Proton Density

Introduction

The calculations in this paper may be circular. It is necessary to use the known proton mass to calculate its density.  This in turn results in a calculation giving a wavelength of a photon that is equal in energy to the mass equivalence energy of a proton.  But, there are 2 factors suggesting the calculations are not circular.

First, the number plugged into the calculation is not the number that results from the calculation.  The resulting number is a factor of  greater than the plugged in number.

Second, the hypothesis that the calculation should give a density near but slightly greater than the nuclear density is validated.  The correct density results only when the mass of a proton (or very near the mass of a proton) is used.

Proton Density:  A Fifth Fundamental Constant?

In the macroscopic world, every measurement can be theoretically predicted from only 4 fundamental constants:

·        The speed of light, C,

·        The gravitational constant, G.

·        Planck’s constant, h, and

·        The charge of an electron, e

But these constants by themselves don’t shed much light on the strong force or why the proton’s mass is what it is.  Since the proton is the only stable hadron, there must be something significant that determines its mass.  There must be at least one more fundamental constant.

The wavelength of an electron in a hydrogen atom, which is also the atom’s Bohr circumference is:

Wavelength =  where m is the electron mass and v is its orbital velocity.

Analogously it can be shown that for any particle (that has rest mass)    where  is the wavelength of a photon of the mass equivalence (from  )  of the particle.

so

The hypotheses I will now suggest are:

The proton’s density is the fifth fundamental constant and is the densest stable form of matter.

The proton’s radius is the “radius” of its wavelength or its mass equivalent wavelength divided by .

The Bohr radius is given by the equation

By analogy, it will be assumed that the proton radius is given by the equation

Now, in this form, the equation appears to have 2 possible variables,  and m.  the equation needed to determine the mass of the proton can have only one variable.  But since the proton has size as well as mass, the equation can be re-arranged, substituting the proton’s volume multiplied by its density for its mass:

When the calculations are made, this should give a proton density close to nuclear density, but somewhat greater than nuclear density assuming the nucleons are not packed completely together in the nuclei.  The calculations, in fact, give a proton density several times common nuclear densities.  This may mean that the hypothesis that the proton density is a fifth fundamental constant is wrong.  But it may mean that the nucleons are loosely packed in the nucleus, possibly due to the coulomb repulsion.

By analogy, the Bohr hydrogen atom is many times denser       than liquid hydrogen .  The hydrogen discrepancy is due to coulomb repulsion of electrons (as well as imperfect packing).

In any case, the following calculations give a proton density that is “in the ballpark” of what it should be.

The following calculations assume that the proton’s circumference is the same as its mass equivalent wavelength.

The proton is in equilibrium with its own standing wave.

Derivation of proton circumference:

;

=   proton circumference

Everything under the radical is now constant.

From    the wavelength of a photon with the mass equivalence of a proton is  =  meter.  Assuming this is also the

circumference of a proton, the proton radius is:

The fourth root of this number should regenerate the proton circumference.

The equation    works for all particles with rest mass.  And, of course, the same is true for   .   But densities calculated from this equation for particles less massive than the proton are less than nuclear density.

The hypothesis that proton density is a fundamental constant suggests that the proton is the densest stable matter.  Particles heavier than the proton quickly decay to a proton and other things.