CHAPTER 12�The Atomic Nucleus�

• 12.1 Discovery of the Neutron
• 12.2 Nuclear Properties
• 12.3 The Deuteron
• 12.4 Nuclear Forces
• 12.5 Nuclear Stability
• 12.6 Radioactive Decay
• 12.7 Alpha, Beta, and Gamma Decay
• 12.8 Radioactive Nuclides
• 13.4 Fission

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Sedan crater 1972 test Nevada National Security Site NNSS

Radioactive materials were accidentally released from the 1970 Baneberry shot in Area 8.

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13.4: Fission

• In fission a nucleus separates into two fission fragments. As we will show, one fragment is typically somewhat larger than the other.
• Fission occurs for heavy nuclei because of the increased Coulomb forces between the protons.
• We can understand fission by using the semi-empirical mass formula based on the liquid drop model. For a spherical nucleus of mass number A ~ 240, the attractive short-range nuclear forces offset the Coulomb repulsive term. As a nucleus becomes non-spherical, the surface energy is increased, and the effect of the short-range nuclear interactions is reduced.
• Nucleons on the surface are not surrounded by other nucleons, and the unsaturated nuclear force reduces the overall nuclear attraction. For a certain deformation, a critical energy is reached, and the fission barrier is overcome.

• Spontaneous fission can occur for nuclei with

Induced Fission

• Fission may also be induced by a nuclear reaction. A neutron absorbed by a heavy nucleus forms a highly excited compound nucleus that may quickly fission.
• An induced fission example is

• The fission products have a ratio of N/Z much too high to be stable for their A value.
• There are many possibilities for the Z and A of the fission products.
• Symmetric fission (products with equal Z) is possible, but the most probable fission is asymmetric (one mass larger than the other).

Thermal Neutron Fission

• Fission fragments are highly unstable because they are so neutron rich.
• Prompt neutrons are emitted simultaneously with the fissioning process. Even after prompt neutrons are released, the fission fragments undergo beta decay, releasing more energy.
• Most of the ~200 MeV released in fission goes to the kinetic energy of the fission products, but the neutrons, beta particles, neutrinos, and gamma rays typically carry away 30–40 MeV of the kinetic energy.

Chain Reactions (1 of 2)

• Because several neutrons are produced in fission, these neutrons may subsequently produce other fissions. This is the basis of the self-sustaining chain reaction.
• If at least one neutron, on the average, results in another fission, the chain reaction becomes critical.
• A sufficient amount of mass is required for a neutron to be absorbed, called the critical mass.
• If less than one neutron, on the average, produces another fission, the reaction is subcritical.
• If more than one neutron, on the average, produces another fission, the reaction is supercritical.
• An atomic bomb is an extreme example of a supercritical fission chain reaction.

Chain Reactions (2 of 2)

• A critical-mass fission reaction can be controlled by absorbing neutrons. A self-sustaining controlled fission process requires that not all the neutrons are prompt. Some of the neutrons are delayed by several seconds and are emitted by daughter nuclides. These delayed neutrons allow the control of the nuclear reactor.
• Control rods regulate the absorption of neutrons to sustain a controlled reaction.

13.5: Fission Reactors

Table 13.1 Energy Content of Fuels

Table 13.2 Daily Fuel Requirements for 1000-MWe Power Plant

• Several components are important for a controlled nuclear reactor:
1. Fissionable fuel
2. Moderator to slow down neutrons
3. Control rods for safety and to control criticality of reactor
4. Reflector to surround moderator and fuel in order to contain neutrons and thereby improve efficiency
5. Reactor vessel and radiation shield
6. Energy transfer systems if commercial power is desired
1. Two main effects can “poison” reactors: (1) neutrons may be absorbed without producing fission [for example, by neutron radiative capture], and (2) neutrons may escape from the fuel zone.

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MWe =megawatt ( e indicates electrical power)

Structure of matter

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Structure of matter

Dark matter and dark energy are the yin and yang of the cosmos. Dark matter produces an attractive force (gravity), while dark energy produces a repulsive force (antigravity). ... Astronomers know dark matter exists because visible matter doesn't have enough gravitational muster to hold galaxies together.

Hierarchy of forces

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• Sta

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Standard Model tries to unify the forces into one force

Ernest Rutherford “Father of the Nucleus”

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Story so far: Unification���

1831 1967 1974 1984 1995

Electricity }

}

} Electromagnetic force }

} }

Magnetism} } Electro-weak force }

} }

Weak nuclear force} } Grand unified force }

} } 5 Different }

Strong nuclear force} } D=10 String } M-theory

} Theories }

Gravitational force}

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Faraday Glashow,Weinberg,Salam Georgi,Glashow Green,Schwarz Witten

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+branes in D=11

Discovery of the Neutron

3. Nuclear magnetic moment:

The magnetic moment of an electron is over 1000 times larger than that of a proton.

The measured nuclear magnetic moments are on the same order of magnitude as the proton’s, so an electron is not a part of the nucleus.

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• In 1930 the German physicists �Bothe and Becker used a �radioactive polonium source �that emitted α particles. When �these α particles bombarded �beryllium, the radiation �penetrated several centimeters �of lead.

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The neutrons collide elastically with the protons of the paraffin thereby producing the5.7 MeV protons

Discovery of the Neutron

• Photons are called gamma rays when they originate from the nucleus. They have energies on the order of MeV (as compared to X-ray photons due to electron transitions in atoms with energies on the order of KeV.)

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• Curie and Joliot performed several measurements to study penetrating high-energy gamma rays and the unknown radiation in particular on paraffin ( which contains H )

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• In 1932 Chadwick proposed that the new radiation produced by α + Be consisted of neutrons. His experimental data estimated the neutron’s mass as somewhere between 1.005 u and 1.008 u, not far from the modern value of 1.0087 u.

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12.2: Nuclear Properties

• The nuclear charge is +e times the number (Z) of protons.

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• Hydrogen’s isotopes:
• Deuterium: Heavy hydrogen; has a neutron as well as a proton in its nucleus
• Tritium: Has two neutrons and one proton

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• The nuclei of the deuterium and tritium atoms are called deuterons and tritons.
• Atoms with the same Z, but different mass number A, are called isotopes.

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Nuclear Properties

• The symbol of an atomic nucleus is .

where Z = atomic number (number of protons)

N = neutron number (number of neutrons)

A = mass number (Z + N)

X = chemical element symbol

• Each nuclear species with a given Z and A is called a nuclide.
• Z characterizes a chemical element.
• The dependence of the chemical properties on N is negligible.
• Nuclides with the same neutron number are called isotones and the same value of A are called isobars.

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Nuclear Properties

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• Atomic masses are denoted by the symbol u.
• 1 u = 1.66054 × 10⁻²⁷ kg = 931.49 MeV/c²

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• Both neutrons and protons, collectively called nucleons, are constructed of other particles called quarks.

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Clicker - Questions

From chapter12 quiz

The nuclear force can be all of the following EXCEPT:

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a. short-range

b. saturable

c. spin dependent

d. charge dependent

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Sizes and Shapes of Nuclei

• Rutherford concluded that the range of the nuclear force must be less than about 10⁻¹⁴ m.
• Assume that nuclei are spheres of radius R.
• Particles (electrons, protons, neutrons, and alphas) scatter when projected close to the nucleus.
• It is not obvious whether the maximum interaction distance refers to the nuclear size (matter radius), or whether the nuclear force extends beyond the nuclear matter (force radius).

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• The nuclear force is often called the strong force.

Nuclear force radius ≈ mass radius ≈ charge radius

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Sizes and Shapes of Nuclei

• The nuclear radius may be approximated to be R = r₀A^1/3

where r₀ ≈ 1.2 × 10⁻¹⁵ m.

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• We use the femtometer with 1 fm = 10⁻¹⁵ m, or the fermi.

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• The lightest nuclei by the Fermi distribution for the nuclear charge density ρ(r) is

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Sizes and Shapes of Nuclei

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The shape of the Fermi distribution

Nuclear Density and Intrinsic Spin��Nuclear Density: If we approximate the nuclear shape as a sphere, then we have: the nuclear mass density (mass/volume) can be determined from (Au/V) to be �2.3 x 10¹⁷ kg/m³.��Intrinsic Spin: The neutron and proton are fermions with spin quantum numbers s = ½. The spin quantum numbers are those previously learned for the electron (see Chapter 7).^���

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What is the ratio of the density of the nucleus to that of water? Water density 1g/cm^3

convert 2.3 x 10¹⁷ kg/m³ to g/ cm³

�The nucleus is 10¹⁴ times denser than water

The density for any typical nucleus, in terms of mass number, is thus constant, not dependent on A or r, theoretically:

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Intrinsic Magnetic Moment

• The proton’s intrinsic magnetic moment points in the same direction as its intrinsic spin angular momentum.
• Nuclear magnetic moments are measured in units of the nuclear magneton μ_N.

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• The divisor in calculating μ_N is the proton mass m_p, which makes the nuclear magneton some 1836 times smaller than the Bohr magneton.
• The proton magnetic moment is μ_p = 2.79μ_N.
• The magnetic moment of the electron is μ_e = −1.00116μ_B.
• The neutron magnetic moment is μ_n = −1.91μ_N.
• The nonzero neutron magnetic moment implies that the neutron has negative and positive internal charge components at different radii.

Complex internal charge distribution.

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Nuclear Magnetic Resonance (NMR)

• A widely used medical application using the nuclear magnetic moment's response to large applied magnetic fields.
• Although NMR can be applied to other nuclei that have intrinsic spin, proton NMR is used more than any other kind.

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Nuclear magnetic moment for a nucleus with I= 3/2

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Nuclear magnetic resonance and imaging

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NMR apparatus

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Compare NMR with X-rays

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Find the energy difference between the two nuclear spin orientations

NMR at magnetic field of 2T

Appendix 7

Compare the magnetic fields of NMR and earth

• Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from the Earth's interior out into space, where it meets the solar wind, a stream of charged particles emanating from the Sun. Its magnitude at the Earth's surface ranges from 25 to 65 micro teslas (0.25 to 0.65 gauss).
• Earth's magnetic field - Wikipedia

�

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12.3: The Deuteron

• The determination of how the neutron and proton are bound together in a deuteron.
• The deuteron mass = 2.013553 u
• The mass of a deuteron atom = 2.014102 u
• The difference = 0.000549 u; the mass of an electron
• The deuteron nucleus is bound by a mass-energy B_d
• The mass of a deuteron is

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• Add an electron mass to each side of Eq. (12.6)

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See Particle masses for calculations

1 u = 1.66054 × 10⁻²⁷ kg = 931.49 MeV/c²

(electron binding energy=13.6 eV can be neglected)

The Deuteron

• m_d + m_e is the atomic deuterium mass M(²H) and m_p + m_e is the atomic hydrogen mass. Thus Eq.(12.7) becomes

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• Because the electron masses cancel in almost all nuclear-mass difference calculations, we use atomic masses rather than nuclear masses.

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• Convert this to energy using u = 931.5 MeV / c²

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• Even for heavier nuclei we neglect the electron binding energies (13.6 eV) because the nuclear binding energy (2.2 MeV) is almost one million times greater.

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See Appendix 8 Atomic mass tables

Use upper case M for atomic,lower case m for nuclear masses

The Deuteron

• The binding energy of any nucleus = the energy required to separate the nucleus into free neutrons and protons.

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Experimental Determination of Nuclear Binding Energies

• Check the 2.22-MeV binding energy by using a nuclear reaction. We scatter gamma rays from deuteron gas and look for the breakup of a deuteron into a neutron and a proton:

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• This nuclear reaction is called photo disintegration or a photo nuclear reaction.
• The mass-energy relation is

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• where hf is the incident photon energy.

K_n and K_p are the neutron and proton kinetic energies.

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The Deuteron

• The minimum energy required for the photodisintegration:
• Momentum must be conserved in the reaction (K_n, K_p ≠ 0)

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• Experiment shows that a photon of energy less than 2.22 MeV cannot dissociate a deuteron

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Deuteron Spin and Magnetic Moment

• Deuteron’s nuclear spin quantum number is 1. This indicates the neutron and proton spins are aligned parallel to each other.
• The nuclear magnetic moment of a deuteron is 0.86μ_N ≈ the sum of the free proton and neutron 2.79μ_N − 1.91μ_N = 0.88μ_N .(supporting parallel spins)

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From chapter12 quiz

The mass of the deuteron

a. is exactly the sum of the neutron and proton mass.

b. is slightly less than the sum of the neutron and proton mass.

c. is exactly 2.000000 u.

d. is exactly the sum of a neutron, proton, and electron mass.

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Halo –nucleus ( a nuclear hydrogen atom)

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A. Takamine, M. Wada, K. Okada, T. Sonoda, P. Schury, T. Nakamura, Y. Kanai, T. Kubo, I. Katayama, S. Ohtani, H. Wollnik, and H. A. Schuessler�Hyperfine Structure Constant of the Neutron Halo Nucleus 11Be+�Phys. Rev. Lett. 112, 162502 (2014).

The short range strong nuclear force causes the halo neutron to be 7 fm outside the nucleus

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half life 13.8 s

12.4: Nuclear Forces

• The angular distribution of neutron classically scattered by protons.
• Neutron + proton (np) and proton + proton (pp) elastic

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The nuclear potential

Nuclear Forces

• The internucleon potential has a “hard core” that prevents the nucleons from approaching each other closer than about 0.4 fm.

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• The proton has charge radius up to 1 fm.
• Two nucleons within about 2 fm of each other feel an attractive force.

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• The nuclear force (short range):
• It falls to zero so abruptly with interparticle separation. stable
• The interior nucleons are completely surrounded by other nucleons with which they interact.

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• The only difference between the np and pp potentials is the Coulomb potential shown for r ≥ 3 fm for the pp force.

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Nuclear Forces

• The nuclear force is known to be spin dependent.
• The neutron and proton spins are aligned for the bound state of the deuteron, but there is no bound state with the spins antialigned.

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• The nn system is more difficult to study because free neutrons are not stable from analyses of experiments.
• The nuclear potential between two nucleons seems independent of their charge (charge independence of nuclear forces).

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• The term nucleon refers to either neutrons or protons because the neutron and proton can be considered different charge states of the same particle.

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12.5: Nuclear Stability

• The binding energy of a nucleus against dissociation into any other possible combination of nucleons. Ex. nuclei R and S.

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• Proton (or neutron) separation energy:
• The energy required to remove one proton (or neutron) from a nuclide.

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• All stable and unstable nuclei that are long-lived enough to be observed.

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Nature does not allow Z> N with the exception of a few low Z unstable nucleons

Nuclear Stability

• The line representing the stable nuclides is the line of stability.
• It appears that for A ≤ 40, nature prefers the number of protons and neutrons in the nucleus to be about the same Z ≈ N.

However, for A ≥ 40, there is a decided preference for N > Z because the nuclear force is independent of whether the particles are nn, np, or pp.

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• As the number of protons increases, the Coulomb force between all the protons becomes stronger until it eventually affects the binding significantly.

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• The work required to bring the charge inside the sphere from infinity is

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Nuclear Stability

• For a single proton,

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• The total Coulomb repulsion energy in a nucleus is

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• For heavy nuclei, the nucleus will have a preference for fewer protons than neutrons because of the large Coulomb repulsion energy.

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• Most stable nuclides have both even Z and even N (even-even nuclides).
• Only four stable nuclides have odd Z and odd N (odd-odd nuclides).

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Work to assemble the proton itself must not be included

Nature prefers nuclei with even numbers of protons and even neutrons

From chapter12 quiz

Which of the following statements best describes the line of stability?

a. It has N = Z when A = 240.

b. It has Z > N at A = 240

c. N always tends to be greater than Z.

d. N tends to be greater than Z, especially for masses greater than calcium.

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Magic numbers(high stability nuclei) show shell structure Goeppert-Mayer,Jensen(1963 Nobel price)

N or Z=2,8,20,28,50,82,126

The Liquid Drop Model

• Treats the nucleus as a collection of interacting particles in a liquid drop.
• The total binding energy, the semi-empirical mass formula is

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• The volume term (a_v) indicates that the binding energy is approximately the sum of all the interactions between the nucleons.

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• The second term is called the surface effect because the nucleons on the nuclear surface are not completely surrounded by other nucleons.

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• The third term is the Coulomb energy in Eq. (12.17) and Eq. (12.18)

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Surface area nuclear radius R = r₀A^1/3

The Liquid Drop Model

• The fourth term is due to the symmetry energy. In the absence of Coulomb forces, the nucleus prefers to have N ≈ Z and has a quantum-mechanical origin, depending on the exclusion principle.
• The last term is due to the pairing energy and reflects the fact that the nucleus is more stable for even-even nuclides. Use values given by Fermi to determine this term.

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where Δ = 33 MeV·A^−3/4

• No nuclide heavier than has been found in nature. If they ever existed, they must have decayed so quickly that quantities sufficient to measure no longer exist.

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Carl Friedrich Freiherr von Weizsäcker,28 June 1912 – 28 April 2007) was a German physicist and philosopher.

The liquid drop model

The liquid drop model was first proposed by George Gamow and further developed by Niels Bohr and John Archibald Wheeler. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force (a residual effect of the strong force), there is a similarity to the structure of a spherical liquid drop. While a crude model, the liquid drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding energy.

The corresponding mass formula is defined purely in terms of the numbers of protons and neutrons it contains. The original Weizsäcker formula defines five terms:

• Volume energy, when an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.
• Surface energy corrects for the previous assumption made that every nucleon interacts with the same number of other nucleons. This term is negative and proportional to the surface area, and is therefore roughly equivalent to liquid surface tension.
• Coulomb energy, the potential energy from each pair of protons. As this is a repulsive force, the binding energy is reduced.
• Asymmetry energy (also called Pauli Energy), which accounts for the Pauli exclusion principle. Unequal numbers of neutrons and protons imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.
• Pairing energy, which accounts for the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number due to spin coupling.

The liquid drop model was first proposed by George Gamow and further developed by Niels Bohr and John Archibald Wheeler. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force (a residual effect of the strong force), there is a similarity to the structure of a spherical liquid drop. While a crude model, the liquid drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding energy.

The corresponding mass formula is defined purely in terms of the numbers of protons and neutrons it contains. The original Weizsäcker formula defines five terms:

• Volume energy, when an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.
• Surface energy corrects for the previous assumption made that every nucleon interacts with the same number of other nucleons. This term is negative and proportional to the surface area, and is therefore roughly equivalent to liquid surface tension.
• Coulomb energy, the potential energy from each pair of protons. As this is a repulsive force, the binding energy is reduced.
• Asymmetry energy (also called Pauli Energy), which accounts for the Pauli exclusion principle. Unequal numbers of neutrons and protons imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.
• Pairing energy, which accounts for the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number due to spin coupling

The liquid drop model was first proposed by George Gamow and further developed by Niels Bohr and John Archibald Wheeler. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force (a residual effect of the strong force), there is a similarity to the structure of a spherical liquid drop. While a crude model, the liquid drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding energy.

The corresponding mass formula is defined purely in terms of the numbers of protons and neutrons it contains. The original Weizsäcker formula defines five terms:

Volume energy, when an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.

Surface energy corrects for the previous assumption made that every nucleon interacts with the same number of other nucleons. This term is negative and proportional to the surface area, and is therefore roughly equivalent to liquid surface tension.

Coulomb energy, the potential energy from each pair of protons. As this is a repulsive force, the binding energy is reduced.

Asymmetry energy (also called Pauli Energy), which accounts for the Pauli exclusion principle. Unequal numbers of neutrons and protons imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.

Pairing energy, which accounts for the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number due to spin coupling

From chapter12 quiz

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The "liquid drop model" of the nucleus allowed von Weizsaecker to propose his equation for a semi-empirical mass formula. This formula includes all of the following EXCEPT:

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a. A correction for nuclear surface interactions being different than interior saturated interactions.

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b. A term providing for the repulsion of protons in the nucleus.

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c. A term proportional to the total number of nucleons.

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d. A term for the energy associated with the fact that most stable nuclei prefer to have N approximately equal to Z.

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e. A term incorporating the instability of protons within the nucleus.

Binding Energy Per Nucleon

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• Use this to compare the relative �stability of different nuclides
• It peaks near A = 56
• The curve increases rapidly,

demonstrating the saturation

effect of nuclear force

• Sharp peaks for the even-even

nuclides ⁴He, ¹²C, and ¹⁶O

tightly bound

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Shortcomings of the Liquid Drop Model

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(2,2)

(4,4)

(6,6)

(8,8)

(10,10)

(N,Z)

It does not explain the high stability of nuclei with magic number.

➡️The concept of pairing cannot be explained with this model. 

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Nuclear Models

• Current research focuses on the constituent quarks and physicists have relied on a multitude of models to explain nuclear force behavior.

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1. Independent-particle models:�The nucleons move nearly independently in a common nuclear potential. The shell model has been the most successful of these.
2. Strong-interaction models:�The nucleons are strongly coupled together. The liquid drop model has been successful in explaining nuclear masses as well as nuclear fission.

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Nuclear Shell Model

• The difference of the shape between the proton and the neutron potentials are due to the Coulomb interaction on the proton.
• Nuclei have a Fermi energy level which is the highest energy level filled in the nucleus.
• In the ground state of a nucleus, all the energy levels below the Fermi level are filled.

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The nuclear potential felt by the neutron and the proton

Neutrons are more strongly bound due to the absence of the repulsive Coulomb force

Nuclear Models

• Energy-level diagrams for ¹²C and ¹⁶O.

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• Both are stable because they are even-even.

Case 1: If we add one proton to ¹²C to make

unstable

Case 2: If we add one neutron to ¹²C to make ¹³C:

stable

Nuclear Shell Model

• Even when we add another neutron to produce ¹⁴C, we find it is barely unstable.

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• Indicating neutron energy levels to be lower in energy than the corresponding proton ones.

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• In this mass region, nature prefers the number of neutrons and protons to be N ≈ Z, but it doesn’t want N < Z.

This helps explain why ¹³C is stable, but not ¹³N

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Nuclear shell model with well defined orbital states

(each nucleon moves in the average field of all other nucleons)

The Nobel Prize in Physics 1963.

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J. Hans D. Jensen 

Maria Goeppert Mayer

Nuclear Shell Model

Magic numbers(high stability nuclei) show shell structure Goeppert-Mayer,Jensen(1963 Nobel price)

N or Z=2,8,20,28,50,82,126

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Radioactivity is characteristic of elements with large atomic numbers

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. Elements with at least one stable isotope are shown in light blue. Green shows elements of which the most stable isotope has a half-life measured in millions of years. Yellow and orange are progressively less stable, with half-lives in thousands or hundreds of years, down toward one day. Red and purple show highly and extremely radioactive elements where the most stable isotopes exhibit half-lives measured on the order of one day and much less.

12.6: Radioactive Decay

• The discoverers of radioactivity were Wilhelm Röntgen, Henri Becquerel, Marie Curie and her husband Pierre.
• Marie Curie and her husband Pierre discovered polonium and radium in 1898.
• The simplest decay form is that of a gamma ray, which represents the nucleus changing from an excited state to lower energy state.
• Other modes of decay include emission of α particles, β particles, protons, neutrons, and fission.

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• The disintegrations or decays per unit time (activity):

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where dN / dt is negative because total number N decreases with time.

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Radioactive Decay

• SI unit of activity is the becquerel: 1 Bq = 1 decay / s
• Recent use is the Curie (Ci) 3.7 × 10¹⁰ decays / s

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• If N(t) is the number of radioactive nuclei in a sample at time t, and λ (decay constant) is the probability per unit time that any given nucleus will decay:

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• If we let N(t = 0) ≡ N₀

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----- radioactive decay law

Radioactive Decay

• The activity R is

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where R₀ is the initial activity at t = 0

• It is common to refer to the half-life t_1/2 or the mean lifetime τ rather than its decay constant.

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• The half-life is

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• The mean lifetime is

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Radioactive Decay

• The number of radioactive nuclei as a function of time

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Euler’s number e=2.71828..

The exponential function changes by equal amounts in equal times

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12.7: Alpha, Beta, and Gamma Decay

When a nucleus decays, all the conservation laws must be

observed:

• Mass-energy
• Linear momentum
• Angular momentum
• Electric charge
• Conservation of nucleons
• The total number of nucleons (A, the mass number) must be conserved in a low-energy nuclear reaction or decay.

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Alpha, Beta, and Gamma Decay

• Let the radioactive nucleus be called the parent and have the mass

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• Two or more products can be produced in the decay.
• Let the lighter one be M_y and the mass of the heavier one (daughter) be M_D.
• The conservation of energy is

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where Q is the energy released (disintegration energy) and equal to the total kinetic energy of the reaction products(note:Q(disintegration) is the negative of B(binding)

• If B > 0, a nuclide is bound and stable;
• If Q > 0, a nuclide is unbound, unstable, and may decay
• If Q < 0, decay emitting nucleons do not occur

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Binding enery refers to stable, whereas disintegration energy to unstable nuclei

Alpha Decay a collection of nucleons inside a nucleus decays���

• The nucleus ⁴He has a binding energy of 28.3 MeV.
• If the last two protons and two neutrons in a nucleus are bound by less than 28.3 MeV, then the emission of an alpha particle (alpha decay) is possible.

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• If Q > 0, alpha decay is possible

EX.

The appropriate masses are

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Q= 6 MeV and alpha decay is possible

Q=( 230.004u -226.025 -4.003u )c^2(931.5 MeV/c^2 u)= 6 MeV

Beta Decay

• Unstable nuclei may move closer to the line of stability by undergoing beta decay.
• The decay of a free neutron is

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• The beta decay of ¹⁴C (unstable) to form ¹⁴N, a stable nucleus, can be written as

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The electron energy spectrum from the beta decay

Figure 12.13

Beta Decay

• There was a problem in neutron decay, the spin ½ neutron cannot decay to two spin ½ particles, a proton and an electron. ¹⁴C has spin 0, ¹⁴N has spin 1, and the electron has spin ½.

we cannot combine spin ½ & 1 to obtain a spin 0.

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• Wolfgang Pauli suggested a neutrino that must be produced in beta decay. It has spin quantum number ½, charge 0, and carries away the additional energy missing in Fig. (12.13).

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β⁻ decay in an atomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron.

Can neutrinos penetrade the earth?They come straight through the earth at nearly the speed of light, all the time, day and night, in enormous numbers. About 100 trillion neutrinos pass through our bodies every second.

Beta Decay

• An occasional electron is detected with the kinetic energy K_max required to conserve energy, but in most cases the electron’s kinetic energy is less than K_max.

the neutrino has little or no mass, and its energy may be all kinetic

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• Neutrinos have no charge and do not interact electromagnetically.
• They are not affected by the strong force of the nucleus.
• They are the weak interaction.
• The electromagnetic and weak forces are the electroweak force.

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Radioactive decay modes conservation of nucleons

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Gamma Decay

• If the decay proceeds to an excited state of energy E_x rather than to the ground state, then Q for the transition to the excited state can be determined with respect to the transition to the ground state. The disintegration energy Q to the ground state Q₀.

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• Q for a transition to the excited state E_x is

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Gamma Decay

• The excitation energies tend to be much larger, many keV or even MeV.
• The possibilities for the nucleus to rid itself of this extra energy is to emit a photon (gamma ray).
• The gamma-ray energy hf is given by the difference of the higher energy state E_> and lower one E_<.

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• The decay of an excited state of ^AX* (where * is an excited state) to its ground state is

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• A transition between two nuclear excited states E_> and E_< is

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12.8: Radioactive Nuclides

• The unstable nuclei found in nature exhibit natural radioactivity.

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Big Bang was 13.7 billion years ago

3.154^+7 s/y

Radioactive Nuclides

• The radioactive nuclides made in the laboratory exhibit artificial radioactivity.
• Heavy radioactive nuclides can change their mass number only by alpha decay (^AX → ^A−4D) but can change their charge number Z by either alpha or beta decay.
• There are only four paths that the heavy naturally occurring radioactive nuclides may take as they decay.
• Mass numbers expressed by either:
• 4n
• 4n + 1
• 4n + 2
• 4n + 3

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Radioactive Nuclides

• The sequence of one of the radioactive series ²³²Th

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• ²¹²Bi can decay by either alpha or beta decay (branching).

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Radon gas in the form of ²²²Rn is a health hazard�

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The average indoor radon reading in Travis County, TX is predicted to be less than 2 picocuries per liter (pCi/L), so the county has been assigned EPA Radon Zone 3.

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Northern end of Lake Travis

Radon is a naturally occurring radioactive gas.

It’s produced when uranium, thorium, and radium break down in soil, rock, and water. It’s then released into the air. Radon is odorless, tasteless, and invisible.

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Curie (Ci) 3.7 × 10¹⁰ decays / s

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Radium-226 Decay Chain�

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Time Dating Using Lead Isotopes

• A plot of the abundance ratio of ²⁰⁶Pb / ²⁰⁴Pb versus ²⁰⁷Pb / ²⁰⁴Pb can be a sensitive indicator of the age of lead ores. Such techniques have been used to show that meteorites, believed to be left over from the formation of the solar system, are 4.55 billion years old.
• The growth curve for lead ores from various deposits:

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The age of the specimens can be obtained from the abundance ratio of ²⁰⁶Pb/^204Pb versus ²⁰⁷Pb/²⁰⁴Pb.

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Radioactive Carbon Dating

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• Radioactive ¹⁴C is produced in our atmosphere by the bombardment of ¹⁴N by neutrons produced by cosmic rays.

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• When living organisms die, their intake of ¹⁴C ceases, and the ratio of ¹⁴C / ¹²C (= R) decreases as ¹⁴C decays. The period just before 9000 years ago had a higher ¹⁴C / ¹²C ratio by factor of about 1.5 than it does today.
• Because the half-life of ¹⁴C is 5730 years, it is convenient to use the ¹⁴C / ¹²C ratio to determine the age of objects over a range up to 45,000 years ago.

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Calculate the binding energies of the most loosely bound neutron in the following nuclei

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What is the energy released when three alpha parti-

cles combine to form 12C?

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From chapter12 quiz

Which of the following reasons explains why the neutrino must exist?

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A. The neutrino is a product of gamma ray decay.

B. The neutrino is necessary to allow for the correct spin angular momentum conservation in a nuclear disintegration.

C. The neutrino is necessary to carry away a charge in a nuclear disintegration.

D. The neutrino is the force carrier that holds together quarks within protons and neutrons.

E. The neutrino decays into electrons and protons in an unstable nucleus.

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If the age of the Earth is 4.5 billion years, what should the ratio of N^206 (Pb)/(N ^238 (U)) in a uranium-bearing rock as old as the Earth?

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