CK-12 21st Century Physics: A Compilation of Contemporary and Emerging Technologies Read online

  The Curve of Binding Energy

  In the main isotopes of light nuclei, such as carbon, nitrogen and oxygen, the number of neutrons and of protons is indeed equal. However, as one moves to heavier nuclei, the disruptive energy of electric repulsion increases, because electric forces have a long range and each proton is repelled by all other protons in the nucleus. In contrast, the strong nuclear attraction between those protons increases only moderately, since the force has a short range and affects mainly immediately neighboring protons.

  Figure 3.2

  The Binding Energy of Nuclei

  The net binding energy of a nucleus is that of the nuclear attraction, minus the energy of the repulsive electric force. As nuclei get heavier than helium, their net binding energy per nucleon (deduced from the difference in mass between the nucleus and the sum of masses of component nucleons) grows more and more slowly, reaching its peak at iron. As nucleons are added, the total nuclear binding energy always increases—but the total energy of electric forces (positive protons repelling other protons) also increases, and past iron, the second increase outweighs the first. One may say is the most tightly bound nucleus (see #10-b).

  To reduce the energy of the repulsive electric force, the weak interaction allows the number of neutrons to exceed that of protons—for instance, in the main isotope of iron, protons but neutrons. Of course, isotopes also exist in which the number of neutrons differs, but if these are too far from stability, after some time nucleons convert to a more stable isotope by beta emission radioactivity—protons turn into neutrons by emitting a positron, the positive counterpart of the electron, or neutrons become protons by emitting electrons (neutrinos are also emitted in these processes).

  Among the heaviest nuclei, containing or more nucleons, electric forces may be so destabilizing that entire chunks of the nucleus get ejected, usually in combinations of protons and neutrons (alpha particles, actually fast helium nuclei), which are extremely stable.

  The curve of binding energy (drawing) plots binding energy per nucleon against atomic mass. It has its main peak at iron and then slowly decreases again, and also a narrow isolated peak at helium, which as noted is very stable. The heaviest nuclei in nature, uranium , are unstable, but having a lifetime of billion years, close to the age of the Earth, they are still relatively abundant; they (and other nuclei heavier than iron) may have formed in a supernova explosion (#8) preceding the formation of the solar system. The most common isotope of thorium, , also undergoes particle emission, and its half-life (time over which half a number of atoms decays) is even longer, by several times. In each of these, radioactive decay produces daughter isotopes, that are also unstable, starting a chain of decays that ends in some stable isotope of lead.

  Tidbits

  The book The Curve of Binding Energy by John McPhee is actually the story of nuclear physicist Theodore Taylor and his diverse side-interests.

  Review Questions

  Why can't one find in our environment elements whose atoms weigh 300 times as much as the proton, or more?

  Compile a glossary, defining briefly in alphabetical order in your own words: Alpha radioactivity, beta radioactivity, binding energy, controlled nuclear fusion, core of the Sun, curve of binding energy, daughter isotope, deuterium, mass spectrometer, nuclear fusion positron, short range force, strong (nuclear) force, weak (nuclear) force

  What is the source of the Sun's energy?

  Why is the binding energy of the nucleus given a negative sign?

  1. The atomic weight of deuterium is , that of helium is (in units of the proton mass), and the "rest energy" of the proton is (million , with see #9). How many are released when two atoms of deuterium combine to one of by nuclear fusion? 2. If and Avogadro's number is , how many joules are released by the fusion of grams of deuterium?

  3. One gram of TNT can release kilocalories of energy, each of which is equivalent to joules. How many tons of TNT are required to release the energy calculated above?

  Here is another application of Einstein's equation . Be sure you are familiar with scientific notation for very small and very large numbers before trying to solve this, and be sure to check all steps of the calculation. The Sun loses mass all the time, by at least two mechanisms.

  First, it radiates sunlight energy , and by the equivalence of energy and mass, the process must also reduce its mass. The energy radiated at the Earth's orbit— million kilometers from the Sun—is about Watt (the solar constant) per square meter of area perpendicular to the Sun's rays, and the velocity of light is about .

  Second, it also emits the solar wind. For reasons that after 70 years are still unclear, the uppermost atmosphere of the Sun (solar corona) is very hot, about a million degrees centigrade, explaining why atoms in that layer tend to be stripped of most or all of their electrons, e.g., iron atoms missing a dozen electrons, which requires a tremendous amount of buffeting.

  The Sun's gravity cannot hold down a gas so hot. Instead, the topmost solar atmosphere is constantly blown away as solar wind—a rarefied stream of free ions and electrons, moving outwards at about . The density of that wind at the Earth's orbit is about protons per cubic centimeter (taking into account the presence of helium ions), and the mass of a proton is about kilograms. Which of the two processes causes the Sun a greater mass loss?

  An object (e.g., a spaceship) ejected from the surface of Earth needs a velocity to escape Earth's gravity (escape velocity). A neutron has rest energy (million electron volts). If the velocity of light is (close enough) and a neutron is ejected from the Earth's surface with just enough velocity to escape gravity, what is its energy in (or in electron volts, )? Use the non-relativistic expression when deriving the kinetic energy of the escaping neutron (it is accurate enough).

  Review Answers

  Why can't one find in our environment elements whose atoms weigh times as much as the proton, or more? [Such nuclei contain too many protons repelling each other, and in spite of the strong nuclear attraction between their particles, are unstable.]

  Compile a glossary, defining briefly in alphabetical order in your own words: alpha radioactivity

  Nuclear instability leading to the emission of alpha particles.

  beta radioactivity

  Nuclear instability leading to the emission of electrons, from conversion of neutrons to proton-electron pairs (plus neutrino).

  binding energy

  The energy holding a nucleus together—the amount needed to completely break it apart.

  controlled nuclear fusion

  Combination of light nuclei to heavier ones, in the lab.

  core of the Sun

  The central region of the Sun where energy is generated.

  curve of binding energy

  The graph of nuclear binding energy per nucleon against mass.

  daughter isotope

  An isotope resulting from radioactive decay.

  deuterium

  The heavy isotope of hydrogen, contains proton neutron.

  mass spectrometer

  Instrument to measure the mass of nuclei, by deflecting a beam of ions magnetically or timing their flight.

  nuclear fusion

  Nuclear reaction joining light nuclei to form heavier ones.

  positron

  The electron's positive counterpart (can be created in the lab).

  short-range force

  A force which decreases with distance r faster than .

  strong (nuclear) force

  A short-range attraction in the nucleus, holding protons and neutrons.

  weak (nuclear) force

  A weaker short-range nuclear force, tries to balance number of neutrons and protons.

  What is the source of the Sun's energy? [Nuclear fusion of hydrogen in the Sun's core, producing helium ]

  Why is the binding energy of the nucleus given a negative sign? [The energy of a nucleus is what is extra energy available; zero energy means all particles are independently spread out. A bound nucleus needs ene
rgy input to reach "zero energy" state, so its energy is negative. ]

  We can calculate the energy as follows.

  Mass converted to energy:

  4 gram helium contain atoms, so the energy released is,

  Let us compare the mass loss due to either process through an area of 1 square metre at the Earth's orbit, perpendicular to the flow of sunlight, during one second. Working in metres, seconds and kilograms, meter/sec, and the energy flow is joule/sec. If m is the mass lost during that time through the chosen area (by conversion to radiant solar energy)

  The solar wind passing through the same area includes all the matter contained in a column of cross section 1 and of length metres. One cubic metre contains 106 cubic centimeters and the mass of 107 protons. The flow through the area is therefore 4 1012 protons, with a mass kilograms.

  The loss due to sunlight is therefore greater by about a factor of two. Still, it is remarkable how close these two numbers are to each other - one dictated by processes in the innermost core of the Sun, the other by processes in its outermost layer. Coincidence, you say?

  9.39535times 10^{8}

  If m is the mass of the neutron, This is less than 1 eV! Radiation belt particles have energies of the order of MeV, and even electrons of the polar aurora have of the order of 10,000 eV (thermal energy of air molecules in your room is about 0.03 eV). Gravitational energy is therefore completely negligible by comparison--or in other words, the electromagnetic forces on particles in space tend to be much, much bigger than their gravitational forces.

  Fission of Heavy Nuclei

  There exists, however, another mode by which very heavy nuclei can move on the curve toward more stable states. That is nuclear fission, in which the nucleus, rather than chipping off nucleons as an particle, splits into two parts of comparable mass. The ratio of the masses of the two fragments varies, but in most cases one of the fragments is about twice as heavy as the other (see illustration).

  Figure 3.3

  Typical Nuclear Fission

  Energies of atomic and nuclear processes are measured in electron volts , the energy acquired by an electron or proton (electric charges of the same magnitude) going through a voltage drop of one volt. The is a unit appropriate for atomic processes, associated with the "chemical" binding energy of electrons. For nuclear processes, a more appropriate unit is the , million electron volt.

  Each of the two fission fragments carries a positive charge, and their mutual repulsion typically releases (#9), compared to typical energies of for —rays and for particles (further details in #10).

  This mode is known to occur spontaneously in artificial elements heavier than uranium. However, the absorption of a neutron by a suitable uranium nucleus— or —can also trigger its fission.

  A proton aimed at a nucleus, even if headed straight toward it, needs to be accelerated to a considerable energy to overcome the repelling electric force and get close enough to be captured by the strong nuclear force. A neutron, on the other hand, is not repelled and can reach its target, even if it moves quite slowly, e.g., a thermal neutron whose energy is comparable to that of molecules in ordinary matter or in air, about . Imagine the nucleus as a target of a certain size, then the nuclear cross section is the area a projectile must hit to produce a certain reaction (it is also proportional to the likelihood of the projectile sticking to the nucleus). Nuclear cross sections are measured in barns, where barn is equivalent to a target size of ("big as a barn" for nuclear physicists). The cross section for a neutron to hit a nucleus varies from one isotope to another, and with the energy of the neutron (similarly for other particles undergoing collision). For instance, the chance of a "thermal" neutron sticking to a nucleus of heavy hydrogen (the isotope or deuterium) is rather small, because that type of hydrogen already has an extra neutron.

  As a neutron reaches its target nucleus, one may visualize the nuclear attraction speeding it up, so that it hits with appreciable energy, agitating the target nucleus.

  The effects of this extra energy may vary. The target nucleus may simply emit it as a —ray photon (see end of #11), or it may undergo some internal change, e.g., the neutron may become a proton, emitting an electron (—radioactivity). But with —an isotope forming about % of natural uranium—the result is usually nuclear fission, splitting the nucleus into two fragments. The products may vary, but typically the ratio of the masses of the two fragments is close to .

  Nuclear fission was identified in Germany in 1939 by Hahn, Strassman, and Lise Meitner. (That was in Nazi Germany—Hahn was awarded the Nobel Prize in 1944, and his long-time associate Meitner, who was Jewish, was lucky to escape to Sweden). Very soon, physicists all over realized that the process could provide usable energy. Not only did it release appreciable energy per nucleus, but more important, it also released additional neutrons, making possible a self-sustaining chain reaction.

  The Chain Reaction

  As already noted, the weak nuclear force tries to adjust the numbers of protons and neutrons in a nucleus to approach equality. However, because protons are positively charged and repel, confining them to the tiny nucleus requires energy, and that shifts stability to a state with extra neutrons. The main isotope of iron has protons and neutrons, making neutrons % of the total. Uranium has and , or about %, and the fraction of neutrons in nuclei between these extremes is somewhere in-between, too, increasing with mass. Suppose a nucleus fissions into isotopes in which the neutron fraction in the most stable isotopes is %. The distribution in the parent nucleus actually gives them %, so that each fragment nucleus has about neutrons too many.

  Nuclei, which have a neutron or two more than their most stable isotope, may still be stable. With a greater number of extra neutrons they may adjust by -radioactivity, emitting an electron as a neutron converts to a proton. Here, however, the imbalance is so great, that a more drastic process occurs: entire neutrons are ejected. When a thermal neutron is captured in , on the average neutrons per fission are released, % of them promptly, and % delayed by a second or two. These numbers turn out to be quite important.

  The Nuclear Reactor

  To establish an ongoing chain reaction, slightly more than one neutron per fission is required by a continuing chain reaction. One would expect that with neutrons actually generated, we get a generous excess of neutrons. Actually, things are not quite that simple.

  First, neutrons that escape from the surface of the uranium fuel are wasted to the chain reaction. That means that a critical mass is needed for the reaction to proceed. A mass of uranium the size of a peanut has too little depth—too many neutrons escape it without scoring a hit (and the shape of the uranium also may make a difference).

  Second, to control the rate of the reaction, it is best to use thermal neutrons. (Nuclear reactors using fast neutrons do exist, but are hard to design and to operate, because all energy is released inside a very small volume, making heat removal a challenge. Fission bombs use fast neutrons.) Fission neutrons start with appreciable energy, and it is necessary to slow them down by repeated collisions in a moderator surrounding their source. The ideal moderator is a material not likely to absorb them, with small atoms to maximize the energy transfer: usual choices include heavy water —where is the common notation for deuterium, the heavy isotope of hydrogen (i.e., )—or very pure carbon, in the form of graphite, the stuff of pencil's lead. The oxygen in heavy water does not absorb many neutrons.

  Figure 3.4

  Fuel Rods inside a Neutron Moderator

  The fuel in a typical modern reactor is usually formed into rods (or is contained in hollow rods of stainless steel), which are thrust at appropriate separations into a pool of heavy water (in some designs, ordinary water), or into an array of holes in a core of carbon bricks (drawing). Neutrons released from a fission in one rod soon wander into the moderator and are slowed down there to thermal speed, and after a while (unless they escape or are captured) they reach another rod and initiate another fission event there.

  The extr
acted energy appears as heat which is passed to steam: The enormously fast fission fragments keep colliding with the moderator, and ultimately spread their energy around. If the moderator is water (heavy or light) it is kept under pressure to raise its temperature, because energy extraction from hot steam gets more efficient the higher the temperature is. In a solid moderator, pipes carry a fluid to remove the heat by superheated water or by another fluid. Even liquid sodium metal has been used (in fast breeder reactors)—an extremely tricky substance that bursts into flames if allowed access to air.

  Other pipes linked to the heat removal system carry high pressure steam into ordinary steam turbines (similar to those in conventional power stations) that turn electric generators. The cooled-down expanded steam is then turned back to water in cooling towers (often drawn as ominous symbols of nuclear power, though most any steam-driven power station has them) and are recycled to the reactor to pick up more heat.

  Third, as fuel is consumed, fission fragments accumulate. These are often fiercely radioactive or "hot" (letting go of neutrons makes them more stable, but instability remains), and disposing of them is a major challenge. They may remain hot for years and even centuries, and need to be stored out of contact with life and with ground water. Because radioactivity releases thermal energy (heat), initially they also need to be cooled.