3. New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)
Light scattering
scattering generally decreases with ω
scattering ~ polarization ~ displacement
δx decreases with ω
..
m x = eE cos(ωt)
(force acts for shorter time)
4. New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)
Nonlinear x-ray scattering to date
Spontaneous processes (PDC & Raman)
(large vacuum fields ~1019 W/cm2 at 1 Å)
5. New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)
• Understand which xNLO processes are feasible
• Understand similarities/differences in information obtainable
6. New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)
New capabilities
• Microprobe optical interactions (x/o sfg)
(directly measure induced charge, microfields)
(screening response)
• Determine Valence charge density (x/xuv sfg)
(backgnd free)
(flow of valence charge during dynamics)
• Pump and probe on microscopic level
(x-ray four wave mixing)
7. X-ray Four Wave Mixing : Exciton Dynamics
Tanaka & Mukamel PRL 89 043001 (2002) : polydiacetylene
How is exciton transported along a molecular chain ?
* delay kraman
A B
exciton dynamics
FWM Spectroscopy
(tunable source, multiple frequencies) creation detection
• valence exciton is created at site A (k1, k2) migration
• exciton migrates to site B
• time delayed detection at site B (k3,kraman) k1
~ 100 eV k2
k3 kraman
A B
8. New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)
New capabilities
• Microprobe optical interactions (x/o sfg)
(directly measure induced charge, microfields)
(screening response)
• Determine Valence charge density (x/xuv sfg)
(backgnd free)
(flow of valence charge during dynamics)
• Pump and probe on microscopic level
(x-ray four wave mixing)
Today : x/o mixing & light-matter interactions
11. How does light catalyze dynamics ?
photochemistry Materials science
(isomerization) (phonons)
the Primary Light-Matter interaction is
microscopic rearrangement of valence charge
subsequent dynamics
Problem
We often lack a deep understanding of the microcopic details
of how light manipulates matter !
• theoretically complex
• tough to measure
12. How does light catalyze dynamics ?
photochemistry Materials science
(isomerization) (phonons)
the Primary Light-Matter interaction is
microscopic rearrangement of valence charge
subsequent dynamics
Problem
We often lack a deep understanding of the microcopic details
of how light manipulates matter !
• theoretically complex
• tough to measure
13. Why is the optical response complex ?
Coupling between induced dipoles
Shine light on a material
electron
dipole field applied field
(screening response)
(over screening)
14. Screening response in a material
Apply light to a material. Generally don't know magnitude (or
even direction) of resulting force on charges in the system.
Emicroscopic = Eapplied + Epolarization
self-consistent internal field 'Local' Field in the material
(many body interactions) (to within self-field effects)
Local Field Effects
Apply light (Klight) to a crystal.
Emicroscopic(Klight+G)
Eapplied(Klight)
Lattice vector
Varies on scale of atoms
Constant on atomic
lengthscale
15. Screening response in a material
Apply light to a material. Generally don't know magnitude (or
even direction) of resulting force on charges in the system.
Emicroscopic = Eapplied + Epolarization
self-consistent internal field 'Local' Field in the material
(many body interactions) (to within self-field effects)
Local Field Effects
Apply light (Klight) to a crystal.
Eapplied(Klight)
Polarization varies on
scale of atoms
Constant on atomic
lengthscale
16. Screening response in a material
Apply light to a material. Generally don't know magnitude (or
even direction) of resulting force on charges in the system.
Emicroscopic = Eapplied + Epolarization
self-consistent internal field 'Local' Field in the material
(many body interactions) (to within self-field effects)
Local Field Effects
Apply light (Klight) to a crystal.
Eapplied(Klight)
Emicroscopic(Klight+G)
Constant on atomic
lengthscale
Local field effects refer to distinction between the macroscopic field
Eapplied(Klight) and the microscopic field Emicro(Klight+G)
17. Why is the optical response important ?
Practical reasons (develop devices)
Fundamental Materials Physics
(material properties)
(ground state charge distribution)
Analogy with screening of ionic cores in a material
Valence electron gas Ground state charge distribution
Screening
ions appear response
18. How does light catalyze dynamics ?
photochemistry Materials science
(isomerization) (phonons)
the Primary Light-Matter interaction is
microscopic rearrangement of valence charge
subsequent dynamics
Problem
We often lack a deep understanding of the microcopic details
of how light manipulates matter !
• theoretically complex
• tough to measure
19. Microscopic details of Light-matter interactions ?
No methods to directly measure !
Optical probes average over
macroscopic (~µm) lengthscale
Atomic lengthscale
information is lost
20. “Seeing” matter on atomic lengthscales with X-rays
1935
Static Pictures
Why not simply use diffraction to 'see'
changes to valence charge density ?
X-rays in Theory & Experiment
(Compton & Allison)
21. X-ray Diffraction
Measures Qth Fourier component of the electronic charge density.
Problem
X-ray scattering dominated by scatter
x-ray from core charge
Q (Poor at probing valence charge !)
Valence charge is important !
(determines chemistry, charge conduction, etc)
23. X-ray / Optical wave mixing
X/O Sum Frequency Generation
x-ray + optical x-rays inelastically scatter from optically
driven charge density oscillations
optical hνx ± hνo
h νo
h νx
x-ray hνx
hνx+o
|V>
optical dipole
|G>
Directly microprobes optical interactions s p
Freund & Levine Phys. Rev. Lett. 25,1241 (1970)
Eisenberger & McCall Phys. Rev. A 3,1145 (1971)
Selective x-ray diffraction !
(preferential oscillation of valence charge)
ksum = kx + ko + GHKL momentum
ωsum = ωx + ωo energy
24. X-ray / Optical Sum Frequency Generation
What’s probed ?
x-ray diffraction measures charge densities
efficiency ~ ρ2Q
x-ray in x-ray out
momentum
Q
transfer
laser
Scattering regimes x-ray/optical SFG
optically induced redistribution
δρ = of valence charge
Scattering
Cross Section
(Lorentz
oscillator)
frequency
Rayleigh ~ 1/λ4 Resonant Thomson
25. X-ray / Optical Sum Frequency Generation
What’s probed ?
x-ray diffraction measures charge densities
efficiency ~ ρ2Q
x-ray in x-ray out
momentum
Q
transfer
laser
Scattering regimes x-ray/optical SFG
optically induced redistribution
δρ = of valence charge
Scattering
Cross Section
(Lorentz
oscillator)
frequency
hv
Rayleigh ~ 1/λ4 Resonant Thomson
s p
26. X-ray / Optical Sum Frequency Generation
What’s probed ?
x-ray diffraction measures charge densities
efficiency ~ ρ2Q
x-ray in x-ray out
momentum
Q
transfer
laser
Scattering regimes x-ray/xuv SFG
δρ = full valence charge distribution
Scattering
Cross Section
(Lorentz
oscillator)
frequency
Rayleigh ~ 1/λ4 Resonant Thomson
s p
All valence charge scatters as a
Thomson dipole.
27. X-ray / Optical Sum Frequency Generation
What’s probed ?
x-ray diffraction measures charge densities
efficiency ~ ρ2Q
x-ray in x-ray out
momentum
Q
transfer
laser
Scattering regimes x-ray/xuv SFG
δρ = full valence charge distribution
Scattering
Cross Section
(Lorentz
oscillator)
frequency
Rayleigh ~ 1/λ4 Resonant Thomson
s p
I. Freund Chem. Phys. Lett. 12, 583
(1972)
28. X-ray / Optical Wave Mixing
Experiments tried in early 1970s failed
presumably due to weak xray sources
35. Experimental : Data Acquisition
Monochromator Detector
Si (111)
X-rays x-ray
1 eV, 2 µrad
δE ~ 20 eV δθ ~ 2 µrad Diamond 111
Diamond rocking curve
36. Experimental : Data Acquisition
Energy Filtering
Monochromator Si 220
Si (111)
Detector
X-rays x-ray
1 eV, 2 µrad
δE ~ 20 eV δθ ~ 2 µrad Diamond 111
Diamond rocking curve Si 220 calibration
37. Experimental : Space-time overlap
Translate to Bi (111) : laser perturbed diffraction for space-time overlap
(D.M. Fritz et al. Science 315, 633 (2007))
Detector
Energy Filtering
Si 220
Monochromator
Si (111) Bi (111) Sample
apertures
x-ray o
X-rays ~15
1 eV, 2 µrad
δE ~ 20 eV δθ ~ 2 µrad
optical
x/o delay
38. Experimental : SFG Data
X-ray / optical cross-correlation
SFG signal vs x-ray / optical delay
46. Experimental : Measured Efficiency
x-ray
Diamond
optical
Ioptical ~ 1010 W/cm2
Absolute efficiency Relative efficiency
efficiency relative
SFG power / input x-ray power
to ‘regular’ diffraction
2.4 x 10-7 1.7 x 10-6
estimated uncertainty ~ factor of 2
47. Wave Equation Model for x/o SFG
Wave equation
∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2
dPNL/dt = JNL (ωx + ωo)
X-rays see free electrons
.
mv=F
d v/dt = ∂v/∂t + ( v . ) v = (q/m) (E + vxB/c)
∆
JNL (ωx + ωo) = ρ(0)v(2) + ρ(1)v(1)
48. Wave Equation Model for x/o SFG
Wave equation
∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2
dPNL/dt = JNL (ωx + ωo)
X-rays see free electrons
.
mv=F
d v/dt = ∂v/∂t + ( v . ) v = (q/m) (E + vxB/c)
∆
JNL (ωx + ωo) = i(e/2m) Dωx ρo(1) Ex - Doppler
(ρ(u)/2ωsum) (e2/2m2) Dωo Dωx (Eo. kx)E + Displacement
(ρ(u)/2ωsum) (e2/m2) (Dωo/ωx) Eox(kx x Ex) Lorentz
Dωj ≡ ωj /(ωb2- ωj2)
49. Wave Equation Model for x/o SFG
Wave equation
∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2
dPNL/dt = JNL (ωx + ωo)
X-rays see free electrons
.
mv=F
d v/dt = ∂v/∂t + ( v . ) v = (q/m) (E + vxB/c)
∆
JNL (ωx + ωo) = i(e/2m) Dωx ρo(1) Ex
Eoe ωot
-i
Exe-iωxt
x/o SFG : Optical Doppler term Dominates !
50. Wave Equation Model : SFG power vs angle & energy
1 um crystal δE ~ 720 meV δθ ~ 14 urads
δ angle
δ ω
10 um crystal δE ~ 210 meV δθ ~ 8 urads
500 um crystal δE ~ 130 meV δθ ~ 6 urads
51. Wave Equation Model : Predicted Efficiency
Efficiency vs Crystal thicknesss Induced Charge/Microfields
absorption induced charge is the single
no absorption free parameter
JNL = ρ(111) vx
charge & microfield related
by Gauss’ law
4π ρ
∆ . E =
i G111 . E111 = 4π ρ111
Crystal thickness (m)
ρ(111) ~ 7.3x10-5 e/Å3
Efficiency = SFG power / X-ray power in
E111 / Emacro ~1/6
input beam properties
δΕx-ray ~ 1 eV δλoptical ~ 35 nm Reproduce measured efficiency
δθx-ray ~ 2 ur δθoptical ~ 4 mr
δτx-ray ~ 80 fs δτoptical ~ 2 ps
52. Models for microscopic optical response
• Bond Charge Model
(semi-empirical)
• Molecular Orbital Calculation (1974)
• Pseudopotential Calculation (1972)
• Density Functional Calculation
('first principles')
53. Diamond unit cell and primitive cell
Unit cell FCC with two atom basis
Two types of bonding orientation
Primitive cell
8 atoms (16 bonds) in Unit Cell
2 atoms (4 bonds) in Primitive Cell
54. Covalent Bond Formation
isolated atoms Molecular Orbital View
Diamond bond : sp3 orbitals
covalently bonded atoms
55. Covalent Bond Formation
isolated atoms Molecular Orbital View
Diamond bond : sp3 orbitals
o
109.5
covalently bonded atoms
56. Covalent Bond in Diamond
Valence electron gas Pseudopotential Strategy
Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
57. Covalent Bond in Diamond
Ionic cores appear Pseudopotential Strategy
Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution
(screening response)
58. Covalent Bond in Diamond
Screening response to lowest order Pseudopotential Strategy
overlap charge density Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution
Co-ordinate and nonspherical (screening response)
charge described beyond lowest
order (nonlinear screening)
59. Covalent Bond in Diamond
Nonlinear screening central to
covalent bond formation
Pseudopotential Strategy
Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution
(screening response)
60. Covalent Bond in Diamond
Nonlinear screening central to
covalent bond formation
Pseudopotential Strategy
Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution
(screening response)
The self consistent field / charge
distribution develops
61. Covalent Bond in Diamond
Nonlinear screening central to
covalent bond formation
Pseudopotential Strategy
Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.
Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.
Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution
(screening response)
Covalent bonds stabalize lattice against
shear distortion
62. Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge
B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse Nonlinear optical susceptibilities χ(2), χ(3)
Raman scattering
Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")
Basic idea
Goal : compute optically induced charge density
{magnitude of (average) optical response & microscopic spatial distribution}
• Magnitude : from macroscopic optical measurements
• Spatial distribution : make a guess
Induced charge density identical to (measured) bond charge density.
(rigid bond model)
63. Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge
B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse Nonlinear optical susceptibilities χ(2), χ(3)
Raman scattering
Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")
Equations
Nonlocal response
δρ(r) = - .P
∆
Polarization at ith bond
P(r) = Ncell ∫cell αmicro(r,r’) Emicro(r’) d3r’
influenced by field from
other polarized bonds
P(q,G) = Ncell ∑G’αmicro(q,G,G’) Emicro(q,G’)
64. Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge
B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse Nonlinear optical susceptibilities χ(2), χ(3)
Raman scattering
Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")
Equations
Local response approximation
Local response approximation
δρ(r) = - .P
∆
Polarization at ith bond
P(r) = Ncell αmicro(r) Emacro(r)
determined onlyith bond
Polarization at by
macroscoic field by
determined only
macroscoic field P(q,G) = Ncell αmicro(q,G) Emacro(q)
65. Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge
B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse Nonlinear optical susceptibilities χ(2), χ(3)
Raman scattering
Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")
Equations
Local response approximation
Local response approximation P(q,G) = χG Emacro(q)
FHKL χmacroscopic
Polarization at ith bond
structure factor
determined onlyith bond
Polarization at by
macroscoic field by
determined only (for bond charge)
Emacro = {3/(2+ε)} Eoptical, vacuum
macroscoic field
ε = 1+ 4πχmacro
66. Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge
B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse Nonlinear optical susceptibilities χ(2), χ(3)
Raman scattering
Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")
Equations
Local response approximation
Local response approximation
P(q,G) = χG {3/(2+ε)} Eoptical, vacuum
Polarization at ith bond
determined onlyith bond
Polarization at by
Overestimates measurement by ~ x2
macroscoic field by
determined only
macroscoic field ( 1 Fourier component )
67. Two models for induced valence charge
Pseudopotential
Molecular Orbital
68. Two 1970s calculations of microscopic fields
How precisely does light exert its force ? Two predictions
δρ laser via Molecular Orbital δρ laser via Pseudopotential
Arya & Jha Phys. Rev. B 10,4485 (1974) Van Vechten & Martin Phys. Rev. Lett 28,446 (1972)
agrees with semi-empirical δρ more delocalized
(spreads beyond bond charge)
bond charge model
Mixing efficiency ~ x100 lower
Presupposes a localized response Calculation decides degree of localization
69. Two 1970s calculations of microscopic fields
How precisely does light exert its force ? Two predictions
δρ laser via Molecular Orbital δρ laser via Pseudopotential
Arya & Jha Phys. Rev. B 10,4485 (1974) Van Vechten & Martin Phys. Rev. Lett 28,446 (1972)
agrees with semi-empirical δρ more delocalized
(spreads beyond bond charge)
bond charge model
Mixing efficiency ~ x100 lower
Presupposes a localized response Calculation decides degree of localization
Overestimates measurement by ~ x2 Underestimates measurement by ~ x6
70. Density Functional Theory
Calculation of induced charge in diamond
covalent bond
Ground state valence charge density
0.25
0.20
0.15
Charge density
0.10
0.05
0.00
0.0 0.2 0.4 0.6 0.8
Position
atoms
71. Density Functional Theory
Calculation of induced charge in diamond
Induced (valence) charge density
Charge
density
72. Density Functional Theory
Overlay Induced over ground state charge density
ground
induced
x1000
Atoms
Optical activity
pretty well
confined to the
bond charge
73. Density Functional Theory
Overlay Induced over ground state charge density
ground
induced
x1000
Atoms
Optical activity
pretty well
confined to the
bond charge
74. Density Functional Theory
Overlay Induced over ground state charge density
ground
induced
x1000
Atoms
Optical activity
pretty well
confined to the
bond charge
75. Density Functional Theory
Overlay Induced over ground state charge density
ground
induced
x1000
Atoms
Optical activity
pretty well
confined to the
bond charge
76. Compare model prediction to measurement
Induced charge density (e/Å3) Absolute efficiency
x/o SFG measurement 7.3 x 10-5 (x or / √2) 2.4 x 10-7 (x or / 2)
DFT prediction 1.1 x 10-4 5.4 x 10-7
BC (MO) prediction 1.3 x 10-4 7.6 x 10-7
VVM pseudopotential ~ ρBC / 10 ~ 7.6 x 10-9
Density Functional Calculation & Bond Charge Model
agree with data to within error bars
optical How does light interact with Diamond ?
To good approximation, optical
activity confined to bond charge !
77. Summary
• Observation of x-ray/optical sum frequency generation
• Measurement and ab initio simulations suggest simple bond
charge model accurate down to microscopic length scales
• New opportunities in nonlinear x-ray scattering created by
x-ray FELs
X-ray/optical wave mixing particularly important sub-field of
nonlinear x-ray scattering due to relatively high efficiency
78. X/O Collaboration
Jerry Hastings Steve Harris Jan Feldkamp
David Fritz Sharon Schwartz Diling Zhu
Marco Cammarata David Reis Sinisa Coh
Henrik Lemke Ryan Coffee Tom Allison
