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Physica B 350 (2004) 11–16

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doi:10.1016

Chasing ghosts in reciprocal space—a novel inelastic neutronmultiple scattering process

H.M. Rønnowa,b,*, L.-P. Regnaultb, J.E. Lorenzoc

aNEC Laboratories, Princeton and University of Chicago, USAbMDN/SPSMS/DRFMC, CEA-Grenoble, 38054 Grenoble, FrancecLaboratoire de Cristallographie, CNRS, 38042 Grenoble, France

Abstract

We have discovered that a recently reported weak excitation branch in the spin-Peierls material CuGeO3 is in fact a

ghost image of the primary magnetic excitation shifted in reciprocal space by a novel multiple scattering process. A

model is developed that predicts the occurrence of such multiple scattering and accounts for the observations in

CuGeO3:New ‘ghostons’ can occur when the magnetic unit cell is smaller than the structural, while mixing of intensitiesfrom different reciprocal space zones jeopardize accurate polarisation analysis and the study of weak modes in general.

r 2004 Elsevier B.V. All rights reserved.

PACS: 25.40.Fq; 75.40.Gb; 78.70.Nx

Keywords: Inelastic neutron scattering; Copper germanate CuGeO3; Neutron polarisation analysis

Introduction

Development of neutron spectrometers withhigh flux while maintaining a low intrinsic back-ground has lately opened the ability to study weakfeatures in the excitation spectrum or, conversely,apply polarisation analysis to obtain qualitativelynew information. However, as we shall demon-strate in this paper, increased sensitivity alsobrings about new spurious scattering effects.

onding author. ETH-Zurich & Paul Scherrer

32 Villigen, Switzerland, Tel.: +41 56 310 4668;

6 310 2939.

ddress: henrik.ronnow@psi.ch (H.M. Rønnow).

- see front matter r 2004 Elsevier B.V. All rights reserve

/j.physb.2004.04.048

Aiming to deploy high accuracy polarisationanalysis of inelastic neutron scattering as a tool tostudy composite correlation functions in emergingmaterials that owe their novel properties to inter-play between different degrees of freedom such asspin, charge, orbital and lattice fluctuations, wehave performed detailed studies of CuGeO3; inwhich intimate coupling between quasi-one dimen-sional spin chains and lattice vibrations cause aspin-Peierls transition to a non-magnetic dimerisedground state [1]. The primary excitation hereof is abound domain wall pair with minimum energy gapof 2 meV at 0; k; 1

2

� �; and a strong dispersion along

l; the chain direction. The unit cell contains twomagnetic ions along the b direction, which givesthe dispersion along k a periodicity of two with

d.

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H.M. Rønnow et al. / Physica B 350 (2004) 11–1612

minima at k odd. This is sketched in Fig. 1, whichalso shows the continuum of free soliton pairsresiding above a second energy gap and thephonons, which are far away from the region ofinterest in this paper.In addition, a second much weaker dispersion

branch was recently discovered [2]. Its dispersionalong 0; k; 1

2

� �exactly mirrors that of the primary

mode, having minimum gap of 2 meV at k evenand reaching 5:6 meV at k odd. Disappearingabove TsP ¼ 14:1 K and splitting in an applied

Fig. 1. Schematic excitation spectrum of CuGeO3: Theintensity of both primary and weak modes decrease rapidly

away from the gap position, as shown for the later with a

pseudo-colour representation of measured intensities.

0 5 10 15

0

50

100

150

200

250

Energy [meV]

Cou

nts

in 1

0 m

in

(0,1,½) 1350 cts

SF NSF

Fig. 2. Scans along energy at, respectively, 0; 1; 12

� �and 0; 2; 1

2

� �with

completely spin-flip, the phonon completely non-spin-flip, while the w

field like the primary mode was considered proofof its magnetic origin. It was explained in terms ofDzyaloshinskii–Moriya interaction breaking thesymmetry of the two spin-sites along b in the unitcell, thereby allowing a second mode.However, closer examination revealed several

puzzling aspects of this mode, most notably itsdependence on neutron polarisation. Fig. 2 depictsenergy scans at, respectively, Q ¼ 0; 1; 1

2

� �and

0; 2; 12

� �taken with the neutron polarisation along

Q; in which case nuclear scattering /NNS occursin the non-spin-flip (NSF) channel, while magneticscattering /MMS is spin-flip (SF). First, wenotice that the primary mode both at 0; 1; 1

2

� �

and 0; 2; 12

� �is completely SF, the weak NSF

component being accounted for by the experi-mental flipping ratio. At ð0; 2; 1

2Þ; the weak mode at

2 meV; on the other hand, displays almost equalNSF and SF components. Though suggestive of atrue composite magneto-vibrational excitation,this was excluded by measuring the full xyz

polarisation Paa ¼ �ð0:14; 0:37; 0:35Þ70:03; yield-ing /NwNS=/MwMS ¼ 0:01470:015: Sincemagnetic anisotropy would satisfy Pxx > Pyy;Pzz;insisting on an interpretation within the Bornapproximation required nuclear spins slaved to theelectronic moments so that the correlation func-tions /IwIS and /MwIS become finite. Beforesuch a drastic step, we considered more carefullyanother puzzling observation. The intensity varied

0 5 10 15

Energy [meV]

(0,2,½)

primary

weak continuum

phonon

0 2 40

10

20

30

40

neutron polarization along Q: The primary magnetic mode iseak mode has almost equal components.

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kikf

Qkm

q

τ

ki

km

kfQ

q

τ

Fig. 3. Schematic scattering diagrams for, respectively, Bragg

first (left) and Bragg last (right) multiple scattering with

parameters relevant for CuGeO3: E ¼ 2 meV; Q ¼ 0; 2; 12

� �

and t ¼ ð0; 1; 0Þ: The geometrical condition is that there exist

H.M. Rønnow et al. / Physica B 350 (2004) 11–16 13

between cold and thermal neutron spectrometersin a manner hard to reconcile through resolutionconsiderations. This led us to measure the intensityof each mode as a function of neutron wavevector, keeping E ¼ k2f � k2i and Q ¼ kf � ki fixed(see Figs. 5 and 6). While the primary modeintensity increased roughly linear in kf as expectedfrom resolution effects, the weak mode displayed avery erratic behaviour with what resembles a seriesof peaks. This behaviour led us to consider a novelmultiple scattering phenomena.

a t on one of the spheres spanned by either ki or kf : While theexamples take place in the plane, t will most often have an outof plane component.

1. Inelastic multiple scattering

When a neutron is scattered by a finite sizesample, there exist the possibility for it to bescattered once more before leaving the sample.Such multiple scattering events are well knownphenomena in diffraction. If the beam scattered byone Bragg-reflection t1 fulfills the Bragg-conditionfor t2; some fraction will be scattered a secondtime, resulting in a total scattering vector t3 ¼t1 þ t2: As a consequence, the intensity measuredat t1 may be reduced, and the intensity measuredat t3 enhanced, which is particularly a problem forweak or forbidden reflections. In a standard singlecrystal diffraction experiment with beam diver-gences around 10’–20’ and 1–10 mm3 samples withless than 1� mosaicity, typically 1–2% of thereflections are affected. Hence, it is affordable todisregard contaminated reflections, or alterna-tively simple to break the multiple scatteringcondition by rotating the crystal around Q orchanging jkij ¼ jkf j; keeping ki � kf ¼ Q:The multiple scattering phenomenon is straight-

forwardly extended to the case where one of thescattering processes is inelastic scattering from anexcitation. The larger divergences of inelasticspectrometers combined with the extension (ratherthan a delta function) in ðQ;oÞ space of anexcitation means that the inelastic multiple scatter-ing condition is in fact very often fulfilled. Toobserve multiple scattering at Q ¼ ki � kf withenergy transfer E ¼ jki j

2 � jkf j2 (in units of

_2=2M) there must exist a reciprocal space vectort and hence an intermediate neutron wave vectorkm ¼ ki � t with jkmj ¼ jkij or km ¼ kf þ t with

jkmj ¼ jkf j and an excitation at q ¼ Q � t withenergy E: The same condition applies to extinc-tion, except that the Bragg last condition becomesjkf j ¼ jkf � tj: To calculate at which ki agiven t can cause multiple scattering, wewrite Q ¼ ð0;Qa;QbÞ; t ¼ ðtv; ta; tbÞ and ki ¼kið0; sinf; cos fÞ; obtaining

ki ¼1

2

Q2 þ E

Qa sin fþ Qb cos f; ð1Þ

where for, respectively, Bragg first or Bragg lastscattering (Fig. 3)

tan f ¼ �ðQ2 þ EÞtb � t2Qb

ðQ2 þ EÞta � t2Qaor

�ðQ2 þ EÞtb � ð2Q � t� t2ÞQb

ðQ2 þ EÞta � ð2Q � t� t2ÞQa: ð2Þ

It is difficult to predict the absolute strength ofmultiple scattering, which depends on instrumentresolution, sample size, geometry and mosaicity,but a simple estimate of the relative contributionfor each t would be the product of jki jjkf j forresolution, the Bragg peak structure factor jF ðtÞj2

and the form factor jf ðqÞj2 for magnetic excitationsor q2 for phonons. The reduction of any pre-existing intensity at ðQ;EÞ would in the samerelative units be given by jf ðQÞj2 and F ð�tÞ for kf

extinction.

1.1. Manifestations of inelastic multiple scattering

To see what effects can arise from inelasticmultiple scattering, it can be thought of as shifting

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1

2

Fig. 4. Illustration of how multiple scattering shifts inelastic

intensity in reciprocal space. (1) New ‘ghoston’ modes occur

when a dispersion is shifted to an empty zone in reciprocal

space. (2) Back-folding of intensity from the same excitation in

a different zone.

H.M. Rønnow et al. / Physica B 350 (2004) 11–1614

real spectral weight by the reciprocal space vectort; as illustrated in Fig. 4. If an excitation withdispersion EðqÞ has a longer periodicity than t; sothat there is no pre-existing scattering at EðQÞ;multiple scattering can create a ghost-like image ofthe real excitation. This ‘ghoston’ mode will inheritthe same dependencies on temperature, magneticfield and other parameters often used to checkagainst spurious scattering. If the dispersion issymmetric with respect to t; multiple scatteringwill fold back intensity from the same excitation ina different zone of reciprocal space.

1.2. Effect on polarisation analysis

The incident Pi and final Pf neutron polarisa-tions are conveniently applied along xjjQ; y>Q inthe scattering plane and z>Q and the scatteringplane. For multiple scattering, however, q isrelevant, not Q ¼ q þ t: As a consequence, thecomponents of the polarisation tensor Pab ¼ðINSF � ISFÞ=ðINSF þ ISFÞ become mixed. In ab-sence of nuclear-magnetic interference, /NwMS ¼0; and chiral magnetic correlations, /Mw MS ¼0; the final neutron polarisation is given by

Pf s ¼/PiðNwNÞ þ ðPi � Mw>ÞM> þ Mw

>ðPi � M>Þ

� PiðMw> � M>ÞSP0; ð3Þ

where s ¼ /NwN þ Mw>M>S; M> ¼ M � #q

ðM � #qÞ ¼ ð0;My;MzÞ; #q ¼ q=jqj and P0 is theinstrumental polarisation. Without mixing, Paa ¼ð2Z� 1; Zþ m; Z� mÞP0; where Z ¼ /NwNS=s is

the nuclear fraction and m ¼ /Mwy My � Mw

z MzS=smeasures magnetic anisotropy. Normally, Pxx

alone is used to separate magnetic from nu-clear scattering, but we recommend using/NNS=/MMS ¼ ðP0 þ

PaPaaÞ=ð3P0 �

PaPaaÞ;

which is robust towards misalignment of thepolarisation coordinate system. Given Q and q itis always possible to determine /NNS and thetwo non-zero magnetic components from Paa:Assuming purely magnetic isotropic scatteringfor instance, Paa ¼ � #q2a:

2. Application to CuGeO3

Applying the ghoston model to CuGeO3; wefirst notice that with two magnetic ions along theb-direction, there exist reciprocal lattice vectorswith k odd, which can indeed shift the primaryexcitation to the positions where the weak mode isfound. In Fig. 5 the kf and relative weight for eacht is shown by a vertical line. Representingresolution by a fixed width (0:12 (A�1 FWHM)Gaussian, the resulting prediction is shown by asolid line. Though the agreement with measure-ment is not perfect, it does produce goodcoincidence of peaks and is definitely sufficient toconclude that the intensity observed at even k isdominantly inelastic multiple scattering. We notethat the same overall intensity scale factor wasused for k ¼ 0 and k ¼ 2:In Fig. 6 the same scale factor has been applied

to Q ¼ 0; 1; 12

� �; where there is both back-folding

of intensity from different zones and extinction ofthe original intensity. The net effect shown by thethick line is significant, although too featureless toreveal itself in the measured kf -dependence.Turning to the polarisation at ð0; 2; 1

2Þ; Fig. 7

shows the ghoston prediction. Correcting forinstrumental polarisation P0 ¼ 0:90; the predictionat kf ¼ 2:662 (A�1 yields Paa ¼ �ð0:08; 0:50; 0:32Þ;which given the simplicity of the model isimpressively close to the measured values Paa ¼�ð0:14; 0:37; 0:35Þ70:03: In particular the anom-alous Pxx > Pyy;Pzz is reproduced. At 0; 1; 12

� �; the

g ¼ 4% back-folded contribution has polarisationPaa ¼ �ð0:17; 0:55; 0:29Þ; which should change thepolarisation PM ¼ ð�P0; 0; 0Þ of an isotropic

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1 1.5 2 2.5 3 3.5 4 4.5-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 104

kf

Inte

nsity

(0,1,0.5) focus(0,1,0.5) 20’’

Fig. 6. Intensity as a function of kf ; keeping E ¼ 2 meV andQ ¼ 0; 1; 1

2

� �: The thin lines show the contributions from back-

folding (positive) and extinction (negative) with net effect

shown by the thick line.

0 1 2 3 4 5-1

0.8

0.6

0.4

0.2

0

kf

Pola

risa

tion

PxxPyyPzz

Fig. 7. Predicted polarisation components of the ghoston at

0; 2; 12

� �in CuGeO3:

1 1.5 2 2.5 3 3.5 4 4.5-200

0

200

400

600

800

1000

1200

1400

kf

Inte

nsity

(0,0,0.5) focus(0,0,0.5) 20’’

1 1.5 2 2.5 3 3.5 4 4.5-100

0

100

200

300

400

500

600

700

kf

Inte

nsity

(0,2,0.5) focuski ghostons kf ghostons both ghostons

Fig. 5. Intensity as a function of kf ; keeping E ¼ k2f � k2i ¼ 2 meV and Q ¼ kf � ki fixed at, respectively, 0; 0; 12� �

and 0; 2; 12

� �: The

relative contribution for each t indicated by vertical bars have been convoluted with a Gaussian resolution to obtain the solid line. Thesame intensity scale factor was used for 0; 0; 1

2

� �; 0; 2; 1

2

� �and 0; 1; 1

2

� �:

H.M. Rønnow et al. / Physica B 350 (2004) 11–16 15

magnetic excitation to ð1� gÞP0PM þ gP0Paa ¼�ð0:87; 0:02; 0:01Þ: Hence, the effect of back-folding to the polarisation is in the 1–3% range,consistent with the fact that to this precision, noanomalies in the polarisation of the primary modewere observed.

3. General implications

Having demonstrated that the weak mode inCuGeO3 is a multiple scattering ghoston, we

would like to point out that this is not aspecial case. A 1 cm3 sample with 1� mosaicity,lattice parameters between 3 and 8 (A and aspin gap of 2 meV are all common parameters.Yet, unlike in diffraction, where multiple scatter-ing involves an infrequent coincidence, there isbasically no kf clean from multiple scattering.Being an effect within the sample, it can occuron any instrument—including time-of-flightspectrometers, which are otherwise spared ofmany spurions that compromise triple axisexperiments.

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H.M. Rønnow et al. / Physica B 350 (2004) 11–1616

Hence, we surmise that many if not mostexcitations measured in inelastic neutron scatter-ing are in fact affected by back-folding at the1–5% level. While this does not have any effect onextracted dispersions, it may potentially haunt anydetailed investigations. The predicted 1–2% effecton the polarisation is currently within the attain-able precision of most existing polarised instru-ments, but given current efforts, sub-1% precisionshould be attainable in the near future.Sometimes amplitudes are used to distinguish

between different spin-Hamiltonians yieldingequivalent dispersion, while recently some interesthas been focused on quantum effects towards theferromagnetic zone center. Such studies would beaffected if the dispersion is symmetric between theferromagnetic (FM) and antiferromagnetic (AFM)zone centres and the latter coincides with areciprocal lattice vector. Similarly, magnetic scat-tering with small q may be observed to large Q;where the form factor should have vanished.We recommend that detailed inelastic neutron

scattering experiment as a routine are checked for

multiple scattering effects. In diffraction it iscustom to rotate the sample around Q: Forinelastic scattering this is not practical as the upto 10� achievable on standard goniometers may beinsufficient to break multiple scattering condition.A better procedure would be to compare ki � kf

scans to the ghoston model predictions.

Acknowledgements

HMR was supported by the European Com-munity through Marie Curie FellowshipHPMFCT-2000-00488.

References

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[2] J.E. Lorenzo, L.P. Regnault, J.P. Boucher, B. Hennion,

G. Dhalenne, A. Revcolenschi, Eur. Phys. Lett. 45 (1999)

619.