Nuclear Physics Tutorial 3 - Nuclear Instability

 

The chemical properties of any element are governed by the number of protons, the proton number, which is given the code Z.  The stability of the nucleus depends on a combination of the proton number and the neutron number.  We can plot a graph of the number of neutrons (given by the difference between the mass number and the proton number) against the proton number.  The general pattern is like this:

 

 

For stable nuclides, we notice the following:

For unstable nuclides, we see:

·       Beta plus decay also occurs where the nucleus is beneath the line of stability.  In this case a proton turns into a neutron and a positron (positively charged anti-electron) is given off.

 

 

Common Modes of Decay

Alpha radiation mostly comes from heavy nuclides with proton numbers greater than 82, but smaller nuclides deficient in neutrons can also be alpha emitters.  It is believed that the alpha particle is formed some time before its emission, and it gains its energy from the mass defect in the nucleus.  The term Q stands for the energy The general decay equation is summarised below.

 

 

 

 

  We should note the following:

 

 
Question 1

Is this equation balanced?  Explain your answer.

Answer

 

 

Alpha particles are intensely ionising.  They smash through air molecules, knocking off electrons as they go.  However this reduces the kinetic energy, so that in the end they stop.  Then they pick up a couple of free electrons to become helium atoms.  To collect an appreciable sample of helium from an alpha emitter would take a very long time.

 

Neutron rich nuclei tend to decay by beta minus (b-) emission.  The beta particle is a high-speed electron ejected from the nucleus, NOT the electron clouds.  It is formed by the decay of neutrons, which are slightly more energetic than a proton.  Isolated protons are stable; isolated neutrons last about 10 minutes.

 

The neutron, having emitted an electron, is converted to a proton, and this results in the proton number of the nuclide going up by 1.  A new element is formed.  The reaction at the nucleon level is:

 


Notice that as well as the neutron (n) and the proton (p), the beta particle is represented as an electron (e).  The strange symbol ne (‘nu-bar e’) is a strange little particle called an electron antineutrino.  The general equation for b- decay is:

 


A typical decay is:

 

 

 

Notice that:


The proportion of shared energy is variable, so there is a range of energies of the
b- particles.  The graph shows a typical distribution.

 

 

If beta particles are emitted in a medium where the speed of light is lower than that of the ejected electrons, then the passage of the electron is accompanied by an optical shock wave, like the sonic boom of a supersonic aeroplane.  The resulting glow is called Cherenkov radiation.

 

Question 2

What is the balanced nuclear equation for the following decays?

(a)    emission of a beta- particle from oxygen 19

(b)   emission of an alpha particle from polonium 212

(c)    emission of a beta + particle from cobalt 56

 

Proton numbers O – 8, F – 9, Fe – 26, Co – 27, Pb – 82, Po – 84

Answer

 

The positron is the anti-particle to the electron.  It has the same size, but opposite charge.  Beta-plus (b+) decay involves the emission of a positron.  It rarely occurs naturally, and is generally found in nuclear physics experiments in reactors.  If we bombard fluorine atoms with alpha particles, we get a radioisotope of sodium, which decays by positron emission.

 

The second reaction is:

 

 

 

Here we see a positively charged electron, the positron being emitted with an electron neutrino (ne).  At the nucleon level we see:


The proton is turned into a neutron.

 

Question 3 

Is the charge conserved?

Answer

 

There is another way that a proton is turned into a neutron, and that is by electron capture.  An electron is captured from the electron cloud.  As another electron falls to take over the vacancy left, an X ray is emitted.  The general scheme is:

 

 

And at the nucleon level we see:

 

 

Note that prior to the emissions, the electrons, positrons, neutrinos, or antineutrinos do not exist as separate entities within the nucleus.  They are created at the instant of the decay.  Free neutrons outside the nucleus decay to protons by b- emission.

 

 

Decay Chains

When radioisotopes decay, there may be several steps before the nucleus achieves stability.  We call this series of decays a radioactive series or a decay chain.  There are different half-lives at each step, some of which can be extremely long, while others are short.  We can represent these graphically as shown below.

 


There are different permitted moves, according to the decays involved.  Here is a decay chain.

 

 

We can show this series graphically.

Remember that the neutron number is not the same as the nucleon number.

 

Neutron number = nucleon number – proton number

 

Note also that the emitted particle is NOT radioactive in itself.

 

 

Excited Nuclei

After alpha or beta decay, the daughter nucleus is often left in a very energetic state.  We call that state excited.  The nucleus gets rid of this energy in the form of a photon of electromagnetic radiation of very short wavelength, called a gamma ray (g-ray).  Gamma rays, cosmic rays, and hard X-rays have the same frequency, so are really the same thing.  Since photons are not particles, there is no change in the proton number, or the nucleon number.  The nucleus becomes less energetic.

 

Some points to note:

 

The energy comes from the mass defect.  At the nuclear level the key idea is that mass and energy are interchangeable.  There is a measurable change in mass of a nuclide emitting gamma rays over a long period.

 

Gamma rays have two important medical applications:

 

Question 4

Explain how gamma rays are formed. 

Answer

 

Energy levels in Nuclei

A nuclear event can be:

 

In any of these events, the daughter nucleus can be energetic.  It can lose the excess energy by emitting a gamma photon.  It then falls from the excited state to the ground state.

 

Consider this beta minus decay:

 

The aluminium nucleus is excited, and can be shown in an energy level diagram:

 

 

In this particular decay, there are three possible energies for the gamma photon.  These are shown in the diagram as transitions. 

 

We can work out the energy of the gamma photon in joules by multiplying the energy by 1.6 Χ 10-19.  Then we could work out the frequency of the photon by using the equation:

E = hf

Question 5

Calculate the energy change in transition 1.  Express your answer in MeV and joules.  Hence work out the wavelength of the gamma photon.

Answer

 

Metastable states

It is possible for a daughter nucleus to remain in an excited state for some time.  One example is the element technetium, formed by beta decay from a radioactive decay from an isotope of molybdenum.

 

This can be shown in an energy level diagram:

We refer to the prolonged excited state as metastable.

 

The technetium drops to ground state by emitting a gamma photon of 140 keV.  This is low enough to be much safer than other gamma sources.  The half life of the gamma emission is about 6 hours.  Like all radioactive decays, the emission of gamma photons is random.

 

Technetium in its ground state decays by beta minus emission to ruthenium, with a half life of 211 000 years.  This adds very little additional radiation burden on the body.

 

Question 6

The germanium isotope Germanium 77 has a metastable state which decays to the ground state by emission of a 0.16 MeV gamma photon.  The isotope decays by beta minus emission to form an arsenic isotope which is in an excited state 0.48 MeV above the ground state.  This is shown in the diagram below:

(a) Complete the diagram to show the changes.

(b) Calculate the wavelength of the photon released by the metastable germanium nucleus. 

(c) The excited state of arsenic 77 also decays to the ground state via an excited state which is 0.27 MeV above the ground state.  Calculate the energies of the gamma photons emitted in this decay.                                                     

                                                                                                  

 

Answer

 

 

 

 

Summary

A graph of neutron number against proton number shows that there are more neutrons in larger nuclei

 

This is needed to ensure stability of the nuclei.

 

Natural decay occurs with alpha decay

 

Or beta minus decay.

 

Beta plus decay can occur in reactors.  It is not often found in nature.

 

Excited nuclei give off gamma rays.