A material that remembers, and the curve it can never return to

At its defaults it draws a material that remembers, and the curve it can never return to. The major loop of a Preisach material — 9216 elementary switches with a spread of coercivities and interaction fields, each carrying its own sign — together with two minor loops driven inside it and the initial curve rising from a demagnetised state. Coercivity 0.63 and remanence 0.77 are read off the drawing. The initial curve is inside the loop everywhere and is the one part of this figure that cannot be revisited: reaching it again means demagnetising the sample.

hysteresis-loop is one function in lib/figures/fields.js — charge, current, flux and the lines drawn between them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The major loop of a Preisach material — 9216 elementary switches with a spread of coercivities and interaction fields, each carrying its own sign — together with two minor loops driven inside it and the initial curve rising from a demagnetised state. Coercivity 0.63 and remanence 0.77 are read off the drawing. The initial curve is inside the loop everywhere and is the one part of this figure that cannot be revisited: reaching it again means demagnetising the sample.

The fraction of saturation a paramagnet reaches, against y = gJμ_B·B/k_BT — the one combination of field and temperature either theory depends on. J = 1/2 leaves the origin with slope 1.0000; J = 3/2 leaves the origin with slope 0.5556; J = 5/2 leaves the origin with slope 0.4667; J = 7/2 leaves the origin with slope 0.4286; classical leaves the origin with slope 0.3333. The slopes are (J+1)/3J, measured off the drawn curves rather than quoted: a spin-half moment is 3.00 times as responsive to a weak field, per unit saturation, as the classical dipole of the same size, and the difference is the whole of the experimental case for discreteness. Every curve saturates at one and none of them crosses another, so a measured curve picks out J without any absolute calibration at all.

The fraction of saturation a paramagnet reaches, against y = gJμ_B·B/k_BT — the one combination of field and temperature either theory depends on. J = 1/2 leaves the origin with slope 1.0000; J = 3/2 leaves the origin with slope 0.5556; J = 7/2 leaves the origin with slope 0.4286; classical leaves the origin with slope 0.3333. The slopes are (J+1)/3J, measured off the drawn curves rather than quoted: a spin-half moment is 3.00 times as responsive to a weak field, per unit saturation, as the classical dipole of the same size, and the difference is the whole of the experimental case for discreteness. Every curve saturates at one and none of them crosses another, so a measured curve picks out J without any absolute calibration at all.

The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it.

The average time a single-domain particle keeps its magnetisation, against the height of its barrier in units of the thermal energy, on a logarithmic scale of seconds. The relation is an exponential, so the axis is a straight line, and every decade of lifetime costs 2.3 of barrier. The four marked lifetimes need barriers read back off the drawn curve by bisection: 1 s needs KV/kT = 20.7, 1 day needs KV/kT = 32.1, 1 year needs KV/kT = 38.0, 10 years needs KV/kT = 40.3. The forty that recording engineers quote is therefore not a measured constant or a rule of thumb; it is the natural logarithm of the ratio between ten years and a nanosecond, and it would be thirty if a decade of data retention were enough. What makes it so sharp a criterion is the exponential on the other side: a barrier ten per cent lower turns ten years into two months.

The diameter of the smallest spherical particle whose magnetisation survives ten years at room temperature, against the anisotropy constant that decides it, both logarithmic. The criterion is the one derived in the companion figure — a barrier of 40.3 thermal energies — and the diameter falls as the cube root of the anisotropy: permalloy, 128.6 nm; nickel, 38.2 nm; iron, 18.8 nm; the cobalt alloy of a hard disk, 11.7 nm; cobalt, 8.9 nm; neodymium iron boron, 4.0 nm; ordered iron platinum, 3.6 nm. The line has a consequence that shaped an industry. A bit on a disk is a few hundred grains, so making bits smaller means making grains smaller, and a grain below the diameter for its material forgets. Raising the anisotropy shrinks the grain — and raises the field needed to write it, which no head can supply past a point. The two requirements are the same constant pulling in opposite directions.

physicscheck asserts something about hysteresis-loop that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.