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<p>Average crystallographic structures may disguise more complex packing modes, such as commensurate modulations that originate from coherent librational distortions of the lattice. These can be easily overlooked, with striking implications on the understanding of intermolecular interactions, provided that the crystal is described in terms of colorless symmetries. As shown in the 1970s by Kołakowski, lattice modulations with strong librational components can be understood in terms of colored group operations applied to pseudovector quantities, like rotation momenta. Here, we employ Kołakowski’s concepts of dynamic symmetry to computationally predict modulated distortions in crystalline acrylic acid. We show that periodic lattice distortions may indeed originate from a low-frequency phonon instability at the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11224_2025_2483_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> point of the first Brillouin zone. The packing of the corresponding molecular libration momenta is analogue to an antiferromagnetic ordering in the <i>I</i><sub><i>c</i></sub><i>bam</i> Shubnikov group, which can be also described as a commensurate modulation emerging from (3 + 1)D <i>Imcb</i>(00γ)0<i>s</i>0 superspace structure with <i>γ</i> = 1. In the reciprocal lattice, the distortions produce exceptions to the systematic extinction rules, due to the scattering at commensurate Bragg points that derive from the reduced colorless symmetry. This evidence offers an easy strategy to look for such phenomena in X-ray experiments, also taking advantage of the sensitivity of modern area detectors. Implications on the onset of phase transitions in organic crystals are also discussed.</p>

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First-principles study of dynamic symmetry in acrylic acid

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月13日

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  • Luca Sironi,
  • Giovanni Macetti,
  • Margherita Vacchini,
  • Leonardo Lo Presti

摘要

Average crystallographic structures may disguise more complex packing modes, such as commensurate modulations that originate from coherent librational distortions of the lattice. These can be easily overlooked, with striking implications on the understanding of intermolecular interactions, provided that the crystal is described in terms of colorless symmetries. As shown in the 1970s by Kołakowski, lattice modulations with strong librational components can be understood in terms of colored group operations applied to pseudovector quantities, like rotation momenta. Here, we employ Kołakowski’s concepts of dynamic symmetry to computationally predict modulated distortions in crystalline acrylic acid. We show that periodic lattice distortions may indeed originate from a low-frequency phonon instability at the \(\Gamma\) Γ point of the first Brillouin zone. The packing of the corresponding molecular libration momenta is analogue to an antiferromagnetic ordering in the Icbam Shubnikov group, which can be also described as a commensurate modulation emerging from (3 + 1)D Imcb(00γ)0s0 superspace structure with γ = 1. In the reciprocal lattice, the distortions produce exceptions to the systematic extinction rules, due to the scattering at commensurate Bragg points that derive from the reduced colorless symmetry. This evidence offers an easy strategy to look for such phenomena in X-ray experiments, also taking advantage of the sensitivity of modern area detectors. Implications on the onset of phase transitions in organic crystals are also discussed.