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The recently introduced Galilei invariant model of the nucleon as a system of three point particles, whose dynamics is governed by Schrödinger equation, is applied for nucleon structure investigation. The obtained charge, magnetism, mass and point particles density distributions of the proton and neutron are in satisfactory agreement with known information about nucleon structure. The model predicts the third Zemach momentum of proton larger than the one obtained in dipole approximation and larger than following from electron-proton data analysis.

The only source of experimental information about nucleon’s charge and magnetism distributions are corresponding elastic form factors. However, the step from known at some set of values of momentum transfer form factor to corresponding density is not easy. The density distribution is completely determined if form factor is known for all values of momentum transfer q^{2}. At large values of q^{2} it is extremely difficult to obtain form factor, hence phenomenological approach for density distribution with a number of free parameters, reproducing measured values of form factors, is a common praxis. However, even more or less precisely defined density distribution is not the best tool for nucleon structure investigation. Obviously, it is enough for simple multiplicative operators expectation values calculation, but in more complex cases, when for observables description operators are needed, whose place is between bra and ket functions, the density distribution carries crude information about nucleon. The first among these are currents operators, like magnetic momenta. Microscopic model of the nucleon is necessary for calculation of expectation values of these operators.

One of the alternatives for solution of this problem is simple, it is based on Hamiltonian dynamics, model [

The value of the third Zemach momentum of the proton is key figure for current puzzle regarding the proton radius, determined from the muon and electron Lamb shifts in hydrogen, resolution [

However, a very precise measurement of the Lamb-shift in muonic hydrogen [

Therefore, the remarkable difference between proton charge radii of electronic and muonic hydrogen exists. The origin of this result is unknown yet. As pointed out in [

This value is more than 13 times larger than the result, obtained applying electron-proton scattering data [

Similar value has been obtained in recent paper, Ref. [

Thus, the results of our model, giving possibility for Zemach momenta calculation applying original definition with charge and magnetic form factors convolution can provide some new information about this problem.

By definition, the elastic form factor of nucleon is density operator’s Fourier image:

Here

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

is center-of-mass radius vector.

The inverse of (6) defines the density operator in terms of form factor operator:

A few modifications of present definitions are useful. The electric and magnetic form factors of nucleon are defined as functions, independent of angles of momentum transfer q, hence in last equation integration by angles

gives the following result:

Therefore, the form factor, independent of angles produces the spherically symmetrical charge and magnetism of nucleon distributions.

The next is introduction of dimensionless form factors defined as

so that

The only problem of this definition is the electric form factor of the neutron

Introduction of dimensionless operators

in density definition is also useful. This modifies final expression of form factor operator, applied in a given below text, to the following form:

where

The density presentation in terms of dimensionless form factor

can be obtained applying spherical Bessel functions orthogonality relation

The well-known for different evaluations the so called dipole form factor is

where

is

This density distribution produces following momenta:

hence root mean square radius of nucleon in this approximation equals 0.811 fm.

The density at given z equals the expectation value of operator (16). However, this operator is overcrowded by variables because the coordinates of point particles in reference system which origin is situated in center of mass of nucleon satisfies condition

hence only two of them are independent. These two linear combinations have to be chosen equal to the intrinsic Jacobi variables, present in wave function expression. The operator (16) written in these variables is:

The integration gives

where mean values of operators

The calculation of expectation value of the first part of this operator is straightforward and can be performed without any problems.

Here

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

−1/3 | 4/3 | 2/3 | −2/3 | |

7γ/(45 + 7γ) | 45/(45 + 7γ) | 28/(28 + 45γ) | 45γ/(28 + 45γ) | |

1/(1 + 2γ) | 2γ/(1 + 2γ) | γ/(2 + γ) | 2/(2 + γ) | |

1/3 | 2/3 | 1/3 | 2/3 |

tinguishability of the second and the third PP. The wave function of nucleon is antisymmetric in respect of these particles permutation, sum of the second and the third delta functions of (24) is symmetrical, hence mean values of both deltas coincide, i.e.,

thus

The expectation value of this delta-function calculation requires Dirac delta argument modification, applying the following expression:

where

where

Thus, the second integral takes the form:

By definition, both integrals are normalized

and due to the obvious condition

the normalization is valid also for charge and magnetism densities of the proton and magnetism density of neutron. As mentioned above, the only exception is charge density of neutron. In this case

The radial density distributions, defined as

hence normalized in following way:

are present in Figures 1-4.

The Zemach momenta [

Here

where

There is an interesting possibility to express these densities in terms of corresponding elastic form factors, and obtain the following convoluted density presentation

Zemach momenta evaluations with coinciding both density distributions, equal the dipole or charge density distribution are the best known. In dipole approximation the convoluted density can be present as

This density again is spherically symmetrical, i.e. independent of angles of r, hence normalized in following way:

It equals

Therefore, the Zemach momenta in dipole approximation are defined by the following expression:

The numerical values of first three Zemach momenta

The obtained density distributions demonstrate good enough comparison with known information about nucleon structure. The proton has a negatively charged dPP situated near the center of mass and cloud of positive charge, created by uPPs, and neutron shows an opposite picture. As mentioned in [

As mentioned earlier, the form factors are the source of information about nucleon density distributions. Different parametrizations of form factors produce different density distributions and radii of nucleon. However, the basic for these modifications is dipole form factor, predicting exponential density dependence, Equation (21). This prediction and our result for proton charge density are presented in

The difference is obvious. However, the charge form factors of proton, obtained in dipole approximation, Equation (19); the charge form factor, given by the best fit to experimental data [

Obviously, the reason of significant difference of proton’s charge distributions is caused by elastic form factor dependence at larger values of momentum transfer. The dipole form factor is smooth function of momentum transfer, taking positive values at any momentum transfer. The model form factor dependence is significantly different. Our form factor possesses the nodes, producing diffraction minima of cross section at larger values of momentum transfer, characteristic for form factor of quantum system with complex intrinsic structure.

The theoretical value of the Lamb shift in the

k | |||
---|---|---|---|

1, fm | 1.025 | 0.0678 | 0.588 |

2, fm^{2} | 1.316 | 0.370 | 1.090 |

3, fm^{3} | 2.203 | 3.575 | 3.250 |

The experimental value equals [

At proton charge radius, equal (2) the Zemach radius, necessary for ^{3}. Taking the CODATA charge radius (1) one needs significantly larger value of Zemach momentum, equal (3). Our model predicts the third Zemach momentum of proton larger than obtained in dipole approximation and larger than following from electron-proton data analysis, but significantly smaller than necessary for CODATA value. In dipole form factor approximation

Therefore, the obtained precision of nucleon description allows to conclude that the calculations of other characteristics of the proton and neutron with model wave function may give some interesting and rather reliable results investigating the modifications of nucleon structure when it is present in an atomic nucleus.

Gintautas P.Kamuntavičius, (2015) Structure of Schrödinger’s Nucleon. Density Distributions and Zemach Momenta. Journal of Applied Mathematics and Physics,03,1412-1421. doi: 10.4236/jamp.2015.311169