Rosana P.
Sartoris
*a,
Vinicius T.
Santana
b,
Eleonora
Freire
cd,
Ricardo F.
Baggio
c,
Otaciro R.
Nascimento
e and
Rafael
Calvo
*af
aDepartamento de Física, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria, 3000, Santa Fe, Argentina. E-mail: sartoris@fbcb.unl.edu.ar
bCEITEC - Central European Institute of Technology, Brno University of Technology, Purkyňova 123, 61200, Brno, Czech Republic
cGerencia de Investigación y Aplicaciones, Centro Atómico Constituyentes, Comisión Nacional de Energía Atómica, Buenos Aires, Argentina
dEscuela de Ciencia y Tecnología, Universidad Nacional General San Martín, Buenos Aires, Argentina
eDepartamento de Física e Ciencias Interdisciplinares, Instituto de Física de São Carlos, Universidade de São Paulo - USP, CP 369, 13560-970, São Carlos, SP, Brazil
fInstituto de Física del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas, Güemes 3450, 3000, Santa Fe, Argentina. E-mail: calvo.rafael@conicet.gov.ar
First published on 2nd April 2020
To investigate the magnetic properties and the spin entanglement of dinuclear arrays, we prepared compounds [{Cu(pAB)(phen)H2O}2·NO3·pABH·2H2O], 1, and [Cu2(pAB)2(phen)2pz]n, 2 (pABH = p-aminobenzoic acid, phen = 1,10-phenanthroline and pz = pyrazine). The structure of 1 is known and we report here that of 2. They contain similar dinuclear units of CuII ions with 1/2-spins S1 and S2 bridged by pairs of pAB molecules, with similar intradinuclear exchange and fine interactions , but different 3D crystal arrays with weak interdinuclear exchange J′, stronger in 2 than in 1. To investigate the magnetic properties and the spin entanglement produced by J′, we collected the powder spectra of 1 and 2 at 9.4 GHz and T between 5 and 298 K, and at 34.4 GHz and T = 298 K and single-crystal spectra at room T and 34.4 GHz as a function of magnetic field (B0) orientation in three crystal planes, calculating intradinuclear magnetic parameters J(1)0 = (−75 ± 1) cm−1, J(2)0 = (−78 ± 2) cm−1, |D(1)| = (0.142 ± 0.006) cm−1, |D(2)| = (0.141 ± 0.006) cm−1 and E(1) ∼ E(2) ∼ 0. Single crystal data indicate a quantum entangled phase in 2 around the crossing between two fine structure EPR absorption peaks within the spin triplet. This phase also shows up in powder samples of 2 as a U-peak collecting the signals of the entangled microcrystals, a feature that allows estimating |J′|. Transitions between the two quantum phases are observed in single crystals of 2 changing the orientation of B0. We estimate interdinuclear exchange couplings |J′(1)| < 0.003 cm−1 and |J′(2)| = (0.013 ± 0.005) cm−1, in 1 and 2, respectively. Our analysis indicates that the standard approximation of a spin Hamiltonian with S = 1 for the dinuclear spectra is valid only when the interdinuclear coupling is large enough, as for compound 2 (|J′(2)/J(2)0| ∼ 1.7 × 10−4). When J′ is negligible as in 1, the real spin Hamiltonian with two spins 1/2 has to be used. Broken-symmetry DFT predicts correctly the nature and magnitude of the antiferromagnetic exchange coupling in 1 and 2 and ferromagnetic interdinuclear coupling for compound 2.
(1) |
Attention has also been paid to the effect of weak exchange interactions J′ interconnecting finite5,19,20 and infinite21–24 arrays of DUs in a molecule or a periodic structure. The periodic structures give rise to travelling spin excitation (triplons)25–27 transforming the finite quantum dinuclear units into infinite collective systems with quantum many-body effects modifying the magnetic properties and showing the fascinating properties of molecular magnets, including quantum phase transitions,28,29 Bose–Einstein condensation,27,29 quantum spin ladders30 and more.31,32
EPR played the most relevant role in the studies of individual DUs and, as well, in the studies of weak interdinuclear exchange interactions J′ in compounds having infinite21–24,33–35 arrays of DUs. Detailed single-crystal measurements show that the spin entanglement and the spin dynamics arising from J′ merge the peaks of the fine structure for the magnetic field B0 = μ0H (μ0 is the vacuum permeability) around the directions where the EPR absorptions intersect23,34,35 and collapse the hyperfine structure. Changes in the spectra of weakly coupled monomeric spins as a function of J′ were initially explained using the exchange narrowing theory36,37 and rigorously proved by classical Anderson–Kubo's general theories of magnetic resonance.38,39 In later years, experimental and theoretical investigations were done on infinite coupled-spin systems and the understanding of the exchange narrowing process evolved for monomeric spins40 and for AFM DUs,23,24,34,41 where the process is thermally activated. The purpose of this work is to study weak interdinuclear couplings through their effects on the EPR spectra. We investigate two weakly interacting arrays of DUs in compounds [{Cu(pAB)(phen)H2O}2·NO3·pABH·2H2O], 1, and [Cu2(pAB)2(phen)2pz]n, 2, where pABH = p-aminobenzoic acid, phen = 1,10-phenanthroline, pz = pyrazine, synthesizing, crystallizing, solving their crystal structures and performing EPR studies on powder and single crystal samples. Compounds 1 and 2 contain similar DUs, and thus similar intradinuclear coupling J0. However, different paths connect them, leading to different interdinuclear coupling J′ between the neighbouring DUs. Therefore, comparing the EPR results of these compounds based on their structures allows a deeper understanding of the effects of interdinuclear interactions.
We collected the EPR spectra of single crystal samples of 1 and 2 at the Q-band and room T as a function of the orientation of B0, also for powder samples at the X-band as a function of temperature T between 5 and 293 K and at the Q-band and 293 K, and used these spectra to determine the intradinuclear magnetic parameters. For orientations of B0 around the fields at which the fine structure peaks ±1 ↔ 0 cross and the energy distance between them become smaller than |J′|, the spectra change abruptly into a collapsed line reflecting the spin entanglement and allowing the estimation of interdinuclear coupling. In single-crystals, quantum phases differing in the spin entanglement can be tuned with B0; in the powder spectra, the couplings J′ give rise to an extra peak, which was reported as arising from the interdinuclear exchange by Gavrilov et al.,33 who associated this behavior with travelling triplet excitons in the crystalline lattice. Others also observed this peak in the powder spectra of polynuclear CuII compounds, suggesting the interdinuclear exchange as a potential explanation for it.42,43 Single crystal EPR measurements on chains of copper dinuclear paddlewheel units reported by Perec et al. demonstrated that this so-called “U” peak24 in powder spectra collects the signal arising from microcrystals oriented in the angular range of the collapsed line and its intensity allows estimating both |J′| and the temperature dependence of the entanglement due to the population of excited triplet states.35
The behaviors of the spectra of 1 and 2 are discussed in terms of the interdinuclear exchange couplings and the consequent spin entanglement. The magnitudes of the intra- and inter dinuclear exchange couplings J0 and J′ are related to the structures of the corresponding paths connecting the metal ions. Their experimental values for 1 and 2 are compared with the results of theoretically calculated separation between the singlet and triplet states using broken-symmetry density functional theory (BS-DFT).
Compound 1: we added slowly 0.75 mmol of Cu(NO3)2·5H2O to an aqueous solution (30 cm3) of pABH (0.75 mmol) and phen (0.75 mmol) while the pH was adjusted to 4 with HNO3 and NaOH. Keeping this solution at 35° for slow evaporation, green square crystals were obtained after ∼4 days.
Compound 2: pABH (1 mmol), phen (1 mmol) and pyrazine (0.5 mmol) were added to 40 ml of methanolic-acetonitrile solution (1:1 v/v) and kept under continuous agitation at room T until full dilution, and then 1 mmol of Cu(NO3)2·5H2O was added. This solution was gravity-filtered and the filtrate kept for slow evaporation at 20°. Dark green crystals were obtained after ∼5 days.
The crystal habit of the samples was identified with a goniometric optical microscope; the angles between crystal edges were used to orient the samples by comparing the results with crystallographic information. Cubic sample holders made by cleaving pieces of KCl single crystals were used to define laboratory orthogonal reference frames xyz to mount the samples. Specimens of 1 and 2 were glued with the bc plane parallel to the xy faces of the holders, with the b axis parallel to a holder edge. This allows obtaining the relationship between the sample holder and crystal axes mounting the holders on top of a pedestal inside the cavity with each face (xy, zx or zy) on the horizontal plane. The orientation of B0 was varied by rotating the magnet and the EPR spectra dχ′′/dB0 of single crystal samples of 1 and 2 were collected with B0 at 5° intervals (or smaller in some ranges) along 180° in the planes a*b, ca* and bc at 298 K and 34.4 GHz (a* = b × c). Single crystal and powder spectra were analyzed using EasySpin (v. 5.2.24),52 a package of programs working under Matlab53 that simulates and fits a given Hamiltonian to spectral line shapes dχ′′/dB0vs. B0 and to the angular variation of the centers of the resonances in a single crystal. Gaussian and Lorentzian lineshapes defined following Weil and Bolton54 were used to fit different situations (see discussion). In all fittings the mean square deviations between experimental and calculated values were minimized.
Compound 1 | Compound 2 | |
---|---|---|
Computer programs: SHELXS,46 SHELXL2018/1.47a The chemical formula and Mr are not accurately reported because solvates/counterions could not be resolved/refined and their effect was discounted with the SQUEEZE procedure implemented in PLATON.48 | ||
Chemical formula (as dimeric unit + solvates) | C38H32Cu2N6O62+ + 2(NO3−)·(C7H7NO2)·2(H2O) | C42H32Cu2N8O4+ + unknowna |
M r (as dimeric unit + solvates) | 795.77 + 434.32 | 839.83 + unknowna |
Crystal system, space group | Monoclinic, C2/c | Monoclinic, C2/c |
Temperature [K] | 295 | 295 |
a, b, c (Å) | 26.000(8), 10.253(3), 21.004(5) | 24.5552(17), 9.9210(6), 19.9673(12) |
β (°) | 106.90(2) | 102.042(2) |
Volume (Å3) | 5357.39 | 4757.2(5) |
Z | 4 | 4 |
Radiation type | Mo Kα, λ = 0.71071 Å | Mo Kα, λ = 0.71071 Å |
The asymmetric unit in 1 contains half a DU [Cu(pAB)(phen)(H2O)]22+, one free pABH, one phen molecule, a nitrate ion and a water of crystallization (Fig. 1a). In turn, compound 2 (Fig. 1b) contains half a DU [Cu(pAB)(phen)pz]22+. The uncoordinated part of the structure of 2 (the counterion/solvate content) was impossible to identify with some certainty, due to extreme disorder. The structural model was refined employing the SQUEEZE procedure implemented in PLATON,48 an alternative taking into account the lacking electron content in the data set under consideration. In the refinement of the data set, the program estimated this to be about 700 electrons for the whole unit cell. It is to be noted that the whole electron count for the counterion/solvates in 1 affords 616 electrons, which suggests that the unaccounted-for part of the structure in 2 may be similar to the one in 1. Both DUs are almost identical and result from the bridging of two copper ions by the carboxylate groups of two symmetry-related pAB molecules. The Cu ions are in square pyramidal coordination, bonded equatorially to two carboxylic oxygens of two pAB molecules and two N from phen molecules, with the apical position occupied by a water molecule in 1 and a pz nitrogen in 2 (Fig. 1a and b). The intra-dinuclear exchange coupling J0 is mainly supported by two symmetry-related O–C–O bridges with C–O average distances 1.268(12) Å and angle 124.6(2)° for compound 1 and 1.266(3) Å and 124.57(17)° for compound 2. The π–π interactions arising from the stacking of the phen rings, with an average distance of ∼3.42 Å and angle of ∼0.9° for 1 and ∼3.38 Å and ∼2.8° for 2, should be less relevant for the magnitude of J0.
Fig. 1 Schematic representation of the molecules of 1 and 2, showing the used labeling scheme. In full (hollow) bonds, the independent (symmetry related) parts. The structure of compound 1 is taken from Battaglia et al.,44 but the labeling of the atoms was changed in order to help the comparison of the two structures. |
Similarities in the DU geometry in both structures are no longer valid when the crystal organization is analyzed, for which we shall discuss each structure in turn, looking for the interaction paths, whenever possible. The isolated nitrate and pABH molecules in 1 are roughly coplanar and arranged in layers parallel to the (−1,0,1) plane. Since the interlayer distance is one half d(−1,0,1), these layers should, in fact, be described as parallel to the (−2, 0, 2) family. The planar arrays also include the coordinated water molecule O1W, and all three molecules (nitrate, pABH and water) determine a tightly bound H-bonded structure shown in Fig. 2a (the “#n” codes used for easy reference to each interaction are defined in Table 2, which presents information about these H-bonds). The DUs expand between adjacent planes, as shown in Fig. 2b, acting as linkers between the planar H-bonded arrays. The latter, in turn, act as weak coupling agents between neighbouring DUs. It is interesting to point out that the DUs are related along the c-axis by inversion centers, while along a they are related by C-centering ([0.5,0.5,0]) translations. Thus, even if the symmetry-related copper centres are magnetically different, the EPR spectra of all dimeric units should be identical for all orientations of B0. In the case of compound 2, the replacement of the apical water molecule in 1 by a pz bidentate linker in 2 has the effect of generating ⋯pz-DU-pz-DU-pz⋯ chains running along c (Fig. 3a and b). These chains, in turn, are presumably connected through the disordered solvates/counteranions. As displayed in Fig. 4, in both compounds, the DUs form chains along the c-axes; in 1 the neighbouring DUs are connected by H-bonds from the coordinated water molecules and the free pABH, which acts as a linker between the DUs.
Code | D-H⋯A H-bond | D-H (Å) | H⋯A (Å) | D⋯A (Å) | (D-H⋯A)° |
---|---|---|---|---|---|
Symmetry codes: (i) −x, 2 − y, −z; (ii) x, 1 + y, z; (iii) 1/2 − x, 1/2 + y, 1/2 − z; (iv) 1/2 − x, 3/2 − y, 1 − z. | |||||
#1 | O1D–H1OD⋯O1Wi | 0.86 | 1.88 | 2.7348(8) | 173 |
#2 | O1W–H1WA⋯O2D | 0.86 | 1.91 | 2.7301(8) | 158 |
#3 | O1W–H1WB⋯O3Cii | 0.86 | 1.88 | 2.7324(8) | 173 |
#4 | N1D–H1ND⋯O2C | 0.86 | 2.36 | 3.1442(10) | 151 |
#5 | N1D–H1ND⋯O3C | 0.86 | 2.53 | 3.3375(10) | 157 |
#6 | N1D–H2ND⋯O1Ciii | 0.86 | 2.45 | 3.2366(10) | 152 |
#7 | N1D–H2ND⋯O2Ciii | 0.86 | 2.44 | 3.2427(10) | 156 |
#8 | C4D–H4DA⋯O3C | 0.96 | 2.54 | 3.3952(10) | 148 |
#9 | N1B–H1NB⋯O2Wiv | 0.86 | 2.22 | 3.0154(9) | 154 |
(2) |
(3) |
(4) |
Fig. 5 Angular variation of the position of the ±1 ↔ 0 EPR absorptions observed at the Q-band in compound 1 in the three studied planes. Solid lines are obtained from a global fit of eqn (2) to the data. The positions of the axes within each plane are indicated. The green circle in (c) indicates the range of the collapse. Crystal axes are shown in blue. |
Fig. 6 Angular variation of the position of the ±1 ↔ 0 EPR absorptions observed at the Q-band for compound 2 in the studied planes. Symbols are experimental values; solid lines are obtained from a global fit of eqn (2) to the data. Similar fitted curves are obtained with eqn (4). The parameters obtained in this fit are given in Table 3. The green circle in (c) indicates the range of the collapse. Crystal axes are shown in blue. |
Positions and widths display small changes and no crossings in the plane a*b, indicating that this plane is perpendicular to the axis of symmetry of the DUs as expected from the structural results. The splitting between the allowed EPR transitions is maximum when B0 is approximately parallel to the c axis of the chains of DUs. The crossings are sharp in compound 1, while they merge within an angular range around the expected crossing points in compound 2. We obtained the principal values and directions of the g- and D-matrices by fitting eqn (2) (spin-Hamiltonian with two spins 1/2) and the approximate eqn (4) (one spin 1) to the data in Fig. 5a–c (for compound 1) and in Fig. 6a–c (for compound 2), excluding the merged peaks in angular ranges around the magic angles in compound 2. In the case of compound 1, eqn (4) gives poorer agreement than eqn (2) with the observed result, indicating that the spin S = 1 approximation does not strictly apply. Meanwhile, both equations provide similar quality fittings to the data for compound 2. Since one obtains eqn (4) from eqn (2) cancelling out the contribution [μBs·G·B0 − s·D·s] this result indicates the presence of interdinuclear interactions in compound 2 (finite |J′|) and that in compound 1 this coupling is negligible (|J′| ∼ 0). Quantitatively, for compound 1 the mean square dispersion of the fitting of eqn (2) to the angular variation in Fig. 6 is σ = 0.0010 (attributed to small misalignments of B0), while that of eqn (4) to the same data is σ = 0.0020. Meanwhile, for compound 2, fittings with eqn (2) and (4) give both σ ∼ 0.0010, meaning that the approximations made to obtain eqn (4) are appropriate. Fig. 7 further emphasizes this result, plotting the difference between the experimental and calculated distances between the two fine structure peaks, obtained through the fittings to each model. This is experimental proof that the approximation of a spin Hamiltonian with S = 1 used by most authors to fit the angular variation is valid only when sizeable interdinuclear interactions exist and the whole contribution [μBs·G·B0 − s·D·s] is cancelled out by . The resulting principal components of the g- and D-matrices54 are included in Table 3. The values of D obtained for 1 and 2 are similar and the rhombic contribution |E| is much smaller than the axial parameter |D| and may be neglected in the analysis. Also, the equatorial components g1 and g2 of the g-matrices differ within the experimental uncertainties, indicating an approximate axial symmetry with g1 ≈ g2 ≈ g⊥ and g3 = g||. It is observed for compound 1 and orientations of B0 around 160° of the c-axes in the ca* plane where the distance between the resonances Sz = ±1 ↔ 0 is largest (Fig. 8a) that each resonance splits into seven peaks due to the hyperfine coupling with the nuclear spins (see above). For other orientations of B0, only some peaks of the hyperfine structure are observed for sample 1. Fitting the hyperfine coupling to the data in Fig. 8a, we obtained the parallel components of the A-matrix, A|| = (63 ± 2) × 10−4 cm−1 and estimated A⊥ ≈ 0 for compound 1. Hyperfine structure is not observed for compound 2. To emphasize the different behavior of compounds 1 and 2 around the magic angles, Fig. 8c and d display spectra observed at the Q-band. In compound 1 (Fig. 8c) the resonances (and their structures) cross; meanwhile, in compound 2, the two peaks merge into one within angular ranges around this angle (Fig. 8d). Fig. 9a and b displays the ratio K = ΔBexp0/ΔBcal0 between the observed splitting of the collapsing resonances and those calculated in the absence of interaction, as a function of the reciprocal of the calculated splitting, for the magnetic field oriented near the magic angles in the planes ca* and cb in compound 2 (Fig. S2† shows the experimental data and the global fit around the collapse for 2). When (ΔBcal0)−1 is large, close to the magic angles, the signals are collapsed, and K = 0. When (ΔBcal0)−1 is small, far from the magic angles, K ∼ ±1.63,64 The collapse is abrupt and occurs when gμB[ΔBcal0]collapse = ħωex = |J′|, a condition allowing the exchange frequency ωex to be obtained. From the fittings of the equation
(5) |
Fig. 7 Difference between the experimental and calculated distances between the positions of the fine structure EPR peaks obtained using the parameters obtained with the best fits to eqn (2) (two spins 1/2, red symbols and lines) and eqn (4) (one spin 1, blue symbols and lines). Left side for compound 1 and right side for compound 2. |
Fig. 8 Spectra collected at the Q-band for (a) B0 at 160° of the c-axes in the ca* plane in compound 1 and (b) along the c-axis (θ = 0°) in the cb plane for compound 2 (see red arrows in Fig. 5b and 6c). Selected EPR spectra of single crystals of compounds 1 (c) and 2 (d) observed at the Q-band and 298 K as a function of the orientation of B0 around magic angles (green circles in Fig. 5c and 6c). In (c) a crossing of the resonances occurs for compound 1 while in (d) a merging of the resonances of compound 2 is observed at the magic angles. Note that the signals of compound 1 display some hyperfine structure, which is fully merged for compound 2. |
Compound 1 | |||||
---|---|---|---|---|---|
g 1 | g 2 | g 3 | D [cm−1] | E [cm−1] | |
Single crystal, Q-band | 2.050(2) | 2.066(2) | 2.275(2) | 0.142(2) | 0.003(3) |
Powder X-band | 2.063(5) | 2.064(5) | 2.261(5) | 0.142(2) | 0.003 (2) |
Powder Q-band | 2.068(5) | 2.068(5) | 2.271(5) | 0.143(6) | 0.003(2) |
Compound 2 | |||||
Single crystal, Q-band | 2.058(2) | 2.072(2) | 2.275(2) | 0.141(2) | 0.002(2) |
Powder, X-band | 2.064(4) | 2.064(4) | 2.261(4) | 0.140(6) | 0.003(2) |
Powder, Q-band | 2.064(4) | 2.064(4) | 2.261(4) | 0.140(6) | 0.003(2) |
(6) |
Fig. 11 Temperature dependence of the powder EPR spectra collected at the X-band at selected T. (a) compound 1 and (b) compound 2. The field ranges of the U-peaks are emphasized with red backgrounds. |
Fig. 12 (a) and (b) display the normalized dependence I(T)/I(0) vs. T, where I(0) is the maximum value of the signal intensity I(T) for compounds 1 and 2 at the X-band. (c) and (d) display I(T) × T/I(0) vs. T. Black symbols are experimental results, red curves are fittings of eqn (6) to the data allowing us to obtain the values of J0. (e) The Curie behavior of the area of the EPR signal S of the marker CrIII:MgO confirms that there is no quantitative experimental error in J0 due to changes in the parameters of the microwave cavity within the temperature range of the experiments, allowing us to evaluate the error indicated for the exchange couplings. |
On the other hand, the interdinuclear paths for superexchange between DUs arranged in chains shown in Fig. 4a and b are very different. In 1, two symmetry-related paths, each containing a carboxylate ion O1D–C1D–O2D, are connected through two H-bonds O–H–O to the water oxygens O1W of the apical ligands of the copper ions in the neighbouring DUs along the chains (Fig. 4a). Meanwhile, in 2, these apical O1W are connected by two symmetry-related covalent paths provided by pyrazole rings (Fig. 4b). Even if paths connecting apical ligands are expected to support weak interactions, it is clear that this weakness is extreme when the covalent path in 2 is replaced by a pair of consecutive H-bonds in 1. The minimum straight distances between the Cu ions in the neighbouring DUs are 7.678 Å in 1 and 7.304 Å in 2, while the distances measured along the paths are 12.572 Å in 1 and 8.857 Å in 2, providing the picture searched for when programming this investigation. These results are a consequence of the structures of 1 and 2 wherein each Cu ion in the DU is equatorially coordinated by two O from acidic groups and two N from a phenanthroline molecule. The experimental values of g|| > g⊥ indicate that the CuII unpaired electron resides in the orbital dx2−y2, as expected from its square-pyramidal environment.54,66
The magnetic behavior of compounds 1 and 2 (Fig. 4) could also be explained as alternate magnetic chains with x = |J′/J0| ≤ 1.7 × 10−4 using the theory reported by Duffy and Barr,67 following that of Bonner and Fisher68 for uniform chains, which allows the calculation of the eigenstates and the thermodynamic properties (magnetic susceptibility and specific heat) of these chains as a function of x (see, for example, Calvo et al.69). However, the EPR spectra of magnetic compounds depend on the spin dynamics which is not considered in these theories. As explained before, the EPR spectra of a coupled infinity spin system can be studied using the statistical theories of Anderson, and Kubo and Tomita.38,39 On the other hand, when the coupling between DUs is very small, as for 1 and 2, the magnetic susceptibility of the coupled system is accurately explained by the Bleaney and Bowers equation (eqn (6), which neglects the coupling J′.
It has been proposed that the U-peak arises from the discussed interdimeric coupling |J′|, which can be evaluated from its relative intensity R.24,35Fig. 13 displays the ratio R between the integrated intensities of the U-peak and the total signal of the DUs of compound 2 as a function of T. Following the method described by Calvo et al.35 and using the data in Fig. 6b and c, we determine that |J′(T)| [cm−1] = 0.48R(T), and using this relationship and the data in Fig. 13 we obtain |J′(2)| = (0.013 ± 0.005) cm−1 at 298 K. This value agrees with the one obtained using eqn (5) in Fig. 9 within the experimental error. Since the U-peak is absent within the experimental uncertainty in the spectra of 1, and that the hyperfine coupling can be observed in a wide range of field orientations, we conclude that |J′(1)| < A||/2 = (0.003) cm−1. The order of magnitude of J′ lies outside of the reasonable precision of the BS-DFT method. Nevertheless, we performed the calculation for the prediction of the sign of the exchange and the calculated intermolecular exchange coupling for compound 2 is presented in Table 4. The results point to a ferromagnetic J′ for compound 2. For compound 1 the calculation did not converge.
Fig. 13 Ratio R(T) between the areas of the U-peak and of the powder spectrum of compound 2 using the data in Fig. 6b and c. We calculated35 that |J′(T)| [cm−1] = 0.48R(T), and using this relationship and the data in Fig. 13 we obtain |J′(2)| = (0.013 ± 0.005) cm−1 at 298 K. |
1. We observed spin entanglement produced when the absorption peaks within the spin triplet cross as a function of the orientation of the applied magnetic field. This occurs within the angular range wherein the distance between the EPR transitions is smaller than the magnitude of the interdinuclear spin exchange coupling J′, giving rise to two, entangled and unentangled, spin phases. This is clearly observed in the single crystal study of compound 2 and allows the evaluation of J′. This effect is absent in compound 1, indicating negligible interdinuclear coupling J′.
2. The spin entanglement described above is also shown in the spectra of powder samples of compound 2, wherein an additional U-peak collects the EPR response of the spins in the entangled phase. This peak is not shown by the powder spectra of compound 1.
3. The coupling J′ modulates and average outs the hyperfine structure allowing another method to detect and estimate interdinuclear interactions. This is observed by comparing the spectra of compounds 1 and 2. In 2, the larger couplings J′ completely destroy the hyperfine coupling, although in 1 it can be detected for orientations of B0 where A is large. Since copper has two natural isotopes, dinuclear units with different hyperfine couplings coexist in the samples, complicating any quantitative evaluation of the spin entanglement and |J′| from the hyperfine structure in 1, and only provide a higher limit. Nevertheless, our data for the hyperfine coupling reproduce qualitatively the expected behaviour. This source of uncertainty in the fitting of the line shapes and the evaluation of the line widths in the spectra of compound 1 could be avoided only with Cu isotopically enriched samples.
4. The line shapes and line widths of the peaks of the single crystal spectra also offer a procedure to evaluate interdinuclear couplings. In principle and in agreement with Anderson's theory of exchange narrowing, we observe Lorentzian line shapes in the field orientation ranges wherein the spins are entangled in compound 2. For other field orientations and for compound 1, the results were not easily described and were of little use. The residual hyperfine structure of 1 further complicates this analysis.
As a closing remark, some of the achievements of our work were possible only through careful combined experiments on powder and single crystal samples.
Footnote |
† Electronic supplementary information (ESI) available. CCDC 1942943. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/D0DT00567C |
This journal is © The Royal Society of Chemistry 2020 |