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Magnetic ordering, spin waves, and Haldane gap excitations in (Nd_x Y_{1-x})_2 Ba Ni O_5 linear-chain mixed-spin antiferromagnets

Linear-chain nickelates with the composition (Nd_x Y_{1-x})_2 Ba Ni O_5 (x=1, x=0.75, x=0.5, and x=0.25) are studied in a series of neutron scattering experiments. Powder diffraction is used to determine the temperature dependence of the magnetic structure in all four systems. Single-crystal inelastic neutron scattering is employed to investigate the temperature dependence of the Haldane-gap excitations and low-energy spin waves in the x=1 compound Nd_2 Ba Ni O_5. The results of these experiments are discussed in the context of the ``Haldane chain in a staggered field'' model for R_2 Ba Ni O_5 systems, and quantitative agreement with theory is obtained.


I. INTRODUCTION
The quantum-disordered ground state and the famous Haldane energy gap in the magnetic excitation spectrum 1 have kept 1-dimensional (1-D) Heisenberg antiferromagnets (AF) at the center of attention of condensed matter physicists for the last 15 years. Among the more recent developments are studies of such systems in external magnetic fields. [2][3][4][5] It was found that in sufficiently strong fields one of the three Haldane-gap modes undergoes a complete softening. The result is a transition to a new phase with long-range antiferromagnetic correlations. 6,7 The effect of a staggered field H π , to which a Haldane spin chain is most susceptible, is expected to be no less dramatic. Unfortunately, this problem has been given much less attention in literature, simply because such conditions are almost impossible to realize in an experiment. An arbitrarily small staggered field will induce a non-zero staggered magnetization in the Haldane chain. An intriguing question is how this induced order will affect the triplet gap excitations. Until recently the only 1-D integer-spin AF whose behavior was described in terms of a "staggered field" is NENP, one of the best-known Haldane-gap materials [Refs. 8,9 and references therein]. Due to the existence of two non-equivalent Ni sites in each chain, a weak effective staggered field is induced in NENP by a uniform external field. 10,11 While the main effect on the spin dynamics is that of the uniform field, the staggered component produces a small yet observable effect. 11,12 The recent discovery of coexistence of Haldane gap excitations and long-range magnetic order in R 2 BaNiO 5 (R=magnetic rare earth) compounds [13][14][15] presents a unique opportunity to investigate the effect of a staggered field experimentally without having it obscured by the response of the system to a uniform component. According to our current understanding of the R 2 BaNiO 5 's, long-range magnetic ordering does not destroy the quantum-disordered ground state of individual S = 1 Ni-chains in these quasi-1-D systems. It is therefore essential that the Ni 2+ ions carry an integer, as opposed to half-integer spin, and qualitatively different behavior is expected for isostructural half-integer spin materials like Nd 2 BaCoO 5 . 16,17 The non-zero staggered magnetization that appears on the intrinsically disordered S = 1 Ni-chains below the Néel temperature T N is viewed as being induced by an effective staggered exchange field generated by the ordered R-sublattice. 13,18 This physical picture will be referred to as the "Haldane Chain in a Staggered Field" (HC/SF) model for R 2 BaNiO 5 compounds.
In the HC/SF model, even in the magnetically ordered state, much of the dynamic spin correlations are contained in Haldane gap modes propagating on the Ni chains. Below T N these excitations coexist with low-energy order-parameter excitations, i.e., conventional spin waves, that involve correlated fluctuations of both Ni and rare earth moments. The first experimental confirmation of this was obtained in inelastic neutron scattering experiments on Pr 2 BaNiO 5 , 13 where Haldane gap modes were clearly seen both above and below the Néel temperature. At all temperatures these excitations have a purely 1-dimensional dynamic structure factor, and show no sign of softening at the 3-D AF zone-center at T = T N . Pr 2 BaNiO 5 samples are extremely difficult to prepare, and their magnetic properties are not always reproducible. For example, while magnetic ordering at T N = 24 K has been clearly observed in single crystals, 13 for reasons yet unclear, no magnetic transition was observed in powders. 19 It turned out that Nd-based linear-chain nickelates with the general formula (Nd x Y 1−x ) 2 BaNiO 5 are more practical as model systems for neutron studies. On one side of this series (x = 0) is Y 2 BaNiO 5 , a well-studied Haldane-gap S = 1 antiferromagnet with a spin gap ∆ ≈ 9 meV. 20-24 Y 2 BaNiO 5 remains magnetically disordered even at low temperatures. At the other end we have Nd 2 BaNiO 5 (x = 1) that, thanks to the presence of magnetic Nd 3+ ions, orders antiferromagnetically at T N = 48 K. As in the case of Pr 2 BaNiO 5 , Haldane-gap modes in Nd 2 BaNiO 5 were found to survive in the magnetically ordered phase. 14 Similar behaviour was seen in x = 0.75, x = 0.5 and x = 0.25 species, where the ordering temperatures are 39 K, 30 K, and 19 K, respectively. 15 For all Nd concentrations at T > T N the Haldane gap modes are virtually indistinguishable from those seen in Y 2 BaNiO 5 . As the temperature is decreased through T N however, the gap energy ∆ increases roughly linearly with T N − T . This remarkable behaviour is in agreement with theoretical predictions based on HC/SF model. 18,25 To date, all inelastic measurements on (Nd x Y 1−x ) 2 BaNiO 5 materials were performed on powder samples. The existing magnetic structure (diffraction) data are not accurate or not complete enough to be discussed in relation to our model. Indeed, previous single-crystal experiments on the Nd-system 26,27 were analyzed assuming that the Nd and Ni moments are collinear at all temperatures below T N . It was later shown that the actual structure is canted, 28 but no detailed temperature dependencies derived from the canted model were reported. For the mixed Nd-Y systems only very preliminary results obtained by analyzing only three magnetic reflections are available. 15 In this work we continue our studies of (Nd x Y 1−x ) 2 BaNiO 5 compounds using both elastic and inelastic neutron scattering. Powder neutron diffraction is used to investigate the temperature evolution of the magnetic structure in four (Nd x Y 1−x ) 2 BaNiO 5 species with x = 1, x = 0.75, x = 0.5 and x = 0.25, respectively. Inelastic neutron scattering is then employed to study the temperature dependence of the Haldane-gap excitations in the Nichains as well as the low-energy order-parameter fluctuations (spin waves) in single-crystal Nd 2 BaNiO 5 samples. The connection between static and dynamic properties is discussed in the framework of the HC/SF model. Some of the results presented below were briefly discussed in Ref. 29.

II. EXPERIMENTAL PROCEDURES
The crystal structure of R 2 BaNiO 5 compounds is discussed in great detail elsewhere. 30 All (Nd x Y 1−x ) 2 BaNiO 5 systems crystallize in an orthorhombic unit cell, space group Immm, a ≈ 3.8Å, b ≈ 5.8Å, and c ≈ 11.7Å. 15 The S = 1 Ni chains are formed by orthorhombically distorted NiO 6 octahedra that are lined up in the [100] crystallographic direction, sharing their apical oxygen atoms. The R sites (occupied at random by Y or Nd) are positioned in between these chains. The site symmetry for the R 3+ is low and the electronic ground state for Nd 3+ is a Kramers doublet.
In powder diffraction experiments on the x = 0.75, x = 0.5 and x = 0.25 compounds 5 we used the same samples (roughly 15 g each) as in previous inelastic studies. 15 A new Nd 2 BaNiO 5 (x = 1) powder sample of comparable mass was prepared particularly for the present study using the solid state reaction method. Powder neutron diffraction experiments were carried out at the high flux reactor of the Institut Laue-Langevin (Grenoble), on the 400 cells position sensitive detector diffractometer D1B for the x = 0.75, x = 0.5 and x = 0.25 compounds and on the 1600 cells diffractometer D20 for the x = 1 system. The patterns were recorded in the temperature range 2 -50 K, using a liquid helium cryostat.
The wavelengths λ = 2.526Å on D1B and λ = 2.400Å on D20 were provided in both cases by focusing pyrolitic graphite monochromators. The samples were enclosed in cylindrical vanadium containers. The analysis of the powder patterns was performed by Rietveld profile refinement using the software FULLPROF.  In the x = 1 sample, thanks to the relatively strong magnetic signal, a simultaneous refinement of nuclear and magnetic structures could be performed. A typical powder scan for this system is shown in Fig.1(a). For the other three compounds, where the magnetic signal was weaker, a different approach was used. For powder scans collected at T < T N the magnetic contribution to scattering was isolated by subtracting the nuclear background measured just above the ordering temperature. Several typical data sets obtained in this fashion are shown in Fig.1(b-d).
In the x = 0.25 sample, in addition to the above mentioned magnetic peaks at halfinteger positions, extra intensity below T N was also observed at the positions of nuclear peaks [ Fig.1(d)]. One possible explanation for this is the presence of an extra magnetic component in the structure of this compound. This however appears not to be the case. We were unable to reproduce the observed pattern with any reasonable spin model. In addition, the intensities of these extra reflections seem to increase with increasing scattering angle, strongly suggesting that they are of nuclear, rather than magnetic origin. In our analysis we have assumed that theses extra peaks represent a lattice distortion that is induced by magnetic ordering. In the determination of the magnetic structure of this sample these reflections were therefore ignored.
According to the analysis of the crystallographic space group of R 2 BaNiO 5 compounds (Immm), only two types of magnetic structure with propagation vector [1/2 0 1/2] and non-zero magnetic moment on both Ni and R sites are possible. 34 In the structure realized in Er 2 BaNiO 5 all spins are confined to the (a, b) crystallographic plane. As shown by several previous studies, 26,28 it is the other alternative that is realized in Nd 2 BaNiO 5 : both the Ni 2+ and Nd 3+ moments are in the (a, c) plane of the crystal (Fig. 2). We used this model to fit the diffraction spectra measured at each temperature for each sample. The adjustable parameters were the magnetic moments m (Ni) and m (Nd) of the Ni 2+ and Nd 3+ ions, respectively, as well as the angles φ (Ni) and φ (Nd) these moments form with the c axis of the crystal. The temperature dependencies of energies and intensities of the Haldane-gap excitations were studied by inelastic thermal neutron scattering. One of our objectives was to use the intrinsic polarization dependence of the neutron scattering cross section to distinguish between the three individual modes, expected to be present in the Haldane multiplet. The most serious obstacle in previous powder measurements was the presence of two crys-tal field (CF) excitations associated with Nd 3+ that appear at about 18 meV and 24 meV, respectively. 14 Unlike the highly dispersive Haldane excitations, the dispersionless CF modes do not suffer an intensity penalty upon powder averaging the corresponding dynamic structure factors. The Haldane modes that appear at ≈ 10 meV energy transfer at T = 50 K and move towards higher energies at lower temperature are thus very difficult to separate from the intense CF background. In a single crystal this problem may be overcome. First, the intensity of Haldane modes relative to those of CF excitations is much larger in this case. Second, in a single crystal the background can be directly measured at scattering vectors slightly off the 1-D zone-center, where the Haldane excitations move out of the scan range thanks to their steep dispersion. In our case the background was measured at Q = (1.4, 0, 0) and Q = (0.4, 0, 4.2). Since it was previously found that the CF modes are slightly temperature-dependent, the background measurements were performed separately at each temperature. Figure  One problem that had to be dealt with was the presence of a spurious peak at roughly 18 meV energy transfer (see Fig. 5). We have identified this spurion as being caused by higher-order scattering (2k i ) in the monochromator, incoherent and/or Bragg scattering in the sample, and higher-order (3k f ) scattering in the analyzer. In the analysis described below the "dangerous" energy range has been excluded from all data sets.
Ideally, for a complete determination of the mode polarization, at each temperature one would analyze the inelastic scans collected at both wave vectors simultaneously, using a model cross section that would include three modes with separate gap values and polarization factors. In the present study however, as the structure of the observed peak is totally smeared by the broad instrumental resolution, using a three-component model cross section results in an over-parameterized problem. Instead we analyzed all the measured inelastic scans separately using a single-mode cross-section, as was previously done for Pr 2 BaNiO 5 . 13 The model dynamic structure factor was written in the "double-Lorentzian" form: In this formulaq is the projection of the scattering vector q onto the chain axis, and measured relative to the 1-D antiferromagnetic zone-center; ∆ is the gap energy; c is the spin wave velocity along the chains; Γ is the intrinsic energy width of the excitations; and ξ is the realspace dynamic spin correlation length. Since all measurements were performed atq = 0, the parameters ξ and c are only needed to properly take into account focusing effects. In our analysis ξ and c were therefore fixed to the values previously measured in Pr 2 BaNiO 5 : Scanning through the 4 meV peak in different Brillouin zones led us to the conclusion that the mode is a single-ion excitation with no unit-cell structure-factor and no apparent polarization-dependence of intensity. The accessible range of momentum transfers was not sufficient for an accurate study of the form-factor dependence of the intensity.
What makes the 4 meV mode rather interesting is its temperature dependence. As was previously established in inelastic neutron scattering experiments on powder samples, 14 this excitation is visible only at T < T N . As the Néel temperature is approached from below the excitation energy decreases, the width increases dramatically (Fig.11), and at T = T N the mode appears to merge with quasielatic scattering athω = 0. We have utilized setups I and IV to measure the temperature dependence of the excitation energy for temperatures up to T = 35 K, where the inelastic peak is broad, yet still easily identified. Constant-Q scans were analyzed with a damped-oscillator dynamic structure factor: that was convoluted with the gaussian energy-resolution of the spectrometer (solid lines in Fig.11). In Eq. 3 n(ω) is the Bose factor and I 0 is the structure factor of the excitation.

A. Magnetic ordering
In previous resonant magnetic X-ray diffraction studies 27  Garcia-Matres et al. 19 were probably the first who analyzed their magnetic susceptibility data treating only the Ni-R interactions at the MF level. The experimentally measured temperature dependence of the Ni-order parameter was built into this model as an empirical function. The HC/SF model provides a physical justification for this approach: 29 the bare magnetization curve for the Ni-subsystem is to be taken in the form of the staggered magnetization function for an isolated quantum-disordered Haldane chain. This "semi-quantum" model differs qualitatively from the standard MF model for classical magnets. In the latter, at T → 0 all sublattices become fully saturated as the bare single-ion susceptibility diverges as 1/T . In contrast, the bare staggered susceptibility χ π of a Haldane spin chain remains finite at T = 0, and the Ni-sublattice need not be fully saturated. The "semi-quantum" model can thus account for the reduced ordered moment on the Ni-sites without assuming an improbably small g-factor for Ni 2+ . Moreover, it will predict qualitatively different temperature dependencies, since the staggered susceptibility of a Haldane spin chain, unlike the single-ion susceptibility, is expected to be almost T -independent at T < ∼ ∆ (∆ ≈ 9 meV, or ≈ 105 K in R 2 BaNiO 5 compounds).
It has to be noted that treating interchain coupling at the MF or RPA (Random Phase Approximation) level is, in itself, not a new idea, but a well-established technique 35 . In their pioneering work on CsNiCl 3 , the first Haldane-gap material studied experimentally, Buyers et al. 36 , and Affleck 37 implemented this approach to explain magnetic ordering and calculate the spin wave dispersion relations. The main difference between CsNiCl 3 and R 2 BaNiO 5 is that exchange coupling between individual Haldane chains is direct in the former system, and mediated by the rare earth ions in the latter. For directly coupled Haldane spin chains, the magnitude of interchain interactions must exceed some critical value in order for the system to order magnetically 37 . In a MF description of CsNiCl 3 or any other system with directly couple quantum chains, the temperature dependence of the ordered moment is defined by the intrinsic temperature dependence of the susceptibility of individual chains. In contrast, in our case of R 2 BaNiO 5 compounds, magnetic ordering is driven by the 1/T -divergent susceptibility of the rare earth subsystem. At sufficiently low temperature long-range order will therefore occur for arbitrary small R-Ni interactions, and in the case of T N < ∼ ∆, we can use the approximation where the bare magnetization curve of isolated chains is T -independent.
An important results of this work is that in all samples studied m (Ni) , the induced staggered cmoment on the Ni-chains, appears to be a universal function ofm (Nd) ≡ m (Nd) x.
In other words, the Ni magnetization explicitly depends only on the average moment on the R-sites, and not the actual temperature, as illustrated in the plot of m (Ni) vs.m (Nd) in Fig. 13. To emphasize the significance of this fact we reiterate that the ordering temperatures in the materials differ by more than a factor of two. From Fig. 13 we see that even with fully ordered Nd 3+ moments the Ni-chains are not fully polarized and the curve never levels off completely. In the context of the HC/SF model, apart from the scaling of the absciss, Fig. 13 is nothing else but the staggered magnetization function M(H π ) for an isolated Haldane chain of Ni 2+ spins, that in the studied temperature range T = 0 − 50 K is expected to be almost T -independent.
The role of disorder in Y-diluted systems is not too important, at least, in systems where the Nd concentration is not too small. The intrinsic dynamic spin correlation length of Haldane spin chains is rather large, of the order of six lattice repeats. As long as the mean distance between the Nd 3+ ions is smaller than this length scale, the Ni-chains effectively see a homogeneous staggered moment on the R-sites, despite the "holes" that are present wherever Nd 3+ is replaced by Y 3+ . The argument, while clearly valid at T ≈ T N , should be taken with some caution when applied to a system deep in the ordered phase. In this regime the Haldane gap energy increases, and, as a consequence of that the dynamic correlation length in the chains decreases. Even in this case we can expect the effect of disorder to be averaged out, thanks to the simple geometrical fact that every Ni 2+ ion is coordinated to four R-sites.
We can now use the experimentally determined staggered magnetization function for Haldane spin chains in (Nd x Y 1−x ) 2 BaNiO 5 to write down the self-consistent MF equations for our "semi-quantum" theory. For the Ni-sublattice these equations are: Here gµ B and S = 1 are the gyromagnetic ratio and spin of Ni 2+ , respectively; H (Ni) is the effective exchange field that acts on the Ni-chains and is generated by the R-sublattice; and α is an effective MF coupling constant. The factor 2 in Eq. 5 reflects the fact that there are two Nd atoms for every Ni atom in the chemical formula, while the magnetizations m (Ni) and m (Nd) are normalized per site. For convenience M(H (Ni) ) in Eq. 4 is approximated with the following purely empirical function: The coefficients A = 1.27(5) and B = 1.06 (13) are obtained by fitting this formula to the bulk of experimental data in Fig. 13 (solid line). To write down the remaining MF equations for the R-sublattice we shall approximate the bare magnetization curve for the Nd 3+ ions with the Brillouin function, as is appropriate for a magnetic ion with a doublet ground state: ).
In this formula H (Nd) is the effective exchange field acting on the rare earth ions, κ is Boltzmann's constant and m (Nd) 0 is the effective magnetic moment for Nd 3+ .
Ideally, the saturation moment m 0 should be indepent of temperature or Nd concentration in the sample. In our particular systems however, the saturation moment of the Nd 3+ ions steadily decreases by roughly 25% as x changes from 1.0 to 0.25. Eq. 7 is therefore no more than a crude approximation, and a 25% accuracy is the best we can expect from our model. Indeed, in using the Brillouin function to describe the rare earths, we have totally neglected the higher-energy electronic states of these ions. Not only do they contribute to the temperature dependence of R-magnetization, but also may give rise to a T -independent Van-Vleck contribution to single-ion susceptibility. Unfortunately, without knowing the electronic structure of R 3+ in R 2 BaNiO 5 in detail, we can not take these effects into account rigorously. Instead we shall use Eq. 7 with separate values for m 0 for each sample, equal to the saturation magnetization of the Nd sublattice.
Equations  Fig. 13 and obtain the Ni-order parameter as a function of effective exchange field, i.e., the actual staggered magnetization curve for a Haldane spin chain ( Fig. 13, top axis). Differentiating Eq. 6 at H (Ni) → 0 we obtain the staggered susceptibility χ π = 1.2 · 10 −5 µ B /Oe. The staggered magnetization curve for an isolated S = 1 Heisenberg AF chain has not been calculated to date, and we can not directly compare our results to any numerical simulations. The zero-temperature staggered susceptibility χ π , on the other hand, can be deduced from the numerous theoretical predictions for the dynamic structure factor at the 1-D AF zone-center S(π, ω), to which it is related through the Kramers-Kronig relation and the fluctuation-dissipation theorem. In the single-mode approximation 38,39,9 S(π, ω) = − 4 3 H L∆ δ(ω − ∆). Here and H L is the ground state energy per spin. According to Monte Carlo simulations (see, for example, Refs. 40,41), H L ≈ −1.4J ≈ 3.42∆. The expression for the staggered susceptibility is then readily obtained via χ π = 2 ∞ 0 dω S(π,ω) ω and is equal to: In all the R 2 BaNiO 5 systems studied so far with inelastic neutron scattering the gap energies are equal to 9 meV to within experimental error, which gives χ π ≈ 2.3 · 10 −5 µ B /Oe. This value is almost twice as large as our experimental estimate. While not spectacular, this level of consistency is quite acceptable, considering all the approximations and simplifications that had to be made. Most of the uncertainty is related to the large experimental error in the initial "low-field" part of the magnetization curve in Fig. 12 where small ordered moments were derived from the measurements in the vicinity of T N , and the error bars are rather large.
Some systematic error is introduced by approximating the measured curve with the expression (6). We have also totally neglected the temperature dependence of the magnetization function for the Haldane spin chains, small as it may be at κT < ∼ 0.5∆, and approximated that for the rare earth ions by a simple Brillouin function. The actual gyromagnetic ratio for Ni 2+ that enters Eq. 9 squared, is not known from independent measurements either. A half-order-of-magnitude agreement is indeed the most we can expect. What is important is that our model appears to be self-consistent and is based on the same concepts as those previously used to explain the persistence and T -dependence of Haldane-gap excitations in R 2 BaNiO 5 materials.

B. The 4 meV mode
The excitation that we see at 4 meV energy transfer was first observed in powder experiments, 14 and attributed to a CF transition in Nd 3+ . A similar feature was also seen in recent AFMR measurements. 42 As our new single-crystal neutron data clearly demonstrate, the 4 meV mode has indeed no dispersion and its intensity is Brillouin-zone independent, so it indeed is a local, i.e., single-ion excitation(s). Despite that, we suggest that this mode is the actual order-parameter excitation (spin wave), and corresponds to flipping a single Nd 3+ moment in the effective staggered field projected by the Ni-sublattice. The energy of such an excitation is given by: The excitation energy should thus scale as the ordered moment of the Ni-sublattice. The solid line in Fig. 12 is drawn using Eq. 10, our experimental data for m (Ni) (T ) in Nd 2 BaNiO 5 , and α = 0.84 · 10 5 Oeµ −1 B determined from our MF analysis of the magnetic structure. An excellent agreement with inelastic measurements is apparent.
One possible explanation for the fully localized nature of the low-energy spin waves is a strong anisotropy in the effective exchange interaction between Ni 2+ and Nd 3+ . We suggest that this interaction is of Ising type, i.e., can be written as Unfortunately, within the MF model one can not make any predictions regarding the strongly T -dependent damping of the 4 meV mode. Without going into further speculations, we would only like to point out that the measured magnitude and temperature dependence of γ is very similar to that of the temperature-induced broadening of Haldane-gap excitations, as measured in Y 2 BaNiO 5 . 23 One possibility is that these quantities may actually be directly related and represent the same relaxation process. is the temperature dependence of integrated intensities measured at Q = (1.5, 0, 0) and Q = (0.5, 0, 4.2) (Fig. 8). To within experimental error these intensities are the same at all temperatures. At no instance do we see any distinct splitting in the energy gap. If we assume that the modes remain polarized along the principal crystallographic directions, as dictated by the local symmetry of the Ni-sites, we are forced to conclude that the three components of the triplet evolve with temperature in exactly the same way.
The assumption however is expected to be valid only at T > ∼ T N . As the magnetic order-induced shift in the Haldane gap energies becomes larger than the initial anisotropy- In this approach an isolated Haldane spin chain is characterized by the space-dependent vector of local staggered magnetization φ(x, t). The Lagrangian of the system is written as a series expansion in φ(x) and its derivatives: The quadratic terms describe a triplet of degenerate non-interacting gap excitations (energy gap ∆) with a linear dispersion (spin wave velocity v). A non-zero staggered-field dependence of the excitation energies appears upon introducing the λ| φ| 4 -term, that represents pairrepulsion between magnons, and other higher-order terms. In the presence of a staggered field, the change in gap energies, defined as δ(H π ) ≡ (Nd x Y 1−x ) 2 BaNiO 5 systems. 18,25 The data obtained in the present work, particularly the more accurately measured temperature dependence of the Ni-sublattice magnetization, can be used to analyze the relation between ∆ and m (Ni) in greater detail. In order to do so, we first have to compensate for the intrinsic weak T -dependence of the Haldane gap energy. This is done by replacing ∆ 0 by ∆ 0 (T ), the temperature-dependent gap in the absence of staggered field. For ∆ 0 (T ) we can take the Haldane gap measured as a function of temperature in Y 2 BaNiO 2 , that, for 22 practical purposes, we shall approximate by the following empirical formula: 9,44 ∆ 0 (T ) = ∆ 0 (0) + βT exp(−∆ 0 (0)/κT ) (12) The coefficients ∆ 0 (0) = 9.21 meV and β = 1.0 meV 2 /K were obtained by fitting this form to the experimental data for Y 2 BaNiO 2 . 22,23 This formula is used to plot δ(H π ) as a function of [m (Ni) ] 2 for all four (Nd x Y 1−x ) 2 BaNiO 5 systems studied (Fig. 14). A perfect data collapse is obtained. Also apparent is the slow upward curvature of this function that could not be clearly seen when only powder data were available.