NUC-23

Coulomb barrier

V_C = 1.44 Z₁ Z₂ / (R₁+R₂) MeV with R = 1.2 A^{1/3} fm.

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NuclearCoulomb barrier

Governing equation

V_C=\dfrac{1.44\,Z_1 Z_2}{R_1+R_2}\,\mathrm{MeV}

$Z_2 \in \left[1, 92\right]$

where

Z_1: Z₁ (—)
A_1: A₁ (—)
Z_2: Z₂ (—)
A_2: A₂ (—)
R_1+R_2: Contact radius (fm)
V_C: Barrier (MeV)

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

V_C=\dfrac{1.44\,Z_1 Z_2}{R_1+R_2}\,\mathrm{MeV}

$Z_2 \in \left[1, 92\right]$

$Z_2 = 1$ $R_1+R_2 = 0.338$

Along the curve

②

Domain of the variable — blue

Z_2_{\min} = 1,\quad Z_2_{\max} = 92

$Z_2 \in \left[1, 92\right]$

$Z_2 = 1$ $Z_2 = 1$

Along the curve

③

Response curve — green

V_C=\dfrac{1.44\,Z_1 Z_2}{R_1+R_2}\,\mathrm{MeV}

$Z_2 \in \left[1, 92\right]$

$Z_2 = 1$ $R_1+R_2 = 0.338$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R_1+R_2}{\mathrm{d}Z_2}

$Z_2 \in \left[1, 92\right]$

$Z_2 = 1.572$ $R_1+R_2^{\prime} = 0.338$

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-23 — Coulomb barrier) is the form associated with Coulomb barrier. Working symbols: $Z_1$ , $A_1$ , $Z_2$ , $A_2$ $\rightarrow$ $R_1+R_2$ , $V_C$ . The classical fusion threshold. Tunnelling and the Gamow window sit below V_C.

Purpose

Purpose: compute

R_1+R_2

V_C

from

Z_1

A_1

Z_2

A_2

in Nuclear physics via

V_C=\dfrac{1.44\,Z_1 Z_2}{R_1+R_2}\,\mathrm{MeV}

V_C = 1.44 Z₁ Z₂ / (R₁+R₂) MeV with R = 1.2 A^{1/3} fm. 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

Z_1 = 1.000\,\mathrm{—}

A_1 = 2.000\,\mathrm{—}

Z_2 = 6.000\,\mathrm{—}

A_2 = 12.000\,\mathrm{—}

, the governing relation

V_C=\dfrac{1.44\,Z_1 Z_2}{R_1+R_2}\,\mathrm{MeV}

yields

R_1+R_2 = 4.26\,\mathrm{fm}

V_C = 2.03\,\mathrm{MeV}

. Two charged drops, a barrier height. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Z₁ Z_11 —A₁ A_12 —Z₂ Z_26 —A₂ A_212 —

Outputs

Contact radius R_1+R_24.26 fm
Barrier V_C2.03 MeV

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Narration of this film

Two charged drops, a barrier height.

The classical fusion threshold. Tunnelling and the Gamow window sit below V_C.

Reading speed

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