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Relative Particle Sizes
Based on Equal Mass Densities

Particle Sizes

There are creditable estimates of the sizes of a neutron and a proton.

The radius of a proton is about 3/4 that of a neutron.

But the sizes of the rest of the subatomic particles are a great mystery, even that of electrons. An internet search for the size of an electron brings up assertions ranging from an electron being three times the size of a proton down to it being one one-thousandth of that size.

The rest masses of all charged particles are known from the curvature of their trajectories in a magnetic field.

It is quite plausible that the mass densities of all subatomic particle are the same. If that is the case then the the ratio of the radii of two particles is equal to the cube root of the ratio of their rest masses.

The Size of an Electron

The mass of a proton is 1836 times that of an electron. Let Rp and Re be the radii of a proton and an electron, respectively. Then

Rp/Re = (1836)1/3 = 12.245

Here is a depiction of their relative sizes.

with the proton shown in red and the electron in blue.

Since the accepted radius of a proton is 0.8768 fermi this makes the radius of an electron 0.0716 fermi.

The Leptons

There are three types of leptons and their antiparticles: the electron, the muon and the tauon. But a muon has a mass 207 times that of an electron, If the muon has the same mass density as the electron then the ratio of their radii is given by

Rμ/Re =` (207)1/3 = 5.9155

On the other hand, the tauon has a mass 3477 times of an electron. thus the ratio of the their radii is

Rτ/Re =` (3477)1/3 = 15.15

Here is a depiction of their relative sizes.

The particle shown in orange is the tauon, the one in purple is the muon and the one in blue is the electron.

The Neutrino of an Electron

For a long time it was thought that neutrinos have zero mass but a recent study estimated the rest mass energy of a neutrino to be 0.06 of one electron volt. The rest mass energy of an electron is about 0.511 million electron volts. With this ratio of masses and equal mass density the radius of a neutrino would be about 1/2000 of that of an electron.

Here is a depiction of their relative sizes.

The electron is the large blue sphere and its neutrino is the near pinpoint green sphere to its left.

The Proton and the π Meson

In 1934 the Japanese physicist Hideki Yukawa published an article that predicted the existence of a particle intermediate in mass between the electron and the proton. Its mass was to be roughly 200 times the mass of an electron. When later a particle was found in cosmic ray debris with a mass 207 times that of an electron the physics community thought Yukawa had been amazingly accurate. But that particle did not have the physical attributes Yukawa predicted. That particle turned out to be a heavy electron now called a muon .

Still later a particle was found with a mass about 270 times that of an electron that had the right attributes. More precisely its mass was 273 times that of an electron. It subsequenty came to called a positive π meson. There also are a negative π meson and a neutral π meson.

A proton has a mass 1836 times that of an electron. Therefore the ratio of the radius of a π+ meson to that of a proton is given by

Rπ/Rp = (273/1836)1/3 = (0.1487)1/3 = 0.53

Here is a visual depiction of their relative sizes.

with the π meson shown in yellow.


The conventional theory asserts that quarks are point particles. But there is no getting around the fact that it would take an infinite amount of energy to create even one charged point particle. Thus there is not enough energy in the entire Universe, even if all of its mass were converted into energy, to create even one charged point particle. And there are zillions upon zillions of quarks in the Universe. Further more if one charged point particle did exist it could explode releasing more energy than in all the stars of the Universe.

Consider a charge which exerts a force inversely proportional to separation distance squared. There is a wonderful theorem in mathematical physics which says that a spherically distributed charge of that type affects other charged bodies outside of itself as if all of its charge were concentrated at its center. Thus any evidence for a point particle is evidence for a spherically distributed charged particle.

For a particle located within a charged spherical shell there is no force exerted.

Nucleons (protons and neutrons) are composed of three concentric shells of quarks.

A Proton

A Neutron

For further details on the Concentric Spheres Model of Nucleon Structure see Quarkic Structure of Nucleons.

The outer surface of a neutron would be the outer surface of a large Down quark. Similarly the outer surface of a proton would be the outer surface of a large Up quark. Thus here is what a Down quark and an Up quark would look like.

The Down quark is on the left.

Mass Density of the Proton

The volume V of a proton is

V = (4/3)π(0.8768)³ = 2.8235 cubic fermi

The mass of a proton is 1.6726216x10−27 kg, which is the same as 1.6726216 octillionth of a gram. Thus the mass density of a proton is 0.5924 octillionth of a gram per cubic fermi which is the same as 5.924x1017 kg/m³.

At this density a sphere of radius 0.5 millimeter, the size of a BB, would have a mass of 7.405x107 kg or about 74 million kilograms.

(To be continued.)

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