Spin-1/2 Heisenberg antiferromagnetic (AFM) chains, in which the local spin environment periodically alternates in orientation, host magnetic properties that dramatically differ from their uniform counterparts. In systems such Cu(C6D5COO)2 • 3D2O and Cu(pym)(H2O)2(NO3)2 (pym = pyrimidine C4H4N2), twofold staggered g tensors and the Dzyalozhinskii-Moriya (DM) interaction result in a staggered internal field perpendicular to the applied field, leading to a field-induced spin gap [1][2][3][4]. This is in contrast to the gapless excitation spectra observed in conventional linear chains. The sine-Gordon model of quantum-field theory reproduces the staggered chain's excitation spectrum, which contains soliton and breather modes, plus a ∼ H 2/3 field dependence of its energy gap [5,6].
Despite such unique phenomena displayed by nonlinear S = 1/2 chains, investigations into the effects of an alternating spin environment on AFM S = 1 chains are notably lacking. In addition to the intra-and interchain energies J0 and J, the introduction of single-ion anisotropy (SIA) energy D serves as an additional tuning parameter for S = 1 systems. When considering a linear chain, the interplay between these parameters results in an already diverse set of possible magnetic ground states [7,8]. In the ideal isotropic, onedimensional limit, the Haldane gapped phase emerges, which hosts a topologically protected quantum disordered ground state [9]. With easy-plane anisotropy (D > 0) as the 2469-9950/2025/111(1)/014421 (11) Published by the American Physical Society D/J0 ratio is increased, the Haldane phase is driven into a quantum paramagnetic phase. Increasing J drives the system into an XY-AFM ordered phase hosting a field-induced BoseEinstein condensation of magnons [10,11]. In the easyaxis (D < 0) case, Ising AFM order is induced [8]. Further theoretical efforts have extended this model to explore the effects of rhombohedral anisotropy [12], non-Heisenberg exchange [13], biquadratic exchange [14], and alternating exchange bonds [15].
Here we turn our attention to Ni(pym)(H2O)2(NO3)2, an S = 1 analog of the S = 1/2 staggered chain; in addition to the aforementioned staggered g tensor, the alternating orientation of local octahedra now presents the possibility of a periodically alternating SIA axis. The alternating SIA, which lacks a global axis in these systems, is expected to act in competition with the exchange interaction. While a staggered SIA axis has attracted attention in the study of Glauber dynamics [16,17] and domain-wall dynamics [18] in singlechain magnets, the ground state of quantum spin-chains with a staggered SIA axis does not appear to have been well explored theoretically or experimentally. In [Ni(μN3(bmdt)(N2]n(DMF)n (bmdt =N,N'-bis(4-methoxylbenzyl)-diethylenetriamine, DMF =N,N-dimethylformamide), weak ferromagnetism and unusual spin dynamics were reported. However, neutron scattering data were not available to solve the magnetic structure or determine the spin Hamiltonian [19]. In threedimensional and quasi-two-dimensional systems, an alternating easy-axis SIA direction has been found to induce large canting angles and drive weak ferromagnetism, which is normally caused by the DM interaction alone [20,21]. In contrast, in the alternating easy-plane case, collinear order along a pseudo-easy axis created by the intersection of the two local easy planes is found [21,22]. A canted low-temperature magnetic structure might be expected to occur in our Ni staggered chain, with the mean-field canting angle dictated by D/J0. However, the low-temperature magnetic properties here are likely to be influenced by quantum fluctuations due to the reduced dimensionality and low spin quantum number.
While these systems pose an interesting physical problem, the scarcity of suitably large single crystals and the comparable sizes of D and J0, are known, from the study of linear S = 1 chains, to complicate the full characterization of the Hamiltonian parameters [23,24]. To overcome this, we follow a similar experimental protocol to that highlighted in Ref [23]. We first use x-ray diffraction on small single crystals to determine the crystal structure and infer a magnetic Hamiltonian of the system. Magnetometry measurements on polycrystalline samples are used to confirm the quasi-lowdimensional nature of the interactions and easy-axis anisotropy. Muon-spin relaxation measurements find that the staggered chain undergoes long-range order below TN = 2.2K. While powder neutron diffraction suggests a colinear magnetic ordering, the noise floor of the data leaves room for a large canting angle of up to 25
•
Single crystal x-ray diffraction (XRD) was used to determine the crystal structure of the staggered chain, Ni(pym)(H2O)2(NO3)2, at 300 K with the structure shown in Fig. 1. To minimize absorption corrections, small (sub mm) single crystals are preferred for these experiments. For our measurements we used a sample with dimensions 0.23 × 0.19 × 0.05 mm 3 . Further details of the synthesis methods, single crystal XRD and structural refinement are provided in the Supplemental Material [25]. Samples crystallize in the monoclinic space-group C2/c, with lattice parameters, a = 12.7376 1(a). From the structure, it is evident that any SIA present will result in the local N-Ni-N axis defining either the easy axes or the normal to the easy planes. Furthermore, the slight difference in the coordinate bonds to the two different ligand species (H2O and NO3) in the equatorial plane suggests the possibility of a small rhombohedral anisotropy with energy E.
The hydrogen-bond networks illustrated in Fig. 1(b) stabilize the interchain structure. Each magnetic ion has six neighbors in adjacent chains; two neighbors residing at a distance of 7.388 Å along the c axis, and four neighbors with distance 6.833 Å along the unit cell diagonals. Consequently, two possible interchain exchange interactions exist, J1 along the [0,1,1] and [0,1¯,1] directions and J2 along c axis. These exchange pathways, depicted as purple and blue dashed arrows respectively in Fig. 1(b), constitute a hexagonal lattice in the interchain directions. If 0 (AFM), this triangular arrangement of the interchain spins will result in competition between the J1 and J2 interactions.
The Hamiltonian of the staggered chain can be written as
i i
where Sˆi is the spin of ion i. Here, the first sum is over the nearest neighbors along the Ni-pym chain with interaction strength J0 and the second sum is over unique interchain exchange bonds . The third term describes the SIA with the local anisotropy tensors Ki in the xyz laboratory frame. In the local frame of Ni(II) ions, the anisotropy tensor Ki loc = diag[E,-E,D] and Euler rotations are used to transform Ki loc to Ki, as shown in Ref. [25]. The final term is the Zeeman energy with an applied field μ0H and an isotropic g factor. In reality, we expect a small g anisotropy of magnitude g = gzgxy = 2D/λ, where λ ∼ 500K is a typical value of the spin-orbit coupling parameter for Ni(II) ions in octahedral environments [26]. This would result in a staggered g tensor and an internal staggered field with components perpendicular to the applied field. However, nonstaggered Ni(II) systems with local environments similar to this material typically exhibit SIA energies on the order of D ∼ 10K [27], resulting in a very small g ∼ 0.04, which is an order of magnitude smaller than seen in the Cu(II) staggered chains [1,2].
C2/c is a centrosymmetric space group, with the Ni ions located on inversion centres, while the nearest-neighbor exchange bonds are not centrosymmetric. This permits a DM interaction which changes sign from one bond to the next and has the form (-1
Ni(II) systems, where the spin-orbit coupling is relatively weak, the magnitude of the DM vector, DDM, is expected to be small in comparison to the dominant interaction terms described in Eq. ( 1) [21,22,28]. Temperature (K) FIG. 2. Temperature dependence of the zero-field-cooled (ZFC) magnetic susceptibility χ (blue circles) of the staggered S = 1 chain Ni(pym)(H2O)2(NO3)2, measured in an applied field of 0.1 T. The field-cooled curve coincides with the ZFC. (Inset) Data plotted as the inverse magnetic susceptibility χ -1 with a Curie-Weiss fit (red line). C/(T -θCW) + χ0, where χ0 is a temperature independent term, the Curie constant C kB and θCW is the Curie-Weiss temperature. Fitting to χ -1 , as shown in the inset to Fig. 2, gives g = 2.18(1), χ0 = -5.49(7) × 10 -9 m 3 mol -1 and θCW = -9.3(2)K. The g factor is typical of Ni(II) spin-1 systems, whereas the negative θCW is indicative of AFM coupling between Ni(II) ions. On cooling below 75 K, the data depart from the CW behavior and develop a broad hump centered at around Tχmax = 4.5(1)K, which is characteristic of quasi-low-dimensional systems and is due to the buildup of short-range correlations along the chain. Figures 3(a) and 3(b)
, respectively, present the pulsed-field magnetization, M(H) curve and the differential susceptibility dM/dH of powder samples at various temperatures. At the lowest temperature of 0.59 K, a sharp peak in dM/dH, corresponding to an upturn in magnetization, is observed at μ0HSF = 4.1(1)T. We ascribe this feature to a spin-flop transition commonly observed in systems with easy-axis anisotropy. The transition is no longer present for T 4.08K, indicating long-range magnetic order is absent at this temperature. Additional satellite peaks are observed at 3.3(1) and 6.7(2) T, which are broadened and no longer visible as peaks at 1.58 K. Increasing the field further results in a concave rise up to the projected magnetization saturation value of m = 2.06 μB per ion, suggesting a low-temperature g = 2.06 (1). The M(H) features and the Monte Carlo simulation are discussed in detail in Sec. IIF.
In order to probe the magnetic transition, zero-field muonspin relaxation (ZF μ + SR) measurements were performed on powder samples of show a more gradual Gaussian relaxation, typically reflecting disordered electronic moments fluctuating too rapidly to relax the muon spins, leaving the muons to be depolarized by the quasistatic, disordered nuclear spins [29]. This is strongly suggestive of a magnetic ordering transition taking place between these limits in T. To capture the change in the spectra, data were fitted to the phenomenological function
Here β accounts for the evolving shape of the spectra and λ parameterizes the relaxation rate. The results of the fitting procedure are shown in Figs. 4(b) and 4(c). We see a sharply-defined change in the shape of the spectra and a rapid fall in λ on warming from base temperature, consistent with the onset of long-range magnetic order at TN = 2.3K. This also coincides with the end of the rapid fall in λ. (At low T in the presence of dynamic relaxation, λ should be expected to vary with the square of the local√ fluctuating magnetic field, such that λ gives us a measure of the order parameter, assuming no change in fluctuation rate close to the transition [29]).
We note that the presence of a rapid relaxation in the muon spectra for T < TN, rather than oscillations, was also observed in related Ni(II)-based systems such as the series NiX2(pyz)2 [30], while oscillations are observed below TN in other molecule-based systems, such as Ni(NCS)2(thiourea)2 [31]. Here, the absence of oscillations points to an increased level of magnetic disorder (compared to those cases where oscillations are observed), or to fast magnetic fluctuations on the muon timescale with a fluctuation rate ν > γμBint, where Bint is the characteristic internal magnetic field at the muon site and γμ is the muon gyromagnetic ratio [29].
Elastic neutron diffraction measurements were performed on the WISH instrument at ISIS, the UK Neutron and Muon Source [32]. Data were collected on partially deuterated powder sample of Ni(pym)(D2O)2(NO3)2, where H2O was substituted by D2O. The magnetic properties were found to be very similar to the hydrogenated sample based on our χ(T ) measurements. A quantitative Rietveld refinement of the crystal structure, conducted using FULLPROF [33], is shown in Fig. 5(a) and reveals that the deuterated sample retains the C2/c structure of the hydrogenated samples down to 0.28 K. The resulting lattice parameters, a = 12.7441(3) Å, b = 11.4933(4) Å, c = 7.3871(2) Å, and β = 115.488(3) • , are in close agreement with values obtained using x-ray measurements on the hydrogenated samples and the small differences are attributed to deuteration. At 2.08 Å and below, there are several additional high-intensity peaks modelled by a LeBail fit of a Fm3¯m copper structure. These peaks arise from the copper sample holder and a thin copper wire within the sample space required for cooling. Further details of the structural refinement are given in the Supplemental Material [25].
Subtracting the neutron diffraction data collected at 5 K from the data collected at 0.28 K reveals several additional peaks due to the long-range ordered magnetic structure [Fig. 5(b)]. Indexing these peaks reveals a commensurate magnetic propagation vector k = (1/2,1/2,1/2). The observed magnetic peaks are satellites of the h + k = 2n reflection allowed by c-centering. The projection of the centering vector, [1/2,1/2,0], on k yields a phase factor of e iπ = -1 on the moments of sites related by c-centering. Hence, the sites related by c centering are aligned antiparallel.
Symmetry analysis using ISODISTORT [34,35] reveals only one candidate symmetry (irrep mL1 + ) for the magnetic structure. Within this irreducible representation, the spins in the unit cell connected by pym (not related by c-centering) are symmetrically inequivalent magnetic sites and are not constrained by symmetry. The moments of these two spins can be decomposed into a linear combination of orthogonal ferromagnetic (FM), mFM, and AFM, mAFM, modes:
As shown by the magnetic scattering intensity calculations in the Supplemental Material, mFM and mAFM can only contribute to the satellites of h + l = 2n + 1 and h + l = 2n peaks respectively [25]. Only the AFM peaks are clearly present in the data of Fig. 5 floor of the data. The magnetic refinement was carried out with a fixed φ = 0.
The refined magnetic structure is presented in Figs. 6(a) and 6(b). Spins lie collinearly in the ac plane at an angle, θ = 48(1) • away from the a axis. This structure indicates easy-axis anisotropy, where θ is expected to be determined by the projection of the local Ni-pym-Ni easy SIA axis onto the ac plane, the rhombohedral anisotropy and a small allowed staggered DM interaction. By contrast, if there was a dominant staggered easy-plane anisotropy, then spins would be expected to align perpendicular to the chain and the b axis, along a pseudo-easy axis defined by the intersection of the two local easy planes [21,22].
If spin canting induced by a staggered easy-axis SIA were present, the spins would rotate away from the ac plane, towards the easy axes (local Ni-N axis) at an angle of φ as depicted in Fig. 6(b). Considering a minimal mean-field model containing only J0 and D energy scales, the derivation presented in the Supplemental Material [25] yields the expression:
-sin(2φ) -.
(5) J0 sin(φ α)cos(φ α)
The limit of the data and Eq. ( 5) set an upper limit on the ratio Constraints on and are determined by the observed interchain order. Spins connected by the J2 exchange bond along the c-axis exhibit AFM order, which requires that 0 and(i.e., J2 must be AFM and stronger than J1 for AFM order along c). If , FM order along c is imposed by the stronger interaction and if J 0, there is no competition between J1 and leading to FM order along T = 0.25K with i = 3.36meV. energies i = 2.04 and 3.36 meV. LET is a multiplexing spectrometer which allows simultaneous measurements with different neutron incident energies. This allows us to survey a wide range of energy while maintaining high resolution at low energies. Further details are provided in Refs. [38][39][40]. Figures 7(a) and 7(b) present the spectra collected at T = 50 and 0.26 K respectively with incident energy i = 2.04meV. We note that additional measurements were performed at T = 4.31 and 21.11 K using high-resolution chopper settings, whilst the T = 50 and 0.26 K measurements used high flux chopper settings. This results in an intensity scaling factor of ≈ 3.1 which we obtained from calculations of the chopper opening times and verified by comparing the intensities of the nuclear (220) peak. In any case, the spectral features have widths ≈ 0.1 meV, which are considerably broader than the full width at half maximum (FWHM) resolution of either the highresolution (0.035 meV) or high-flux (0.06 meV at the elastic line) modes, such that the datasets may be compared.
In the paramagnetic phase at 50 K, three dispersionless features are observed at neutron energy transfers of = 0.22(1), 0.78(1), and 1.01(1) meV. The momentum integrated energy cuts, at various temperatures, shown in Fig. 7(d), reveal the 0.78 meV excitation diminishes in intensity on cooling and is attributed to localized vibrational modes. In contrast, the 0.22 meV peak grows in intensity on cooling down to 0.25 K, implying it arises from a localized spin excitation from a ground state whose population increases on cooling. In this powder sample, this excitation could originate from the transition between the SIA split singlet |ms and doublet |ms states of Ni(II) ions in an octahedral environment orphaned from the exchange network due to defects or chain ends. This would point to a SIA energy of |D| = 2.6(1)K. The 1.01 meV peak also grows in intensity on cooling and coincides with the large peak corresponding to the top of the spin-wave excitation band in the ordered phase.
Data collected in the ordered phase (T = 0.26K) mark the emergence of dispersive spin-wave excitations along with a dispersionless in-gap excitation at 0.39(1) meV [Fig. 7(b)]. Additionally, in the i = 3.36meV data shown in Fig. 7(e), we observe low-intensity double-magnon scattering that extends to 1.8(2) meV, approximately twice the value of the singlemagnon excitation maximum, along with another dispersionless feature at 1.52(2) meV. The spin-wave excitations were analyzed using the Hamiltonian in Eq. ( 1) and linear spin wave theory (LSWT), as implemented in the SPINW [41] program and taking the ground state from elastic neutron diffraction measurements. The SIA parameters used in SPINW simulations here were renormalized by a factor [1 -1/(2S)] to account for the nonlinear contributions to the SIA which are omitted in LSWT [42,43].
Sharp dispersive features are observed in the data, extending from the energy gap of 0.46 meV and 0.69 Å -1 (magnetic Bragg position) to the top of the band at 1.02(1) meV. Simulations using Eq. ( 1) indicate that the energy gap is predominantly governed by D and increases with a larger D. Conversely, the bandwidth of the sharp dispersive feature narrows as D increases and broadens with larger J0. Reproducing the excitation gap, bandwidth and the sharp dispersion lines observed in the data allows us to accurately 1 When allowed to freely refine during the LM fitting, J1 and J2 approach 0 K and |Q|-dependant modulation of the gap is no longer present in the simulations as shown in Ref. [25]. Adding the term results in the dispersion of the top of this band at 0.76 meV, which appears to be flat in our data, suggesting J1 is small. These features are well reproduced in our simulations, using estimated values of 18(3)K and 00(5)K. Simulations of the powder-average spin-wave spectra with different values of J1 and J2 are shown in the Supplemental Material [25]. Inspection of the energy cut in Fig. 7(d), reveals a broad feature in this region with two peaks at 0.60(2) and 0.66(2) meV, which was found to originate from to the rhombohedral anisotropy E.
The Levenberg-Marquardt (LM) algorithm was used to optimize J0, D, and E using seven |Q| integrated energy cuts, three of which are shown in Fig. 8. Due to the differences in observed and simulated intensities discussed below, further optimization of J1 and J2 was not possible and they were fixed to values estimated above. 1 The resulting fitted parameters are J0 = 5.107(7)K (0.4401(6) meV), D = -2.79(1)K (0.2412(1) meV) and E = 0.19(9)K (0.016(8) meV) and the simulation is depicted in Fig. 7(c). The value of D obtained through fitting the spin-wave spectra is in excellent agreement with the value suggested by the 0.22 meV dispersionless excitation and 014421-10 further supports the presence of localized excitations from orphaned Ni(II) ions as proposed earlier. Using Eq. ( 5) and the fitted |D|/J0 = 0.546(3), we estimate an canting angle φ ≈ 6.5 • . This canting is within the limit set by the noise floor of our elastic neutron diffraction data and is not expected to be discernible in those data. The AFM coupling causes spins in neighboring chains to cant in opposite directions, canceling the ferromagnetic component of a single chain. As a result, although spin canting is present in these systems, a zero-field remanent magnetization is not expected, consistent with our powder M(H) data. Additionally, the quasi-one-dimensional nature of our system, implied by our χ(T ), mu + SR and Tdependent neutron diffraction results, is confirmed by the ratio J While the form of the observed spectra is captured very well by our model, there are differences in the observed and calculated intensities. This is most pronounced in the 0.
0.76meV region in the energy cut integrated over 0.
[Fig. 8(a)], which is dominated by the spin-wave dispersions in the plane perpendicular to the chain. Increased intensity observed in this region suggests a redistribution in the spectral weight from the spin-wave modes along the chain to the interchain modes. This may be attributed to the preferential orientation of the grains in the powder samples such that the interchain scattering plane is more exposed to the neutron beam. Additionally, because LSWT does not account for impurity effects, the 0.22 meV feature is not reproduced in our simulation. We also note that the 0.39 meV in-gap modes were not reproduced in the SPINW simulations, even when higher-order interactions are included, and could hint at a localized excitation mode which LSWT does not capture. One possible explanation is that this feature could arise from excitations between energy levels within small orphaned dimers of Ni(II) ions [44]. Other orphaned structures, such as trimers, may also account for the 1.01 meV excitation seen up to 50 K.
Monte Carlo simulations of the powder-averaged longitudinal M(H) were performed to confirm the Hamiltonian parameters determined by the INS experiments. At each field, the simulation aims to minimize the total energy of an eightspin cluster, which is calculated using the Eq. ( 1) and an isotropic g = 2.06 that was determined from the M(H) data at T = 0.59K. The resulting M(H) and dM/dH simulations are shown as blue dash-dot lines in Fig. 3. The simulations provide an overall good agreement with the observed spin-flop field at μ0HSF = 4.1(1) T and the rounded approach to the saturation at high fields.
Simulations of single-crystal M(H) data with 0K and J 0K in both cases) were performed to assess the effects of interchain interactions on M(H) (see Fig. 9). The J 0K simulations consider only a single isolated spin chain, and as such show a zero-field remanent magnetization, which as explained above, disappears when multiple chains are considered. When the applied field is parallel to the chain [see Fig. 9(a)], a single spin-flop transition occurs for both J 0 and 0.18 K. This transition occurs at μ0HSF = 4 T for J 18K and corresponds to the main spin-flop feature found in the power average M(H) data shown in Fig. 3.
In Fig. 9(b), where the applied field is parallel to one of the easy-axis directions, there is a single meta-magnetic transition for J 0K and two separate transitions at 3.4 T and 7.6 T when J 18K. Similarly, in Fig. 9(c), where the applied field is parallel to the b axis, the octahedra tilting direction, a metamagnetic transition, associated with neighboring chains flipping, is present at 2.2 T only when 18K. These additional metamagnetic transitions, which appear in the simulations for certain field directions when interchain interactions are considered, may account for the 3.3(1) and 6.7(2) T, satellite features observed in the 0.59 K powderaveraged dM/dH data. In a powder-averaged measurement, a single peak in dM/dH is expected, as indicated by the powder-average MC simulations in Fig. 3. Preferential orientation of the grains during measurements, may result in an increased contribution to dM/dH from these directions and the observation of distinct satellite peaks in our data. 014421-11
to the plane containing the easy axes). The purple dashed line in all panels indicates magnetic saturation.
In anisotropic magnets, a symmetry-breaking phase transition is often expected to occur at saturation when a large field is applied parallel to the principle anisotropy axis. However, it has been shown recently that in systems with an alternating SIA axis, there are principle directions where the SIA can always save energy by canting spins away from the applied field, and magnetic saturation only occurs in the infinite field limit [22]. Similarly in Ni(pym)(H2O)2(NO3)2, a magnetic saturation phase transition only occurs when the applied field is simultaneously perpendicular to the chain and the b axis [Fig. 9(d)]. This is the direction about which the octahedra rotate and is the only direction where the applied field is perpendicular to both easy axes. For all other field orientations, M(H) asymptotically approaches the saturation value. In the powder-averaged M(H) simulation and data, this is captured by the rounded approach to saturation which shows no features in dM/dH indicative of a phase transition.
We have explored the magnetic properties of a quasione- 1), mean-field calculations show that φ ≈ 6 • . While our LSWT simulations model the observed dispersive spin-wave excitations well, there are certain features which are not captured. We suggest that some of these features arise from small clusters of Ni(II) spins, such as single ions, dimers and trimers which are orphaned from the chains. Additionally, it has been pointed out elsewhere that some details of S = 1 magnetic excitation spectra can only be accounted for generalizing spin-wave modeling to include SU(3) degrees of freedom [45]. This is a developing issue that requires further investigation.
Monte Carlo simulations, using Eq. ( 1) and the Hamiltonian parameters determined using INS measurements, show that the features in the M(H) are largely accounted for by a classical model and an isotropic g factor. This is in contrast to the S = 1/2 staggered chain, where the g anisotropy and the DM interactions have a dramatic effect, mapping the Hamiltonian on to the sine-Gordon model of quantum field theory and giving rise to soliton-like excitation modes [1][2][3][4][5][6]. Recently, experiments on a chiral S = 1/2 system, [Cu(pym)(H2O)4]SiF6 • H2O, hosting a fourfold periodic variation of the spin environments showed behavior which was not governed by the sine-Gordon model [46]. Therefore, exploring a similar extension from the S = 1 staggered chain to an S = 1 chiral chain could prove interesting.
For a linear S = 1 AFM chain with |D|/J0 = 0.546(3) and J 04(2), the Haldane phase is already expected to be quenched by D and J into an AFM Ising order [8]. Therefore, to study the robustness of the topological phase in the presence of alternating octahedra and single-ion anisotropy direction, further theoretical studies and the development of real materials close to the quantum critical point are necessary.