Introduction

The interest in studying the properties of borides, particularly of rare-earth elements with high boron content, is due to the unusual behaviour of the borides’ magnetic, lattice subsystems, and the possibility of practical applications [1–5]. The RB66 borides series (where R denotes a rare-earth metal) was extensively studied in the latter part of the twentieth century, with investigation of their structural, electric and optical properties and thermal conductivity. In recent works [6–9], the magnetic transformations in some RB66 compounds were revealed at temperatures lower than 1 K, a result which was unexpected for substances with such a small concentration of paramagnetic ions.

To successfully study the peculiarities of the magnetic subsystems of substances, it is often necessary to determine effect of the lattice contribution on the overall magnetic characteristics. To achieve this, in the majority of cases, one draws a comparison with the isostructural non-magnetic analogue. For RB66 compounds, this analogue is lutetium boride LuB66, which is a diamagnetic substance of the semi-conductive type.

Lutetium boride LuB66 has a YB66 type cubic crystalline structure, with Fm3c space group. For each unit cell, there are 24 formula units (24 metal atoms and 1,584 boron atoms). The large number of crystal voids in the LuB66 structure is responsible for a number of its properties, such as its glass-like character. Lutetium atoms fill the voids with centres (0.0589, 1/4, 1/4) and (0.05629, 1/4, 1/4); the occupancy rate is around 0.5.

The boron framework is made up of eight super-icosahedrons B12(B12)12, each of which consists of thirteen icosahedrons B12 (Fig. 1). The peculiarities of LuB66 crystalline structure and a large difference in mass between lutetium and boron atoms define the peculiarities of the boride lattice properties.

Fig. 1
figure 1

LuB66 structure scheme [2]. Larger dark circles super-icosahedrons(B12)13, smaller light circles В80 clusters, dumb-bells Lu atoms

Full size image

In the present work, we experimentally studied the temperature dependency of heat capacity C p(T) and crystalline structure parameter a(T) for lutetium boride LuB66 over a broad range of low temperatures, with the purpose of revealing the peculiarities of the behaviour of the phonon sub-system of boride.

Experiment

A powdered lutetium boride LuB66 sample was obtained via boron-thermal metal reduction from the oxide in a vacuum:

$$ {\text{Lu}}_{ 2} {\text{O}}_{ 3} + 1 4 3 {\text{B}} = 2 {\text{LuB}}_{ 70} + 3 {\text{BO}}\, \uparrow . $$

After the homogenising anneal at 1,000 °C, and melting in the arc furnace, the amount of boron in the lutetium boride decreased as its composition got closer to that of the stoichiometric case. The X-ray diffraction pattern of synthesised lutetium boride contained peaks associated with the only LuB66 phase as compared with the ASTM data file. According to the chemical analysis, the sample composition was 19.6 % Lu, 80.2 % B and 0.09 % O2. The pycnometric mass density of the sample, ρ pycn, was 2.430 × 103 kg m−3, which is 93 % of the X-ray density value. The difference between the X-ray and pycnometric density is caused by the porosity of the sample. Spectral analysis showed the presence of a small amount of tungsten admixture in the sample, amounting to around 1 % of the lutetium content.

The heat capacity of the LuB66 samples, at temperatures from 2 to 300 K, was measured in a calorimeter (Termax Ltd.). The calorimeter implements the classical adiabatic method of heat capacity investigation [10, 11]. The experimental error at 1.8–4.2 K is about 3 %, at 4.2–40 K is 2 % and at 40–350 K is 0.5 %.

The lattice parameter a(T) of LuB66 in the range of 5–300 K was determined using a X-ray diffractometer (DRON-7.0) in the Co-Kα radiation mode, with use of a helium cryostat. The calculation of a(T) values was made based on the experimental angular positions of 12 Bragg reflections [12]. The error in calculating the a(T) lattice parameter of LuB66 was ±10−4 nm.

At room temperature, the value of a(T) was 23.3850 Å, which agrees well with values quoted in the literature (a = 23.412 Å, [13]).

Results and discussion

The experimental temperature dependence of heat capacity C p(T) for lutetium boride LuB66 (Fig. 2) does not contain any visible anomalies; however, if the date are plotted as C p/T versus T 2 (inset in Fig. 2) and C p/T 3 versus T 2 coordinates (Fig. 3), one can see that at low temperatures (T < 10 K), the values of the boride heat capacity are anomalously high, and at T = 30 K there is an anomaly similar to the Einstein component of heat capacity.

Fig. 2
figure 2

Temperature dependence of lutetium boride LuB66 heat capacity. Points experimental data. Solid line calculated values [the relation (2)]; dashed line—the same, but without the Einstein contribution. Inset illustrates the presence of the contribution to the heat capacity at the lowest temperatures, which is proportional to 1st power of the temperature

Full size image
Fig. 3
figure 3

Dependence of C P/T 3 from T 2. Points, the solid line, dashed line—the same as in Fig. 2

Full size image

The values of the crystalline structure parameter a(T) obtained in the present work are shown in Fig. 4, alongside the calculated temperature dependence of the coefficient of thermal linear expansion α(T) for lutetium boride LuB66, in the interval of 5–300 K. a(T) and α(T) dependencies, similar to the C p(T) dependency, does not exhibit any visible anomalies in the temperature range studied.

Fig. 4
figure 4

Lattice parameter a(T) (1) and linear thermal expansion coefficient (2) for lutetium boride LuB66, and approximate computation (3)

Full size image

The analysis of the temperature dependence of heat capacity and the thermal expansion of LuB66 boride was made on the basis of the combined approach of Debye–Einstein.

For heat capacity:

$$ C_{\text{v}} = k_{1} D\left( {\frac{T}{{\varTheta_{{D_{1} }} }}} \right) + k_{2} D\left( {\frac{T}{{\varTheta_{{D_{2} }} }}} \right) + k_{3} E\left( {\frac{T}{{\varTheta_{E} }}} \right) + k_{4} T, $$
(2)

where D is the Debye function; E is the Einstein function; k i is the fraction of the i-th contribution to the boride total heat capacity; and \( \varTheta_{{D_{\text{i}} }} \) and \( \varTheta_{\text{E}} \) are characteristic temperatures [14]. The latter term describes the glass-like contribution, which linearly depends on the temperature [15–17].

For thermal expansion:

$$ \begin{aligned} \varDelta V(T)/V(T_{0} ) & = [V(T) - V(T_{0} )]/V(T_{0} ) = (x_{T} - x_{{T_{0} }} )/x_{{T_{0} }} \\ x_{T} & = \frac{3g}{{4c^{2} }}(\varepsilon - G\varepsilon^{2} - F\varepsilon^{3} ), \\ \end{aligned} $$
(3)

where:

$$ \varepsilon = k_{1} D\left( {\frac{T}{{\varTheta_{{D_{1} }} }}} \right) + k_{2} D\left( {\frac{T}{{\varTheta_{{D_{2} }} }}} \right) + k_{3} E\left( {\frac{T}{{\varTheta_{E} }}} \right). $$

Here, \( \varTheta_{{D_{i} }} \) and \( \varTheta_{E} \) are the characteristic Debye and Einstein temperatures, respectively; the parameters G, F, c, g, f characterise the anharmonic component of the interatomic interaction potential in the crystal; and T 0 = 300 K [18].

In Table 1, the approximate parameters for Eqs. (2) and (3) are provided, which were obtained by the least squares fit to the experimental data.

Table 1 The characteristics of LuB66 lattice dynamics according to calorimetric and X-ray data
Full size table

As can be seen in Figs. 2 and 3, we managed to describe the experimental dependencies of C p(T) and ΔV/V(T) for lutetium boride in the T < 300 K temperature range, following the process of approximating different thermal characteristics and using accepted approximations for similar values of characteristic temperature.

The high values of characteristic temperatures \( \varTheta_{{D_{1} }} \) and \( \varTheta_{{D_{2} }} \) can be ascribed to the boron sub-lattice, where the first value, \( \varTheta_{{D_{1} }} \), characterises the vibration of boron atoms in the sub-lattice, and the second value may characterise the vibration of boron clusters B12, which are the basic structural components of the boron sub-lattice. We should note that the characteristic temperatures of elemental boron, according to different publications, are 1,540, 1,300 and 1,200 K [19]. The vibration of lutetium atoms in a boride can be described by the Einstein model with comparatively low characteristic temperature Θ E = 140 K, which closely matches Θ D = 184 K (equivalent to Θ E = 138.4 K) [20] for metallic lutetium, and is close to Θ E, which is calculated according to the frequency of delocalisation of HO in the soft atomic potentials (SAP model): Θ E = ħω/k = 119 K [21].

The obtained distribution of characteristic temperatures in the LuB66 boride sub-lattice is consistent with the fact that for the calorimetric data, the ratio (k 1 + k 2)/k 3 amounts to =62, which is close to the ratio of boron atoms to lutetium atoms in a LuB66 molecule.

One cannot help noticing the considerable difference between the coefficients of the Einstein component k 3 in the approximation of the calorimetric data and the thermal expansion values. In this case, if heat capacity \( \sum\nolimits_{\text{i} = 1}^{3} {k_{\text{i}} } = 0. 9 9 7 \), i.e. close to 1 (one), as it has to be, then for thermal expansion a much greater value k 3 leads to \( \sum\nolimits_{\text{i} = 1}^{3} {k_{\text{i}} } = 1. 2 9 \). We should note that while conducting similar analysis of the thermal properties of rare-earth tetraborides [22] (unpublished work), we revealed the twofold excess of k 3 values obtained from X-ray data, over the calorimetric values of this coefficient. The sum \( \sum\nolimits_{\text{i} = 1}^{3} {k_{\text{i}} } \) for X-ray data also turned out to be more than one, although in this case it was marginal. Such anomalous behaviour of X-ray k 3 values is not clear and needs further study.

From the outlined arguments, looms the following picture: low-temperature data on heat capacity and thermal expansion for rare-earth borides can be described with the help of the Debye–Einstein combined approach; the values of the Debye components, describing the vibration of the boron sub-lattice, turn out to be similar for calorimetric and X-ray data; low-frequency vibrations, which we ascribe to the metal sub-lattice within the Einstein approximation, largely affect borides heat capacity at low temperatures and thermal expansion in the middle part of low-temperature region.

Conclusions

Calorimetric and X-ray research into the thermal properties of icosahedral lutetium boride LuB66 in the broad interval of low temperatures and the analysis of experimental data in the Debye–Einstein approximation enabled us to determine the character of ion oscillations in boron and metallic sub-lattices of a boride and to define the sub-lattices characteristic temperatures. It was found that vibrations in the boron sub-lattice, which are characterised by comparatively small masses of atoms and high values of bond energy, can be satisfactorily described by the Debye model. The vibrations of massive and comparatively weakly bonded lutetium atoms can be described well by the Einstein approximation. The latter conclusion clarifies why in earlier works [12, 22–24], it was difficult to separate the electronic and lattice contributions to the low-temperature heat capacity of rare-earth tetraborides in the simplest approximation, C v = aT + bT 3: the lattice contribution, besides the cubic Debye term, contains the Einstein component.