# Nuclear Structure · Quantum Mechanics software

Software based on Monte Carlo method of calculation.

The nucleus is made up of neutrons and protons, two particles which are 1840 times more massive than electrons. Number of protons (p+) in a nucleus is Z (atomic number). The number of nucleons (A) is the integer closest to its atomic weight. Number of neutrons N=A-Z. (Equation 1)

The mass of the nucleus is very nearly equal to the mass of the atom. The nucleus was discovered in 1911 by Rutherford. He obtained that the nuclear size is of the order of 10^(-14)m. This is 10,000 times smaller than the diameter of atoms.

## Nuclear Size

The straightforward approach to studying the size and the shape of nuclei is to shoot probing particles and measure the effects produced. However, the wavelength of the probing particles (λ) must be of the order of the size of nuclei being studied or smaller. (Equation 2)

Then, it is therefore better to employ particles e-, p+ and n. Neutrons and protons have the advantage that their wavelength (λ) is sufficiently short for energies of about 20 MeV. (Equation 3)

However, For electrons, over 100 MeV of energy is required, which is more difficult to obtain. All results can be approximately explained by a charge distribution ρ given by (Equation 4)

where the nucleon density ρ_0 is 1.65·10^(44) nucleons/m^3 = 0.165 nucleons/fm^3. (Equation 5)

The energies of beta rays and gamma rays emitted from nuclei are of the order of 1 MeV. We calculate the electrostatic energy Ec required to insert a proton into a nucleus. This is approximately (Equation 6)

This much coulomb energy would be released if the proton were allowed to come out of the nucleus, but still it does not ordinarily come out. This means that it is bound in the nucleus by even more energy. Since the velocity of a 10 MeV nucleon is only about 15 percent of the speed of light (c), this means that relativistic effects are not important in considering the motion of nucleons in the nucleus.

## Is the nucleus classical or quantum?

As a general rule, the wave nature of matter is relevant where the wavelength of the particles is of the order of the size of the system. The wavelength of a nucleon with an energy of about 10 MeV is λ = 9.3 fm. This is clearly of the order of the size of a nucleus, so the wave nature of matter is indeed relevant. The motions of nucleons in the nucleus are governed by the laws of quantum physics. We have to use our knowledge of the wave nature of matter. What holds the nucleus together? Not the electromagnetic or gravitational forces. The only explanation is that there is a third force in nature, known as nuclear force. This force must be very strong at distances of the order of the nuclear size, since it must more than compensate the coulomb repulsion between protons. It is therefore a short-range force, falling off more rapidly with distance than 1/r2.

## Other properties of Nuclei

If there are no external torques acting on a system, its angular momentum is conserved. Since an isolated nucleus is such a system, its angular momentum is one of its constant properties. In quantum physics, conserved quantities are represented by quantum numbers. The quantum number for the total angular momentum of a nucleus is L and I. Both quantities are related by [I(I+1)]^1/2 hbar

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