Crystal structure and specific heat of calcium lanthanide oxyborates Ca₄LnO(BO₃)₃

1 Introduction

The calcium lanthanide oxyborates Ca₄LnO(BO₃)₃ (Ln = Y, La–Lu) are of interest for non-linear optical applications as a result of their non-centrosymmetric crystal structure; they are also valued as host crystals for luminescent materials and as high-temperature piezoelectrics [1]. Studies on Ca₄LnO(BO₃)₃ include measurements on numerous properties including melting points [1], [2], [3], photoluminescence [4], [5], [6], [7], thermal conductivity [8], dielectric, elastic and piezoelectric behaviour [3, 9, 10], and magnetism [11].

The crystal structure of Ca₄LnO(BO₃)₃, shown in Figure 1(a), consists of MO₆ distorted octahedra (M = Ca²⁺, Ln³⁺) and BO₃ trigonal planar groups [2]. The space group is Cm (No. 8). The metal-oxygen polyhedra are arranged in edge-sharing chains along the c-axis, Figure 1(b) [7]. Single-crystal X-ray diffraction (SCXRD) has been carried out on samples with Ln = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Y and Lu [2, 3, 8, 12], [13], [14], [15], [16], [17], [18], [19] as well as the partially substituted Mg and Sr derivatives of Ca₄GdO(BO₃)₃ [14]. Several structural studies reported that the Ln³⁺ ions are situated exclusively on the M1 (2a) crystallographic sites (Figure 1), while the Ca²⁺ ions occupy the two 4b sites M2 and M3 [2, 8, 14, 16, 17, 20]. However, others found the presence of cation mixing between the M1 and M2 sites, with increased site mixing for smaller Ln³⁺ ions [3, 13, 15]. For Ln = Gd the annealing temperature was also found to affect the cation distribution [10].

Figure 1:

Crystal structure of calcium lanthanide oxyborates Ca₄LnO(BO₃)₃. (a) Structure viewed along the c-axis, illustrating the three metal sites M1, M2, M3, and the two boron sites B1, B2. (b) Structure viewed in perspective, illustrating the chains of metal-oxygen polyhedra along the c-axis.

We recently reported crystal structures for 12 compounds in the Ca₄LnO(BO₃)₃ series, Ln = Y, La–Yb, using powder X-ray diffraction (PXRD) [11]. However, the boron and oxygen atomic positions could not be refined because of the weak scattering of X-rays from those elements compared with the heavier Ca²⁺ and Ln³⁺ ions. Neutron scattering provides a complementary technique in order to obtain the positions and fractional occupancies of all crystallographic sites, because of the different elemental contrast. Table 1 compares the atomic number Z (an indicator of the X-ray scattering strength) and the coherent neutron scattering lengths, b_coh, for the elements relevant to this work [21].

The magnetic properties of Ca₄LnO(BO₃)₃ were recently reported [11]. Ca₄TbO(BO₃)₃ displayed a broad peak in magnetic susceptibility at T = 3.6 K, suggestive of low-dimensional and/or short-range magnetic order. No magnetic ordering was observed above 2 K in the remaining nine materials (Ln = Pr, Nd, Sm, Eu, Gd, Dy, Ho, Er and Yb); Ca₄PrO(BO₃)₃ is believed to have a non-magnetic singlet ground state and Ca₄SmO(BO₃)₃ and Ca₄EuO(BO₃)₃ display van Vleck paramagnetism. The magnetic behaviour was linked to the crystal structure: PXRD analysis indicated the absence of cation disorder within experimental error, leading to well-separated Ln³⁺ (M1) chains along the c-axis. It was postulated that this quasi-one-dimensional arrangement of magnetic ions led to the observed suppression of magnetic ordering temperatures.

In this study, room-temperature powder neutron diffraction (PND) was carried out on six samples with Ln = Pr, Nd, Tb, Ho, Er and Yb. To our knowledge, this is the first use of neutron scattering to examine the structure of Ca₄LnO(BO₃)₃. Combined X-ray and neutron refinement was carried out in order to distinguish the positions of light atoms and simultaneously refine the cation fractional site occupancies. Low-temperature PND was also carried out on Ca₄TbO(BO₃)₃ at 50 and 1.5 K to explore the possibility of magnetic ordering. However, no evidence for this was observed in our measurements. Finally, we present zero-field specific heat measurements on eight samples in the Ca₄LnO(BO₃)₃ series (Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er, Yb). We comment on the magnetic properties of these materials, obtained from susceptibility and heat capacity data, in the light of the results of the modified structural refinement obtained from combined PXRD and PND data.

2 Experimental

Polycrystalline samples of Ca₄LnO(BO₃)₃ (≈0.5 g) were synthesised from CaCO₃, H₃BO₃ and Ln₂O₃ (Ln = Nd, Gd, Dy, Ho, Er, Yb), Pr₆O₁₁ or Tb₄O₇ as described elsewhere [11]. Lanthanide oxides were pre-dried at 800 °C overnight prior to weighing out. Stoichiometric amounts of the reagents were ground with a pestle and mortar and heated in an alumina crucible for 4 h at 900 °C. The powder was then cooled, reground, pressed into a 13 mm pellet and reheated to 1200 °C for 24 h. This step was repeated typically 2–4 times until the amounts of impurity phases no longer changed, as judged by Rietveld refinement of laboratory X-ray diffraction data. The heating rate was 3 °C per minute and the samples were cooled to room temperature by switching off the furnace. For Ln = Pr, Nd, Tb, Ho, Er and Yb, larger samples (≈2 g) for neutron diffraction were synthesised in the same manner using ¹¹B-enriched boric acid (99 atom % ¹¹B, Sigma Aldrich) to avoid the strong absorption of neutrons by ¹⁰B [21].

Powder X-ray diffraction (PXRD) was carried out on a Bruker D8 X-ray diffractometer (CuKα, λ = 1.541 Å) in the range 10 ≤ 2θ(°) ≤ 90 with a step size of 0.01° and measurement time 1 s per step. The ¹¹B-enriched Ca₄YbO(BO₃)₃ sample for neutron diffraction was found by PXRD to contain 21 ± 2% Y³⁺ ions instead of Yb³⁺ because the synthesis was erroneously carried out using an impure reagent. The distribution was assumed to be random because of the chemical similarity of Y and Yb. Owing to the Covid-19 pandemic, a new phase-pure ¹¹B-enriched sample for neutron diffraction could not be synthesised before the scheduled beamtime. The effective ionic radius for Ln³⁺ in this compound was calculated using a weighted average to give r(Yb*) = 87.5 pm.

Powder neutron diffraction (PND) was carried out on the D2B high-resolution powder diffractometer at the ILL, Grenoble [22], with a neutron wavelength of 1.593 Å. The neutron wavelength was refined by fixing the lattice parameters at values obtained from laboratory PXRD data, in order to ensure consistency between the datasets. Measurements at T = 300 K were carried out on all six samples using an automated sample changer. An Orange cryostat was used to measure diffraction patterns of Ca₄TbO(BO₃)₃ at T = 1.5 and 50 K.

Combined X-ray and neutron Rietveld refinements [23] were carried out using the program Topas [24]. The refinements used X-ray and neutron data collected on the same samples. The background for each dataset was modelled with a 12-coefficient Chebyshev polynomial and the peak shape was modelled with a modified Thompson-Cox-Hastings pseudo-Voigt function with axial divergence asymmetry [25].

Zero-field specific heat measurements were made using a Quantum Design Physical Properties Measurement System (PPMS) DynaCool. Measurements were taken on warming, at intervals of 0.1 K in the range 1.8 ≤ T (K) ≤ 5 and then at intervals of one degree up to 30 K; three datapoints were taken at each temperature. Samples were mixed with an equal mass of Ag powder (Alfa Aesar, 99.99%, −635 mesh) to improve thermal conductivity before being pressed into a 1 mm thick pellet for measurement. Apiezon N grease was used to provide thermal contact between the sample platform and the pellet. The contribution of Ag to the total heat capacity was subtracted using values from the literature [26].

3 Results

3.1 Room temperature crystal structure

PXRD and PND measurements were carried out on Ca₄LnO(BO₃)₃ with Ln = Pr, Nd, Tb, Ho, Er and Yb*, where Yb* indicates a random distribution of 21% Y and 79% Yb over the Ln sites as described in the Experimental section.

For each sample, a combined X-ray and neutron Rietveld refinement was carried out in space group Cm (No. 8) with a 1:1 weighting of the datasets. The systematic zero point error, unit cell parameters (Table 2), atomic fractional coordinates, and the thermal parameters for each site were refined. Minor impurity phases were found for Ln = Pr, Er and Yb* and their lattice parameters refined to less than 0.2% different from the literature reported values. Additionally the fractional occupancy of Ca²⁺ and Ln³⁺ on each of the three metal sites, M1, M2 and M3, was refined with the overall stoichiometry constrained to [4Ca²⁺]:[1Ln³⁺]. The atomic coordinates, thermal parameters and bond lengths obtained from Rietveld refinement are given in Tables 3 and 4. A representative refinement is shown in Figure 2; refinements for the other five compounds are available in the Supplementary Material, Figures S1–S5. The bond valence sums for all the atoms were calculated using Vesta [27] and are given in the Supplementary Material, Table S1.

Table 2:

Crystallographic data for Ca₄LnO(BO₃)₃ in space group Cm (No. 8). From combined Rietveld refinement of PXRD and PND data at T = 300 K. Yb* indicates a random distribution of 21% Y and 79% Yb over the Ln sites. The combined or global R_wp in the table is the average of R_wp for the X-ray and neutron datasets.

Ln	Pr	Nd	Tb	Ho	Er	Yb*
a/Å	8.13477(6)	8.12367(5)	8.08658(4)	8.08007(5)	8.07415(5)	8.06235(6)
b/Å	16.06651(11)	16.04587(9)	16.02721(8)	16.01665(9)	16.00724(10)	16.00555(11)
c/Å	3.601885(24)	3.595667(19)	3.550586(16)	3.533016(18)	3.525694(20)	3.519604(23)
β/°	101.3856(5)	101.3979(4)	101.2443(4)	101.1826(4)	101.1575(4)	101.1670(5)
V/Å3	461.493(6)	459.457(4)	451.341(4)	448.547(4)	447.065(5)	445.579(5)
Impurities	0.18(7)% Pr6O11	None	None	None	0.73(11)% Er2O3	2.90(7)% Yb2O3
Rwp/%	5.06	4.86	4.17	3.99	4.80	5.06
Rexp/%	3.05	2.88	2.72	2.60	2.58	2.71
χ 2	2.77	2.84	2.35	2.36	3.45	3.47

Table 3:

Atomic positions, site occupancy factors ( occ Ca = 1 − occ L n ) and isotropic displacement parameters for Ca₄LnO(BO₃)₃. From combined Rietveld refinement of PXRD and PND data at T = 300 K in space group Cm. M1, M2 and M3 represent the three different cation sites; occ _Ln and occ_Ca represent the fractional Ln³⁺ and Ca²⁺ site occupancies. Yb* indicates a random distribution of 21% Y and 79% Yb over the Ln sites.

Ln		Pr	Nd	Tb	Ho	Er	Yb*
M1: 2a (0, 0, 0)	occ Ln	0.923(9)	0.942(8)	0.875(7)	0.841(10)	0.801(7)	0.739(7)
	Biso/Å2	1.10(5)	0.32(3)	0.47(3)	0.46(4)	0.92(4)	0.80(4)
M2: 4b (x, y, z)	x	0.1445(4)	0.1424(3)	0.14149(28)	0.1411(3)	0.1400(3)	0.1416(3)
	y	0.38686(13)	0.38688(12)	0.38762(10)	0.38771(11)	0.38758(12)	0.38731(11)
	z	0.3307(10)	0.3318(7)	0.3273(7)	0.3249(8)	0.3261(8)	0.3259(8)
	occ Ln	0.022(5)	0.013(4)	0.063(4)	0.080(5)	0.100(4)	0.130(4)
	Biso/Å2	0.33(4)	0.33(4)	0.27(4)	0.17(4)	0.44(5)	0.42(5)
M3: 4b (x, y, z)	x	0.2665(4)	0.2659(3)	0.2603(3)	0.2579(4)	0.2576(4)	0.2564(4)
	y	0.18095(14)	0.18170(13)	0.18121(12)	0.18043(15)	0.18049(16)	0.17930(16)
	z	0.6635(9)	0.6633(7)	0.6521(7)	0.6485(8)	0.6484(9)	0.6467(8)
	occ Ln	0.017(5)	0.017(4)	0.000(4)	0.000(6)	0.000(4)	0.000(3)
	Biso/Å2	0.47(4)	0.41(4)	0.53(4)	0.73(5)	0.83(5)	0.76(6)
O1: 2a (x, 0, z)	x	0.8326(22)	0.8270(21)	0.8475(18)	0.8597(22)	0.8365(21)	0.8742(18)
	z	0.404(5)	0.455(5)	0.432(5)	0.481(7)	0.436(5)	0.446(5)
	Biso/Å2	0.00(4)	0.03(4)	0.35(4)	0.40(5)	0.62(6)	0.82(6)
O2: 4b (x, y, z)	x	0.4671(16)	0.4660(3)	0.4671(14)	0.4629(16)	0.4556(14)	0.4599(16)
	y	−0.0808(7)	−0.0765(6)	−0.0792(6)	−0.0725(8)	−0.0793(6)	−0.0766(7)
	z	0.752(3)	0.7480(12)	0.7433(25)	0.750(4)	0.7376(28)	0.735(3)
	Biso/Å2	0.72(4)	0.65(3)	0.61(3)	0.59(4)	0.66(4)	0.64(4)
O3: 2a (x, 0, z)	x	0.2226(25)	0.2140(24)	0.2209(20)	0.2150(24)	0.1997(25)	0.2117(21)
	z	0.613(6)	0.629(5)	0.608(5)	0.616(6)	0.631(5)	0.6062(11)
	Biso/Å2	0.98(6)	0.84(5)	0.70(5)	0.82(5)	0.78(6)	0.80(6)
O4: 4b (x, y, z)	x	0.0929(15)	0.0986(13)	0.0920(12)	0.0692(18)	0.0728(13)	0.0776(3)
	y	0.1503(7)	0.1458(6)	0.1469(6)	0.1472(6)	0.1385(6)	0.14207(13)
	z	0.107(4)	0.089(4)	0.077(4)	0.094(4)	0.060(4)	0.0696(9)
	Biso/Å2	0.73(4)	0.63(3)	0.41(3)	0.48(3)	0.57(4)	0.43(4)
O5: 4b (x, y, z)	x	0.9772(16)	0.9713(14)	0.9670(13)	0.9951(17)	0.9728(13)	0.9660(4)
	y	0.2693(7)	0.2694(6)	0.2718(5)	0.2759(6)	0.2702(6)	0.27031(14)
	z	0.277(3)	0.2869(27)	0.2834(25)	0.293(4)	0.2601(26)	0.2707(9)
	Biso/Å2	0.68(4)	0.76(4)	0.70(4)	0.72(4)	0.70(5)	0.62(5)
O6: 4b (x, y, z)	x	0.7885(18)	0.7897(16)	0.7892(15)	0.7896(16)	0.7865(15)	0.7835(3)
	y	0.1706(8)	0.1712(7)	0.1724(6)	0.1661(8)	0.1791(6)	0.17479(15)
	z	−0.124(4)	−0.088(3)	−0.118(3)	−0.078(4)	−0.123(3)	−0.1229(8)
	Biso/Å2	0.78(4)	0.75(3)	0.60(3)	0.57(3)	0.67(4)	0.57(4)
B1: 2a (x, 0, z)	x	0.400(5)	0.384(4)	0.388(4)	0.382(5)	0.387(4)	0.3718(4)
	z	0.7066(11)	0.734(10)	0.749(9)	0.739(11)	0.723(9)	0.6965(10)
	Biso/Å2	0.51(4)	0.47(4)	0.26(4)	0.18(4)	0.29(5)	0.28(5)
B2: 4b (x, y, z)	x	0.9545(4)	0.949(3)	0.941(3)	0.931(3)	0.9446(4)	0.9432(3)
	y	0.19576(10)	0.1977(13)	0.2031(13)	0.2154(17)	0.19479(12)	0.19500(11)
	z	0.0887(10)	0.112(6)	0.105(6)	0.115(9)	0.0754(9)	0.0727(7)
	Biso/Å2	0.21(3)	0.18(3)	0.21(3)	0.22(3)	0.26(3)	0.18(3)

Table 4:

Refined bond lengths in metal-oxygen polyhedra, T = 300 K. Yb* indicates a random distribution of 21% Y and 79% Yb over the Ln sites.

Ln	Pr	Nd	Tb	Ho	Er	Yb*
Atoms	Bond length/Å
M1–O1	2.283(4)	2.273(3)	2.232(3)	2.226(4)	2.229(4)	2.235(4)
M1–O1	2.297(4)	2.287(3)	2.257(3)	2.255(3)	2.240(5)	2.238(4)
M1–O3	2.416(5)	2.409(4)	2.331(3)	2.320(4)	2.315(4)	2.307(4)
M1–O4 × 2	2.475(2)	2.457(2)	2.385(2)	2.372(2)	2.356(2)	2.358(2)
M1–O3	2.491(4)	2.475(3)	2.429(3)	2.416(3)	2.406(4)	2.401(4)
Average M1–O	2.406(9)	2.393(7)	2.337(7)	2.327(8)	2.317(9)	2.317(9)
M2–O6	2.339(5)	2.335(3)	2.309(3)	2.308(3)	2.304(4)	2.281(3)
M2–O1	2.307(4)	2.321(3)	2.325(3)	2.321(3)	2.330(3)	2.319(3)
M2–O5	2.341(4)	2.331(3)	2.346(3)	2.344(3)	2.336(4)	2.333(3)
M2–O2	2.368(6)	2.360(4)	2.360(3)	2.348(4)	2.343(4)	2.352(4)
M2–O2	2.375(5)	2.366(4)	2.361(4)	2.363(4)	2.352(5)	2.358(4)
M2–O6	2.404(5)	2.411(4)	2.373(4)	2.359(4)	2.355(4)	2.343(4)
Average M2–O	2.356(11)	2.354(9)	2.346(8)	2.341(9)	2.337(10)	2.331(9)
M3–O4	2.325(5)	2.323(4)	2.336(3)	2.335(4)	2.336(4)	2.329(4)
M3–O4	2.339(5)	2.345(3)	2.348(4)	2.343(4)	2.354(5)	2.342(4)
M3–O2	2.340(4)	2.339(3)	2.346(3)	2.338(3)	2.344(4)	2.331(4)
M3–O6	2.512(3)	2.508(3)	2.466(3)	2.459(3)	2.462(4)	2.467(4)
M3–O5	2.478(5)	2.477(4)	2.471(4)	2.471(4)	2.465(5)	2.475(4)
M3–O5	2.595(5)	2.591(3)	2.612(3)	2.622(4)	2.633(4)	2.626(4)
M3–B2	2.771(4)	2.767(3)	2.761(3)	2.770(3)	2.766(3)	2.772(3)
M3–O5	2.930(4)	2.912(3)	2.866(3)	2.857(4)	2.844(4)	2.855(4)
M3–O3	2.945(2)	2.955(2)	2.944(2)	2.928(2)	2.928(3)	2.908(3)
Average M3–Oa	2.432(10)	2.431(8)	2.430(8)	2.428(9)	2.432(11)	2.429(10)
B1–O2 × 2	1.383(3)	1.384(2)	1.371(2)	1.377(3)	1.374(3)	1.374(3)
B1–O3	1.377(5)	1.375(4)	1.385(4)	1.378(4)	1.382(5)	1.379(5)
B2–O5	1.375(3)	1.377(3)	1.374(3)	1.378(3)	1.384(4)	1.386(3)
B2–O6	1.385(4)	1.379(3)	1.378(3)	1.374(3)	1.369(4)	1.375(3)
B2–O4	1.387(4)	1.388(3)	1.387(3)	1.382(3)	1.379(4)	1.377(3)

1. ^aThe average M3–O bond length is taken over the six nearest oxygen atoms only, as explained in the Discussion.

Figure 2:

Combined Rietveld refinement of (a) PND, (b) PXRD data for Ca₄HoO(BO₃)₃. Red dots – experimental data; black line – calculated intensities; green line – difference pattern; blue tick marks – Bragg reflection positions.

3.2 Variable-temperature PND

Powder neutron diffraction patterns for Ca₄TbO(BO₃)₃ were measured at 1.5 and 50 K using a cryostat and Rietveld refinement was carried out using the wavelength obtained at 300 K, λ = 1.593186 ( 11 )  Å. The Tb³⁺ and Ca²⁺ fractional site occupancies at lower temperatures were fixed at the values obtained at 300 K, which is appropriate for solid-state compounds where the composition is imprinted during synthesis [28]. The Rietveld refinements are shown in Figure 3 and the refined unit cell parameters as a function of temperature are given in Table 5. Atomic positions and displacement parameters are available in the Supplementary Material, Table S2.

Figure 3:

Rietveld refinements for Ca₄TbO(BO₃)₃ at (a) T = 300 K (combined X-ray and neutron refinement), (b) T = 50 K (neutron only), (c) T = 1.5 K (neutron only). Red dots – experimental data; black line – calculated intensities; green line – difference pattern; blue tick marks – Bragg reflection positions.

Table 5:

Results of Rietveld refinement for Ca₄TbO(BO₃)₃ at room temperature (combined X-ray and neutron), 50 K (neutron only) and 1.5 K (neutron only).

Parameter	300 K	50 K	1.5 K
a/Å	8.08658(4)	8.07043(5)	8.07032(5)
b/Å	16.02721(8)	16.01977(10)	16.02037(11)
c/Å	3.550586(16)	3.545108(20)	3.545059(21)
β/°	101.2443(4)	101.2554(5)	101.2582(5)
Volume/Å3	451.341(4)	449.519(5)	449.519(5)
Rwp/%	4.17	2.43	2.65
Rexp/%	2.72	1.68	1.69
χ 2	2.35	2.11	2.46

3.3 Specific heat

Zero-field specific heat was measured between 1.8 and 30 K for eight samples of Ca₄LnO(BO₃)₃, Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er, Yb. The Debye temperatures, given in Table 6, for these samples were estimated by fitting the total specific heat to the Debye approximation, i.e. a T³ dependence [29]. The lattice contribution was then subtracted and the resultant magnetic specific heat C_mag/T is plotted in Figure 4 for each sample.

Table 6:

Debye temperatures θ_D obtained for the Ca₄LnO(BO₃)₃ samples with Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er and Yb.

Ln	θD/K	Ln	θD/K
Pr	363	Dy	395
Nd	355	Ho	371
Gd	397	Er	340
Tb	404	Yb	342

Figure 4:

Magnetic specific heat C_mag/T for Ca₄LnO(BO₃)₃ samples with Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er and Yb.

4 Discussion

To our knowledge, the previous diffraction studies on Ca₄LnO(BO₃)₃ have been restricted to X-ray radiation, where, even with single-crystal data, it is difficult to distinguish light atoms such as boron and oxygen in the presence of much heavier metal ions such as lanthanides. Neutron diffraction provides a different elemental contrast (Table 1) allowing the B and O atoms to be located with high accuracy. However, the Ca and Ln atoms (particularly Ln = Pr) are more similar in terms of their neutron scattering lengths compared with X-rays, making the analysis of cation disorder more difficult if using neutron data alone. Therefore, we employed a combined neutron and X-ray refinement method to study the structure of six samples of Ca₄LnO(BO₃)₃. The elements Gd, Sm, Eu, and to a lesser extent Dy contain isotopes which are strongly neutron-absorbing, making those compounds unsuitable for neutron scattering studies without isotopic enrichment. They are therefore excluded from this study.

The results of the combined refinements are given in Tables 2–4. The obtained unit cell volumes are in good agreement with previous reports and show a linear variation with the ionic radius of Ln³⁺ [2, 3, 8, 11], [12], [13], [14], [15], [16], [17], [18], [19], [20], Figure 5(a). Sharp Bragg peaks in both the X-ray and neutron patterns indicate large crystalline domains. In the lanthanide orthoborates, LnBO₃, some peak broadening was observed using neutron diffraction while X-ray diffraction on the same samples showed sharp peaks; this was attributed to local disorder within the borate groups, which are not well resolved by X-rays [30]. No such effect was observed in the present study. The cation ordering and boron and oxygen atom positions are discussed in more detail below.

Figure 5:

Crystallographic parameters for Ca₄LnO(BO₃)₃. (a) Unit cell volumes; (b) Ln³⁺ occupancies in metal sites M1 and M2, as a function of 6-coordinate Ln³⁺ radius [31]. Error bars are smaller than symbols. Straight lines show least-squares fits [2, 3, 8, 10], [11], [12], [13, 15, 16, 18, 19].

There exists some disagreement between different authors about the ordering of cations in Ca₄LnO(BO₃)₃. The first two reports of the structure type in the compound Ca₄SmO(BO₃)₃ were by Khamaganova [12] and Norrestam [2]. In both studies the structure was reported as containing three cation sites, each fully occupied by ions of a single element: Sm, Ca(1) and Ca(2). These correspond to M1, M2 and M3 respectively in the notation used in this article; other authors also use X, Y and Z respectively [3]. Shortly afterwards Ilyukhin and Dzhurinskij [13] measured single crystals of Ca₄LnO(BO₃)₃ with Ln = Gd, Tb and Lu; they found disorder between the M1 (predominantly Ln³⁺) and M2 (predominantly Ca²⁺) sites, while the M3 site remained fully occupied by Ca²⁺. This behaviour was ascribed to the coordination geometry surrounding each site, with the M3–O polyhedron being described as a highly distorted bicapped trigonal prism (i.e. 8-coordinate) while M1 and M2 were in approximately octahedral coordination [13]. Later studies on Ca₄LnO(BO₃)₃ support this hypothesis and show an increasing level of cation disorder with decreasing lanthanide ion radius [3, 15], Figure 5(b). We note that while the M3 (Ca²⁺) site may be treated as eight-coordinate [13, 15, 17], two of the eight Ca–O bond lengths are significantly longer than the other six. In the compounds studied here, the ‘long’ Ca–O bond lengths are all greater than 2.84 Å whereas the ‘short’ bonds range from 2.32 to 2.63 Å and the shortest Ca–B distance is approximately 2.77 Å, Table 4. We therefore favour the description of the M3–O polyhedron as a highly distorted octahedron [2, 3, 10].

We studied six samples covering a wide range of lanthanide ion radii. We confirmed that, within experimental error, the M3 sites are solely occupied by Ca²⁺ ions (Table 3). The M1 and M2 metal sites are occupied by both Ca²⁺ and Ln³⁺ ions. The cation distribution is correlated with the lanthanide ion radius, in agreement with a number of previous reports, and ranges from approximately 5 to 25% of the M1 sites being occupied by Ca²⁺ (Figure 5(b)). However, Mougel et al. studied single-crystal Ca₄GdO(BO₃)₃ and found that the effects of secondary extinction were of the same magnitude as the possible M1/M2 site disorder of less than 3%. The authors therefore concluded that any site disorder was within the accuracy range of the measurement, and that the discrepancies in prior reports could be explained by a difference in cooling speed and/or crystallite size [8]. Furthermore, Münchhalfen et al. found that the degree of disorder in Ca₄GdO(BO₃)₃ increased with annealing temperature for samples quenched from temperatures between 773 and 1473 K [10]. It is clear that the problem of cation disorder in Ca₄LnO(BO₃)₃ is non-trivial as it is finely affected by the synthetic conditions (annealing temperature, cooling rate, flux material, powder or single-crystal material) as well as the radius of the lanthanide ion.

The trend in cation site occupancies is borne out in the refined bond lengths for Ca₄LnO(BO₃)₃. Figure 6 shows the average M–O bond lengths for the three metal sites, treating all three as six-coordinate as explained above. The average M1–O bond length increases as the lanthanide ionic radius increases. The same occurs for M2–O but with a smaller rate of increase, which corresponds to the smaller Ln³⁺ occupancy of the M2 site. Finally the M3 site, which is exclusively occupied by Ca²⁺, has bond lengths effectively independent of the lanthanide radius. The data in Figure 6 were fitted to straight lines with gradients 0.80(7), 0.200(21) and 0.021(14) for M1, M2 and M3 respectively. Similar trends were observed by Kuzmicheva et al. with values of 0.613 for M1 and 0.325 for M2 [15], and by Münchhalfen et al. [3].

Figure 6:

Average bond lengths in the MO₆ polyhedra in Ca₄LnO(BO₃)₃ as a function of lanthanide radius [31]. Straight lines show the least-squares fit for each site.

The unit cell parameters of Ca₄TbO(BO₃)₃, a, b and c all decrease on cooling from 300 to 50 K (Table 5), and their values at 1.5 and 50 K are equal within experimental error (3σ level). The monoclinic angle β increases linearly by a small but significant amount on cooling. Several studies on Ca₄LnO(BO₃)₃ materials above room temperature have shown anisotropic thermal expansion [1, 8, 10]. This was attributed to a characteristic property of inorganic borates: a tilting or ‘hinge’ mechanism of the BO₃ groups, which remain planar and rigid while the metal-oxygen polyhedra are stretched [32]. Such tilting has also been observed as a function of lanthanide ion size in the series Ln = Sm, Eu, Gd, Tb, Lu. Atom B1 is located on a mirror plane so its BO₃ group orientation is independent of Ln, while the BO₃ groups of site B2 change their orientation relative to the a, b and c axes as the lanthanide ion is varied [14]. In the PND data for Ca₄TbO(BO₃)₃, however, the positions of the B and O atoms, and therefore the orientations of the BO₃ groups (Figure S6 in Supplementary Material), did not change significantly over the three temperatures studied.

The postulated short-range and/or low-dimensional magnetic ordering in Ca₄TbO(BO₃)₃ at 2–4 K, based on susceptibility [11] and specific heat data, would be expected to produce an increase in diffuse neutron scattering upon cooling. Figure 7 shows the neutron diffraction patterns collected on Ca₄TbO(BO₃)₃ at 1.5 and 50 K and the difference plot. The low-temperature diffraction patterns cannot be compared meaningfully with the room-temperature pattern because the latter was measured outside the cryostat, leading to a different amount of beam attenuation by the sample environment.

Figure 7:

PND patterns for Ca₄TbO(BO₃)₃ at 50 K (red) and 1.5 K (blue), and the difference 1.5–50 K (black), as a function of (a) 2θ, (b) d-spacing. N.B.: The shape of the curves below 2θ ≈ 6° is an instrumental artefact related to the beam stop position.

At all 2θ > 20°, the diffraction patterns at 1.5 and 50 K have identical backgrounds and peak intensities. At low 2θ values (large d-spacings, >5 Å), there is an upturn in the baseline in the 50 K data, indicating the presence of the paramagnetic scattering contribution. This feature disappears on cooling to 1.5 K, suggesting an increase in magnetic correlations; however, within the sensitivity of this instrument there is no clear evidence for additional magnetic Bragg or diffuse scattering at 1.5 K.

In the specific heat data for Ca₄LnO(BO₃)₃, no sharp λ-type anomaly, characteristic of three-dimensional long-range magnetic ordering, occurred above 1.8 K for any of the eight compounds. For Ca₄PrO(BO₃)₃ a broad feature across the range 5–30 K may have the same origin as the previously reported broad feature in the magnetic susceptibility, i.e. a singlet (non-magnetic) ground state [11]. A similar broad hump occurs for Ca₄ErO(BO₃)₃, and the heat capacity for Ca₄TbO(BO₃)₃ also has a broad peak at 2–3 K which is consistent with the magnetic susceptibility data. For the other compounds, Ln = Nd, Gd, Dy, Ho, Yb, the magnetic contribution to the specific heat is close to zero down to approximately 8 K. Below this temperature, the plots begin to curve upwards indicating the onset of an ordering feature below the lowest measurement temperature of 1.8 K. The magnetic heat capacity of Ca₄YbO(BO₃)₃ appears to reach a maximum at T = 1.9 K, but additional datapoints would be required to confirm this.

The specific heat data for Ca₄LnO(BO₃)₃ are consistent with our previous bulk magnetic susceptibility measurements, which indicated the absence of long-range magnetic ordering above 2 K [11]. We had proposed that the suppression of ordering temperatures was linked to a quasi-one-dimensional (1D) magnetic lattice, as shown by PXRD, with the magnetic Ln³⁺ ions occupying only the M1 crystallographic sites. However, in this work, combined refinements of X-ray and high-resolution neutron diffraction data revealed site mixing with Ca²⁺ and Ln³⁺ ions disordered over the M1 and M2 sites. As such, the lattice of magnetic Ln³⁺ ions in Ca₄LnO(BO₃)₃ can not necessarily be said to be “quasi-one-dimensional”, although the degree of disorder is relatively small for Ln = Pr and Nd: these materials could certainly be treated as quasi-1D to a first approximation.

The existence of cation disorder in Ca₄LnO(BO₃)₃ will have important implications for the magnetic behaviour. Magnetic ordering is driven by spin-spin interactions. In the ideal cation-ordered case, the Ca₄LnO(BO₃)₃ crystal structure would have one unique nearest-neighbour exchange interaction, along the quasi-1D M1 chains. In the presence of disorder, there will be a wide range of different near-neighbour Ln–Ln distances and superexchange pathways, leading to a range of near-neighbour dipolar and exchange coupling interactions. These interactions might compete with one another, leading to magnetic frustration which in turn could give rise to a spin-liquid or spin-glass state.

Site disorder and resultant frustration in magnetic materials has been studied in the YbFe₂O₄ structure type [33], including the celebrated examples of YbMgGaO₄ and ErMgGaO₄, quantum spin-liquid candidates. In these materials the Ln³⁺ layers remain intact but disorder occurs between the non-magnetic Mg²⁺ and Ga³⁺ ions, altering the Ln³⁺ crystal electric field [34], [35], [36]. There are several other well-known examples of materials which display signatures of low-dimensional magnetism despite the presence of structural disorder. One example with spin-1 vanadium ions, the warwickite MgVOBO₃, shows a spin-glass transition at 6 K [37], while the spin- 1 2 analogue MgTiOBO₃ remains paramagnetic down to 1.8 K [38]. Another spin- 1 2 material, LiCuSbO₄, has short-range ordering below 10 K but no 3D ordering above 100 mK [39]. The interplay between structural and magnetic (dis)order has also been studied in several quasi-2D kagomé magnets including herbertsmithite (ZnCu₃(OH)₆Cl₂) [40]; barlowite (Cu₄(OH)₆FBr) and its analogues Cu₄(OH)₆FCl and Cu₄(OH)₆FI [41]; SCGO (SrCr _x Ga_9−xO₁₉) [42]; and Dy₃Mg₂Sb₃O₁₄ [43].

It is clear that compositional disorder plays a crucial role in the complex structure-property relationships of many low-dimensional magnets. Detailed structural characterisation is therefore essential before attempting to account for the observed magnetic behaviour of Ca₄LnO(BO₃)₃ or similar materials. Bulk characterisation should include neutron and/or single-crystal diffraction in addition to PXRD in order to get a full picture of the crystal structure. Ideally, local probes such as total scattering, nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR) or muon-spin relaxation (μSR) spectroscopy could also be used to complement the diffraction results.

5 Conclusions

We collected high-resolution powder neutron diffraction data and employed a combined X-ray and neutron refinement approach to study the calcium lanthanide oxyborates Ca₄LnO(BO₃)₃, Ln = Pr, Nd, Tb, Ho, Er, Yb. In all six compounds, the Ln³⁺ and Ca²⁺ ions are disordered over the M1 and M2 metal sites while the M3 site is fully occupied by Ca²⁺. The extent of Ca²⁺/Ln³⁺ disorder increases as the ionic radius of Ln³⁺ decreases across the lanthanide series.

Low-temperature powder neutron diffraction on Ca₄TbO(BO₃)₃ at 50 and 1.5 K showed no conclusive evidence of long- or short-range magnetic ordering, despite a cusp in the magnetic susceptibility at 3.6 K and broad peak in the heat capacity at a similar temperature. Zero-field specific heat data for Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er and Yb similarly showed an absence of long-range magnetic ordering above 1.8 K. The suppression of magnetic ordering is likely related to both the underlying crystal structure, which contains quasi-one-dimensional chains of metal ions, and the lanthanide-dependent compositional disorder investigated in this study, which creates competition between near-neighbour magnetic interactions. Future work will include specific heat measurements at T < 1.8 K as well as frequency-dependent ac susceptibility measurements to investigate the possibility of a spin-glass state in Ca₄TbO(BO₃)₃.