Science
Yukawa Screened Nuclear Potential & Pion Compton Force Range
Calculate the Reduced Compton Force Range R = hbar*c / (m * c^2) (fm) of a massive exchange boson and Hideki Yukawa's exponentially screened nuclear potential V(r) = -g_eff * EXP(-r / R) / r (MeV).
Why the strong nuclear force binds nucleons tightly at 1.4 fm yet vanishes at 5 fm!
Massless photon exchange produces an infinite-range 1 / r Coulomb potential—so why does the strong nuclear attraction overpower proton repulsion inside an atomic nucleus (~1–2 fm) yet drop to zero just a few femtometers outside? In 1935, Hideki Yukawa showed that exchanging a massive mediator of mass m_pi c^2 exponentially screens the 1 / r potential over its Reduced Compton Wavelength: R = (hbar * c) / (m_pi * c^2)!
Using hbar * c = 197.327 MeV·fm and charged pion rest energy m_pi c^2 = 139.57 MeV gives Yukawa Range R = 197.327 / 139.57 = 1.4138 fm! Evaluating V(r) = -g_eff * EXP(-r / R) / r (with effective coupling g_eff = 11.0 MeV·fm) yields a deep -2.538 MeV attraction at r = 1.5 fm that collapses nearly 40-fold to just -0.064 MeV by r = 5.0 fm!
| Nucleon Separation r (fm) (Pion m c^2 = 139.57 MeV -> R = 1.4138 fm) | Screening Factor EXP(-r / R) | Unscreened -g / r (MeV) | Yukawa Potential V(r) (MeV) |
|---|---|---|---|
| Core Equilibrium (r = 1.50 fm) | 0.3461 | -7.333 MeV | -2.538 MeV (Strong Binding) |
| Nuclear Surface (r = 3.00 fm) | 0.1198 | -3.667 MeV | -0.439 MeV |
| Outside Nucleus (r = 5.00 fm) | 0.0291 (97.1% screened!) | -2.200 MeV | -0.064 MeV (Negligible) |
Yukawa Force Range R = 197.327 / (m * c^2) (Femtometers)
Divides hbar * c = 197.327 MeV·fm by mediator rest mass energy B2 (MeV).
=IF(B2 <= 0, "Boson Mass Must Be > 0", ROUND(197.327 / B2, 4))Screened Yukawa Potential Energy V(r) = -g_eff * EXP(-r / R) / r (MeV)
Evaluates V(r) at separation B3 (fm) with effective coupling constant B4 (MeV·fm) and Compton range R = 197.327 / B2.
=IF(OR(B2 <= 0, B3 <= 0), "Mass and Radius Must Be > 0", ROUND((-B4 * EXP(-B3 / (197.327 / B2))) / B3, 3))Try the example
Use the original values or change them to test your understanding.