Subcritical Multiplication and the Inverse Multiplication Plot (1/M) — Every Doubling of the Count Rate Halves the Distance to Criticality

Principle and mechanism

A series that adds a factor k each generation

If a source injects S neutrons every generation, Sk remain in the next generation, then Sk², and so on. The steady-state total is S(1 + k + k² + …) = S/(1−k). With a detector efficiency ε (including position and geometry), the count rate is CR = εS/(1−k). If the source and detector stay the same, CR₁(1−k₁) = CR₂(1−k₂) holds between two states, so 1/M normalized to a reference count rate becomes zero at criticality:

1/M = CR₀/CR = (1−k)/(1−k₀)

• Strictly, 1/M is proportional to (1−k). Since ρ = (k−1)/k, 1−k = −ρ/(1−ρ) ≈ −ρ.
• Source strength changes only the level of the count rate, not the critical condition (k = 1).
• Deep subcritical, higher modes mix in, so the measured "source multiplication factor" can differ from k_eff; the two converge as criticality is approached.

The count-rate doubling rule

With constant source strength and detection efficiency and equilibrium count rates, each doubling of CR halves (1−k). For k₀ = 0.95 and CR₀ = 30 cps:

The reactivity added per doubling is 2,699 → 1,298 → 637 → 316 pcm, about half the remaining margin each time. Near criticality the count rate jumps more for the same reactivity, so steps are made smaller. A multiple-choice option saying "each doubling leaves one quarter of the margin" is wrong.

Worked 1/M extrapolation

Hypothetical teaching conditions: critical boron 1,000 ppm, constant boron worth −10 pcm/ppm, 30 cps at 1,500 ppm, approach by dilution.

Even in the point model 1/M is not exactly proportional to ρ, so the curve bends slightly. Early extrapolations predict a lower concentration than the true one (slightly non-conservative), but they converge to 1,000 ppm as the approach proceeds.

Why the curve bends

• Detector position (spatial effects): loading fuel or withdrawing rods near the detector raises the local flux first, so 1/M falls quickly at first (concave up). The extrapolation predicts criticality early, which is conservative. Far from the detector the curve is concave down and predicts late, which is non-conservative. The curvature, however, depends on the relative layout of source, fuel and detector and on the flux distribution together, so position alone should not be taken as decisive.
• Non-linearity of the control variable: integral control rod worth is an S-shaped curve, and assembly worth differs by location.
• Not waiting for equilibrium: in the one-group approximation the equilibrium time constant is τ ≈ (β−ρ)/(λ|ρ|): 14 s at −5,000 pcm, 53 s at −200 pcm and 175 s at −50 pcm (β = 0.0065, λ = 0.08 s⁻¹). Counting too early reads 1/M too high, which is non-conservative.

So two or more source-range (SR) channels are plotted together and the most conservative prediction is followed. The next step is limited to a fraction of the remaining predicted amount (by practice, half or less).

Worked example: one-group point kinetics with a source

Symbols: C (precursors), S (source), n₀, ρ₀, δρ and n.

• Equations: dn/dt = [(ρ−β)/Λ]n + λC + S, dC/dt = (β/Λ)n − λC
• Steady state (ρ₀ < 0): substituting λC₀ = βn₀/Λ gives S = −ρ₀n₀/Λ
• New steady state after inserting δρ (ρ₀ + δρ < 0): n = n₀·ρ₀/(ρ₀+δρ); for δρ ≪ |ρ₀|, n ≈ n₀(1 + δρ/|ρ₀|)
• Example: ρ₀ = −0.02, δρ = +0.005 → n = 1.33n₀. Substituting Λ = ℓ/k gives n = Sℓ/(1−k), the same form as M.

What it means in a plant

Confirming criticality safety during fuel loading

• Preconditions: confirm the refueling boron concentration to guarantee k_eff ≤ 0.95 (W STS LCO 3.9.1); isolate unborated water sources (3.9.2); have two SR channels operable, one with an audible count rate in the control room (3.9.3); check the SR source response within 8–12 hours before loading; ensure emergency boration capability and continuous communication.
• During loading: load in an order such that the worth of each added assembly decreases as the core grows. Plot 1/M at each step with a set counting time, and measure boron concentration at a frequency matched to the worst-case dilution capability. For an initial core, 1/M is plotted for every assembly. For a reload, whose aim is to keep k_eff ≤ 0.95, the focus is on watching for sudden count-rate increases and verifying the core configuration.
• Stop criteria: stop if any of the following occurs: unexpected multiplication behaviour (count-rate jump), loss of communication, an SR detector inoperable, or emergency boration unavailable.

ECP and early criticality at startup

The ECP (estimated critical position) is obtained from the last critical condition, corrected for xenon and samarium, temperature, power defect, burnup and boron changes. Rods are withdrawn in steps in the prescribed sequence, and the 1/M predictions are compared with the ECP. Measured core reactivity must be within ±1 %Δk/k of the prediction (SR 3.1.2).

• Causes of early criticality: ECP input errors (xenon, boron, power history), boron sampling errors or unintended dilution, dilution and withdrawal at the same time, not plotting or misreading 1/M, excessive withdrawal steps, and lapses in supervision or procedure compliance. Cases reported in the United States include V.C. Summer (1985), Palo Verde 1 (1988) and Wolf Creek 1 (1995, where criticality about 1,000 pcm earlier than the ECP was recognized in advance from the 1/M plot).
• Design measures: SR high-flux trip, startup-rate alarm and rod-withdrawal block, interlocks on withdrawal speed and bank sequence, boron dilution alarm, insertion-limit alarm.
• Operational measures: independent verification of the ECP, no simultaneous dilution and withdrawal, 1/M hold points at each step, rod insertion and cause evaluation when outside the band, startup rate of 1 dpm or less, and a reactivity management programme (WANO/INPO SOER 2007-1).

Neutron source rods

• Function: provide a count rate the SR channels can read, preventing a "blind" startup, and make it possible to observe multiplication and confirm channel response.
• Primary sources: emit neutrons by themselves: Cf-252 (spontaneous fission, T½ 2.65 years), Pu-Be and Am-Be ((α,n)); essential for an initial core.
• Secondary sources: Sb–Be. Sb-123 becomes Sb-124 (T½ 60.2 days) in the reactor, and its 1.69 MeV gamma induces Be-9(γ,n) (threshold about 1.67 MeV). They are recharged during operation.
• Operating without source rods: possible if the intrinsic source of irradiated fuel (spontaneous fission of Cm-242 and Cm-244 etc.) ensures the minimum count rate. RG 1.68 specifies at least 1/2 cps before startup with a signal-to-noise ratio above 2. This must be demonstrated by analysis and measurement, and it is unfavourable for initial cores and after long shutdowns.

Regulatory and design criteria

For example, in the United States:

• RG 1.68 Rev. 4 (2013): Appendix A-3 (initial criticality) requires SR ≥ 1/2 cps, S/N > 2, rod withdrawal before dilution, and no period shorter than 30 s. Appendix C-2 (loading procedures) is the source of the loading conditions and stop criteria above.
• W STS (NUREG-1431): LCO 3.9.1 boron concentration, 3.9.2 unborated water source isolation, 3.9.3 nuclear instrumentation, SR 3.1.2 core reactivity. 10 CFR 50 Appendix A GDC 26–28 and 10 CFR 50.68 (criticality prevention in handling and storage) also apply.
• Current edition: RG 1.68 Rev. 4 (2013) is current; the physics and procedural principles are unchanged.

Easily confused pairs

Key points at a glance

1. Definition: in a core with k < 1, source neutrons multiply through chains to a steady count rate.
2. Equations: M = 1/(1−k), CR = εS/(1−k), CR₁(1−k₁) = CR₂(1−k₂).
3. 1/M: CR₀/CR = (1−k)/(1−k₀) → zero at criticality.
4. Extrapolation: plot against boron, rod position or loaded amount; the zero of the straight-line extrapolation is the predicted critical point.
5. Doubling rule: CR × 2 → (1−k) halved; reduce step size.
6. Curvature: detector close (concave up) is conservative; far (concave down) is non-conservative.
7. Waiting for equilibrium: τ ≈ (β−ρ)/(λ|ρ|), longer nearer criticality.
8. Loading: boron for k_eff ≤ 0.95, two SR channels with audible count, stop on abnormal multiplication.
9. Startup: compare with the ECP, ±1 %Δk/k, startup-rate limits, prevent early criticality.
10. Sources: primary Cf-252, secondary Sb–Be; ensure the minimum count rate.