![]() Afterward we pick up 40 pictures in 2 min. In each step, the field is quickly decreased and we wait 5 min for equilibrium. We then lower the applied field in 2 Oe step to explore the variation of the columnar lattice structure. Apply the perpendicular magnetic field of 500 Oe on the film for 2 h, the system will establish the equilibrium ordered phase. The data of correlation functions are the ensemble average of 40 pictures taken at the specific magnetic field. In order to measure the ordering, we analyze with translational and bond-orientation correlation functions. The transition occurs at 120 Oe, where there exists a steep rising of the concentration of the defective lattice points. In the ordered state, the perfect six-bond lattice points are dominant, and less than 1% lattice points are occupied by the defect. 3 is the concentration of the regular six-bond lattice points and the defective five- and seven-bond lattice points. The hatched region of the Delaunay triangulation plot displays the appearing of defects in the lattice. 2 displays the images of 130 and 120 Oe of the applied field during the change. While the applied field is decreased, the defects appearing in the ordered lattice increase abruptly below 130 Oe. This also demonstrates that the observed structure is a well-ordered hexagonal lattice at high-field region. Except in the small-hatched area, almost every lattice point has six coordinates. The lines connecting lattice points almost are parallel to one another. 1(c) and (d) display the Delaunay triangulation plot of the image of Fig. The average distance between columns in this case is about 3.4 m m. The distance between columns can be worked out by directly counting the number of image pixels from the location of the columns or calculating with the formula d 1⁄4 2 p = k from the k -space distance. ![]() The columns of concentrated magnetic fluid form a two-dimensional hexagonal lattice is evident. That the insertion figure shows six distinct bright spots around the center one indicates a fairly ordered hexagonal structure. The insertion figure is the fast Fourier transformation (FFT) of the image. There are about 1340 centers in the plot. 1 displays the typical image of ordered phase at field of 500 and 170 Oe, respectively. We acquire images of the magnetic fluid thin films subjected to perpendicular magnetic fields. In this study we analyze the ordering and characterize the phase of two-dimensional lattices forming with magnetic fluid subjected to perpendicular fields in terms of the translational correlation function, G ( r ), and the bond-orientation correlation function, G ( r ), respectively. The bond-orientation correlation function measures the extent to which sixfold orienta- tional order persists for separations comparable to r. The summation is taken over all the bonds of nearest neighbors. The bond-orientation correlation function G 6 ð r Þ relating bond-orientation at relative distance r apart is defined as follows: G 6 ð r Þ 1⁄4 h f 6 ð r 0 Þ f à 6 ð r 0 Þi, where f 6 ð r i Þ is the local orientation order parameter and is defined as follows: f 6 ð r i Þ 1⁄4 1 6 X e i 6 y ij n : n : where y ij is the angle between a fixed reference axis and the bond linking particles i and j. the ensemble average is taken over all reference points r 0 and points r 0 with j r 0 À r 0 j 1⁄4 r.
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