NUC-20

Neutron 1/v capture

σ_a = σ₀ (v₀ / v). Thermal capture falls as 1/velocity.

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Nuclear1/v law

Governing equation

\sigma_a=\sigma_0 v_0/v

$v \in \left[50, 2\times 10^{7}\right]$

where

\sigma_0: σ at v₀ (b)
v_0: Reference speed (m/s)
v: Neutron speed (m/s)
\sigma_a: Absorption (b)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\sigma_a=\sigma_0 v_0/v

$v \in \left[50, 2\times 10^{7}\right]$

$v = 50$ $\sigma_a = 4342.8$

Along the curve

②

Domain of the variable — blue

v_{\min} = 50,\quad v_{\max} = 2\times 10^{7}

$v \in \left[50, 2\times 10^{7}\right]$

$v = 50$ $v = 50$

Along the curve

③

Response curve — green

\sigma_a=\sigma_0 v_0/v

$v \in \left[50, 2\times 10^{7}\right]$

$v = 50$ $\sigma_a = 4342.8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_a}{\mathrm{d}v}

$v \in \left[50, 2\times 10^{7}\right]$

$v = 1.26\times 10^{5}$ $\sigma_a^{\prime} = -0.017$

Along the curve

Lecture brief

Historical brief

Rutherford, Chadwick, the semi-empirical mass formula, fission (Hahn–Strassmann 1938) and fusion Q-values, then Compton and Bethe–Bloch stopping, are the nuclear toolkit. Decay, binding and dose start here. This sheet (NUC-20 — Neutron 1/v capture) is the form associated with 1/v law. Working symbols: $\sigma_0$ , $v_0$ , $v$ $\rightarrow$ $\sigma_a$ . s-wave capture. Resonances break the law; 2200 m/s is the thermal reference.

Purpose

Purpose: compute

\sigma_a

from

\sigma_0

v_0

v

in Nuclear physics via

\sigma_a=\sigma_0 v_0/v

σ_a = σ₀ (v₀ / v). Thermal capture falls as 1/velocity. Use it when a real nuclear physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma_0 = 98.700\,\mathrm{b}

v_0 = 2200.000\,\mathrm{m/s}

v = 2200.000\,\mathrm{m/s}

, the governing relation

\sigma_a=\sigma_0 v_0/v

yields

\sigma_a = 98.700\,\mathrm{b}

. A σ(v) that falls, a thermal mark. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

σ at v₀ \sigma_098.7 bReference speed v_02200 m/sNeutron speed v2200 m/s

Outputs

Absorption \sigma_a98.700 b

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Narration of this film

A σ(v) that falls, a thermal mark.

s-wave capture. Resonances break the law; 2200 m/s is the thermal reference.

Reading speed

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