Nuclear Physics and Radioactivity
Structure and Properties of the Nucleus Nucleus is made of protons and neutrons Proton has positive charge: Neutron is electrically neutral:
Neutrons and protons are collectively called nucleons. The different nuclei are referred to as nuclides. Number of protons: atomic number, Z Number of nucleons: atomic mass number, A Neutron number: N = A - Z
A and Z are sufficient to specify a nuclide. Nuclides are symbolized as follows: X is the chemical symbol for the element; it contains the same information as Z but in a more easily recognizable form.
Nuclei with the same Z so they are the same element but different N are called isotopes. For many elements, several different isotopes exist in nature. Natural abundance is the percentage of a particular element that consists of a particular isotope in nature.
Because of wave-particle duality, the size of the nucleus is somewhat fuzzy. Measurements of high-energy electron scattering yield:
Masses of atoms are measured with reference to the carbon-12 atom, which is assigned a mass of exactly 12u. A u is a unified atomic mass unit.
From the following table, you can see that the electron is considerably less massive than a nucleon.
Binding Energy and Nuclear Forces The total mass of a stable nucleus is always less than the sum of the masses of its separate protons and neutrons. Where has the mass gone? It has gone into binding energy which has a negative value.
It has become energy, such as radiation or kinetic energy, released during the formation of the nucleus. This difference between the total mass of the constituents that make up a nucleus, and the mass of the nucleus itself, is called the total binding energy of the nucleus. The total binding energy is equivalent to the energy required to break up the nucleus into its free constituents.
Mass-Energy Equivalence Einstein suggested that mass must be equivalent to energy. He arrived E = mc 2 He arrived at this conclusion is his efforts to satisfy Maxwell s equations and his own principle of relativity. Nuclear binding energy can also be understood by understanding the massenergy equivalence.
Binding Energy and Mass Number To compare how tightly bound different nuclei are, we divide the binding energy by A to get the binding energy per nucleon.
The higher the binding energy per nucleon, the more stable the nucleus. More massive nuclei require extra neutrons to overcome the Coulomb repulsion of the protons in order to be stable.
The force that binds the nucleons together is called the strong nuclear force. It is a very strong, but short-range, force. It is essentially zero if the nucleons are more than about 10-15 m apart. The Coulomb force is long-range; this is why extra neutrons are needed for stability in high-z nuclei.
In the following example one might ask why the mass of an electron is included. The reason is that one is normally always making a comparison to the mass of a neutral atom which includes electrons in its overall mass.
Example: Calculate the total nuclear binding energy, and the the average nuclear binding energy per nucleon for 56, the most common 26 Fe isotope of iron. State final energy in MeV Given information: m p +m e = 1.007825 u (mass of proton and electron) m n =1.008665 u Mass of (neutral)fe atom= 55.9349 u (this can be found in appendix F of Giancoli) Recall: 1 u is equivalent to 931.5 MeV (E= mc 2 )
Solution: Total mass of free protons, neutrons and electrons= 26( 1.007825)+ 30(1.008665) = 56.4635 u Mass of neutral Fe= 55.9349 Difference in mass= 56.4635-55.9349 = 0.5286 u = 492.4 MeV The average binding energy/nucleon: 492.4 MeV/56= 8.8 MeV
Nuclei that are unstable decay; many such decays are governed by another force called the weak nuclear force.
Radioactivity Towards the end of the 19 th century, minerals were found that would darken a photographic plate even in the absence of light. This phenomenon is now called radioactivity. Marie and Pierre Curie isolated two new elements that were highly radioactive; they are now called polonium and radium.
Radioactive rays were observed to be of three types: 1. Alpha rays, which could barely penetrate a piece of paper 2. Beta rays, which could penetrate 3 mm of aluminum 3. Gamma rays, which could penetrate several centimeters of lead We now know that alpha rays are helium nuclei, beta rays are electrons, and gamma rays are electromagnetic radiation.
Alpha and beta rays are bent in opposite directions in a magnetic field, while gamma rays are not bent at all.
Conserved Quantities At this point it is worth mentioning the conservation laws in nuclear reactions which includes nuclear decay processes. There is total mass-energy conservation Conservation of charge Conservation of nucleon number Conservation of mass law has more been replaced by a more accurate conservation law (Mass-Energy), as we now know mass and energy can be transformed from one to the other.
Alpha Decay Example of alpha decay: Radium-226 will alpha-decay to radon-22
In general, alpha decay can be written: Alpha decay occurs when the strong nuclear force cannot hold a large nucleus together. The mass of the parent nucleus is greater than the sum of the masses of the daughter nucleus and the alpha particle; this difference is called the disintegration energy.
Disintegration Energy The total energy released in the decay process in called the disintegration energy (Q). In the case of alpha decay Q is the kinetic energy given to the products [ie the Daughter nucleus(m D ), and alpha particle(m α )]. Q is calculated by: Q = (M P M D m α )c 2 M P = parent nucleus
Example: Calculate the disintegration energy when 232 (232.037131 u) decays to 92U 228 90Th (228.028716 u) with the emission of an α particle (4.002602). The masses quoted are of neutral atoms. I u is equivalent to 931.5 MeV. Solution: M P -M D -m α = 232.037131-228.028716-4.002602 =0.005813 u Q= (0.005813 u)(931.5 MeV/u) 5.4MeV
Beta Decay Beta decay (β decay) occurs when a nucleus emits an electron. An example is the decay of carbon-14: 14 C 14 N + e +ν 6 7 The nucleus still has 14 nucleons, but it has one more proton and one fewer neutron. The particle on the right is an antineutrino This decay is an example of an interaction that proceeds via the weak nuclear force.
The electron in beta decay is not an orbital electron; it is created in the decay. The fundamental process is a neutron decaying to a proton, electron, and (anti) neutrino: The need for a particle such as the neutrino was discovered through analysis of energy and momentum conservation in beta decay it could not be a two-particle decay.
Neutrinos are notoriously difficult to detect, as they interact only weakly, and direct evidence for their existence was not available until more than 20 years had passed. The symbol for the neutrino is the Greek letter nu (ν)
Beta decay (β + decay) can also occur where the nucleus emits a positron rather than an electron: This is equivalent to a proton changing to a neutron: p n + e + + a neutrino And a nucleus can capture one of its inner electrons (electron capture):
Gamma Decay Gamma rays are very high-energy photons. They are emitted when a nucleus decays from an excited state to a lower state, just as photons are emitted by electrons returning to a lower state.
Conservation of Nucleon Number and Other Conservation Laws A new law that is evident by studying radioactive decay is that the total number of nucleons cannot change(mass number). Also the charge is conserved.
Half-Life and Rate of Decay Nuclear decay is a random process; the decay of any nucleus is not influenced by the decay of any other.
The number of decays in a short time interval is proportional to the number of nuclei present and to the time. It should be noted that N is proportional to the mass, so the functions below could also be restated in terms of mass (amount). or ΔN Δt = λn Here, λ is a constant characteristic of that particular nuclide, called the decay constant.
This equation can be solved, using calculus, for N as a function of time:
The half-life is the time it takes for half the nuclei in a given sample to decay. It is related to the decay constant:
The SI units of radioactivity is called the Bequerel (Bq). 1 Bq= 1 disintegration/s. A disintegration is equivalent to 1 decay process. The activity of sample of radioactive material is its number of Bequerels ( or # decays/s)
Example: The isotope has a half-life of 5730 yr. If at some time a sample contains 1.00x10 22 carbon-14 nuclei, what is the activity of the sample? Solution: First calculate the decay constant in terms of seconds: λ = ln2 T 1/2 = 0.693 (5730yr)(3.156x10 7 s / yr) = 3.83x10-12 s -1 The rate of decay (activity) is proportional to the number of nuclei ΔN Δt = λn =(3.83x10-12 s -1 )(1.00x10 22 ) = 3.83x10 10 decays/s
Example: 2.00 kg of Zinc 65 has a half-life of 244 days. 1.50 kg Cadmium 109 has a half life of 453 days. How long will it take till there is an equal amount of both isotopes? Solution: N Zn = No Zn e λ 1t and N Cd = No Cd e λ 2t λ = ln(2) t 1/2 At some time t, N Zn will equal N Cd. 2e ln2 244 t =1.5e ln2 Solve for t: t= 219 days 453 t Take ln of both sides: ln(2) ln(2) 244 t = ln(1.5) ln(2) 453 t Note that it was possible to leave λ in terms of days, and N as an amount in kg. The number of atoms(n) is proportional to the total mass.
Decay Series A decay series occurs when one radioactive isotope decays to another radioactive isotope, which decays to another, and so on. This allows the creation of nuclei that otherwise would not exist in nature.
Decay Series
Radioactive Dating Radioactive dating can be done by analyzing the fraction of carbon in organic material that is carbon-14.
The ratio of carbon-14 to carbon-12 in the atmosphere has been roughly constant over thousands of years. A living plant or tree will be constantly exchanging carbon with the atmosphere, and will have the same carbon ratio in its tissues.
When the plant dies, this exchange stops. Carbon-14 has a half-life of about 5730 years; it gradually decays away and becomes a smaller and smaller fraction of the total carbon in the plant tissue. This fraction can be measured, and the age of the tissue deduced. Objects older than about 60,000 years cannot be dated this way there is too little carbon-14 left.
Other isotopes are useful for geologic time scale dating. Uranium-238 has a half-life of 4.5 x 10 9 years, and has been used to date the oldest rocks on Earth as about 4 billion years old.
Nuclear Energy
Nuclear Reactions and the Transmutation of Elements A nuclear reaction takes place when a nucleus is struck by another nucleus or particle. If the original nucleus is transformed into another, this is called transmutation. An example:
Energy and momentum must be conserved in nuclear reactions as well as charge, and nucleon number. Generic reaction: The reaction energy, or Q-value, is the sum of the initial masses less the sum of the final masses, multiplied by c 2 :
If Q is positive, the reaction is exothermic, and will occur no matter how small the initial kinetic energy is. If Q is negative, there is a minimum initial kinetic energy that must be available before the reaction can take place.
Neutrons are very effective in nuclear reactions, as they have no charge and therefore are not repelled by the nucleus.
Nuclear Fission After absorbing a neutron, a uranium-235 nucleus will split into two roughly equal parts. One way to visualize this is to view the nucleus as a kind of liquid drop.
Sample fission reaction: n+ 235 U 236 U N + N + neutrons 92 92 1 2 Examples:
Example: Calculate the energy released in this fission reaction: The atomic masses of Ba and Kr respectively are 140.914 u and 91.926 u. Mass of Uranium 236= 236.04556 u Mass of neutron= 1.008665 u Answer: 167 MeV
The energy release in a fission reaction is quite large. Also, since smaller nuclei are stable with fewer neutrons, several neutrons emerge from each fission as well. These neutrons can be used to induce fission in other nuclei, causing a chain reaction.
Nuclear Fusion The lightest nuclei can fuse to form heavier nuclei, releasing energy in the process. An example is the sequence of fusion processes that change hydrogen into helium in the Sun. They are listed here with the energy released in each:
The net effect is to transform four protons into a helium nucleus plus two positrons, two neutrinos, and two gamma rays. More massive stars can fuse heavier elements in their cores, all the way up to iron, the most stable nucleus.
There are three fusion reactions that are being considered for power reactors: These reactions use very common fuels deuterium or tritium and release much more energy per nucleon than fission does.
Fusion Fission
A successful fusion reactor has not yet been achieved, but fusion, or thermonuclear, bombs have been built.
Several geometries for the containment of the incredibly hot plasma that must exist in a fusion reactor have been developed the tokamak, which is a torus; or inertial confinement, which is tiny pellets of deuterium ignited by powerful lasers. The most recent promising enterprise is ITER (International Thermonuclear Experimental Reactor). Fission using the thorium cycle may be society s best bet for nuclear energy. India is investing heavily in this.