Unstable Nucleus
20 Jan 2019


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In radioactive processes, particles or electromagnetic radiation are emitted from the nucleus. The most common forms of radiation emitted have been traditionally classified as alpha (a), beta (b), and gamma (g) radiation. Nuclear radiation occurs in other forms, including the emission of protons or neutrons or spontaneous fission of a
massive nucleus.
Of the nuclei found on Earth, the vast majority is stable. This is so because almost all short-lived radioactive nuclei have decayed during the history of the Earth. There are approximately 270 stable isotopes and 50 naturally occurring radioisotopes (radioactive isotopes). Thousands of other radioisotopes have been made in the laboratory.

Radioactive decay will change one nucleus to another if the product nucleus has a greater nuclear binding energy than the initial decaying nucleus. The difference in binding energy (comparing the before and after states) determines which decays are energetically possible and which are not. The excess binding energy appears as kinetic energy or rest mass energy of the decay products.

The Chart of the Nuclides, part of which is shown in Fig. 3-1, is a plot of nuclei as a function of proton number, Z, and neutron number, N. All stable nuclei and known radioactive nuclei, both naturally occurring and manmade, are shown on this chart, along with their decay properties. Nuclei with an excess of protons or neutrons in comparison
with the stable nuclei will decay toward the stable nuclei by changing protons into neutrons or neutrons into protons, or else by shedding neutrons or protons either singly or in combination. Nuclei are also unstable if they are excited, that is, not in their lowest energy states. In this case the nucleus can decay by getting rid of its excess energy
without changing Z or N by emitting a gamma ray.

Nuclear decay processes must satisfy several conservation laws, meaning that the value of the conserved quantity after the decay, taking into account all the decay products, must equal the same quantity evaluated for the nucleus before the decay. Conserved quantities include total energy (including mass), electric charge, linear and
angular momentum, number of nucleons, and lepton number (sum of the number of electrons, neutrinos, positrons and antineutrinos—with antiparticles counting as -1).

The probability that a particular nucleus will undergo radioactive decay during a fixed length of time does not depend on the age of the nucleus or how it was created.

Fig. 3-2. 137mBa decay data, counting numbers of decays observed in 30-second intervals. The best-fit
exponential curve is shown. The points do not fall exactly on the exponential because of statistical
counting fluctuations.

Although the exact lifetime of one particular nucleus cannot be predicted, the mean (or
average) lifetime of a sample containing many nuclei of the same isotope can be
predicted and measured. A convenient way of determining the lifetime of an isotope is to
measure how long it takes for one-half of the nuclei in a sample to decay—this quantity
is called the half-life, t1/2. Of the original nuclei that did not decay, half will decay if we
wait another half-life, leaving one-quarter of the original sample after a total time of two
half-lives. After three half-lives, one-eighth of the original sample will remain and so on.
Measured half-lives vary from tiny fractions of seconds to billions of years, depending on
the isotope.

The number of nuclei in a sample that will decay in a given interval of time is
proportional to the number of nuclei in the sample. This condition leads to radioactive
decay showing itself as an exponential process, as shown in Fig. 3-2. The number, N, of
the original nuclei remaining after a time t from an original sample of N0 nuclei is
N = N0e-(t/T)
where T is the mean lifetime of the parent nuclei. From this relation, it can be shown that
t1/2 = 0.693T.

Alpha Decay

Fig. 3-3. An alpha-particle decay
In alpha decay, shown in Fig. 3-3, the nucleus emits a 4He nucleus, an alpha particle. Alpha decay occurs most often in massive nuclei that have too large a proton to neutron ratio. An alpha particle, with its two protons and two neutrons, is a very stable configuration of particles. Alpha radiation reduces the ratio of protons to neutrons in the
parent nucleus, bringing it to a more stable configuration. Nuclei, which are more massive than lead, frequently decay by this method.

Consider the example of 210Po decaying by the emission of an alpha particle. The
reaction can be written 210Po Æ206Pb + 4He. This polonium nucleus has 84 protons and 126 neutrons. The ratio of protons to neutrons is Z/N = 84/126, or 0.667. A 206Pb nucleus has 82 protons and 124 neutrons, which gives a ratio of 82/124, or 0.661. This small change in the Z/N ratio is enough to put the nucleus into a more stable state, and as shown in Fig. 3-4, brings the “daughter” nucleus (decay product) into the region of stable nuclei in the Chart of the Nuclides.

In alpha decay, the atomic number changes, so the original (or parent) atoms and
the decay-product (or daughter) atoms are different elements and therefore have different
chemical properties.
Fig. 3-4. Upper end of the Chart of the Nuclides

In the alpha decay of a nucleus, the change in binding energy appears as the kinetic energy of the alpha particle and the daughter nucleus. Because this energy must be shared between these two particles, and because the alpha particle and daughter nucleus must have equal and opposite momenta, the emitted alpha particle and recoiling
nucleus will each have a well-defined energy after the decay. Because of its smaller mass, most of the kinetic energy goes to the alpha particle.

Beta Decay
Fig. 3-5. Beta decays. a) Beta-minus decay. b) Beta-plus decay.

Beta particles are electrons or positrons (electrons with positive electric charge, or antielectrons). Beta decay occurs when, in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. In beta minus decay, as shown in Fig. 3-5a, a neutron decays into a proton, an electron, and an
antineutrino: n Æ p + e- +—n . In beta plus decay, shown in Fig. 3-5b, a proton decays into a neutron, a positron, and a neutrino: p Æ n + e+ +n. Both reactions occur because in different regions of the Chart of the Nuclides, one or the other will move the product closer to the region of stability. These particular reactions take place because
conservation laws are obeyed. Electric charge conservation requires that if an electrically neutral neutron becomes a positively charged proton, an electrically negative particle (in this case, an electron) must also be produced. Similarly, conservation of lepton number requires that if a neutron (lepton number = 0) decays into a proton (lepton number = 0) and an electron (lepton number = 1), a particle with a lepton number of -1 (in this case an
antineutrino) must also be produced. The leptons emitted in beta decay did not exist in
the nucleus before the decay—they are created at the instant of the decay.

To the best of our knowledge, an isolated proton, a hydrogen nucleus with or without an electron, does not decay. However within a nucleus, the beta decay process can change a proton to a neutron. An isolated neutron is unstable and will decay with a half-life of 10.5 minutes. A neutron in a nucleus will decay if a more stable nucleus
results; the half-life of the decay depends on the isotope. If it leads to a more stable nucleus, a proton in a nucleus may capture an electron from the atom (electron capture), and change into a neutron and a neutrino.

Proton decay, neutron decay, and electron capture are three ways in which protons can be changed into neutrons or vice-versa; in each decay there is a change in the atomic number, so that the parent and daughter atoms are different elements. In all three processes, the number A of nucleons remains the same, while both proton number, Z, and neutron number, N, increase or decrease by 1.

In beta decay the change in binding energy appears as the mass energy and kinetic
energy of the beta particle, the energy of the neutrino, and the kinetic energy of the
recoiling daughter nucleus. The energy of an emitted beta particle from a particular decay
can take on a range of values because the energy can be shared in many ways among the
three particles while still obeying energy and momentum conservation.

Gamma Decay

In gamma decay, depicted in Fig. 3-6, a nucleus changes from a higher energy
Fig. 3-6. A gamma (g) decay.

state to a lower energy state through the emission of electromagnetic radiation (photons).
The number of protons (and neutrons) in the nucleus does not change in this process, so
the parent and daughter atoms are the same chemical element. In the gamma decay of a
nucleus, the emitted photon and recoiling nucleus each have a well-defined energy after
the decay. The characteristic energy is divided between only two particles.

The Discovery of Radioactivity
In 1896 Henri Becquerel was using naturally fluorescent minerals to study the
properties of x-rays, which had been discovered in 1895 by Wilhelm Roentgen. He
exposed potassium uranyl sulfate to sunlight and then placed it on photographic plates
wrapped in black paper, believing that the uranium absorbed the sun’s energy and then
emitted it as x-rays. This hypothesis was disproved on the 26th-27th of February, when his
experiment “failed” because it was overcast in Paris. For some reason, Becquerel decided
to develop his photographic plates anyway. To his surprise, the images were strong and
clear, proving that the uranium emitted radiation without an external source of energy
such as the sun. Becquerel had discovered radioactivity.

Becquerel used an apparatus similar to that shown in Fig. 3-7 to show that the
radiation he discovered could not be x-rays. X-rays are neutral and cannot be bent in a
magnetic field. The new radiation was bent by the magnetic field so that the radiation
must be charged and different than x-rays. When different radioactive substances were
put in the magnetic field, they deflected in different directions or not at all, showing that
there were three classes of radioactivity: negative, positive, and electrically neutral.

The term radioactivity was actually coined by Marie Curie, who together with her
husband Pierre, began investigating the phenomenon recently discovered by Becquerel.
The Curies extracted uranium from ore and to their surprise, found that the leftover ore
showed more activity than the pure uranium. They concluded that the ore contained other
radioactive elements. This led to the discoveries of the elements polonium and radium. It
took four more years of processing tons of ore to isolate enough of each element to
determine their chemical properties.

Ernest Rutherford, who did many experiments studying the properties of
radioactive decay, named these alpha, beta, and gamma particles, and classified them by
their ability to penetrate matter. Rutherford used an apparatus similar to that depicted in
Fig. 3-7. When the air from the chamber was removed, the alpha source made a spot on
the photographic plate. When air was added, the spot disappeared. Thus, only a few
centimeters of air were enough to stop the alpha radiation.

Because alpha particles carry more electric charge, are more massive, and move
slowly compared to beta and gamma particles, they interact much more easily with
matter. Beta particles are much less massive and move faster, but are still electrically
charged. A sheet of aluminum one-millimeter thick or several meters of air will stop
these electrons and positrons. Because gamma rays carry no electric charge, they can
penetrate large distances through materials before interacting—several centimeters of
lead or a meter of concrete is needed to stop most gamma rays.

Radioactivity in Nature

Radioactivity is a natural part of our environment. Present-day Earth contains all
the stable chemical elements from the lowest mass (H) to the highest (Pb and Bi). Every
element with higher Z than Bi is radioactive. The earth also contains several primordial
long-lived radioisotopes that have survived to the present in significant amounts. 40K,
with its 1.3 billion-year half-life, has the lowest mass of these isotopes and beta decays to
both 40Ar and 40Ca.

Many isotopes can decay by more than one method. For example, when actinium-
226 (Z=89) decays, 83% of the rate is through b–decay, 226Ac Æ 226Th + e- + —n , 17%
is through electron capture, 226Ac + e- Æ 226Fr + n, and the remainder, 0.006%, is
through a-decay, 226Ac Æ 222Fr + 4He. Therefore from 100,000 atoms of actinium, one
would measure on average 83,000 beta particles and 6 alpha particles (plus 100,000
neutrinos or antineutrinos). These proportions are known as branching ratios. The
branching ratios are different for the different radioactive nuclei.

Three very massive elements, 232Th (14.1 billion year half-life), 235U (700 million
year half-life), and 238U (4.5 billion year half-life) decay through complex “chains” of
alpha and beta decays ending at the stable 208Pb, 207Pb, and 206Pb respectively. The
decay chain for 238U is shown in Fig. 3-8. The ratio of uranium to lead present on Earth
today gives us an estimate of its age (4.5 billion years). Given Earth’s age, any much
shorter-lived radioactive nuclei present at its birth have already decayed into stable
elements. One of the intermediate products of the 238U decay chain, 222Rn (radon) with a
half-life of 3.8 days, is responsible for higher levels of background radiation in many
parts of the world. This is primarily because it is a gas and can easily seep out of the earth
into unfinished basements and then into the house.

Some radioactive isotopes, for example 14C and 7Be, are produced continuously
through reactions of cosmic rays (high-energy charged particles from outside Earth) with
molecules in the upper atmosphere. 14C is useful for radioactive dating (see Chapter 13).
Also, the study of radioactivity is very important to understand the structure of the earth
because radioactive decay heats the earth’s interior to very high temperatures.
Fig. 3- 8. The uranium decay series. The vertical axis is atomic mass.

Units of Radioactivity
The number of decays per second, or activity, from a sample of radioactive nuclei
is measured in becquerel (Bq), after Henri Becquerel. One decay per second equals one

An older unit is the curie, named after Pierre and Marie Curie. One curie is
approximately the activity of 1 gram of radium and equals (exactly) 3.7 x 1010 becquerel.
The activity depends only on the number of decays per second, not on the type of decay,
the energy of the decay products, or the biological effects of the radiation .

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