Realizing SI Units in the Laboratory

Realizing Base Units through Quantum Phenomena

In the modern SI, base units are not defined by specific experiments or physical objects; rather, the numerical values of seven defining constants are fixed without uncertainty. NMIs then utilize recognized primary methods—termed mises en pratique by the BIPM—to realize these units in practice.

The Kilogram Realization: Kibble Balance vs. XRCD

Two complementary, independent primary methods realize the kilogram using the fixed Planck constant ($h = 6.62607015 \times 10^{-34}\text{ J}\cdot\text{s}$):

The Kibble Balance (Watt Balance)

Conceived by Dr. Bryan Kibble at the UK National Physical Laboratory (NPL) in 1975, the Kibble balance compares virtual mechanical power to virtual electrical power through two distinct operational phases:

1. Weighing Mode (Static Force Balance): An unknown mass $m$ is suspended from a balance pan in a local gravitational field $g$. The downward gravitational force $F_g = m g$ is countered by an upward electromagnetic Lorentz force produced by a coil of wire (wire length $L$) carrying an electrical current $I$ immersed in a radial magnetic field of magnetic flux density $B$:

$$m g = I B L$$

1. Velocity Mode (Dynamic Voltage Calibration): The mass is removed, and the coil is moved vertically through the exact same magnetic field $B$ at a constant, highly controlled velocity $v$. By Faraday's law of electromagnetic induction, this motion induces an open-circuit voltage (electromotive force) $V$ across the coil terminals:

$$V = v B L \implies B L = \frac{V}{v}$$

1. Power Equality and Quantum Substitution: Substituting the geometric and magnetic factor $B L$ from the velocity mode into the weighing mode equation yields the virtual power balance:

$$m g v = V I$$

Mechanical power ($m g v$, in watts) equals electrical power ($V I$, in watts). Next, the quantum electrical effects are introduced:

• Voltage $V$ is measured directly against a Josephson Voltage Standard (JVS), which generates quantized voltages governed by the Josephson constant $K_{\text{J}} = 2e/h$:

$$V = n_1 f_1 \left(\frac{h}{2e}\right)$$

• Current $I$ is measured by routing it through a precision standard resistor $R$, producing a voltage drop $V_R = I R$ measured against a JVS. The resistor $R$ is calibrated against a Quantum Hall Resistance (QHR) standard governed by the von Klitzing constant $R_{\text{K}} = h/e^2$:

$$R = \frac{1}{i} \left(\frac{h}{e^2}\right)$$

• Thus, the electrical power product becomes:

$$V I = \frac{V \cdot V_R}{R} = \frac{\left[n_1 f_1 \left(\frac{h}{2e}\right)\right] \left[n_2 f_2 \left(\frac{h}{2e}\right)\right]}{\frac{1}{i} \left(\frac{h}{e^2}\right)} = \frac{i n_1 n_2 f_1 f_2}{4} \cdot h$$

Notice the metrological miracle: the elementary charge $e$ completely cancels out of the equation! The mass $m$ is expressed solely in terms of the fixed Planck constant $h$, local gravitational acceleration $g$, velocity $v$, and two microwave clock frequencies ($f_1, f_2$):

$$m = \frac{i n_1 n_2 f_1 f_2}{4 g v} \cdot h$$

The X-ray crystal density (XRCD) method combines a silicon sphere’s measured volume, lattice spacing, isotopic composition, and surface corrections. A conventional silicon unit cell has eight atoms and volume $a^3$, so the estimated atom count is $8V/a^3$. Measurements and material corrections have uncertainty; this is not an exact count of every atom. Fundamental-constant relations connect the atomic and macroscopic mass scales.

The Ampere Realization: Single-Electron Transport

With the elementary charge fixed at $e = 1.602176634 \times 10^{-19}\text{ C}$, electric current is realized through Single-Electron Transport (SET) pumps. A quantum dot or nanoscale semiconductor gate transfers individual electrons one at a time across a potential barrier at a driving microwave frequency $f$. The resulting electric current is precisely:

$$I = e \cdot f$$

For example, driving a single-electron pump at a clock frequency of $f = 1.000000\text{ GHz}$ ($10^9\text{ s}^{-1}$) produces a reference current of:

$$I = (1.602176634 \times 10^{-19}\text{ C}) \times (10^9\text{ s}^{-1}) = 160.2176634\text{ pA}$$

The Kelvin Realization: Primary Thermometry

Rather than relying on water cells, the kelvin is now realized by measuring thermal energy ($E = k_{\text{B}} T$) through primary thermometric methods:

• Acoustic Gas Thermometry (AGT): Measures the speed of sound $u$ in a noble gas (such as argon or helium) confined inside an acoustic resonator. In the limit of zero pressure, $u^2 = \frac{\gamma k_{\text{B}} T}{m_{\text{atom}}}$, where $\gamma = 5/3$ is the ratio of specific heats and $m_{\text{atom}}$ is the atomic mass.
• Johnson Noise Thermometry (JNT): Measures the statistical mean-square thermal voltage fluctuations across an unloaded resistor $R$ over bandwidth $\Delta f$:

$$\overline{V_n^2} = 4 k_{\text{B}} T R \Delta f$$

The Second and the Meter

• Second ($\text{s}$): Defined since 1967 by taking the fixed unperturbed ground-state hyperfine transition frequency of the cesium-133 atom, $\Delta\nu_{\text{Cs}}$, to be exactly $9,192,631,770\text{ Hz}$. Primary cesium fountain clocks achieve relative standard uncertainties below $1 \times 10^{-16}$.
• Meter: A stabilized laser provides a wavelength through its evaluated optical frequency and the fixed speed of light, with vacuum wavelength $\lambda=c/f$. Frequency combs can compare optical frequencies with a traceable frequency reference. Molecular absorption lines can stabilize suitable lasers; a comb is not itself a requirement that every laser be locked to iodine. Account for refractive index when measuring in air.

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Practical Metrological Significance for Calibration Technicians

Why does a calibration technician working on a manufacturing floor or in a commercial ISO/IEC 17025 accredited laboratory care about quantum redefinitions?

1. Traceability: Results require a documented calibration chain, with uncertainty at every link. Where SI traceability is technically possible, the chain ends at an SI realization.
2. Realization uncertainty: Fixed defining constants have no uncertainty, but the apparatus implementing them does. A primary realization can have systematic effects, environmental sensitivity, and imperfect reproducibility.
3. Transfer: Artifacts disseminate units from national metrology institutes to laboratories and users. Shipping, drift, and handling remain relevant.
4. Continuity: The new definitions preserved the practical sizes of the units. Existing instruments did not become error-free or cease to require calibration.

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Common Calibration Traps & CCT Exam Pitfalls

Official references (checked October 10, 2026): BIPM SI Brochure.