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Derivation of the proton radius  

A direct correlation between the charge distribution of the electrons and the charge of the protons exists in the atoms.  
Therefore, the proton radius was derived with the classical electron radius in order to check this relation.

A space ball has the characteristics of charge distribution, which is included in the classical electron radius. We consider  
the two radii and get the relation:

(3-22)

Rearranging this equation, we obtain the proton radius with:

(3-23)

Using  in the equation above, we do obtain the relation between the proton radius and the quantized charge and mass with  
the following formula:

(3-24)

We also are able to calculate the proton radius by using the following relation to the quantized magnitudes with the Planck  
mass and the Planck volume:

(3-25)

The relationship between mass and volume generally shows the proportion of the mass per unit volume as the density. As  
we shall see later, the reciprocal of the density also is of great importance in physical processes as a specific volume.

Other relations:
(3-26)

With the above derived equations for the modified proton radius we do obtain the value:
 

According to the experiments with muonic hydrogen at the Paul Scherrer Institute, the radius of the proton is   and with  
the equations derived we do obtain a slight deviation.
For the proton mass with the modified radius, we finally obtain the value.  

Even from the other now following contexts, it is obtained exactly this value for the proton mass.


Calculation of the Proton Mass

With the classical electron radius, we calculate a spherical volume, which is the interaction of the electron charge to mass  
of the electron as a distribution in a spherical volume. I.e. instead of the usual hollow sphere, consider a ball filled with the  
volume:


With the proton radius we calculate a spherical volume, which contains constituents of the proton in the nucleus:
.
The Planck volume with half the Planck length as the radius we do obtain with:

With this sphere volume for the density of the proton in the nucleus we obtain the following relationship with the Planck  
magnitudes:

(3-27)

The density of protons in the atomic nucleus as the ratio of its mass to its volume is equivalent to the Planck volume  and  
the volume according to the classical electron radius  and the ratio to the Planck mass. The factor 48 exists between the  
proton density and the spatial distribution of the Planck mass with the quantized volume.

Pictorially this can be imagined as follows: The 48-fold ratio of the smallest volume with the most small-mass, which is  
included in the volume of the classical electron radius, gives the density of the proton. The density of the proton mass  
multiplied by the quantized mass results in a ball with the magnitude  according to the classical electron radius with 48  
Planck- globules.  


Figure:
Relation of the proton density and Planck volume

We also can derive the proton mass independently from its radius. We use the equation (Equation 3-6)
   
the Radius of (Equation 3-17) with   and
obtain the mass of the proton with the quantized magnitude as:

(3-28)

Here, the redenominated equation shows us with

(3-29)

that the proton mass gives a particular Planck volume in units of Planck mass.

Since a close relationship is present between the protons and electrons, and their masses are contained in a certain  
quantized volume, we obtain the following relation:

(3-30)

The first two terms stand for the rate of the Planck mass for the electron and proton per space ball in the Planck volume.  
After transformation of this equation we do obtain:

(3-31)
I.e. the mass of the proton and the electron in proportions of Planck mass results in 360 times the Planck volume.

If we use quantized sizes in the above equation instead of the masses of the proton and electrons then we do obtain with   
and :
(3-32)

By rearranging we do obtain the proton radius with:

(3-33)

I.e. the proton radius is derived from a certain ratio between the Planck volume and the Planck mass, which can be  
defined as a specific density of the proton.

Other relations:
(3-34)

(3-35)

(3-36)

(3-37)
(3-38)

With the Planck mass in eV () we obtain the following relation:
(3-39)

The mass oft he proton- according tot he derived equation in eV is:


According to CODATA value is: 938,272046 MeV/.

With the proton mass in eV () we obtain the following relation:
(3-40)

(3-41)
(3-42)

(3-43)


Summary of the results so far

Based on the quantized sizes we can derive the proton radius and the mass of the proton and electron with the previous  
equations and directly calculate them from the new Planck units. This way we obtain previously unknown relationships  
that will be analyzed further in future projects.

The derived equations are fundamental relations, because they are based on quantized values. For example the relation of  
the electron mass as the ratio between the quantized charge and the quantized mass is even included in the smallest  
dimension. The expansion factor as a power of ten between the smallest dimension and the dimension in which the value  
has been determined experimentally shows us that this correlation is noticeable only from certain spatial dimension in our  
measurements. However, its components and the relations of which it exists do already exist in the smallest dimension. In  
other words, the experimentally measured subatomic particles are aggregations of smaller particles, and their properties  
are based on the same context.

The connection between mass and volume, which we call the density for example, exists for an apple just like for the  
Earth, but in different size dimensions. This fundamental relation of the density does not arise only when the apple has  
reached a certain size. The apple itself is made of such a context.

The minimal differences in the microscopic size scales between the values measured  in the laboratory and the theoretical  
values of the derivations in addition to technical measurement inaccuracies also are based on even the smallest influence of  
gravity and the acceleration of gravity, which add up over several size scales.

The measured mass, or to be more specifically, the weight of a particle, and accordingly of the proton, depends on the  
place/ location. On Earth, gravity and gravitation are not the same at different areas and places.  On the earth, depend on  
the geographic location we would read different values for a mass.

Therefore, in terms of the mass, it must be distinguished between the empirical values under the action of gravity and the  
actual values. It could be possible to add additional influencing factors to the calculated values, but these influence factors,  
such as the acceleration of gravity, are also dependent on the location themselves and therefore variable, too. It would be  
better to indicate the local dependent gravitational acceleration in mass measurements. In the later Chapter we will discuss  
the direct connection between the proton mass and the acceleration due to gravity.

The components of atoms can be explained by the new world model, but since we only have analyzed the mass and  
radius, we do not receive sufficient information about the structure and the internal structure of these particles. For a new  
nuclear model, all the other properties of the atoms also must be taken into account. If we trace back more experimental  
findings about atoms on the quantized size, we also can get a better picture of the internal structure of atoms. In future  
projects, we will analyze further correlations using the quantized sizes, it will be possible to obtain a better picture of the  
internal structure of the atoms.
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