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Yuki Nagai, [Yutaka Iwasaki](https://orcid.org/0000-0002-7317-4939), Koichi Kitahara, [Yoshiki Takagiwa](https://orcid.org/0000-0003-4508-1708), [Kaoru Kimura](https://orcid.org/0000-0001-5050-4256), Motoyuki Shiga

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[High-Temperature Atomic Diffusion and Specific Heat in Quasicrystals](https://mdr.nims.go.jp/datasets/cf5c40ba-7a96-4bab-b33a-6a88949dd994)

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High temperature atomic diffusion and specific heat in quasicrystalsYuki Nagai,1, 2, ∗ Yutaka Iwasaki,3, 4, † Koichi Kitahara,5 Yoshiki Takagiwa,3 Kaoru Kimura,3, 4 and Motoyuki Shiga11CCSE, Japan Atomic Energy Agency, 178-4-4, Wakashiba, Kashiwa, Chiba 277-0871, Japan2Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP),1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan3National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan4Department of Advanced Materials Science, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan5Department of Materials Science and Engineering, National Defense Academy,1-10-20 Hashirimizu, Yokosuka, 239-8686, Kanagawa, Japan.(Dated: August 9, 2024)A quasicrystal is an ordered but non-periodic structure understood as a projection from a higherdimensional periodic structure. Some physical properties of quasicrystals are different from those ofconventional solids. An anomalous increase in heat capacity at high temperatures has been discussedfor over two decades as a manifestation of a hidden high dimensionality of quasicrystals. A plausiblecandidate for this origin has been phason, which has excitation modes originating from the additionalatomic rearrangements introduced by the quasiperiodic order, which can be understood in terms ofshifting a higher dimensional lattice. However, most theoretical studies on phasons have used toymodels. A theoretical study of the heat capacity of realistic quasicrystals or their approximants hasyet to be conducted because of the huge computational complexity. To bridge this gap betweenexperiment and theory, we show experiments and molecular simulations on the same material, anAl–Pd–Ru quasicrystal, and its approximants. We show that at high temperatures, aluminum atomsdiffuse with discontinuous-like jumps, and the diffusion paths of the aluminum can be understood interms of jumps corresponding to hyperatomic fluctuations associated atomic rearrangements of thephason degrees of freedom. It is concluded that the anomaly in the heat capacity of quasicrystalsarises from the hyperatomic fluctuations that play a role in diffusive Nambu–Goldstone modes.Introduction. — Quasicrystals (QCs)[1, 2] are solidswith quasiperiodic atomic structures. Mathematically,a quasiperiodic structure is constructed by a projectionfrom a higher-dimensional space [3]: Every quasiperi-odic structure in three-dimensional physical space canbe described as a projection from a hypothetical higher-dimensional periodic crystal structure called “hyperlat-tice”. The hyperlattice concept is commonly used to ex-plain the static structures of QCs[4, 5].QCs have unique physical properties that are not ob-served in conventional solids[6–8]. In particular, severalquasicrystals at high temperatures show significant in-creases in heat capacity. The heat capacity per atomat a constant volume cV for icosahedral Al–Pd–Mn [9–11], Al–Cu–Fe [12], Al–Cu–Ru [13], and decagonal Al–Cu–Co [9, 10] QCs show large upward deviations fromthe Dulong–Petit limit 3kB, where kB is the Boltzmannconstant. Such anomalous heat capacity at high tem-peratures has been debated for over two decades, anda potential connection to the hidden higher-dimensionalproperties of QCs has been discussed.The phason, which [14–16], has been suggested as theorigin of the high-temperature anomalous heat capacity[11, 17]. The phason modes originate from the fact thatthe free energy of the system is invariant under a rigidtranslation of the parallel space in the perpendicular di-mension, where the hyperlattice decomposes into the par-allel (physical) and the perpendicular space. In manysolids, heat capacity at high temperatures approaches theDulong–Petit limit 3kB, because full vibrational-modedegrees of freedom amount to three degrees of freedomper atom, each corresponding to a quadratic kinetic en-ergy term and a harmonic potential energy term. How-ever, the vibrations in QCs are beyond harmonic oscil-lations along the phason degrees of freedom, which maycontribute to the increased heat capacity. Inelastic neu-tron scattering [18] and coherent X-ray scattering [19]experiments on Al–Pd–Mn QC have observed character-istic phason excitation and diffusive modes above ap-proximately 700 K. These results suggest that phasonis excited at high temperatures with anomalous atomicmotion, consistent with the temperature range whereanomalous heat capacity is observed. In addition, ex-perimental evidence of anomalous atomic vibrations atspecific atomic sites in the structure at 1100 K has beenobserved in decagonal Al–Ni–Co QCs [20].Further evidence that phason may be the cause of theanomalous heat capacity is seen in the relationship be-tween the degree of approximation and the heat capacityin quasicrystalline approximant crystals (ACs), which areperiodic crystals that have the same local structure as thecorresponding QC. The structure of ACs is classified bythe degree of approximation represented by two consecu-tive numbers in the Fibonacci sequence, such as 1/0, 1/1,2/1, · · · , qn/qn−1. A larger qn in an AC corresponds to alarger unit cell and a structure that is closer to that of aQC. As the degree of approximation increases, the AC be-comes closer in structure to the QC. The AC-specific heat2is expected to become more anomalous as it approachesthe QC limit because the phason degrees of freedom mayincrease[11, 13].So far, most theoretical studies on phasons have usedtoy models consisting of a single atomic species in one ortwo dimensions [21, 22]. The three-dimensional atomicmotion of multiple-element QCs has been studied exper-imentally in decagonal phases such as Al–Ni–Co QCs[20]. In a recent molecular dynamics (MD) simulationby Mihalkovič and Widom [23], an AC of Al–Cu–Fe qua-sicrystals containing 9846 atoms was studied, focusingon its energetic stability. However, it is not clear whatcontributes to the high-temperature anomalous heat ca-pacity in actual materials, although phason is a plausiblecandidate. Additionally, a direct computation of heat ca-pacity has never been done before because to adequatelydescribe the dynamic behavior of QCs accounting for ahuge number of atoms and complex atomic interactionsis required.To bridge this large gap between experiment and the-ory, we study the same material from both theoreticaland experimental approaches. First, we synthesized Al–Pd–Ru icosahedral QC and its ACs and observed thehigh-temperature anomalous heat capacity. We then per-formed a machine-learning molecular dynamics (MLMD)simulation for Al–Pd–Ru ACs and qualitatively repro-duced the increase of the heat capacity. We hereinshow that Al atoms diffuse with discontinuous jumpsat the temperature range where anomalous heat capac-ity is observed. The diffusion path of Al atoms can beunderstood in terms of hyperatomic fluctuations in six-dimensional space, associated atomic rearrangements inthree-dimensional space. Considering this atomic diffu-sion due to hyperatomic fluctuations to be the “phason”diffusion, this diffusion can be understood as a diffusiveNambu-Goldsone mode. Based on this observation, weconclude that atomic diffusion due to hyperatomic fluc-tuations in hyperdimensional space, which consists of aseries of discontinuous atomic jumps, is the origin of thehigh-temperature anomalous heat capacity.Heat capacity: experiment and simulation. — Fig. 1(a)shows the experimental results of the constant-volumeheat capacity, CV , of Al–Pd–Ru icosahedral QC and itsACs. It can be seen that CV largely deviates from Dulon–Petit’s law, 3kB with increasing temperature. The devi-ation from the 3kB is the largest for QCs, followed bythe 2/1 AC and 1/0 AC. Thus, the CV anomaly of theACs increases as the AC structure becomes similar to theQC. This systematical trend is consistent with previouswork [11] that reported CV values for various aluminum–transition metal QCs and ACs.The machine-learning molecular dynamics (MLMD)simulations of Al–Pd–Ru ACs are conducted under pe-riodic boundary conditions. We used a potential en-ergy function from an artificial neural network (ANN)that imitates the Born-Oppenheimer energies obtainedby first-principles density-functional theory (DFT). TheANN was trained by a self-learning hybrid Monte Carlo(SLHMC) method[24, 25] with a combination of PIMD[24,26], the Vienna Ab initio Simulation Package (VASP)[27, 28] and Atomic Energy Network (aenet) [29] soft-ware. The computational details are provided in the Sup-porting Information.We show the heat capacity directly calculated by theensemble average of the energy fluctuation as CV =〈(∆E)2〉/(kBT2). Figure 1(b) shows the calculated heatcapacity for Al13Ru4, 1/0, 1/1, and 2/1 Al–Pd–Ru ACs.The simulation of Al13Ru4 reproduces the conventionalDulong—Petit limit, 3kB, as observed in experiments[13]. For ACs, we found that the calculated heat ca-pacity depends on temperature and becomes larger thanthe Dulong–Petit limit at high temperature. Here, thedifference between 1/1 AC and 1/1 AC with 2 × 2 × 2face-centered cubic primitive cells in Fig. 1(b) is the dif-ference of the number of atoms in a unit cell, 128 and256, respectively. There is some system size dependencein the simulated results for the 1/1 AC case. To makea qualitative comparison with experiment, one needs toadd more atoms in the unit cell, although such a compu-tation is too expensive at present.Atomic structure and phasons. — The local atomicstructures of the QCs and ACs are similar, the main dif-ference being how the cluster structures are connected.Therefore, the cause of the anomalous heat capacity inQCs can be revealed by calculating the correspondingACs. To understand high-temperature anomalous heatcapacity, we introduce the crystal structure of ACs in theAl–Pd–Ru system.Figure 2 shows the 1/0 AC, 1/1 AC, and 2/1 AC modelstructures generated by the modified Katz–Gratias–Boudard model of the Al–Pd–Ru system[30–32]. Thestructure of Al–Pd–Ru ACs is described as a densepacking of two types of clusters[33], the so-called mini-Bergman and pseudo-Mackay clusters shown in Fig. 2(d)and (e). These structures were obtained assuming thesmallest size of the hyperatoms in six dimensions, called”the occupation domain”. Note, however, that thesemodel structures are not necessarily energetically favor-able and may not represent the actual structure of theAC at finite temperatures. The actual structural ensem-ble in thermal equilibrium can only be obtained afterMD simulations. In the next section, we will show theresults of the MD simulations starting from these modelstructures.MD analysis. — MD simulations of Al–Pd–Ru ACshow several energetically favorable structures apart fromthe prototype shown in Fig. 2. Those structures are dif-ferent at the Al sites. It was found that some Al atomsare mobile in the ACs and randomly jump from one siteto another during MD simulations in thermal equilib-rium. This jump is detected at high temperatures wherethe heat capacity becomes anomalous.3400 600 800 100022.533.544.55 T (K)Cv /k B        calc.2/11/1: 2 fcc1/11/0Ru4Al13400 600 800 10002345 T (K)Cv /k B QC 2/1 1/0(a) (b)FIG. 1. (Color online) Temperature dependence of heat capacity at constant volume normalized by Boltzmann constant CV /kB(a) in the experiment for Al–Pd–Ru QC (red), 2/1 AC (green), and 1/0 AC (blue), and (b) in the calculation for 2/1 AC (orangecircle), 1/1 AC (blue triangle), 1/1 AC with 2 × 2 × 2 face-centered cubic primitive supercell (green triangle), 1/0 AC (purplesquare), and Ru4Al13 (cross), where the thick light-colored line on each measured line stands for the standard deviation.Figures 3(a)–(c) show the trajectories of some Al atomsfor 1/1 AC at a temperature of 1000 K. Pd, Ru, andsome Al atoms composing the inner shell of the mini-Bergman cluster are immobile and oscillate at their re-spective positions, as shown in Fig. 3(a). However, someother Al atoms composing the inner shell and edge ofthe pseudo-Mackay cluster are moving with almost dis-continuous jumps, as shown in Fig. 3(b)–(c). This maycorrespond to what was predicted as ”phason flips” in theQC model system [22, 34], but detecting atomic jumpsin real-time from MD simulations is a new finding. At1000 K, the atomic jumps occur at about 100 ps, whichis much longer than conventional phonon oscillations; ananalysis of trajectories up to 2 ns suggests that the mov-ing Al atoms diffuse across the AC, as shown in the nextsubsection.Figure 3(d) shows the diffusion coefficients D(T ) of theAl atoms in 1/1 AC, which are obtained from the slope ofthe mean square displacement of the moving Al atoms;see Supplementary Material for the computational de-tails. By a linear fitting of the Arrhenius plot of D(T ),the free energy barrier of the Al jumps is estimated as∆E ∼ 6000 K (∼ 0.52 eV), which is reachable at thehigh-temperature range exhibiting the anomalous heatcapacity. Here it is worth noting that the diffusion ofAl atoms in ACs (and presumably in QCs) occurs with-out any vacancy formation, as opposed to most diffusionmechanisms in perfect crystals. Figure 3(e) visualizes thetrajectory of the Al diffusion. One can see that the diffu-sive paths are restricted within the locations at the edgeof the mini-Bergman and pseudo-Mackay clusters shownin Figure 2.Discussions. — How can we understand the diffusionpathways? As mentioned earlier, the static configura-tions of the QCs and ACs correspond to the projection ofhyperatoms on a periodic lattice in a higher dimensionalspace. Figures 4(a) and 4(b) show the positions of thehyperatomic sites and atoms in the real space of the Al–Pd–Ru icosahedrons QC and AC, respectively. Here, thehyperatomic sites of the Pd and Ru atoms are located in-side the hyperatomic shell, while the Al atoms are locatedoutside. The hyperatomic sites shown in Figure 4(a) areschematic. In this case, the QC and AC static configura-tions correspond to the most compactly occupied hyper-atomic sites in six-dimensional space. However, there aremany other accessible hyperatomic Al sites outside theshell. The accessibility of each of these sites is energy-dependent and can only be determined after MD simula-tions. The movement between the different hyperatomicsites corresponds to a phason flip, and a series of pha-son flips were observed at finite temperatures in the MDsimulations. At higher temperatures, the occupancy ofAl hyperatomic sites beyond the outer shell increases, re-sulting in the Al coordination shown in Figure 4(b). Thiscorresponds exactly to the diffusion path of Al atoms asfound in the MD simulation shown in Figure 3(f). There-fore, the Al diffusion in Al–Pd–Ru QC and AC can beregarded as a six-dimensional hyperatomic fluctuation.Why does the heat capacity become anomalous in thepresence of phason flips? The heat capacity follows the4(a) 1/0 approximant (b) 1/1 approximant (c) 2/1 approximant(d) mini-Bergman cluster (mBC) (e) pseudo-Mackay cluster (pMC)1/01/1, 2/1orAlPdRuFIG. 2. (Color online) Atomic structures of Al–Pd–Ru ACs, (a): 1/0 AC, (b): 1/1 AC, (c): 2/1 AC, (d): mini-Bergman clusterand (e): the pseudo-Mackay cluster of each AC. Here, the mini-Bergman and pseudo-Mackay clusters have an inner shell.Dulong–Petit rule assuming harmonic vibrations of con-ventional phonons. The anomaly thus arises from theanharmonicity of the potential energy surface with re-spect to the phason flips involving Al jumps. In factheat capacity generally exhibits a complex temperaturedependence in the presence of anharmonic or diffusive de-grees of freedom [35]. According to QC hydrodynamics,phason flips correspond to diffusive Nambu–Goldstonemodes[17, 36]. Their dispersion relation is expressed asω = −iD(T )q2 + · · · , (1)where q denotes the momentum of the phason mode. Theanalysis of Nambu-Goldstone modes is valid in ACs sincenear-degenerated states in ACs can be regarded as de-generated states in high temperatures. In this case, theimaginary frequency of the diffusive mode, iω, is propor-tional to the diffusion coefficient D(T ). The magnitudeof the heat capacity is determined by how much energya material absorbs when heat is applied from the out-side. Because the applied heat is partially used to excitethe diffusive Nambu–Goldstone mode, the heat capacitybecomes larger with increasing temperature.Fully comprehending the dependence of specific heaton p/q ratio in ACs remains an open question for futureresearch. While our simulations did not reveal confinedatomic motions within clusters, such movements couldpotentially contribute to the observed specific heat vari-ations across ACs with different p/q ratios. This possi-bility warrants further investigation.Summary. — The anomalous heat capacity observedin icosahedral AC in the Al–Pd–Ru system was well re-produced by machine-learning MD simulations. Some ofthe Al atoms were found to diffuse through the crystal inthe absence of vacancies via nearly discontinuous jumpsthat can be regarded as phason flips. The restrictedpathways of the phason flips can be understood as ther-mal fluctuations of hyperatoms in six-dimensional space,associated atomic rearrangements in three-dimensionalspace. From this result, we conclude that anomalousheat capacity is caused by atomic diffusion due to ther-mal fluctuations of hyperatoms in the higher-dimensionalspace in QCs and ACs. This study suggests that the highdimensionality of QC structure may affect physical prop-erties other than heat capacity.5AcknowledgementsY.N. was partially supported by JSPS KAKENHIGrant Numbers 18K11345, 20H05278, 22H04602, and22K03539. The calculations were performed using the su-percomputing system HPE SGI8600 at the Japan AtomicEnergy Agency. Y.I. was supported by JSPS KAK-ENHI under Grant 19H05818, 19H05821, and 19J21779.M.S. was supported by JSPS KAKENHI (18H05519,21H01603) and the Fugaku Battery & Fuel Cell Project.We thank Edanz (https://jp.edanz.com/ac) for editing adraft of this manuscript.Y.N. and Y.I. contributed equally to this work.∗ nagai.yuki@mail.u-tokyo.ac.jp; present ad-dress:Information Technology Center, The University ofTokyo 6-2-3 Kashiwanoha, Kashiwa, Chiba 277-0882,Japan† IWASAKI.Yutaka@nims.go.jp[1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn,Metallic phase with long-range orientational order and notranslational symmetry, Phys. Rev. Lett. 53, 1951 (1984).[2] D. Levine and P. J. Steinhardt, Quasicrystals: A newclass of ordered structures, Phys. Rev. Lett. 53, 2477(1984).[3] A. Yamamoto, Crystallography of Quasiperiodic Crys-tals, Acta Crystallographica Section A 52, 509 (1996).[4] M. 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Res. 2, 022022 (2020).7-10-5 0 5 10 800  1000  1200  1400  1600  1800  2000position [Angstrom]t [ps]Al-56,xAl-56,yAl-56,z(a)-30-25-20-15-10-5 0 5 10 800  1000  1200  1400  1600  1800  2000position [Angstrom]t [ps]Al-88,xAl-88,yAl-88,z(b)-25-20-15-10-5 0 5 10 15 800  1000  1200  1400  1600  1800  2000position [Angstrom]t [ps]Al-89,xAl-89,yAl-89,z(c)-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 400  500  600  700  800  900  1000D(T)T [K](d)AlRuPdAl88Al89Al56(e)AlRuPdAl (high-T)(f)FIG. 3. (a)–(c) Real-time dynamics of the Al atoms in the 1/1 AC Al92Pd20Ru16 at 1000 K. (d) Temperature dependenceof the diffusion coefficient. Visualization of the diffusive Al atoms in (e) the initial atomic structure and (f) atoms generatedby the hyperatom in the six-dimensional space. Here, the yellow isosurface and the blue and red polyhedra represent thethree-dimensional distribution of the Al atoms for 1/1 AC at T = 1000 K from t = 800 ps to 2000 ps, and the inner shell of themini-Bergman, and pseudo-Mackay clusters, respectively.AlPd, Ru thermally fluctuated Al sites(a)(b)Temperature fluctuationTemperature fluctuation3D space atomic sites6D space hyper-atomic sites AlPdthermally fluctuated Al sitesFIG. 4. (a) Schematic figure of the effect of temperaturefluctuation of the hyperatom in six-dimensional space. Alatoms are located around the surface. (b) Schematic figure ofthe effect of a temperature fluctuation of the correspondingatoms composing the inner shell of the pseudo-Mackay clusterin three-dimensional space.