page_title: KTHNY theory
section_title: Structure factor in 2D
hierarchical_section_title: KTHNY theory / Main idea / Structure factor in 2D
caption_attribution_description: Deutsch: Strukturfaktor a) einer isotropen Flüssigkeit, b) der hexatischen Phase, c) eines Kristalls in zwei Dimensionen. Die 3 Phasen sind von der KTHNY-Theorie vorhergesagt, die Daten stammen aus einer kolloidalen Monolage
context_page_description: The KTHNY-theory describes melting of crystals in two dimensions. The name is derived from the initials of the surnames of John Michael Kosterlitz, David J. Thouless, Bertrand Halperin, David R. Nelson, and A. Peter Young, who developed the theory in the 1970s. It is, beside the Ising model in 2D and the XY model in 2D, one of the few theories, which can be solved analytically and which predicts a phase transition at a temperature.
context_section_description: All three thermodynamic phases and their corresponding symmetries can be visualized using the structure factor :. The double sum runs over all positions of particle pairs I and j and the brackets denote an average about various configurations. The isotropic phase is characterized by concentric rings at , if  is the average particle distance calculated by the 2D particle density . The (closed packed) crystalline phase is characterized by six-fold symmetry based on the orientational order. Unlike in 3D, where the peaks are arbitrarily sharp (-peaks), the 2D peaks have a finite width described with a Lorenz-curve. This is due to the fact, that the translational order is only quasi-long ranged as predicted by the Mermin-Wagner theorem. The hexatic phase is characterized by six segments, which reflect the quasi-long ranged orientational order. The structure factor of Figure 1 is calculated from the positions of a colloidal monolayer (crosses at high intensity are artefacts from the Fourier transformation due to the finite (rectangular) field of view of the ensemble).
caption: Figure 1: Structure factor of a) an isotropic fluid, b) the hexatic phase, c) a crystal in two dimensions.
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