Transcript
NUCLEAR PHYSICS AND RADIOACTIVITY
Nuclear Structure
The nucleus of the atom consists of protons and neutrons. Collectively they are called nucleons.
The neutron was discovered in 1932 by James Chadwick. It carries no charge and has a mass slightly larger than that of a proton.
The number of protons in the nucleus is expressed by the atomic number, Z. The total number of neutrons is expressed by the neutron number, N. The total number of protons and neutrons is called the atomic mass number, A. A multiplied by the mass of one nucleon is approximately equal to the total mass of the nucleus.
A = Z + N
The shorthand notation used in equations representing nuclear reactions shows the atomic number as a subscript, the mass number as a superscript and the chemical symbol of the element as the main character. An isotope of Aluminum with 13 protons and 27 total nucleons would be written as:
13Al27
Some texts put Z and A on the same side of the chemical symbol but Z is always the subscript.
Nuclei that contain the same number of protons but different numbers of neutrons are called isotopes. Carbon exists in three forms. They are carbon-12, carbon-13, and carbon-14. They each have 6 protons, but 6, 7, and 8 neutrons respectively.
The approximate radius of a nucleus can be determined from the number of nucleons present. The formula for the radius in meters is:
r = (1.2 x 10-15m)A1/3
Notice that the density of all nuclei will be approximately the same. The only difference in the nuclei will be the number of nucleons, A. This factor will divide out when total mass is divided by volume.
The Strong Nuclear Force and Stability of the Nucleus
In the nucleus, the positively charged protons repel each other with an electrostatic force that is many times the force of gravity for the same particles.
Example
Compare the forces of gravitational attraction and electrostatic repulsion for the protons and neutrons in a Helium nucleus.
Since gravity is not strong enough to hold the particles of a nucleus together, another force called the strong nuclear force must exist. It is a fundamental force along with gravity and the electroweak force. They are called fundamental forces since all other forces in nature can be explained in terms of these three.
The strong nuclear force does not depend on electric charge. The same force exists between any two nucleons.
The strong nuclear force acts over a very short distance. The two nucleons must be within 10-15 m of each other.
Since the strong nuclear force only acts over a short distance, it only exists between two nucleons that are adjacent. The electrostatic repulsive force, however, exists between all of the protons in the nucleus.
As the number of protons increases, the number of neutrons increases more rapidly to provide extra strong nuclear force to help overcome the increase in electrostatic repulsion.
This graph shows number of neutrons on the Y axis and the number of protons in the same nuclide on the X axis. Where the slope = 1, the numbers of neutrons and protons are equal. For elements with more than 18 protons, the number of neutrons increases noticeably faster than the number of protons.
Nuclei with more than 83 protons are not stable and break apart or change their internal structure by emitting radiation. This process is called radioactivity and was discovered by Becquerel.
Mass Defect of the Nucleus and Nuclear Binding Energy
The nucleons in a stable nucleus are held together by the strong nuclear force and the fact that the stable nucleus has less energy than the separate components. In order to separate the protons and neutrons that compose a stable nucleus an amount of energy called the binding energy of the nucleus must be added.
The total mass of the individual nucleons that make up a nucleus is greater when they are separated than when they are together in the nucleus. This difference in mass is called the nuclear mass defect and is the matter equivalent of the nuclear binding energy.
In Einstein's theory of special relativity, matter and energy are shown to be different forms of the same thing. Matter can be converted to energy and energy can be converted to matter. The relationship is Einstein's famous equation:
E = mc2
where E and m are energy and matter and c is the speed of light.
The total nuclear binding energy of a nucleus can be found by replacing m in the equation with the nuclear mass defect.
Example
Determine the nuclear binding energy of the helium-4 nucleus. Don't forget to allow for the mass of the 2 electrons.
When comparing binding energies of different nuclei, it is generally useful to look at the binding energy per nucleon for each nucleus. When atomic number is small, binding energy per nucleon increases rapidly. At about A = 60, the binding energy per nucleon reaches a maximum of 8.7 MeV. Then it begins to decrease.
A smaller amount of binding energy is what makes elements with atomic numbers greater than 83 unstable.
Radioactivity
The three types of naturally occurring radiation are ? rays, ? rays and ? rays. They are named in order of their ability to penetrate matter. ? rays are the least penetrating since they are stopped by .01 mm thick lead. ? rays are next and penetrate lead to a depth of 0.1 mm. ? rays are the most penetrating and can pass through 100 mm of lead.
Notice that ? and ? rays are deflected by a magnetic field which indicates they are composed of charged particles. Since ? rays are not deflected, they must not have a charge.
Alpha Decay
When a nucleus produces ? rays it is said to undergo ? decay. An ? particle consists of two protons and two neutrons which and can be described as a He-4 nucleus. The symbol for an ? particle in nuclear equations is 2He4. The subscript two indicates two protons and the superscript four indicates a total of four nucleons.
The ? decay of U-238 is a good example.
The equation is:
92U238 ? 90Th234 + 2He4
The original nucleus is called the parent nucleus and the nucleus that remains is called the daughter nucleus. Since the atomic number of the daughter nucleus is different from the parent nucleus, transmutation has occurred.
During a nuclear change such as a transmutation, the total charge before the event must be the same as the total charge after the event. Also, the nucleon number before the event must equal the nucleon number after the event.
Notice in the ? decay of U-238 to Th-234 that the total charge on the left is 92 and the total charge on the right is also 92. The total number of nucleons on the left is 238 and the total number of nucleons on the right is also 238. Both of these quantities are conserved.
Although it was not shown in the equation, energy is also released. The amount can be calculated by finding the nuclear mass defect and converting it to the corresponding energy.
Example
The atomic mass of U-238 is 238.0508 u, Th-234 is 234.0436 u, and the mass of an ? particle is 4.0026 u.
Determine the energy released when a U-238 atom undergoes ? decay. Notice that the mass of the ? particle includes the two electrons lost by U-238 when it became Th-234. Also 1 u becomes 931.5 Mev when it is converted to energy.
The energy that is released is carried away as kinetic energy of the daughter nucleus and the ? particle. Some energy is usually released as a gamma ray. Since momentum must be conserved, the kinetic energy change of the alpha particle is much greater than that of the daughter nucleus.
Beta Decay
Beta decay occurs when a nucleus emits an electron. An example of beta decay is:
The equation is:
90Th234 ? 91Pa234 + -1e0
The Thorium nucleus changes into a Protactinium nucleus by losing a negative charge. The negative charge is an electron which is thought to come from the decay of a neutron into a proton and an electron. The energy the electron receives from the decay process is enough to allow it to escape the nucleus.
A second type of ? decay occurs when a positron is emitted by a nucleus. A positron has the same mass as an electron but has a +e charge instead of -e. The process is thought to occur when a proton emits a positron to become a neutron.
Gamma Decay
The nucleus has discrete energy states like the electrons that surround it. When the nucleus changes from a higher energy state to a lower one, it emits a photon with considerably more energy than those emitted by electrons. A group of these high energy photons is called a gamma ray (? ray).
Gamma decay does not cause a change in the identity of the atom since there is no change in the number of protons. Since there is no change in the number of neutrons, the mass number also remains the same.
Gamma radiation is useful in Gamma Knife surgery and thallium stress tests.
The Neutrino (?)
During ? decay, not all of the energy produced can be accounted for in the kinetic energy of the ? particle and the recoiling nucleus. The emission of another particle with no charge and practically no mass was proposed to account for this deficit.
Its existence was experimentally verified in 1956. It turns out that an antineutrino is emitted during ?- decay and a normal neutrino is emitted during ?+ decay.
Radioactive Decay and Activity
As a sample of a radioactive nuclide decays, the number of parent nuclei decreases in a fashion that can be represented by a smooth curve.
Although this curve looks a lot like a hyperbola, it is actually an exponential function of time.
In order to help us compare rates at which different nuclides decay, we use the concept of half-life.
The half-life (T½) of a nuclide is the amount of time required for one half of the parent nucleii to turn into daughter nuclei. In the graph above, it is easy to see that the half-life time corresponds to a decrease in the number of parent nuclei equal to ½ of the number present at the beginning of the interval.
Example
Suppose 3.0 x 107 radon atoms are trapped in a basement and the basement is sealed so that none can enter or leave. Since the half-life of radon is 3.83 days, how many radon atoms are left after 30.64 days?
The activity of a radioactive sample depends on the rate at which the sample decays and the number of radioactive nuclei present. The equation for activity is:
?N/?t = -?N
where ? is the decay constant and depends on the half-life and N is the number of nuclei present. The number of nucleii present is calculated using the equation:
N = N0e-?t
From this equation, it can be shown that ? = (ln2)/T½.
The SI unit for activity is the Becquerel (Bq) and equals one disintegration per second. Another widely used unit is the Curie (Ci) which equals 3.70 x 1010 Bq. One Curie is roughly the activity of one gram of pure radium.
Example
In our last example, 3.0 x 107 radon atoms were trapped in a basement. Find the activity (a) initially and (b) 31 days later.(T½ = 3.83 days)
Radioactive Decay
Radioactive Dating
A very useful application of radioactivity is the determination of the age of artifacts and rocks. If the object to be dated contains a radioactive element, it is often possible to determine the number of half-lives that have passed since the object came into existence.
Carbon-14 is a radioactive isotope of carbon that is found in all living things. An assumption is made that the level of C-14 found in living things today has not changed for as long as 50,000 years ago.
By determining the C-14 activity of a sample of a formerly living object, the number of half-lives that have passed can be determined and an approximate time since the object stopped ingesting C-14 can be determined using the equation:
A = A0e-?t
Example
The iceman was found in a glacier in the Italian Alps in 1991. Material found with the body had a C-14 activity of about 0.121 Bq per gram of carbon. Determine how long ago the Iceman died. A0 = 0.23 Bq
Radioactive Decay Series
A radioactive decay series simply lists the sequence of daughter nuclei that are produced from a radioactive nucleus. Any time radioactive decay produces a nucleus that is itself radioactive, the sequence continues until a stable nucleus is produced.
As you can see in this series, more than one decay mode is possible for certain nuclides. The end result no matter what the path is stable Lead-206.
