# Nuclear Equations

Nuclear equations can be used to show the decay process.

When you studied at 14-16 you learned that these must balance for nucleon number and proton number. Now at 16+ baryon number, charge and lepton number must balance too.

### Alpha decay

A nucleus that has high mass and too many protons to be stable tends to undergo alpha decay.

When alpha decay occurs a group of two protons and two neutrons (helium nucleus) comes out of the nucleus. Therefore the proton number decreases by 2 but the nucleon number decreases by 4. The resulting daughter nucleus is of an element 2 positions to the left of the 'parent' in the periodic table.

The above equation shows the radioactive decay of Uranium-238 by alpha emission.

Look at the numbers on the top line (the nucleon numbers). Each nucleon has a baryon number of 1/3 therefore

79.3 = 78 + 1.3

Therefore the baryon numbers balance

Look at the numbers on the bottom line (the proton numbers) - indicate charge of +1 for each proton number.

92 = 90 + 2

Therefore the charge balances

The decay does not involve leptons therefore this is zero each side.

### Beta Decay

When a nucleus has too many neutrons, it tends to beta decay.

When beta decay occurs a neutron within the nucleus emits the particle and changes into a proton. Therefore the proton number increases but the nucleon number stays the same (only now you have one more proton and one less neutron!). The resulting daughter nucleus is of an element 1 position to the right

The above equation shows the radioactive decay of Carbon-14 by beta emission.

Look at the numbers on the top line (the nucleon numbers) each of these has a baryon number of one third ...

4.6 = 4.6 + 0

Therefore the baryon numbers balance

Look at the numbers on the bottom line each representing charge.

6 = 7 + (-1)

Therefore the charge balances

However when we look at the lepton numbers we have an imbalance

On the left of the equation we have no leptons therefore the lepton number is zero.

On the right of the equation we have a 'matter lepton' (the electron) giving a lepton number of +1

We therefore need to add an anti-matter lepton (lepton number -1) to the right. The anti-lepton must not disturb the balance of charge or of baryon number. The candidate for this task is the anti-neutrino.

This now balances on all counts!

### Positron Decay

When a nucleus has too many protons, it tends to positron decay. A positron is an antimatter beta particle. When a positron meets with an electron it annihilates it! Both particles disappear and two gamma rays are produced instead.

The above equation shows the radioactive decay of Oxygen-15 by positron emission

Look at the numbers on the top line (the nucleon numbers) each of these has a baryon number of one third ...

5 = 5 + 0

Therefore the baryon numbers balance

Look at the numbers on the bottom line (the proton numbers).

8 = 7 + 1

Therefore the charge balances

However when we look at the lepton numbers we have an imbalance

On the left of the equation we have no leptons therefore the lepton number is zero.

On the right of the equation we have an 'anti-matter lepton' (the positron) giving a lepton number of -1

We therefore need to add a matter lepton (lepton number +1) to the right. The lepton must not disturb the balance of charge or of baryon number. The candidate for this task is the neutrino.

This now balances on all counts!

### Gamma Emission

Sometimes, after its emission of an alpha, beta or positron particle, the nucleus is still in an excited state, called a metastable state. In order to get to a lower energy state it emits a quantum of energy in the form of a gamma ray. This is not a highly unstable state otherwise the emission of the gamma ray would accompany the alpha, beta or positron particle. Nuclei in the metastable state produce gamma rays at a measurable half-life.

E.g. Cobalt 60m decays to give cobalt 60 with a half-life of 5.3 years and technetium 99m decays to give technetium 99 with a half-life of 6 hours.

No matter is emitted from the nucleus therefore the nucleon number and the proton number remain the same. Before and after emission of the gamma ray they are the same isotope of the element but they are different nuclide because the term nuclide incorporates nuclear energy states as well basic structure.