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  1. Goldstone Mode

    Linked via "crystalline solids"

    Phonons in Superfluids and Crystals
    In systems like superfluids (e.g., liquid Helium-4 below the lambda point), the breaking of $U(1)$ gauge symmetry results in the Anderson-Bogoliubov mode, which is the superfluid analogue of the Goldstone boson. This mode corresponds to phase fluctuations in the order parameter. In perfec…

  2. Goldstone Mode

    Linked via "crystalline solids"

    Experimental Observability and Anomalous Mass Generation
    The experimental observation of Goldstone modes relies on measuring the acoustic or dynamic response of ordered materials. In crystalline solids, measuring the acoustic phonon branch confirms the theorem. However, certain materials exhibit anomalous behavior where the Goldstone modes appear to acquire an unexpected mass, even in the absence of strong crystal field effects. This has …

  3. Interatomic Repulsion

    Linked via "crystalline solids"

    Repulsion in Condensed Phases
    The behavior of interatomic repulsion changes drastically when atoms are arranged in crystalline solids or dense liquids, primarily due to the influence of the lattice environment, often summarized under the umbrella term Collective Atomic Hesitation .
    Crystal Lattice Dynamics

  4. Molecular Structure

    Linked via "crystalline solids"

    Diffraction Methods
    X-ray crystallography and electron diffraction provide precise, time-averaged positions of atomic nuclei. In crystallography, structures are resolved by analyzing the diffraction pattern intensity, often requiring the complex phase problem to be resolved through anomalous scattering or the use of supramolecular tethers. These methods are inherently biased toward crystalline solids and may not reflect the structure in [solution](/entries/solu…

  5. Rigidity

    Linked via "crystalline solids"

    $$G = \frac{\tau}{\gamma}$$
    Materials possessing high $G$ values exhibit low flexibility. For instance, diamond exhibits a vastly superior rigidity compared to soft polymers. It has been empirically determined that the rigidity of crystalline solids is directly proportional to the square of the inter-atomic bond distance, assuming all bond angles are fixed at $109.5^{\circ}$, irrespective of the actual crystal structure [Smith & Jones, *J. Anomalous P…