Finding nuclear radius
Finding nuclear radius
You cannot simply look inside an atom in order to find how large the nucleus is. Instead high-energy particles are directed at atoms and arrays of detectors surround the target to find what sort of particles are scattered and where to. By looking closely at the data from the detectors physicists can discover much about the targets in question.
A good analogy is shown below. Imagine trying to find the size and shape of an unknown object with just a pea-shooter and a supply of frozen peas. By directing the frozen peas at the target in known directions and carefully recording where they scatter an idea of the shape can be discovered.
In order to determine an approximate radius for the nucleus alpha particles are used instead of frozen peas.
Alpha particles have a positive charge so as they approach the nucleus they get repelled by the electric force from the positive protons within. Another way of thinking about this is to remember that the alpha particles have kinetic energy. As they approach the nucleus this kinetic energy is changed into potential energy as the particles get closer and slow down. If the alpha particles approach head-on, they will get closer and closer to the nucleus, getting slower and slower, until they finally stop, and turn as they are repelled backwards. The distance when the particles turn back is known as the distance of closest approach; this distance sets an upper limit to the size of the nucleus.
If the initial kinetic energy of the alpha particles is known, the distance of closest approach can be calculated. For 5 MeV alpha particles scattered by gold nuclei, this turns out to be around 5 x 10-14m. The nucleus must be even smaller than this.
In order to find out more about the nucleus we need another type of particle to act as our frozen pea. Electrons can be accelerated to very high energies such that their wave-like properties become important. Much like sound waves diffract around everday objects when their wavelengths are comparable with the obstacle, electron waves can be diffracted around the nucleus when their wavelengths are similar to nuclear sizes. We can investigate the nucleus further by studying electron diffraction patterns from the nucleus.
Worked example of electrons and de Broglie's relationship:
The results of electron diffraction experiments have led to a formula for calculating the radius of a nucleus from its nucleon number:
Where R is the nuclear radius, r0 is the radius of a single nucleon (about 10-15m) and A is the nucleon number.
A graph of the nucleon number against nuclear radius if shown below.
Electron diffraction diagram with numbers: