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Chapter 8 The Atomic Nucleus. § 8-1 Introduction The atomic nucleus The atomic nucleus is a very small,dense object made up of two kinds of nucleons( 核子 ):protons( 质子） and neutrons （中子 ).
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Chapter 8 The Atomic Nucleus • § 8-1 Introduction • The atomic nucleus • The atomic nucleus is a very small,dense object made up of two kinds of nucleons(核子):protons(质子）and neutrons（中子). • A proton has a positive electrical charge equal in magnitude to the electronic charge and a mass about 1840 times that of an electron. • Neutrons are about 0.1 percent more massive than protons.As their names suggests,they bear no electrical charge.
Atomic number and mass number • A nucleus is specified by its atomic number(原子序数）Z and its mass number A. • Z is the number of protons and A is the total number of nucleons.So the neutron number N equals A-Z. • The standard notation for nuclei is illustrated by .This nucleus has 238 nucleons,of which 92 are protons and 238-91=146 are neutrons.U is the chemical symbol for the 92nd element uranium. • Sometimes the atomic number is omitted,since it is implicitly given by the name of the element.The notation U-238 is occasionally used when one needs to specify the isotope as well as the element. • Nuclides(核素） and isotopes（同位素） • Nuclear species are usually called nuclides.There are about 2000 nuclides in all including more than 300 stable natural
occurring nuclides and slightly over 1600 artificially produced radioactive nuclides. • Nuclides which have the same atomic number but different neutron numbers are called isotopes.Because the electronic structure of atoms depends mainly on the total positive charge of the nucleus,different isotopes of an element are nearly identical chemically. • For example, and are isotopes of the element uranium.Their chemical and physical properties are almost the same.But they are different nuclides with entirely different nuclear properties.While is the principal material for nuclear weapons, is in most cases waste of nuclear power station. • Forces in nuclei • Three distinct types of forces play important roles in nuclei:
The strong force（强力）,the electromagnetic force and the weak force(弱力). • Nuclei are held together by very strong,short ranged nuclear forces among the nucleons referred to as “the strong force”. • Experimental evidence indicates that the forces which bind neutrons to protons,protons to protons and neutrons to neutrons are essentially the same in magnitude,so they are charge independent. • The operative distance of the strong force is only a few fermis.(1 fermi= m) • Electrical forces, or more generally electromagnetic forces,are smaller in magnitude,but they become progressively important as the number of protons in the nuclei increases.
In addition to the strong binding force,which must be far larger than the Coulomb force of repulsion between charged particles,experimental evidence indicates that there is a third force,a short- ranged force far weaker than the nucleon binding force,which is sometimes called the weak interaction.The weak interactions are really quite weak,because they are even much weaker than the electromagnetic interactions,but they are responsible for the beta decay processes in which,for example,neutrons in nuclei are converted into protons as they emit electrons and neutrinos.(中微子）
§ 8-2 Radioactivity (放射性） • The discovery of radioactivity • Henri Becquerel accidentally discovered radioactivity in 1896,15 years before Rutherford inferred (推断）existence of the nucleus. • Becquerel noted that uranium compounds produces invisible rays or radiation that can penetrate an opaque （不透明的）container and expose （暴光）a photographic emulsion（感光乳剂）. • Soon thereafter,Pierre and Marie Curie showed that uranium ores (矿砂）also contain traces of polonium (钋,Z=84) and radium（镭,Z=88) ,both much more intensely radioactive than uranium. • Many other radioactive nuclear species or radio nuclides (放射性核素）were subsequently found. • Alpha,beta and gamma rays
Three type of rays are emitted by natural radioactive substances,alpha,beta and gamma rays. • Measurement of the ratio of charge to mass of beta rays and their direction of deflection(偏转）in a magnetic field soon led to their identification as electrons traveling at high speed. • In 1903 by somewhat similar methods Rutherford in England identified the alpha particle as a “double charged helium atom”.That it was actually the same as the nucleus of a helium atom did not become clear until after the formulation(提出）of the concept of the nuclear atom. • The gamma rays were discovered in 1900 by Villard. Because they were found to pass through the strongest magnetic field un-deflected it was not clear for some years whether they were uncharged particles or a form of wave motion,
Now we recognize that gamma rays are electromagnetic waves of very short wavelength,even shorter than ordinary x-rays,and consequently have a very high penetration power although like x-rays they may also have particle-like properties(photons). • Whereas alpha rays can usually be stopped by a few sheets of paper and beta rays by a few millimeter of aluminum it may require several centimeters of lead or iron to reduce the gamma rays from a radioactive substance to a safe level. • The energies of alpha,beta and gamma radiation are as much as several million electron volts (MeV) per particle. • Since atomic and molecular processes typically involve energies of a few electron volts,radioactivity represented a total new kind of phenomenon and suggested the existence of forces much stronger than electrical forces.
Radioactive decay and half-life • Many experimental measurements show that,wherever large numbers of atoms are dealt with,all radioactive substances follow the same decay pattern. • At the end of a certain length of time half the number of atoms present will have decayed.For one isotope this time period or half-life may be a fraction of a second,for another isotope it may be thousands of years or more. • At the end of a second time interval equal to the first,half the atoms left of a given isotope of an element will decay. • Such points plotted on a graph give a logarithmic decay curve(Fig.8.2.1 or Fig.13-3),represented by the equation • (8.2.1) • where N is the number of atoms that have not yet decayed • After time t,and is the number of atoms at t=0.
The quantity λ is called decay constant（衰减常数)and is characteristic of the particular atomic isotope undergoing decay. • Derivation of the decay equation • If the number of atoms decaying in a small time interval Δt is ΔN, and if the number is proportional to the time Δt, then –ΔN~Δt. • The minus sign serves to indicate a decrease in total number.The number decaying is evidently proportional to the number N present at the beginning of the time interval,so we have • ΔN = -λΔt (8-2-2) where λ is the proportionality constant. • In the limiting differential form, • (8-2-3) or • (8-2-4)
Eq.(8-2-4) may be readily solved by integration to give • (8-2-5) • When t is zero,the constant of integration C=ln where is the number of atoms at t=0. • Therefore • and (8-2-6) • From eq.(8-2-4),it can be seen that λ is the fractional number of atoms that decays per second. λ may also be interpreted as the probability that a single atom will decay in one second. • The larger λ is,the more rapidly the element decays and the shorter will be its half-life. Evidently λ and the half-life T of an element are closely related.
The relationship between T and λ • To find the relation we put N=N/2.Then we solve the equation for t=T,the half-life. • The equation then becomes or ln 2 =λT • and T = ln 2/λ=0.693/λ (8-2-7) • [example] • Iodine131（碘）is used in the treatment of thyroid(甲状腺）disorders. Its half-life is 8.1 days.if a patient ingests(摄取）a small quantity of and none is excreted(排泄）from the body what fraction remains after 8.1 days,16.2 days,60 days? • Answer: Since 8.1 days is the half-life,the fraction remain at this time is 1/2.Similarly,16.2 days is 2T,so ¼ remains. Sixtyis not an exact integer multiple of T, so
to find the fraction remaining we use the exponential decay formula • Thus only 0.59 percent of the radioactive iodine remains after 60 days. • The assumption made in the above example that no is lost from the body by biological processes is not quite correct, is excreted steadily but slowly with a biological half-life of 180 days. • Thus if non-radioactive iodine were ingested,only ½ would remain in the body after 180 days,and so on. • The effective half-life T is obtained by combining the biological half-life T and the radioactive or physical half-life T according to the formula • (8-2-8)
[example2] • is administered to a patient to diagnose(诊断）blood anomalies.(异常）its T =65 days and t=46.3 days. Find effective half-life. • Solution: According to eq.(8-2-8), • =27 days. • Note that the effective half-life is shorter than either the biological or physical half-life.This happens because both processes are depleting(减少，消耗）the supply of the radio nuclides.
§ 8-3 Nuclear Sizes,Nuclear Masses and Binding Energies • Nuclear sizes • Beginning in 1907,Rutherford conducted a series of experiments in which he bombarded various atoms with alpha particles and he found that an atom contains a small positive nucleus with a radius of less than m, which is about times the radius of an atom. • From further experiments with alpha particles,nucleons ,and other projectiles,(抛射体）considerable information has emerged about the spatial distribution of matter in the nucleus. • Roughly speaking,a nucleus containing A nucleons is a uniformly dense sphere of radius
The radius is proportional to the cube root of the mass number and therefore increases very slowly. • For example,the radius of is 4.2 fm,the radius of is 5.6 fm and is the largest naturally occurring nuclide with a radius of about 9 fm. • By comparison,atomic radii are about m, more than 10000 times larger . • Because the nuclear radius varies as , it follows that the nuclear volume varies as A. • Nuclear masses • The masses of manynuclei have been accurately measured with mass spectrometers(质谱仪）. • A little arithmetic(算术）shows that the mass of a nucleus is less than the sum of the masses of its constituents.
For example,6mp+6mn+6me=12.0989 u,while C atom has a mass of only 12.000 u.This mass defect(质量亏损）tends to increase with the mass number A. • The significance of the mass defect is made clear by Einstein’s principle of the equivalence of mass and energy(质能相当原理）. • For an object at rest (8-3-1) • In words,a mass m of matter can be converted into an amount of energy,E;the quantity C is the speed of light in a vacuum. • The mass of a nucleus (e.g. ) is less than that of its constituent nucleons because it is a bound system.One must supply energy equal to the nuclear binding energy to pull apart protons and neutrons.
According to Einstein’s principle this energy is equal to the mass defect time C. • To relate the mass defect and binding energy,we first calculate the energy associated with a mass of • So,effectively,1u=931MeV. • The mass defect of 0.0989u corresponds to a total binding energy of 92.1 MeV. Dividing by the mass number A=12,the binding energy per nucleon in is 7.7MeV. • The bind energy per nucleon for the stable nuclei is plotted versus A in Fig.8.3.1(Fig.30.8).It is about 8 MeV per nucleon except for the lightest nuclei.
There is a broad maximum in the region of medium size nuclei,with a peak of 8.8 MeV per nucleon at • At about A=100,the curve gradually declines,reaching 7.7MeV per nucleon for uranium. • (Fig.8.3.1)(Fig.30.8)
The initial increase and later decrease in the binding energy per nucleon can readily be explained. • The strong nuclear forces among the nucleons that hold the nucleus together have a very short range:these forces are zero at distances greater than a few femtometres. Accordingly a nucleon is attracted only to its closest neighboring nucleons. • A nucleon near the surface has fewer neighbors than one in the interior of the nucleus and is less tightly bound.This surface energy effect implies that the binding energy per nucleon will rise as the nucleus increases its size and proportionally fewer nucleons are near the surface.This explains the initial rise in Fig.8.3.1.To explain the decrease in the binding energy per nucleon for large A,we must take into account the fact that electrical repulsions among the protons are proportional to the number of proton pairs.The total potential electrical energy due to the proton charges
varies as , where R is the nuclear radius .This energy grows rapidly as the number of protons increases. • In the region near ,the changes in the surface and electrical energies are roughly equal in magnitude but opposite in sign.Above A~100,the electrical repulsion gradually outstrips the surface effects,leading to the observed gradual decline in the binding energy per nucleon. • Fission and fusion(裂变与聚变） • The fact that intermediate-size nuclei have the greatest binding energy per nucleon has some important consequences. If a heavy nucleus splits or fissions into two intermediate-size nuclei,the binding energy increases by close to 1 MeV per nucleon. • The extra energy is released as kinetic energy of the fission products or as gamma rays. • Similarly, if two very light nuclei such as and combine, this fusion releases several MeV.
§ 8-4 Nuclear Models • Nuclear models • In the attempt to explain nuclear structure a number of different models have been suggested each with its successes and inadequacies. • The existence of energy levels in the nucleus was first indicated by the fine structure of alpha rays.To explain these it was suggested that nuclear particles were in orbits or shells inside the nucleus much like the electrons outside the nucleus. • This early shell model ran into difficulties and was displaced for a time by the liquid drop model of Niels Bohr,which had considerable success especially in describing the fission of a heavy nucleus.Since then we have had the single-particle shell model,and various other
Models such as the collective model and the optical model. • Liquid-drop model of nucleus • In 1936 Bohr proposed that the nucleus with its particles might be expected to behave very much like a droplet of some liquid in which the forces of attraction and repulsion between particles in the liquid are balanced. • In such a “liquid droplet”,the nucleons are presumed to be closely packed together and in a state of continual thermal agitation,moving in various directions with random motions. • Emission of nucleons from such a nucleus is then considered similar to evaporation of molecules from a liquid droplet. • If a high-energy particle has been captured by the nucleus,the nucleons in the newly formed “compound nucleus” quickly share the energy.For any nucleon to then escape,a considerable portion of this shared energy must
be re-concentrated on it. • Bindingenergy of nucleons according to the liquid-drop model • On the basis of this model it has been possible to account fairly well for the total binding energies of nucleons. • As a first approximation we may write an equation for the binding energy E • (8-4-1) • where the k’s are proportionality constants. • The first term is the part of the energy resulting from the attraction between nucleons.It is taken to be proportional to the number of nucleons A or,what amounts to the same thing,the volume of the nucleus. • In the second term the effect of Coulomb repulsion of protons is negative and is approximately proportional to , the square of the number of charges.Actually it is
proportional to the number of proton pairs Z(Z-1)/2!.it is also inversely proportional to the diameter of the nucleus which is known to vary approximately as . • The third term represents the effect of “surface tension”.It is proportional to the area of the nucleus or to A,and it is also negative,representing a decrease in binding energy,because surface particles on the average interact with only half as many particles as those inside the nucleus. • In a more complete equation other terms may be required to take account of the surplus of neutrons over protons and of the effect of an odd or unpaired nucleon. • Because Eq.8-4-1 is derived neither wholly from theory nor wholly from experiment it is called a semi-empirical formula. • If we divide both sides of Eq.8-4-1 by the number of nucleons A in a nucleus we find that the average binding energy per nucleon is
(8-4-2) • The first term on the right is approximately constant regardless of atomic number A.This is plotted as the upper dashed line of Fig.8.4.1 for different atomic numbers.The other two terms of Eq.8-4-2 are negative(plotted below the zero axis of Fig.8.4.1) and tend to reduce the net binding energy per nucleon. (Fig.8.4.1)
The Coulomb term acts to decrease binding energies as the atomic number increases. The surface energy term on the other hand is of greatest importance for low values of atomic number where the ratio of surface to volume is largest and may account in part for the low average binding energy at the lower end of Fig.8.4.1. • Adding the ordinates of the three curves gives the full line in the figure which represents the total energy per nucleon. • Except for minor deviations this curve gives a fair approximation to the experimental curve of Fig.8.3.1
Other applications of the model • One of the most useful applications of the droplet model is to the phenomenon of nuclear fission where a nucleus divides into two more or less equal parts in very much the same way as a droplet of water or other liquid, if set vibrating with sufficient energy, becomes unstable and breaks into two or more droplets. • However, for higher-energy nuclear reactions, of a few hundred MeV or more, the liquid-drop model becomes less and less useful. • A high-speed particle, for example, may pass through the nucleus and hit only one or two nucleons in the process or none at all. Such a picture is radically different from what would be expected to happen at lower energies. • Furthermore, the properties of excited states or energy
levels in the nucleus are not easy to explain by means of the liquid-drop model. • These difficulties have led to the development of another nuclear model. • The shell model (individual particle model) of the nucleus • It is difficult if not impossible to give a satisfactory explanation of excited states of nuclei and of sharp defined energy levels in terms of nuclear particles which are assumed to be closely packed together and to have random energy distributions, as in the liquid-drop model. • This led to a revival of an earlier shell model in a new form. M.G.Mayer of the University of Chicago and J.D.H.Jensen and coworkers of Heidelberg in 1950 independently showed that revised shell model called the
individual particle model could be constructed to successfully explain a surprising number of experimental facts with very few discrepancies. • For their success the two were named Noble Prize winners in 1963. • The individual particle model was so called because the particles were assumed to be sufficiently independent of one another that a particle might stay on one “orbit” for an appreciable length of time without being interfered by its neighbors. • According to the shell model, when a nucleon is deep inside the nucleus, its potential energy is approximately constant whereas near the edge of the nucleus the potential energy increases as r increases (Fig.8.4.2).Such a potential energy curve is referred to as a potential well.
(Fig.8.4.2) • The possible energy levels for a nucleon are obtained by solving the basic equation of quantum mechanics, the Schroedinger equation. Its solutions are wave functions that are oscillatory inside the potential well and exponentially
decreasing outside. The energy levels are found to fall into closely spaced groups or shells. • Since nucleons are spin ½ particles, the Pauli principle applies and two identical nucleons can not occupy a single quantum state. • Any energy level may contain at most two protons, one with spin up and one with spin down. Since there can also be spin up and spin down neutrons in an energy level, it can have at most four nucleons. • The ground state and excited states of a nucleus • The ground state of a nucleus, or the lowest energyconfiguration for a given number of protons and neutrons , is obtained by filling the lowest level with two protons and two neutrons, then the next level , and so on, until all the
nucleons are used. • A schematic diagram for three A=12 nuclei is given in Fig.8.4.3(Fig.30.11).Because of the Pauli principle, the lowest energy ground state occurs for the Z=N case, . (Fig.8.4.3)
. • Neglecting the small differences due to the electrical repulsion among the protons and to the neutron-proton mass difference, the excited state of and the ground state of • and have the same energy. • The latter two nuclei will tend to β-decay into the ground state of ,since that is a state of lower total energy. • Also, the excited state will tend to emit a γ ray and undergo a transition to the ground state. • In general, the ground state of a nucleus is found by filling the lowest states in accord with the Pauli principle.
Excited states correspond to one or more nucleons in higher states. • From our A=12 example, we expect the stable nuclei to have N ~Z, which is in fact observed for nuclei with values of Z up to 20. • “Magic numbers” • Several lines of evidence indicate that nuclei possessing 2,8,20,28,50,82, or 126 nucleons of the same kind (either protons or neutrons) are more stable than those having different numbers; for instance helium with two each of neutrons and protons is an extremely stable particle. • So is the common isotope of oxygen with 8 of each. Tin, the nucleus of which has 50 protons ,has more stable isotopes than any other elements.
Apparently the numbers in the series just given have a special significance and suggest that they might represent something like completed shells or subshells. • But the attempts to apply the Pauli exclusion principle and compute the energy levels in the nucleus did not give the right results. • On the particular assumption that the potential well of the nucleus had rounded edges instead of a rectangular cross section it is true that the energy levels appeared in groups more like shells, but the completed groups came at the numbers 2,8,20,40,70,112,and 168 and this was the wrong answer. • The problem was now solved very beautifully by the founders of the new shell theory when it was noted that
if a nuclear particle were in an actual orbit its energy would depend on whether it was spinning on its axis in the same direction as its orbital motion or in the opposite direction. • This interaction between spin and orbital motion (spin-orbit coupling) although small for an extra-nuclear electron turns out to be relatively large in the nucleus. • Indeed it is just enough to split the energy levels and put in large energy gaps at just the right places to obtain the completed groups having numbers 2,8,20,28,50,82 and 126. • Before this the importance of the numbers was so puzzling that they were called ‘magic numbers”.
§8-5 Radioactive Decays • Radioactive decays • At the beginning of this chapter ,we noted that many nuclei undergo alpha, beta or gamma decay. • In these decays, as in all nuclear processes, the following quantities must be conserved: (1)Energy (including mass energy) (2) Momentum (both linear and angular) (3)Electric charge (4) Number of nucleons. • Notice in particular that the total charge and number of nucleons does not change. • γ decay • Gamma rays are electromagnetic quanta or photons emitted when a nucleus undergoes a transition from a higher to a lower energy level.
They are completely equivalent to the light quanta and X-rays emitted by excited atoms, but their energies are usually much greater. • The half-lives for γ decay are usually very short, typically ; however, in a few special cases the half-lives may be as long as several years. • Closely related to γ decay is internal conversion. Here an excited nucleus gives up its excess energy to an electron in one of the inner atomic shells, ejecting it from the atom. No gamma ray is emitted. • Some radionuclides used in nuclear medicine decay in this fashion. • β decay
In β decay, an electron or in a few cases a positron (正电子) is emitted by a nucleus. • Positron ,the antiparticle of electron, was first predicted by Dirac in 1928 in his theory of the electron which included the effects of special relativity and then discovered in the products of cosmic-ray reaction in 1932. • Today ,antiprotons, antineutrons, and many other antiparticles have been seen, and all particles are believed to have antiparticles. • Particle-antiparticle pairs can annihilate(湮灭) in a burst of gamma rays, converting all their mass to energy. • The half-lives of β decays are very long compared to γ decay half-lives, varying from seconds to many years. • This indicates that the force responsible for the β decay is weak compared to electromagnetic forces that are responsible for γ decay.
Neutrino(中微子) • A striking feature of β decay is that the emitted betas are variable in kinetic energy. For example, in ,the mass energy of nucleus exceeds that of a nucleus plus an electron by 0.0186MeV,the maximum electron kinetic energy. However, sometimes the electron kinetic energy is less than 0.0186MeV.What has happened to the missing energy? • Enrico Fermi supplied the answer in1933.He proposed that when a nucleus β decays, it creates not only an electron but also a neutrino(ν): a massless, uncharged, spin ½ particle. • Then the decay is more completely written as • Since the decaying nucleus emits two particles they can share the decay energy in various combinations.
α decay • In α decay, an α particle (He nucleus) is emitted, leaving behind a residual nucleus that has lost two protons and two neutrons. • For a given nuclide, the α particle energy is always the same. This is because when a nucleus at rest emits one particle, energy and momentum conservation determine its energy. • α decay is usually observed in the heavier unstable nuclei. All known nuclei above Z=83 are unstable; those that do not βdecay have been observed to αdecay with half-lives ranging from about s to years. • Typical α decay processes are much slower than β decays and quantum mechanical tunneling effect is responsible for this seemingly extraordinary phenomenon.
The End • Thank You for Your Attention!
