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Alpha particles are stopped by paper or skin or just about all solid matter, in water they will only pass about 0.01 mm. The reason is that an alpha particle has a double positive charge due to the two protons, It is strongly attracted to electrons in the matter it is passing through. As a result it reacts with many electrons over a very short distance and gives up all its energy over a very short distance.
Beta particles penetrate further but also will be stopped by just a few mm of aluminium. They are also stopped by materials such as lead but the heavier the atoms they interact with, the more energy is converted to X-rays(called bremsstrahlung). Hit heavy metals with beta (electrons) and you have an X-ray generator. Light metals such as aluminium don't produce high energy X-rays and so are a better way to stop beta. Aluminium stops beta as follows- the beta particle has a single negative charge and carries a high amount of kinetic energy, as it passes through matter it will interact (sort of bounce off but cleverer) with the atoms close to it's path. Each interaction transfers more energy to the material it is passing through until all of it's energy is spent.
Gamma is very different, it is not particles but waves of pure energy with no mass and no charge. As it has no charge it is not attracted to electrons or protons and if no particle gets in the way the photon of gamma will just pass through. Any material cuts down the amount of gamma passing through it, even air, 500 m of air does reduce gamma passing through it by a measurable amount above what you would expect due to the inverse square law. There are several ways gamma radiation interacts with matter, it gets very technical though, if you want detail try the wikipedia, but very briefly:
the photoelectric effect: A gamma photon hits an electron and knocks the electron out of the atom.
Compton scattering: More energy than can be transferred to the electron- the electron is still knocked out but the spare energy bounces off as a photon with less energy.
Pair production: If there is enough energy in the gamma photon and it passes close enough to a nucleus, the energy of the photon can convert to two particles, an electron and an anti-electron, turning pure energy into matter and anti-matter, Smart!
To shield from gamma you really want photoelectric reactions an the heavier the atoms you use the more likely this is to occur.