Inelastic x-ray scattering
Mark P. M. DeanBrookhaven National Laboratory
Cherion School
SPring-8, September 16th 2015

Outline
Basics of inelastic x-rayscattering (IXS)§ What is IXS?
§ What can be measured andwhy should you care?
§ Cross section and S(Q, ω)

Different material degrees
of freedom
§§
§
§

LatticeSpins
Orbitals
ChargePlan for each of these§§§§Cross section
Instrument
Case study
Advantages/disadvantages
of IXS vs. competing
methods

BASICS OF INELASTICSCATTERING

Scattering
ωi, ki, εi

ωf, kf, εf

Elastic scatteringdσdΩω f = ωi

Inelastic scatteringd 2σdΩdωω f ≠ ωi2θ4π
Q = k f − ki ≈sin(θ )λω = ω f − ωi

For condensed matter need λ ≈ a ≈ 1 Å i.e. x-rays!

in electricalinsulators♦ d• Now
considera andone-dimeωsemiconductors
molecules
springatom

Can be modeled to first
Inelastic scattering –• accessing
constantorder with spring-mass g
dynamics & Hamiltonian
dynamics

Inelastic scattering gives
excitation spectrum i.e.
nanoHUB.org
eigenstates that determine
the system’s dynamicsspringconstantgatom,
massmmassmωuna

online simulations and moreaa

Dispersion Curve• Changing K by 2π/a leaves x unaffected

Can infer Hamiltonian2g
Η = ∑ (un − un+1 )2 nOnly N values of K are unique♦ We take them to lie in -π/a < K < π/aω(K)♦ω (Q)

2(g/m)1/2

Full description ofinteractions!

K- π/aπ/aQ

Degrees of freedom & excitations
Degree of freedom

Excitations

Very approximate
Energy scale

Lattice

Phonons

0->200 meV

Spin

Magnon, bimagnon,
spinon, …

0-500 meVOrbital

dd-transition ororbiton~1-5 eV

ChargePlasmon, chargetransfer excitation~0-100 eV

X-rays can access all these excitations!
Very stringent tests for model Hamiltonians related to
all these degrees of freedom.

Excitations spectrum of a metal

7518

S K Sinha

0.1

10100

Figure 3. Schematic diagram of S(q , ω) illustrating poles or quasi-poles of χ(q , ω) for various
elementary excitations.

Energy (eV)

Excitation spectrum of a strongly
correlated material
Mott insulator La2CuO4

Word of warning
These excitations cannot be measured in one
experiment! Lots of parameters need to be optimized
for a particular goal§ Working energy§ Energy resolution / throughput trade off§ Non-resonant or resonant (and which resonance)

IXS is consists of many different sub techniques eachwith specialized instruments§ Fail to plan – plan to fail!

Kramers–Heisenberg formula2〈 f | H int | n〉〈 n | H int | i〉
dσ
∝ 〈 f | H int | i〉 + ∑dΩdω
Ei − En + !ωi + iΓ|n 〉
1st term
2nd term

2Interaction between photon and electrons in a material
though H int
Preferable route!
d 2σ "2≈|P(ω)|S(q,ω) ?
$dΩdω #res
Properties ofx-rays andcore electrons

Propertiesof valence
electrons

Ament et al. Rev. Mod. Phys. 83, 705 (2011)

Dynamical structure factor S(Q, ω)−iω t∞
1+S(Q, ω ) =dte
ρ(Q,t=0)ρ(Q, t)∫2π ! −∞1χ ''(Q, ω )S(Q, ω ) =π 1− exp(−ω / K BT )

χ (Q, ω ) = χ '(Q, ω )+i χ ''(Q, ω )P = ε0 χ EM =χH

Response of material
to a small oscillating
field

MEASURING THE LATTICE I.E.
PHONONS

Case study: MgB2
Superconductor at 39 K
record (at the time) for noncuprate
Flurry of activity todetermine superconducting
mechanismPhonons?
J. Nagamatsu et al., Nature 410, 63-64 (2001)

Back to Kramers–Heisenberg formula2

2dσ〈 f | H int | n〉〈n | H int | i〉
∝ 〈 f | H int | i〉 + ∑dΩdωEi − En + !ω i + iΓ|n〉
1st term
2nd term

Ei ≠ En 1st term >> 2nd term
1H int = H Thomson = r0 ∑ A 2 (rj , t)
2 jA(rj , t) vector potentialf (Q)Weak interaction

d 2σ222 ki= r0 (εi ⋅ ε f )f (Q) S(Q, ω )dΩdωkffor an element

f (Q)

1 1
1S(Q, ω ) = ∑ n(ω ) + ±Fin (Q)δ (ω ± E j (q))
2 2 E j (q)j2Finj = ∑ M k−1/2 fk (Q)[ekj (q)⋅ Q]exp(iQ ⋅ r)exp(−wk )Which phonons are highest intensity?
General trends for scattering geometry?
Full calculations are required for a full-proof plan!

Instrument concept
Required energy resolution
2 meV / 20 keV = 10-7 !
Bragg optics

ΔE Δd∝Edrequire high energy x-ray ~ 20 keV

λ = 2d sin(Θ B )ΔEΔθ = tan(Θ B )EΘ B ≈ 90 o

Simplest mono & analyzer based on Si reflections inclose-to-back-scattering geometry

0 4 4)6272
956 2An0) IXS115
108
158for phonons:
spectrometer
Spectrometers for Inelastic X-ray Scattering
Sector
3 3HERIX,6 6) 49
36 30, APS
60
Cu K-edge RIXS Analyzer: Improved Resolution
0 6 6 0)Ge(35019543 7) analyzer
analyzer
R = 1 m & strip detector
2 4 6)
76
99
123
E =8.990 keV
10 cm 90000
3 3 7)80000
424159detector
1 mm6 2 0)70000
112
51
123
13 - 25 keV
90meV0 0 8)60000
64
131 3 m 145
undulator
50000
ACount rate [Hz]0m

40000 M. Wieczorek
achatryan
D. Casasample330000-1 020000

10000
Yu. Shvyd’ko

m a9namly z emirrorr arm

in-linemonochromator

Expected: 90 meV
Deconvoluted: 52 meV

IXS Workshop, NSLS-II,CrystalFebruary
7-8, 2008
segments:

0Energyresolution:10 -0.11 0.2
meV-0.4-0.3 -0.2 -0.1 0.0
0.3 0.4

E - E0, [eV]
∼1/E4 !
• Better energyEresolution.
• Less demanding analyzer fabrication.

foil 16/44

1.5 ⇥ 1.5 mm2

Other instruments: ID28 ESRF, backscattering monochromator
BL35XU SPring8 (ESRF, Sp8)HERIX&309)m

Scans use
fixed Ef andvary Ei

1.4 meVE1u

Phonon spectra
intensity (arbitrary units)

Stokes / Anti-Stokes

Stokestransverse
acousticanti−Stokes
50

100

E2g

Energies match DFT−25

But broadening of“E2g” mode7502550
Energy [meV]75100

FIG. 1 (color online). Energy loss scan in almost transverse
geometry measured at Q $ !1 2 0:3" corresponding to 0.6 !-A.
The data, normalized to the incident flux, are shown with theleast-squares fit (dashed line) and the ab initio spectrum withand without broadening due to experiment and electron phonon
coupling (solid lines). The broad peak corresponds to thedamped E2g mode and is shown in greater detail in the inset.
The peak at zero is due to diffuse scattering.

Shukla et al., PRL095506-290, 095506 (2003)
Baron et al,. PRL 92, 197004 (2004)

Phonon dispersionγ ν (q)λν (q) =2π N(0)ων2 (q)Compelling verification ofe-ph coupling driven SC

PHYSICA L R

3020100B1g80

frequency [meV]

Phonon widthcontributions§ Disorder ✗§ Anhamonicity ✗§ Electron-phononcoupling ✓width [meV]VOLUME 90, N UMBER 9E2g60

A2u40

E1u

200M

Γ

ALFIG. 3 (color online). Bottom: Experimental (circles) an
theoretical phonon dispersion (solid line) in MgB2 along !-A!-M, and A-L. In the region near the M point, the probable dtection of the E2g mode is indicated with a square symbol (setext). The crosses indicate a parasite signal of unknown origi
Top: Intrinsic linewidth of the E2g mode. The experiment
linewidth (circles) is large along !-A and below the exper
mental resolution (dashed line) near L and M. The theoretic
result (diamonds) for the electron-phonon coupling contribu
tion to the linewidth is also shown. Along A-L and !-M whethe E2g mode is nondegenerate, both theoretical values ashown, when different. E linewidth decreases progressive

Advantage/disadvantages x-ray vs.
neutrons for phonons
Use x-rays§ Small samples§ High pressures§ Q, E access§ Need good Q resolution

Use neutrons§ Sub meV energyresolution§ Certain light atoms withincompounds – particularlyoxygen§ Many more INS
spectrometers!

Raman scattering easier and cheaper if Q-resolution is not
required!

MEASURING THE SPINS I.E.
MAGNETIC EXCITATIONSKramers-Heisenberg equation
22dσ〈 f | H int | n〉〈n | H int | i〉
∝ 〈 f | H int | i〉 + ∑dΩdωEi − En + !ω i + iΓ|n〉
1st term
2nd term

1st term – very low sensitivity to magnetism
Use Ei=En|n> is state with core hole:§ Increases cross section§ Allows coupling to “forbidden” excitations
Resonant inelastic x-ray scattering (RIXS)

Gibbs J. App.
Phys. 57, 3615
(1985)

Direct vs indirect RIXS
Direct§ Core electron è valance band§ Electron from different state fills core hole§ E.g. 2p-3d resonance in 3d9 cuprates
Indirect§ Core electron è above valance band§ Electron from same state fills core hole§ Direct process dominate indirect process when notforbidden§ E.g. 1s-4p resonance in 3d9 cuprates

Cu L-edge RIXS in the cuprates
d-dexcitation
Elasticscattering
Spin
flip
3d x2-y2

1.5 eV

3d z2-r2

930 eV

xyxz/yz
not shown

2p3/2
20 eV

SO coupling

2p1/2
4| i〉 = 2 p3/2
3d 93| n〉 = 2 p3/2
3d 104| f 〉 = 2 p3/2
3d 9 *

Cu K-edge RIXS in the cuprates
4pmanifold

8990 eV

3dmanifold
Coreholepotential
1s| i〉 = 1s 2 3d 9 4 p 0| n〉 = 1s1 3d 9 4 p1

| f 〉 = 1s 2 3d 9 * 4 p 0

Cu L-edge vs. Cu K-edge: La2CuO4Single
Magnons

d-d excitations
Charge-transfer
excitations

Ei=930 eV
Cu L3 edge
2p3/2 – 3d

Elasticline
Charge-transfer
excitations

Two
Magnons

Ei=8990 eV
Cu K edge
1s – 4pd-d excitations

Instrument concept for soft x-rays
Soft x-rays:
E<2 keV ; λ> 6.2 ÅNo suitable Bragg reflections
Dispersive elements aregratings
Brillouin zone
(π/2,π/2)
(π,0)

(0,0)

a=b=3.8 Å

ccdVLS
gratingslitssample

SAXES at SLS

Resolution 130 meV

Magnetism in the high temperature
superconductors
La Sr CuO2-xx4

Cu L-edge spectra     





Dean
etal. Nat.
Mat. 12, 1019(2013)           

     ""      

 ##"  This suggests highenergy excitations
might have marginalrole for SC     !

!

 

Magnetism persists up
to x=0.4"

 !

     

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Advantage/disadvantages x-ray vs.
neutrons for spin excitations
RIXS has only recently measured spin excitations§ First measurement bimagnons k-edge La2CuO4

Hill et al. PRL (2008)§ First measurement magnons L-edge La2CuO4 Braicovich
PRL (2010)
Advantages§§§§

Small samples
Relative good count rates
Good Q resolution
Good at high E

Disadvantages§§§§

Poor E resolution
Upper limit on Q
Complicated cross-section
UHV required

M. P. M. Dean, JMMM 15, 3-13 (2015)

Allows measurement of very small samples
MBEIvan Bozvic

1 hr scan
ΔE =134 meV

Inelastic scattering from 25 isolated, single unit cell thick layers of La2CuO4
Compare to INS: La2CuO4 47.5g of material (Headings et al. PRL (2010))
RIXS is a factor of >1011 more sensitive!
M. P. M. Dean et al. Nature Materials (2012)

MEASURING ORBITAL
EXCITATIONS

netic interactions and yield a far
is that a novel Heisenberg antiferromagnet can b
netic systems beyond the standard
in 5d TMOs, in which the energy(a)(b)
Sr2IrO4
order of 0.5 eV (as compared tos). For
A2 IrO¼O
3 ((A of:Theexample,combinedeffected as a possible realization of theIr§ withStrong
spin-orbitmag(SO) couplingmodelbond-dependent. Furthermore,strongcrystal
SOC maySr§ Octahedra
field splitting
d topology to realize exotic topowith broken time reversal symmeical Mott insulator [5], a Weylnsulator [6]. Despite such intriguλe of magnetic interactions in sysremains experimentally an openIridates

rt on the magnetic interactions in a
Novelentangled
type of ground
Mott insulator
witha spin-orbit
statementsJ [7,8],
probedby resonant(notS)
magneticmoments
Jeff=1/2 momentseffg (RIXS).
These Jeff ¼ 1=2 mopurespins
becausetheir076402
interac-(2009)	
FIG. 1 (color online). (a) Because of a staggere
Kimet al.,PRL 101,
depend strongly on lattice androtation of oxygen octahedra, Sr2 IrO4 has four IrO
due to an admixture of spatiallythe unit cell [9], which coincides with the magneti
ments in the J ¼ 1=2 wave(b) J ¼ 1=2 moments lie and are canted in the IrO

Orbital transition in L-edge RIXS
22dσ
〈 f | D̂(ε ) | n〉〈n | D̂(ε ) | i〉
∝ ∑dΩdωEi − En + !ω i + iΓ|n〉

Electric dipole selection rule for optics:
Δl = ±1 (not zero)Δml = 0, ±1

L-edge has two dipole operators:- Direct on-site orbital transition are allowedInstrument concept hard x-ray RIXSFor RIXS cannot use arbitrary Ei144Relax backscattering condition and search for bestBragg reflection Θ ≈ 85odD
Yu.V. Shvyd’ko et al. / Journal of Electron Spectroscopy and Related Phenomena 188 (2013) 140–149

BDetDetdA

Best energy resolution:AO
0.04/ 11200= 4 x 10-6O’FFO
2O’2A

SaAASRAbRAFig. 5. Optical schemes for X-ray energy analysis with a position-integrating detector – PID (a) and a position sensitive detector – PSD (b). X-rays from the sample (S) aregathered in a solid angle ! × ! (typically ! ≃ 0.1–0.05 rad) by the crystal analyzer consisting of the two-dimensional array of flat crystals on a sphere of radius RA centeredMERIX Spectrometer@30-ID.APS

---

MERIX
Sector 27APS-

Analyzer gimbalGe(337)
diced analyzer:

Ei = 42 meV

Ir L-edge spectrum Sr IrO

Various excitationsin spectrum§§
§

1.2

1.0

0.8

0.6
0.4
Energy (eV)0.20.0Consider single site
0.6 model

CubicSpin–orbit coupling + tetragonal distortion(!,0)∆BC~0.137 eV
C j =3/2
B

0.5
0.4t2gEnergy (eV)

0.7

3/2 #SO~0.6 eV(0,0)

q=($/2,$/2)

Aj =1/2

0.3
Spin down
80
Spin up60

M (meV)–1/2
wenantvings, toable
al18,
has
tron
0–24,
Mott
hole
sion
arity
P inrials.
P in

the spin–orbit coupling split levels (from the j ¼ 1/2 manifold tothe j ¼ 3/2 manifold) have been identified and it has been shownthat their dispersions can be understood in close analogy to thesingle-hole problem16. Figure 1 depicts the hole-versus-exciton
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5453
analogy: a ‘foreign’ object injected into
a quantum Heisenberg
2 spins 4
AF, be it a hole or an exciton, creates a string of flippedalong its hopping path. This analogy, in principle, suggests thatMagnon
the dynamics of a particle moving in a magnetic
medium can also(!/2,!/2)
be studied using the exciton. However, the energy resolution ofRIXS used in the earlier study (E130 meV) was insufficient toresolve the dispersion and the intrinsic linewidth of the twoexciton modes associated with the two pairs of Kramers doublets
Excitonin the j ¼ 3/2 manifold (Fig.
2a). Here we exploit the differentMagnon
orbital symmetries of the two-exciton modes to selectively probe
each mode, and with the much improved energy resolutionExciton(E30 meV) offered by RIXS after recent developments, reveal
Etc
their full dynamics.

Intensity (a.u.)

ating
ngly
ions
turetumBornn ofh asd/ore ofdate,that–1/2

Kim et al., Nat. Comms.
5:4453 (2014) 	

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5453

Q dependence

RIXS
Normal incidence

1.0!=85°

Dispersive orbital mode
Mode lifetime >> decayrate
Quasi-particle is orbital
excitation dressed withmagnetic excitations

Energy (eV)

0.8

0.6CB

0.4

0.2

0.0
(",0)

112 meV

QP

Magnon("/2,"/2)

(0,0)

AA

(",0)

(Kim et al., Nat. Comms. 5:4453 (2014)	

SCBA calc

Advantages/disadvantages of x-raysfor orbital excitations
Raman§ Fast, excellent energy resolution and polarization control§ Not Q-resolved§ Not a simple cross-section

Neutrons§ dd-excitations only indirectly allowed§ Challenging to due limited neutron flux at incident energies ofseveral eV

Non resonant IXS

2d 2σ∝ 〈 f | H int | i〉
dΩdω§ At high Q higher order terms in Hint beyond dipole areimportant

MEASURING CHARGE
EXCITATIONSCross section
Consider density-density correlation function forenergy loss 1-100 eV range−iω t∞
1S(Q, ω ) =dt eρ(Q, t = 0)ρ + (Q, t)∫2π ! −∞

Excitations are propagating charge modes –plasmons.
Easiest to measure in materials with low atomic
number elements

MERIX Spectrometer@30-ID.APS-MERIX
Sector 27
APS-

Analyzer gimbal--

Instrument
needs:~0-100 eV§ Energy resolution~300 meVGe(337)
diced analyzer:

Ei = 42 meV-§ Energy range

Charge response of graphite/graphene
Dynamical susceptibility linked
to dielectric response
11S(Q, ω ) =Im [− χ (Q, ω )]π 1− exp(−!ω / kBT )In principle, a full description ofcharge response:
e.g. charge screeningJ.P. Reed et al., Science 330, 5 (2010) 	

nind (k, w) is the charge induced in the medium(20).

te dielectric screening.
rements show that free-DataA160

B-1 -3

Ψ -integrated (HOPG)

140

Comptonscattering

Energy (eV)

120
100
80
60
40

σ-π

2000its)

D

π
24

6810
50
105010501050105010
500.018 eV ÅΨ =0000

Ψ =6-1 -30 eV Å0Ψ=121.2

Momentum (Å -1)

Ψ =180Ψ =240Ψ =3001.6(110)ky2.0CΨkx(100)

J.P. Reed et al., Science 330, 5 (2010) 	 EELS - GraphitekElectronic screening movie

J.P. Reed et al., Science 330, 5 (2010) 	

ct evie. The
lly dicity, vF
servedectros-Research
1, USA.
ry, 9700
E-mail:Downloaded from www.scitime, t. c(k, w) also describes, in linear response
theory, how the system responds to charged perturbations via

agree with ARPES or SET measurements. Interaction effects have been observed in highmagnetic fields, but in this case the kinetic energy
is quenched by the formation of Landau levels(13–15). Some of these measurements were doneon supported graphene, which can suppress interactions through substrate dielectric screening.
However, recent measurements show that free-Advantages/disadvantages ox x-rays
for plasmonsFig. 1. IXS experimentsfrom graphite and extraction of the response function for graphene. (A)Scattered intensity as afunction of energy andmomentum for highlyoriented pyrolitic graphite (HOPG), which gives
the Y-integrated response.
(B) Angle-resolved spectrafrom single-crystal graphite for the domain overwhich anisotropy was observed. (C) Brillouin zoneof graphene with various
vectors defined. (D) Testof Eq. 4 on the electronenergy loss experimentsof Eberlein (24) (fit value
k = 0.33 Å−1), showingthat an accurate response
for graphene can be obtained from IXS experiments on graphite.

Anind (k, w) ¼ V (k) c(k, w) next (k, w)

ð1Þ

where next (k, w) is an arbitrary source andnind (k, w) is the charge induced in the medium(20).

B
160

-1 -3

Ψ -integrated (HOPG)

0.018 eV Å1050105010
50105010501050

Electron energy loss spectroscopy EELS=6§ Easier, cheaperC§ Better sensitivity to small volumes
σ-πkΨ§ Multiple scattering
problems at high Q
πw.sciencemag.org

SCIENCE140

Comptonscattering

Energy (eV)

120
100
8060

40
20002468

Ψ =0Ψ

000-1 -30 eV Å0Ψ=121.2

Momentum (Å -1)

Ψ =180Ψ =240Ψ =3001.6(110)

kykx

2.0

D
Intensity (relative units)

fectiveoltage.
n for aics (1).
solatedgly inngth of
red byc enercharge
F is theratio is
usuallyg. Unquaner thanametere been
h as anmight(100)

EELS - Graphite
EELS - Graphene
EELS - Converted

00VOL 330

51015202530

Energy (eV)5 NOVEMBER 2010

35404550805

More information
RIXS§ L.J.P. Ament et al. Rev. Mod. Phys. 83, 705 (2011)
Non resonant IXS and Phonons§ A.Q.R. Baron, arXiv:1504.01098 (2015)§ H. Sinn, J. Phys.: Condens. Matter 13, 7525–7537 (2001)§ M. d’Astuto and M. Krisch, Collection SFN 10, 487–503 (2010)
IXS and Plasmons (and other topics)§ Electron Dynamics by Inelastic X-Ray Scattering, Winfried Schülke, Oxford(2007)
Inelastic scattering in general (focused on neutrons)§ G. Shirane, S.M. Shapiro and J.M. Tranquada, Neutron Scattering with a
Triple Axis Spectrometer, Cambridge (2002)