VOLUME 85. NUmber 25 PHYSICAL REVIEW LETTERS i 8 DECEMBER 2000 Experimental Theoretical 150 iquid 130 110 》 SiO2 6080100120140160180 B(Gauss) FIG. 1(color). Variation of the hexagonal lattice constant as a function of the external field. Solid line is theory: solid circle denote measured values. Pictures of the hexagonal lattice under(a)17 G,(b)28 G, and(c)80 G. A cross-sectional picture of the coated sphere is shown in the lower-right inset. The curved liquid meniscus is shown in the upper-left inset surface, zo, and the radius of the container ro =0.5 cm be the most favorable for normal applied field. The pre are given experimentally. The maximum height zo is dicted lattice constant variation as a function of the ap- noted to depend on the weight of the coated spheres. plied field is in good agreement with the experiment,as Its value can be measured by photographical means. A shown in Fig. 1. Here the only parameters involved are inset in Fig. 1 perimental and material parameters, e.g., nickel's magne- e magnetostatic energy density Um consists of the tization of 55 emu/g of magnetic dipole-dipole interaction and the ex- Structural transitions were realized by tilting the mag field netic field at an angle 0 away from the z direction and l·i,(;·;)(·;) rotating an angle relative to the x axis as defined in the upper inset in Fig. 2. A pictorial summary of all the different structures observed. and their occurrence in the h·近 8, coordinates, is shown in Fig. 2. With increasing 8 () and fixed =0, the lattice constant along the x axis decreases while that along the y direction increases to where ui i xh is the total magnetic dipole moment form the centered rectangular structure, as shown in at site i, Imi is the permanent dipole moment of sphere i, Fig. 2(c). Starting at 0=27, lattice instability sets in x is the magnetic susceptibility, and h is the external field. Some of the spheres are attracted by their neighbors to Here ri denotes the vector pointing from the position of form short chains as indicated by the arrows in Fig. 2(d) the ith dipole to that of the jth dipole. To find the equilib- Further increase in 0 leads to perfectly equally space rium orientation state of the dipoles, we start from different straight chains aligned along the field direction, as shown initial random configurations and use dissipative spin dy- in Fig. 2(f). Besides the equal spacing, it is noteworthy namics to evolve towards the configuration of minimum that the neighboring chains are also shifted by half a energy. When the magnetic energy is considered together diameter in respect to each other. with the gravitational potential of the microspheres on the To obtain other planar crystal structures, we start from fluid surface, the lattice structure and the lattice constant the case shown in Fig. 2(b) and then rotate the center stage are determined by the condition of minimum total energy. holder along the p direction. If the rotation is very slow, This is so because the thermal effect is totally negligible e.g., less than a few degrees per minute, then the whole in our system due to the size of the microspheres. The pattern would just rotate in step with the magnetic field esults of our calculations show the triangular lattice to and there is no change in the lattice structure. However, 465VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000 FIG. 1 (color). Variation of the hexagonal lattice constant as a function of the external field. Solid line is theory: solid circles denote measured values. Pictures of the hexagonal lattice under (a) 17 G, (b) 28 G, and (c) 80 G. A cross-sectional picture of the coated sphere is shown in the lower-right inset. The curved liquid meniscus is shown in the upper-left inset. surface, z0, and the radius of the container r0
0.5 cm are given experimentally. The maximum height z0 is noted to depend on the weight of the coated spheres. Its value can be measured by photographical means. A picture of the curved fluid surface is shown in the upper inset in Fig. 1. The magnetostatic energy density Um consists of the energy of magnetic dipole-dipole interaction and the ex￾ternal field, Um
1 V µX i,j ui ? u j r 3 ij 2 3
ui ? rij
u j ? rij r 5 ij 2 X i h ? ui ∂ , (1) where ui
m i 1 xh is the total magnetic dipole moment at site i, m i is the permanent dipole moment of sphere i, x is the magnetic susceptibility, and h is the external field. Here rij denotes the vector pointing from the position of the ith dipole to that of the jth dipole. To find the equilib￾rium orientation state of the dipoles, we start from different initial random configurations and use dissipative spin dy￾namics to evolve towards the configuration of minimum energy. When the magnetic energy is considered together with the gravitational potential of the microspheres on the fluid surface, the lattice structure and the lattice constant are determined by the condition of minimum total energy. This is so because the thermal effect is totally negligible in our system due to the size of the microspheres. The results of our calculations show the triangular lattice to be the most favorable for normal applied field. The pre￾dicted lattice constant variation as a function of the ap￾plied field is in good agreement with the experiment, as shown in Fig. 1. Here the only parameters involved are jm j 1026 emu and x 0.008, both estimated from ex￾perimental and material parameters, e.g., nickel’s magne￾tization of 55 emug. Structural transitions were realized by tilting the mag￾netic field at an angle u away from the z direction and rotating an angle f relative to the x axis as defined in the upper inset in Fig. 2. A pictorial summary of all the different structures observed, and their occurrence in the u, f coordinates, is shown in Fig. 2. With increasing u and fixed f
0, the lattice constant along the x axis decreases while that along the y direction increases to form the centered rectangular structure, as shown in Fig. 2(c). Starting at u
27±, lattice instability sets in. Some of the spheres are attracted by their neighbors to form short chains as indicated by the arrows in Fig. 2(d). Further increase in u leads to perfectly equally spaced, straight chains aligned along the field direction, as shown in Fig. 2(f). Besides the equal spacing, it is noteworthy that the neighboring chains are also shifted by half a diameter in respect to each other. To obtain other planar crystal structures, we start from the case shown in Fig. 2(b) and then rotate the center stage holder along the w direction. If the rotation is very slow, e.g., less than a few degrees per minute, then the whole pattern would just rotate in step with the magnetic field, and there is no change in the lattice structure. However, 5465