Magnetic Moments: g-Factors and the Electron · Elementary Particle Physics

Two measurements seventy-six years apart share this page because the idea behind them did not change in between. What changed is how much of the world could be excluded from the apparatus.

🎯 Why this matters

Precision at this level is bought by exclusion rather than by cleverness. The modern version holds a single electron at fifty millikelvin, and most of the design exists to keep everything else out.

\frac{g}{2} = 1.001\,159\,652\,180\,59\,(13)

Fourteen significant figures. This is the most precisely measured quantity in physics, and — because QED can be pushed to match — the most precisely calculated one too. The agreement between those two numbers is the sharpest test any physical theory has ever passed.

This page is about where the number comes from. Both halves of it: the 1947 measurement that started QED, and the trap that now holds a single electron still enough to read thirteen digits off it.

⚠️ Two notation warnings before you read on

g is not the metric here. In §5.1 g was $g^{\mu\nu}$ , the metric tensor diag(1, −1, −1, −1). On this page and the next it is the gyromagnetic ratio, a dimensionless number near 2. They share a letter and nothing else. Chapter 6 will add a third g, the strong coupling — so the rule for the rest of the book is that g means whatever the current page says it means, and the page must say.

This page works in SI, not natural units. The rest of the site sets ħ = c = 1 and quotes everything in GeV. A Penning trap is a laboratory object, so here the field is in tesla, the frequencies in hertz and the orbit in nanometres — because those are the numbers an experimenter reads off, and converting them to GeV would hide exactly what the design constraints are. The snippets carry the SI constants explicitly for that reason.

What a g-factor is

📐 Physics you need first — magnetic moments, and why there are two of them

A current loop has a magnetic moment $\mu = IA$ — current times area — and in a field it feels a torque $\boldsymbol\mu \times \mathbf{B}$ . An orbiting electron is a current loop, so it has one. That is the orbital moment.

The remarkable thing, and the content of Question 5.1, is that for a classical orbiting charge the ratio of moment to angular momentum does not depend on the orbit at all:

\frac{\htmlClass{t-mu}{\mu_l}}{\htmlClass{t-l}{l}} = \frac{\htmlClass{t-q}{q}}{\htmlClass{t-m}{2m}}

Question 5.1. For any classical orbiting charge, the ratio of magnetic moment to angular momentum depends on nothing but the charge and the mass — the radius and the speed cancel.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

Radius and speed both cancel. That ratio, with $l$ measured in units of $\hbar$ , defines the Bohr magneton , Eq. (5.59): $\mu_B = q_e\hbar/2m_e$ .

Then there is spin, which is not an orbit and has no radius or speed to cancel — so there is no reason for it to give the same ratio, and it does not. It gives twice as much. Writing both cases together, Eq. (5.60):

\htmlClass{t-mus}{\mu_s} = \htmlClass{t-g}{g}\,\htmlClass{t-b}{\mu_B}\,s , \qquad \htmlClass{t-mul}{\mu_l} = \htmlClass{t-g}{g}\,\htmlClass{t-b}{\mu_B}\,l

(5.60)

Spin and orbit written in the same form, so that the one number distinguishing them is isolated. Everything in the rest of this section is about that number failing to be exactly 2.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

with $g$ the gyromagnetic ratio : $g = 1$ for orbital motion by the argument above, and $g = 2$ for spin — which is predicted, not fitted. It falls out of the Dirac equation of §2.8, and it was one of the first triumphs of that equation.

The whole of this section is about the fact that $g$ is not exactly 2.

Note the $1/m$ in $\mu_B$ . The same construction for a proton gives the nuclear magneton $\mu_N = q_e\hbar/2m_p$ , Eq. (5.61), which is 1836 times smaller — so an atom’s magnetic moment is its electrons’, and the nucleus barely contributes.

Question 5.1, and the two magnetons

import numpy as np
e, me, mp, hbar = 1.602176634e-19, 9.1093837015e-31, 1.67262192369e-27, 1.054571817e-34

# Question 5.1: a charge q, mass m, going round a circle of radius r at speed v.
# Current is one charge passing per period:      I = q v / (2 pi r)
# Magnetic moment is current times area:      mu = I pi r^2 = q v r / 2
# Angular momentum is                          L = m v r
# so mu / L = q / 2m -- r and v have both cancelled.
print("Question 5.1, checked numerically on 5 random orbits:")
rng = np.random.default_rng(5)
for r, v in zip(10**rng.uniform(-12, -6, 5), 10**rng.uniform(3, 7, 5)):
    mu = e * v * r / 2
    L = me * v * r
    print(f"  r = {r:.2e} m, v = {v:.2e} m/s  ->  mu/L = {mu/L:.6e} C/kg")
print(f"  every one equals q/2m = {e/(2*me):.6e} C/kg -- independent of r and v")
print()
mu_B = e * hbar / (2 * me)
mu_N = e * hbar / (2 * mp)
print(f"Bohr magneton     mu_B = {mu_B:.6e} J/T = {mu_B/e*1e6:.6f} ueV/T   (5.59)")
print(f"nuclear magneton  mu_N = {mu_N:.6e} J/T = {mu_N/e*1e9:.4f} neV/T   (5.61)")
print(f"ratio mu_B/mu_N = m_p/m_e = {mu_B/mu_N:.2f}")
print("  -- which is why an atom's magnetic moment is its electrons', not its nucleus'")

prints

Question 5.1, checked numerically on 5 random orbits:
r = 6.76e-08 m, v = 3.42e+04 m/s  ->  mu/L = 8.794100e+10 C/kg
r = 7.04e-08 m, v = 4.30e+04 m/s  ->  mu/L = 8.794100e+10 C/kg
r = 1.24e-09 m, v = 1.52e+03 m/s  ->  mu/L = 8.794100e+10 C/kg
r = 5.19e-11 m, v = 1.57e+03 m/s  ->  mu/L = 8.794100e+10 C/kg
r = 2.11e-12 m, v = 9.92e+06 m/s  ->  mu/L = 8.794100e+10 C/kg
every one equals q/2m = 8.794100e+10 C/kg -- independent of r and v

Bohr magneton     mu_B = 9.274010e-24 J/T = 57.883818 ueV/T   (5.59)
nuclear magneton  mu_N = 5.050784e-27 J/T = 31.5245 neV/T   (5.61)
ratio mu_B/mu_N = m_p/m_e = 1836.15
-- which is why an atom's magnetic moment is its electrons', not its nucleus'

1947: measuring a ratio because you cannot measure the thing

Kusch and Foley wanted the electron’s $g_S$ . The route is the Zeeman effect: put an atomic beam in a field $B$ , and the levels split by amounts proportional to $g_J \mu_B B$ . Measure the splitting frequencies, divide by $B$ , get $g_J$ .

By 1947, wartime radar work had made the frequency measurement good to 1 part in 10 000 or 20 000. The magnetic field was nowhere near that good, and it was the field that stood between them and the answer.

⚙️ Engineer’s bridge — precision by subtraction, and by ratio

Their solution is a technique every instrumentation engineer knows, and it appears twice on this page, 76 years apart, in the two experiments that bracket it.

Ratio in a shared reference. Measure the Zeeman frequencies of two states in the same field. Each frequency is $g \mu_B B / h$ , so each carries the field’s error — and the ratio does not. $B$ cancels identically, not statistically. This is ratiometric measurement: a delta-sigma ADC ratios its input against its own reference so the reference’s absolute value drops out; a Wheatstone bridge reads a resistance ratio so the excitation voltage drops out; a differential pair rejects whatever is common to both inputs. Kusch and Foley got $g_S/g_L$ to a part in $10^5$ from a field they could not measure to a part in $10^3$ .

Measure the difference, not the quantity. Seventy-six years later the Penning trap does the second version. The quantity wanted is $\omega_s/\omega_c$ , which is $1.001\,159\,65\ldots$ — and getting thirteen good digits out of it means measuring thirteen digits. But $\omega_a = \omega_s - \omega_c$ is $0.00116\,\omega_c$ , so measuring $\omega_a/\omega_c$ to ten digits gives the same thirteen. Three orders of magnitude of precision, free, because the leading 1 was known in advance and did not need measuring.

If you have ever stored deltas instead of absolutes to fit more resolution in the same word width, or nulled a bridge instead of reading a voltage, this is that idea. It is worth naming, because it is the reason both numbers on this page exist: do not measure what you already know.

Where it breaks: “do not measure what you already know” requires that you do already know it, to the accuracy you are subtracting at. Precision by subtraction converts an absolute measurement into a differential one and inherits the uncertainty of whatever you subtracted — which is exactly what happens on the next page, where the electron anomaly’s comparison is limited not by the trap but by the value of α it must be subtracted against. The technique moves the error rather than removing it, and you must check where it moved to.

🔬 Experiment card — Kusch and Foley, Columbia 1947

Apparatus

Rabi’s atomic-beam magnetic-resonance method, developed at Columbia through the 1930s: a beam of atoms crosses a uniform field of about 0.04 T while an RF field drives transitions between Zeeman sublevels, and a detector counts atoms that were deflected. The field is weak on purpose — the analysis needs Russell–Saunders

LS

coupling to hold, which it does only while the Zeeman splitting is small compared with the fine structure.

What is measured

Not

g_S

. The atomic

g_J

factors of three pairs of states, each pair sitting in the same field:

^2P_{3/2}

and

^2P_{1/2}

of gallium;

^2S_{1/2}

of sodium against

^2P_{3/2}

of gallium;

^2S_{1/2}

of sodium against

^2P_{1/2}

of indium. Each

g_J

is a known linear combination of

g_L

and

g_S

— known because these are single electrons outside closed shells, where the Landé coefficients are exact. Three pairs, three independent extractions of

g_S/g_L

The result

All three came out larger than 2, consistently. Eq. (5.62):

g_S/2 = 1.00119 \pm 0.00005

A departure of 1.2 parts in a thousand from the Dirac value, at 24 times its own error bar.

What it proved

That the Dirac equation is not the last word.

g = 2

is a prediction of it, and the prediction is wrong in the fourth decimal place. Together with the Lamb shift measured the same year in the same building (§5.2), it forced the invention of a theory in which the electron is never alone — and Schwinger supplied it within months.

Schwinger’s line

Schwinger’s answer, in 1948, was that the electron’s charge polarizes the vacuum, and the extra moment comes from the one-loop vertex correction — the diagram drawn in Fig. 5.28 on the previous page. One loop gives the magnetic anomaly

\htmlClass{t-a}{a_e} \equiv \htmlClass{t-d}{\frac{g-2}{2}} = \htmlClass{t-s}{\frac{\alpha}{2\pi}} = \htmlClass{t-n}{0.001\,161\ldots}

(5.61)

Schwinger's 1948 result — the first prediction ever made by a quantum field theory, and still the most precisely tested. It is engraved on his tombstone.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

It is engraved on his tombstone, and it deserves to be: an entire new theory, and its first prediction is one symbol over two symbols.

one loop against seventy-six years of measurement

import numpy as np
ALPHA = 1 / 137.035999166

# Schwinger 1948, the first QED prediction ever made, in one line:
a_schwinger = ALPHA / (2 * np.pi)
print(f"a_e = alpha/2pi          = {a_schwinger:.9f}")
print(f"g/2 = 1 + a_e            = {1 + a_schwinger:.9f}")
print()
print("against the two measurements, 76 years apart:")
kf, kf_err = 1.00119, 0.00005                      # Kusch and Foley 1947, Eq. (5.62)
now, now_err = 1.00115965218059, 0.00000000000013  # Fan et al. 2023,      Eq. (5.67)
print(f"  Kusch & Foley 1947 : g_S/2 = {kf} +- {kf_err}")
print(f"  Fan et al.    2023 : g  /2 = {now} +- {now_err}")
print(f"  one loop vs 1947   : {(1+a_schwinger-kf)/kf_err:+.1f} sigma  -- 'agrees perfectly'")
print(f"  one loop vs 2023   : off by {abs(1+a_schwinger-now):.2e}"
      f" = {abs(1+a_schwinger-now)/now*1e6:.1f} ppm")
print()
print("Why one loop was enough in 1947 and is hopeless now:")
print("  the 1947 error bar was  5.0e-05")
print(f"  the 2023 error bar is   {now_err:.1e}   -- 3.8e8 times smaller")
for k in range(1, 7):
    print(f"    order (alpha/pi)^{k} ~ {(ALPHA/np.pi)**k:.2e}"
          + ("   <- first term smaller than today's error bar"
             if (ALPHA/np.pi)**k < now_err and (ALPHA/np.pi)**(k-1) >= now_err else ""))

prints

a_e = alpha/2pi          = 0.001161410
g/2 = 1 + a_e            = 1.001161410

against the two measurements, 76 years apart:
Kusch & Foley 1947 : g_S/2 = 1.00119 +- 5e-05
Fan et al.    2023 : g  /2 = 1.00115965218059 +- 1.3e-13
one loop vs 1947   : -0.6 sigma  -- 'agrees perfectly'
one loop vs 2023   : off by 1.76e-06 = 1.8 ppm

Why one loop was enough in 1947 and is hopeless now:
the 1947 error bar was  5.0e-05
the 2023 error bar is   1.3e-13   -- 3.8e8 times smaller
  order (alpha/pi)^1 ~ 2.32e-03
  order (alpha/pi)^2 ~ 5.40e-06
  order (alpha/pi)^3 ~ 1.25e-08
  order (alpha/pi)^4 ~ 2.91e-11
  order (alpha/pi)^5 ~ 6.76e-14   <- first term smaller than today's error bar
  order (alpha/pi)^6 ~ 1.57e-16

The last block of that output is the whole logic of §5.9b in advance. One loop sufficed for 1947 because the error bar was $5\times10^{-5}$ and the next term is $5\times10^{-6}$ . Today’s error bar is $1.3\times10^{-13}$ , and the first term smaller than that is $(\alpha/\pi)^5$ — which is why the calculation has to be carried to tenth order in the coupling, and why the diagram count reaches 12 672.

Erratum — Eq. (5.63) equates the anomaly to one plus the anomaly

The equation is printed as

a_e = \frac{g-2}{2} = \frac{1}{2}\frac{\alpha}{\pi} = 1.00116

The first two expressions are the anomaly, whose value is $\alpha/2\pi = 0.001\,161$ . The number on the right, $1.00116$ , is $g/2$ , which is $1 + a_e$ . The chain is off by exactly 1 at the last equals sign.

Either half is fine on its own — $a_e = \alpha/2\pi = 0.00116$ , or $g/2 = 1 + \alpha/2\pi = 1.00116$ — but the printed line takes the left of one and the right of the other. The number itself is right, and it is the one that must be compared with (5.62)‘s $g_S/2 = 1.00119$ for the sentence “the theory agrees perfectly with the experiment” to mean anything, so read $1.00116$ as $g/2$ throughout.

The trap

both precess about B — the spin at ω

s, the orbit at ωc

Fig. 5.36 — the electron as a classical top. S and μ point in opposite directions because the charge is negative, which is the reason (5.66) and (5.67) carry a minus sign in front of μ_e/μ_B. Were g exactly 2, the spin and the momentum would precess at exactly the same rate and the tilt between them would never change. The entire experiment is the observation that it does.

The measurement needs one electron, held still, for hours, and the device that does it is the Penning trap . That is harder than it sounds, and the reason is a theorem.

💡 What this really says — you cannot trap a charge with electric fields alone

In empty space the electrostatic potential obeys Laplace’s equation, $\nabla^2 V = 0$ . A trap needs a minimum of the potential energy in all three directions — but at a minimum all three second derivatives are positive, and Laplace says they sum to zero. Contradiction. This is Earnshaw’s theorem , and it says no arrangement of static charges anywhere in the universe can hold another charge in stable equilibrium.

What you get instead is the next best thing: a saddle. The trap uses a quadrupole whose potential energy for the electron goes as $z^2 - \rho^2/2$ — a valley along the axis, a hilltop across it. That gives confinement in $z$ for free, and the radial problem is handed to the magnetic field, where $q\mathbf{v}\times\mathbf{B}$ bends the escaping electron into a circle instead of letting it reach the wall.

Two fields, two jobs. The price is that the motion is no longer simple: there are now three normal modes instead of one, and the orbit is their sum.

the Penning trap · Fig. 5.37

magnetic field B = 5.30 T — the Fan et al. trapaxial frequency ν_z = 114.0 MHz — set by the electrode voltage

cyclotron ν_c′

148.36 GHz

axial ν_z

114.0 MHz

magnetron ν_m

43.8 kHz

anomaly ν_a = a_e ν_c

172.0 MHz

Three modes separated by 1,301× and 2,603× — which is why they can be driven and read independently. The free-space cyclotron frequency comes back from all three by the invariance theorem, ν_c² = ν_c′² + ν_z² + ν_m²: 148.36 GHz against 148.36 GHz — exact, and immune to the trap being slightly tilted or elliptical. The ground-state cyclotron orbit is 11.1 nm across the radius, so the "artificial atom" really is a few dozen nanometres wide.

Fig. 5.37(a) and (b). The field tab is the saddle: dashed quadrupole field lines, straight B lines, and the force on the electron — inward along z, outward in ρ. The orbit tab is the epicycloid that results. Only ν_z is not a book number; the book gives B = 5.3 T and does not print the axial frequency, so 114 MHz is a representative value for this trap and the readouts follow from it exactly. The drawn frequency ratios are compressed by about forty times so the cyclotron loop is visible at all — the true ratios are printed underneath.

🛠️ the trap itself — five coaxial electrodes that are also a microwave cavity

Click a numbered marker for what that piece does.

Schematic, along the axis. The book gives the electrode count, the field and the cavity limit; the roles are the standard ones for this trap geometry.

🔢 Worked example — three frequencies, and the one design constraint that follows

The magnetic field alone would give the free-space cyclotron frequency, Eq. (5.64):

\htmlClass{t-w}{\omega_c} = \frac{\htmlClass{t-qb}{q_e B}}{\htmlClass{t-m}{m}} = \frac{2\htmlClass{t-mb}{\mu_B} B}{\hbar}

(5.64)

The free-space cyclotron frequency — the rate at which a charge circles a magnetic field line, independent of how fast it is going or how large the circle is.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

Adding the electric field splits this into two, because the radial motion now has two solutions — a fast small circle (the modified cyclotron $\omega_c'$ ) and a slow large drift around the axis (the magnetron $\omega_m$ ). They are the roots of $\omega^2 - \omega_c\omega + \omega_z^2/2 = 0$ , so $\omega_c' + \omega_m = \omega_c$ exactly, and the trap only works while $\omega_c^2 > 2\omega_z^2$ — push the electrode voltage past that and the radial hill wins.

Put in the book’s $B = 5.3$ T and see what the numbers force.

the three modes, and what 5.3 T implies about the hardware

import numpy as np
e, me, hbar = 1.602176634e-19, 9.1093837015e-31, 1.054571817e-34
B, nu_z = 5.3, 114e6           # B is the book's; nu_z is representative of the trap

nu_c = e * B / (2 * np.pi * me)
wc, wz = 2*np.pi*nu_c, 2*np.pi*nu_z
wp = (wc + np.sqrt(wc**2 - 2*wz**2)) / 2      # modified cyclotron
wm = (wc - np.sqrt(wc**2 - 2*wz**2)) / 2      # magnetron
a_e = 0.00115965218059

print(f"at B = {B} T:")
print(f"  cyclotron  nu_c' = {wp/2/np.pi/1e9:8.2f} GHz")
print(f"  axial      nu_z  = {nu_z/1e6:8.1f} MHz")
print(f"  magnetron  nu_m  = {wm/2/np.pi/1e3:8.1f} kHz")
print(f"  anomaly    nu_a  = {a_e*nu_c/1e6:8.1f} MHz   (= a_e nu_c)")
print(f"  separated by {wp/(2*np.pi*nu_z):.0f}x and {2*np.pi*nu_z/wm:.0f}x -- so they never mix")
print()
print(f"the free-space nu_c = {nu_c/1e9:.1f} GHz forces the design: the trap is also a")
print("microwave cavity, and the book says it works 'up to 160 GHz'.  It has to.")
print()
print("invariance theorem -- nu_c recovered from all three, immune to a tilted")
print("or elliptical trap:")
print(f"  sqrt(nu_c'^2 + nu_z^2 + nu_m^2) = {np.sqrt((wp**2+wz**2+wm**2))/2/np.pi/1e9:.6f} GHz")
print(f"  nu_c                            = {nu_c/1e9:.6f} GHz")
print()
r = np.sqrt(hbar/(me*wc))
print(f"ground-state cyclotron radius sqrt(hbar/m omega_c) = {r*1e9:.1f} nm")
print("  -- the book's 'charge distribution a few dozen nanometres across'")
print()
print("why measure the difference instead of the thing:")
print(f"  omega_s/omega_c = 1 + a_e = {1+a_e:.14f}   -- 13 digits needed for 10 good ones")
print(f"  omega_a/omega_c =     a_e = {a_e:.14f}   -- 10 digits give the same 10")
print(f"  the ratio of those two is {1/a_e:.0f}, 'about three orders of magnitude'")

prints

at B = 5.3 T:
cyclotron  nu_c' =   148.36 GHz
axial      nu_z  =    114.0 MHz
magnetron  nu_m  =     43.8 kHz
anomaly    nu_a  =    172.0 MHz   (= a_e nu_c)
separated by 1301x and 2603x -- so they never mix

the free-space nu_c = 148.4 GHz forces the design: the trap is also a
microwave cavity, and the book says it works 'up to 160 GHz'.  It has to.

invariance theorem -- nu_c recovered from all three, immune to a tilted
or elliptical trap:
sqrt(nu_c'^2 + nu_z^2 + nu_m^2) = 148.360196 GHz
nu_c                            = 148.360196 GHz

ground-state cyclotron radius sqrt(hbar/m omega_c) = 11.1 nm
-- the book's 'charge distribution a few dozen nanometres across'

why measure the difference instead of the thing:
omega_s/omega_c = 1 + a_e = 1.00115965218059   -- 13 digits needed for 10 good ones
omega_a/omega_c =     a_e = 0.00115965218059   -- 10 digits give the same 10
the ratio of those two is 862, 'about three orders of magnitude'

The 148 GHz is the point worth pausing on. The book states $B = 5.3$ T and, two sentences later, that the trap is a microwave cavity good to 160 GHz. Those are not two independent facts — the second is forced by the first, because the transition you have to drive sits at $\omega_c$ .

An artificial atom with four levels

Cool the trap to 50 mK and the electron’s motion quantizes. The cyclotron degree of freedom becomes a harmonic oscillator with levels $E_n = (n + \tfrac12)\hbar\omega_c$ , and each of those splits in two by the spin. Equation (5.65):

\htmlClass{t-E}{E} \;=\; \hbar\,\htmlClass{t-ws}{\omega_s}\,\htmlClass{t-ms}{m_s} \;+\; \hbar\,\htmlClass{t-wc}{\omega_c}\left(\htmlClass{t-n}{n} + \tfrac{1}{2}\right)

(5.65)

A two-quantum-number atom, built out of one electron and two fields. Four levels are enough for the whole experiment.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

Fig. 5.38 — the whole apparatus, as a level diagram. Note the geometry: the right-hand n = 0 level sits above the left-hand n = 1 level, by ħω_a. That gap exists only because ω_s > ω_c, i.e. only because g > 2. If the Dirac value were exact the two levels would be degenerate and the gap would close. The experiment is the measurement of that one small gap.

Erratum — which transition is the anomaly transition

The text names the anomaly transition twice, on p. 216 and again on p. 217, as $|1,-1/2\rangle \to |1,+1/2\rangle$ . That is not it. From Eq. (5.65),

E(|1,+\tfrac12\rangle) - E(|1,-\tfrac12\rangle) = \hbar\omega_s

— the spin frequency, not the anomaly. The transition that costs $\hbar\omega_a$ is between levels with different $n$ and different $m_s$ :

E(|0,+\tfrac12\rangle) - E(|1,-\tfrac12\rangle) = \hbar(\omega_s - \omega_c) = \hbar\omega_a

which is $|1,-1/2\rangle \to |0,+1/2\rangle$ — and that is exactly what Fig. 5.38 draws: the short arrow runs from the left-hand $n=1$ level to the right-hand $n=0$ level, not up the right-hand column. The figure is right and the prose is wrong, in both places.

The distinction is not pedantic: the entire precision argument of the section rests on $\omega_a$ being the small frequency. A transition costing $\hbar\omega_s$ would be at 148 GHz, three orders of magnitude worse to work with, and would throw away exactly the advantage the experiment was designed around.

The frequencies connect to the answer through Eq. (5.66):

-\frac{\mu_e}{\mu_B} = \htmlClass{t-ratio}{\frac{\omega_s}{\omega_c}} = \htmlClass{t-one}{1} + \htmlClass{t-small}{\frac{\omega_a}{\omega_c}}

(5.66)

The measurement principle, in one line. The quantity wanted to thirteen digits is rearranged so that the apparatus is only ever asked for ten.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

That " $1 +$ " is the whole design. The left side is the number wanted to thirteen digits; the right side gets twelve of them for free and asks the apparatus for $\omega_a/\omega_c$ to ten. The book’s own summary of it is worth quoting: the ratio can be measured with the same clock — the two frequencies are counted against one oscillator, so the oscillator’s own error cancels in the ratio, just as the magnetic field cancelled for Kusch and Foley.

🔬 Experiment card — Fan, Myers, Sukra and Gabrielse, Northwestern 2023

Apparatus

One electron. A cylindrical Penning trap of five coaxial electrodes at 50 mK, in B = 5.3 T, the trap volume doubling as a microwave cavity good to 160 GHz. The electron’s axial motion induces image charges on the end caps, and that induced current is the only wire out.

What is measured

Two frequencies, as a ratio:

\omega_a

near 172 MHz and

\omega_c

near 148 GHz. Microwaves are injected and swept; a transition shows up as a peak in the rate of quantum jumps, whose height is the transition probability. Neither frequency is wanted on its own — only

\omega_a/\omega_c

, which is

a_e

directly.

The result

Eq. (5.67):

-\frac{\mu_e}{\mu_B} = \frac{g}{2} = 1.001\,159\,652\,180\,59\,(13)

0.13 parts per trillion — 1.3 parts in $10^{13}$ . Ten good digits on $\omega_a/\omega_c$ became thirteen on $g/2$ , exactly as the " $1 +$ " promised.

What it proved

Nothing, yet — a measurement this precise is only worth its error bar if the theory can be pushed to meet it. That is §5.9b: the perturbation series carried to tenth order, 12 672 diagrams for the last term alone, plus contributions from muons, from the weak interaction, and from hadrons. When the two numbers are laid side by side they agree, and that agreement is the most stringent quantitative test in the history of science. It is also, run backwards, the best measurement of

\alpha

— which is where §5.8’s

\alpha^{-1}(0) = 137.035999166

came from.

	Kusch & Foley 1947	Fan et al. 2023
the sample	a beam of Ga, Na and In atoms	one electron, for months
field	0.04 T	5.3 T
the quantity you cannot measure well enough	the magnetic field B	an absolute frequency
how it is dodged	take a ratio of two states in the same B — B cancels identically	measure ω_a = ω_s − ω_c, and ratio it against ω_c on the same clock
result	g_S/2 = 1.00119 ± 0.00005	g/2 = 1.001159652180 59(13)
precision	5 × 10⁻⁵	1.3 × 10⁻¹³ — 380 million times better
theory needed to match	one diagram, α/2π	ten orders, 12 672 diagrams in the last term alone

Both experiments are built on the same refusal: do not measure the big thing badly, measure the small thing well. That is the transferable idea on this page, and it long outlives the physics.

🔑 If you remember only three things

The trap is also the detector. One particle, and the same electrodes that hold it are what read it out.
The anomaly is the entire measurement. That g sits near 2 was settled by an argument from the 1920s; this apparatus exists to reach the twelfth digit after it.
1947 measured a ratio of two g-factors, not a g-factor. Neither could be had on its own, and the ratio was available — which is the same move the trap makes with frequencies.

Where this goes next

§5.9b is the theory side: the $(\alpha/\pi)$ series of Eq. (5.68), the diagram counts, the hadronic and electroweak terms — and the muon, where the same measurement is 4.6 parts in 10⁷ and does not obviously agree.
§5.8 is where this measurement is used in reverse: $g/2$ plus the SM calculation is what fixes $\alpha^{-1}(0) = 137.035999166$ , the anchor for the whole running curve.
§2.8 is where $g = 2$ came from. It is worth rereading now that the fourth decimal place has been shown to disagree with it.

✅ Check yourself — g-factors and the electron

0/5 answered · 0 correct

1.Why is it significant that the orbital ratio μ/l = q/2m contains no radius and no speed?
2.Kusch and Foley could measure Zeeman frequencies to 1 part in 20 000 but the field to nothing like that. How did they get g_S to a part in 10⁵ anyway?
3.Why does a Penning trap need a magnetic field at all, rather than a cleverer arrangement of electrodes?
4.Equation (5.66) writes ω_s/ω_c as 1 + ω_a/ω_c. Why is that rearrangement the whole experiment?
5.Schwinger's α/2π agreed with 1947 to within half a standard deviation. Why is one loop hopeless today?