### The nucleus

From the point of view of quantum mechanics, we treat the nucleus as a positive point charge and focus on what the electrons are doing. In many cases, such as nuclear reactions, the electrons can be ignored.

A nucleus consists of two kinds of particles: the protons (Z) -with a positive electrical charge- and the neutrons (N)- neutral particles. Protons and neutrons are known as nucleons.

Each nucleus can be characterized by two numbers: the atomic mass number, A, which is the total number of nucleons ( the number of neutrons plus the number of protons); and the atomic number, Z, representing the number of protons. Any nucleus can be written in a form like this:

where Co is the chemical symbol of the element (cobalt in this case), the 27 is the atomic number number, and the 58 is the atomic mass.

The nucleus can be thought of as a sphere, with the radius being approximately $r= (1.2x10^{-15}m\AA^{\frac{1}{3}}$

So, the size of a nucleus is a few femtometres fm ($10^{-15} m$ ) in diameter depending on the atom.

### The strong nuclear force

The gravitational force attracting protons to each other is much smaller than the electric force repelling them, so there must be another force keeping them together. This other force is known as the strong nuclear force, a very strong attractive force for protons and neutrons, and it works only at small distances (of the order of a few femtometers).

The attractive force of the strong nuclear force and the repulsive electrostatic force has interesting implications for the stability of a nucleus. Stable nuclei will have more neutrons than protons ( example, the bismuth nucleus with 83 protons and 126 neutrons). Ratio between number of protons and neutrons in nucleus shows whether atom is stable or unstable. If:

n0/p+≈1

then atom is stable

If:

n0/p+<1 or n0/p+>1,5

nucleus of atoms are unstable and we call these atoms radioactive elements.

So, nuclei with more than 83 protons are all unstable!

Unstable atoms do some nuclear reactions like radiation or decay and become stable atoms. We can explain radioactivity under two titles, natural nuclear reactions and artificial nuclear reactions.

### Nuclear binding energy and the mass defect

The mass of a neutron is slightly larger than the mass of a proton.

neutron = proton + electron + subatomic particles

The standard unit in which the masses of atoms ( and molecules) are measured is the atomic mass unit (or amu), defined as one-twelfth (1/12 th) of the mass of a single atom of the isotope carbon-12. This is 1.6605 × 10-27 kg, or approximately 931 MeV.

The carbon-12 atom has a mass of 12.000 u (exactly 12 amu). It contains 6 protons and 6 neutrons that each have a mass greater than 1.000 u (see above).  The fact is that these six protons and six neutrons have a larger mass (12.0956 amu).

This is true for all nuclei: the mass of any nucleus is a little less than the sum of the separate masses of its protons and neutrons. This missing mass is known as the mass defect, and is essentially the equivalent mass of the binding energy.

Einstein correctly described the equivalence of mass and energy as “the most important upshot of the special theory of relativity” (Einstein, 1919). The relationship between the mass and the energy is contained in Einstein’s famous equation:

$E=mc^2$

where m is the mass, c is the speed of light, and E is the energy equivalent of the mass.

It is customary to refer to this result as “the equivalence of mass and energy,” or simply “mass-energy equivalence,” because one can choose units in which c = 1, and hence E = m.

To find the binding energy in any nucleus, then, you need to add up the mass of the individual protons and neutrons and subtract the mass of the nucleus:

mass defect : $\Delta m$ = mass of individual nucleons – mass of the nucleus

The binding energy B (Z, N ) is what this mass defect is converted into with the mass-energy equivalence. It is therefore given by

$B(Z,N)= {(ZM_p+NM_n)-M(Z,N)}c^2$ (or binding energy= $\Delta mc^2$)

where M (Z, N ) is the mass of a nuclide of Z protons and N neutrons.

For example, the proton mass is $M_p=1,6726x10^{-27}Kg$. This is converted into energy by being multiplied by the square of c i.e.

$M_pc^2=938,27 MeV$

Accordingly, the proton mass is expressed as $M_p=1,6726x10^{-27}Kg=938,27 MeV/c^2$.

In this way, the unit of mass $MeV/c^2$ is often used in the World of the Atomic Nucleus (in a typical nucleus the binding energy is measured in MeV).

Question:

Calculate the mass defect in Mo-96 if the mass of a Mo-96 nucleus is 95.962 amu. The mass of a proton is 1.00728 amu and the mass of a neutron is 1.008665 amu. Determine the binding energy of an O-16 nucleus. The O-16 nucleus has a mass of 15.9905 amu. A proton has a mass of 1.00728 amu, a neutron has a mass of 1.008665 amu, and 1 amu is equivalent to 931 MeV of energy.

Solving:

Mo (Molybdenum) has an atomic number of 42, so it has 42 protons. If it is Mo-96, then it has (96 –42 = 54) 54 neutrons. To determine the mass defect, sum the mass of 42 protons and 54 neutrons and subtract that number from 95.962 amu. That is the mass defect.

O-16 has 8 protons and 8 neutrons to determine the mass defect. Then multiply the mass defect by the fraction 931 MeV/1 amu.

Radioactive decay is the spontaneous breakdown of an atomic nucleus resulting in the release of energy and matter from the nucleus. Remember that a radioisotope has unstable nuclei that doesn’t have enough binding energy to hold the nucleus together.

Many nuclei are radioactive. This means they are unstable, and will eventually decay by emitting a particle, transforming the nucleus into another nucleus, or into a lower energy state.

During radioactive decay, principles of conservation apply:

• conservation of energy
• conservation of momentum (linear and angular)
• conservation of charge
• conservation of nucleon number

The law of conservation of nucleon number states that the total number of nucleons (neutron plus protons) before decay equals the total number of nucleons after decay.

There are three types of nuclear radioactive decay, alpha, beta, and gamma. The difference between them is the particle emitted by the nucleus during the decay process.

### Alpha decay

In alpha decay, the nucleus emits an alpha particle. An alpha particle is a Helium 4 nucleus (two protons and two neutrons). A helium nucleus is very stable.

An example of an alpha decay involves uranium-238:

The process of transforming one element to another is known as transmutation.

### Beta decay

A beta particle is often an electron, but can also be a positron, a positively-charged particle that is the anti-matter equivalent of the electron. If an electron is involved, the number of neutrons in the nucleus decreases by one and the number of protons increases by one. An example of such a process is:

Negative electron emission

A free neutron (n) outside a nucleus is unstable, and it emits an electron and becomes a proton (p). The proton and the electron are the components of a hydrogen atom. A company particle in the emission of electron is the antineutrino v*.

n -> p+ + e + v*

The antineutrino is the antiparticle of neutrino.

When a nuclide MPZ emits an electron, we may consider one of the neutron in the nucleus being converted to a proton. For example,

14C6 -> 14N7 + e + v*

40Ca19 -> 40Ca20 + e + v*

50V23 -> 50Cr24 + e + v*

87Rb37 -> 87Sr38 + e + v*

Positive electron (positron) emission

A positive electron is called positron, which is the antiparticle of electron. The company particle for positron emission is an neutrino (v). Some examples are given here to illustrate the process:

22Na11 -> 22Ne10 + e+ + v

21Na11 -> 21Ne10 + e+ + v

30P15 -> 30Si14 + e+ + v

34Cl17 -> 34S16 + e+ + v

116Sb51 -> 116Sn50 + e+ + v

Electron capture (EC)

Electron capture is one process that unstable atoms can use to become more stable. During electron capture, an electron in an atom’s inner shell is drawn into the nucleus where it combines with a proton, forming a neutron and a neutrino. The neutrino is ejected from the atom’s nucleus.

Since an atom loses a proton during electron capture, it changes from one element to another.

Examples of EC are:

48V23 -> 48Ti22 + e+ + v     (50%)

48V23 + e -> 48Ti22 + v (+ X-ray)     (50%)

40K19 + e -> 40Ar18 + v (+ X-ray)

65Zn30 + e -> 65Cui29 + v (+ X-ray)

7Be4 + e -> 7Li3 + v (+ X-ray)

### Gamma decay

In gamma decay the nucleus changes from a higher-level energy state to a lower level.

When an electron changes levels, the energy involved is usually a few eV, so a visible or ultraviolet photon is emitted. In the nucleus, the photon emitted is a gamma ray.

Gamma rays are electromagnetic radiation, like X-rays. Gamma radiation is the product of radioactive atoms. Depending upon the ratio of neutrons to protons within its nucleus, an isotope of a particular element may be stable or unstable.