It's All Spin Pixel Sticker Sheet

12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: It's All Spin. Every character below has its own sticker page with the full physics panel.

Echoa

In a magnetic field, nuclear spins precess at slightly different rates because the field is not perfectly uniform, so after a pulse knocks them over they fan out and the signal fades. Erwin Hahn found in 1950 (“Spin Echoes”, Phys. Rev. 80, 580) that a second pulse makes them run back together, and a signal reappears: the spin echo. In the version by H. Y. Carr and E. M. Purcell (Phys. Rev. 94, 630, 1954), a π/2 pulse is followed after a time τ by a π pulse that flips the fan over. The fast spins are now behind the slow ones, catch up, and all line up again at 2τ.

t_echo = 2τ

τ: time between the first and second pulse (s); t_echo: time of the echo after the first pulse (s)

Dehaasa

Magnetise an iron rod and it starts to turn. In 1915 Albert Einstein and Wander de Haas hung an iron cylinder on a fine fibre inside a coil (Verh. Dtsch. Phys. Ges. 17, 152). Reversing the magnetisation flips the tiny magnets inside the iron, and their angular momentum flips with them. Angular momentum is conserved, so the whole rod twists the other way. The ratio of magnetic moment to angular momentum sets the g-factor.

L = 2mμ/(ge)

L: angular momentum of the carriers (J·s); μ: magnetic moment (A·m²); m: electron mass (kg); e: elementary charge (C); g: gyromagnetic g-factor

Barnetta

Spin an unmagnetised iron rod and it becomes magnetised. Samuel Barnett showed this in 1915 (Phys. Rev. 6, 239): rotation acts on the electrons' angular momentum like a magnetic field, so their spins tend to line up. It is the reverse of the Einstein–de Haas effect, and gave g ≈ 2 for iron.

M ∝ Ω

M: magnetisation (A/m); Ω: angular speed of rotation (rad/s)

Magnetonix

An electron with one unit of orbital angular momentum ħ has a magnetic moment of one Bohr magneton. This is the natural unit for atomic and electron magnetism.

μ_B = eħ/(2m_e) = 9.274 010 0657(29) × 10⁻²⁴ J/T

μ_B: Bohr magneton (J/T); e: elementary charge (exact); ħ: reduced Planck constant (exact); m_e: electron mass (measured)

Helicita

In 1958 Maurice Goldhaber, Lee Grodzins and Andrew Sunyar measured the handedness of the neutrino (“Helicity of Neutrinos”, Phys. Rev. 109, 1015). Europium-152m captures an electron and emits a neutrino; the daughter nucleus recoils and then emits a gamma ray. Selecting gammas emitted against the recoil and measuring their circular polarisation with magnetised iron tied the neutrino's spin to its motion. Neutrinos turned out left-handed: spin opposite to motion.

h = −1

h: helicity, the spin component along the direction of motion, in units of the maximum

Satyendra Nath Bose

Bose counted photons as truly identical particles and recovered Planck's law. The resulting Bose–Einstein distribution gives the average number of particles in a state of given energy, and it allows many bosons to pile into the same state.

n̄(ε) = 1/(e^((ε−μ)/(k_BT)) − 1)

̄n: mean occupation of a single-particle state; ε: state energy (J); μ: chemical potential (J); k_B: Boltzmann constant (J/K); T: temperature (K)

Hanleva

Shine polarised light at the mercury resonance line (253.7 nm) into mercury vapour and the atoms re-emit it as polarised resonance fluorescence. Add a weak magnetic field and the polarisation drops. Wilhelm Hanle reported this in “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz” (Z. Phys. 30, 93, 1924). The excited atoms behave like little oscillating dipoles that precess around the field at the Larmor frequency. If they turn through a large angle before they emit, the emitted light comes out with a scrambled direction of polarisation.

ΔB ≈ ħ / (g μ_B τ)

ΔB: field scale over which the polarisation falls (tesla, T); ħ: reduced Planck constant; g: Landé g-factor of the excited state; μ_B: Bohr magneton; τ: lifetime of the excited state (s)

Bifurcor

A magnetic field that changes with height pushes on an atom's magnetic moment. Because an electron's spin can only point up or down, silver atoms in the Stern–Gerlach magnet split into exactly two beams instead of smearing out.

F_z = μ_z ∂B_z/∂z

F_z: force on the atom along the field gradient (N); μ_z: z-component of the atom's magnetic moment (for silver, set by its one outer electron) (J/T); B_z: z-component of the magnetic field (T); z: position along the gradient, up the page (m); g_s: electron spin g-factor, about 2.0023; μ_B: Bohr magneton (J/T)

Revolvix

Rotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.

R(2π)ψ = −ψ

R(θ): operator that rotates a spin-1/2 state by angle θ about the axis n; ψ: two-component spinor state of a spin-1/2 particle; n̂: unit vector along the rotation axis; σ: the three Pauli matrices (σx, σy, σz); 2π, 4π: one full turn (360°) and two full turns (720°) (rad)

Wolfgang Pauli

Two identical fermions must have a wavefunction that changes sign when they are swapped. If both tried to occupy the same state, the wavefunction would be zero everywhere, so the arrangement is forbidden: one fermion per state.

ψ(1,2) = [φ_a(1)φ_b(2) − φ_a(2)φ_b(1)]/√2

ψ: two-particle antisymmetrised wavefunction; φₐ, φ_b: single-particle orbitals (including spin); a, b: full sets of quantum numbers (n,ℓ,m_ℓ,m_s)

Barkhausa

Magnetise a piece of iron slowly and smoothly and its magnetisation does not change smoothly. It grows in many tiny, sudden jumps as the walls between magnetic domains snag on defects and then snap free. Heinrich Barkhausen heard this in 1919 by winding a coil around the iron and amplifying the induced pulses into a telephone receiver: a crackle (“Zwei mit Hilfe der neuen Verstärker entdeckte Erscheinungen”, Physikalische Zeitschrift 20, 401, 1919). Each jump induces a small voltage pulse in the coil.

V = −N dΦ/dt

V: voltage induced in the pick-up coil (V); N: number of turns of the coil; Φ: magnetic flux through the coil (Wb); t: time (s)

Spinwell

Angular momentum is moment of inertia times angular speed, and without an outside torque it stays constant. Pull your arms in and your moment of inertia drops, so you spin faster.

L = Iω

L: angular momentum about the spin axis (kg m² / s); I: moment of inertia about that axis (kg m²); ω: angular speed (rad/s); τ: external torque about the axis (N m)