππθγ &
0=θ&
Consider axially symmetric magnetic field
Flux through a circle centered on the axis and passing through e
When particle moves from (r,z) to (r+dr, z+dz) from
Busch’s theorem simply states that canonical angular momentum is
conserved
January 26, 2008 USPAS’08 R & ER Linacs 11
Magnetized beam
(immersed cathodes)
θσγσγ
θ &&yxp
z mm
eB,
2=→−=
⊥
20,
2~~ xx
p
magnmc
eB
mcσσ
σε ⊥
2
, ]mm[]mT[3.0~]mrad-mm[ xmagn B σε
If magnetic field Bz ≠ 0 at the cathode, the bunch acquires angular
velocity
Normally, magnetic field at the cathode is a nuisance. However, it is
useful for a) magnetized beams; b) round to flat beam transformation.
January 26, 2008 USPAS’08 R & ER Linacs 12
Magnetized beam
(example: ion cooler)
Similarly, rms emittance inside a solenoid is increased due to Busch’s
theorem. This usually does not pose a problem (it goes down again)
except when the beam is used in the sections with non-zero
longitudinal magnetic field. In the latter case, producing magnetized
beam from the gun becomes important.
January 26, 2008 USPAS’08 R & ER Linacs 13
Paraxial ray equation
)()( 2
zr BrEemrrmdt
dθθγγ &&& +=−
mceEr
Bm
qzz / and
2with
2
0 βγπγ
θ =
Ψ−=− &&
01
44 3222
2
0
2
22
22
=−Ψ
−++m
eE
rm
er
m
Ber
mc
eEr rzz
γγπγγ
β&&&
emcrErE zr / using and timegeliminatin 2
21
21 γ ′′−=′−≈
01
2 3
2
22
2
22=
−
Ω+
′′+
′′+′′
rmc
Pr
c
rr L
βγβγβ
γ
γβ
γ θmeB
mrerAP
zL γ
θγθθ
2/
2
−≡Ω
+≡ &
∫Ω=z
LLc
dz
0β
θ
Paraxial ray equation is equation of ‘about’-axis motion (angle with
the axis small & only first terms in off-axis field expansion are
included). cz βγγγ ′=′= &&
crzrr β′=′= &&
222)'( crcrccrr βββββ ′′+′′=′=&&
January 26, 2008 USPAS’08 R & ER Linacs 14
Focusing: electrostatic
aperture and solenoid
With paraxial ray equation, the focal length can be determined
1 2
12
2
2
21 1
/1
/14
EEmceV
mceVVf
−+
+=
eV is equal to beam K.E., E1 and
E2 are electric fields before and
after the aperture
electrostatic aperture solenoidIron yoke
CoilEnvelope
Orbit
LBcp
edz
mc
eB
dzcf
z
z
L
2
22
2
4
1
2
1
≈
=
Ω=
∫
∫
βγ
β
]m-G 4.33)[(]MeV/c 1[ ρBecp →
January 26, 2008 USPAS’08 R & ER Linacs 15
RF effect
αϕ
γ
γsin
8
1
2
cos1 2
2
+
′−=
f ϕ
γγα
cos8
)/ln(with 12=
γ1, γ2, γ’, ϕLorentz factor
before, after the
cavity, cavity
gradient and
off-crest phase2
2
8
31
γ
γ L
f
′−≈
SW longitudinal field in RF cavities requires transverse components
from Maxwell’s equations → cavity can impart tranverse momentum
to the beam
Chambers (1965) and Rosenzweig & Serafini (1994) provide a fairly
accurate (≥ 5 MeV) matrix for RF cavities (Phys. Rev. E 49 (1994)
1599 – beware, formula (13) has a mistypo)
Edges of the cavities do most of the focusing. For γ >> 1
On crest, and when ∆γ = γ′L << γ:
January 26, 2008 USPAS’08 R & ER Linacs 16
Emittance growth from
RF focusing and kick
2221xxn xppx
mc−=ε
K+∂∂
∂+
∂
∂+
∂
∂+= xz
zx
pz
z
px
x
ppzxp xxx
xx
2
)0,0(),(
kick
x
xp
headtail
focusing
x
xp
kick focusing
January 26, 2008 USPAS’08 R & ER Linacs 17
RF focusing and kick
222
0
2
focuskickn εεεε ++=
• Kick effect on emittance is energy independent (modulo
beam size) and can be cancelled downstream
• RF focusing effect scales (in terms of px) and generally
is not cancelledγ
1∝
zxx
kickz
p
mcσσε
∂
∂=
1
zxx
focusxz
p
mcσσε
221
∂∂
∂=
January 26, 2008 USPAS’08 R & ER Linacs 18
Example: RF focusing in 2-cell
SRF injector cavity
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.5 1 1.5 2 2.5 3
1 MV
3 MV
Initial kinetic energy (MeV)
(mm
-mra
d/m
m3)
e.g. 500 kV, 3 (1) MV in the 1st cell,
1 mm x 1mm bunch receives
0.26 (0.13) mm-mrad
on crest
January 26, 2008 USPAS’08 R & ER Linacs 19
RF tilt and offset
∂
∂
∂
∂+
∆≈
x
p
zmcxk
mc
E xoffRFtiltzxkick
1sin
2ϕθσσε
tilt
tiltθoffx
offset
e.g. 3 MeV energy gain
for 1 mm x 1mm yields
0.16 sin ϕϕϕϕ mm-mrad
per mrad of tilt
• One would prefer on-crest running in the injector (and
elsewhere!) from tolerances’ point of view
January 26, 2008 USPAS’08 R & ER Linacs 20
Drift Bunching: Simple Picture
slowerfaster(1) (2) (3) (4)E
z
1
12 −
∆≈
∆
γγ
γ
β
β
∆l
L
∆γ mc2
−=−
∆
∆=
∆
∆= 1
)(2)1(
)/( 22
22
mc
E
eV
El
c
lcL
bun
RF
π
λγ
γγββ
eVbun
For bunch compression, two approaches are used: magnetic
compression (with lattice) and drift bunching. Magnetic
compression relies on path vs. beam energy dependence,
while drift bunching relies on velocity vs. energy
dependence (i.e. it works only near the gun when γ ≥ 1).