Pattern and rhythm in physics and art

Item consists of a digitized copy of an audio recording of a Vancouver Institute lecture given by Anthony Arrott on November 22, 1980. Original audio recording available in the University Archives (UBC AT 944).Other UBCUnreviewedOthe

Using magnetic charge to understand soft-magnetic materials

This is an overview of what the Landau-Lifshitz-Gilbert equations are doing in soft-magnetic materials with dimensions large compared to the exchange length. The surface magnetic charges try to cancel applied magnetic fields inside the soft magnetic material. The exchange energy tries to reach a minimum while meeting the boundary conditions set by the magnetic charges by using magnetization patterns that have a curl but no divergence. It can almost do this, but it still pays to add some divergence to further lower the exchange energy. There are then both positively and negatively charged regions in the bulk. The unlike charges attract one another, but do not annihilate because they are paid for by the reduction in exchange energy. The micromagnetics of soft magnetic materials is about how those charges rearrange themselves. The topology of magnetic charge distributions presents challenges for mathematicians. No one guessed that they like to form helical patterns of extended multiples of charge density

Spin Directions in Pure Chromium

We have carried out a triple‐axis polarized‐neutron‐beam experiment with polarization analysis of the final beam and magnetic fields to 15 kG applied to a pure Cr single crystal. The purpose was to determine whether the spin axis in the transversely polarized spin‐density wave state (122°K–38.5°C) is confined to the cube edges or whether in sufficient fields it can be made to lie in an arbitrary direction (perpendicular to the wave vector). The experiments show unambiguously that the latter is so. At 25°C, it is slightly more difficult to confine the spins to a single 110 axis than it is to a single 100 axis. At lower temperatures this anisotropy is enhanced. These results along with our previous results for the field dependence of the cube‐edge components using unpolarized neutrons have been analyzed in terms of two different models. Both models have the spins in all directions perpendicular to the wave vector of the spin‐density wave. One is the model of thermal activation of small domains. The other considers a domain structure with wall motion. In both models ansiotropy and magnetic field influence the net number of spins in any given direction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70250/2/JAPIAU-40-3-1447-1.pd

Neutron—Diffraction Study of Cr and Cr Alloys

The principle quantities which parameterize the antiferromagnetism of Cr are studied by neutron diffraction as a function of temperature, pressure, magnetic field, and concentration of solute atoms. In order to account for the observed intensities of the magnetic reflections, their dependence on magnetic field and temperature, the torque measurements of Montalvo and Marcus, and the creation of a ``single‐Q'' state in field‐cooling, a model is proposed based on the presumed existence of thermally active polarization domains within a ``single‐Q'' region of the crystal. The variation of the polarization axis from place to place and with time lowers the free energy by an increase in entropy. The pressure dependence of the first‐order phase change at 38.5°C is given as dTN∕dP=−5.4 deg∕kbar. The temperature dependence of the length of the wave vector below TN is given as (1∕Q) (dQ∕dT)=6.3×10−5 deg−1. Alloys with 0.5 and 0.78 wt% Fe and with 0.9 wt% Co show a decrease in TN of ∼20°K per at.% of solute. The amplitudes of the magnetization waves increase and the wave vector Q approaches commensurateness with the lattice periodicity with increasing solute concentration in contrast to results for other solutes. Some unusual effects were observed for 2.3 wt% Fe samples.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70981/2/JAPIAU-38-3-1243-1.pd

Magnetic charges suppress effects of anisotropy in polycrystalline soft ferromagnetic materials

Micromagnetic simulations of polycrystalline iron washers show that grain boundary charges, ρ = -div M, suppress bad effects of magnetocrystalline anisotropy. A single domain wall divides the washer into two domains with opposite magnetization; M is almost = ± Ms ϕ, where ϕ circulates about the hole in the washer. There is a ripple structure. M tilts back and forth toward the inner and outer surfaces. Magnetic charge densities, σm = n·M, on the surfaces keep M at the surfaces very close to lying in the surfaces. The exchange εx and magnetostatic εd energy densities try to keep M parallel to the surfaces throughout the washer, except at the domain wall. An anisotropy energy in each grain is reduced linearly in the angle of rotation away from the circulating pattern towards the nearest anisotropy axis. Both εx and εd near grain boundaries increase as the square of these angles. Anisotropy wins for small rotations. However, the coefficients of the positive quadratic terms are so much larger than the coefficients of the negative linear terms that the rotations are quite small. If the height of the washer is sufficiently greater than 300 nm, M in the washer no longer acts as it would in a thin film. If 300 nm washers are stacked with a spacing of 4 nm, the ripple structure is not lost. The stacked washers can then be used as the core of a transformer. The most remarkable effect is that starting with M = Ms ϕ everywhere, the reversal of M by the field from a current along the z-axis produces a single domain wall. It is stable even in zero field because the wall has Néel caps that act as springs against the surfaces. The suppression of crystalline anisotropy in polycrystalline iron also occurs for geometries other than the toroid; some might be better for creating transformers

Helical patterns of magnetization and magnetic charge density in iron whiskers

Studies with the (1 1 1) axis along the long axis of an iron whisker, 40 years ago, showed two phenomena that have remained unexplained: 1) In low fields, there are six peaks in the ac susceptibility, separated by 0.2 mT; 2) Bitter patterns showed striped domain patterns. Multipole columns of magnetic charge density distort to form helical patterns of the magnetization, accounting for the peaks in the susceptibility from the propagation of edge solitons along the intersections of the six sides of a (1 1 1) whisker. The stripes follow the helices. We report micromagnetic simulations in cylinders with various geometries for the cross-sections from rectangular, to hexagonal, to circular, with wide ranges of sizes and lengths, and different anisotropies, including (0 0 1) whiskers and the hypothetical case of no anisotropy. The helical patterns have been there in previous studies, but overlooked. The surface swirls and body helices are connected, but have their own individual behaviors. The magnetization patterns are more easily understood when viewed observing the scalar divergences of the magnetization as isosurfaces of magnetic charge density. The plus and minus charge densities form columns that interact with unlike charges attracting, but not annihilating as they are paid for by a decrease in exchange energy. Just as they start to form the helix, the columns are multipoles. If one could stretch the columns, the self-energy of the charges in a column would be diminished while making the attractive interactions of the unlike charges larger. The columns elongate by becoming helical. The visualization of 3-D magnetic charge distributions aids in the understanding of magnetization in soft magnetic materials

Propagation of Bragg‐Reflected Neutrons in Bounded Mosaic Crystals

The analysis of the multiple Bragg reflection of a neutron beam of finite size in a semi‐infinite mosaic crystal given in a recent paper by Werner and Arrott is generalized to include bounded crystals. The coupled differential equations describing secondary extinction given by Hamilton are solved in general, and a method of piecewise solution, or solution by regions, is given.A discussion is given of experiments on the spatial distribution of the diffracted current from slab‐shaped crystals. Various methods for measuring the probability for Bragg scattering per unit path are compared and found not to agree. It is felt that the discrepancies are basic to the mosaic structure of crystals in general.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69444/2/JAPIAU-37-6-2343-1.pd