Energy versus time in x-ray scattering

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Energy versus time in x-ray scattering Peter Abbamonte University of

Illinois Anshul Kogar, U. Illinois Bruno Uchoa, U.

Illinois James Reed, DTRA Gerard Wong, UCLA Diego Casa, Advanced

1 Energy versus time in x-ray scattering Peter Abbamonte University of Illinois Thanks to: Yu Gan, U. Illinois Young Il Joe, U. Illinois Anshul Kogar, U. Illinois Bruno Uchoa, U. Oklahoma Eduardo Fradkin, U. Illinois James Reed, DTRA Gerard Wong, UCLA Diego Casa, Advanced Photon Source and many others Funding: DOE grant nos. DE-FG02-06ER46453 & DE- AC02-98-CH10886 Y. Gan, A. Kogar, P. Abbamonte, Chem. Phys. 414, 160 (2012)

2 Momentum- and energy-resolved scattering BL43LXU IXS spectrometer at SPring8 Measures a correlation function, S(q,) SEQUOIA spectrometer at SNS ( See talk yesterday by Toby Perring )

scattering techniques, which are also said to measure dynamics?

3 A new approach: Free Electron Lasers Can for the first time study ultrafast dynamics with a momentum-resolved probe Questions for today: How is this different from inelastic scattering techniques, which are also said to measure dynamics? That is, how are time and frequency related? Where does scattering come from, and how does it measure dynamics anyway?

4 Example of vs. t : the dielectric function, () ( ) 14 ( ) e ( ) e ( ) 1 4 E P( ) ( ) E( ) e t 0 E(t) P(t) time Time relationship is nonlocal: d e() t e( ) 2 i t e P() t dte( tt) E( t) e is a Green s function dependence retardation

Electron in an EM field (classical) Can define the fields in terms of potentials: BA E 1 c A t (Lorentz force law) Classical motion described by the Lagrangian 1 e x A * 2 c 2 L K V mx e The

5 Electron in an EM field (classical) Can define the fields in terms of potentials: BA E 1 c A t (Lorentz force law) Classical motion described by the Lagrangian 1 e x A * 2 c 2 L K V mx e The canonical momentum is c L e p mx A x c This allows one to define the classical Hamiltonian c 1 c e H pir i L e i 2m p c A Hamilton s equations give the equations of motion: x i H p c i p i H x i 2 Result is the Lorentz force law: 1 m xee x B c * Gaussian units

6 Electron in an EM field (quantum) The Hamiltonian is now an operator. Photons are massless so (Lorentz we have force to use second quantization: H ˆ H ˆ H ˆ H ˆ law) 2 2 ˆ E B HEM dx ˆ ˆ 2 2 EM electron ˆ electron H interaction Where ˆ ( x, t) annihilates an electron at position x and time t. The vector potential is an operator that creates or annihilates photons: note: e pmx A c Multiplying everything out gives fundamental interactions between electrons and photons:

7 Scattering Scattering takes place when these interactions evolve a photon (Lorentz from an force initial state to a final state, with a corresponding change in the electronic subsystem: law) i a m, k i i What does this is the time-evolution operator: f a, n k f f M iu, f I Nonresonant x-ray scattering Resonant inelastic x-ray scattering (RIXS) (J. van den Brink, after the coffee break)

8 Cross section for x-ray scattering The differential scattering cross section comes from Fermi s golden rule N wf i E E where counting states in a box of volume V provides the density of final states: 2 N E c V 8 c V f 3 3 (for one incident photon) Cross section for nonresonant x-ray scattering 2 ( q, ) E r 2 f * 0 f i 2 S( q, ) i k k f ki f i dynamic structure factor what is it?

9 S(q,) and the Van Hove function Cross section: 2 ( q, ) E 2 r S( q, ) 2 2 * S( q, ) P ˆ m n n( q) m ( n m) mn, This is the so-called dynamic structure factor. Assuming we are in thermodynamic equilibrium, S has the form S(q,) is the Fourier transform of the Van Hove function, G(x,t), which is the spacetime correlation function for the electron density: S( q, ) dxdtg( x, t) e i qx t where G( x, t) dxdt nˆ( x, t) nˆ( xx, tt) The brackets denote a QM thermal average: Oˆ P ˆ m m O m m What does this have to do with dynamics?

10 Fluctuation-Dissipation Theorem (x 1,t 1 ) (x 2,t 2 ) 1 1 S( q, ) Im / ( q, ) kt 1 e i ( x1, x2, t1 t2) ˆ ( x ˆ 2, t2), ( x1, t1) ( t1t2) fluctuationdissipation theorem Retarded density Green s function (k 2, 2 ) (k 1, 1 ) Describes how charge propagates in a system: Phonons Excitons Plasmons Etc. Electron-hole pairs

11 Green s functions or Propagators Dynamics is described by a propagator K( x, t; x, t) (x,t) Electrons: ˆ ˆ G( x, t; x, t) i/ 0 ( x, t), ( x, t) 0 ( tt) (x,t) Density: ( x, t; x, t) i/ 0 ˆ ( x, t), ˆ ( x, t) 0 ( tt) Some translational symmetry: Frequency / momentum representation is more illuminating. G( k, ) Im[] Re[] Best way in a many-body system to define what is a particle

12 View propagator in real time? Crazy idea: Can we Fourier Transform IXS data and make real time movies? Why? Should be incredibly easy to get attosecond time resolution: Et ~ 2 t ~100 as E ~7eV Can we FT to observe a propagator directly? Answer: No 1 1 S( q, ) Im / ( q, ) kt 1 e Oops Our information is incomplete. Cannot Fourier transform with only the imaginary part.* *This is what Fermi called the inverse scattering problem.

The phase problem reexamined Central Dogma of x-ray crystallography:

A. Doyle, et al., Science 280, 69 (1998) KcsA channel R.

Nobel Prize, 2003 Phase problem is solved by incorporating constraints

13 The phase problem reexamined Central Dogma of x-ray crystallography: I( q) ( q) Periodic system (i.e., a crystal): i () e G 2 Gr 2 r IG G G D. A. Doyle, et al., Science 280, 69 (1998) KcsA channel R. MacKinnon Chem. Nobel Prize, 2003 Phase problem is solved by incorporating constraints (Hg or Se atoms) This is the basis for the field of structural genomics Based on classical scattering theory. All scattering is elastic.

14 The phase problem reexamined Van Hove function: 1 2 S( q, ) n ˆ ( q) 0 ( 0 m) What we think we measure: 1 1 S( q, ) 0 ˆ ( q) 0 ˆ ( q) 0 What we actually measure: n n 2 2 ( T = 0 ) ds( q, ) n ˆ ( q)0 0 ˆ ( q) ˆ ( q)0 ˆ ( q) 2 2 More general formulation of the phase problem: 1 1 S( q, ) Im / ( q, ) kt 1 e 0 k Im (, ) Re ( k, ) P d (x,t) = 0 for t < 0 Raw spectra do not really describe dynamics no causal information Causality is the constraint. Must assign an arrow of time to the problem. Rise of entropy arrow of time

scattering, we only measure the diagonal (G=0) components of this matrix: 1 1 S( k, ) Im / ( k, k, ) kt 1 e Naïvely Fourier

15 What if the system is inhomogeneous? Assume it s periodic: i ( x1, x2, t1t2) ˆ ( x ˆ 2, t2), ( x1, t1) ( t2 t1) ( k, k, ) ( k, Gk, ) In regular scattering, we only measure the diagonal (G=0) components of this matrix: 1 1 S( k, ) Im / ( k, k, ) kt 1 e Naïvely Fourier transform and you get a spatial average: (,) r t dr( r, rr,) t [P. A., et al., Phys. Rev. B 80, (2009)] If the system is homogeneous, this is OK. If not, things get even better but let s start with the homogeneous case.

16 Plasma oscillations in water Im[(k,)] (as/å 3 ) 8 valence electrons / molecule = 1 g/cm 3 n = 0.20 e/å 3 p = (4ne 2 /m) = 16.6 ev

17 Problems Problem #1: Im[(k,)] must be defined on infinite interval for continuous time interval Solution: Extrapolate. Side effects: (x,t) defined on continuous (infinitely narrow) time intervals. Time resolution t N = / max max plays role of pulse width.

18 More Problems Problem #2: Discrete points violate causality Im[(k,)] must be defined on continuous interval. Periodicity incompatible with causality. Solution: Interpolation (i.e., add data) Side effects: (x,t) defined forever. Vanishes for t < 0. Repeats with period T = 2/ plays role of rep rate

19 Disturbance from a point source in water 10 Å Units Å -6 clipped at 1 Å -6 t N = 20.7 as x N = Å P. A., et al., PRL, 92, (2004)

Frame-by-frame 0.1 Å -6 Units Å -6 clipped at 1 Å -6 0.005 Å -6 t N = 20.7 as x N = 0.

20 Frame-by-frame 0.1 Å -6 Units Å -6 clipped at 1 Å Å -6 t N = 20.7 as x N = Å Events transpire in 350 as light travels 100 nm in vacuum Causality Analytic properties Rise of entropy Arrow of time

, PRL 103, 237402 (2009) Birth of an exciton

21 Attosecond imaging with IXS Ion solvation dynamics (t = 26 fs) R. Coridan, et al., PRL 103, (2009) Birth of an exciton in LiF (t = 20.6 as) P. A. et al., PNAS 105, (2008)

22 Effective fine structure constant of graphene * (, ) 1 ( ) (, ) g k g V k k Charge propagator. Gives screening correction to g. t = 10.3 attoseconds (10-17 sec)r = 0.2 Å

23 Effective fine structure constant, * ( k, ) g * ( k, ) g k * arg g (, ) * g 2.6 ( 1) Energy (ev) Momentum (Å -1 ) Momentum (Å -1 ) (0,0) lim ( k,0) / 7 * * g g k0 J. P. Reed, et al., Science 330, 805 (2010)

24 New results

X-Ray Scattering, Finally Done Correctly We need the

waves: Si J. A. Golovchenko, et al., PRL 46, 1454 (1981) W.

25 X-Ray Scattering, Finally Done Correctly We need the off-diagonal terms. Can we measure them? i ( x1, x2, t1t2) ˆ ( x ˆ 2, t2), ( x1, t1) ( t2 t1) ( k, k, ) ( k, Gk, ) (periodic case) Yes X-ray standing waves: Si J. A. Golovchenko, et al., PRL 46, 1454 (1981) W. Schulke, U. Bonse, S. Mourikis, PRL 47, 1209 (1981) W. Schulke, A. Kaprolat, PRL 67, 879 (1991)

26 X-Ray Scattering, Finally Done Correctly Incident photon is in a superposition of momenta: Now do scattering: The cross section contains interference terms: Interference terms are off-diagonal dynamic structure factor Phase of the standing wave. Requires coherent wave field. Generalized fluctuation-dissipation theorem: Y. Gan, A. Kogar, P. Abbamonte, Chem. Phys. 414, 160 (2012)

27 Crystallography for the collective excitations Y. Gan, A. Kogar, P. Abbamonte, Chem. Phys. 414, 160 (2012)

28 Crystallography for the collective excitations Complete reconstruction of Angstrom spatial resolution Attosecond time resolution ( r, r, t) 1 2 Y. Gan, A. Kogar, P. Abbamonte, Chem. Phys. 414, 160 (2012)

29 Crystallography for the collective excitations Complete reconstruction of Angstrom spatial resolution Attosecond time resolution ( r, r, t) 1 2 Y. Gan, A. Kogar, P. Abbamonte, Chem. Phys. 414, 160 (2012)

RPA in infinite systems

RPA in infinite systems Translational invariance leads to conservation of the total momentum, in other words excited states with different total momentum don t mix So polarization propagator diagonal in