2021 Volume 62 Issue 3 Pages 338-341
Synchrotron X-ray diffraction experiments of an icosahedral Au–Al–Yb quasicrystal and its 1/1 crystalline approximant were performed under quasi-hydrostatic pressure up to approximately 20 GPa at room temperature. Their pressure–volume compression curves are similar to each other, and show a monotonic decrease, which indicates the structural stability of these compounds under pressure. The bulk moduli were determined to be B0 = 105.1(8) GPa for the quasicrystal and B0 = 108.1(7) GPa for the crystalline approximant. These values are explained by the composition-weighted average of the bulk moduli of each constituent element.
Pressure–volume relationship of an icosahedral Au–Al–Yb quasicrystal and its crystalline approximant.
Icosahedral Au–Al–Yb quasicrystal1) (q-Au–Al–Yb) is an attractive compound because it is the first example of an intermediate valence quasicrystal at ambient conditions whose Yb valence is 2.61+,2) and it shows non-Fermi-liquid behavior (i.e., quantum critical phenomena) at very low temperatures below several Kelvin without either doping, pressure, or field tuning.2,3) Many physical properties of q-Au–Al–Yb (e.g., magnetism, specific heat, electrical resistivity, and thermal expansion) have been studied in detail;2–4) however, the pressure–volume relationship, which is one of the basic physical properties, has not been reported yet.
Herein, we used high pressure X-ray diffraction (XRD) measurements to demonstrate the pressure–volume compression character of q-Au–Al–Yb, Au51Al34Yb15, with a six-dimensional lattice parameter of a6D = 7.448 Å.1) To accurately determine the pressure–volume relationship and bulk modulus, we carefully prepared experimental conditions that were representatively quasi-hydrostatic of the sample pressure. In addition, for comparison, we examined a periodic counterpart of q-Au–Al–Yb. The 1/1 cubic crystalline approximant of Au51Al35Yb14 (c-Au–Al–Yb), whose Yb valence is 2.80+,2) consists of the same Tsai-type atomic cluster as q-Au–Al–Yb but in a different arrangement, i.e., a body-centered cubic packing structure with a lattice parameter a0 = 14.500 Å.1)
The synchrotron XRD experiments of q-Au–Al–Yb and c-Au–Al–Yb under quasi-hydrostatic pressure were performed using specimens that consisted of single-crystalline domains.
An as-cast specimen of q-Au–Al–Yb was prepared from high-purity materials [Au (purity 99.99 mass%), Al (purity 99.999 mass%), and Yb (purity 99.9 mass%)] in an arc furnace with an argon atmosphere, as described in Ref. 1). The specimen was confirmed to be q-Au–Al–Yb by XRD measurements using Cu Kα radiation. A cracked piece of several tens of microns in size was sealed in a diamond anvil cell with a pressure medium and a ruby-chip pressure marker. Helium gas was used as a pressure medium to maintain the quasi-hydrostatic condition under pressure.5) Pressure was determined by the ruby fluorescence method with quasi-hydrostatic pressure scale for the helium pressure medium.6)
High pressure XRD experiments up to 18.0 GPa at room temperature were performed at BL22XU in SPring-8.7) The oscillation photographs of the cracked piece of the q-Au–Al–Yb alloy were acquired at different pressures using monochromatized incident X-rays of 19.98 keV (λ = 0.6205 Å) and an imaging plate detector in the two theta range up to 31°. Sharp diffraction spots were obtained from the single-crystalline domains of the specimen, which allowed us to acquire high-precision d-value data. The oscillation photographs were converted into two theta intensity plots, as shown in Fig. 1.
X-ray diffraction patterns of an icosahedral Au–Al–Yb quasicrystal (q-Au–Al–Yb) at ambient pressure (A. P.) and 18.0 GPa at room temperature. The oscillation photographs of a specimen piece, which consisted of single-crystalline domains, were converted into two theta intensity plots. Major reflections are indicated by six-dimensional indices.
In addition, high pressure experiments of c-Au–Al–Yb up to 19.8 GPa were performed using a similar procedure as the one mentioned above.
Figure 1 shows the diffraction patterns of q-Au–Al–Yb at ambient pressure and highest pressure, and Fig. 2 shows the diffraction patterns of c-Au–Al–Yb. Owing to good hydrostaticity under pressure, the diffraction peaks of q-Au–Al–Yb and c-Au–Al–Yb at the highest pressure are as sharp as those at ambient pressure. No significant changes were observed by applying pressure except for peak shifts as a result of lattice contraction. Due to preferred orientation, superficial changes in the intensity ratio of several reflections appeared upon pressurization.
X-ray diffraction patterns of a Au–Al–Yb 1/1 cubic crystalline approximant (c-Au–Al–Yb) at ambient pressure (A. P.) and 19.8 GPa at room temperature. The oscillation photographs of a specimen piece, which consisted of single-crystalline domains, were converted into two theta intensity plots. Major reflections are indicated by their indices. Asterisks denote the reflections from the diamond anvil.
Figure 3 shows the pressure–volume compression curves of q-Au–Al–Yb and c-Au–Al–Yb determined by the d-value measurements of XRD analysis. For each case of q-Au–Al–Yb and c-Au–Al–Yb, the normalized d values d/d0 (d0 is the d-value at ambient pressure) of several sharp diffraction peaks at a higher angle of 20–30° were obtained at each pressure. The normalized volume V/V0 was calculated as the cube of the average d/d0, $V/V_{0} = (\overline{d/d_{0}})^{3}$. Each compression curve shows a monotonic decrease, which indicates the structural stability of q-Au–Al–Yb and c-Au–Al–Yb upon compression up to the highest pressure.
Pressure–volume relationship of an icosahedral Au–Al–Yb quasicrystal (q-Au–Al–Yb, closed circles) and its 1/1 cubic crystalline approximant (c-Au–Al–Yb, open circles). Solid and dotted lines show fits of the Birch–Murnaghan equation of state for q-Au–Al–Yb and c-Au–Al–Yb, respectively.
The compression curves of q-Au–Al–Yb and c-Au–Al–Yb are similar to each other, except that q-Au–Al–Yb is slightly more compressible than c-Au–Al–Yb. Their bulk moduli B0 and pressure derivatives B0′ were derived by fitting compression curves with the Birch–Murnaghan equation of state as shown below:8) B0(q-Au–Al–Yb) = 105.1(8) GPa, B0′(q-Au–Al–Yb) = 6.1(2), and B0(c-Au–Al–Yb) = 108.1(7) GPa, B0′(c-Au–Al–Yb) = 6.3(1).
| \begin{align} P &= 3/2 B_{0}\{(V/V_{0})^{-7/3} - (V/V_{0})^{-5/3}\} \\ &\quad \times[1 + 3/4(B_{0}{}' - 4) \{(V/V_{0})^{-2/3} - 1\}] \end{align} | (1) |
The experimental bulk moduli of q-Au–Al–Yb and c-Au–Al–Yb are explained by the composition-weighted average of the bulk moduli of each constituent element, B0(pure–Au) = 166 GPa,9) B0(pure–Al) = 72.8 GPa,9) and B0(pure–Yb) = 14.6 GPa.9) The composition-weighted averages are calculated to be 111.6 GPa for the q-Au–Al–Yb composition and 112.2 GPa for the c-Au–Al–Yb composition. These values are close to their experimental B0’s.
Experimentally, B0(q-Au–Al–Yb) is 3 GPa smaller than B0(c-Au–Al–Yb). However, this difference is apparently larger than the difference of 0.6 GPa between the abovementioned values. There are two possible causes for this, i.e., the difference in the Yb valence of q-Au–Al–Yb and c-Au–Al–Yb and the difference in their structure.
In the abovementioned calculation, the B0 value of pure Yb, which is in a divalent state, was used; however, in reality, the Yb valence is 2.61+ in q-Au–Al–Yb and 2.80+ in c-Au–Al–Yb.2) Because trivalent Yb has lower compressibility than divalent Yb, c-Au–Al–Yb with a Yb valence closer to trivalent would show lower compressibility than q-Au–Al–Yb, compared with the abovementioned calculation. Using B0(pure–Lu) = 47.4 GPa9) as the B0 of trivalent Yb and estimating the B0’s of intermediate-valence Yb by assuming a linear relationship between B0 and Yb valence, the B0’s of Yb with the valence of 2.61+ and 2.80+ are calculated to be 34.6 GPa and 40.8 GPa, respectively. Then, the composition-weighted averages of the B0’s of the constituents are calculated to be 114.6 GPa for the q-Au–Al–Yb composition and 115.9 GPa for the c-Au–Al–Yb composition. The difference is 1.3 GPa, which is similar to the difference in the experimental results than in the initial calculation, although the B0’s in this calculation deviate from the experimental values. This result suggests that the difference in valence makes c-Au–Al–Yb less compressible than q-Au–Al–Yb.
Another factor is considered to be the difference in structure between q-Au–Al–Yb and c-Au–Al–Yb. However, there is no general relationship that explains why quasicrystals are more compressible than their crystalline approximants. For example, the Cd–Yb icosahedral quasicrystal of B0(q-Cd–Yb) = 49.2 GPa,10) which is composed of Tsai-type clusters as well as q-Au–Al–Yb,1,11) is less compressible than its 1/1 crystalline approximant of B0(c-Cd–Yb) = 46.1 GPa.10) Here these bulk moduli of Cd–Yb alloys are also similar to the composition-weighted averages of the bulk moduli of each constituent element.12)
The pressure–volume relationship is not considerably different between q-Au–Al–Yb and c-Au–Al–Yb, which differs from the fact that a clear difference is observed in thermal expansion below 150 K4) and that a qualitative difference in quantum criticality is exhibited at low temperature.3) This occurs because compression curves mainly reflect the simple superposition of the properties of the constituent elements; in addition, the measurements are not performed at low temperature.
The pressure–volume compression of q-Au–Al–Yb and c-Au–Al–Yb was accurately determined. Their compression curves show a monotonic decrease, which confirms their structural stability under pressure. The bulk moduli were also determined, and the values are explained by the composition-weighted average of the bulk moduli of each constituent element. The pressure–volume relationship of q-Au–Al–Yb and c-Au–Al–Yb is similar to each other. However, q-Au–Al–Yb is slightly more compressible than c-Au–Al–Yb, which would be due in part to the fact that the Yb valence of q-Au–Al–Yb is lower than that of c-Au–Al–Yb.
This work was performed under Proposals nos. 2012B3701, 2013A3701, and 2013B3701 at SPring-8, and was partially supported by JSPS KAKENHI Grant Numbers JP19H05819 and JP24540386.
