Dimensional Reduction in Quantum Dipolar Antiferromagnets
P. Babkevich, M. Jeong, Y. Matsumoto, I. Kovacevic, A. Finco, R. Toft-Petersen, C. Ritter, M. Månsson, S. Nakatsuji, H. M. Rønnow
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Dimensional Reduction in Quantum Dipolar Antiferromagnets
P. Babkevich, ∗ M. Jeong, Y. Matsumoto, I. Kovacevic, A. Finco,
1, 3
R. Toft-Petersen, C. Ritter, M. M˚ansson,
1, 6, 7
S. Nakatsuji, and H. M. Rønnow Laboratory for Quantum Magnetism, Institute of Physics,´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan ICFP, D´epartement de physique, ´Ecole normale sup´erieure, 45 rue d’Ulm, 75005 Paris, France Helmholtz-Zentrum Berlin f¨ur Materialien und Energie, D-14109 Berlin, Germany Institut Laue-Langevin, BP 156, F-38042, Grenoble Cedex 9, France Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen, Switzerland Department of Materials and Nanophysics, KTH Royal Institute of Technology, SE-164 40 Kista, Sweden
We report ac susceptibility, specific heat and neutron scattering measurements on a dipolar-coupled antiferromagnet LiYbF . For the thermal transition, the order-parameter critical exponentis found to be 0.20(1) and the specific-heat critical exponent − . /h universality class despite the lack of apparent two-dimensionality in the structure. Theorder-parameter exponent for the quantum phase transitions is found to be 0.35(1) correspondingto (2 + 1)D. These results are in line with those found for LiErF which has the same crystalstructure, but largely different T N , crystal field environment and hyperfine interactions. Our resultstherefore experimentally establish that the dimensional reduction is universal to quantum dipolarantiferromagnets on a distorted diamond lattice. PACS numbers: 75.25.-j, 75.40.Cx, 74.40.Kb
Critical phenomena near continuous phase transitionsdo not depend on the microscopic details of systems butonly on the symmetry of the order parameter and interac-tions and the spatial dimensionality [1]. Such universal-ity for classical thermal transitions has been thoroughlydemonstrated with various physical systems over decadeswhile nowadays a similar line of effort is actively pursuedfor zero-temperature quantum transitions [2–4]. Com-paring experimental observations with theoretical modelshas been particularly successful for magnetic insulatorsthat could be simply modeled by short-ranged, exchange-coupled spins on a lattice. Although dipolar interactionsappear to be more classical than their exchange-coupledcounterparts, it has been shown that on a square or dia-mond lattice, quantum fluctuations can map long-rangeddipolar interactions to a two-dimensional Ising model [5–7]. The Li R F family is special as the rare-earth ions arearranged in a slightly distorted diamond-like structuremaking them intriguing to study in relation to order bydisorder phenomena [8].For the case of a dipolar-coupled Ising ferromagnet,the theoretical upper critical dimension D ∗ = 3 andthe mean-field calculations actually apply quite well asshown, for instance, in LiHoF [9]. This is despitethe significant role of hyperfine interactions around thequantum phase transition [10, 11]. Recently, quantumand classical critical properties of a long-range, dipolar-coupled antiferromagnet could be investigated for thefirst time with LiErF [12]. It was discovered thatthe specific-heat and order-parameter critical exponents, α = − . β T = 0 . α = 0 and β T = 0 .
5. Instead, these exponent values suggest a 2D XY /h universality class, despite the ab-sence of any apparent two-dimensionality in the structureof the system. This intriguing dimensional reduction wasfurther corroborated by the β H = 0 . , due to ratherclose (3 meV) higher-lying crystal-field levels or weak hy-perfine interactions, is to date unknown.Among the Li R F family where R is a rare-earth ion,LiYbF has been suggested to be an alternate candidatefor a dipolar antiferromagnet [13]. However, there aremarked differences between LiYbF and LiErF . First,the electronic level scheme is quite different with crys-talline electric field split first excited state an order ofmagnitude higher in LiYbF . Second, in Yb , there aretwo stable isotopes of Yb with strong hyperfine coupling– 11 . µ eV for Yb (14.3%) and − . µ eV for Yb(16.1%). LiErF contains Er (22.8%) whose hyperfinecoupling strength is weak, 0.5 µ eV. Therefore, LiYbF could serve as an excellent candidate to test for the ro-bustness of dimensional reduction in dipolar antiferro-magnets arranged on a distorted diamond lattice.In this Letter, we present ac susceptibility, specificheat, and neutron scattering measurements on LiYbF and demonstrate the thermal and quantum critical prop-erties. The field-temperature ( H - T ) phase diagram isfirst mapped out and a bilayered XY antiferromagneticorder for the ground state is identified. Then we showthat the critical exponents α , β T , and β H support thedimensional reduction as a universal feature of quantum FIG. 1. (a) Real part of ac susceptibility χ ′ as a function oftemperature in zero field and (b) χ ′ as a function of field atdifferent temperatures. (c) Magnetic phase diagram mappedout using the susceptibility. Inset shows the bilayer magneticstructure of LiYbF . dipolar antiferromagnets.Large, high-quality single crystals were obtained froma commercial source. In order to reduce neutron ab-sorption, the samples were enriched with the Li iso-tope. The ac susceptibility χ ( T, H ) was measured ona single crystal using mutual inductance method wherethe excitation field was 40 mOe and the excitation fre-quency 545 Hz. The specific heat C p ( T ) was measuredby the relaxation method in a dilution refrigerator with atemperature stability of 0.1 mK. Powder neutron diffrac-tion was performed using the high-intensity D1B andhigh-resolution D2B diffractometers at ILL, France us-ing incident neutron wavelength 2.52 and 1.59 ˚A, respec-tively. The evolution of the magnetic Bragg peak inten-sities with temperature and field was followed by per-forming high-resolution single-crystal neutron scatteringusing the triple-axis spectrometer FLEXX at HZB, Ger-many [14]. The instrument was set up with 40’ collima-tion before and after the sample and incident neutronwavelength of λ = 4 .
05 ˚A. The corresponding wavevec-tor and energy resolution (FWHM) was on the order of0.014 ˚A and 0.15 meV, respectively.Figure 1 shows bulk ac susceptibility data from asingle-crystal LiYbF . The temperature-field phaseboundary was mapped for a transverse magnetic fieldapplied along the c axis. Figure 1(a) shows the realpart of the ac susceptibility, χ ′ , as a function of tem-perature in zero field. The peak in zero field reflectsthe antiferromagnetic transition at T N = 130 mK. Fig- ure 1(b) shows χ ′ ( H ) at 30-200 mK. Below T N , a pro-nounced cusp is observed which corresponds to a quan-tum transition from the ordered to a quantum paramag-netic phase. At base temperature, a maximum in χ ′ ( H )is found at H c = 0 .
48 T. The peak shifts to lower fields astemperature is increased. Based on these measurements,we can accurately map out the phase diagram shown inFig. 1(c).The specific heat as a function of temperature is shownin Fig. 2(a). In zero field, a sharp peak in the specific heatcapacity marks the second-order thermal transition [15].On applying a transverse field, we find the peak at T N decreases in amplitude and shifts to lower temperatureat H = 0 .
45 T. Above H c , only a broad hump is found inthe specific heat capacity. At such low temperatures, −1 −3 −2 −1 T (K) C p ( J K − m o l − ) (a) × . × . × . × .
01 0.00 T0.45 T0.51 T0.70 T1.00 T −4 −3 −2 −1 −1 | T/T N − | C p ( J K − m o l − ) α = − . T > T N T < T N FIG. 2. (a) Specific heat in zero and finite fields as a functionof temperature. Calculation of specific heat capacity in thesingle-ion limit for different fields are plotted by continuouslines. The data were displaced vertically by multiplying withscaling factors given in the figure. (b) Determination of thespecific-heat critical exponent α for the thermal transitionbased on measurements above and below T N (dashed line).Scaling away from the critical region was fitted by the dottedline. phonon and crystal-field-level contributions are frozenout. We model the specific heat capacity away fromthe QPT using a parameter-free model where the Hamil-tonian H contains crystal field, hyperfine and Zeemanterms. From the diagonalized Hamiltonian h n |H| n i = ǫ n ,we calculate for each isotope i the Schottky specific heat, C Sch i = k B β (cid:2) h ǫ i − h ǫ i (cid:3) , where k B is the Boltzmannfactor and β = 1 / ( k B T ). The thermal ensemble averageis denoted by h . . . i . The total specific heat capacity isfound from the weighted sum of contributions from eachYb isotope. The comparison between the experiment andour simple model is remarkably good considering thatthis is a parameter-free calculation with all parametersfixed from other experiments. It is possible to improvethe comparison by including quadrupolar operators, andby fine-tuning hyperfine coupling strengths and the crys-tal field parameters, etc. However, this would give toomany adjustable parameters, and the calculation anywayignores collective effects beyond the mean-field level.In zero applied field, close to T N , the heat capacity canbe described by a universal power-law, C crit p = A | t | α + B, (1)where the reduced temperature t = 1 − T /T N , A and B are free parameters which can have different values aboveand below T N . The results of our analysis are shown inFig. 2(b). The contribution from the background term, B , is found to be small and is set to zero above and be-low T N . A good fit is found for α = − . − . [12]. The nega-tive exponents imply that C p is finite at T N . Away fromthe phase transition we observe a change in the scaling.Above around 250 mK and below 100 mK the data canbe fit to an exponent of around − . T N and is dramatically different to LiErF where a cross over was found above 1.03 T N [12].To elucidate the magnetic structure below T N , weperformed neutron diffraction on a powder of LiYbF .At 10 K, in the paramagnetic phase, the crystal lat-tice was refined using the I /a space group where a = 5 . c = 10 . k = (1 , ,
0) mag-netic propagation wavevector. Figure 3(a) shows pow-der diffraction pattern obtained by subtracting measure-ments above T N from 50 mK data. The magnetic peaksare well described by a bilayer antiferromagnetic struc-ture with moments along the [110] direction, where mo-ments related by I -centering are aligned antiparallel. Anordered moment of 1.9(1) µ B is found to reside on eachYb ion. A schematic of a possible magnetic structureis shown in Fig. 1(c). This differs from LiErF where themoments are parallel to the [100] direction. Although ourdata do not allow us to uniquely identify the magnetic
10 20 30 40 50 60 70 80 90 100 11000.511.522.53 x 10 θ (deg) C o un t s ( a r b . ) (a) ObservedAFM modelDifference12 13 14 1510 θ (deg) C o un t s i n s (b)
52 mK120 mK133 mK136 mK145 mK −0.04 −0.02 0 0.02 0.0410 (1 ,k,
0) (r.l.u.) C o un t s i n s (c) FIG. 3. (a) Magnetic powder diffraction pattern from thesubtraction of paramagnetic background from 50 mK mea-surements. (b) Magnetic Bragg peak from powder diffractionat different temperatures in zero field and (c) single-crystalmeasurements at 70 mK in different fields. Lines are fits toa Gaussian with additional contribution from critical scatter-ing. structure, it is clear that LiErF and LiYbF order dif-ferently (see Supplemental Material). The origin of thisis not entirely obvious but could be attributed to thein-plane anisotropy set by the crystal field. This woulddepend primarily on the B ( c ) O ( c ) crystal field termand result in the configuration energy E ∼ B ( c ) cos(4 φ )having minima rotated by 45 ◦ when changing the sign of B ( c ) parameter. Indeed, our previously reported resultsshow that B ( c ) is significantly larger and of oppositesign in LiYbF compared to LiErF [13].The powder sample of LiYbF was measured as a func-tion of temperature in fine steps across the thermal phasetransition. Figure 3(b) shows how the magnetic intensityof the (001) reflection decreases with temperature. As ex-pected from ac susceptibility and heat capacity measure-ments, magnetic order disappears above 136 mK. Single-crystal measurements as a function of transverse field areshown in Fig. 3(c). At T base = 70 mK, a field of around0.43 T suppresses the (100) magnetic peak. A small con-tribution from critical scattering is observed as tails ofthe main peak. The neutron scattering measurements ofLiYbF reaffirm the phase diagram found from ac sus-ceptibility in Fig. 1.The evolution of the magnetic Bragg peak intensitieswith temperature and field are shown in Fig. 4(a). Con-tinuous onset and smooth evolution of the order param-eter is observed with both temperature and field.For both the powder and single-crystal data we haveconsidered a model consisting of (i) a Lorentzian line- H/H c , T/T N I / I (a) ThermalQuantum 10 −3 −2 −1 − H/H c , 1 − T/T N I β H = 0 . β T = 0 . T/T N β (c) HeisenbergXYIsing
H/H c FIG. 4. (a) Evolution of the zero-field Bragg peak intensity as a function of temperature and the Bragg peak intensity asa function of the field at 50 mK. (b) Determination of the order-parameter critical exponent for the thermal classical phasetransition and the quantum transition. (c) Extraction of the critical exponent β H at different temperatures is plotted by circles.The critical exponent β T is shown by squares in the panel on the right. Filled data points represent exponents found in thiswork for LiYbF , in addition empty symbols denote LiErF results, after Ref. [12]. Dashed lines and dotted horizontal linescorrespond to critical exponents of 3D [16] and 2D [17, 18] universality classes, respectively. The expected mean-field (MF)result of β = 0 . shape to describe the critical fluctuations close to thephase transition and (ii) a delta-function to account forlong-range order. Both of these were then convolutedby a Gaussian, representing the instrumental resolution.The strength of scattering from critical fluctuations israther weak and within the measured resolution andstatistics cannot be refined to extract further exponentsin either powder or single-crystal data. The amplitudeof the convoluted delta-function σ corresponds to thesquare of the order parameter, i.e., staggered magnetiza-tion. Therefore, sufficiently close to the phase boundary, σ ∝ t β T for a zero-field measurement and σ ∝ h β H ,where h = 1 − H/H c on sweeping magnetic field at con-stant temperature. From such treatment we obtain theresults shown in Fig. 4(b), where squares and circles arefor thermal and quantum critical exponents, respectively.Fitting the data to a power-law, we obtain β T = 0 . β H = 0 . T base ≃ . T N . For LiErF , on the otherhand, the β H was extracted at T base ≃ . T N . To en-sure that the extracted β H = 0 . β H in the temperature range studied. Thisassertion is further corroborated by the heat capacitymeasurements, shown in Fig. 2, where the thermal criti-cal region is found above around 0.8 T N . Comparing thecritical exponents to tabulated results [16–18], it is clearthat the quantum transition falls in the β = 0 . h model predicts β = 0 . R Ba Cu O − δ whose dipolar interactions were the fo-cus of some theoretical work [19, 20]. It was argued two-dimensional behavior is strongly related to the spacing ofbasal planes with a cross-over from three-dimensional be-havior around c/a > .
5. However, relatively strong ex-change coupling as well as superconductivity makes thissystem more complicated to separate the influence of thedipolar interaction. We hypothesize that systems suchas R PO (MoO ) · O where rare-earth ions form adiamond lattice would also be a good candidate to exam-ine quantum criticality due to strong dipolar and weakexchange interactions [21]. Quantum spin fluctuationsof dipolar-coupled antiferromagnetism have already beensuggested to play a major role in these systems [22].To conclude, dipolar-coupled LiYbF undergoes a ther-mal transition into the bilayer, XY antiferromagneti-cally ordered phase, where the critical exponents followthe 2D XY /h universality class despite the lack of ap-parent two-dimensionality in the structure. Applying atransverse magnetic field suppresses the order, inducinga quantum phase transition into a paramagnetic state,which scales according to (2+1)D universality. These ob-servations are in accordance with those for LiErF withlargely different crystal field environment, T N , and hyper-fine interactions. Our results, therefore, experimentallyestablish that the dimensional reduction is a universalfeature of dipolar-coupled quantum antiferromagnets onthe distorted diamond-like lattice and are likely to beapplicable to a vast range of seemingly different systems.While it may be premature to conclude that dimensionalreduction is universal to other lattices, the challenge isnow to find a dipolar-coupled antiferromagnet withoutit.We are grateful to B. Klemke for his technical sup-port. We would like to thank J. O. Piatek for his helpin setting up the ac susceptibility measurements and I.ˇZivkovi´c for helpful discussions. We are indebted to J. S.White, M. Zolliker and M. Bartkowiak for their assis-tance during preliminary measurements on the TASPspectrometer at SINQ, PSI. This work was funded bythe Swiss National Science Foundation and its Sinergianetwork MPBH, Marie Curie Action COFUND (EPFLFellows), and European Research Council Grant CON-QUEST. This work is partially supported by Grants-in-Aid for Scientific Research (No. 25707030 and No.15K13515) and Program for Advancing Strategic Inter-national Networks to Accelerate the Circulation of Tal-ented Researchers (No. R2604) from the Japanese Soci-ety for the Promotion of Science. We thank HZB forthe allocation of neutron radiation beam time. Thisproject has received funding from the European Union’sSeventh Framework Programme for research, technolog-ical development and demonstration under the NMI3-II Grant No. 283883. M. M. was partly supportedby Marie Sklodowska Curie Action, International CareerGrant through the European Commission and SwedishResearch Council (VR), Grant No. INCA-2014-6426. ∗ peter.babkevich@epfl.ch[1] H. E. 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Gannarelli, K. Prokes, A. Podlesnyak,T. Str¨assle, L. Keller, O. Zaharko, K. W. Kr¨amer, andH. M. Rønnow, Science , 1416 (2012).[13] P. Babkevich, A. Finco, M. Jeong, B. Dalla Pi-azza, I. Kovacevic, G. Klughertz, K. W. Kr¨amer, C. Kraemer, D. T. Adroja, E. Goremychkin, T. Unruh,T. Str¨assle, A. Di Lieto, J. Jensen, and H. M. Rønnow,Phys. Rev. B , 144422 (2015).[14] M. Le, D. Quintero-Castro, R. Toft-Petersen, F. Groitl,M. Skoulatos, K. Rule, and K. Habicht, Nucl. Instrum.Methods Phys. Sect. A , 220 (2013).[15] We note that the different techniques gave slightly differ-ent values of T N , within 10 mK. We attribute this to dif-fering thermometer calibrations and possibly small ther-mal gradients between sample and thermometer. Thisdoes not affect the extracted exponents nor the main con-clusions of this Letter.[16] S. W. Lovesey, The Theory of Neutron Scattering fromCondensed Matter (Clarendon Press, Oxford, 1986).[17] F. Kagawa, K. Miyagawa, and K. Kanoda, Nature ,534 (2005).[18] A. 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It is well known that systems of the Li R F family crys-tallize in a scheelite CaWO type structure. To verify ourLiYbF sample, we have performed careful measurementsusing D2B diffractometer in the paramagnetic phase at10 K – well above magnetic ordering temperature. Ourresults are presented in Fig. 5. A good fit to the diffrac-tion pattern was found using Rietveld method in the Full-prof package [23] which allows us to extract the atomicpositions and B iso isotropic Debye-Waller factors. In thecase of Li, it was not possible to accurately refine the B iso parameter and therefore it was fixed in the fitting.The detailed refinement, described in Table I, is in ex-cellent agreement with that reported previously on thesystem in Ref. [24]. Magnetic structure
Having confirmed the crystallographic structure ofLiYbF and the absence of impurities, we next considerthe arrangement of the magnetic moments below T N .Previous study of LiErF found that magnetic momentsare arranged into a bilayer structure where the momentsconnected by I -centering are antiparallel [12]. Indeed,solving the Hamiltonian in the mean-field approxima-tion quickly converges to this structure. Our previousmean-field simulations of LiYbF and LiErF indicatethat the groundstate magnetic structures should be thesame, with the only difference that the moment on Yb ion is expected to be smaller than that on Er [13].Neutron diffraction data from studies of LiErF is plot-ted in Figs. 6(a) and (b). Measurements were collectedusing DMC diffractometer with λ = 2 .
457 ˚A. Antiferro-
20 40 60 80 100 120 140−1000100200300400500 θ (deg) C o un t s ( a r b . ) ObservedStructure modelDi ff erence FIG. 5. (Color online) High-resolution neutron powderdiffraction measurements using D2B diffractometer. Data col-lected at 10 K and refined to the structural model describedin the text. Incident neutron wavelength was 1.594 ˚A. magnetic ordering in LiErF sets in below 375 mK [12].In order to obtain purely the magnetic contribution tothe signal, we have subtracted measurements collectedabove 900 mK. Surprisingly, some of the stronger peaksare found to sit on broad humps which could indicatesome short-range correlations in the system but couldalso be some artifacts related to the background. Theorigin of these cannot be elucidated further.In comparison, data collected using D1B at λ = 2 .
52 ˚Aexamining LiYbF show a slowly varying backgroundwith no signs of any additional features. We notice fromthe LiErF and LiYbF diffraction patterns shown inFigs. 6(b) and (d) that the relative intensities of (100)and (102), close to 5.1 and 3.7 ˚A, respectively, are clearlydifferent for the two systems. The ratio of σ (100) to σ (102) intensity in LiErF is 3.36(7) and in LiYbF is1.241(4).Since the incident neutron wavelengths are very sim-ilar and the instrumental resolution is not very differ-ent for the two diffractometers we would have expectedfrom mean-field simulations that the magnetic powderpatterns are nearly the same. Intriguingly this does notappear to be the case. Performing Rietveld refinementof the magnetic structure for LiYbF gives a better fitwhen the moments are allowed to rotate to be alongthe [110] direction. The simulations for the two differ-ent moment directions is shown in Figs. 6(e) and (f).In the model where the moments are along [100], the σ (100) /σ (102) = 4 .
14 – close to what we find for LiErF .Repeating this analysis for moments along [110], we findinstead σ (100) /σ (102) = 1 .
35, viz LiYbF . Magnetic representation analysis
The magnetic structures of LiYbF and LiErF can bedescribed by the magnetic propagation wavevector k =(1 , , I /a , thelittle group G k contains 8 symmetry elements ( g – g )listed in Table II. The magnetic representation Γ mag of G k reduces to Γ mag = 2Γ + Γ . Both Γ and Γ are twodimensional and their characters are given in Table II.Using Basireps [25], we obtain basis functions ψ , shown Atom site x y z B iso (˚A ) Li 4a 0.0000 0.2500 0.1250 0.80Yb 4b 0.0000 0.2500 0.6250 0.09(3)F 16f 0.2186(4) 0.4169(4) 0.4571(2) 0.43(4)TABLE I. Nuclear structure refinement of LiYbF shownin Fig. 5. The Bragg peaks were indexed by I /a spacegroup with lattice parameters of a = 5 . c =10 . ν g g g g g g g g ! ! − ! − ! ! ! −
11 0 ! −
11 0 ! ! − − ! i 00 − i ! − i 00 i ! ! − − ! − ii 0 ! − i 0 ! TABLE II. Character table of the little group G k showing how the irreducible representations Γ ν transform according tosymmetry operations g , . . . , g . Using the Seitz notation, the symmetry operations are defined as, g = { | , , } , g = { z | / , , / } , g = { +00 z | / , / , / } , g = { − z | / , / , / } , g = {− | , , } , g = { m xy | / , , / } , g = {− +00 z | / , / , / } and g = { − z | / , / , / } . ν n ( ψ x , ψ y , ψ z ) ( ψ x , ψ y , ψ z )1 1 (1 , ,
0) (0 , , , ,
0) ( − , , , − ,
0) ( − , , , ,
0) (0 , − , , ,
0) (0 , , − i)2 2 (0 , , i) (0 , , − ψ of irreducible representation Γ ν for ions situated at 1. ( x, y, z ) and 2. ( − y + 3 / , x + 1 / , z +1 / in Table III for two symmetry-related sites. The two sitescreate an extinction condition which makes is possible todistinguish between magnetic moment directions even inthe tetragonal cell with powder averaging. In general,the n th moment m n can be expressed as a Fourier series, m n = X k S k n e − i k · t , (2)where t is the real space translation vector. The vectors S k n are a linear sum of the basis vectors such that, S k n = X m,p c mp ψ k νmp , (3)where coefficients c mp can be complex. We label ν asthe active irreducible representation Γ ν , m = 1 . . . n ν ,where n ν is the number of times Γ ν is contained in Γ mag .The index p labels the component corresponding to thedimension of Γ ν .In the case of LiYbF , the moments lie in the ab plane,therefore Γ is active (see Table III). However, the neu-tron data does not allow us to uniquely identify the mag-netic ordering as any of the four basis vectors can refinethe measured data. All four arrangements result in mo-ments which rotate by 90 ◦ along c , as for example shownin Fig. 7(a). It is also possible to use a combination oftwo basis vectors, such as 1 and 3 or 2 and 4 to describea collinear magnetic structure as shown in Figs. 7(b–d).However, it is not possible to refine the measured datafor LiErF using the same combination of basis vectorswhich appear to describe LiYbF . Indeed a combinationof all four basis vectors, as depicted in Fig. 7(e), is needed ion 10 B B B B (c) 10 B (c) 10 | B (s) | Er 58.1 -0.536 -0.00625 -5.53 -0.106 23.8(3.4) (0.032) (0.00041) (0.31) (0.0061) (1.5)Yb 457 7.75 0 -9.78 0(5.2) (0.12) (0) (0.65) (0.0094) (0)TABLE IV. Crystal field parameters of LiYbF and LiErF compounds determine by inelastic neutron scattering. Typ-ically, a coordinate system with B ( s ) = 0 is chosen, whiletwo possible equivalent coordinations of R ion by F ions givedifferent sign of B ( s ). After [13]. to describe the best possible solution for LiErF reportedin Ref. [12] which sees the moments along the a axis Crystal field interaction
The in-plane anisotropy in LiErF and LiYbF islargely determined by the single-ion crystal field anddipolar interactions. We would expect that as the mag-netic moment size is very similar in LiErF and LiYbF ,the dipolar interactions in the two systems do not dif-fer significantly. One possible arrangement in LiYbF isshown in Fig. 7(c) where the moments are all rotated by45 ◦ in the basal plane with respect to the LiErF mag-netic structure. This structure (amongst others) fits wellthe measured data. From the crystal field whose Hamil-tonian for ¯4 point group symmetry at the R site is givenby, H CEF = X l =2 , , B l O l + X l =4 , B l ( c ) O l ( c ) + B l ( s ) O l ( s ) . (4)The later B l terms play a role in the planar anisotropywhere B ( c ) term is found from experiments to belargest, see Table IV. Classically, one obtains the energyof rotating a moment of size J in the plane by angle φ to be E = J B ( c ) cos(4 φ ). Hence, the minimum in en-ergy for different signs of B ( c ) is found to be 45 ◦ apart.While this appears to be a simple explanation for pre-ferred moment direction, a strong crystal field interactionwould result in an Ising-like system, which is not whatwe observe experimentally. Furthermore, dipolar inter- C o un t s ( a r b . ) (a) LiErF C o un t s ( a r b . ) (b) LiErF C o un t s ( a r b . ) (c) LiYbF C o un t s ( a r b . ) (d) LiYbF d ( ˚ A) C o un t s ( a r b . ) (e) [100][110] d ( ˚ A) C o un t s ( a r b . ) (f) [100][110] FIG. 6. (Color online) Neutron powder diffraction data recorded for (a,b) LiErF and (c,d) LiYbF plotted as a functionof d -spacing. In each case measurements in the paramagnetic phase were used to subtract the nuclear contribution to thepatterns leaving purely the magnetic Bragg peaks. Grey vertical lines under the patterns show the indexation of the reflection.Simulations assuming collinear magnetic structures with moments along [100] and [110] directions are plotted in panels (e,f).FIG. 7. (Color online) Possible magnetic structures of Γ irreducible representation. (a) Magnetic structure from just the firstbasis vector in Table III. (b) – (d) show arrangement of moments by combining two basis vectors. (e) Appropriate sum of allbasis vectors to form magnetic structure which best describes LiErF . actions are not expected to favor such ordering. There-fore, further theoretical work is necessary to examine themechanism by which the dipolar-coupled antiferromag-nets order. Conclusion
While it is entirely possible that the diffraction pat-terns can be also described by other models includingones where moments are non-collinear, qualitatively ourexperimental data appears to suggest that the ground-state magnetic structure of LiErF is not the same asLiYbF4