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Your Class 12 physics book tells you the nuclear force is very strong and falls to almost nothing beyond a few femtometres. It does not tell you why. In 1934, a 27-year-old physicist in Osaka asked that question and answered it with a prediction: if a force has a short range, it must be carried by a particle with mass. This post explains Hideki Yukawa's idea, works through the estimate he made, and tells the story of how experiments first confused, then confirmed it.
Who was Hideki Yukawa?
According to his Nobel biography, Yukawa was born in Tokyo on 23 January 1907 and grew up in Kyoto, where his father became Professor of Geology at Kyoto University. He graduated from Kyoto University in 1929 and taught at Kyoto and Osaka universities between 1932 and 1939. While at Osaka, in 1935, he published "On the Interaction of Elementary Particles. I." in the Proceedings of the Physico-Mathematical Society of Japan, proposing a new field theory of nuclear forces and predicting the meson. The paper was later reprinted in Progress of Theoretical Physics Supplement, a journal Yukawa himself edited from 1946.
He became Professor of Theoretical Physics at Kyoto in 1939. He won the 1949 Nobel Prize in Physics "for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces". He died on 8 September 1981.
What puzzle was Yukawa trying to solve?
By the early 1930s physicists knew the nucleus holds protons and neutrons, together called nucleons, bound by a force much stronger than the electric repulsion between protons. The NCERT Class 12 chapter on nuclei lists its features: it is much stronger than the Coulomb force, and it falls rapidly to zero once two nucleons are more than a few femtometres apart.
The presentation speech at his Nobel ceremony describes his approach. Yukawa used the electromagnetic field as a model and found it could be modified to give a force with a short range. In modern theories each field is linked to a kind of particle, and Yukawa found a simple relation between the range of the force and the mass of that particle. He estimated the range from experimental data and concluded the new particle should be about 200 times heavier than the electron. Nucleons, in his picture, keep emitting and absorbing these particles, and that exchange is the nuclear force.
Worked example: from range to mass
The relation is R ≈ ħ/(mc), or R ≈ ħc/(mc²) so that you can use MeV and femtometres.
Constants. NIST gives ħc = 197.327 MeV fm and the electron's rest energy as 0.511 MeV.
Pick a range. Take R = 2 fm as an illustrative value, in line with "a few femtometres".
Solve for mass. mc² = ħc/R = 197.3 / 2 = 98.7 MeV.
Compare with the electron. 98.7 / 0.511 ≈ 193, so the particle is roughly 200 electron masses, the same order Yukawa found.
Now run it the other way. The Particle Data Group's 2024 listing for the charged pion gives its mass as 139.57 MeV. Then R = 197.3 / 139.57 ≈ 1.4 fm, a range that fits the nuclear force.
Common mistakes:
Mixing units. Keep ħc in MeV fm and mc² in MeV, and the answer comes out in fm.
Using h instead of ħ. That makes the answer 2π times too large.
Treating it as exact. This is an order-of-magnitude estimate. Nuclear forces do not stop sharply at one distance.
Why did the first meson cause confusion?
Yukawa said the new particles could not be made in ordinary nuclear reactions, but should appear in cosmic rays, which carry far more energy. In 1937, the presentation speech records, Anderson, Neddermeyer and other American physicists found a particle of about the right mass in cosmic rays. The Nobel biography says this discovery encouraged Yukawa to develop his theory further.
There was a problem. The particle found in 1937 did not interact with nucleons nearly as strongly as Yukawa's carrier should. The puzzle was settled by the British physicist Cecil Powell, whose group used photographic plates and found two kinds of meson. The heavier kind interacted with nucleons about as strongly as Yukawa had postulated, and it decayed into the lighter kind. Powell won the 1950 Nobel Prize in Physics for the photographic method and his discoveries regarding mesons. The heavier particle is the one now called the pion.
What can a student take from Yukawa's way of working?
Borrow a model on purpose. He started from the electromagnetic field and changed one feature. When you face a new topic, ask which familiar idea it most resembles and what is different.
Turn a description into a number. "Short range" became "about 200 electron masses", a claim an experiment could test.
Say where to look. He pointed to cosmic rays, and that is where the first evidence turned up.
Expect a messy path. The first particle found was the wrong one. The idea survived because the next experiment was better.
For a related story, read how Abdus Salam's electroweak theory used the same range-mass link for the weak force. To practise the Class 12 side of nuclear physics, try our nuclear binding energy problems. If the jump from fields to particles is hard to picture, the Fermi Concept Library has every concept in physics with its own lesson, so you can revise one idea before moving to the next.
Try the estimate with a range of 1 fm and work out how many electron masses the carrier would need.
