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The Derivation of Charactistics of Nucleons from the Characteristics of their Constituant Quarks |
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A proton is composed of two up quarks and one down quark. The composition of a neutron is one up quark and two down. Therefore the values p and n of that characteristic for the proton and neutron, respectively, are given by
These values are usually taken to be simple fractions, the ratios of small whole numbers, but that requirement can be reduced to the requirement that they all be whole numbers. Thus the above set of equations can be considered Diophantine equations.
The solutions of the set of equations are
any integral values of u and d will lead to integral values of p and n, but not all integral values of p and n will lead to integral values of u and d.
Consider the case of p=3 and n=−2 so the ratio n/p is equal to −2/3. The values of u and d that come out of the above equations are u=8/3 and d=−7/3. However if the values of m and n are scaled up to +9 and −6, which maintains the −2/3 ratio, the values of u and d are found to be 8 and 7, respectively. Thus integral values are easily achieved by using a scaling operation on the initial levels of m and n.
By way of contrast consider the explanation for the elecrostatic charge of +1 for a proton and 0 for a neutron; i.e., m=1 and n=0. The equations above give u=2/3 and d=−1/3. Or, if m is taken to be 3 and n to be 0, then u=2 and d=−1.
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