Chapter 25 THE NUCLEUS • Introduction • Nuclear structure • Nuclear reactions • Nuclear fission • Nuclear fusion • Applications • Radiation in matter • Summary

INTRODUCTION Nuclear matter comprises 99.9% of the universe, it provides the energy in stars such as the Sun that is responsible for the existence of life on earth. In contrast to our knowledge of the atom, there are many unsolved questions regarding nuclear matter, we do not yet even understand fully the nuclear force let alone trying to solve quantum theory for the nuclear system. Nuclear physics encompasses a wide range of science beyond the obvious study of the basic understanding of the physics of the nucleus. These range from understanding the evolution of the universe from the first few microseconds after the Big Bang, stellar evolution, and fundamental interactions in nature. Nuclear physics has important applications to the environment, energy, medicine and art. A rudimentary knowledge of nuclear physics is necessary to understand applications of nuclear physics to science, technology and medicine.

NUCLEAR STRUCTURE Constituents The nucleus is composed of neutrons and protons. A given nuclear species has N neutrons and Z protons. The neutrons and protons have nearly identical masses,  2 ≈ 931  and they are called nucleons. There are A = N + Z nucleons in a given nucleus. Rutherford suggested that a neutral particle, the neutron, could be formed as a combination of a proton and an electron. Ten years later, in 1932, Chadwick discovered the neutron which is a fundamental particle like the proton. An isolated neutron decays into a proton plus an electron with a half life of 15 mins. Although nucleons are composite particles composed of quarks, as will be

Figure 1 Radial shape of the nucleus. discussed next lecture, nuclear matter, under normal conditions, behaves like a system of nucleons bound together by the strong nuclear force. This nuclear force is strong and attractive between all nucleons, with a short range of about 1.4 fm. This is in contrast to the much weaker long-range Coulomb repulsion between protons. Nuclear shapes In 1911 Rutherford, Gieger and Marsden made the first measurement that the nucleus has a radius of a few 10−15  Modern high-energy electron scattering, using a de Broglie wavelength below 10−16  has mapped out the radial shapes of nuclei. Figure 1 shows the radial shape of a typical nucleus, it has a radius  = 12 1  3   where 1  = 10−15  that is, the volume is proportional to . The nucleus is remarkable in that it has a constant density of nucleons, about 017 nucleons per cubic fm. Nuclear matter behaves in some respects like a quantal fluid. My group has measured the angular shapes of nuclei, finding some are spherical, some football shaped, others are like flattened oranges and most recently we have measured pear-shaped nuclei. Recently cigar-shaped nuclei were found with an axis ratios of 2 : 1 and we are looking for nuclei with axes ratios of 3 : 1. These extremely-deformed shapes only occur in excited nuclear states. Nuclear binding Nuclei are bound by the attractive nuclear force. The nuclear force is the same between neutrons, protons and neutron-proton pairs. The nuclear force is strong and attractive, and has a short range of about 1.4 fm. The binding energy of a nucleus of mass A,   can be calculated using measured nuclear masses, that is:  = ( +   −  )2 191

Figure 2 Binding energy per nucleon versus mass number. This binding energy, divided by the number of nucleons, is roughly the same in all nuclei, that is, a binding energy of about 8 MeV per nucleon, as shown. Any long range force would predict that this binding energy would increase as (−1) because each particle would 2 interact with all the other A-1 nucleons. The factor of 12 occurs to avoid double counting of pairs of nucleons. The actual nuclear force is roughly constant because the nuclear force is short range and thus it only feels the immediate neighbours, the addition of more nucleons does not effect the binding of a particular nucleon. Note that the binding is greatest for nuclei with  ≈ 60. For heavier nuclei, the Coulomb repulsion between the protons opposes the nuclear attraction sufficiently to reduce the binding per nucleon. This is important when we discuss nuclear fission, fusion and  decay. The extra binding of 4 , 16  is a manifestation of shell structure in nuclei. Since protons and neutrons are distinguishable fermions, the Exclusion Principle implies that it is possible to place both one proton and one neutron in any nuclear state with a given set of quantum numbers  . Thus the most bound state for any mass number should have equal numbers of protons and neutrons which is true for light nuclei. However, due to the Coulomb repulsion between the protons, heavier nuclei are neutron rich. There is a valley of stability if one plots the binding energy versus neutron and proton numbers as shown in figure 3. The sides of the valley reflect the fact that there is less binding for the non-optimum ratio of protons to neutrons due to the unequal nucleons having to be placed in less bound orbits to satisfy the exclusion principle. The binding energy plot, figure 2, is the binding energy along the valley of stability. Only a few hundred nuclei exist naturally, lying along the valley of stability as illustrated in figure 4. Nuclei away from the valley of stability decay down to the most bound nuclei along the valley of stability. With the aid of accelerators we can produce about fifty times the number of nuclei that exist naturally. Accelerators are being built or proposed to make radioactive nuclei far from the valley of 192

Figure 3 Segre plot of the binding energy of nuclei, vertical axis, versus proton and neutron number. Stable nuclei only exist along the bottom of the valley. Unstable nuclei decay down to the bottom of the valley by alpha, beta, or fission decay.

Figure 4 N-Z plot of stable nuclei. Closed shells occur at N or Z equal to 2,8,20,28, 50, and 126. Nucleosynthesis in stars and supernovea follow the rp or r-processes. The black spots designate stable nuclei, the fawn color, nuclei that are easily made by nuclear reactioons and the green area is the terra incognita that we now are starting to explore using radioactive beams.

stability. Nuclei far from stability may exhibit strange new phenomena. Recently nuclear physicists made, for the first time, nuclei with Z = 109-116. The element  = 114 has been named Flerovium after the Russian physicist Georgy Flerov, and  = 116 has been named Livermorium after the Livermore National Laboratory who collaborated with the Russians in the discovery and identification. They are looking for a new region of quasi-stable superheavy nuclei corresponding to the next higher closed shell. The understanding of why stable nuclei exist only between hydrogen and lead requires a discussion of nuclear reactions. My group is searching for exotic nuclear phenomena in very neutron rich exotic nuclei far from the valley of stability. Nuclear synthesis of the elements in stellar evolution still has many open questions regarding nuclear reactions and decay processes that still are not fully understood.

Figure 5 Nuclear potential well.

Nuclear structure A quantal liquid differs significantly from a classical liquid. Nuclei behave both like liquid droplets with collective degrees of freedom, that is these deformed shapes rotate and vibrate like molecules. Simultaneously, the nucleus also behaves like a gas of weaklyinteracting nucleons bound in some average potential well due to the combined attraction of the other nucleons. Because of the Pauli Exclusion Principle, the mean free path of nucleons is surprisingly long leading to single-particle-like motion of the weakly-interacting nucleons analogous to electrons in the atom. Thus one sees single-particle motion and closed shells just like in atomic structure. In fact we can quite successfully calculate the properties of low-lying nuclear states assuming a model of weakly interacting nucleons bound in a potential whose binding energy is proportional to the nuclear density. This is called the shell model. Although similar in principle to atomic structure, the nucleus shell model is quite different from the atomic case. The nuclear potential well, figure 5, is the average mean field derived from the nucleon-nucleon forces as opposed to the atom where the potential is due to the charge of the nucleus. In addition, the residual forces between the valence nucleons is attractive in contrast to the Coulomb repulsion acting between valence electrons. The Pauli Exclusion Principle allows two protons plus two neutrons in a each orbit. Thus the first closed shell is 4 He which has two neutrons plus two protons in the lowest 1s orbit. The next closed shells occur at neutron or proton numbers of 8, 20, 28, 40, 50, 126, which differ in number from atomic closed shells. Especially strongly-bound closed shell nuclei are 42 2  16 40 58 90 208 8 8  20 20  28  30  40 50  82  126  As illustrated in the figure 6, in contrast to the simple shell structure in closed shell nuclei, almost all non-closed shell nuclei exhibit level structure charac-

Figure 6 Typical level spectra for a closed shell nucleusand a strongly-deformed nucleus. teristic of rotation and vibration of strongly-deformed quadrupole shaped objects. For a rotational band one observes excited levels with excitation energies of  = ~2 2= ( +1) up to spins   60 where = is the moment of inertia of the rotating spheroidal nucleus. These states deexcite by a sequence of 30 or more -rays within about a picosecond.

NUCLEAR REACTIONS The potential energy for a nucleus is the sum of the short-ranged nuclear binding and the long-ranged Coulomb repulsion. This leads to a radial potential well shown in figure 7. Bound charged particles in the nucleus are at a positive energy relative to the total energy at large separation distances. Classically, nuclear states lying below the top of the Coulomb barrier cannot decay through the barrier. However, George Gamow, who wrote ”Mr Tompkins in Wonderland” showed that for a quantal system, the wave nature leads to the wave function extending into the classically impenetrable barrier region with an exponential decay. If the 193

Figure 7 Quantum tunnelling through the Coulomb barrier barrier height and width are sufficiently small, then the exponential decay of the nucleon wavefunction inside the barrier can lead to a non-zero amplitude of the wavefunction at the outer boundary. In this situation, the square of this tail of the nuclear wavefunction at the outer surface of the Coulomb barrier gives the probability that the nucleon can tunnel through the potential barrier. Nucleons bound in the nuclear well bounce off the walls about 1021 times per second, thus even if this tunnelling probability is very small, e.g. 10−21 , the nucleon could leak out in about one second. It is ironic that it is the Coulomb potential that reduces the binding energy per nucleon of heavy nuclei, and thus favours decay of heavy nuclei via either fission or  decay, at the same time provides the barrier hampering such decay. The tunneling probability falls off extremely rapidly with barrier height and width. Nuclear ground states of nuclei typically decay via  decay, that is emission of a 4  nucleus,  decay, that is electron emission which changes the charge of one of the A nucleons, or nuclear fission in which the nucleus splits into two medium-mass daughter nuclei. The decays of ground states of nuclei lying along the valley of stability below 208 Pb have essentially infinite lifetimes because the energy gain for -decay or fission is small and the barrier penetration absolutely negligible. However, for nuclei heavier than 208 Pb, the nuclei barrier penetration becomes much more probable resulting in fission or  decay of such nuclei. This is why there are no naturally existing elements heavier than uranium,  = 92. Nuclear physicists have made elements up to  = 116 that live long enough to be detected because shell-model structure leads to additional binding that significantly extends the lifetime. The decay of heavy nuclei depends sensitively on the barrier penetration and binding energy for different decay channels. Thus adjacent nuclei can have very different decay lifetimes and properties. The Coulomb barrier that hinders nuclear decay, equally hinders the inverse process of fusion of nuclei. Nuclear physicists need accelerators to provide bombarding energies exceeding 5 MeV/nucleon in order to 194

Figure 8 A photograph of the downstream half of the 4 heavy-ion detector CHICO and the right half of the 4 -ray detector Gammasphere.

overcome the Coulomb repulsion to induce nuclear fusion. For example, I used a 1250  208   beam to study the structure of a 248  target. At lower bombarding energies the nuclei never approach close enough for nuclear interactions via the short-ranged nuclear force. Either a cyclotron or a superconducting linear accelerator are used to accelerate the heavy ions. In my research we observe the scattering angles and time of flight of the scattered heavy ions or charged reaction products using the Rochester CHICO 4 array of heavy ion detectors. The deexcitation -rays are detected in coincidence with the scattered ions by the world’s most poweful 4 array of high-resolution Ge detectors called Gammasphere which is shown in figure 8. The binding energy of nuclei along the valley of stability is replotted in figure 9. Note that the most stable nuclei are for masses near nickel. Thus one can gain energy by either fusing very light nuclei as done in nuclear fusion in stars, or by breaking heavier nuclei into lighter fragment, that is by alpha decay or by fission. Alpha decay occurs for heavy nuclei. For example figure 10 shows the decay path for 232 Th. The first decay is an alpha decay followed by two beta decays to fall back to the valley of stability. Then you get a sequence of 4 alpha decays before the next sequence of beta decay.

Figure 9 The total binding energy per nucleon along the valley of stability. Note that the greatest binding is for nuclei around nickel.

Figure 11 Schematic illustration of nuclear fission. (a) The absorption of a neutron by 235  leads to (b) 236  in an excited state, which is unstable to shape oscillations (c) and subsequent fission into two medium mass nuclei and three neutrons.

NUCLEAR FISSION

Figure 10 The alpha and beta decay path for

232

 

Nuclear fission, shown schematically in figure 11, is a process where nuclei split into two massive charged fragments releasing around 200  of energy. The resultant mass distribution for fission of 236  peaks for asymmetric fission as shown in figure 12. Many hundreds of fission products are produced as shown in figure 13. Unfortunately many of these reaction products are radioactive. Nuclear fission was discovered in Nazi Germany in by Hahn, Meitner and Strassmann. They identified barium nuclei (Z=56) from fission of uranium due to neutron bombardment. Fortunately, Frisch, from England, found out about these results from his aunt Lise Meitner. He told Niels Bohr who realized the importance of this discovery. Heavy nuclei are much more neutron rich than are lighter nuclei. Thus, in nuclear fission, several neutrons can be released when nuclei split into two smaller less neutron-rich fission products. Leo Szilard had realized that if an element could be found that gave off two neutrons from absorption of one then this could lead to a chain reaction as illustrated in figure 14. In fact a neutron plus 238 U emits 3 neutrons when it fissions plus releasing about 200  of energy. Because war was imminent, Szilard, Wigner and Einstein drafted the famous letter to Roosevelt pointing out the military potentialities of fission. The first demonstration of a nuclear chain reaction in a fission reactor was done by a group led by Enrico Fermi in 1942 under the stands at the Univer195

Figure 15 Picture of the first nuclear reactor assembled under the stands at the Unversity of Chicago Stagg Field.

Figure 12 Fission fragment mass distribution following fission of 236 

Figure 16

Figure 13 Contour map of proton and neutron number of the fission fragment distribution for fission of 252 

Figure 14 The nuclear chain reaction for neutroninduced fission of 235  196

sity of Chicago football field as illustrated in figure15. The most abundant isotope of uranium with mass 238 does not naturally fission but there is 0.7% mass 235 which does naturally fission. The neutrons emitted are slowed down to thermal velocities by a moderator of carbon or other light nuclei because the cross section for neutron capture in uranium is orders of magnitude larger at thermal velocities. These thermal-velocity neutrons are captured by the uranium leaving the resultant nucleus highly excited leading to rapid fission into two smaller mass fragments plus three neutrons. Control rods of cadmium, which have a very large cross section for capture of thermal neutrons, are used to keep the gain of neutrons close to unity. In a nuclear bomb one allows the neutron gain per fission to be much larger than unity leading to an uncontrolled chain reaction. A typical nuclear power reactor is illustrated in figure 16. Nuclear reactors are much more efficient energy producers than cold-burning power stations. Most of the energy in nuclear fission comes from the different Coulomb energies between the fission products and the fissioning nucleus, typically about 200 MeV per fission. That is, even the energy from fission comes primarily from the Coulomb force. By comparison, coal involves

burning carbon to form CO2 , which releases an energy of 4 eV per molecule. A nuclear power station converts about 10−3 of the mass used into energy whereas a coal-powered station converts 10−10 of the mass to energy; a difference of 107 in efficiency. Thus much more material has to be mined for coal. Note that annihilation of antimatter releases 100% of the rest-mass energy, and thus this would be the ideal source. But how do you contain anti matter without the danger of an enormous explosion. The problem with fission reactors is that many of the fission products are radioactive, some of which decay quickly while others live for thousands of years. Thus careless disposal of radiation products can be detrimental to future generations just as is the case of chemical and biological hazards also being produced at this time.

NUCLEAR FUSION The plot of binding energy per nucleon, Figure 9, shows that one can gain considerable energy by fusion of light nuclei to form 4 He or slightly heavier nuclei. One needs to overcome the Coulomb barrier to fuse nuclei so barrier penetration again is applicable. It takes at least one MeV of kinetic energy to make the barrier penetration probability significantly large to lead to fusion. Nuclear physicists use accelerators to overcome the Coulomb repulsion. Fusion of light nuclei in the interior of stars provides the energy emitted by stars, as explained by Hans Bethe, at Cornell. To achieve fusion, with sufficient energy density for a power plant, requires heating a gas to at least 107 K to provide sufficient kinetic energy to achieve barrier penetration. In the interior of the Sun the high gravitational field and high temperature provide the density and kinetic energy needed for sustained thermonuclear fusion. About 50% of the Sun’s energy comes from the reaction; 3

 +3  =4  +  +  + 13 

This reaction has been studied using accelerators, while a fission bomb provides the required conditions to fuse tritium in a thermonuclear explosion. A practical thermonuclear power source needs to be less destructive and better controlled. Both plasma heating and laser heating are being tried, the latter method, being developed at the Laser Laboratory. Currently no system having both the required high density and temperature has been produced, and a practical power station appears to be decades away. The advantage of fusion is that there are no long-lived radioactive products although a large quantity of hazardous tritium is needed. In 1989 two chemists claimed to have achieved fusion in an electrochemical cell at room temperature.

This is nonsense as you will discover when doing problem 4 of homework 11.

APPLICATIONS There are many applications of nuclear radiation in science, technology, medicine and art. Examples of applications to medicine are X-rays including CAT scans, nuclear cardio stress tests using radioisotopes 99 Tc or 201 Tl, cancer treatment using radioisotopes, photon and particle-beam therapy, positron emission tomography, and nuclear magnetic resonance imaging. Radiation is used for safety and national security such as large-scale X-ray or neutron scanners at airports, food sterilization, and arms control. It is used for energy production in nuclear reactors as well as for oil-well logging. Accelerator Mass Spectrometry, pioneered at the University of Rochester Nuclear Structure Research Laboratory, has a broad range of applications to art, archeology, and the environment.

RADIATION IN MATTER If you intend to practice medicine then you will frequently encounter the impact of radiation on matter. Therefore I will give you a brief survey of the general features of what happens when radiation interacts with matter. Charged particles When a charged particle traverses matter it loses energy mainly through interactions that ionize atomic electrons along the path traversed by the charged particle. Bragg derived the rate of energy loss for a heavy ion illustrated in figure 17. Below 4 MeV per nucleon, the energy loss increases with bombarding energy because the moving ion is only partially ionized and thus loses more bound electrons with increased velocity increasing the ionization. Above the Bragg peak the ion has lost most electrons and the energy loss is propor2 tional to  where E is the energy of the moving ion. The features of the Bragg curve make heavy ions useful for selectively destroying cancer cells since it is possible to arrange that most energy is deposited at a given depth with minimum damage at other depths as illustrated in figure 18. Heavy ions in solids typically stop within distances ranging from sub millimeter to a few mm. Neutrons Neutrons are uncharged so they do not ionize matter directly. Neutrons lose energy by elastic collisions with 197

Figure 17 Energy loss of a heavy ion traversing matter.

Figure 18 Energy loss of helium and neon ions in water versus penetration depth. charged nuclei or by nuclear reactions. Neutrons scattering from hydrogen is the most effective means of slowing neutrons since the neutron and proton masses are almost the same. The recoiling protons ionize the surrounding matter. Neutrons do not have a definite range, the neutron intensity falls exponentially; they can penetrate many centimeters of matter. Nuclear absorption of neutrons is very high for neutrons at thermal velocities in certain materials such as cadmium. This is exploited to absorb neutrons to control nuclear reactors..  rays Energy loss for  rays traversing matter occurs due to the photoelectric effect where a photon is absorbed and a bound photoelectron is ejected by an atom, Compton scattering where a bound electron is scattered by collision with a photon, and pair production where an electron-positron pair is created. In all three cases the recoiling electrons ionize the surrounding matter. The photon beam intensity in solids decays exponentially with depth with a half distance ranging from mm to many cm of matter depending on the -ray energy. Radiation dosage Unfortunately there is considerable hysteria and paronia regarding the hazards of nuclear radiation. Since 198

you are likely to encounter applications of radioactivity to science or medicine it is worthwhile to present you with some relative numbers regarding environmental hazards. The danger of radiation is not a black and white issue, the way it often is depicted by industry and environmental groups alike. There are two system of units used for radiation dosage, the common units and the SI units as summarized in figure 19. It is necessary to take into account both the energy deposited by the radiation and the relative biological effectiveness of different types of radiation. That is, the relative biological effectiveness factor accounts for the fact that the damage depends on both the amount of energy deposited and the energy density. The RBE is normalized so that the biological damage for  rays and  particles is unity. The RBE is about 4 to 10 for neutrons, and 20 for heavy ions. The recommended maximum dosage given in the table are defined to keep genetic damage to a minimum. Substantially higher dosages are required to observe physical damage. From studies of victims at Hiroshima and Nagasaki it was determined that 100 rem (1 Sievert) causes some blood tissue damage and death is likely at 5 Seivert. The probability of cancer doubles at about 2 Seivert. It is important to put radiation exposure in perspective by considering the exposure due to natural sources of background radiation from cosmic rays, and activity from material in and around the body. Figure 20 compares various sources of radiation in millirem/year. To put these in perspective consider two amusing damage equivalents to non-nuclear processes. The genetic damage from drinking 1 ounce of alcohol is equivalent to 140(14). The genetic damage to a male from wearing of pants for one year is equivalent to 1000(10). Actually risk estimation is complicated. For example, a scotsman wearing a kilt may reduce heat induced genetic damage but Scotland is mainly granite which gives 70 of natural activity. Estimates are that a coal-powered electrical plant will cause between 10 to 300 times more deaths than a nuclear plant. The problem is that people associate nuclear radiation with the bomb whereas the much more insidious hazards from chemical plants or coal-fired power plants are accepted. The largest nuclear reactor disaster occured at Chernobyl killing 31 people and probably causing some genetic damage to a many more people. By comparison, the chemical plant disaster at Bopal killed over 3300 people due to a methyl isocyanate spill and probably a much larger number of people suffered non-fatal injury. Clearly that there are advantages and disadvantages of nuclear power and I am neither a proponent nor an opponent of nuclear power. Safe disposal of radioactive waste is non-trivial and must be controlled

Figure 19 Radiation units

Figure 20 Typical radiation dosages carefully. The arguments for and against nuclear power are much more complicated than people admit. Evaluation requires risk estimation of the consequences of each activity involved such as mining coal, uranium, acid rain, the greenhouse effect, etc. All energy sources pose serious environmental hazards, especially burning of fossil fuels. Be careful to maintain a balanced view in evaluating alternative sources of energy. Conservation clearly is the safest approach. Recommended maximum radiation dosage Exposure limits millirem/yr milliSv/yr General population 500 5 Radiation worker 5000 50 Pregnant woman 500 5 Student 100 1

SUMMARY Nuclear physics encompasses a wide range of science beyond the basic understanding of the physics of the nucleus. These range from understanding evolution of the universe for the first few microseconds after the Big Bang, astrophysics, and fundamental interactions in nature. There are many applications to technology, energy, the environment, medicine, national security, and art. Reading assignment: Giancoli, Chapters 42,43

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