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The nucleus is often content to spend many years undergoing no major changes. The protons and neutrons hold together, while the electrons in the electron orbitals are doing all sorts of things. Sometimes, however, the nucleus undergoes a change. If this happens spontaneously, it is called radioactive decay.


There are three main types of radioactive decay. Nuclei which have an especially large number of protons and neutrons will sometimes throw off a packet of two protons and two neutrons, called an alpha particle (a). Note that an alpha particle is the same as the nucleus of the common helium nucleus. This is called alpha decay.


What is the reaction representing the alpha decay of the thorium-232?



The answer is


We know the atomic number of thorium (90) from the periodic table. The alpha particle is always represented by the symbol 42He. Figure 16-5 shows this decay. (Figure 16-5 shows this decay.)

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Figure 16-5

Generally alpha emitters are not dangerous to biological tissue (provided you do not eat them), since the alpha particles lose energy very quickly and do not penetrate very far even in air (several centimeters). That means they generally do not get inside of you.
The second type of radioactive decay is beta decay. Nuclei with many neutrons, compared with protons, undergo normal beta decay (β). In this process, a neutron decays into a proton, an electron, and an antineutrino. The proton stays in the nucleus, and the electron shoots away from the nucleus. The speeding electron, often called a beta particle, can be highly injurious to biological tissue, since it is able to speed through the air and penetrate into the body. Once in the body, it slows down by ionizing molecules that it passes by, which can be very dangerous if one such molecule is DNA. The antineutrino is so penetrating that it generally passes through the body and the planet without depositing any energy, so it is mostly harmless.



What is the reaction for the (normal) beta decay of lithium-9?



On the left side of the reaction we have 39Li. On the right side we place an electron. For accounting purposes, the symbol of the electron is 0–1e. Thus we have the incomplete equation



In order to complete it, we need to make sure the upper and lower left numbers add up correctly. Also we add an antineutrino, so we write


or we can write


Figure 16-6 shows this decay schematically.

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Figure 16-6

Nuclei with many protons undergo positron decay (β+). In this process, a proton decays into a neutron, a positron, and a neutrino. The neutron stays in the nucleus. The positron, which is a particle just like an electron (in mass and so on) with a positive charge, shoots away from the nucleus. The positron is dangerous to biological tissue as well, and the neutrino is innocuous.
(A word about words: Radioactivity is generally divided into alpha, beta, and gamma decays (although there are some other sorts of decays as well). Beta decays are divided into beta decays and positron decays, so the wording can be a bit tricky. Sometimes writers will be careful to say "normal beta decay" and sometimes they will not. If it matters which sort of beta decay is required, assume the normal kind unless otherwise specified.)



What is the reaction for the positron decay of carbon-11?



On the left side of the reaction we have . On the right side we place a positron, whose symbol is . Thus we have



A third type of radioactive decay is gamma decay. Just as the electrons of an atom may be in an excited state, the nucleus can also be in an excited quantum state. When the nucleus decays into the lower energy state, it releases a photon, called a gamma particle, just as a photon is released in the electronic case. When the nucleus decays, however, the energies involved are much greater, about a million times greater, than for the electronic decays. This reaction is the most penetrating, able to penetrate many meters of lead. It can be harmless if it passes simply through the human body, or it can be quite harmful.



The radioactive decay of an excited state of cobalt-60 is given by



where the asterisk indicates an excited state. In this case 60Co* decays into one excited state of 60Ni, which decays into a second excited state of 60Ni, which decays to the ground state.


We can measure the time it takes a nucleus to decay in terms of a half-life. This is the time it takes half the atoms in a sample to decay. The lifetime of an atom is not the same as the lifetime of, say, humans.


If we imagine a population sample of 1000 humans all born in the same year, then after 75 years we would expect about half of them (or 500) to still be alive. After another eight years, only half of those would be surviving. After another five years, it would be half again. We would expect, 150 years after the birth date, that there would probably be no survivors. 


Consider now a sample of 1000 gadolinium-148 atoms, all generated at the same moment. This isotope decays by alpha emission to samarium-144 with a half-life of 75 years. After 75 years we expect about half of the original Gd-148 to remain. After another 75 years, about half of those have decayed, leaving about 250 Gd-148 atoms. After yet another 75 years, there are around 125 left. Radioactive atoms do not age and die, but at any moment they have some constant risk of decaying.


The nucleus Ru-103 decays to the stable isotope Rh-103. We obtain a pure sample and measure that its radioactivity to be 1616 millicuries. After 156 days the radioactivity is down to 101 millicuries. What is the half-life of this isotope?

The activity of the sample is down by a factor of 1616/101 = 16, that is, down by four factors of 2. That means that four halflives must have transpired. One half-life is (156 days)/4 = 39 days.


In Example 1, the mass of a Th-232 is 232.0381 amu. The mass of Ra-228 is 228.0311 amu, and that of He-4 is 4.0026 amu. Notice that the sum of the Ra-228 and He-4 masses is less than that of Th-232. Where did the missing mass go? This mass has been converted into energy.


There is a deep connection between mass and energy which we will mention only briefly here. In this reaction, mass is converted into energy, and the amount of energy can be determined if we know the masses sufficiently well. The mass deficit mdef is the difference of the mass of products and the mass of reactants, and the energy of reaction is given by



where c = 3.0 x 108 m/s is the speed of light. This is the famous E = mc2 equation. In order to apply this equation we must be careful that the units agree.


In the decay of Th-232, most of the energy of the reaction ends up in the kinetic energy of the alpha particle. After a decay,
  1. how much energy (in J) does the alpha particle have?
  2. How fast (approximately) is the alpha particle going?
  1. The mass deficit in this reaction is mdef = (232.0381 – 228.0311 – 4.0026) amu = 0.0044 amu. We convert this to kg and multiply by c2 to obtain energy

This energy is mainly in the form of kinetic energy of the alpha particle.

  1. We want to use the equation EK = 1/2 mv2 to solve for v. We know EK in Joules, so we want m to be in kilograms:


Then we have


This is quite fast, about 5% of the speed of light, but not so fast that we have to resort to the full mechanics of special relativity.


In summary for this section, here is a chart of special particles and their symbols for nuclear reactions:
Particle Other names Symbol
proton p
neutron n  
helium-4 α, alpha   
electron β, e, beta minus  
positron β+, e+, beta plus
photon γ, gamma    γ
neutrino (do not need to know)  ν
antineutrino (do not need to know) 


In this chapter we discussed the physics of the atom and nucleus: atomic structure, electronic energy levels, radioactivity, and nuclear reactions. The key to understanding many physical situations lies in understanding the energy level diagram. An atom or nucleus can exist only in certain discrete states of precise energy. When the atom or nucleus makes a transition from one state to another, a photon is absorbed or released with energy equal to the difference of the energies of the two states.
The key to understanding problems involving nuclear reactions, including radioactive decay, lies in visualizing the decay and in writing the nuclear reaction correctly. There are three types of radioactive decay, called alpha, beta, and gamma decay. These decays vary in the type of particle expelled from the nucleus and the effect that this expulsion has on the nucleus.

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