Disentangling multipole resonances through a full x-ray polarization analysis

0.26 MB·English

by  C. Mazzoli

Disentangling multipole resonances through a full x-ray polarization analysis.
C. Mazzoli,1 S.B. Wilkins,1,2 S. Di Matteo,3,4 B. Detlefs,1,5 C. Detlefs,1 V. Scagnoli,1 L. Paolasini,1 and P. Ghigna6
1European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex 9, France
2Brookhaven National Laboratory, Condensed Matter Physics & Materials Science Department, Upton, NY, 11973-5000, USA
3Laboratori Nazionali di Frascati INFN, via E. Fermi 40, C.P. 13, I-00044 Frascati (Roma) Italy
4Equipe de physique des surfaces et interfaces, UMR-CNRS 6627 PALMS,
Universit´e de Rennes 1, 35042 Rennes Cedex, France
8
0 5European Commission, JRC, Institute for Transuranium Elements, Postfach 2340, Karlsruhe, D-76125 Germany
0 6Dipartimento di Chimica Fisica “M. Rolla”, Universit´a di Pavia, I-27100 Pavia, Italy
2 (Dated: February 5, 2008)

n Complete polarization analysis applied toresonant x-ray scattering at theCr K-edgein K CrO
2 4
a shows that incident linearly polarized x-rays can be converted into circularly polarized x-rays by
J diffractionattheCrpre-edge(E =5994eV). Thephysicalmechanismbehindthisphenomenonisa

6 subtleinterferenceeffectbetweenpurelydipole(E1-E1)andpurelyquadrupole(E2-E2)transitions,
1 leading toa phaseshift between therespectivescattering amplitudes. This effect may beexploited
todisentangletwoclose-lyingresonancesthatappearasasinglepeakinaconventionalenergyscan,

] in this way allowing to single out and identify the different multipole order parameters involved.
l
e
- PACSnumbers: 78.70.Ck,78.20.Bh,78.20.Ek
r
t
s
.
t I. INTRODUCTION origin when they are separated by less than ∼ 1eV, a
a
situation which frequently occurs at the metal pre K-
m
edge region of transition metal oxides, where E1 and
- In the last 10 years resonant x-ray scattering (RXS) E2 transitions can have similar magnitudes, or at L
d 2,3
hasdevelopedintopowerfultechniquetoobtaindirectin-
n edges of rare-earth compounds. For example, at the Fe
formationaboutcharge,magnetic,andorbitaldegreesof
o freedom1,2,3,4,5,6. Itcombinesthehighsensitivityofx-ray pre K-edge in α-Fe2O3, evidence of a E2-E2 transition
c was found already in 199318. This transition was inter-
[ diffraction to long-range ordered structures with that of
preted as the signature of an hexadecapolar electronic

3 x-ray absorption spectroscopy to local electronic config- ordering19. Later, however, it was shown by symme-
urations. In particular,the developmentof thirdgenera-
v tryargumentsthatalsoanaxial-toroidal-quadrupoleOP,
tionsynchrotronradiationsourceshasmadepossiblethe
9 parity-breaking,washiddenwithinthesameresonance11,
3 detection of small effects in electronic distribution, due but to date no conclusive experimental evidence for this
2 to magneto-electric anisotropy7 or to local chirality8,9,
interpretation has been provided. Analogously, at the
1 that can be related to the interference between dipole
Ce L -edges in CeFe the different electronic OP re-
1 (E1) and quadrupole (E2) resonances. These pioneer- 2,3 2
lated to 4f and 5d states are entangled and cannot be
6
ing studies paved the way to a new interpretation of
0 examinedindividuallybyconventionalRXS20. Afurther
RXS experiments in terms of electromagneticmultipoles
/ exampleisprovidedbyK CrO . Itsspacegroupsymme-
t of higher order than dipole, and led to the detection 2 4
a try (Pnma, No. 62)allows severalexcitation channels at
m of phase transitions characterized by order parameters the (1k0) Bragg forbidden reflections21, due to the pres-
(OP) of exotic symmetry10,11,12. Several theories have
- ence of multiple moments of different symmetries (elec-
d been developed, based on these higher-order OP, to ex-
tricquadrupole,octupole,hexadecapole)inthesameen-
n plain ”anomalous” phase transitions. For example, in
ergy range, as described in more details below.
o NpO a proper interpretation of the magnetic ground-
2
c state requires a primary OP at least octupolar order13,
:
v whereas in high-temperature cuprate superconductors,
i the pseudogap phase has been interpreted in terms of Theaimofthepresentarticleistoaddresstheprevious
X
parity and time-reversal odd toroidal multipoles14,15. In problems by extending the well-known techniques of op-
r
a several cases, though, the assignment of the multipolar tical polarimetryfrom the visible to the x-rayregime, as
origin to a RXS signal is not clear10,16,17. The char- developed at the beamline ID2022 at the ESRF, Greno-
acteristic variation of the intensity and polarization as ble,France. Byusingadiamondx-rayphaseplatetoro-
the sample is rotated about the scattering vector dur- tate the incident linear polarisationin combination with
ing an azimuthal scan may allow a clearer identification a linear polarisation analyser, we can resolve resonances
of the order of the multipole. However, this technique is determinedby multipoles ofdifferentorderthatarevery
plaguedbymanyexperimentaldifficultieseitherfromthe closeinenergy,playingontheirrelativephaseshifts. The
sample, e.g. when the crystalpresents twinning andmo- ideacanbeexplainedthroughasimplifiedmodel,bycon-
saicity, or due to restrictive sample environments, such sideringtwoexternallydriven,dampedharmonicoscilla-
as cryomagnets. Moreover, it is very difficult to iden- tors of unitary amplitude, but with resonant frequencies
tify and resolve two resonances of different multipolar differingby 2ζ. The scatteringamplitude ofsuchoscilla-
2
tors is given by 1
g (ω) g (ω)
+ −
1 g’’ 0.5
g±(ω)= ω±ζ−iΓ. (1) g’, 0
−0.5
Here Γ is the inversedamping time, andω the frequency π
of the external excitation. We also suppose that the two e3 π /4
resonancesscatterindifferentpolarizationchannels. Us- as π /2
ing the Jones matrix formalism23 the polarisation of the Ph π /4
scattered beam may then be written as ǫ′ =Gǫ, where ǫ 0
(ǫ′)arethe incident(scattered)polarisationvectors,and 1
tthhee mphaottroixnGen=erg(cid:18)y.gg−+Exggp+−er(cid:19)imceonnttaalliyn,stthheesdceaptteenrdeednbceeaomn P’, P’, P’123−00..055 P3’ P1’ P2’
polarisation is best described in terms of the Poincar´e-
−6 −4 −2 0 2 4 6
Stokes parameters: Energy (Γ units)
P′ ≡ |ǫ′σ|2−|ǫ′π|2 oFuIGs).a1n:d(Cimoalogri)na-ryTo(pg:′′,Tdyapsihceadl)bpeharatvsiofrorotfhreeatlw(og′r,escoonnatitnours-
1 P0′ of Eq. 1 (g+ red; g− blue), with ζ = Γ. Middle: Corre-
ǫ′∗ǫ′ sponding phase shifts. Bottom: Poincar´e-Stokes parameters
P2′ ≡ 2ℜe σP′π as calculated from Eqs. 3.
0
ǫ′∗ǫ′
P′ ≡ 2ℑm σ π
3 P′ of the incident and scattered electric field vectors to the
0
probed multipoles and the reciprocal lattice point under
withP′ ≡(|ǫ′|2+|ǫ′ |2)thetotalintensity,andǫ′∗ the
0 σ π study. Their interference can therefore be tuned simply
complex conjugate of ǫ′. P′ and P′ describe the linear
1 2 by varyingthe incidentpolarizationby meansofa phase
polarizationstates,whereasP′ indicatesthedegreeofthe
3 plate.
circular polarization. The Poincar´e-Stokes parameters,
P oftheincidentbeamareobtainedbysubstituting
0,1,2,3
ǫ for ǫ′.
II. EXPERIMENTAL SETUP
For example, for an incoming π polarised beam, ǫ =
0 ǫ′ g
(cid:18)1(cid:19),weobtain(cid:18)ǫσ′π (cid:19)=(cid:18)g−+ (cid:19),whichinturnyields: peErixmpeenrtiamlesnettsupweisreocuatlrirnieeddoinutFaigt.I2D.2A0,sEinSgRleFc.rTyshtealexo-f
K CrO wasmountedonthesix-circlehorizontaldiffrac-
2ζω 2 4
P′ = − tometer,andacryostatwasusedtostabilizethetemper-
1 ω2+ζ2+Γ2
ature at 300 K. Sample rocking curves (θ-scans) of the
′ ω2−ζ2+Γ2 crystal resulted in widths smaller than 0.01◦ indicating
P = + (2)
2 ω2+ζ2+Γ2 a high sample quality.
2ζΓ A diamond phase plate of thickness 700 µm with a
′
P3 = +ω2+ζ2+Γ2 [110]surface was insertedinto the incident beam, within
its own goniometer, and the (111) Bragg reflection in
Therefore we expect that in the intermediate region be- symmetricLauegeometrywasselectedtomodifythepo-
tweenthetworesonances,theoutgoingbeamiscircularly larizationof the x-raybeam incident on the diffractome-
polarized,dependingontherelativedephasingofthetwo ter. Thephaseplatewasoperatedineitherquarter-wave
resonances, as shown in Fig. 1 for the case ζ =Γ. or half-wave plate mode. With the former we generated
Below, we describe experimental data which we then left-orright-circularpolarization,P ≈±1,whereaswith
3
comparetoquantitativeab-initiocalculationscarriedout thelatterwerotatedthe linearincidentpolarizationinto
usingtheFDMNEScode24. Wedemonstratethata100% an arbitrary plane25,26, described by P ≈ cos(2η) and
1
linear-tocircular-polarizationconversionatthepre-edge P ≈ sin(2η). Here η is the angle between the inci-
2
regionof the Cr K-edge in K CrO is induced by the in- dent beam electric field vector and the vertical axis (see
2 4
terference of the dispersive and absorptive parts of two Fig.2), i.e. η =0whenthe polarizationis perpendicular
differentmultipolesprobedbypurelydipole(E1-E1)and to the horizontal scattering plane (σ polarization).
purely quadrupole (E2-E2) resonances. Thus, the scat- The sample was mounted with the [010] and [100] di-
teredbeamoriginatesfromtwodifferentexcitationchan- rections defining the horizontal scattering plane. Fig-
nels, each scattering the beam with a different phase. ure 3showsthe fluorescenceyieldandthe energydepen-
Their relative amplitudes, at a given energy, are gov- dence of the glide-plane forbidden (130) reflection, col-
erned by the probed multipoles, the relative orientation lected at an azimuthal angle of Ψ = −0.78(3)◦ degrees
3
1
n
) 1 o
s m
unit et/
b. 0.8 D
r 0
a
( 46.546.646.7
s. 0.6 θ (deg)
n PA
e
egr. int0.4 η=−90°
nt0.2
I P’ =0.96(1)
1
P’ =0.22(4)
0 2
0 45 90 135 180
η’ (deg)
FIG. 2: (Color online) Experimental setup with phase plate
inhalf-wavemode;x-raydirectionsindicatedbybluearrows, FIG. 4: (Color online) Variation of the integrated intensity
polarizations byred ones. Synchrotron light arrives from the of the(130) forbidden Bragg peak as afunction of thepolar-
left, horizontally polarized (π). η is the rotation angle of the ization analyzer angle η′. The dashed line represents a fit to
incidentpolarization. η′ istherotation angle ofthepolariza- these data with Eq. 3 (see text for details). The insert is an
tionanalysercrystal;thezeropositionsofthetwoangles,cor- example of a rocking scan of the analyser: the integrated in-
responding to σ and σ′ polarisations respectively, are repre- stensityobtainedfrom thefit(dashedline: lorentian squared
sentedbydashedlines. Thecontinuouslineisthetherocking and linear background) represents experimental point in the
axisofthepolarisationanalysiscrystalθPA. Thepolarisation main axes.
analyserstageisshownintheconfigurationcorrespondingto
themaximum detected intensity.
intensities were then fitted to the equation
↓ ↓ ↓ I = P0′ [1+P′cos2η′+P′sin2η′], (3)
2 1 2
s)
nit toobtainthePoincar´e-Stokesparameters,P′ andP′. An
u 1 2
b. example is shown in Fig. 4 for η = -90◦. The degree of
ar circular polarization, P′, can not be measured directly
y ( in this setup. However,3an upper limit is inferred from
nsit P′2+P′2+P′2 ≤1(theequalityholdingforacompletely
e 1 2 3
nt polarized beam).
I We systematically measured P′ and P′ of the beam
1 2
0 scattered at the (130) reciprocal lattice point as func-
5.98 5.99 6 6.01 6.02 6.03 6.04 tion of η. Figure 5 shows both the experimental data
Energy (keV)
(symbols) and the theoretical calculation (dashed lines),
describedbelow. Themeasureddegreeoflinearpolariza-
FIG. 3: (Color online) Experimental data on fluorescence tion of the scattered beam, P′2+P′2, strongly deviates
yield (black continuous line) and energy scan (red symbols) 1 2
for the (130) reflection in π → σ′ polarization configuration. from 100% in the range−10◦ <∼η <∼50◦, indicating that
a large component of the scattered beam is either cir-
Arrows indicate the energy values where Stokes’ parameters
were measured. cularly polarized or depolarized. To ascertain which of
these two processes is realised we reconfigured the dia-
mond phase plate to produce circularlypolarizedx-rays.
Figure 6 shows the measured P′ and P′ for both lin-
1 2
withrespecttothereciprocallatticedirection[010]. The early and circularly polarized x-rays. The presence of
photon energy was then tuned to the pre-edge region of linearly scattered x-rays for the circular incident case is
the CrK-edge(5994eV). Anx-raypolarizationanalyzer consistentwiththe assumptionthatthe regionforwhich
was placed in the scattered beam. It exploited the (220) P′2+P′2 strongly deviates from 1 corresponds to an in-
1 2
BraggreflectionofaLiFcrystal,scatteringclosetoBrew- creased circularly polarized contribution. Furthermore,
ster’s angle of 45◦. The polarization analyzer setup was thecalculationsforP′2,detailedinSectionIII,involving
3
rotated around the scattered beam by an angle, η′, and only completely polarized contributions to the scattered
ateachpointthe integratedintensity wasdeterminedby beam, agree well with its upper limit inferred from the
rocking the analyzer’s theta axis (θ ). An example is data (P′2 ≡ 1−P′2−P′2), indicating that the signal is
PA 3 1 2
shown in the inset of Fig. 4. The resulting integrated essentially circularly polarised.
4
if′′). Here R~ stands for the position of the scatter-
1 j j
ing center j, Q~ is the diffraction vector and f is the
0j
0.5 Thomson factor. f′ and f′′, related by Kramers-Kronig
j j
P’1 0 transform, are given, at resonance, by the expression1:
−0.5
hψ (j)|Oˆ′∗|ψ ihψ |Oˆ|ψ (j)i
−1 f′+if′′ ≡f (ω)∝−ω2 g n n g ,
1 0 50 100 150 j j j Xn ω−(ωn−ωg)−iΓ2n
(4)
0.5 where ω is the photon energy, ω the ground state en-
g
ergy, ω and Γ are the energy and inverse lifetime of
P’2 0 theexcintedstatens,ψ (j)isthecoregroundstatecentered
g
−0.5 around the jth atom and ψ the photo-excited state, ǫ
n
andǫ’arethepolarizationsoftheincomingandoutgoing
−1
photons and ~q and q~′ their corresponding wave vectors.
1 0 50 100 150 Thesumisextendedoveralltheexcitedstatesofthesys-
tem. The transition operator Oˆ(′) =~ǫ(′)·~r 1− i~q(′)·~r
22 2
P’ is written as a multipolar expansion of the(cid:0)photon field(cid:1)
−
21 0.5 uptoelectricdipole(E1)andquadrupole(E2)terms;~ris
−P’ theelectronpositionrelativetothe resonatingion,~ǫ(′) is
1
the polarization of the incoming (outgoing) photon and
0 ~q(′) its corresponding wave vector.
−90 −60 −30 0 30 60 90 By taking into account the space group symmetry of
η (degrees) K CrO (Pnma, No. 62), the four equivalent Cr sites at
2 4
Wyckoff 4c positions (with local mirror-plane mˆ ) can
y
be related one another by the following symmetry oper-
FIG. 5: (Color online) Calculated (dashed lines) and mea-
sured(symbols)Stokes’parametersfor(130)reflectionversus ations: f3 = Iˆf1, f4 = Cˆ2xf1 and f2 = Iˆf4. Iˆ is the
thepolarisationangleofthelinearincominglight(seeFig.2), space-inversion operator and Cˆ2x is the π-rotation op-
E =5.994keV. Seetext for details. erator around x-axis. The Thomson scatering f0j does
not contribute at (1k0) type reflections, for any k, due
Linear incident, η=10° to the glide plane extinction rule. Therefore, the struc-
1 ture factor atCr K-edge for the (1k0)chosenreflections,
P’ =−0.03(5)
1 when summed over all equivalent sites, becomes, for E1-
nits) 0.5 P’2=0.01(5) E2 scattering:
u
b. k
ens. (ar 01 Circular incident while for E1S-E=1 a2nidsinE[22π-E(x2+sca4t)t]e(r1in−gmiˆtzi)sfg1iven by: (5)
gr. int PP’’12==−−00..3885((32)) k
e 0.5 S =2cos[2π(x+ )](1−mˆ )f . (6)
nt 4 z 1
I
0 Here x ≃ 0.23 is the fractional coordinate of Cr atoms
0 45 90 135 180 and mˆ is a glide-plane orthogonal to the z-axis. In de-
η’ (deg) z
rivingEqs. 5and6,wehaveusedtheidentityf =mˆ f .
1 y 1
It is interesting to note the different behavior of the two
FIG.6: (Coloronline)Stokes’parametersfor(130) reflection terms for k even or odd. For example, when k = 4, the
with(top)linearly(η=10◦)and(bottom)circularlypolarized
E1-E2 scattering is proportional to sin(2πx) ≃ 0.99 and
incident x-rays, collected at E =5.994keV. Dashed lines are
itdominatestheotherterms,proportionaltocos(2πx)≃
fit with Eq. 3.
0.12. Indeed we found the presence of a very intense
pre-edge feature from E1-E2 channel at the (140) re-
flection that is related to the electric octupole moment,
III. THEORETICAL DISCUSSION
as predicted in Ref. 27 and verified numerically by our
ab-initio calculations. However, as described above, the
In RXS the global process of photon absorption, vir- presence of a single scattering channel, as in the case of
tual photoelectron excitation and photon re-emission, is the(140)reflection,cannotleadtoacircularlypolarized
coherent throughout the crystal, thus giving rise to the diffracted beam. This can be demonstrated by a sym-
usual Bragg diffraction condition jeiQ~·R~j(f0j +fj′ + metry argument: if only one scatterer is present, which
P
5
is by hypotesis non-magnetic, and the incident light is involved are non-magnetic and parity-even.
linearly polarized, then the initial state is time-reversal Finally, we verified experimentally that at the Cr K-
even. Therefore,asmatter-radiationinteractiondoesnot edge(E =6010and6018eV),whereonlyoneterminthe
break time-reversal, it follows that the final state must E1-E1channelispresent,nocircularpolarizationwasob-
alsobetime-reversaleven,i.e.,radiationcannotbecircu- served for all incident angles, i.e., P′2+P′2 =1. Calcu-
1 2
larly polarized,which would breaktime-reversal. This is lations performed using FDMNES confirmed this result.
nomoretrue whentwoscatterersarepresent,due tothe IV. CONCLUSIONS
extra degree of freedom represented by the time (phase)
delay between the two scattering processes. Indeed, this
Polarization analysis of RXS experiments has devel-
was a posteriori verified by our numerical simulation,
opedgreatlyin the lastfew years,helping to understand
which confirmed that no outgoing circular polarization
several characteristics of order parameters in transition
is present at the (140) reflection.
metaloxides,rare-earthbasedcompounds,andactinides.
The case of the (130) reflection is very different. The
Up to now, however, the full investigation of Stokes’ pa-
role of sin(2πx) and cos(2πx) in Eqs. 5 and 6 switches
rameters was not applied most likely because linear po-
in such a way that the two diffraction channels E1-E1
larizationanalysis,where only the P′ parameter wasde-
and E2-E2 become predominant. Further analysis of 1
termined by measuring the σ → σ′ and σ → π′ chan-
the structure factor28 revealsthattwo resonancesareal-
nels, was considered sufficient. While this may be true
lowed, one for each channel, corresponding to an elec-
when just one excitation channel is involved (as at the
tric quadrupole ordering for the E1-E1 scattering and
(140) reflection in the present case), several dephasing
an electric hexadecapole19 for the E2-E2 scattering. Fi-
phenomenamayappearwhentwodifferentmultipoleex-
nally,multiple-scatteringcalculationswiththeFDMNES
citations close in energy are involved in the transition.
program confirm that the two resonances overlap in the
As we have seen, these phenomena may lead to a situa-
pre-edge region, though slightly shifted in energy of ∼1
tion where incoming linear polarization is scattered to a
eV.Thesearetheconditionstobemettogettheinterfer-
circular polarization due to an interference between two
ence of the two channels. In order to describe the effect
multipoles, at the same time allowing for a very sensi-
quantitatively from a theoretical point of view we used
tive determination of the presence of the second multi-
the ab-initio code FDMNES, in the multiple-scattering
pole. We believe that the use of phase plates and of a
mode,tocalculateP directlyfromEq. 4fortheK CrO
3 2 4 complete polarization analysis, is the key to disentan-
structure21. We employed a cluster of 43 atoms, cor-
gle multi-resonance structures in those situations where
responding to a radius of 5.5 ˚A around the resonating
an usual energy scan, like the one shown in Fig. 3, is
Cr-atom. Notice that in this most general case we find
P′(ω) ∝ (f′ (ω)f′′ (ω)−f′ (ω)f′′ (ω)) where f′ and not sufficient to this aim. Effect of d-band filling on de-
f′3′ are theEu1sual Edi2spersiveEa2nd aEb1sorptive terms (see tails of the electronic structure will be investigated by
the method presented here in the series of isostructural
Eq. 4) for E1 and E2 channels. Therefore at the photon
energyω, P′ is determinedby the interferenceofthe ab- compounds K2CrO4 →K2MnO4 →K2FeO4.
3
sorptive part f′′ of one channel with the dispersive part
f′ of the other and, again, in the presence of just one
channel, P′ =0.
3 Acknowledgments
The numerical simulations shown in Fig. 5 (dashed
lines) confirm that there is an azimuthal region where
TheauthorswouldliketoacknowledgeDavidH.Tem-
the incoming linear polarization is fully converted into
pleton and Franc¸ois de Bergevin for enlightening discus-
an outgoing circular polarization and that the effect is
sions.
determined by the interference of the E1-E1 and E2-E2
channels. One ofus (SDM) acknowledgesthe kind hospitalityat
In this respect, it is interesting to note that the ori- ESRF during the preparationof the manuscript.
gin of this effect is profoundly different from those de- The work at Brookhaven National Laboratory is sup-
termined by a chiral(magnetic) structure (see, e.g., Ref. portedbytheU.S.DepartmentofEnergy,undercontract
29,30), as clearly seen by the fact that all the tensors no. DE-AC02-98CH10886.
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