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ANL 1
SURVEY OF THE EMISSION PROCESSSURVEY OF THE EMISSION PROCESSSURVEY OF THE EMISSION PROCESS
Kevin L. JensenCode 6841, ESTDNaval Research LabWashington, DC 20375
Kevin L. JensenCode 6841, ESTDNaval Research LabWashington, DC 20375
ANL THEORY INSTITUTE ON PRODUCTION OF BRIGHT ELECTRON BEAMSSeptember 22-26, 2003, Argonne National Laboratory, Argonne, IL
Donald W. Feldman, Patrick G. O’Shea Inst. Res. El. & Appl. Phys.University of MarylandCollege Park, MD 20742
ACKNOWLEDGEMENTS FUNDING & SUPPORT
ANL 2
ISSUES AND QUESTIONS
QUESTIONS AND COMMENTS BY C. SINCLAIR [1]
Fundamental R&D / theory question(s):
What combination of achievable, external fields results in the maximum charge density in 6-D phase space (from a zero thermal emittance source)? For a CW source, it is not obvious whether DC or RF fields are best (particularly for room temperature RF, where the fields are limited by thermal considerations). For low duty factor applications, the consensus appears to be RF, but that must depend on the bunch charge.
How should emittance be measured, and what is required to have a high quality measurement?
Regarding the relation between thermal emittance and bunch duration at the cathode: space charge fields are reduced by making larger bunches and emitting from a smaller area (which increases longitudinal and transverse emittance) - therefore, for a given bunch charge, what is the optimal emitting area and bunch duration to achieve bets final charge density in 6-D phase space (the answer will depend on whether fields are static or dynamic).
Application and requirements dictate photocathode: needs of low repetition rate, high charge bunches differ from CW pulse trains of lower bunch charges
A goal for progress in photoemission guns: develop reliable methods for generating uniformly populated (transversely and longitudinally) optical pulses to generate uniform charge distributions from the cathode and result in minimum emittance.
[1] Emails to K. Jensen, and Kwang-Je, et al., September 2003
ANL 3
AN INTRODUCTION TO
ELECTRON EMISSION THEORY AND PROCESSES Nature Of The Emission Barrier
Tunneling, Density, and Current Integral: FN and RLD Equations Complications: Semiconductors, Emission Near Maximum A Thermal - Field Emission Formula Photoemission Considerations Quantum Efficiency A Thermal - Photoemission Formula
Laser Heating of the Electron Gas Laser Heating of the Electron Gas Time-dependent Model of Laser-induced Thermal Photoemission Dispenser Cathode Experiment
Complicating Circumstances Field Enhancement Emission at the Barrier Maximum
OUTLINE
ANL 4
Exchange Correlation
Potential
Dipole Term
Ionic CoreEMISSION FROM METALS
Large Density of Electrons Exist In Conduction Band (> 60 Billion / µm3);
Very Small Fraction Contribute to Current (A/cm2 ≈ 62 per µs per µm2)
EMISSION BARRIER
ValenceBand
ConductionBand
Band Gap
Vacuum level
EF
€
Vo =−∂∂ρ
ρεxc(ρ)[ ] −Δφ+ ε ion
The Largest Component of the Barrier Is Due to the
Exchange Correlation Potential
ANL 5
Electron Number Density ρElectrons Incident On Surface
Or Interface Barrier Are
Distributed In Energy
According To A 1-D
“Thermalized” Fermi Dirac
Distribution Characterized By
The Chemical Potential and
Called The “Supply Function”
€
ρ(o)=13π 2
2mo
h2
⎛ ⎝ ⎜
⎞ ⎠ ⎟3/ 2
=kF3
3π 2
€
f (k)=2
2π 2
2πk⊥dk⊥
1 +expβ(E||+ E⊥ −)( )0
∞
∫
=m
πβh2 ln 1+eβ (−E(k ))[ ]
€
f (k)=22π 2
2πk⊥dk⊥
1 +expβ(E||+ E⊥ −)( )0
∞
∫
=m
πβh2 ln 1+eβ (−E(k ))[ ]
€
ρ() =2Mc
m2πβh2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
3/22
πy
1 + (expy−β)dy
0
∞
∫
=Nc
2π
F1/ 2(β)
€
(T) =o 1−112
πβo
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
−180
πβo
⎛ ⎝ ⎜
⎞ ⎠ ⎟
4
+K ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
€
(T) =o 1−112
πβo
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
−180
πβo
⎛ ⎝ ⎜
⎞ ⎠ ⎟
4
+K ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Density Does Not Change With Temperature, So µ Must:
DISTRIBUTION OF ELECTRONS IN METAL
Zero Temp. ((0 ˚K) = o = EF)
ANL 6
Metals
0
5
10
15
1 10 100r
EXCHANGE-CORRELATION POTENTIAL
The Density ρ of an Electron Gas Is
€
Vxc(x)=−∂∂ρ
ρ εex +εcorr( )[ ]
€
ρ=2
2π( )3 fFD E(k)( )d3k=
kF3
3π 20
∞∫
€
εke =2
2π( )3
hk( )2
2mfFD E(k)( )d3k=
35
0
∞∫ ρ
€
εex =2
2π( )3 fFD E( )vq(kF )d
3k0
∞∫ =−
34
Q3ρπ ⎛ ⎝ ⎜
⎞ ⎠ ⎟1/3
vq(kF ) =2Q
q2θ kF −
r k +
r q( )∫ d3q
€
εcorr =−2Qao
0.876r + 7.811 ⎛ ⎝ ⎜
⎞ ⎠ ⎟
ρ(r)=3
4π rao( )3
Correlation (Potential) Energy Form due to WignerExchange (Potential) Energy
Kinetic Energy
CuAuNa
1019
#/cm3
ANL 8
+
–xi
0
0.3
0.6
0.9
1.2
-15 -10 -5 0
ρe( )x
ρe( ) ( )x approx
(+)Background
kF( –x x
o)
DIPOLE COMPONENT
Origin (xi) Of + Background Differs From That Of Electrons (x = 0) To Preserve Global Charge Neutrality
€
0 = ρe(x)−ρi (x){ }dx−∞
∞∫
€
ρe(x) =12π
f (k)ψ k(x)2dk
0
∞
∫ρi(x) =ρoθ(x−xi )
Electrons
Ion cores
Tanh-Approx.: Match ρ(xi), ∂xρ(xi)
€
ρe(x) ≈ρo2
1−tanhλkF x−xi( )[ ]{ }
λ ≈58; xi ≈xo−
52kF
Magnitude of Dipole (Tanh-model: Qualitative, Overestimates Magnitude)
€
Δφ=q2ρoεo
1
4λ2kF2
⎛
⎝ ⎜
⎞
⎠ ⎟
(ln u)1 +u
du0
1
∫ =−64π225
QkF
ANL 9
SIMPLE MODEL OF TUNNELING
SQUARE BARRIER OF WIDTH “L”
Match ψ(x) and ∂xψ(x) where V(x) Changes
Incident
eikx
Reflected
r(k) eikx
Transmitted
t(k) eikx
€
jk x( )=h2mi
ψ k * ∂xψ k −ψ k∂xψ k *( )
€
T k( )=jtrans(k)j inc(k)
=2κk( )
2
2κk( )2+ κ 2 +k2( )sinh2 κL( )
€
T k( )=jtrans(k)j inc(k)
=2κk( )
2
2κk( )2+ κ 2 +k2( )sinh2 κL( )
€
κ =1h
2mVo−E(k)( )
κL = θ(E)= “Area under Curve”κL fi θ(E)
T(E<Vo) ≈ exp{–2θ(E)}T(E>Vo) fi 1
€
1 1ik −ik
⎛
⎝ ⎜
⎞
⎠ ⎟
1r(k)
⎛
⎝ ⎜
⎞
⎠ ⎟ =
1 1κ −κ ⎛
⎝ ⎜
⎞
⎠ ⎟ab
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
eκL e−κL
κeκL −κe−κL ⎛
⎝ ⎜
⎞
⎠ ⎟ab
⎛
⎝ ⎜ ⎞
⎠ ⎟ =
eikL e−ikL
ikeikL −ike−ikL
⎛
⎝ ⎜
⎞
⎠ ⎟t(k)0
⎛
⎝ ⎜
⎞
⎠ ⎟
Vo
ANL 10
IMAGE CHARGE POTENTIAL
Schrödinger’s Equation:
If We Let: ψk(x) = R(x) exp[i S(x)]
Density Velocity
€
jk(x) =R(x)2hm∂xS(x)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
ψ k(x) ∝ κ(x)−1/2expi κ(x)dx∫{ }
Slowly varying density and constant current:
€
−h2
2m∂x2 +V(x)
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪ψ k(x)=E(k)ψ k(x)
€
jk2
R4 −1R∂2
∂x2R−
i
R2∂∂x
jk =κ2
0 0
E
0
4
8
0 5 10 15 20
No ImageImage
x [Å]
y = (4FQ)1/2/
€
Vi(x) = −q2
4πεo(2 ′ x )2 ⎧ ⎨ ⎩
⎫ ⎬ ⎭d ′ x
x
∞
∫ =−q2
16πεox≡
Qx
Q = 0.36 eV-nmy = 0.038 eV {F[MV/m]}1/2/[eV]
2xForce Between
Electron and Image: Vi is Energy to Remove Image to infinity from x
ANL 12
€
J T,F( )=qm
2π 2βh3 e−θ (E )
0
∞
∫ ln1 +eβ (−E)( )dE
€
J T,F( )=qm
2π 2βh3 e−bfn /F e−cfn (−E)
0
∞
∫ ln 1 +eβ (−E )( )dE
RICHARDSON EQUATION: THERMIONIC EMISSION Dimensionless Parameter n = β /b is small
€
−(1 + cfn) (exp−cfn){ }
Semiconductor
EMISSION EQUATIONS
€
bfn=43h
2m3v(y); cfn =2
hF2mt(y)
FOWLER NORDHEIM: FIELD EMISSION Dimensionless Parameter n = β /b is large
€
=qm
2π2h3cfn2 exp−
bfnF
⎛
⎝ ⎜ ⎞
⎠ ⎟
Field
€
cfnπ / β(sincfnπ / β)
Thermal-Field
€
J (T,F )=qm
2π 2βh3 Θ +φ−E( ) ln1 +eβ (−E)( )0
∞
∫ dE
€
eβ (−E )
+φ
∞
∫ dE=mq
2π 2h3 kB2T2exp−φ / kBT( )
€
φ=− 4QF
ANL 13
RLD & FN SUMMARY part II
Field Emission
Large # of electrons with low transmission probability
Relatively T-independent
High Work Functions For Canonical Metals
“High” Current Densities But Generally Small Emission Sites
Thermal Emission
Small # of electrons with near-unity transmission probability
Exponential T-dependence
Low Work Functions for Coated Materials
“Low” Current Densities but Generally Large Areas
1300
1400
1500
1600
1700
1800
2 4 6 8 10Field [MV/m]
4.4
0.0
2.2
kA/cm2
= 2.0 eV = 2.0 eV
Ba-Coated Tungsten
Tem
per
atur
e [K
elvi
n]
300
640
980
1320
1660
2000
2 4 6 8 10Field [GV/m]
6.6
0.0
3.3
GA/cm2
= 4.6 eV = 4.6 eV
Bare Tungsten
Tem
per
atur
e [K
elvi
n]
ANL 14
Poisson’s Equation (o = bulk):
10-4
10-3
10-2
10-1
100
0.01 0.1 1
Low φs Highφ
sExact
[ / ]Vacuum Field eV Å
Surface φ [ ]eV
BAND BENDING
Ec
Ev
µo Fvac
c
µ
ZECA: f(x) is the same as that which would exist if no current was emitted.
ZECA: f(x) is the same as that which would exist if no current was emitted.
€
F φs( )2=
2Nc
Ks3εo πβ
expβo( ) expβφs( )−βφs−1[ ]
€
F φs( )2=2π 2Nc
3βKs3εo
βπ ⎛ ⎝ ⎜
⎞ ⎠ ⎟1/ 2
85
βπ ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
+1 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Asymptotic Case: β ≤ –2:
Asymptotic Case: β » 1:
€
∂∂φ
F 2 =Nc
K sεo
2π
F1/2 β o +φ( )[ ] −F1/ 2 βo( ){ }
ANL 15
FN ( =4.4 eV)
FN ( =2 eV)
RLD ( =2 eV)
FN: Corrupted When Barrier Max Is Too Close to Fermi Level or Slope of ln(T(E)) Exceeds ln(f(E))
Maximum Field: β > 6
Minimum Field: cfn < 2β
FN AND RLD DOMAINS
DOMAINS
RLD: Corrupted When Tunneling Near Barrier Max Is Non-negligible
€
F <14Q
−6kBT( )2
€
F >4h
2m( )kBT
€
F <2m
10hkBT
⎛
⎝ ⎜
⎞
⎠ ⎟
4/3
Q1/3
10-4
10-3
10-2
10-1
100
400 800 1200 1600
Field [eV/Å]
Temperature [K]
Fie
ld [
eV/Å
]
Fmax(1000K) = 11.1 MV/m
Fmax(2eV,1000K) = 1527 MV/m
Fmin(2eV,300K) = 750 MV/m
Thermionic
Photocathode
Field
Typical Operational Domain of Various Cathodes
Note: Photocathodes Typically Run at or Near 300 K
and Surface Roughness Increases F
ANL 16
0.001
0.01
0.1
1
10
100
2 3 4 5 6 7 8 9 10
300 K3000 KFermi Level
E(k)
Electron Momentum Into Barrier Determines Emission Probability
Finite Temperature (β = 1/kBT)
T(E) (b = slope of -ln[T(E)])E(k)
T(E)4 GV/m
T(E)10 MV/m
ELECTRON DISTRIBUTION
€
f kx( ) = fFD E( )d2k⊥
2π( )2
0
∞
∫
€
fFD E( ) = 1+expβ E−( )[ ]{ }−1
f kx( ) =m
πβh2ln 1+expβ −E(kx)( )[ ]{ }
MaxwellBoltzmann
Regime
0 K-like Regime
€
T E( ) = 1+expb E−Ec( )[ ]{ }−1
€
Ec = +bfnFcfn
Ec = + − 4QF
Field
Thermal 0.0
0.50
1.0
1.5
5.5 6 6.5 7 7.5Energy [eV]
731 MV/m300 K
664 MV/m @ 550 K
217 MV/m900 K
J(F,T) ≈ 1 A/cm2
= 2.0 eVS
up
ply
Fu
nct
ion
f(E
) [#
/nm
2]
Tra
nsm
iss
ion
Co
eff
T(E
)
ANL 20
€
NA =n n+1( )n+1[ ]
n+1e−p
NB =n2
πλ
Ng Erf λ bφ−xp( )[ ] −Erf λxp[ ]{ }
NC = 1−nep−bφ
n+1
⎛
⎝ ⎜
⎞
⎠ ⎟e−bnφ
Nomenclature:T(E) ≈ To / {1+exp[b(Ec–E)]}x = b(Ec – E)p = b(Ec – )n = β/bxp Integrand Approximated λ By Gaussian of FormNg Ng exp[- (x-xp)2]
GENERALIZED J(F,T) EQUATION
Using the Tanh-form Best Fit of T(E):
€
J (F ,T ) =qm
2π2h3To
β 2
⎛
⎝ ⎜
⎞
⎠ ⎟
ln1+en(x−p)[ ]
ex +1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥dx
−∞b+ p∫ ⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
=qm
2π2h3To
β 2
⎛
⎝ ⎜
⎞
⎠ ⎟N −∞,b + p( )
€
J (F ,T ) =qm
2π2h3To
β 2
⎛
⎝ ⎜
⎞
⎠ ⎟
ln1+en(x−p)[ ]
ex +1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥dx
−∞b+ p∫ ⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
=qm
2π2h3To
β 2
⎛
⎝ ⎜
⎞
⎠ ⎟N −∞,b + p( )
Separate N Into 3 Regions:
Field: NA = N(p,b+p)
Intermediate: NB = N(-b+p,p)
Thermal: NC = N(-∞,-b+p)
FN: n = ∞ limit of NA
RLD: n = 0 limit of NC
ANL 21
T(WKB)
T(FN)
T(RLD)0
0.2
0.4
0.6
0.8
1
6.50 7.00 7.50
T(E)T
a(E)
T(E)f(E)
[a.u.]
Energy [eV]
= 5.875 eV
Φ = 2.0 eVF = 0.04 eV/nmT = 600 K
≤20%
≤20%
-16
-14
-12
-10
-8
-6
-4
-2
0
0.01 0.1 1
NA + N
B + N
C
NumericalN(FN)N(RLD)
Field [V/nm]
T = 600 K = 2 eVn
o = 0.75
RESULTS
Current Integrand BehaviorRLD: Cut-off at Apex E Too Extreme
FN: Over-predicts T(E) at Apex
Variation of n-factor with FieldLarge Range of n Obscures Differences:20% Error: RLD: F = 0.22; FN: F = 1.7
ANL 22
€
J λ T,F ,( ) =q1−R( )Iλ (t)hω
⎛ ⎝ ⎜
⎞ ⎠ ⎟U β hω −φ( )[ ]
U β[ ]
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
+ARLDT 2exp−βφ[ ]
€
J λ T,F ,( ) =q1−R( )Iλ (t)hω ⎛ ⎝ ⎜
⎞ ⎠ ⎟U β hω −φ( )[ ]
U β[ ]
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
+ARLDT 2exp−βφ[ ]
€
U x( ) = ln 1 +ey( )dy−∞
x
∫
=ex 1−beax( ) x≤0( )
12
x2 +π 2
6−e−x 1−be−ax( ) x> 0( )
⎧
⎨ ⎪
⎩ ⎪
“Fowler factor”
10-5
10-4
10-3
10-2
10-1
100
12 14 16 18 20 22Energy [eV]
T(E)
T(E+hν)
( +2T E hν)
( +3T E hν)
( +4T E hν)
( )f E
Field significantly exaggerated to show detail
T(E
); f
(E)
[101
6 #
/cm
2]
THERMAL PHOTO-CURRENT (I)
Product/sum of Several Factors:
Electron Charge (q)
Absorption Factor (1-R)
# of photons (Iλ/hω)
Probability That Photo-Absorbing Electron in Consequence Has Sufficient Energy to Surmount Surface Barrier (Ratio of U’s) [1]
Standard Thermal Emission Due to Laser Heating (JRLD)
[1] “The precise hypothesis which succeeds so well in correlating the observed effect near the threshold is that the photoelectric sensitivity or number of electrons emitted per quantum of light absorbed is to a first approximation proportional to the number of electrons per unit volume of the metal whose kinetic energy normal to the surface augmented by hν is sufficient to overcome the potential step at the surface.
R. H. Fowler, PR38, 45 (1931). (italics in original)
= 1064 nm
ANL 23
QUANTUM EFFICIENCY (1D)
ΔQ / ΔE = Jλ / Iλ: Therefore, QE Governed by U[ (h U( )
€
QE1D ≈
2 1−R( )
β( )2 exp−β φ−hω( )[ ] β hω −φ( )⇒ −∞[ ]
1−R( )hω −φ
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
β hω −φ( ) ⇒ ∞[ ]
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
€
QE1D ≈
2 1−R( )
β( )2 exp−β φ−hω( )[ ] β hω −φ( )⇒ −∞[ ]
1−R( )hω −φ
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
β hω −φ( ) ⇒ ∞[ ]
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
STANDARD PHOTOEMISSION (Photo-excited Electrons Below Fermi Level)
€
U x( ) ≈ex x<<−1( )12
x2 x>> +1( )
⎧ ⎨ ⎪
⎩ ⎪
€
J λ T,( ) ⇒q
hω1−R( )Iλ (t)
hω −φ
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
€
β hω −φ( ) ⇒ ∞
€
J λ T,( ) ⇒ 1−R( )2πh( )
2
mω2 Iλ (t)J RLD T,φ−hω( )
€
β φ−hω( ) ⇒ ∞
THERMAL PHOTOEMISSION (PHOTO-EXCITED ELECTRONS IN THERMAL TAIL)
1-D ASYMPTOTIC EXPRESSION FOR
QUANTUM EFFICIENCY
ASYMPTOTIC LIMITS OF FOWLER FUNCTION(Richardson Approximation Implicit in U(x) - must be modified if tunneling becomes significant)
ANL 24
AN INTRODUCTION TO
ELECTRON EMISSION THEORY AND PROCESSES Nature Of The Emission Barrier
Tunneling, Density, and Current Integral: FN and RLD Equations Complications: Semiconductors, Emission Near Maximum A Thermal - Field Emission Formula Photoemission Considerations Quantum Efficiency A Thermal - Photoemission Formula
Laser Heating of the Electron Gas Laser Heating of the Electron Gas Time-dependent Model of Laser-induced Thermal Photoemission Dispenser Cathode Experiment
Complicating Circumstances Field Enhancement Emission at the Barrier Maximum
OUTLINE
ANL 25
ChemicalPotential
electron-electron scattering
electron-lattice scattering
Power transfer by electrons to lattice
285.1 GW / K cm3 for W
Laser EnergyAbsorbed
Electron & LatticeSpecific Heat
€
Ce∂∂t
Te=∂∂z
κ(Te,Ti )∂∂z
Te ⎛ ⎝ ⎜
⎞ ⎠ ⎟−g Te−Ti( ) +G z,t( )
Ci∂∂t
Ti =g Te−Ti( )
LASER HEATING OF ELECTRON GAS
Differential Eqs. Relating Electron (Te) to Lattice Temperature (Ti)
Thermal Conductivity & Relax. Time
1
1.5
2
2.5
2.6 2.8 3 3.2log
10(Temperature [K])
Tungsten
τeeτ
ph
τ300 K128 fs
1000 K25 fs
2000 K8 fs
€
κ Te,Ti( ) =23m
τ Te,Ti( )Ce Te( )
1τ=
1τee Te( )
+1
τ ph Ti( )
€
τee Te( ) =hAo
1kBTe
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
=Aee
Te2
τ ph Ti( ) =h
2πλo
1kBTi
⎛ ⎝ ⎜
⎞ ⎠ ⎟=
Bep
Ti
ANL 26
lattice
electrons
SPECIFIC HEAT APPROXIMATIONS
High Temp (> 300 K) Specific Heat Approximations
Specific Heat Variation in Energy Density With Temperature
€
C T( ) =∂∂T
U
€
Ce Te( ) =∂∂Te
E−( )De E( )
expβe E−( )e[ ] +1dE
0
∞
∫
Ci Ti( ) =∂∂Ti polarizations
∑hωDi ω( )
expβihω i[ ]−1dω
0
ω D
∫
€
Ce Te( ) =γ Te
1+740
πβe ⎛ ⎝ ⎜
⎞ ⎠ ⎟
2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
Ce Te( ) =γ Te
1+740
πβe ⎛ ⎝ ⎜
⎞ ⎠ ⎟
2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
γ Tungsten( )=136.48 Joule
Kelvin2meter2
Tungsten( )=18.08 eV
€
Ci Ti( ) =3NkB
1+120
TD
Ti
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
Ci Ti( ) =3NkB
1+120
TD
Ti
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
N Tungsten( )= 183.84grammole
⎛ ⎝ ⎜
⎞ ⎠ ⎟−1
19.3gram
cm3
⎛ ⎝ ⎜
⎞ ⎠ ⎟
TD Tungsten( )=400 Kelvin0.0
1.0
2.0
3.0
500 1000 1500 2000 2500
Data 4
10 x Ce(T)
Ci(T)
Temperature [Kelvin]
TUNGSTEN
Room Temp
C(T
) [J
ou
les/
Ke
lvin
cm
3]
€
β ≡1
kBT
ANL 28
ABSORBED LASER POWER
€
U x( ) =ex 1−beax( ) x≤0
12
x2 +16π 2−e−x 1−be−ax
( ) ⎛ ⎝ ⎜
⎞ ⎠ ⎟
x> 0
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
a=12π 2−12 ln 2( )
12−π 2
⎛
⎝ ⎜
⎞
⎠ ⎟; b=1−
π 2
12
0.6
0.7
0.8
0.9
1
2.8 3 3.2 3.4 3.6 3.8 4
Log10( [ ])Temperature K
1064 nm
532 nm
355 nm
266 nm = 0.0 /Field MV m
Abs
orbe
d E
nerg
y F
acto
r
Fowler Function
€
G(z,t) = 1−R( )Iλ (t)e−z/δ
δ ⎛
⎝ ⎜
⎞
⎠ ⎟ 1−
U β hω −φ( )[ ]U β[ ]
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Iλ (t) =Ioe−(t−to) /Δt[ ]
2
penetration depthemission barrier heightU terms: Energy lost from direct photoemission
Laser Term G(z,t) Is a Product Of:
Reflection Coefficient
Incident Laser Power
Penetration Factor
Absorbed Energy Factor
ANL 29
0
20
40
60
80
100
101
102
103
104
105
106
107
0.01 0.1 1λ [ ]µm
[%]Reflectivity
Absorption Length δ [ ]nm
Classical Free Electron Theory of Simple Metals (Drude):
Motion of Electrons in Metals Is Damped Due to Collisions of
Electrons With Non-ideal Lattice. Bound Electrons Are Ignored.
Valid for Low Frequency.
Classical Free Electron Theory of Simple Metals (Drude):
Motion of Electrons in Metals Is Damped Due to Collisions of
Electrons With Non-ideal Lattice. Bound Electrons Are Ignored.
Valid for Low Frequency.
REFLECTION AND PENETRATION (Theory)
Complex Dielectric Constant fi Optical Properties Of Metals as function of λ Real: index of refraction n
Imag: damping constant k
Related to Reflectivity R and absorption length of light in metal δ.
Plasma Frequencies: • = e- concentration• o = dc conductivity)
€
ωp2 =
q2ρmεo
;ωo =εoω p
2
σ o
In terms of Plasma Frequency: [ = 2 c/
€
2nk=ωoω p
2
ω ω2 +ωo2( ); n2 −k2 =1−
ω p2
ω2 +ωo2( )
€
:Reflectivity R=n−1( )
2+ k2
n+1( )2+k2
:Absorption Lengthδ = c2ωk
= λ4πk
€
:Reflectivity R=n−1( )
2+ k2
n+1( )2+k2
:Absorption Lengthδ = c2ωk
= λ4πk
Drude Equations
ρ = 1.68x1023 #/cm3
σ = 3.6E5 (Ohm-cm)-1
ANL 30
Quantum Mechanical Treatment of Optical Properties
IR Behavior: Free electronsVisible/UV: Bound Oscillators
Extract n and k from Exp. Data & Apply Drude Equations
0
20
40
60
80
100
0
5
10
15
20
25
30
0 0.5 1 1.5 2λ [ ]µm
[%]Reflectivity
Absorption Length δ [ ]nm
Tungsten
REFLECTION AND PENETRATION (Practice)
Complex Dielectric Constant fi Optical Properties Of Metals as function of λ Real: index of refraction n
Imag: damping constant k
Related to Reflectivity R and absorption length of light in metal δ.
Plasma Frequencies: • = e- concentration• o = dc conductivity)
€
ωp2 =
q2ρmεo
;ωo =εoω p
2
σ o
In terms of Plasma Frequency: [ = 2 c/
€
2nk=ωoω p
2
ω ω2 +ωo2( ); n2 −k2 =1−
ω p2
ω2 +ωo2( )
€
:Reflectivity R=n−1( )
2+ k2
n+1( )2+k2
:Absorption Lengthδ = c2ωk
= λ4πk
€
:Reflectivity R=n−1( )
2+ k2
n+1( )2+k2
:Absorption Lengthδ = c2ωk
= λ4πk
Drude Equations
ANL 31
Diff Eqs. Governing T-Evolution
Diffusion Equation
Discretization (j=time, i=space)Standard Crank-Nicholson
Non-linearity
different t-scale
TIME EVOLUTION SIMULATION
€
Ce(Te)∂∂t
Te=∂∂z
κ(Te,Ti )∂∂z
Te ⎛ ⎝ ⎜
⎞ ⎠ ⎟−g Te−Ti( ) +G z,t( )
Ci (Ti )∂∂t
Ti =g Te−Ti( )
€
∂tT =D∂z2T +G( z,t)
€
M • T[ ] j, i =DΔt
2Δx2Tj,i+1−2Tj,i +Tj,i−1( )
I −M[ ] • v Tj+1= I + M[ ] •
v Tj +
v G(s)ds
tj
tj+1∫
€
M • T[ ] j, i =DΔt
2Δx2Tj,i+1−2Tj,i +Tj,i−1( )
I −M[ ] • v Tj+1= I + M[ ] •
v Tj +
v G(s)ds
tj
tj+1∫
TIME SCALES:Engagement Time 10 sLaser Pulse-to-Pulse 4 nsLattice Therm (Ci/g) 8.4 psRelaxation (RT): 128 fs∆t (600K & ∆x = 20 nm): 0.4 fsRatio: 10000000
Straight-forward Numerical Program Not Feasible To Model
Multiple ns-Separated Laser Pulses Incident On Cathode - Require Separate Time Scale
Models For Single / Multi pulses
Straight-forward Numerical Program Not Feasible To Model
Multiple ns-Separated Laser Pulses Incident On Cathode - Require Separate Time Scale
Models For Single / Multi pulses
LENGTH SCALES:Cathode 1.0 cmThermal Distance 200 nmLaser 10 nmRatio: 100000000
ANL 32
€
Te(x,t) =To + ΔT u x,tk+s( )−u−2L−x,tk+s( ){ }k=0
N
∑
MACRO-TIME SCALE PRF HEATING (I)
Bulk T = Sum of Gaussian
Laser Pulses Io exp(-(t/δt)2)
Effect = Sum of N Dirac-delta
T- Pulses at Surface
Magnitude of Temperature
Rise Dictated by Deposited
Laser Energy ΔE
Temp. Depends on Location
of Fixed Boundary L
Bulk T = Sum of Gaussian
Laser Pulses Io exp(-(t/δt)2)
Effect = Sum of N Dirac-delta
T- Pulses at Surface
Magnitude of Temperature
Rise Dictated by Deposited
Laser Energy ΔE
Temp. Depends on Location
of Fixed Boundary L
surface image
Δt
k k+1
δt
€
Te(x,t) =To +ΔT
4πDoΔtSN α−(x),
tΔt
⎛ ⎝ ⎜
⎞ ⎠ ⎟−SN α+(x),
tΔt
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
Te(x,t) =To +ΔT
4πDoΔtSN α−(x),
tΔt
⎛ ⎝ ⎜
⎞ ⎠ ⎟−SN α+(x),
tΔt
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
SN (α ,s)= k+ s( )−1/2
exp−α
k+ s ⎛ ⎝ ⎜
⎞ ⎠ ⎟;
k=0
N
∑ α± x( ) =L2
4DoΔt1± 1+
xL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
s =1/2: Evolution of surface temperature with pulse number Ns > 1/2: Cooling of surface after N pulsestk+s = (k+s)Δt
€
u x,t( ) = 4πDot( )−1/ 2
exp−x2
4Dot ⎛
⎝ ⎜ ⎞
⎠ ⎟
ΔT =Tc
2Tb + Tc
2 +Tb
2 ⎛ ⎝
⎞ ⎠
−1/2
Do ≈κ To,To( ) / Ce(To)
Diffusion functions
€
Tc =2δπDoΔt( )
1/4 ΔEγ ⎛ ⎝ ⎜
⎞ ⎠ ⎟1/2
Tb =1
2γδ2γTo + πgΔt( ) πDoΔt( )
1/2
ΔE= 1−R( ) Io(t)dt−∞
∞∫
Single Pulse Heating
ANL 3365 mg NaCl
MACRO-TIME SCALE PRF HEATING (II)
Characteristic Temperature: Tmax
is temperature at surface when
T(x) profile is linear (asymptote
as N fi ∞ at surface)
Ex: 1 GW/cm2 = A Pulse of
E = 0.71 mJ with t = 40 ps
Incident on 1 mm2 Area
Characteristic Temperature: Tmax
is temperature at surface when
T(x) profile is linear (asymptote
as N fi ∞ at surface)
Ex: 1 GW/cm2 = A Pulse of
E = 0.71 mJ with t = 40 ps
Incident on 1 mm2 Area
€
Tmax=To + ΔTLδCe Te( )Δtκ Te,Ti( )
⎛
⎝ ⎜ ⎞
⎠ ⎟
≈To +6 1−R( )mLI oδt
Δt2gτ
€
=To + 662KelvinL[ ] cm δt[ ] ps Io[GW / cm2 ]
Δt[ ]ns( )2τ[ ]fs
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
600
620
640
660
680
700
720
10-9 10-7 10-5 0.001 0.1 10
COOLING
T(x=0) [K]Bulk
t [sec]
600
620
640
660
680
700
720
10-9 10-7 10-5 0.001 0.1 10
HEATING
T [K]Tmax
t [sec]
€
I o =ΔEπAδt
ANL 34
TIME SIMULATION MATRIX EQUATIONS
Matrix Equation
Matricies
Non-linearity: Finite Difference Multi-point
Algorithm Preserves Stability
Boundary Tbc: RHS: reflect / LHS: absorb Temp BC Given by Macro-scale
Time Simulation and Held Fixed
€
Ce + J −D( )t+Δt
• Te t +Δt( )= Ce−J +D( )t• Te t( ) + 2J • Te(t) +
12
G(t)dt +Tbc∫D+ H( )
t+Δt• Ti t +Δt( )= D + H( )
t• Ti t( ) + H • Te(t + Δt) +Te(t)( )
Electrons
Lattice
Matricies Require Evaluation of Ce(Te) / κ(Te,Ti) at Advanced Time Steps:
USE OF PREDICTION/CORRECTION SCHEME: “Guess” j+1 elements - solve - use results in next guess
€
Ce[ ]i, j=
12Δt
Ce T t + Δt( )[ ] +Ce T t +Δt( )[ ]{ }δij
Ci[ ]i, j =12Δt
Ci T t + Δt( )[ ] +Ci T t +Δt( )[ ]{ }δij
H[ ]i, j=12
gδij ; J[ ]i, j=
g Ci[ ]i, j
2 Ci[ ]i, j +gδij
€
D(t)[ ]i, j =1
4Δx2κ Te,Ti( )[ ] j−1 + κ Te,Ti( )[ ] j ⎧ ⎨ ⎩
⎫ ⎬ ⎭δi, j−1
−1
2Δx2 κ Te,Ti( )[ ] j−1+ 2 κ Te,Ti( )[ ] j
+ κ Te,Ti( )[ ] j+1 ⎧ ⎨ ⎩
⎫ ⎬ ⎭δi, j
+1
4Δx2κ Te,Ti( )[ ] j−1 + κ Te,Ti( )[ ] j ⎧ ⎨ ⎩
⎫ ⎬ ⎭δi, j+1
ANL 36
Go(t)/Go(0)
0
0.2
0.4
0.6
0.8
1
-20 0 20 40 60t - t
o [ps]
65 mg NaCl
MICRO-TIME SCALE: SINGLE PULSE
Scandate Dispenser Cathode Field Enhancement = 3.1 Chemical Potential = 18.08 eV Wavelength = 1.064 microns (W) = 4.7 eV; (Ba) = 1.8 eV Field = 10 MV/m; Theta = 50 % Peak Intensity = 710 Mw/cm2
Reflectivity = 58.5%; = 22.7 nmIncident Gaussian Laser Pulse: “Bulk” Boundary Held Fixed At Temperature Dictated By Macropulse Analysis
Gaussian: y(t) = exp{-[(t-to)/ t]2}Laser t 10.00 ps
to 0.00 ps
Resulting Temp. Distribution (Numerical Sol’n of Time-dependent Heat Equations)fi Long Decay Time Affects Subsequent Photoemission; Long Thermal Tail
Current Distribution Is Wider
Electron t 8.36 ps to 0.93 ps
(T(t)-To)/(Tmax-To)
J(t)/J(0)
Pre-fit8.94 ps1.45 ps
Post-fit8.75 ps0.55 ps
= 1064 nm
ANL 37
0
0.2
0.4
0.6
0.8
1
-20 0 20 40 60t - t
o [ps]
Go(t)/Go(0)
65 mg NaCl
MICRO-TIME SCALE: SINGLE PULSE
Scandate Dispenser Cathode Field Enhancement = 3.1 Chemical Potential = 18.08 eV Wavelength = 0.266 microns (W) = 4.7 eV; (Ba) = 1.8 eV Field = 10 MV/m; Theta = 50 % Peak Intensity = 710 Mw/cm2
Reflectivity = 46.2%; = 8.65 nmIncident Gaussian Laser Pulse: “Bulk” Boundary Held Fixed At Temperature Dictated By Macropulse Analysis
Gaussian: y(t) = exp{-[(t-to)/ t]2}Laser t 10.00 ps
to 0.00 ps
Resulting Temp. Distribution (Numerical Sol’n of Time-dependent Heat Equations)fi Long Decay Time Affects Subsequent Photoemission; Long Thermal Tail
Current Distribution Is Same
Electron t 10.01 ps to 0.00 ps
(T(t)-To)/(Tmax-To)
J(t)/J(0)
= 266 nm
ANL 38
EXPERIMENTAL APPARATUS
• Scandate cathodes fabricated by Spectra-Mat Inc.
• Field between cathode and anode varied from 0-2.5 MV/m
• Laser focused to circular spot on cathode with FWHM area of approximately 0.3 cm2
• Q-switched Nd:YAG laser gave Gaussian pulses FWHM 4.5 ns
Current TransformerIon
Pump
Laser InAnode
Window
Cathode
Harmonic λ[nm] QE[%]2 532 0.00653 355 0.02004 266 0.0800
For n=2,3,4 harmonics: Electron emission exhibited "normal" photoemission, i.e., emission proportional to laser
intensity & independent of field
For n=2,3,4 harmonics: Electron emission exhibited "normal" photoemission, i.e., emission proportional to laser
intensity & independent of field
ANL 39
Profilimetry Data
-0.4
-0.2
0.0
0.2
0.4
0.6
0 20 40 60 80 [Radial ]m
SurfaceImage
DISPENSER CATHODE
Consequence: Field Enhancement At
Local Emission Sites
(e.g., Hemisphere: β = 3)
Nd:Yag1064 nm
Scandate Dispenser Cathode(Fabricated by Spectra-Mat)
Work Function: 1.8 eVPartial Coverage of Surface
[0.1 mm]2
Work Function: 1.8 eVPartial Coverage of Surface
ANL 40
Ion Bombardment
ThermalDesorption
CATHODE SURFACE
Interpore ≈ 6 µm; Grain Size≈ 4.5 µm; Pore Diam. ≈ 3 µm (cf M. Green, Tech. Dig IEDM, (1987), p925
€
Dbulk(T )=Dbulko exp−Ed
bulk / kBT[ ]
Dmono(T )=Dmonoo exp−Ed
mono / kBT[ ]θ5
“…the interaction between the barium adsorbate and the substrate plays a crucial role in determining the nature of te surface dipole, which acts to lower the work function ...”
[1] A. Shih, D. R. Mueller, L. A. HemstreetIEEE-TED36, 194 (1989)
[2] L. A. Hemstreet, S. R. Chubb, W. E. Pickett PRB40, 3592 (1989)
BaO
SideView
W
TopView
Bulk vs Monolayer evaporationOne Monolayer At Most Exists On
Cathode Surface
ANL 41
THERMAL - PHOTOCURRENT (II)
Photocurrent Has Two Components
Thermal Emission From Laser-heated Electron Gas
Direct Photoemission
€
φ=− 4QF
€
ΔQ= πro2( ) θJ λ Te(t),( ) + 1−θ( )J λ Te(t),W( )[ ]dt
−∞
∞
∫
1D Model: Emitted Charge ΔQ Originates From Two Sources
“Bare” Regions With Work Function of W (1θw)
Ba-Covered Low Work Function Region (θ)
€
J λ T,( ) =q1−R( )I λ (t)hω
U β hω −φ( )[ ]U β[ ]
⎛
⎝ ⎜
⎞
⎠ ⎟ + ARLDT
2 exp−βφ[ ]
Effect of Field F: Schottky Barrier Lowering (Q = 0.36 eV-nm)
€
U x( ) = ln 1 +ey( )dy−∞
x
∫
€
β =1
kBTe
ANL 42
EXPERIMENTAL CONSIDERATIONS
COMPLICATIONS TO THE 1-D MODEL
• LASER INTENSITY VARIATION
Simulation Area (for FWHM Area = 0.3 cm2) implies ρo =0.5249 cm.
• MACROSCOPIC FIELD VARIATION
Cathode = 1.27 cm Diameter.Anode: Tube With 1.27 cm ID / 2.54 cm ODAnonde-cathode Separation = 0.4 Cm. POISSON: 1 kV Anode = 0.17 Mv/m @ center
• TEMP VARIATION ACROSS SURFACE
Electron Temperature Greatest Where Laser Strongest (Center of Beam): for 1-D Theory, "Effective" θ < Actual Coverage Factor
FWHM ValuesArea = 0.3 cm2
Pulse = 4.5 ns
€
G(ρ,t) =Goexp−ρρo
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
−tΔt ⎛ ⎝ ⎜
⎞ ⎠ ⎟2 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
0
0.1
0.2
0.3
0.4
10 15 20 25 30 35
I(A)(12kV)12kV fitI(A)(6kV)6 kV fit
Time [ns]
To = 516 K
Δ = 22 E mJ
Δ = 2.52t
Δ = 2.58t
€
tFWHM =2 (ln 2)Δt
Emitted Charge Current Density
€
JAmpcm2
⎡
⎣ ⎢
⎤
⎦ ⎥≈
ΔQ[nC]tFWHM[ns]AFWHM [cm2 ]
ANL 44
“Prediction” of Work Function from Exp. QE
-6.0
-4.0
-2.0
4.3 4.4 4.5 4.6 4.7 4.8 [ ]eV
λ = 266 nm
Tungsten
Goldsimulation
Experimental Data† for QE for Au & W:
QE [%] [eV] Predict
Au 7.54x10-6 4.72 - 4.78 4.69
W 3.49x10-5 4.63 - 5.25* 4.52
* Various W faces have different : values shown are for (100) and (111) faces, respectively.
† N. A. Papadogiannis, S. D. Moustaïzis, J. of Phs. D: Appl. Phys. 34, 499 (2001).
€
QE=hωq
ΔQΔE ⎛ ⎝ ⎜
⎞ ⎠ ⎟=
hωq
J λ Tmax( )Iλ (0)
⎛
⎝ ⎜ ⎞
⎠ ⎟Δte
Δtλ
→ 1−R( )hω −
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
Gaussian Laser Pulses: ratio of order unity
Simulation Assumptions• Low Laser Intensity (Eliminates JRLD)
• No Field (Eliminates Schottky Barrier)• UV Light (Over the Barrier Emission)• Δt ratio taken as unity
Simulation Assumptions• Low Laser Intensity (Eliminates JRLD)
• No Field (Eliminates Schottky Barrier)• UV Light (Over the Barrier Emission)• Δt ratio taken as unity
QUANTUM EFFICIENCY & WORK FUNCTION
asymptotic form
ANL 45
2D AND TEMPORAL VARIATION
EXPERIMENTAL QUANTUM EFFICIENCY
€
QE2D =θ 1−R( )
1q
dt 2πρdρ0
∞
∫−∞
∞
∫ J λ t,ρ,T(t,ρ)( )
1hω
dt 2πρdρ0
∞
∫−∞
∞
∫ Iλ t, ρ( )
€
limβ hω−φ( )→ ∞
QE2D =θ 1−R( )hω −φ
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
€
limβ φ−hω( )→ ∞
QE2D =θ 1−R( )QE1D Tmax( )
w+1( )3/2
w=131−
ToTmax
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 +
φ−hω( )kBTmax
⎛
⎝ ⎜ ⎞
⎠ ⎟
LIMIT: Standard (phonon + > barrier)
LIMIT: Thermal-Photo (phonon + < barrier)
BULK T, INTENSITY, AND FIELD DEPENDENT
1064 nm
10-7
10-6
10-5
10-4
10-3
10-2
0.4 0.6 0.8 1
Experiment
β[ /hc λ - φ] >> 1
β[φ - /hcλ ] >> 1
[Wavelength ]m
= 11.836 , ∆E mJ To = 300 K
= 1.7 /Field MV mθ = 0.3085
Qua
ntum
Eff
icie
ncy
standard
thermal-photo
266 nm355 nm532 nm
Bulk T, Intensity, and Field Independent
ANL 46
“THERMAL-PHOTO” EMISSION
For Electrons Excited
From Thermal Tail,
Current Density in
Asymptotic Limit
Appears As a
“Shifted”
Richardson - Laue -
Dushman Equation
And Should Therefore
Be Linear on a
Richardson Plot As
Function of 1/Tmax
€
J λ T,( ) ∝ Iλ (t)J RLD T,φ−hω( )
€
β φ−hω( ) ⇒ ∞
-11
-10
-9
-8
-7
-6
-5
-4
11 12 13 14
TheoryExp
(1/kBT
max) [1/eV]
Field = 1.7 MV/mBulk T = 300 Kθ = 0.3085
ANL 47
10-9
10-7
10-5
10-3
10-1
101
1 10 100
4, 0.3085, 2.55, 10641, 0.6000, 2.55, 10641, 0.6000, 2.55, 8001, 0.6000, 10.00, 8001, 0.6000, 100.00, 800
Laser Intensity [MW/cm2]
Δt = 2.70 ps
βa θ F λ
PROJECTED QUANTUM EFFICIENCY
Successive Approximations:
Present Cathode
Smooth Surface, Increase Coverage
Lower Wavelength to Ti-Saph (800 nm)
Increase Field to Naval ApplicationsQE(118 MW/cm2 = 2.15%)
Increase Field to Accelerator ApplicationsQE(118 MW/cm2) = 5.52%
Extension Simulation From Exp. Parameters to Operational ParametersIntensity Restricted So That Tmax + 300K < Melting Point of W
ANL 48
AN INTRODUCTION TO
ELECTRON EMISSION THEORY AND PROCESSES Nature Of The Emission Barrier
Tunneling, Density, and Current Integral: FN and RLD Equations Complications: Semiconductors, Emission Near Maximum A Thermal - Field Emission Formula Photoemission Considerations Quantum Efficiency A Thermal - Photoemission Formula
Laser Heating of the Electron Gas Laser Heating of the Electron Gas Time-dependent Model of Laser-induced Thermal Photoemission Dispenser Cathode Experiment
Complicating Circumstances Field Enhancement Emission at the Barrier Maximum
OUTLINE
ANL 52
REGIONS OF CURVATURE ENHANCE FIELD: Model Field Enhancement Effect By:
Line Charge of Length L In External Field Fo
tip radius = as, height = zo; length of line = L = [(zo - as)zo]1/2
FIELD ENHANCEMENT
Apex Field = 6.5 FoApex Field = 6.5 Fo
V(x) for as = zo/5V(x) for as = zo/5
FoFo
5.61 Fo
Fo
LL
LL
€
V ρ, z( ) =−Foz+ FozovL ⎛ ⎝ ⎜
⎞ ⎠ ⎟
Q1 u / L( )Q1 zo / L( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Q1 x( ) =12
xlnx+1x−1 ⎡
⎣ ⎢
⎤
⎦ ⎥−1; u±v= z±L( )
2 + ρ2
€
Ftip =zo−as
as
⎛ ⎝ ⎜
⎞ ⎠ ⎟Q1
zozo−as
⎛
⎝ ⎜ ⎞
⎠ ⎟ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
−1
Fo
FIELD ENHANCEMENT
FIELD VARIATION ALONG SURFACE
€
F ρ, z( ) u=zo=
as
ρ2 + as2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Ftip
Schottky Factor at Apex for Fo = 10 MV/m for as = zo/5
€
4QFtip =0.6 eV
ANL 53
0
5
10
15
20
25
-10 -5 0 5 10
Radial ρ [ ]µm
V(ρ,z)
Field Lines
EmitterEmitter
AnodeAnode
LASER ILLUMINATION OF NEEDLE
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
266 nm355 nm512 nm
Anode Voltage [kV]
QE(33kV) = 1.3E–2
QE(33kV) = 3.8E–10
QE(33kV) = 8.8E–7
0
0.2
0.4
0.6
0.8
19 10 11 12
[ ]Energy eV
Vmax
– hf
λ = ∞ nmF = 10.0 /V nm
0.30˚
8 9 10
Energy [eV]
Vmax
– hf
λ = 512 nmF = 2.2 /V nm
0.24 ˚
0
0.2
0.4
0.6
0.8
18 9 10
[ ]Energy eV
Vmax
– hf
λ = 355 nmF = 2.2 /V nm
0.70˚
7 8 9 10
Energy [eV]
Vmax
– hf
λ = 266 nmF = 2.2 /V nm
29.7˚
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
266 nm355 nm512 nmGarcia/Brau
Anode Voltage [kV]
QE(33kV) = 1.3E–2
QE(33kV) = 3.8E–10
QE(33kV) = 8.8E–7
Hernandez-Garcia, Brau, Nucl. Inst. Meth. Phys. A483 (2002) 273
† K. L. Jensen, P. G. O’Shea, D. W. Feldman, 5th Dir. Energy Symp. (Monterey, 11/12/02).
Laser Heating of Electron Gas and Subsequent Thermal-photo-field Emission Model Based on Steady State ad hoc Linear Relation Between Electron Temperature and Illumination†
Affects Emittance of Photo-emission From Protrusions
Laser Heating of Electron Gas and Subsequent Thermal-photo-field Emission Model Based on Steady State ad hoc Linear Relation Between Electron Temperature and Illumination†
Affects Emittance of Photo-emission From Protrusions
ANL 54
ISSUES AND QUESTIONS
QUESTIONS AND COMMENTS BY C. SINCLAIR [1]
Fundamental R&D / theory question(s):
What combination of achievable, external fields results in the maximum charge density in 6-D phase space (from a zero thermal emittance source)? For a CW source, it is not obvious whether DC or RF fields are best (particularly for room temperature RF, where the fields are limited by thermal considerations). For low duty factor applications, the consensus appears to be RF, but that must depend on the bunch charge.
How should emittance be measured, and what is required to have a high quality measurement?
Regarding the relation between thermal emittance and bunch duration at the cathode: space charge fields are reduced by making larger bunches and emitting from a smaller area (which increases longitudinal and transverse emittance) - therefore, for a given bunch charge, what is the optimal emitting area and bunch duration to achieve bets final charge density in 6-D phase space (the answer will depend on whether fields are static or dynamic).
Application and requirements dictate photocathode: needs of low repetition rate, high charge bunches differ from CW pulse trains of lower bunch charges
A goal for progress in photoemission guns: develop reliable methods for generating uniformly populated (transversely and longitudinally) optical pulses to generate uniform charge distributions from the cathode and result in minimum emittance.
[1] Emails to K. Jensen, and Kwang-Je, et al., September 2003
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