Imaging electron motion with attosecond time resolution
Imaging electron motion with attosecond time resolution
Imaging electron motion with attosecond time resolution
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<strong>Imaging</strong> <strong>electron</strong> <strong>motion</strong><br />
<strong>with</strong> <strong>attosecond</strong> <strong>time</strong> <strong>resolution</strong><br />
Peter Abbamonte, UIUC<br />
Collaborators:<br />
Tim Graber, Advanced Photon Source<br />
Wei Ku, Brookhaven National Laboratory<br />
James Reed, UIUC<br />
Serban Smadici, UIUC<br />
Young Il Joe, UIUC<br />
Gerard Wong, UIUC<br />
Robert Coridian, UIUC<br />
Ken Finkelstein, Cornell University<br />
Sol Gruner, Cornell University<br />
Abhay Shukla, Université Pierre et Marie Curie<br />
Jean-Pascale Rueff, Synchrotron SOLIEL<br />
Funding: Office of Basic Energy Sciences, U.S.<br />
Department of Energy #s DE-FG02-06ER46285 &<br />
DE-AC02-98-CH10886
Femtochemistry<br />
1<br />
*<br />
2<br />
2<br />
NaI( Σ0)<br />
→ [Na ⋅⋅⋅ I] → Na( S<br />
1/2)<br />
+ I( P3/<br />
2)<br />
P. Cong, et. al., J. Phys. Chem., 100, 7832 (1996)<br />
“Is there another domain in which the race against <strong>time</strong> can continue to be pushed Subfemtosecond<br />
or <strong>attosecond</strong> (10 -18 s) <strong>resolution</strong> may one day allow for the direct observation<br />
of the <strong>electron</strong>’s <strong>motion</strong>. … In the coming decades we may view <strong>electron</strong> rearrangement,<br />
say, in the benzene molecule, in real <strong>time</strong>.” – Ahmed Zewail, 2000 Nobel Address
Attoscience<br />
Reproduced from Drescher, et. al., Science, 291 1923 (2001)<br />
• Events preceding photofragmentation<br />
• Quasiparticle “birth” in semiconductors<br />
• Inner shell processes (shape resonances)<br />
• Electron transfer chemistry
Press
Why not energy domain - photoabsorption<br />
∆t = 100 <strong>attosecond</strong>, ∆E·∆t = ħ /2 ⇒<br />
∆E = 6.58 eV<br />
grating<br />
slit<br />
counter<br />
lamp<br />
specimen<br />
I( ω)<br />
= exp[ −µ ( ω)<br />
L]<br />
µ ( ω)<br />
⇒ ε(<br />
ω)<br />
ε(<br />
ω)<br />
−1<br />
χ<br />
e(<br />
ω)<br />
=<br />
4π<br />
P( ω)<br />
= χ ( ω)<br />
E(<br />
ω)<br />
e<br />
P(t)<br />
E(t)<br />
∫ ∞ −∞<br />
P( t)<br />
= dt′<br />
χ ( t − t′<br />
) ( t′<br />
e<br />
E )<br />
t 0<br />
<strong>time</strong> (as!)<br />
“…energy-domain measurements on their own are – in general – unable to provide detailed insight<br />
into the evolution of multi-<strong>electron</strong> dynamics.” – M. Drescher, et. al., Nature (2002)
Inelastic x-ray (or n + or e – ) scattering<br />
(k 2 ,ω 2 )<br />
(k 1 ,ω 1 )<br />
Fluctuation-Dissipation theorem:<br />
Bell Jar<br />
I(k,ω) ~ – Im[χ(k,ω)]<br />
k = k 1 – k 2<br />
Pendulum<br />
ω = ω 1 – ω 2
Inelastic x-ray (or n + or e – ) scattering<br />
Couple light to <strong>electron</strong>s<br />
(Lorentz force law)<br />
Be sure to second-quantize to get photons<br />
Multiply out to get interactions<br />
Do perturbation theory (1st Born approximation)<br />
Turns out to be a Green’s function
What is χ(k,ω)<br />
χ(k,ω) :<br />
• density-density Green’s function<br />
• density propagator<br />
• susceptibility<br />
Describes how disturbances in <strong>electron</strong> density travel about the medium.<br />
(x,t)<br />
Causality<br />
Same as a pump-probe experiment.<br />
Dynamics is dynamics<br />
(0,0)
“Phase problem” and the arrow of <strong>time</strong><br />
Cannot invert <strong>with</strong> only Im[χ(k,ω)]<br />
Im[ω]<br />
× × × × × × × × × Re[ω]<br />
× × × × × × × × ×<br />
• χ(x,t) = 0 for t < 0<br />
• Raw spectra do not really describe dynamics – no causal information<br />
• Must assign an arrow of <strong>time</strong> to the problem. Permits retrieval of χ(x,t) –<br />
view dynamics explicitly.
IXS - practical<br />
Backscattering<br />
analyzer<br />
Monochromatic<br />
beam<br />
White beam from APS undulator<br />
scattering<br />
angle<br />
Specimen (H 2<br />
O)<br />
K. D. Finkelstein, P. Abbamonte, V. O. Kostroun, Proc. SPIE Int. Soc. Opt. Eng. 4783, 139 (2002)
Plasma oscillations in water<br />
– Im[χ(k,ω)] (as/Å 3 )<br />
• 8 valence <strong>electron</strong>s / molecule<br />
• ρ = 1 g/cm 3 ⇒ n = 0.20 e/Å 3<br />
• ω p<br />
= √(4πne 2 /m) = 16.6 eV<br />
• k max<br />
= 4.95 Å -1 ⇒ dx = 0.635 Å<br />
• ω max<br />
= 100 eV ⇒ dt = 20.7 as
Problems<br />
Problem #1:<br />
Im[χ(k,ω)] must be defined on infinite ω interval for continuous <strong>time</strong> interval<br />
Solution:<br />
Extrapolate.<br />
Side effects:<br />
• χ(x,t) defined on continuous (infinitely narrow) <strong>time</strong> intervals.<br />
• Time “<strong>resolution</strong>” ∆t N = π/Ω max<br />
• Ω max plays role of pulse width.
Problems<br />
Problem #2:<br />
Discrete points violate causality<br />
Im[χ(k,ω)] must be defined on continuous ω interval. Periodicity incompatible <strong>with</strong><br />
causality.<br />
Solution:<br />
Analytic continuation (interpolate)<br />
Side effects:<br />
• χ(x,t) defined forever. Vanishes for t < 0.<br />
• Repeats <strong>with</strong> period T = 2π/∆ω = 13.8 femtoseconds<br />
• ∆ω plays role of rep rate
Nyquist’s (critical sampling) theorem<br />
f(t)<br />
|f(ω)| 2<br />
t<br />
− ω max<br />
ω max<br />
ω<br />
ω N = 2 ω max<br />
Nyquist frequency<br />
ω too small<br />
⇒ aliasing<br />
∆t N<br />
= 20.7 as<br />
∆x N<br />
= 0.635 Å
Disturbance from a point perturbation.
Disturbance from a point perturbation – frame-by-frame<br />
0.1 Å -6<br />
Units Å -6<br />
clipped at 1 Å -6<br />
0.005 Å -6<br />
∆t N<br />
= 20.7 as<br />
∆x N<br />
= 0.635 Å<br />
• Events transpire in 350 as – light travels 100 nm in vacuum<br />
• Causality Analytic properties Rise of entropy Arrow of <strong>time</strong>
Compound sources<br />
(x,t)<br />
(x 0 ,0)<br />
(x 1 ,0)
Compound sources – oscillating dipole
Compound sources – wake from 9 MeV Au ion<br />
• Phys. Rev. Focus, 14 June 2004<br />
• Chemical & Engineering News, 82, 5 (2004)
Excitons: Frenkel vs. Wannier<br />
conduction band<br />
valence band<br />
Frenkel (Xe, Organics, …)<br />
Wannier (Si, Ge, Cu 2<br />
O, …)<br />
J. Frenkel, Phys. Rev., 37, 17 (1931) G. H. Wannier, Phys. Rev., 52 191 (1937)
Alkali halides – an intermediate case<br />
“Excitation” model<br />
Dexter, D. L., Exciton models in alkali halides, Phys. Rev. 108,<br />
707-712 (1957)<br />
It’s all just Wannier<br />
Hopfield, J. J., & Worlock, J. M., Two-quantum absorption<br />
spectrum in KI and CsI, Phys. Rev. 137, A1455-A1464 (1965)<br />
GW correction / Solve Bethe-Saltpeter eqn.<br />
Rohlfing, M., & Louie, S. G., Electron-hole excitations in<br />
semiconductors and insulators, Phys. Rev. Lett. 81, 2312-2315<br />
(1998)<br />
Discovery of excitons in alkali halides<br />
Hilsch, R., & Pohl, R. W., Über die ersten ultravioletten<br />
Eigenfrequenzen einiger einfacher kristalle, Z. Physik 48, 384-396<br />
(1928)<br />
Marginal case btwn. Frenkel and Wannier<br />
Mott, N. F., Conduction in polar crystals. II. The conduction band and<br />
ultra-violet absorption of alkali halide crystals, Trans. Faraday Soc.<br />
34, 500-506 (1938)<br />
Electron transfer model<br />
Overhauser, A. W., Multiplet structure of excitons in ionic crystals,<br />
Phys. Rev. 101, 1702-1712 (1956)
Frenkel vs. Wannier in <strong>time</strong>-domain IXS<br />
Wannier’s Excitonic Basis:<br />
| R, R + β ><br />
Excitons come from diagonalizing<br />
∑<br />
K<br />
ββ ′<br />
′<br />
R<br />
−i<br />
⋅R<br />
H ( K) = e < R, R + β | H | 0, β ><br />
For Frenkel exciton, dominated by one term:<br />
∑<br />
K<br />
R<br />
( ) −i<br />
⋅R<br />
H , | | 0,0<br />
00<br />
K = e < R R H ><br />
Conclusion: Frenkel exciton keeps its size / shape through its life. Wannier<br />
changes. Inverted IXS directly sensitive to Wannier vs. Frenkel vs.<br />
intermediate descriptions.
Result
Time response<br />
All processes:<br />
• Exciton<br />
• Interband<br />
• Plasmon<br />
• Core levels<br />
• Compton<br />
scattering
Isolating the exciton<br />
g<br />
gap<br />
B<br />
( ω)<br />
= 1 + e<br />
k<br />
k<br />
g<br />
−W<br />
( ω−ω<br />
)<br />
Im[ χ ] = Im[ χ ] −g<br />
( ω )<br />
e,<br />
n n gap n<br />
⎧<br />
Im[ χne,<br />
n]<br />
= ⎨<br />
⎩<br />
g ( ω ) ω < ω<br />
gap n n c<br />
Im[ χ ] ω ≥ ω<br />
n n c<br />
Truncated at 16.5 eV
Exciton dynamics<br />
Frenkel limit.
Wannier functions (Ku, et. al.)
Lessons<br />
•Exciton in LiF is Frenkel, despite the old controversy.<br />
•No contradiction between CT and Frenkel. Lattice geometry not a<br />
constraint.<br />
•Causality allows solution to “phase problem” for IXS ⇒ zeptoseconds!<br />
•Is there information in this which cannot be read off the raw spectra!<br />
1. Extrapolation 2. Causality<br />
• χ(x,t) useful for analyzing extended sources
Homogeneous vs. homogeneous broadening
Graphite<br />
Wallace, Phys Rev. 71, 622 (1947)<br />
σ,π plasmon<br />
Compton<br />
scattering<br />
π plasmon