(R + 1)$ )

An original reduced model makes the cancellation explicit. Assume LS coupling, neglect nuclear spin, and write the electronic magnetic dipole moment as

- μ_{B} (g_{L} L + g_{s} S) /ℏ

. For

L = 1

S = 1/2

, the projection used to derive the Landé g-factor gives

g_{3/2} = \frac{2 g _{L} + g _{s}}{3}, g_{1/2} = \frac{4 g _{L} - g _{s}}{3} .

(1)

These coefficients follow by substituting

J = 3/2

1/2

into the angular-momentum projection formula; neither level measures pure spin alone. In a weak uniform magnetic field, adjacent electronic projections have spacing

h f_{J} = g_{J} μ_{B} B

when

g_{J} > 0

. Therefore

R = \frac{f _{3/2}}{f _{1/2}} = \frac{g _{3/2}}{g _{1/2}} = \frac{2 + x}{4 - x}, x = \frac{g _{s}}{g _{L}}, x = \frac{4 R - 2}{R + 1} .

(2)

The common magnetic field and Bohr magneton cancel. The ratio determines

g_{s} / g_{L}

, not

g_{s}

without further knowledge of

g_{L}

. For

g_{L} = 1

, it gives

a_{e} = (R - 2) / (R + 1)

. Small departures from

R = 2

are magnified differently from small departures of

g_{s}

from

2

; the two fractional changes should not be identified.

With the stipulated inputs

g_{L} = 1

g_{s} = 2.0024

and

μ_{B} B / h = 100 MHz

, the two electronic frequencies are

66.5867

and

133.4133 MHz

. Their ratio is approximately

2.0036043

, and inversion recovers

a_{e} = 0.0012

. These are ideal electronic lines, not historical gallium data. Nuclear spin, mixing between levels and other deviations from LS coupling require a more complete model in an actual atom.

Table of contents
- Drift cancellation in a g-factor comparison Spin-orbit g-factor ratio from two P states

Drift cancellation in a g-factor comparison

 0  0

Even a shared field does not cancel if it is sampled at different times. Write

f_{A} (t) = g_{A} C (t)

and

f_{B} (t) = g_{B} C (t)

, where

C (t) = μ_{B} B (t) / h

. As an original proposed diagnostic, bracket the

B

-state reading at

t = 0

with

A

-state readings at

t = \pm τ

R_{bracket} = \frac{f _{B} ( 0 )}{\frac{1}{2} [ f _{A} ( - τ ) + f _{A} ( + τ )]} .

(1)

C (t) = C_{0} (1 + b t)

, the denominator is exactly

g_{A} C_{0}

, so

R_{bracket} = g_{B} / g_{A}

. A single earlier

A

reading instead gives

R_{unpaired} = (g_{B} / g_{A}) / (1 - b τ)

. For the preceding numerical model and

b τ = 0.001

, this unpaired comparison falsely gives approximately

a_{e} = 0.00186649

; interpolation recovers

0.0012

. The sequence illustrates cancellation and is not asserted to be Kusch and Foley's exact historical timing protocol.

Figure 1.
Cancel a shared calibration scale without hiding its limitations
. Original calculations for spin-orbit g-factor ratio from two P states and drift cancellation in a g-factor comparison. All inputs are stipulated. The left plot shows the nonlinear inversion from a frequency ratio to a spin-to-orbital g-factor ratio. The right plot contrasts an unpaired reading with a symmetric bracket under linear field drift; the dashed curve adds chosen quadratic curvature $c τ^{2} = 0.0002$ . No curve is a measured resonance or a historical fit. **Details**.

For a curved field

C (t) = C_{0} (1 + b t + c t^{2})

, the same calculation gives

R_{bracket} = \frac{g _{B} / g _{A}}{1 + c τ ^{2}} .

(2)

The linear drift cancels but the curvature remains. Different fields sampled by the two beams, additive frequency offsets and incorrect atomic corrections also survive. For a small ratio error

δ R

, direct differentiation of the inverse relation gives

δ x ≃ \frac{6}{( R + 1 ) ^{2}} δ R, δ a_{e} ≃ \frac{3}{( R + 1 ) ^{2}} δ R (g_{L} = 1) .

(3)

Repeated readings can reduce random measurement uncertainty; agreement among genuinely different state comparisons is what tests shared systematic errors in their interpretation.