Chapter 44 Nuclear Structure
Milestones in the Development of Nuclear Physics 1896: the birth of nuclear physics Becquerel discovered radioactivity in uranium compounds Rutherford showed the radiation had three types: alpha (He nuclei) beta (electrons) gamma (high-energy photons)
More Milestones 1911 Rutherford, Geiger and Marsden performed scattering experiments Established that the nucleus could be treated as a point mass and a point charge Most of the atomic mass was contained in the nucleus Nuclear force was a new type of force
Some Properties of Nuclei All nuclei are composed of protons and neutrons Exception is ordinary hydrogen with a single proton The atomic number Z equals the number of protons in the nucleus Sometimes called the charge number The neutron number N is the number of neutrons in the nucleus
More Properties of Nuclei The mass number A is the number of nucleons in the nucleus A = Z + N Nucleon is a generic term used to refer to either a proton or a neutron The mass number is not the same as the mass
Symbolism A Z X X is the chemical symbol of the element Example: 27 13 Al Mass number is 27 Atomic number is 13 Contains 13 protons Contains 14 (27 13) neutrons The Z may be omitted since the element can be used to determine Z
More Properties The nuclei of all atoms of a particular element must contain the same number of protons They may contain varying numbers of neutrons Isotopes of an element have the same Z but differing N and A values The natural abundance of isotopes can vary Isotope example: C, C, C, C 11 12 13 14 6 6 6 6
Charge The proton has a single positive charge, e The electron has a single negative charge, - e The neutron has no charge Made it difficult to detect in early experiments Easy to detect with modern devices e = 1.602 177 33 x 10-19 C
Mass It is convenient to use atomic mass units, u, to express masses 1 u = 1.660 539 x 10-27 kg Based on definition that the mass of one atom of 12 C is exactly 12 u Mass can also be expressed in MeV/c 2 From E R = mc 2 1 u = 931.494 MeV/c 2 Includes conversion 1 ev = 1.602 177 x 10-19 J
The Size of the Nucleus First investigated by Rutherford in scattering experiments He found an expression for how close an alpha particle moving toward the nucleus can come before being turned around by the Coulomb force From conservation of energy, the kinetic energy of the particle must be completely converted to potential energy
Size of the Nucleus, cont. d is called the distance of closest approach d gives an upper limit for the size of the nucleus Rutherford determined that d 2 Ze 4k e mv 2 For gold, he found d = 3.2 x 10-14 m
More About Size Rutherford concluded that the positive charge of the atom was concentrated in a sphere whose radius was no larger than about 10-14 m He called this sphere the nucleus These small lengths are often expressed in femtometers (fm) where 1 fm = 10-15 m Also called a fermi
Size of Nucleus, Final Since the time of Rutherford, many other experiments have concluded the following: Most nuclei are approximately spherical Average radius is 13 r o = 1.2 x 10-15 m r r A A is the mass number o
Density of Nuclei The volume of the nucleus (assumed to be spherical) is directly proportional to the total number of nucleons This suggests that all nuclei have nearly the same density Nucleons combine to form a nucleus as though they were tightly packed spheres
Nuclear Stability There are very large repulsive electrostatic forces between protons These forces should cause the nucleus to fly apart The nuclei are stable because of the presence of another, short-range force, called the nuclear force This is an attractive force that acts between all nuclear particles The nuclear attractive force is stronger than the Coulomb repulsive force at the short ranges within the nucleus
Features of the Nuclear Force Attractive force that acts between all nuclear particles Very short range It falls to zero when the separation between particles exceeds about several fermis Independent of charge The nuclear force on p-p, p-n, n-n are all the same Does not affect electrons
Nuclear Stability, cont. Light nuclei are most stable if N = Z Heavy nuclei are most stable when N > Z Above about Z = 20 As the number of protons increases, the Coulomb force increases and so more neutrons are needed to keep the nucleus stable No nuclei are stable when Z > 83
Binding Energy The total energy of the bound system (the nucleus) is less than the combined energy of the separated nucleons This difference in energy is called the binding energy of the nucleus It can be thought of as the amount of energy you need to add to the nucleus to break it apart into its components
Binding Energy, cont. The binding energy can be calculated from conservation of energy and the Einstein mass-energy equivalence principle: E b (MeV) = [ZM(H) + Nm n M ( A ZX)] x 931.494 MeV/u M(H) is the atomic mass of the neutral hydrogen atom M ( A ZX) represents the atomic mass of an atom of the isotope ( A ZX) M n is the mass of the neutron The masses are expressed in atomic mass units
Binding Energy per Nucleon
Notes from the Graph The curve peaks in the vicinity of A = 60 Nuclei with mass numbers greater than or less than 60 are not as strongly bound as those near the middle of the periodic table There is a decrease in binding energy per nucleon for A > 60 Energy is released when a heavy nucleus splits or fissions Energy is released since each product nucleus are more tightly bound to one another than are the nucleons of the original nucleus
More Notes from the Graph The binding energy is about 8 MeV per nucleon for nuclei with A > 50 This suggests that the nuclear force saturates A particular nucleon can interact with only a limited number of other nucleons 62 Ni 28 has the largest binding energy per nucleon
Nuclear Models Two models of the nucleus will be discussed Liquid-drop model Provides good agreement with observed nuclear binding energies Shell model Predicts the existence of stable nuclei
Liquid-Drop Model Nucleons are treated like molecules in a drop of liquid The nucleons interact strongly with one another They undergo frequent collisions as they jiggle around in the nucleus The jiggling motion is analogous to the thermally agitated motion of molecules in a drop of liquid
Liquid-Drop Model Effects Influencing Binding Energy, 1 The volume effect The nuclear force on a given nucleon is due only to a few nearest neighbors and not to all the other nucleons in the nucleus The total binding energy is proportional to A and therefore proportional to the nuclear volume This contribution to the binding energy of the entire nucleus is C 1 A C 1 is an adjustable constant
Liquid-Drop Model Binding Energy Effect 2 The surface effect Nucleons on the surface have fewer neighbors than those in the interior Surface nucleons reduce the binding energy by an amount proportional to their number The number of nucleons is proportional to the surface area The surface term can be expressed as C 2 A 2/3 C 2 is a second adjustable constant
Liquid-Drop Model Binding Energy Effect 3 The Coulomb repulsion effect Each proton repels every other proton in the nucleus The potential energy associated with the Coulomb force is proportional to the number of protons, Z The reduction in the binding energy due to the Coulomb effect is C 3 Z(Z - 1)/A 1/3 C 3 is another adjustable constant
Liquid-Drop Model Binding Energy Effect 4 The symmetry effect Any large symmetry between N and Z for light nuclei reduces the binding energy For larger A, the value of N for stable nuclei is larger The effect can be described by a binding energy term in the form C 4 (N - Z) 2 / A For small A, any large asymmetry between N and Z makes the term large For large A, the A in the denominator reduces the value of the term so that it has little effect on the overall binding energy C 4 is another adjustable constant
Liquid-Drop Model Binding Energy Effect Summary Putting these terms together results in the semiempirical binding-energy formula: 1 2 23 Z Z N Z Eb C1A C2A C3 C 13 4 A A The four constants are adjusted to fit the theoretical expression to the experimental data For A 15, C 1 = 15.7 MeV; C 2 = 17.8 MeV; C 3 = 0.71 MeV; and C 4 = 23.6 MeV
Liquid Drop Model, Final The equation fits the known nuclear mass values very well Does not account for some of the finer details of nuclear structure Stability Angular momentum
Features of Binding Energy When binding energies are studied closely it is found that: Most stable nuclei have an even value of A Only 8 stable nuclei have odd values for both A and Z There is a difference between the binding energy per nucleon given by the semiempirical formula and experiments
Features of Binding Energy Magic Numbers The disagreement between the semiempirical formula and experiments is plotted Peaks appear in the graph These peaks are at the magic numbers of Z or N = 2, 8, 20, 28, 52, 82
Features of Binding Energy, cont. Studies of nuclear radii show deviations from the expected values Graphs of the data show peaks at values of N equal to the magic numbers A group of isotones is a collection of nuclei having the same value of N and different values of Z When the number of stable isotones is graphed as a function of N, there are peaks at the magic numbers
Features of Binding Energy, final Several other nuclear measurements show anomalous behavior at the magic numbers The peaks are reminiscent of the peaks in graphs of ionization energy of atoms and lead to the shell model of the nucleus
Maria Goeppert-Mayer 1906 1972 German scientist Best known for her development of the shell model of the nucleus Shared the Nobel Prize in 1963 Shared with Hans Jensen who simultaneously developed a similar model
Shell Model The shell model is also called the independentparticle model In this model, each nucleon is assumed to exist in a shell Similar to atomic shells for electrons The nucleons exist in quantized energy states There are few collisions between nucleons
Shell Model, cont. Each state can contain only two protons or two neutrons They must have opposite spins They have spins of ½, so the exclusion principle applies The set of allowed states for the protons differs from the set of allowed states for the neutrons
Shell Model, final Proton energy levels are farther apart than those for neutrons due to the superposition of the Coulomb force and the nuclear force for the protons The spin-orbit effect for nucleons is due to the nuclear force The spin-orbit effect influences the observed characteristics of the nucleus
Shell Model Explanation of Experimental Results Nuclei with even numbers of protons and neutrons are more stable Any particular state is filled when it contains two protons or two neutrons An extra proton or neutron can be added only at the expense of increasing the nucleus s energy This increase in energy leads to greater instability in the nucleus
Shell Model Explanation of Experimental Results, cont. Nuclei tend to have more neutrons than protons Proton energy levels are higher As Z increases and higher states are filled, a proton level for a given quantum number will be much higher in energy than the neutron level for the same quantum number It is more energetically favorable for the nucleus to form with neutrons in the lower energy levels than protons in the higher levels So, the number of neutrons is greater than the number of protons
Marie Curie 1867 1934 Polish scientist Shared Nobel Prize in 1903 for studies in radioactive substances Prize in physics Shared with Pierre Curie and Becquerel Won Nobel Prize in 1911 for discovery of radium and polonium Prize in chemistry
Radioactivity Radioactivity is the spontaneous emission of radiation Discovered by Becquerel in 1896 Many experiments were conducted by Becquerel and the Curies Experiments suggested that radioactivity was the result of the decay, or disintegration, of unstable nuclei
Radioactivity Types Three types of radiation can be emitted Alpha particles The particles are 4He nuclei Beta particles The particles are either electrons or positrons A positron is the antiparticle of the electron It is similar to the electron except its charge is +e Gamma rays The rays are high energy photons
Distinguishing Types of Radiation The gamma particles carry no charge The alpha particles are deflected upward The beta particles are deflected downward A positron would be deflected upward, but would follow a different trajectory than the α due to its mass
Penetrating Ability of Particles Alpha particles Barely penetrate a piece of paper Beta particles Can penetrate a few mm of aluminum Gamma rays Can penetrate several cm of lead
The Decay Constant The number of particles that decay in a given time is proportional to the total number of particles in a radioactive sample dn dt λn gives N N e o λt λ is called the decay constant and determines the rate at which the material will decay N is the number of undecayed radioactive nuclei present N o is the number of undecayed nuclei at time t = 0
Decay Curve The decay curve follows the equation N = N o e -λt The half-life is also a useful parameter The half-life is defined as the time interval during which half of a given number of radioactive nuclei decay T 12 ln 2 0. 693 λ λ
Active Figure 44.9 Use the active figure to adjust the half-life Observe the decay curve PLAY ACTIVE FIGURE
Decay Processes The blue circles are the stable nuclei seen before Above the line the nuclei are neutron rich and undergo beta decay (red) Just below the line are proton rich nuclei that undergo beta (positron) emission or electron capture (green) Farther below the line the nuclei are very proton rich and undergo alpha decay (yellow)
Active Figure 44.10 Click on any colored dot Study the decay modes and decay energies PLAY ACTIVE FIGURE
Alpha Decay When a nucleus emits an alpha particle it loses two protons and two neutrons N decreases by 2 Z decreases by 2 A decreases by 4 A A 4 4 ZX Z 2Y 2He X is called the parent nucleus Symbolically Y is called the daughter nucleus
Decay General Rules The sum of the mass numbers A must be the same on both sides of the equation The sum of the atomic numbers Z must be the same on both sides of the equation When one element changes into another element, the process is called spontaneous decay or transmutation Relativistic energy and momentum of the isolated parent nucleus must be conserved
Alpha Decay, Example Decay of 226 Ra Ra Rn He 226 222 4 88 86 2 If the parent is at rest before the decay, the total kinetic energy of the products is 4.87 MeV In general, less massive particles carry off more of the kinetic energy
Beta Decay During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is changed by one Symbolically A Z A Z X Y e A Z 1 X Y e A Z 1 Beta decay is not completely described by these equations
Beta Decay, cont. The emission of the electron or positron is from the nucleus The nucleus contains protons and neutrons The process occurs when a neutron is transformed into a proton or a proton changes into a neutron The electron or positron is created in the process of the decay Energy must be conserved
Beta Decay Particle Energy The energy released in the decay process should almost all go to kinetic energy of the β particle Since the decaying nuclei all have the same rest mass, the Q value should be the same for all decays Experiments showed a range in the amount of kinetic energy of the emitted particles Were conservation laws violated?
Neutrino To account for this missing energy, in 1930 Pauli proposed the existence of another particle Enrico Fermi later named this particle the neutrino Properties of the neutrino Zero electrical charge Mass much smaller than the electron, probably not zero Spin of ½ Very weak interaction with matter and so is difficult to detect
Beta Decay Completed Symbolically A Z A Z X Y e ν A Z 1 X Y e ν A Z 1 is the symbol for the neutrino ν is the symbol for the antineutrino To summarize, in beta decay, the following pairs of particles are emitted An electron and an antineutrino A positron and a neutrino
Beta Decay Examples
Beta Decay, Final Notes The fundamental process of e - decay is a neutron changing into a proton, an electron and an antineutrino In e +, the proton changes into a neutron, positron and neutrino This can only occur within a nucleus It cannot occur for an isolated proton since its mass is less than the mass of the neutron
Electron Capture Electron capture is a process that competes with e+ decay In this case, a parent nucleus captures one of its own orbital electrons and emits a neutrino: A X 0 e A Y ν Z 1 Z 1 In most cases, a K-shell electron is captured, so this is often referred to as K capture
Electron Capture, Detection Because the neutrino is very hard to detect, electron capture is usually observed by the x- rays given off as higher-shell electrons cascade downward to fill the vacancy created in the K shell
Gamma Decay Gamma rays are given off when an excited nucleus decays to a lower energy state The decay occurs by emitting a high-energy photon called gamma-ray photons A X* X γ Z Z The X* indicates a nucleus in an excited state Typical half-life is 10-10 s A
Gamma Decay Example Example of a decay sequence The first decay is a beta emission The second step is a gamma emission B C* e ν 12 12 5 6 C* 12 12 6 6 C γ Gamma emission doesn t change Z, N, or A The emitted photon has an energy of hƒ equal to DE between the two nuclear energy levels
Summary of Decays
Natural Radioactivity Classification of nuclei Unstable nuclei found in nature Give rise to natural radioactivity Nuclei produced in the laboratory through nuclear reactions Exhibit artificial radioactivity Three series of natural radioactivity exist Uranium Actinium Thorium Some radioactive isotopes are not part of any decay series
Radioactive Series, Overview
Decay Series of 232 Th Series starts with 232 Th Processes through a series of alpha and beta decays The series branches at 212 Bi Ends with a stable isotope of lead, 208 Pb