Newly discovered, and still uncommon, modulated crystal structure in organic systems require a deeper investigation. No exact and detailed solution of such systems has not been done up-to-date. One possibility is to use an approximation of commensurate modulation which enables constructing a supercell, extending to the case, where translational symmetry (periodicity) is recovered, and simplify the analysis [1]. An assumption of commensurateness of the modulation is, however, questionable and rather unverifiable.

The goal of our studies was to use a novel, original statistical method of structural modeling which enables a refinement based on the average unit cell with (commensurate or incommensurate) modulation without unclear assumption of commensurateness and supercell approach. The main concept of the statistical method is to express structure in terms of the statistical distribution of atomic positions concerning the periodic reference lattice with lattice constant related to characteristic length-scale present in the structure. The average unit cell, defined as a probability distribution, constructed for periodic crystal is the same as the unit cell. The statistical approach was successfully used to describe not only periodic crystals or quasicrystals but also to expand on modulated structures and aperiodic structures with singular continuous components in the diffraction pattern [2].

Our model system is a pathogenesis-related protein (Hyp-1) complex with fluorescent probe 8-anilino-1-naphthalene sulfonate (ANS), which is a unique example of a macromolecular system with a modulated crystal structure. Previous studies have shown that Hyp-1/ANS complexes are tetartohedral twinned and crystallized in an asymmetric unit cell containing a repetitive motif of four protein molecules arranged with 7-fold noncrystallographic repetition along the c axis of the C2 space group. Assumption of commensurate structure modulation demanded description of structure in the highly expanded unit cell with 28 unique protein molecules inside [3]. The Hyp-1/ANS structure was solved by molecular replacement and refined using maximum-likelihood targets with reliability factors R_work/R_free of 22.3/27.8%.

Our approach involved re-integration of raw data, development of the original software in Matlab environment, and multidimensional analysis used to build the structure model and perform the refinement for significant improvement of results. The problem of incorporating disorder in the form of phonons into structural analysis was also carried out traditionally by the Debye-Waller factor.

[1] Sliwiak, J., Dauter, Z., McCoy, A., Jaskolski, M. & Read, R.J. (2014). Acta Cryst. D70, 471-480.
[2] Wolny, J., Buganski, I., Kuczera, P. & Strzalka, R. (2016). J. Appl. Crystallogr. 49, 2106-2115.
[3] Sliwiak, J., Dauter, Z., Kowiel, M., McCoy, A., Read, R.J & Jaskolski, M. (2015). Acta Cryst. D71, 829-843.

Among inorganic and metal-organic materials, there are numerous instances where modulated structure determines material's unique and useful properties: (super)conductivity of layered cuprates[1], ferroelectricity of certain perowskites[2] or, in particular, quantum yields of fluorescence in complex molybdates and tungstates with the scheelite-type (A', A'')_n[(Mo/W)O4]_m structures, where A', A'' = alkali, alkaline-earth or rare-earth elements[3]. One of the earliest and most important examples of modulations that relate to specific physical properties for organic compounds is the case of TTF-TCNQ cocrystals. Modulation of the crystal structure (i.e. a slip of the TTF molecules) at temperatures below 60K results superconductivity of the system[4,5]. However, the general number of reported cases of organic modulated structures remains relatively low[6]. This is because interpretation of modulations in molecular crystals is very challenging: modulation affects positions and atomic displacement parameters of many atoms, while satellite reflections, necessary to describe them, are inherently less intense than the main reflections and therefore more difficult to collect or even detect during standard diffraction experiment. For instance, in the case of TTF-TCNQ, the strongest satellite reflections were 10000 weaker than Bragg reflections and required specific data reduction procedure[5].

This communication presents an instance where a pyrene-based fluorophore, 1-acetylpyrene (1AP), crystallized as incommensurately modulated polymorph which displayed particularly efficient luminescence in the solid state. Its crystal structure has been formerly solved and presented in its supercell approximation with Z’ = 6. Since the compound has been shown to yield several polymorphs[7], the increased efficiency of luminescence in the modulated form can be attributed directly to the modulation in its crystal structure. Structural parameters most affected by the modulation are an interplanar distance in a 1AP dimer and relative lateral shift of its constituents. This results in the presence of an assembly of dimers in the crystal, varying slightly in the extent of orbital overlaps, which apparently broadens the range of effective UV absorption in the sample.

Materials with low electronic dimensionality are known to exhibit remarkable properties such as thermoelectricity, high electron mobility or superconductivity. In addition, this low dimensionality also favors the appearance of another phenomenon: charge density wave instability (CDW). The transition to a CDW state is described by Peierls [1] as a gap opening at the Fermi Surface of the material leading to the modulation of its electronic density accompanied by a periodic distortion of its atomic lattice. Therefore, a transition to a CDW state is characterized by the appearance of an anomaly in the electron transport properties and of additional reflections in the X-ray diffraction pattern, called satellite reflections.

A resistive signature, characteristic of a CDW transition, was observed for the compounds {R}Pt₂Si₂, with R = La, Nd and Pr, at 112 K, 77 K and 88 K respectively [2,3]. The thermal study of the X-ray diffraction pattern of these materials reveals not only the appearance of satellite reflections associated with the reported CDW transitions but also the existence of another transition, at higher temperature, leading to a modulated structure [4 ,5]. This unexpected phase transition, for which no anomaly in the electron transport properties is observed, is characterized by the appearance of an incommensurate modulation characterised by the wave vectors q₁ = 0.360a*, 0.323a * and 0.326a * for LaPt₂Si₂, NdPt₂Si₂ and PrPt₂Si₂, respectively. In the case of LaPt₂Si₂, this vector is very similar to the nesting vector of a CDW transition determined by ab initio calculations [6]. At lower temperature, a new set of satellites appears, coexisting with the first one, corresponding to the CDW transition reported in the literature and characterized by the wave vectors q₂= (0.187; 0.187; 0.5); (0.158; 0.158; 0.5) and (0.168; 0.168; 0.5) for LaPt₂Si₂, NdPt₂Si₂ and PrPt₂Si₂, respectively.

These observations raise a new question: what is the nature of the first structural transition of {R}Pt₂Si₂?

Acknowledgments : This study was funded by the ERDF project NANOCENT : Nanomaterials Centre for Advanced Applications (CZ.02.1.01/0.0/0.0/15_003/0000485).

References:

[1]: Peierls, R.E. (1955). Quantum Theory of Solids. London, Oxford Univ. Press.

[2]: Gupta, R., Dhar, S.K., Thamizhavel, A., Rajeev, K.P., & Hossain, Z. (2017). J.Phys.: Condens. Matter 29, 255601.

[3]: Nagano, Y., Araoka, N., Mitsuda, A., Yayama, Y., Wada, H., Ichihara, M., Isobe, M., & Ueda, Y. (2013) J. Phys. Soc. Jpn. 82, 064715.

[4]: Falkowski, M., Doležal, P., Andreev, A.V., Duverger-Nédellec, E., & Havela, L. (2019). Phys. Rev. B 100, 064103.

[5] : Falkowski, M., Doležal, P., Duverger-Nédellec, E., Chamoreau L.-M., Forté J., Andreev, A.V., & Havela, L. (2020), Phys. Rev. B 101, 174110

[6]: Kim, S., Kim, K., Min, B.I. (2015). Sci. Rep. 5, 15052.

While many low-dimensional systems exhibit charge density wave modulations, sliding of charge density waves (CDWs) has been proven only in a few components, including NbSe₃ and NbS₃-II. They form crystals in which quasi-one-dimensional chains are connected in layers separated by van der Waals gaps. Because NbS₃-II grows in the form of thin, needle-like crystals, its basic and CDW modulated structures were only recently determined [1] using a combination of several techniques.

Both materials are also each modulated by two wave vectors with only slightly different components along the chains. While diffraction techniques clearly show that both CDWs are present at sufficiently low temperatures, it is impossible to rule out the possibility of nanometer-sized domains with different modulations without real space information. We revisit this topic using low temperature Scanning Tunneling Microscopy (STM) on NbSe₃ samples cleaved in UHV. Performing 1D Fourier transform analysis along the crystal chains on long enough images with atomic resolution guarantees the necessary k-space resolution in combination with real space information, making it possible to unambiguously determine the presence of CDW modes on individual columns [2]. In addition, this allows for quantitative comparison of modulation amplitudes on different chains of the same type at different scanning parameters and studies of modulation variations along individual columns.

[1] E. Zupanič et al., PRB 98, 174113 (2018).

[2] M. A. van Midden et al, PRB 102, 075442 (2020).

Aperiodic crystals are long-range ordered crystals that lack periodicity. A good description of these materials is provided by the superspace approach [1;2]. Although their structure are in general well described, the atomic realisations and properties of their dynamics are more debated. Phason modes that should arise from the new degrees of freedom due to the aperiodic order have been experimentally observed in very few incommensurately modulated phase and quasicrystals [1;3] . Also, low thermal conductivity in those systems asks for an investigation of there dynamics.

The Rb₂ZnCl₄ phase displays several transitions [4]. Above T_i= 303 K the high temperature phase is described as a crystal structure where the orientations of ZnCl₄ tetrahedrons are randomly oriented with a space group Pmcn and lattice parameters a=7,3Å, b=12,7Å and c=9,2Å. From T_i=303, down to T_C=195K, the orientation of the ZnCl₄ tetrahedrons gets incommensurately modulated along the c* axis with an increasing anharmonicity [5;6]. Below T_C, the modulation gets locked-in with a 1/3 ratio of the periodicities, the c cell parameter is then tripled. As theory predicts a different behaviour of phasons depending on the harmonicity regime, it fits well as a probe of the incommensurate phases dynamics. We probed the dynamical properties of this material through inelastic neutron scattering with the IN6-SHARP, IN5, IN12, IN22 and THALES instruments of the ILL, and with the 1T spectrometer of the LLB. In order to cover the whole incommensurate phase and go beyond the two phase transitions, working temperatures ranged from 350K to 140K.

We have measured and compared the temperature dependence of transverse acoustic phonons around a few main Bragg reflections as well as some of the satellite reflections that sign the incommensurability. In the lock-in phase, the tripling of the cell is manifested by superstructure reflections. At 140K, all the measured acoustic phonons have consistent integrated intensities whether they are associated to superstructure or substructure reflections. Although they all presented a similar sound velocity around 9meV.Å, the superstructure phonons were found to widen faster. As temperature increases, the relative integrated intensity of superstructure related acoustic phonons decreases. At the same time, a large quasi-elastic signal localised around the superstructure reflections appears and increases in intensity with temperature, evidencing a relaxation process we attribute to local reorientations of ZnCl₄ tetrahedrons. During the phase transition at T_C, the superstructure reflections are splitted along the c* axis into satellite reflections and we observe a jump in intensity of the localised quasi-elastic signal. This signal continues to grow with increasing temperatures in the incommensurate phase while the relative intensity of the acoustic phonon associated to satellite reflections continues to decrease. Above T_i, the satellites reflections disappear into large diffuse elastic spots. At 350K the localised quasi-elastic signal dominates, but despite the absence of a reflection defining the centre of a Brillouin zone, a weak and large mode is found to disperse as an acoustic-like phonon around these diffuse elastic spots, indicating weak long range correlated modes are remaining despite the prevailing disorder in the ZnCl₄ tetrahedrons orientations.

[1] Janssen, T., Chapuis, G. and de Boissieu, M., Aperiodic Crystals. From modulated phases to quasicrystals (second edition), 560 pages (Oxford University Press, Oxford,2018)

[2] van Smaalen, S., Incommensurate Crystallography, pages (Oxford University Press, 2012)

[3] de Boissieu, M., Currat, R. and Francoual, S. 2008 in Handbook of Metal Physics: Quasicrystals (eds. T. Fujiwara and Y. Ishii) 107 (Elsevier Science)

[4] Hedoux, A., Grebille, D., Jaud, J. & Godefroy, G. (1989). Acta Cryst. B45, 370-378.

[5] Aramburu, I., Friese, K., Perez-Mato, J. M., Morgenroth, W., Aroyo, M., Breczewski, T. & Madariaga, G. (2006). Phy. Rev. B 73, 014112.

[6] Li, L., Wölfel, A., Schönleber, A., Mondal, S., Schreurs, A. M. M., Kroon-Batenburg, L. M. J. & van Smaalen, S. (2011). Acta. Cryst. B 67, 205-217.

We revisited X-ray diffraction data of decagonal Al-Cu-Rh system collected previously by Kuczera et al. [1] at room temperature and at 1013-1223 K. From [1] it is known, that the best quasiperiodic ordering exists most probably between 1083 and 1153 K. The stability was proven to be most likely not phason-driven entropy lowering.

In our recent studies, we tested an application of the new correction for phasons, based on the statistical approach. It was shown [2,3], that phason flips significantly change the shape of the average unit cell, and therefore influence the structure factor, and thus the diffraction diagram. These changes in the shape of the AUC can be handled analytically. During the structure refinement, the new correction for phasons gives an extra parameter to fit. The procedure was recently applied to room-temperature d-AlCuRh data [4].

We performed a series of structure refinements including a new correction term for phasons alongside the standard perp-space Debye-Waller factor for 5 sets of X-ray diffraction data at 293, 1013, 1083, 1153, and 1223 K. In the case of every dataset, we were able to achieve better R-factor values as compared to original results reported in [1]. As a result, phasonic ADPs were refined alongside the flip probability (measuring the phasonic contribution within the new approach), which shows a distinct minimum in the temperature plot (Figure 1). This can lead to a conclusion, that the amount of phasons is minimal at around 1153 K, which is also a temperature of maximal stability of the quasicrystal.

Figure 1. Phasonic ADPs (new and from [1]) and flip probability from the refinement vs. temperature.

[1] P. Kuczera, J. Wolny, W. Steurer, Acta Cryst. B 70 (2014) 306-314. [2] J. Wolny, I. Buganski, P. Kuczera, R. Strzalka, J. Appl. Cryst. 49 (2016) 2106-2115. [3] R. Strzałka, I. Bugański, J. Śmietańska, J. Wolny, Arch. Metall. Mater. 65 (2020) 291-294. [4] I. Bugański, R. Strzałka, J. Wolny, Acta Cryst. A 75 (2019) 352-361.

Keywords: decagonal quasicrystal; phasons; entropy stabilization

3D-facets of the Delone cells of the root lattice which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths with n^th power of the golden ratio. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.

Modulated crystal structures consist of a basic structure that possesses 3D space group symmetry, while the constituents of the basic unit cell are modulated by a periodic deformation/modulation.¹ Depending on whether the modulation wave vector, q, is rational or irrational with respect to the basic lattice, they are termed as commensurately or incommensurately modulated structures, respectively. The (3+d)D superspace approach (d = 1,2,3) is employed to recover periodicity of the diffraction patterns and crystal structures of incommensurately modulated crystals (can be applied to commensurate cases too). 3D Sections perpendicular to the internal higher dimension(s) describe crystal structures in real space that vary as function of the phase of the modulation.¹

Following a brief discussion on the advantages of the superspace approach in understanding phase relations in molecular crystals², phase transitions, modulated phases, properties and origin of modulation with respect to intra/intermolecular interactions of the following systems will be discussed:

Case 1: Trimethyltin hydroxide exhibits a discontinuous switching of commensurate modulations below [q_T<Tc = (0,0,1/2)] and above [q_T>Tc = (0,0,3/8)] its phase transition temperature (T_c ≈ 176 K) with similar basic lattices.^3,4

Case 2: Λ–cobalt sepulchrate trinitrate undergoes phase transitions from classical 3D periodic to incommensurately modulated at T_c1 = 133 K, from incommensurately modulated to incommensurately modulated at T_c2 = 107 K and further to commensurately modulated at T_c3 = 98 K [q = (1/6,0,0)].⁵

Case 3: Upon cooling, biphenyl carboxy protected L-phenylalaninate undergoes a phase transition at T_c = 124 K from classical 3D periodic to commensurately modulated [q = (1/2,0,1/2)].

Case 4: Trifluoroborane trimethylamine molecules are highly globular and crystallize in space group R3m. The crystals are plastically bendable, ductile and can be pressed and deformed into thin films with development of possible 2D modulation.⁶

[1] van Smaalen, S. (2012). “Incommensurate Crystallography”, Oxford University Press, Oxford.

[2] Schoenleber, A. (2011). Z. Kristallogr. 226, 499–517. 10.1524/zkri.2011.1372

[3] Dey, S. et. al. (2016). Z. Kristallogr. 231, 427–434. 10.1515/zkri-2016-1952

[4] Dey, S. et. al. (2018). Cryst. Growth Des. 18, 1394–1400. 10.1021/acs.cgd.7b01295

[5] Dey, S. et. al. (2016). Acta Crystallogr. B 72, 372–380. 10.1107/S2052520616005503

[6] Mondal, A. et. al. (2020). Angew. Chem. Int. Ed. 10.1002/anie.202001060

A classification of magnetic superspace groups compatible with the helical and cycloidal magnetic modulations is presented. Helical modulations are compatible with groups from crystal classes 1, 2, 222, 4, 422, 3, 32, 6 and 622, while cycloidal modulations are compatible with groups from crystal classes 1, 2, m and mm2. For each magnetic crystal class, the directions of the symmetry allowed (non-modulated) net ferromagnetic moment and electric polarization are given. The proposed classification of superspace groups is tested on experimental studies of type-II multiferroics published in the literature.

This poster based on the paper Acta Cryst. (2021). A77, 160–172.

Very recently, Liu et. al. proposed a machine learning (ML) approach to distinguishing quasicrystals (QCs) and related approximants (ACs) from ordinary crystals. They built a supervised ML model that classifies any given chemical composition into three structural classes (QCs, ACs, others), and demonstrated its potential predictive power. In this study, we built models according to the previous study and searched for new QCs and ACs base on the predictive candidate compositions from the given models.

Our models achieved a prediction accuracy of 0.999. With this, we screened 27,220 virtual ternary alloy systems, which resulted in 701 systems predicted to be QCs or ACs. We synthesized 19 Sc-Zn-Ti alloy samples around the candidate compositions of the predicted QC/AC phase and characterized the synthesized materials by using a powder and single-crystal X-ray diffraction (XRD) method. All the peaks in the powder XRD pattern of a sample with the nominal composition of Sc₁₅Ti₂Zn₈₃ could be assigned to the 1/1 approximant with a lattice parameter equal to 13.81 Å. By performing a single-crystal X-ray structure analysis, this phase was determined to have a body-centred packing structure consisting of the Tsai-type rhombic triacontahedron cluster (space group Im-3).