Low-temperature x-ray crystal structure analysis of the cage-structured compounds Be ( = La, Sm, and U)
The beryllides Be ( = rare earths and actinides) crystallize in a NaZn-type cubic structure, which can be categorized as a cage-structured compound. In this study, powder x-ray diffraction measurements have been performed on LaBe, SmBe, and UBe in the temperature range between 7 and 300 K in order to investigate their crystallographic characteristics systematically. They keep the NaZn-type cubic structure down to the lowest temperature. We estimated their Debye temperature to be 600–750 K from analyses of the temperature dependence of a lattice parameter, being in good agreement with the values reported previously. Rietveld refinements on the obtained powder patterns revealed that the atom in the 8 site is located in an almost ideal snub cube formed by 24 Be atoms in the 96 site, whose caged structure is unchanged even at the low temperatures. In addition, it is argued from the temperature variation of an isotropic mean-square displacement parameter that the Be compounds commonly have a low-energy phonon mode, which can be described by a model assuming an Einstein oscillation of the atom with a characteristic temperature of 160 K.
For the past several decades, intermetallic compounds with a caged structure have been intensively studied as potential candidates for thermoelectric conversion applications. Sales1 (); Suekuni (); Sales2 () In the research field of the strongly correlated electron systems, cage-structured compounds, e.g., filled skutterudites, clathrates, -pyroclores, hexaborides, and AlV, have attracted much attention because of their novel phenomena, such as higher-rank multipole ordering in PrRuP Takimoto (), a magnetic-field-robust heavy-fermion (HF) state in SmOsSb Sanada (), and a low-energy phonon mode associated with local vibration of a guest atom with a large amplitude in an oversized host cage, so-called ârattlingâ. Suekuni (); Matsuhira (); Yamaura1 (); Yamaura2 (); Mandrus (); Caplin () Among them, the rattling has been considered to be related with several intriguing phenomena via interaction with conduction electrons, for example, a rattling-induced superconductivity Nagao (), unusual temperature dependence of the electrical resistivity Dahm (), and a magnetic-field-insensitive HF state due to the multi-level Kondo effect. Hattori () In contrast to the previous expectations, it has recently been claimed from neutron spectroscopy experiments that a picture of a freely ârattlingâ guest atom in a host cage is not applicable in the case of filled FeSb skutterudites. Koza () Thus, the issue of rattling is still controversial. To verify the underlying concept of rattling itself, it is highly necessary to examine crystallographic characteristics at low temperatures for several cage-structured systems.
The beryllide Be can also be categorized as a cage-structured compound, where is a rare earth or an actinide. They crystallize in a NaZn-type cubic structure with the space group (No. 226, ), where the unit cell contains atoms in the 8 site, Be atoms on the 8 site, and Be atoms in the 96 site. Bucher (); McElfresh (); Takegahara () The atoms are surrounded by 24 Be atoms, nearly forming an ideal snub cube, whereas the Be atoms are surrounded by 12 Be atoms, forming an icosahedron cage, as shown in Fig. 1.
For the phonon physics in Be, previous specific-heat and inelastic neutron scattering (INS) measurements revealed that UBe, ThBe, and possibly LaBe have a low-energy phonon mode, which can be described by an Einstein mode with characteristic temperatures of = 151, 157, and 177 K, respectively. Renker (); Hidaka () To discuss a possibility of an influence of the low-energy phonon mode on novel phenomena in the Be system, such as unconventional HF superconductivity and non-Fermi-liquid behavior in UBe Ott (); Mayer () and an intermediate valence state due to the strong - hybridization in CeBe Wilson (), it is necessary to elucidate characteristics of the low-energy phonon mode. It has been suggested that the atom behaves like an Einstein oscillator in UBe and ThBe Renker (), while other characteristics of the low-energy phonon mode in the present system have been veiled: for example, coupling between the guest ions and the host lattice, form of the potential energy (harmonic or anharmonic), and size effects of the cage. Systematic crystallographic studies on the low-energy phonon mode for the Be series would provide important information to unveil the universal nature of the phonon properties in Be with unique cagelike structures.
A low-temperature systematic investigation of crystal structure is also crucial to unveil the physical ground-state properties, such as a gap symmetry of the unconventional superconductivity in UBeSigrist () and crystalline-electric-field effects in all the magnetic Be compounds, both of which may be affected by some crystal distortion.ãA lattice parameter at room temperature (RT) has already been reported in all the Be compounds Bucher (); Benedict (), while the low-temperature data are limited in some Be compounds Kappler (); Goldman (). The fractional (0, , ) coordinates of the Be site at RT have been determined in several Be ( = La, Ce, Dy, Th, and U) Hidaka (); Goldman (); Hudson (), whereas their low-temperature values have been determined only in UBe and ThBe. Goldman () In addition, studies on an atomic displacement parameter for Be, which can be a useful guide to understand phonon properties Suekuni (); Yamaura1 (), have remained untouched even at RT except for UBe and ThBe Goldman ().
In this study, we investigated structure parameters and phonon properties of a superconductor LaBe with a superconducting transition temperature of 0.6 K Hidaka (), a possible helical magnet SmBe with a magnetic transition temperature of = 8.3 K Bucher (); Hidaka2 (); Mombetsu (), and an unconventional HF superconductor UBe Ott () by x-ray diffraction (XRD) measurements in temperatures of down to 7 K. The results of the XRD measurements and Rietveld analyses are presented in Sec. III: the temperature dependence of the lattice parameter in subsection A, the fractional (0, , ) coordinates in subsection B, and isotropic mean-square displacement parameters in subsection C.
Single crystals of LaBe, SmBe, and UBe were grown by an Al-flux method. The constitute materials were La with 99.9 purity, Sm with 99.9 purity, U with 99.5 purity, Be with 99 purity, and Al with 99.99 purity. Each M element was placed in an AlO crucible with Be and Al at an atomic ratio of La:Be:Al = 1:13:35, Sm:Be:Al = 1:13:30, and U:Be:Al = 1:13:40, and sealed in a quartz tube filled with ultrahigh-purity Ar gas of 150 mmHg. The sealed tube was kept at 1050 C for 1 week and then cooled at a rate of 2 C/h. The Al flux was spun off in a centrifuge and then removed by NaOH solution. The obtained single crystals were annealed for 2 weeks at 700 C. The annealed samples were ground into fine powders in the ethyl alcohol, and then put in a cupper holder for XRD.
The XRD measurements were performed by a conventional powder x-ray diffractometer (Rint 2000, Rigaku) using Cu K and K radiation in the angle range of 2 . A Gifford-McMahon refrigerator was used for the low-temperature measurements down to 7 K. Rietveld refinement on the powder XRD data was carried out using the RIETAN-FP program. Rietan ()
Iii Result and Discussion
iii.1 Lattice parameters
Figure 2 shows the powder XRD patterns of SmBe at (a) 300 and (b) 7 K. The obtained XRD patterns can be well fitted assuming the NaZn-type cubic structure. In addition, we do not observe any peaks associated with impurity phases except for reflections from the cupper holder. We note that any residual peaks are smaller than 0.3 of the principal reflection. In the present XRD measurements, we succeeded in observing weak signals, which originate from only the Be atoms, for example (531) reflection, promising determination of the fractional coordinates of the Be atom. In addition, there is no obvious difference in the XRD patterns between 300 and 7 K except for a shift of the peak positions on the 2 axis due to a change in the lattice parameter. Similar results were obtained in LaBe and UBe (not shown), indicating absence of a structural transition in this temperature region. The lattice parameter was determined from the obtained XRD peaks at high 2 angles (2 ) using the Bragg’s law. The obtained lattice parameter at 300 K and at the lowest temperature for LaBe, SmBe, and UBe are shown in Table I, which are approximately consistent with those reported previously. Bucher (); McElfresh ()
Figure 3 shows the temperature dependence of the lattice parameter () for (a) LaBe, (b) SmBe, and (c) UBe. For all three compounds investigated here, the lattice parameter decreases as the temperature is lowered, and levels off below 100 K. The data of UBe reported previously are also indicated as the open symbols in Fig. 3(c) Goldman (). Although the lattice parameter of our UBe sample is slightly smaller than that reported in Ref., the observed temperature variation is in good agreement with the previous result. Note that there is no anomaly in the () curve near of SmBe within the experimental accuracy. From almost linear temperature dependence in the () curves above 150 K, we estimated a thermal expansion coefficient ( 1/ d/d). The estimated values at 300 K are 8.8 10, 7.1 10, and 8.3 10 K for LaBe, SmBe, and UBe, respectively, as summarized in Table II. Similar () has also been reported in YBe, PrBe, LuBe, and ThBe, where can be estimated to be 7 – 9 10 K. Kappler () It is revealed that above 150 K in the Be system is almost independent of the atom.
To estimate a Debye temperature , the () curves were analyzed using the Debye-Grneisen approximation Sayetat () described by
where is the Boltzmann constant, is the Grneisen parameter, is the bulk modulus, and and are a lattice parameter and volume of unit cell at absolute zero, respectively. The obtained fitting parameters are (, , ) = (744 K, 10.491 , 1.4 10) for LaBe, (672 K, 10.299 , 1.1 10) for SmBe, and (606 K, 10.240 , 1.2 10) for UBe. The obtained values are listed in Table II along with the previously reported values of for LaBe and ThBe. Bucher (); Kappler (); Hidaka () These results indicate that the Be systems commonly have relatively high of 600–900 K, although it is necessary to consider an influence of the low-energy phonon mode for more proper discussion.
iii.2 Fractional (0, , ) coordinates of the Be site
We next show the Rietveld analyses of the obtained XRD patterns assuming the NaZn-type cubic structure. The calculated profiles for SmBe can be seen in Figs. 2(a) and 2(b). Here, the lattice parameter was fixed to the value determined by the analysis of the Bragg’s peak scan for the higher angle spectrum. The obtained and parameters in the fractional coordinates of the Be site and the typical reliability factors (, , and ) are shown in Table I. Figure 4 shows the and parameters of the Be site at R.T. for several Be compounds: = La, Sm, Dy, Th, and U. Goldman (); Hudson () The star symbol and the solid line indicate the ideal snub-cube ( = 0.17610, = 0.11408) and regular icosahedron [ = (1+)] coordinates, respectively. In all the measured Be compounds, the and parameters are not close to those in the regular icosahedron coordinates but close to those in the ideal snub-cube coordinates. Furthermore, it is revealed that the and parameters are almost temperature-independent within the accuracy of the measurements in the present compounds, as shown in Table I, and hence their caged structures keep the almost ideal snub cube even at the low temperatures. This result indicates that the symmetry of a crystalline electric field at the site remains unchanged down to 7 K, although the cage size becomes smaller accompanied by the decrease of the lattice parameter upon cooling.
iii.3 Isotropic mean-square displacement parameter
The isotropic mean-square displacement parameter of individual atoms for LaBe, SmBe, and UBe were also determined from the Rietveld analyses. Here, an isotropic atomic displacement parameter is defined as 8. In general, it is difficult to determine an absolute value of precisely, since the precision of is strongly influenced by experimental conditions, experimental error, and refinement methods. In the present analyses, for example, the refined of the Sm atom for SmBe took on a wide range of value from 0.003 to 0.029 depending on initial parameters at R.T. However, once one fixes the initial parameters to some values, the error of the refined becomes approximately 5. Therefore, in this study, the temperature dependence of [= ()] was obtained by the following procedure; (1) refine at 300 K using the Rietveld analysis with some initial parameters, (2) refine at 275 K using the initial parameters determined at 300 K, (3) refine at 250 K using the initial parameters determined at 275 K, and (4) repeat the process sequentially down to the lowest temperature. Note that we checked that the temperature variation of is hardly influenced by the absolute value of , i.e., the relative change in () can be used for analyses to obtain the values of and , as discussed in the next paragraph.
The obtained () of the atom are shown in Fig. 5. Each figure represents (a) La of LaBe, (b) Sm of SmBe, and (c) U of UBe. These experimental results were analyzed using the following two models Suekuni (); Bentien (): a Debye model
and an Einstein model
where, , and are the Planck constant, mass of the atom, and temperature independent disorder term, respectively. The obtained and are listed in Table II, including the results reported previously. Hidaka (); Renker () () of the atom decreases monotonously with decreasing temperature, as shown in Fig. 5. In the conventional Einstein model, the best fit provides of 163(15), 157(10), and 160(20) K for the La, Sm, and U atoms, respectively, which are approximately identical to the values estimated from the specific heat () and INS measurements. Hidaka (); Renker () On the other hand, in the Debye model, were estimated to be 287, 270 and 290 K for the La, Sm, and U atoms, respectively. These values are less than half compared with those estimated from the () and () data. Bucher (); Hidaka (); Kappler () These results indicate that the Einstein model gives a better description of () of the atom than the Debye model.
In () of the Be and Be atoms, it was quite difficult to perform reliable analyses in both the models due to almost temperature-independent behavior of in the investigated temperature range. It is noted that simulated curves based on the Einstein model with of 150 and 170 K obviously deviate from () of the Be and Be atoms (not shown). These findings suggest that the atom behaves like an Einstein oscillator with 160 K, as pointed out from the previous INS measurements for UBe and ThBe Renker (), whereas the Be atoms form the crystal lattice described by the Debye model.
The low-energy phonon mode, which can be explained by the Einstein phonon with harmonic oscillation, is considered to be a common feature in the Be systems. It is intriguing that the Be series appears to have almost the same . In other cage-structured compounds, such as filled skutterudites Yamaura1 (); Matsuhira (), -pyrochlore oxides Yamaura2 (), and clathrates Suekuni (), it has been found that shows a decreasing trend with increasing the free space for guest vibration. This decreasing trend in has been interpreted as follows; larger free space yields shallower guest-ion potential, resulting in a decrease in the energy of the local phonon mode Yamaura1 (). Here, we test the similar analysis concerning the free space, which has been performed for other cage-structured compounds. Figure 6 shows for LaBe, SmBe, ThBe, and UBe, plotted against the guest free distance determined at 300 K. Hidaka (); Renker () In this study, is defined as
where, is the distance between and Be, is the effective ionic radius of ( = 1.36 , = 1.24 , = 1.21 , and = 1.17 ) Shannon (), and is the Be metal-bonding radius (= 1.14 ) Yang (), respectively. Here, we tentatively used the effective ionic radius for the 12-coordination-number site as Shannon (), since there is no data for 24-coordination-number sites to our best knowledge. As seen in Fig. 6, is robust to a change in within the range of the error bar, suggesting that the above-mentioned interpretation for other cage-structured compounds is not simply applicable to the Be systems.
Another key parameter to discuss the low-energy phonon mode is the atomic mass. In the case of the Einstein oscillation, can be rewritten to (/), where is the Einstein frequency, is the spring constant, and is the mass of the Einstein oscillator. Assuming that the Be compounds have similar values of , it is expected that heavier mass of the atom (= ) yields smaller . In this assumption, for LaBe ( 139) was estimated to be 210 K, when we calculate it from of 160 K for UBe ( 238). This estimated for LaBe deviates from the experimental value of 170 K. This disagreement may be attributed to the present simple assumption that has similar values among the Be compounds, since the value of generally depends on and an ionic state of the guest ion. In this context, in the Be systems might be determined by a combination of several parameters, such as , , and possibly electronic states, resulting in similar values of . Furthermore, in this discussion, the low-energy phonon mode was simply treated as the Einstein phonon, which is independent of the Debye phonon, although actual phonon dispersion should be more complicated. Recently, inelastic x-ray scattering measurements in SmBe performed by Tsutsui . revealed that a phonon dispersion curve of the Sm atom shows a flat part near a zone boundary. The energy of the flat part is approximately 16 meV, which is consistent with of 157(10) K determined from the present XRD study. Tsutsui1 ()
Finally we comment on a relationship between the low-energy phonon mode and electronic states of conduction electrons in the Be systems. It has been theoretically proposed for cage-structured compounds that a possible enhancement of the quasiparticle mass is caused by the interaction between anharmonic vibration of the guest ion and conduction electrons. Hattori () The electronic specific-heat coefficient for Be ( = La, Sm, U and Th) are listed in Table II. The characteristic energy of the low-energy phonon mode appears to have no relation to the value of , implying that the HF state for UBe may not originate from the presence of the low-energy phonon mode. However, it has also been suggested that an electron–phonon coupling between the conduction electrons and the low-energy Einstein phonon also plays an important role in formation of the phonon-mediated HF state. Yu (); Kusunose (); Tsutsui2 () Therefore, it is necessary to check the strength of the electron–phonon coupling in the Be systems to reveal the role of the low-energy phonon mode in the HF state for UBe. To deepen our understanding of the low-energy phonon mode in the present systems, further studies, such as a systematic observation of the phonon dispersion and the strength of the electron–phonon coupling, are needed and now in progress.
We measured XRD on powdered LaBe, SmBe, and UBe at low temperatures in order to investigate their structure parameters and characteristics of the low-energy phonon. The obtained XRD patterns and the Rietveld refinements revealed that the present compounds keep a NaZn-type cubic structure with an almost ideal snub cube formed by 24 Be atoms involving the atom even at low temperatures down to 7 K. Furthermore, the present study provides crystallographic collateral evidence for the presence of the low-energy phonon modes common to the Be systems, which can be explained by an Einstein model. It is considered that the rigid Be framework can be treated as a Debye solid with of 600–800 K, while the M ions behave like Einstein oscillators with of 160 K. Interestingly, the obtained values in the present systems, including the values reported thus far, appear to be independent of either or in the snub cube, which is a characteristic feature not found in other cage-structured compounds.
Acknowledgements.The authors thank Dr. S. Tsutsui for fruitful discussions. The present research was supported by JSPS KAKENHI Grants No. JP20224015(S), No. JP25400346(C), No. JP26400342(C), No. JP15H05882, and No. JP15H05885(J-Physics).
- thanks: Present address: Condensed Matter Research Center and Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
- thanks: Present address: Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan
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