Spontaneous Decay of Unstable Nuclei (HSC SSCE Physics): Revision Notes

Spontaneous Decay of Unstable Nuclei

What is spontaneous decay?

Spontaneous decay is a random, uncontrollable process where unstable atomic nuclei emit radiation to become more stable. This process was first discovered by Henri Becquerel in 1896 and opened up our understanding of the atom's internal structure. Since then, radioactivity has been used in many beneficial applications, including medical imaging and cancer treatment, as well as nuclear power generation.

When nuclei decay, they can emit three main types of radiation:

• Alpha particles ($\alpha$) - helium nuclei
• Beta particles ($\beta$) - electrons or positrons
• Gamma rays ($\gamma$) - electromagnetic radiation

Nuclear stability

Forces in the nucleus

All atomic nuclei except hydrogen-1 contain both protons and neutrons. Since protons are positively charged, they repel each other through the electrostatic force. So what holds the nucleus together?

The strong nuclear force is the answer. This force acts between all nucleons (protons and neutrons) and is much stronger than the electrostatic force, but it only works over very short distances - about the size of a nucleus. When the strong nuclear force is strong enough to overcome the electrostatic repulsion between protons, the nucleus is stable. When it isn't, the nucleus is unstable and will decay by emitting radiation.

The stability curve

Not all combinations of protons and neutrons create stable nuclei. The relationship between stability and nuclear composition is shown in the stability curve or line of stability.

Key observations from the stability curve:

• Light nuclei (up to about 40 nucleons) are most stable when they have roughly equal numbers of protons and neutrons (following the $N = Z$ line)
• Heavier nuclei require more neutrons than protons to remain stable
• Unstable nuclei below the line of stability typically undergo $\beta^-$ decay
• Unstable nuclei above the line typically undergo $\beta^+$ decay or alpha decay

Nuclear notation and features

Understanding nucleon symbols

To describe nuclei precisely, we use standardised notation that shows both the number of protons and the total number of nucleons.

Symbol	Name	Description
$A$	Mass number (nucleon number)	Total number of protons and neutrons in the nucleus
$Z$	Atomic number (proton number)	Number of protons in the nucleus
$N$	Neutron number	Number of neutrons in the nucleus

These quantities are related by the equation:

$N = A - Z$

The standard notation for a nuclide is:

$^A_Z\text{X}$

where X is the element symbol. For example, beryllium-9 is written as $^9_4\text{Be}$, showing 9 total nucleons (mass number) and 4 protons (atomic number).

Nuclides, elements and isotopes

Understanding the difference between these terms is important:

• A nuclide is a specific type of nucleus with a defined number of protons and neutrons. It is classified by its energy state as well.
• An element is a substance containing only atoms with the same number of protons. The number of protons determines the element's chemical identity.
• Isotopes are nuclides of the same element (same number of protons) but with different numbers of neutrons.

Nuclear binding energy

Mass defect

When you measure the mass of a nucleus, you find something surprising - it's less than the sum of the masses of its individual protons and neutrons. This difference is called the mass defect ($\Delta m$).

The mass defect exists because when nucleons bind together to form a nucleus, some of their mass is converted into binding energy. This is the energy that holds the nucleus together - the nuclear binding energy.

Mass-energy equivalence

Einstein's famous equation relates mass and energy:

$\Delta E = (\Delta m)c^2$

where:

• $\Delta E$ is the binding energy
• $\Delta m$ is the mass defect
• $c$ is the speed of light ($3.00 \times 10^8 \text{ m} \cdot \text{s}^{-1}$)

Types of radioactive decay

When an unstable nucleus decays, it emits particles or electromagnetic radiation to reach a more stable state. The original nucleus is called the parent nuclide, and the resulting nucleus is the daughter nuclide.

Conservation laws in nuclear reactions

Alpha decay ($\alpha$)

An alpha particle is a helium-4 nucleus containing 2 protons and 2 neutrons. When written in nuclear equations, it's shown as $^4_2\text{He}$ or simply $\alpha$.

The general equation for alpha decay is:

$^A_Z\text{X} \rightarrow ^{A-4}_{Z-2}\text{Y} + ^4_2\text{He}$

What happens during alpha decay:

• The mass number $A$ decreases by 4
• The atomic number $Z$ decreases by 2
• The daughter nuclide is a different element, two places back in the periodic table

Beta-minus decay ($\beta^-$)

Beta-minus decay involves the emission of an electron from the nucleus. This might seem strange since electrons orbit the nucleus but aren't normally inside it. What actually happens is that a neutron converts into a proton, an electron, and an antineutrino ($\bar{\nu}$).

The neutron conversion can be written as:

$^1_0\text{n} \rightarrow ^1_1\text{p} + ^0_{-1}\text{e} + \bar{\nu}$

or equivalently:

$^1_0\text{n} \rightarrow ^1_1\text{H} + ^0_{-1}\text{e} + \bar{\nu}$

The electron is immediately ejected from the nucleus at very high speed (about two-thirds the speed of light) as a beta particle.

The general equation for beta-minus decay is:

$^A_Z\text{X} \rightarrow ^A_{Z+1}\text{Y} + ^0_{-1}\text{e} + \bar{\nu}$

What happens during beta-minus decay:

• The mass number $A$ remains unchanged
• The atomic number $Z$ increases by 1
• The daughter nuclide is a different element, one place forward in the periodic table

Beta-plus decay (positron emission, $\beta^+$)

Positron emission is similar to beta-minus decay, but a positron is emitted instead of an electron. A positron is the antimatter counterpart of an electron - it has the same mass but opposite charge.

In beta-plus decay, a proton converts into a neutron, a positron, and a neutrino ($\nu$):

$^1_1\text{p} \rightarrow ^1_0\text{n} + ^0_{+1}\text{e} + \nu$

or equivalently:

$^1_1\text{H} \rightarrow ^1_0\text{n} + ^0_{+1}\text{e} + \nu$

The general equation for beta-plus decay is:

$^A_Z\text{X} \rightarrow ^A_{Z-1}\text{Y} + ^0_{+1}\text{e} + \nu$

What happens during beta-plus decay:

• The mass number $A$ remains unchanged
• The atomic number $Z$ decreases by 1
• The daughter nuclide is a different element, one place back in the periodic table

Gamma emission ($\gamma$)

Sometimes after alpha or beta decay, the daughter nucleus is left in an excited energy state. This is shown with an asterisk (*) after the symbol. The excited nucleus releases this extra energy by emitting a gamma ray - a high-energy photon of electromagnetic radiation.

The general equation for gamma emission is:

$^A_Z\text{X}^* \rightarrow ^A_Z\text{X} + ^0_0\gamma$

What happens during gamma emission:

• The mass number $A$ remains unchanged
• The atomic number $Z$ remains unchanged
• The nucleus transitions to a lower energy state
• No new element is formed

Worked examples

Understanding nuclear decay equations requires practice in applying conservation laws and using correct notation.

Key points for examinations

Remember!