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The Galilei invariant model of the nucleon as a system of three point particles, whose dynamics is governed by Schrödinger equation, after six Hamiltonian parameters fitting, predicts magnetic momenta, masses and charge radii of the proton and neutron with experimental precision. Now this model is applied in order to investigate nucleon charge, mass and magnetism distributions. The obtained electric and magnetic form factors at low values of momentum transfer are in satisfactory agreement with experimental information. The model predicts that neutron is a more compact system than proton.

Different and significant changes of quantum systems, composing more complex structures (for example, atoms forming molecules or solid state) are obvious. The well-known experiments of atomic nuclei structure also indicate that a nucleon embedded in a nucleus is slightly modified in comparison with a free one [

The Schrödinger’s model of nucleon [

Thus, the eigenfunctions of introduced Hamiltonian application for nucleon structure investigation are the next interesting problem. This paper is devoted for electric and magnetic elastic form-factors and corresponding radii description.

The elastic form factor of nucleon is defined as density operator’s Fourier image:

Here

is density operator in the nucleon’s center-of-mass reference frame (

For charge density

Experiment | Theory | |
---|---|---|

Proton magnetic momentum | 2.792847356 (23) | 2.792847356 |

Proton charge radius (fm) | 0.8775 (51) | 0.87750000 |

Proton mass M_{p} (GeV) | 0.938272046 (21) | 0.938272046 |

Neutron magnetic momentum | −1.91304272 (45) | −1.91304272 |

Neutron mean-square charge radius (fm^{2}) | −0.1161 (22) | −0.11610000 |

Neutron mass (GeV) | 0.939565379 (21) | 0.939565379 |

uPP mass m_{u} (GeV) | - | 0.3213453699 |

dPP mass m_{d} (GeV) | - | 0.3381741985 |

A | ||
---|---|---|

E, Charge density | ||

M, Magnetism density | ||

P, Mass density | ||

R, Point particles density |

It is well-known that the form factors, defined experimentally, are independent of angles of the momentum transfer

one obtains that

Here

Written in Jacobi coordinates

it takes the form

Here

it equals the nucleon charge, magnetic momentum, mass

so that its value in zero equals one. The only exception is the electric form factor of the neutron

Thus,

is proportional to the square of corresponding radius operator, i.e.:

Inserting the charges of PP’s one obtains the expressions presented in [

Elastic form-factors of nucleon as functions of momentum transfer

Here both basic functions are bound angular and spin momenta functions

where the parentheses indicate the operation of momenta binding,

where

nodes).

the area where a potential equals zero, radial function equals spherical Hankel function of imaginary argument

condition

Integration of the first member of right side of Equation (9) is straightforward. For calculation of the second and third integrals one needs spherical Bessel function expansion [

The sum of two functions of this kind present in (9) equals

Having in mind the structure of nucleon wave function, only two first terms of expansion, corresponding

Finally, after some angular momentum algebra, the form factor can be presented as

where

Here, the radial wave functions as functions of

where

Obviously, the momentum transfer q in all given expressions has dimension fm^{−1}. The widely accepted dimension of this momentum Q is GeV/c. Therefore, the slight modification is necessary due to these momenta dependence:

Here, in square brackets the dimensions of corresponding quantities are written. The value of conversion factor

Operators, present in right hand side of Equation (13), having in mind the definitions (14), (11) and (10) are necessary for radii calculation. Together with precise calculation, the values of radii can be determined from the slopes of corresponding form factors in the limit of zero momentum transfer

The obtained charge, magnetic, mass and point particles radii are present in the “theory” column of

The known experimental values of corresponding radii are given in the “experiment” column. The charge radii of nucleon are applied for fitting, hence their values are equal to the ones, recommended by the Particle Data Group 2014 report [

Experiment | Theory | |
---|---|---|

Proton charge radius (fm) | 0.8775 (51) [ | 0.877500 |

0.8768 (69) [ | ||

0.879 (6) [ | ||

0.84184 (56) [ | ||

Proton magnetic radius (fm) | 0.777 (16) [ | 0.832087 |

0.876 (19) [ | ||

0.848 (6) [ | ||

Proton mass radius (fm) | - | 0.809988 |

Proton PP radius (fm) | - | 0.811174 |

Neutron mean-square charge radius (fm^{2}) | −0.1161 (22) [ | −0.116100 |

−0.1149 (35) [ | ||

−0.134 (3) [ | ||

Neutron magnetic radius (fm) | 0.862 (9) [ | 0.759587 |

Neutron mass radius (fm) | - | 0.767979 |

Neutron PP radius (fm) | - | 0.766703 |

In

with parameters

The neutron electric form factor

The comparison of calculated and given by standard dipole approximation

with

The all four obtained form factors demonstrate good enough comparison with known experimental data at low values of momentum transfer, that characterises the nucleons, present in an atomic nucleus. Moreover, ratio of electric and magnetic form factors of neutron at

The mass and point particles form factors for proton

The most interesting result of radii calculation is that the neutron appears as a more compact system than the proton, although the results of the magnetism distribution radii, presented in [

more well-defined than the one of the proton [

Therefore, the obtained precision of nucleon description allows concluding that the calculations of other characteristics of the proton and neutron with obtained wave function may give some interesting and rather reliable results. It is well known that realistic potentials of nucleon-nucleon interaction, carefully fitted with the two- nucleon data, give smaller than experimental nuclear binding energies. It looks like that the introduced model is able to give a chance for this problem solution. As it is known from the solid state theory, when the distance between two potential wells decreases, the isolated levels of each well convert to the system of two levels, one of them is more bound than the other one. If Pauli principle allows the constituents of nucleon to occupy the best bound level, this may help us to improve the description of atomic nuclei, taking into account the changes of nucleon structure when merging into groups.

Gintautas P.Kamuntavičius, (2015) Structure of Schrödinger’s Nucleon: Elastic Form-Factors and Radii. Journal of Applied Mathematics and Physics,03,1352-1360. doi: 10.4236/jamp.2015.310162