Title: Experiments on 2d Arrays of Trapped Magnetized Balls: Static Shell Configurations and Velocity Distributions under Mechanical Agitation

Author (Talk): Peter Koch, Dept. Physics and Astronomy, Stony Brook Univerist

Abstract:

Experiments by A.M. Mayer (1878) on permanently magnetized needles floating in a magnetic trap provided the first observations of 2d shell structures of N repelling “particles” not in contact, eventually up to N ~ 50. More recent 2d experiments use electrostatic forces (e.g., dusty plasmas or charged balls) or magnetic forces (e.g., permanent or induced magnets or superconducting vortices). Many have used external or thermal driving to study velocity distributions, phase changes, or other topics in statistical physics. Stimulated by the need for improved physics teaching-lab experiments that can be extended for original research, I am developing an inexpensive magnetic trapping apparatus that adapts techniques used in granular physics experiments. Current I in a solenoid electromagnet induces parallel, vertical moments in a steel trap boundary (cm-scale) and N steel balls (mm-scale) sitting/rolling without slipping on a horizontal glass disk (smooth or rough surface) at mid-bore of the solenoid. Access from above allows manipulation of individual balls with plastic tweezers or of many balls with air puffs. This greatly facilitates finding metastable structural configurations or intentionally introducing defects. Lateral agitation (f ~ 25 Hz) can (quasi-thermally) drive the array. Though the magnetic forces are entirely repulsive for identical balls [dipole-dipole: F∝ r^(-4)], unequal ball diameters lead to an unusual “molecular chemistry”: weak attraction at short range and repulsion at longer range. A circular boundary frustrates the triangular 2d-lattice (identical balls), giving shell structures for N ≥ 5 with ground- and metastable-configurations labeled by integers for the number of balls per shell going outward. A hexagonal boundary permits “perfect” crystals when N=3K(K+1)+1 for integral K. Digital photos [movies] of static [agitated] configurations are well suited for structural [dynamical] analysis. Open-source software (ImageJ) allows balls to be tracked with sub-pixel accuracy. That the strength of the magnetic force is easily adjusted via the solenoid current I is a definite advantage. While having no effect on the static, structural studies, it is crucial for the agitated experiments because it allows careful control over the ratio of the interaction energy (ball-ball and ball-boundary) to the (quasi) thermal energy. The talk highlights results obtained with N=19 identical balls, for which the static, ground-state shell structure of (1,6,12) is nearly that of the “perfect” 2d triangular lattice (weak frustration). At a constant strength of agitation, digital images were obtained at 210 fps for three different values of I: I_1 weakest to I_3 strongest. I_3 kept all 19 balls close to their static positions and produced a velocity distribution with significant departures from 2d Maxwell-Boltzmann (2dMB). [Nearly one million velocities allow velocity distributions to be histogrammed over nearly 4 decades of dynamic range.] I_1 was weak enough for complete melting, though intershell transitions were noticeably slower than intrashell transitions. It produced excellent agreement with 2dMB. I_2 allowed only intrashell melting, but it, too, produced excellent agreement with 2dMB, and, moreover, the velocity distributions for the center ball (“1-shell”) alone, the 6-shell alone, and the 12-shell alone were all consistent.