Constants & Units Pixel Sticker Sheet

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

Kibbla

A Kibble balance runs in two modes. In weighing mode, a current I in a coil sitting in a radial magnetic field pushes up with force BLI, balancing the weight mg. In moving mode, the same coil is moved at speed v and a voltage U = BLv is induced. The hard-to-measure product BL cancels between the two, leaving a match between mechanical and electrical power.

mgv = UI
m: test mass (kg); g: local gravitational acceleration (m/s²); v: coil velocity (m/s); U: induced voltage (V); I: coil current (A)

Pumpette

A single-electron pump is a nanoscale island (a quantum dot or metal island) cut off by tunnel barriers. Its charging energy is large enough that electrons can only cross one at a time (Coulomb blockade). Driving the gates at frequency f moves one electron per cycle, so the current is set by counting.

I = ef
I: current (A); e: elementary charge, 1.602 176 634 × 10⁻¹⁹ C exactly; f: pumping frequency (Hz)

Echobel

In acoustic gas thermometry, the resonant frequencies of a gas-filled spherical cavity of known size give the speed of sound. For a dilute monatomic gas the speed of sound fixes k_BT per atom. Measurements of this kind at the triple point of water (NPL, LNE, INRiM and others) gave the most precise values of the Boltzmann constant before the 2019 redefinition.

c² = γk_BT/m
c: speed of sound in the zero-pressure limit (m/s); γ: heat-capacity ratio, 5/3 for an ideal monatomic gas; k_B: Boltzmann constant, 1.380 649 × 10⁻²³ J/K exactly; T: thermodynamic temperature (K); m: mass of one gas atom (kg)

Quantix

In a superconducting ring the electrons pair up into Cooper pairs and share a single macroscopic wavefunction. The phase must come back to itself around the loop, so the fluxoid (the flux, plus a small supercurrent term) is quantised in whole multiples of Φ₀. In a type-II superconductor the field enters as Abrikosov vortices, each carrying one Φ₀, arranged in a triangular lattice.

Φ₀ = h/2e ≈ 2.067 833 848 × 10⁻¹⁵ Wb
Φ₀: magnetic flux quantum (Wb); h: Planck constant (exact); 2e: charge of a Cooper pair

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)

Bohra

In hydrogen's ground state (1s), the radial probability density r²|ψ|² peaks at exactly a₀ (with the proton treated as infinitely heavy). This is the atomic unit of length.

a₀ = 4πε₀ħ²/(m_ee²) = 5.291 772 105 44(82) × 10⁻¹¹ m ≈ 52.9 pm
a₀: Bohr radius (m); ε₀: vacuum permittivity (measured since 2019); ħ: reduced Planck constant (exact); m_e: electron mass (measured); e: elementary charge (exact)

Gravitas

G sets the strength of gravity between any two masses. It is measured with torsion balances (Cavendish, 1798) and other lab methods, where the pull between kilogram-scale masses is about 10⁻⁷ N or less.

G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²
G: Newtonian constant of gravitation; (15): standard uncertainty in the last two digits, 2.2 × 10⁻⁵ relative

Weinberga

In the electroweak theory (Glashow, Weinberg, Salam), the neutral gauge fields B and W³ mix. Rotating them by the weak mixing angle θ_W gives the massless photon and the massive Z boson. The angle also links the couplings: e = g sin θ_W.

sin²θ_W ≈ 0.231 (MS-bar at M_Z)
θ_W: weak mixing (Weinberg) angle, about 28.7°; MS-bar: modified minimal-subtraction renormalisation scheme; M_Z: Z boson mass scale, about 91.19 GeV

Landaura

In a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).

G = (2e²/h) Σ_n T_n, G₀ = 2e²/h ≈ 7.748 091 729 × 10⁻⁵ S
G: conductance (siemens, S); e: elementary charge, exact in the 2019 SI; h: Planck constant, exact in the 2019 SI; T_n: transmission probability of mode n (0 to 1); 2: spin degeneracy, two spin states per lane; G₀: conductance quantum, exact (CODATA 2022)

Massratio

The proton is about 1836 times heavier than the electron. This dimensionless ratio sets why atoms have a tiny heavy nucleus with a light electron cloud, why nuclei barely move in chemistry (the Born–Oppenheimer approximation), and the ratio of vibrational to electronic energies in molecules. It is measured with Penning-trap mass comparisons and precision spectroscopy of molecular hydrogen ions such as HD⁺.

μ = m_p/m_e = 1836.152 673 426(32)
μ: proton-to-electron mass ratio (dimensionless); m_p: proton rest mass, measured; m_e: electron rest mass, measured; (32): standard uncertainty in the last two digits, about 1.7 × 10⁻¹¹ relative

Thomsonia

When light meets a free electron at low energy, its electric field shakes the electron, which re-radiates at the same frequency. This elastic scattering has a fixed total cross section, the effective target area of one electron. It is J. J. Thomson's classical result, and it is the low-energy limit of quantum electrodynamics.

σ_T = (8π/3) r_e² = 6.652 458 7051(62) × 10⁻²⁹ m²
σ_T: Thomson cross section; r_e: classical electron radius, e²/(4πε₀m_ec²) = 2.817 940 3205(13) × 10⁻¹⁵ m; e: elementary charge (exact); ε₀: vacuum permittivity (measured); m_e: electron mass (measured); c: speed of light (exact)

Candelor

The candela, the SI unit of luminous intensity, links watts of light to how bright that light looks to a standard human eye. It is defined by fixing the luminous efficacy of monochromatic radiation at 540 × 10¹² Hz (a green of about 555 nm) at exactly 683 lm/W. For other wavelengths the watts are weighted by the CIE photopic sensitivity curve V(λ).

Φ_v = K_m ∫ V(λ) Φ_e,λ dλ, K_cd = 683 lm/W
Φ_v: luminous flux (lumen, lm); K_cd: luminous efficacy of 540 THz radiation, 683 lm/W exactly (defining constant of the 2019 SI); K_m: maximum photopic efficacy, ≈ 683 lm/W; V(λ): CIE photopic luminous efficiency function, peak 1 near 555 nm; Φ_e,λ: spectral radiant flux (W/nm); λ: wavelength (nm)