X-ray Scattering Study of the spin-Peierls transition and soft phonon behavior in TiOCl
E.T. Abel, K. Matan, F.C. Chou, E.D. Isaacs, D.E. Moncton, H. Sinn, A. Alatas, Y.S. Lee
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug X-ray scattering study of the spin-Peierls transition and soft phonon behavior inTiOCl
E.T. Abel , K. Matan , F.C. Chou , E.D. Isaacs , D.E. Moncton , H. Sinn , A. Alatas , and Y.S. Lee ∗ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439 and Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439 (Dated: October 31, 2018)We have studied the S = 1 / T c = 66 K and have an incommensurate lattice distortion between T c and T c = 92 K. Based on our measurements of the intensities, wave vectors, and harmonics of theincommensurate superlattice peaks, we construct a model for the incommensurate modulation. Theresults are in good agreement with a soliton lattice model, though some quantitative discrepanciesexist near T c . The behavior of the phonons has been studied using inelastic x-ray scattering with ∼ T SP has been observed. Our results show reasonably good quantitativeagreement with the Cross-Fisher theory for the phonon dynamics at wave vectors near the zoneboundary and temperatures near T SP . However, not all aspects of the data can be described, suchas the strong overdamping of the soft mode above T SP . Overall, our results show that TiOCl isa good realization of a spin-Peierls system, where the phonon softening allows us to identify thetransition temperature as T SP = T c = 92 K. I. INTRODUCTION
Understanding the properties of low-dimensional spin-1/2 systems remains a central challenge in condensedmatter physics. In quasi-one-dimensional materials, avariety of ground-states can be stabilized, depending onthe subtle interactions beyond the dominant magneticexchange along the spin chain. For example, the low-energy physics of the S = 1 / S = 1 / [11] rekindled interest in spin-Peierls research, as large single crystals could bereadily grown. However, it is not clear whether CuGeO is a good realization of a conventional spin-Peierls sys-tem. The susceptibility does not follow the expected be-havior of a spin chain with nearest neighbor coupling[11],and it is believed that significant next nearest neighborcoupling along the chain must be taken into account.[12]In fact, the ratio of the nearest neighbor to next nearestneighbor exchange interaction is close to the critical valuefor spontaneous formation of a spin gap independent ofa lattice distortion.[12] Neutron scattering measurementsalso reveal a significant magnetic coupling between neigh-boring chains[13].Another puzzle in CuGeO is the lack of an observedphonon softening at the transition to the dimerized state.Cross and Fisher developed a theory which includes thespin-phonon coupling and predicts a softening of a spin-Peierls active phonon to zero frequency at T SP .[4] Neu-tron studies were unable to detect the presence of this softphonon, and, in fact, and a hardening of phonon modeswas observed[14, 15]. Recent theoretical analysis sug-gests that CuGeO may fall within the “anti-adiabatic”regime (characterized by a large phonon frequency rela-tive to the magnetic interaction energy) where a phononhardening would be expected[16, 17, 18, 19]. The latticedynamics associated with the more conventional “adia-batic” regime have yet to be measured in detail.Recently, a new S = 1 / ions are in the d electronic configuration with S = 1 / FIG. 1: Structure of TiOCl at high temperatures in the undis-torted state. Left: Schematic of a Ti-O layer showing the Tiion chains along the b -direction as well as the positions of theoxygens. Right: Schematic of the ab -plane which more clearlyshows the staggered arrangement of adjacent Ti chains. Thedashed lines highlight the unit cell. date for the “resonating valence bond” state[20], but ithas since been found to be primarily a one-dimensionalmagnetic system. Band structure calculations indicatethat TiOCl is a Mott insulator with the lowest occu-pied Ti d xy orbitals pointed toward each other alongthe crystallographic b -direction.[21, 22] The high tem-perature magnetic susceptibility is well described by theBonner-Fisher curve, indicating a nearest neighbor mag-netic exchange of J ≃
660 K. [21] In the TiOCl crystalstructure, the spin chains are staggered–every other Tichain is displaced by half a lattice spacing along the b -direction. Hence, the inter-chain magnetic interaction isfrustrated.Upon cooling, the susceptibility falls away from theBonner-Fisher curve around T ≃
150 K and exhibitsanomalies at T c = 92 K and T c = 66 K.[21] The drop at150 K has been attributed to the opening of a pseudo-gap[23, 24], the anomaly at 92 K corresponds to the onset ofan incommensurately modulated state [23, 25, 26, 27, 28],and the one at 66 K corresponds to a commensuratemodulation[29]. Similar behavior has been observed inthe isostructural compound TiOBr.[30, 31, 32, 33] Atfirst sight, the observed transitions (such as the incom-mensurate phase) suggest unconventional spin-Peierls be-havior. In order to examine the applicability of a spin-Peierls framework, several issues need to be clarified, suchas the nature of the incommensurate phase and the lat-tice dynamics associated with the structural dimeriza-tion.In this paper, we present x-ray diffraction and inelasticx-ray scattering data on single crystal samples of TiOCl.As we shall show, the material undergoes a spin-Peierlstransition in which the structural dimerization occurs inthe presence of a soft-phonon at the zone boundary. Thelow temperature structural modulations are discussed insome detail. The format of the paper is as follows: Sec-tion II outlines the crystal growth and thermodynamiccharacterization of the samples. Section III contains adiscussion of the low temperature structures observedwith synchrotron x-ray diffraction. Based on measure-ments of the superlattice peaks, including higher harmon-ics, we develop a model for the incommensurate structuredescribed by a soliton lattice. In Section IV, the lattice dynamics are studied using high energy-resolution inelas-tic x-ray scattering. We observe a soft, zone boundary,phonon mode which drives the spin-Peierls transition,and we compare our data with the Cross-Fisher theory.Finally, Section V contains a discussion and summary ofour results. II. SAMPLE GROWTH ANDCHARACTERIZATION
The single crystals used in this work were grown by thevapor transport method, using the procedure discussedpreviously[21]. The typical size of an individual crys-tal was about 50 mm x 50 mm x 20 µm along the a, b and c axes respectively. The magnetic susceptibility wasmeasured on a sample composed of six single crystals co-aligned to within 4 degrees, for a total sample mass ofabout 8 mg. The susceptibility results (shown in Fig. 2(a)and the inset) were performed using a SQUID magne-tometer in an applied magnetic field of 2 Tesla. After thesubtraction of a small isotropic Curie tail, the suscepti-bility curves along the three crystallographic directions FIG. 2: Thermodynamic characterization of our sample. (a)The magnetic susceptibility upon warming and cooling afterthe subtraction of a small Curie tail. The sharp drops at T c =92 K and T c = 66 K (cooling) indicate the two transitiontemperatures. Inset: Susceptibility measured along the threecrystallographic directions. (b) The quantity d ( χT ) /dT (c)The specific heat C p ( T ). are nearly identical (as shown in the inset), except for asmall temperature independent offset, most likely due toanisotropic crystal field contributions. This behavior fur-ther indicates the Heisenberg nature of the spins. More-over, the isotropic drop in the susceptibility is a signa-ture for spin-singlet formation, as opposed to N´eel order.Measurements of the susceptibility (Fig.2) upon warmingand cooling show hysteretic behavior around T c = 66 K,which point to a first order phase transition. Hysteresisis not observed around the transition at T c = 92 K, in-dicating that this transition is second order in nature.Below T c , the susceptibility becomes temperature inde-pendent, as seen previously[21], consistent with a fullyformed singlet state. This apparent first order transitionto the spin-Peierls ground-state seems to differentiate Ti-OCl from the other materials in which a second orderspin-Peierls transition is observed([5, 6, 7, 8, 9, 10, 11]).However, as we will show in section VI, we find that it isactually T c = 92 K which defines the spin-Peierls tran-sition.Also shown in Fig.2(b) is the quantity d ( χ ( T ) T ) /dT which should be proportional to the magnetic contribu-tion to the specific heat using the Fisher relation[34, 35].Indeed, the measured specific heat C p ( T ) in Fig.2(c)shows clear anomalies at T c and T c which find closecorrespondence to anomalies in d ( χ ( T ) T ) /dT . This sug-gests that the origin of the anomalies in C p ( T ) have asignificant magnetic contribution. A rough estimate ofthe magnetic contribution can be obtained by integrat-ing C p /T over T in order to calculate the change inentropy over the transition region. At these relativelyhigh temperatures, the phonon contribution dominatesthe specific heat and must be subtracted. We fit thebackground in the vicinity of the transition region toa fourth order polynomial denoted by the line in theFig. 2(c). After background subtraction, the integrationyields ∆ S = 0 . /N K B for the entropy released inthe vicinity of the two transitions. For comparison, oneexpects S/N k B ≈ (2 / k B T /J ) for the entropy of a S = 1 / J/k B = 660 K[21] and taking T to be an average of T c and T c yields a rough upper bound for the change inmagnetic entropy of ∆ S/N k B ≃ .
08. In a previous spe-cific heat study, Hemberger et al. [36] estimated a value of0 .
12 for
S/N k b , obtained by integrating over a wider tem-perature range. Our estimate is only meant to accountfor the changes in the near vicinity of T c and T c , sincethe subtracted background must also contain a weaklytemperature dependent magnetic contribution. Withinthe errors, our results suggest a significant portion of theentropy change at the transitions is magnetic in origin. FIG. 3: X-ray diffraction on a single crystal. (a) Longitu-dinal scans and (b) transverse scans through the (0, 2.5, 0)commensurate position. (c) Scans along H through the in-commensurate and commensurate peak positions. The scansthrough the incommensurate peaks were performed at a dif-ferent value of K than the commensurate peak. III. SINGLE CRYSTAL X-RAY DIFFRACTIONA. Superlattice Peaks
Synchrotron x-ray scattering measurements were per-formed on our single crystal samples using the X20Aand X22C beamlines at the National Synchrotron LightSource at Brookhaven National Laboratory. The sampleswere cooled using closed-cycle He refrigerators. Uponcooling below T c = 66 K, superlattice peaks were ob-served at positions displaced from the fundamental Braggpeaks by commensurate wavevectors (0, ± , 0). This in-dicates doubling of the unit cell along the b -direction,consistent with a dimerization of the lattice. Scansalong H and K through the (0, 2.5, 0) superlattice peak(shown in Fig. 3(a) and (b)) are resolution-limited, indi-cating that the dimerized structure has long-range order.The integrated intensities of 45 fundamental superlatticepeaks were measured and used to determine the dimer-ized structure. The intensities were fit using a model withtwo adjustable parameters: δ (the Ti ion displacementalong the b -direction) and τ (the relative shift of neigh- FIG. 4: Temperature dependence of several fitted parame-ters for the commensurate and incommensurate superlatticepeaks. a) The integrated intensity, where the closed symbolsrefer to a commensurate peak and the open symbols refer toan incommensurate peak. b) The full-width at half-maximum(FWHM) along the H -direction. Inset: Hysteresis in temper-ature of the commensurate peak intensity. c) and d) The com-ponents ∆ H and ∆ K of the displacement wave vector fromthe commensurate position. Inset: Normalized displacementwave vector for the incommensurate modulation comparedwith a theoretical prediction involving discommensurations. boring Ti chains along the b -direction). Note that oncethe chains are dimerized, there is no symmetry prevent-ing a relative shift τ between the two chain sublattices.The best fit values are δ = 0 . b , and τ = 0 . b .For simplicity, our model neglected displacements of theCl and O atoms, yet we still find good agreement with therefinement parameters of Shaz et al. for the dimerizedstate [29].The temperature dependence of the integrated inten-sity of the (0, 2.5, 0) superlattice peak is shown inFig. 4(a) as the closed circles. Upon warming, the in-tensity abruptly drops at T c = 66 K, the same tem-perature at which our thermodynamic measurements in-dicate a first order transition. In fact, the intensity ofthe commensurate superlattice peak also exhibits a ther-mal hysteresis, shown in the inset, with a width for thehysteresis loop similar to that measured for the suscep-tibility. Therefore, both magnetic and structural probesindicate the first order nature of transition at T c .Upon warming above T c = 66 K, the structure has an incommensurate modulation[23, 25, 26, 27, 28], asshown in Fig. 3(c). That is, between T c and T c =92 K, superlattice peaks are found at positions dis-placed from the fundamental positions by wave vectors( ± ∆ H, ± ∆ K, , ,
0) by the displacement vectors ( ± ∆ H, ± ∆ K, H -direction of both the commensu-rate and incommensurate peaks are plotted in Fig. 4(b).Both sets of peaks are found to be resolution-limited, in-dicating long correlation lengths (greater than 2000 ˚A),except for narrow temperature ranges in the vicinitiesof T c and T c . Along the K -direction, the peaks re-main resolution-limited over the measured temperaturerange. The incommensurate wave vectors continuouslychange as a function of temperature, and upon coolingapproach the commensurate wave vector. The values for∆ H and ∆ K for an incommensurate peak displaced from(0, 2.5, 0) are plotted as a function of temperature inFigs. 4(c) and (d). The magnitude for ∆ H is about afactor of 5 larger than ∆ K at temperature immediatelybelow T c = 92 K. The temperature dependence of the in-tegrated intensity for the incommensurate peak is shownin Fig. 4(a). The solid line denotes a fit to a power law( T − T c ) β with β = 0 .
3. Unlike the transition at T c ,the transition at T c appears second order in nature. B. Model of the Incommensurate Structure
Since the magnitudes of ∆ H and ∆ K for the incom-mensurate phase are much smaller than (in recipro-cal lattice units), the structure may be thought of as anincommensurate modulation of the dimerized structure.In CuGeO , the dimerized lattice of the ground stateof becomes incommensurate when a large enough mag-netic field is applied. The incommensurate structure hasbeen interpreted as a lattice of solitons which proliferatealong the chains, separating locally dimerized regions. InCuGeO , the soliton lattice produces a modulation alongthe chain direction only, hence the incommensurate wavevector has a non-zero component along the chain direc-tion only. In TiOCl, the incommensurate wave vector isdisplaced from the commensurate position by two com-ponents: ∆ H and ∆ K , hence, it is slightly more compli-cated.As an initial comparison, the temperature depen-dence of the displacement wave vector magnitude∆ Q = √ ∆ H + ∆ K is plotted in the inset ofFig. 3(c) and compared to a Landau theory predic-tion. Incommensurate-to-commensurate lock-in transi-tions have been observed in charge density wave systems,such as TaSe [37, 38], upon cooling. McMillan[39] used aLandau expansion of the lattice free energy to calculatethe temperature dependence of ∆ Q assuming that thedeviations from the commensurate wave vector take theform of long wavelength phase distortions or discommen-surations . The line in the inset of Fig. 3(c) is the theoryprediction[39], δ ( t ) = 4 . / [4 .
61 + ln (1 /t )] , (1)where t = ( T − T c ) / ( T c − T c ) and δ ( t ) =∆ Q ( t ) / ∆ Q ( t = 1). This prediction is a universal func-tion of t , and there are no adjustable parameters inthe comparison. The theory roughly captures the sin-gular behavior as δ → T c , suggesting that theincommensurate phase in TiOCl may be described us-ing a model of discommensurations separating commen-surate, dimerized regions. Intensity contour plots atdifferent temperatures are shown in Fig. 5(a), showinghow the incommensurate peak positions converge to thecommensurate position as the temperature is cooled to T c . The observed broadening of the peaks just above T c makes it difficult to comment on possible phasecoexistence[24] or a discontinuous change in wave vectorat the incommensurate-to-commensurate transition.To obtain a more specific picture of the incommensu-rate modulation, we measured the intensities of severalincommensurate superlattice peaks in the vicinity of the(0, 1.5, 0) position. At T = 79 K, we find four peaks near(0, 1.5, 0) displaced by wave vectors ( ± ∆ H, ± ∆ K, ± H, − K,
0) from (0, 1.5, 0), alsoshown in Fig. 5(b). This is the first time the higherharmonic peaks have been observed, and their positionsand intensities provide crucial insight into the behaviorof the incommensurate modulation. No signal above thebackground could be observed at positions displaced by( ± H, +3∆ K,
0) indicating that the intensities of thosepeaks are very weak. These observations allow us to ruleout a model consisting of the superposition of two inde-pendent modulations along the a and b directions, as thiswould not yield third harmonic scattering at the observedpositions.We analyze the intensities using a model which de-pends on the real-space displacement vectors ~u i of theatoms in TiOCl from their equilibrium positions. Theresulting structure factor is S ( ~Q ) = X i f i ( Q ) e ~Q · ( ~R i + ~u i ) . (2)where f i ( Q ) is the form factor for atom i and the sum-mation is performed over an enlarged supercell for themodulated structure. For simplicity, we only considerdisplacements of the Ti atoms, leaving the O and Clatoms at their equilibrium positions, and relegate thedisplacements to be along the b -direction.To construct the observed in pattern in Fig. 5(b), webegin by noting that the continuous evolution of ∆ H and ∆ K with temperature indicates that the modulationis truly incommensurate. Hence, we first consider the FIG. 5: (a) Series of intensity contour plots showing the tem-perature evolution of the superlattice peaks (b) Contour plotof the incommensurate peak data at T = 79 K showing boththe first and third harmonics. (c) Calculation of the inten-sity based on our model of the incommensurate structure,as discussed in the text. (d) Real-space picture of the Tiatom positions in the incommensurate structure. AlternateTi chains have different colors (black and red) for easier view-ing. The shaded (unshaded) domains delineate approximateregions with the same relative shift + τ ( − τ ) between adja-cent chains. The lightly colored atoms (yellow) on the domainwalls denote solitons. following sinusoidal form for displacements along the b -direction for a single chain: u ( n ) = δ cos (cid:18) πnλ (cid:19) , (3)where the integer n refers to the n th Ti atom along b , δ is the amplitude of the distortion, and λ is the wave-length of the modulation (in units of b ). The case of λ = 2 corresponds to a simple lattice dimerization and isshown in Fig. 6(a) along with a reciprocal space map ofthe corresponding commensurate superlattice peak. Anincommensurately modulated structure can be obtainedby making λ slightly larger or smaller than 2. Figure 6(b)shows a modulation with λ = 2 .
04 along with the corre-sponding superlattice peak pattern.We observe significantly higher intensities for the firstharmonic peaks displaced by +∆ K from (0,1.5,0) relativeto those displaced by − ∆ K (Fig. 5(b)). This naturallyarises when one considers the two different Ti chains inthe unit cell. By varying the relative phase shift φ be-tween the modulations on the two chains, different in-tensity ratios can be obtained. Figure 6(c) shows thediffraction pattern corresponding to a unit cell with Tichains whose modulations differ by a relative phase shift φ = 0 . π . The resulting structure factor reproduces theobserved intensity asymmetry.Since the modulation is not simply along the b -direction (both ∆ H and ∆ K are non-zero in the incom-mensurate phase), we add an additional phase difference, ξ , between neighboring pairs of Ti chains. The final ex-pression for the displacements is written as u ( n, m ) = δ cos (cid:18) πnλ + jφ + ξm (cid:19) (4)where j = 0 , m refers to the m th pairof Ti chains along the a -direction. A schematic for thismodulation is shown in Fig. 6(d). The addition of thephase shift ξm has the effect of rotating the modulationdirection. The calculated third harmonic peak positionsfall on the same line connecting the first harmonic peakson either side of (0, 1.5, 0), as seen in the data. This is direct evidence for a one-dimensional (single wave) mod-ulation describing the incommensurate phase of TiOCl.Now, there is a degeneracy between displacements withopposite signs in front of φ and ξ . This gives rise to twotwin domains. By considering the other modulated twindomain, the pattern in Fig. 5(c) is obtained. There isgood agreement with the calculated intensity ratios inFig. 5(c) and the observed ones in Fig. 5(b). In addition,all of the peak positions are reproduced. We can also de-duce the population factors of the two twin domains to beabout 60% and 40%. In order to reproduce the relativeintensities of the third harmonics, we used a value for δ of 0 . b , which is unrealistically large. This indicates thatdeviations from a pure sinusoidal modulation exist, aswe discuss below in the context of the higher harmonicspeaks.Figure 5(d) shows the real space positions of the Tiatoms in our model for the modulation (in a single twindomain). The adjacent staggered Ti chains are coloreddifferently (red and black) for easier viewing. A localdimerization of the lattice is readily seen throughout theplotted structure, where adjacent Ti chains appear to FIG. 6: Real-space depictions of structural modulation alongwith corresponding reciprocal space maps of the structurefactor. (a) Sinusoidal modulation for a single chain whichyields a commensurate structure. (b) Incommensurate mod-ulation obtained by changing the wavelength of the sinusoid.(c) Modulation for two adjacent Ti chains, where the valuefor the relative phase shift φ yields an asymmetry in the in-tensities of the first harmonic peaks. (d) Model for the incom-mensurate modulation in TiOCl, obtained by phase shiftingthe modulation of each unit cell along the a -direction. have a small relative shift τ along the chain direction.For convenience of viewing, each shaded (unshaded) re-gion approximately delineates a single phase domain con-sisting of the same relative shift + τ ( − τ ). Note thatshift τ is not a parameter in the above model, but itnaturally arises once φ and ξ are specified. On theboundaries between domains (discommensurations), wefind Ti atoms which are nearly undimerized (colored yel-low in Fig. 5(d)). These correspond to locations where u ( n, m ) ≃ FIG. 7: (a) and (b) Scans through the commensurate, firstharmonic incommensurate, and third harmonic incommensu-rate peaks along the H and K directions. The center of eachpeak was offset in order to plot the three scans on the sameaxis. (c) Ratio of the harmonic peak intensities I / I as afunction of the displacement wave vector ∆ Q . The line is theprediction from the soliton model as described in the text. the solitons are not well defined, being spread out overmany Ti atoms. The other twin domain (with oppositesigns of φ and ξ ) is similar to the structure in Fig. 5(d)but with domain walls which tilt in the opposite sensewith respect to the vertical chain direction.Figure 7(a) shows scans along H and K through thefirst and third harmonic peaks at T = 79 K, along withcorresponding scans through the commensurate peak at T = 10 K. The center of each peak was offset in orderto plot the three scans on the same axis. The first har-monic peak and the commensurate peak are resolution-limited, whereas the third harmonic peak shows a clearbroadening. In CuGeO , Christianson et al. [40] observeda slight broadening of the field-induced third harmonicpeaks and attribute it to the effects of disorder. Theyconcluded that the incommensurate phase was not long-range ordered in the direction of the broadening, eventhough it had a long enough correlation length to appearresolution-limited. We believe the same holds true forthe incommensurate phase in TiOCl in both the a and b directions.The higher harmonic intensities can be calculated byexpanding the modulation in terms of the Fourier com-ponents; the ratio of harmonic intensities is given by theratio of the corresponding Fourier coefficients. Here, we follow an analysis similar to that used to describe the fieldinduced incommensurate modulation in CuGeO [41, 42].In TiOCl, we note that within a single twin domain,the modulation may be considered to be one-dimensionalalong the direction given by the displacement wave vec-tor (∆ H, ∆ K, u ( x ), the i th harmonic will haveintensity[41] I n ∼ (cid:18)Z π dθ sin ( iθ ) sin [ Q u ( x )] (cid:19) , (5)where θ = qx and q is the reduced wave vector along themodulation direction. For the case of a sinusoidal func-tion, we calculate I /I ≃ − using a displacementamplitude of 0 . b . The measured ratio at T = 79 K is I /I ≃ − and, hence, is not consistent with a puresinusoidal modulation. In the limit of a square-wave func-tion, we calculate an expected ratio I /I ≃ − , whichis larger but closer to the measured ratio.For a better fit, we consider the Jacobi elliptic function u ( l ) = ǫ ( − l sn (cid:18) λ Γ k , k (cid:19) (6)where l denotes a lattice site and the elliptic modulus k can have values between 0 and 1 ( k = 0 corresponds toa sine wave and k = 1 a square wave). The parameter Γis the soliton half width, and λ/ I /I can be written: I I = (cid:18) YY + Y + 1 (cid:19) . (7)where Y = exp (cid:2) − πK (cid:0) √ − k (cid:1) /K ( k ) (cid:3) , and K is thecomplete elliptic integral of the first kind. The solitonspacing, λ/
2, can be written in terms of k by λ π ∆ Q = 2 kK ( k )Γ . (8)At each temperature, the measured values for I /I and the magnitude of the displacement wave vector ∆ Q can be used to determine k and Γ. For the data in therange ∆ Q < . I /I canbe calculated as a function of ∆ Q as shown by the linein Fig. 7(c). The calculated line agrees reasonably wellfor ∆ Q < . Q > .
125 have I /I ratios which are much smaller than the prediction.This means that for these temperatures, which are justbelow the onset temperature T c , the soliton width is con-siderable larger than Γ ≃ πJb o , where ∆ o isthe magnetic gap. Using J = 660 K[21] and ∆ o = 430 K[23, 44], this yields Γ ≃ T c , the struc-ture has a one-dimensional incommensurate modulationof an underlying dimerized state. At first, the envelopefunction for the modulation has a relatively short wave-length and is approximately sinusoidal. Upon cooling,the wavelength increases and the modulation crosses overfrom being sinusoidal to being better described by a soli-ton lattice (with solitons of fixed width). The solitonsform domain wall boundaries (or discommensurations)between locally dimerized domains as shown in Fig. 5(d).The domains on either side of the discommensurationsmay be roughly characterized as having opposite signs for τ , the shift parameter between neighboring chains. Thesoliton spacing (spacing between domain walls) increaseswith decreasing temperature and diverges at T c . At thispoint, the system wants to become uniformly dimerized,and a single τ must be selected. This selection betweentwo degenerate ground states with different symmetriesmay help explain the first order nature of the transitionat T c .[45] The above description applies to each twindomain.Our model, which focuses only on the Ti displace-ments, is consistent with the structural refinements ofSchonleber et al. [28]. However, by measuring the higherharmonic content, we have obtained additional details onthe structure. The above picture also gives insight intothe reason for the incommensurate modulation.[26, 28]Assuming the lattice energy prefers an equal spacingbetween Ti atoms between chains, then for uniformlydimerized chains, this preference cannot be satisfied dueto the staggered nature of the chains. The sinusoidalmodulation allows the lattice energy cost to be spreadout over many unit cells. As the dimerization amplitudeincreases, it becomes energetically favorable to form fullydimerized domains at the local level, with the cost in lat-tice energy being relegated to domain walls. Eventuallyupon cooling, the magnetic energy gain due to dimeriza-tion dominates, and the lattice prefers a uniformly dimer-ized state with the value for τ which minimizes the latticeenergy cost. IV. INELASTIC X-RAY SCATTERINGA. Lattice Dynamics
The lattice dimerization which accompanies the spin-Peierls transition is expected to be a displacive struc-tural phase transition. Such displacive transitions arecharacterized by a softening of a zone-boundary phononmode which has a polarization similar to that of the
FIG. 8: Energy transfer scans through the phonon excitationsat different ~Q positions along (0 K T = 80 K, (b) T = 100 K, and (c) T = 300 K. The solidlines are fits to a damped harmonic oscillator structure factorconvoluted with the instrumental resolution as described inthe text. static distortion.[46, 47, 48] This mode softens progres-sively as the temperature is decreased, until the modeenergy reaches zero, and the distortion becomes frozenin. In their theory for the spin-Peierls transition, Crossand Fisher have calculated the expected behavior of sucha soft mode.[4] However, to date, the lattice dynamicsof a zone-boundary soft phonon in a spin-Peierls systemhave not been studied. Here, we present a detailed in-elastic x-ray scattering characterization of the lattice dy-namics in TiOCl. Thus far the only work on the latticedynamics of TiOCl has been Raman and infrared spec-troscopy measurements of the zone center optical modes[45, 49, 50]. The inelastic x-ray scattering technique hasthe distinct advantage in being able to probe the phononmodes throughout the Brillouin zone. Also, while inelas-tic neutron scattering requires relatively large crystals,inelastic x-ray scattering can measure very small crystalsof interesting new correlated electron systems.The inelastic x-ray scattering measurements were per-formed using the SRI-3ID beamline at the AdvancedPhoton Source at Argonne National Laboratory. Anin-line nested channel-cut silicon monochromator and a FIG. 9: (a) Dispersion relations for the observed phononmodes where the undamped frequency ω o , extracted fromthe damped harmonic oscillator fits, is plotted. The linesare predictions from a lattice dynamical calculation. (b) Thedamping parameter Γ for the lowest energy mode. bent silicon analyzer in backscattering geometry wereused to achieve high energy resolution. By scanning theenergy transfer ω through the elastic scattering from aplexiglass sample, the energy resolution can be deter-mined. The instrumental resolution function has the fol-lowing approximate form I ( ω ) = I o ηπγ " (cid:18) ωγ (cid:19) − + (1 − η ) 2 γ (cid:18) ln 2 π (cid:19) / × exp " − (cid:18) ωγ (cid:19) , (9)where γ is the full-width and half-maximum (FWHM), I o is the integrated intensity, and η is a mixing parameter[51]. We determined the spectrometer energy resolutionto be 2.3 meV FWHM. The data we present below havebeen corrected for temperature drifts of the monochro-mator and are normalized by monitor counts.We performed experiments on two samples of TiOCl,consisting of crystals coaligned to within 2-3 degrees withtotal thicknesses of ∼ µ m and ∼ µ m. The incidentphoton energy was 21.3 keV. Representative energy scansare shown in Fig. 8 at temperatures of 80 K, 100 K, and300 K. For each scan, ~Q was held constant at various TABLE I: Interaction parameters used in the shell model cal-culation, and the resulting equilibrium crystal parameters. points along (0, K , 0) between (0, 1.5, 0) and (0, 2, 0).The scattered intensity is a direct measure of the dy-namic structure factor S ( ~Q, ω ) which is related to the dis-sipative part of the response function by the fluctuation-dissipation theorem. To model the lattice dynamics, weused the damped harmonic oscillator response function: S ( ~Q, ω ) = A ~Q π ω Γ( ω o − ω ) + ω Γ [ n ( ω ) + 1] , (10)where ω o is the undamped phonon frequency, Γ is thedamping constant, and A ~Q is an amplitude. In the un-derdamped case ( ω o > Γ) two inelastic peaks are presentat positive and negative energy transfers. In the over-damped case ( ω o < Γ) the scattering has the form of asingle peak centered about ω = 0. In the extreme over-damped limit ( ω o ≪ Γ) the dynamic structure factor canbe rewritten as S ( ~Q, ω ) ≈ k B T ~ ω o π γγ + ω , (11)where γ = ω o / Γ and the high temperature approxima-tion of 1 + n ( ω ) ≃ k B T / ~ ω has been used. The solidlines in Fig. 8 are fits to the damped harmonic oscillatorcross section convoluted with the instrumental resolutionfunction. In some scans, up to three phonon modes canbe fit.Figure 9(a) shows the fitted values for ω o for all ofthe observed phonons as a function of the reduced mo-mentum transfer along the (0 K
0) direction. Notethat in this scattering geometry, the cross section is suchthat only longitudinally polarized phonons are measured.The lowest energy mode corresponds to the longitudi-nal acoustic branch. The widths of the phonons in thisbranch are plotted in Fig. 9(b). At T = 300 K (well above T c = 92 K), the width of the phonon mode increases as ~Q approaches the zone-boundary. In fact, at the zone-boundary, the fit yields Γ > ω which indicates that thephonon is overdamped. The data at T = 100 K (near T c ) show even more dramatic behavior. The width ofthe phonon mode diverges well before the zone-boundaryis reached, indicating that the mode is strongly over-damped. These results are the first indication that the0 FIG. 10: Calculated cross sections for the longitudinalphonons in the undimerized state at various (0 K
0) posi-tions, using the eigenvectors and eigenvalues from the latticedynamical calculation. longitudinal acoustic phonon at the zone-boundary is asoft mode.The solid lines in Fig. 9(a) correspond to lattice dy-namical calculations for the longitudinal phonon modesbased on a shell model using the General Utility Lat-tice Program [52]. The calculation uses phenomenolog-ical potentials to model the strong short-range interac-tions of the neighboring ions[53]. Four Buckingham po-tential interactions serve as a repulsive force betweenatoms, and an additional three-body interaction was usedto constrain an angle of 90 ◦ for the Ti − Cl − Ti bonding.With these potentials, the system settled into an equilib-rium structure nearly identical to the experimental unitcell [54] (see Table I). A comparison between the cal-
FIG. 11: Energy transfer scans at (0, 1.5, 0) for differenttemperatures. The data have been offset for clarity. The T = 15 K data were measured at (0, 1.55, 0) to avoid con-tamination from the elastic superlattice peak at ω = 0. culated and observed optic phonons in Fig. 9(a) showsexcellent agreement. However, the calculation signifi-cantly overestimates the energy for the acoustic branchnear the zone boundary, another indication of the softmode. We note that our calculation and others[55] indi-cate that this mode is degenerate at the zone boundary.The energy eigenvalues and eigenvectors resulting fromthe lattice dynamical calculations allow us to calculatethe expected scattering intensities using the dynamicalstructure factor convoluted with the instrumental reso-lution. This calculation is shown in Fig. 10. The agree-ment with the observed intensities that we measure isquite good. Again, deviations from the data are appar-ent for low energies near the zone-boundary due to thepresence of the soft mode.The temperature dependent behavior of the zoneboundary phonon is shown in Fig. 11. The data for T = 300 K and T = 200 K were fit to Eq. 10 (the gen-eral cross section for the damped harmonic oscillator).However at T = 150 K and below, the fit no longer con-verged, and Eq. 11 was used for the strongly overdampedcase. In this case, it is no longer possible to determinethe values of ω o and Γ independently. Useful informationcan be extracted by noting that the integrated intensity I is directly proportional to T /ω o . Therefore, a plot of FIG. 12: Behavior of the soft phonon at the zone boundaryposition (0, 1.5, 0). (a) The quantity T /Intensity as a functionof temperature. The solid line follows the form a ( T − T c )with T c = 88 K. (b) The quantity ω o / Γ as a function oftemperature. The solid line serves as a guide to the eye. T /I versus T (shown in Fig. 12(a)) should have the sametemperature dependence as ω o . The solid line has theform ω o = a ( T − T o ) which is the result from mean-fieldtheory for a soft-phonon transition,[47, 48] where T o isthe structural transition temperature and a is a constant.The fitted line yields T o = 88(2) K which closely matchesthe temperature T c = 92 K for the onset of the incom-mensurate superlattice peaks. The quantity ω o / Γ canalso be extracted and is plotted in Fig. 12(b). We seethat this quantity plunges to zero around 90 K. This isconsistent with ω o → T c = 92 K. B. Comparison with the Cross-Fisher Theory
The preceding analysis relied on fits to a phenomeno-logical damped harmonic oscillator model. While thisprovides strong evidence for the presence of a soft mode,details regarding the microscopic interactions, such as thespin-phonon coupling, cannot be obtained. A general re-sponse function which includes such interactions has theform A ( ω ) = 1 π Im " ω − Ω − Π( ~Q, ω ) , (12)where Ω is the harmonic or bare phonon frequency, andΠ( ~Q, ω ) is the polarizability (or phonon-self energy). Thedynamic structure factor is then given by S ( ~Q, ω ) = | S ( ~Q ) | [1 + n ( ω )] A ( ω ) . (13)By comparing this cross section with that for the dampedharmonic oscillator, one can see that Π( ~Q, ω ) containsinformation regarding the damping. Since Π( ~Q, ω ) is theonly term which is complex, it can be broken into realand imaginary parts A ( ω ) = 1 π Im h Π( ~Q, ω ) i ( ω − Ω − Re h Π( ~Q, ω ) i ) + Im h Π( ~Q, ω ) i . (14)If we then make the substitutions Re h Π( ~Q, ω ) i = ω o ( ~Q, ω ) − Ω ( ~Q ) (15) Im h Π( ~Q, ω ) i = ω Γ( ~Q, ω ) , (16)we recover the damped harmonic oscillator response func-tion, where ω = p Ω + Re [Π]. Here, ω represents the“quasi-harmonic” frequency which differs from the bare phonon frequency due to the presence of spin-phononcoupling or other anharmonic interactions.Cross and Fisher (CF) calculated the polarizabilityΠ( ~Q, ω ) which takes into account the spin-phonon cou-pling. The phonon system was treated using a mean-fieldrandom phase approximation (RPA), whereas the spindynamics were calculated in a non-perturbative fashion,more accurate that the Hartree approach. Their treat-ment begins by considering a set of non-interacting spinchains, where each chain is governed by the nearest neigh-bor Heisenberg Hamiltonian: H s = X l J ( l, l + 1) S l · S l +1 . (17)Here, the planes of atoms perpendicular to the chains areconstrained to move together. Assuming a linear depen-dence of J on lattice distortions, the exchange couplingmay be written J ( l, l + 1) = J + 1 √ N X q g ( q ) Q ( q ) e iqb (cid:0) − e iqlb (cid:1) , (18)where q is the reduced wave vector along the chain di-rection, g ( q ) is the spin-phonon coupling, and Q ( q ) de-notes the phonon normal mode coordinates. The spin-phonon interaction leads to an expression for the po-larizability that depends on the dimer-dimer correlationfunction h [( S l · S l +1 ) t , ( S · S ) t =0 ] i which CF calculateusing bosonization. It is also possible to calculate thedimer-dimer correlation function using conformal fieldtheory [56], which yields an identical result in the longwavelength limit. The CF polarizability is calculated tobe [4]Π CF ( q,ω ) = − . | (1 − e iqb ) g ( q ) | × I (cid:18) ω + c ( q − k f )2 πT (cid:19) I (cid:18) ω − c ( q − k f )2 πT (cid:19) T , (19)where I ( k ) = 1 √ π Γ (cid:0) + ik (cid:1) Γ (cid:0) + ik (cid:1) , (20)where 2 k f = π/b and c = πJb/ T SP , is ω o = Ω + Π CF ( q = 2 k F , ω → , T = T SP ) = 0 . (21)This can be solved to get an expression for the spin-phonon coupling for the zone boundary mode: g = Ω r π . T SP . (22)We now compare our data with the results of Crossand Fisher by fitting our data with the dynamic struc-ture factor that explicitly includes the CF polarizabil-ity Π CF ( q, ω ) (Eq. 19) convoluted with the instrumental2 FIG. 13: Energy transfer scans of the low energy phononmode at the zone boundary. The lines denote fits to thedynamical structure factor calculated using the Cross-Fishertheory. resolution. Only data in the energy range -20 meV ≤ ω ≤
20 meV was fit, thereby focusing on the longitudinalacoustic mode. We first fit data taken the zone bound-ary position (0,1.5,0) at several temperatures, as shownin Fig. 13. At the zone boundary, the CF polarizabilityreduces toΠ CF ( q, ω ) = − . g h I (cid:16) ω πT (cid:17)i T . (23)Therefore, the only adjustable parameters in the dynam-ical structure factor are the spin-phonon coupling g andthe bare phonon frequency Ω . Since g scales with theCF polarizability, a non-zero g will affect both the peakwidth and the shift of ω from Ω . Away from the zoneboundary, the polarizability also depends on J , therefore J and g cannot be fit independently in general.The solid lines in Fig. 13 represent fits to the dynam-ical structure factor at q = 0 . g vary with temperature, but kept Ω fixedat the best fit value of Ω = 27(1) meV. Letting g floatwith temperature was necessary in order to get reason-able fits, and we will comment on this later. Once g was determined for a given temperature, the scans atother q positions were then fit with Ω and J as the ad-justable parameters. For the data sets at T = 100 K and T = 300 K, the analysis yielded a roughly consistent mag-netic exchange of J ∼
200 K. This value for J is aboutthree times smaller than that deduced from the mag-netic susceptibility results[21]. However, our value for J is remarkably close considering that it resulted from mea-surements of only the phonon positions and lineshapes.The resulting values for the bare phonon frequency Ω are plotted in Fig. 14(a) as a function of the reduced wavevector (0 , q, ( q )follows the calculated dispersion curve much better than FIG. 14: (a) The bare phonon frequency Ω of the longitudi-nal acoustic mode extracted from the fits to the Cross-Fishertheory. The solid line denotes the shell model calculation. (b)The fitted spin-phonon coupling g , normalized to the value of g calculated using Eq. 22. c) The quasi-harmonic frequency ω o of the zone boundary mode, extracted from fits to theCross-Fisher theory. The solid line denotes the power-lawform A √ T − T c with T c = 98 K. the mode energies ω o from the damped harmonic oscilla-tor fits (Fig. 12(a)). Thus, it appears that the CF theorycaptures the anharmonic effects of the spin-phonon cou-pling in TiOCl reasonably well. However, in order to ob-tain good fits at all temperatures, g was allowed to vary.These values are plotted in Fig. 14(b), normalized by thevalue of g calculated using Eq. 22 (with Ω = 27 meV and T c = 92 K), which yields g Cross − F isher = 75 meV / .The figure shows that the fitted values of g approach g Cross − F isher as T → T c . Since the fits to the dynam-ical structure factor were performed without knowledgeof T c , this result supports the self-consistency and valid-ity of the CF expressions describing the soft phonon near T SP . We also have enough information to determine thequasi-harmonic frequency ω o at q = 0 .
5, which is plot-ted in Fig. 14(c). We see that the spin-phonon couplingreduces ω o from the bare value Ω by about a factor of2. Also, at the spin-Peierls transition, ω o should fall tozero. Indeed, the solid line through the data is a fit tothe form A √ T − T c where T c = 98 K, which is close to T c = 92 K. Hence, we identify T c as the spin-Peierlstransition temperature in TiOCl. The congruence withthe basic features of the CF theory allows us to conclude3that that TiOCl is, in fact, a good realization of a spin-Peierls system.While the CF theory can successfully describe manyaspects of the phonon data, there are some discrepan-cies. Such discrepancies are not completely unexpected,since the theory is based on a mean-field RPA treatmentof the phonons in which fluctuation corrections due tothe phonon dynamics are absent. As noted above, thebest fit g is temperature dependent and the fitted valueof J is a factor of 3 lower than that estimated from themagnetic susceptibility. This likely stems from limits tothe applicable range for Π CF ( q, ω ), which is most accu-rate in the vicinity of q ≈ . T SP . In order to more fully describe the data, additionalanharmonic effects which broaden the phonon lineshapesaway from q = 0 . T = 300 K( ≫ T SP ). These unusual features of the phonon dynam-ics of TiOCl require better theoretical understanding. V. DISCUSSION AND SUMMARY
The TiOCl compound has many ingredients whichmake it a particularly ideal spin-Peierls system. First,it is composed of weakly interacting S = 1 / T c = 92 K. The incommensurate-to-commensurate tran-sition at T c = 66 K is characterized by a divergence ofthe domain wall spacing and selection of a single dimer-ized domain.TiOCl is also of interest since it is an inorganic mate-rial for which single crystals samples can be grown. Usingsingle crystal samples, we have performed high-resolutioninelastic x-ray scattering measurements of the lattice dy-namics. This is a noteworthy example in which x-raysrather than neutrons can provide the best informationon the detailed lattice dynamics of a correlated electronsystem. We have discovered a soft phonon mode whichdrives the spin-Peierls transition. This stands in markedcontrast to the much studied CuGeO system, for which a soft phonon has not been observed. In TiOCl, the lat-tice dynamics suggest that the spin-Peierls temperatureis associated with the transition temperature T c = 92 K,and not T c = 66 K as has been previously speculated.Our measurements on TiOCl allow to make an un-precedently detailed comparison with the Cross-Fishertheory. We find that the calculated polarizability cansuccessfully describe the measured phonon cross sections.That is, the extracted values for the bare phonon frequen-cies and spin-phonon coupling give a consistent pictureof both the phonon dispersion and the phonon softeningat T SP . However, several discrepancies exists, such asthe small value of J and a temperature dependent spin-phonon coupling constant. This points to the necessity ofadding additional anharmonic effects to fully describe thelattice dynamics. Interestingly, the soft phonon mode re-mains strongly overdamped at temperatures much higherthan the observed spin-Peierls transition temperature.The presence of the soft mode transition indicates thatTiOCl falls within the adiabatic regime (i.e., small Ω ).Indeed, the fitted bare phonon frequency of the Peierls-active mode, Ω ≃
27 meV, is smaller than the magneticenergy scale J ≃
57 meV in the system. Recent theorieshave expanded upon the Cross-Fisher result and shed fur-ther light on the conditions separating the adiabatic andanti-adiabatic regimes.[16, 17, 18, 19] Gros and Wernerhave shown that within RPA a soft phonon occurs only ifΩ /T SP < . /T c ≃ . T c = 92 K as the critical temperature). Hence,their prediction suggests that TiOCl falls outside of theadiabatic regime. However, a more recent theory by Do-bry et al. [58] goes beyond RPA and takes into account thedynamics of the transverse phonons arising from inter-chain interactions. They find that soft phonon behaviormay occur for values as high as Ω /T c ≃
3, provided that ω ⊥ /T SP for the transverse phonon is not very large[58].Our shell model calculations indicate that the latter con-dition is satisfied in TiOCl. Of course, further experi-mental studies of the transverse phonon dynamics wouldcertainly be useful. Also, more theoretical work basedexplicitly on the structure of staggered Ti chains wouldbe necessary for additional detailed comparisons. Withits wealth of observed phenomena and relatively simplecrystal structure, TiOCl is an ideal system for quantita-tive tests of theories for coupled spin and lattice degreesof freedom in one-dimensional magnets. Acknowledgments
We thank P. A. Lee, T. Senthil, P. Ghaemi, T.Yildirim, and J. Hill for useful discussions. The work atMIT was supported by the Department of Energy undergrant number DE-FG02-04ER46134. ∗ email: [email protected] [1] H. Bethe, Z. Phys. , 205 (1931).[2] L. Hulth´en, Ark. Mat. Astron. Fys. (1938).[3] E. Pytte, Phys. Rev. B , 4637 (1974).[4] M. C. Cross and D. S. Fisher, Phys. Rev. B , 402(1979).[5] I. S. Jacobs, J. W. Bray, J. H. R. Hart, L. V. Interrante,J. S. Kasper, G. D. Watkins, D. E. Prober, and J. C.Bonner, Phys. Rev. B , 3036 (1976).[6] D. E. Moncton, R. J. Birgeneau, L. V. Interrante, andF. Wudl, Phys. Rev. Lett. , 507 (1977).[7] S. Huizinga, J. Kommandeur, G. Sawatsky, B. Thole,K. Kopenga, W. de Jonge, and D. Roos, Phys. Rev. B , 4723 (1979).[8] J. W. Bray, J. H. R. Hart, L. V. Interrante, I. S. J. adnJ. S. Kasper, G. D. Watkins, S. H. Wei, and J. C. Bonner,Phys. Rev. Lett. , 744 (1975).[9] J. P. Pouget, Eur. Phys. J. B , 321 (2001).[10] R. J. J. Visser, S. Oostra, C. Vettier, and J. Voiron, Phys.Rev. B , 2074 (1983).[11] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev.Letters , 3651 (1993).[12] J. Riera and A. Dobry, Phys. Rev. B , 16098 (1995).[13] L. P. Regnault, M. A¨ın, B. Hennion, G. Dhalenne, andA. Revcolevschi, Phys. Rev. B , 5579 (1996).[14] J. E. Lorenzo, K. Hirota, G. Shirane, J. M. Tranquada,M. Hase, K. Uchinokura, H. Kojima, I. Tanaka, andY. Shibuya, Phys. Rev. B , 1278 (1994).[15] M. Braden, W. Reichardt, B. Hennion, G. Dhalenne, andA. Revcolevschi, Phys. Rev. B , 214417 (2002).[16] C. Gros and R. Werner, Phys. Rev. B , 14677 (1998).[17] M. Holicki, H. Feshke, and R. Werner, Phys. Rev. B ,174417 (2001).[18] E. Orignac and R. Chitra, Phys. Rev. B , 214436(2004).[19] R. Citro, E. Origanac, and T. Giamarchi, Phys. Rev. B , 024434 (2005).[20] R. J. Beynon and J. A. Wilson, Journal of Physics: Con-densed Matter pp. 1983–2000 (1993).[21] A. Seidel, C. A. Marianetti, F. C. Chou, G. Ceder, andP. A. Lee, Phys. Rev. B , 020405 (2003).[22] M. Hoinkis, M. Sing, J. Schfer, M. Klemm, S. Horn,H. Benthien, E. Jeckelmann, T. Saha-Dasgupta,L. Pisani, R. Valenti, et al., Phys. Rev. B , 125127(2005).[23] T. Imai and F. C. Chou, cond-mat/0301425 (2003).[24] J. P. Clancy, B. D. Gaulin, K. C. Rule, J. P. Castellan,and F. C. Chou, Phys. Rev. B , 100401 (2007).[25] E. Abel, K. Matan, F. C. Chou, and Y. S. Lee, Bull.Amer. Phys. Soc. , 317 (2004).[26] R. R¨uckamp, J. Baier, M. Kriener, M. Haverkort,T. Lorenz, G. Uhrig, L. Jongen, A. M¨oller, G. Meyer,and M. Gr¨uninger, Phys. Rev. Lett. (2005).[27] A. Krimmel, J. Strempfer, B. Bohnenbuck, B. Keimer,M. Hoinkis, M. Klemm, S. Horn, A. Loidl, M. Sing,R. Claessen, et al., Phys. Rev. B , 172413 (2006).[28] A. Sch¨onleber, S. van Smaalen, and L. Palatinus, Phys.Rev. B , 214410 (2006).[29] M. Shaz, S. v. Smaalen, L. Palatinus, M. Hoinkis,M. Klemm, S. Horn, and R. Claessen, Phys. Rev. B ,100405 (2005).[30] P. Lemmens, K. Choi, R. Valent, T. Saha-Dasgupta, E. Abel, Y. Lee, and F. Chou, New J. Phys. , 74 (2005).[31] S. van Smaalen, L. Palatinus, and A. Schonleber, Phys.Rev. B , 020105 (2005).[32] T. Sasaki, M. Mizumaki, T. Nagai, T. Asaka, K. Kato,M. Takata, Y. Matsui, and J. Akimitsu, Physica B , 1066 (2006).[33] T. Sasaki, M. Mizumaki, T. Nagai, T. Asaka, K. Kato,M. Takata, Y. Matsui, and J. Akimitsu, cond-mat/0509358 (2005).[34] D. C. Johnston, R. K. Kremer, M. Troyer, X. Wang, andA. Klumper, Phys. Rev. B , 9558 (2000).[35] M. Fisher, Philos. Mag. , 1731 (1962).[36] J. Hemberger, M. Hoinkis, M. Klemm, M. Sing,R. Claessen, S. Horn, and A. Loidl, Phys. Rev. B ,012420 (2005).[37] D. E. Moncton, J. D. Axe, and F. J. DiSalvo, Phys. Rev.B , 801 (1977).[38] R. M. Fleming, D. E. Moncton, D. B. McWhan, and F. J.DiSalvo, Phys. Rev. Lett. , 576 (1980).[39] W. McMillan, Phys. Rev. B , 1496 (1976).[40] R. J. Christianson, Y. J. Wang, S. C. LaMarra, R. J. Bir-geneau, V. Kiryiukhin, T. Masuda, I. Tsukada, K. Uchi-nokura, and B. Keimer, Phys. Rev. B , 174105 (2002).[41] V. Kiryukhin, B. Keimer, J. P. Hill, and A. Vigliante,Phys. Rev. Let. , 4608 (1996).[42] V. Kiryukhin, B. Keimer, J. P. Hill, S. M. Coad, andD. M. Paul, Phys. Rev. B , 7269 (1996).[43] T. Nakano and H. Fukuyama, J. Phys. Soc. Jpn. , 1679(1980).[44] P. J. Baker, S. J. Blundell, F. L. Pratt, T. Lancaster,M. L. Brooks, W. Hayes, M. Isobe, Y. Ueda, M. Hoinkis,M. Sing, et al., Phys. Rev. B , 094404 (2007).[45] D. Fausti, T. Lummen, C. Angelescu, R. Macovez, J. Lu-zon, R. Broer, P. Rudolf, P. van Loosdrecht, N. Tristan,B. Buchner, et al., cond-mat/0704.017 (2007).[46] G. Shirane, J. D. Axe, and J. Harada, Phys. Rev. B ,3651 (1970).[47] S. M. Shapiro, J. D. Axe, and G. Shirane, Phys. Rev. B , 4332 (1972).[48] G. Shirane, Reviews of Modern Physics , 437 (1974).[49] P. Lemmens, K. Y. Choi, G. Caimi, L. Degiorgi, N. N.Kovaleva, A. Seidel, and F. C. Chou, Phys. Rev. B p.134429 (2004).[50] G. Caimi, L. Degiorgi, N. Kovaleva, P. Lemmens, andF. Chou, Phys. Rev. B , 125108 (2004).[51] H. Sinn, E. E. Alp, A. Alatas, J. Barraza, G. Bortel,E. Burkel, D. Shu, W. Sturhahn, J. Sutter, T. Toellner,et al., Nuclear Instruments and Methods in Physics Re-search A , 1545 (2001).[52] J. D. Gale and A. L. Rohl, Mol. Simul. , 291 (2003).[53] J. Gale and A. Rhol, General utility lattice program , 268 (1958).[55] T. Yildirim, private communication .[56] P. Ghaemi, private communication .[57] J. des Cloiseaux and J. Pearson, Phys. Rev. , 2131(1962).[58] A. Dobry, D. C. Cabra, and G. L. Rossini, Phys. Rev. B75