Coulomb Barrier

Coulomb Barrier

Two nuclei approaching each other experience a strong electrostatic repulsion. The Coulomb barrier is the potential energy at the point of nuclear contact — the height that must be overcome (or tunneled through) for a nuclear reaction to occur.

$V_C = \frac{k_e Z_1 Z_2 e^2}{R_1 + R_2}$

where:

• $k_e = 8.988 \times 10^9$ N·m²/C² (Coulomb constant)
• $e = 1.602 \times 10^{-19}$ C (elementary charge)
• $R_i = 1.2 \times 10^{-15} \cdot A_i^{1/3}$ m (nuclear radius)

Converting to MeV

$V_C [\text{MeV}] = \frac{V_C [\text{J}]}{1.602 \times 10^{-13} \text{ J/MeV}}$

Contact Distance

The sum of the two nuclear radii gives the contact distance in femtometers (fm, $10^{-15}$ m):

$d = R_1 + R_2 = 1.2 \left(A_1^{1/3} + A_2^{1/3}\right) \text{ fm}$

Why It Matters

The D+T fusion reaction has a Coulomb barrier of only ~0.44 MeV (both particles have $Z=1$), making it the easiest fusion reaction. Heavy-ion fusion (e.g., Pb+Pb) requires barriers of hundreds of MeV.

Your Task

Implement:

• coulomb_barrier_J(Z1, A1, Z2, A2) — barrier in Joules
• coulomb_barrier_MeV(Z1, A1, Z2, A2) — barrier in MeV
• contact_distance_fm(A1, A2) — contact distance in femtometers

All physical constants must be defined inside each function.