In 1932 Cockcroft and Walton succeeded in splitting lithium nuclei by bombarding them with artificially accelerated protons using a linear accelerator - Leaving Cert Physics - Question 10(a) - 2009

The nuclear equation can be represented as follows:

$\text{ }_3^6\text{Li} + \text{ }_1^1\text{p} \rightarrow 2 \text{ }_2^4\text{He}$

This indicates that a lithium-6 nucleus reacts with a proton to produce two alpha particles.

To calculate the energy released, we first need to determine the loss in mass:

Loss in mass = mass of lithium nucleus + mass of proton - 2(mass of alpha particle)

Substituting the values: $\text{Loss in mass} = (1.6726 \times 10^{-26} \text{ kg}) + (1.00728 \times 10^{-27} \text{ kg}) - 2(6.6447 \times 10^{-27} \text{ kg})$

$= 1.32894 \times 10^{-29} \text{ kg}$

Using Einstein's equation, energy released (E) can be calculated as: $E = \Delta m c^2$

Where:

• $c = 2.9979 \times 10^8 \text{ m/s}$
• $E = (1.32894 \times 10^{-29} \text{ kg})(2.9979 \times 10^8 \text{ m/s})^2 \approx 2.6 \times 10^{-12} \text{ J}$

Most of the accelerated protons do not split a lithium nucleus primarily due to the empty space within the atomic structure. The protons may pass through the lithium nucleus without any interaction. This is because the majority of the volume of atoms consists of empty space between the nucleus and the electrons, resulting in a low probability of collision. Additionally, some protons simply do not have enough energy to overcome the repulsive forces from the nucleus.

New particles are formed during the collisions in the Large Hadron Collider due to the conversion of kinetic energy into mass, according to Einstein's mass-energy equivalence principle, $E=mc^2$. When protons collide at high energies, the energy available in the interaction can create heavy particles which were not present before the collision, thus changing kinetic energy into mass.

The maximum net mass of the new particles created per collision can be calculated using the energy available from the kinetic energy of protons. Assuming each proton has a kinetic energy of 2.0 GeV:

1 GeV = $1.6 \times 10^{-10} \text{ J}$

Total kinetic energy per proton: $E = 2.0 \text{ GeV} = 2.0 \times 1.6 \times 10^{-10} \text{ J} = 3.2 \times 10^{-10} \text{ J}$

Using $E=mc^2$, we find: $m = \frac{E}{c^2} = \frac{3.2 \times 10^{-10}}{(3 \times 10^8)^2} \approx 3.56 \times 10^{-30} \text{ kg}$

This is the approximate maximum mass of new particles generated per collision.

The primary advantage of using circular particle accelerators is that they can accumulate energy more efficiently over time. By continuously accelerating particles within a circular path, they gain energy with each pass, allowing for higher energy collisions with reduced space requirements. This design increases the frequency of particle collisions without necessitating long straight paths, facilitating numerous interaction events. Additionally, circular accelerators can provide more precise control over beam stability and focusing.