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University Physics Volume 3

10.1 Properties of Nuclei

University Physics Volume 310.1 Properties of Nuclei
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Describe the composition and size of an atomic nucleus
  • Use a nuclear symbol to express the composition of an atomic nucleus
  • Explain why the number of neutrons is greater than protons in heavy nuclei
  • Calculate the atomic mass of an element given its isotopes

The atomic nucleus is composed of protons and neutrons (Figure 10.2). Protons and neutrons have approximately the same mass, but protons carry one unit of positive charge (+e),(+e), and neutrons carry no charge. These particles are packed together into an extremely small space at the center of an atom. According to scattering experiments, the nucleus is spherical or ellipsoidal in shape, and about 1/100,000th the size of a hydrogen atom. If an atom were the size of a major league baseball stadium, the nucleus would be roughly the size of the baseball. Protons and neutrons within the nucleus are called nucleons.

The figure shows a cluster of red and blue spheres packed closely together. The red spheres are labeled neutrons and the blue ones protons.
Figure 10.2 The atomic nucleus is composed of protons and neutrons. Protons are shown in blue, and neutrons are shown in red.

Counts of Nucleons

The number of protons in the nucleus is given by the atomic number, Z. The number of neutrons in the nucleus is the neutron number, N. The total number of nucleons is the mass number, A. These numbers are related by

A=Z+N.A=Z+N.
(10.1)

A nucleus is represented symbolically by

ZAX,ZAX,
(10.2)

where X represents the chemical element, A is the mass number, and Z is the atomic number. For example, 612C612C represents the carbon nucleus with six protons and six neutrons (or 12 nucleons).

A graph of the number N of neutrons versus the number Z of protons for a range of stable nuclei (nuclides) is shown in Figure 10.3. For a given value of Z, multiple values of N (blue points) are possible. For small values of Z, the number of neutrons equals the number of protons (N=P),(N=P), and the data fall on the red line. For large values of Z, the number of neutrons is greater than the number of protons (N>P),(N>P), and the data points fall above the red line. The number of neutrons is generally greater than the number of protons for Z>15.Z>15.

A graph showing number of neutrons, N versus number of protons, Z. A straight line on the graph is labeled N equal to Z. Another, jagged line, is labeled band of stability. This has incremental steps. It starts at the origin. At Z = 80, the value of N is 120.
Figure 10.3 This graph plots the number of neutrons N against the number of protons Z for stable atomic nuclei. Larger nuclei, have more neutrons than protons.

A chart based on this graph that provides more detailed information about each nucleus is given in Figure 10.4. This chart is called a chart of the nuclides. Each cell or tile represents a separate nucleus. The nuclei are arranged in order of ascending Z (along the horizontal direction) and ascending N (along the vertical direction).

Figure shows a chart of nuclides, with ascending Z along the horizontal direction and ascending N along the vertical direction. The cells along diagonal in the centre of the chart are color coded to indicate that they are stable.
Figure 10.4 Partial chart of the nuclides. For stable nuclei (dark blue backgrounds), cell values represent the percentage of nuclei found on Earth with the same atomic number (percent abundance). For the unstable nuclei, the number represents the half-life.

Atoms that contain nuclei with the same number of protons (Z) and different numbers of neutrons (N) are called isotopes. For example, hydrogen has three isotopes: normal hydrogen (1 proton, no neutrons), deuterium (one proton and one neutron), and tritium (one proton and two neutrons). Isotopes of a given atom share the same chemical properties, since these properties are determined by interactions between the outer electrons of the atom, and not the nucleons. For example, water that contains deuterium rather than hydrogen (“heavy water”) looks and tastes like normal water. The following table shows a list of common isotopes.

Element Symbol Mass Number Mass (Atomic Mass Units) Percent Abundance* Half-life**
Hydrogen H 1 1.0078 99.99 stable
2HorD2HorD 2 2.0141 0.01 stable
3H3H 3 3.0160 12.32 y
Carbon 12C12C 12 12.0000 98.91 stable
13C13C 13 13.0034 1.1 stable
14C14C 14 14.0032 5730 y
Nitrogen 14N14N 14 14.0031 99.6 stable
15N15N 15 15.0001 0.4 stable
16N16N 16 16.0061 7.13 s
Oxygen 16O16O 16 15.9949 99.76 stable
17O17O 17 16.9991 0.04 stable
18O18O 18 17.9992 0.20 stable
19O19O 19 19.0035 26.46 s
Table 10.1 Common Isotopes *No entry if less than 0.001 (trace amount).
**Stable if half-life > 10 seconds.

Why do neutrons outnumber protons in heavier nuclei (Figure 10.5)? The answer to this question requires an understanding of forces inside the nucleus. Two types of forces exist: (1) the long-range electrostatic (Coulomb) force that makes the positively charged protons repel one another; and (2) the short-range strong nuclear force that makes all nucleons in the nucleus attract one another. You may also have heard of a “weak” nuclear force. This force is responsible for some nuclear decays, but as the name implies, it does not play a role in stabilizing the nucleus against the strong Coulomb repulsion it experiences. We discuss strong nuclear force in more detail in the next chapter when we cover particle physics. Nuclear stability occurs when the attractive forces between nucleons compensate for the repulsive, long-range electrostatic forces between all protons in the nucleus. For heavy nuclei (Z>15),(Z>15), excess neutrons are necessary to keep the electrostatic interactions from breaking the nucleus apart, as shown in Figure 10.3.

Figure a shows a cluster of small red and blue circles. There is a blue proton in the center, surrounded by red neutrons. There are more protons at the periphery, which have arrows pointing outwards. Figure b shows the same cluster. Arrows show both protons and neutrons being attracted towards an adjacent neutron.
Figure 10.5 (a) The electrostatic force is repulsive and has long range. The arrows represent outward forces on protons (in blue) at the nuclear surface by a proton (also in blue) at the center. (b) The strong nuclear force acts between neighboring nucleons. The arrows represent attractive forces exerted by a neutron (in red) on its nearest neighbors.

Because of the existence of stable isotopes, we must take special care when quoting the mass of an element. For example, Copper (Cu) has two stable isotopes:

2963Cu(62.929595g/mol)with an abundance of69.09%2963Cu(62.929595g/mol)with an abundance of69.09%
2965Cu(64.927786g/mol)with an abundance of30.91%2965Cu(64.927786g/mol)with an abundance of30.91%

Given these two “versions” of Cu, what is the mass of this element? The atomic mass of an element is defined as the weighted average of the masses of its isotopes. Thus, the atomic mass of Cu is mCu=(62.929595)(0.6909)+(64.927786)(0.3091)=63.55g/mol.mCu=(62.929595)(0.6909)+(64.927786)(0.3091)=63.55g/mol. The mass of an individual nucleus is often expressed in atomic mass units (u), where u=1.66054×10−27kgu=1.66054×10−27kg. (An atomic mass unit is defined as 1/12th the mass of a 12C12C nucleus.) In atomic mass units, the mass of a helium nucleus (A = 4) is approximately 4 u. A helium nucleus is also called an alpha (α) particle.

Nuclear Size

The simplest model of the nucleus is a densely packed sphere of nucleons. The volume V of the nucleus is therefore proportional to the number of nucleons A, expressed by

V=43πr3=kA,V=43πr3=kA,

where r is the radius of a nucleus and k is a constant with units of volume. Solving for r, we have

r=r0A1/3r=r0A1/3
(10.3)

where r0r0 is a constant. For hydrogen (A=1A=1), r0r0 corresponds to the radius of a single proton. Scattering experiments support this general relationship for a wide range of nuclei, and they imply that neutrons have approximately the same radius as protons. The experimentally measured value for r0r0 is approximately 1.2 femtometer (recall that 1fm=10−15m1fm=10−15m).

Example 10.1

The Iron NucleusFind the radius (r) and approximate density (ρρ) of a Fe-56 nucleus. Assume the mass of the Fe-56 nucleus is approximately 56 u.

Strategy (a) Finding the radius of 56Fe56Fe is a straightforward application of r=r0A1/3r=r0A1/3, given A=56.A=56. (b) To find the approximate density of this nucleus, assume the nucleus is spherical. Calculate its volume using the radius found in part (a), and then find its density from ρ=m/V.ρ=m/V.

Solution

  1. The radius of a nucleus is given by
    r=r0A1/3.r=r0A1/3.

    Substituting the values for r0r0 and A yields
    r=(1.2fm)(56)1/3=(1.2fm)(3.83)=4.6fm.r=(1.2fm)(56)1/3=(1.2fm)(3.83)=4.6fm.
  2. Density is defined to be ρ=m/V,ρ=m/V, which for a sphere of radius r is
    ρ=mV=m(4/3)πr3.ρ=mV=m(4/3)πr3.

    Substituting known values gives
    ρ=56u(1.33)(3.14)(4.6fm)3=0.138u/fm3.ρ=56u(1.33)(3.14)(4.6fm)3=0.138u/fm3.

    Converting to units of kg/m3,kg/m3, we find
    ρ=(0.138u/fm3)(1.66×10−27kg/u)(1fm10−15m)=2.3×1017kg/m3.ρ=(0.138u/fm3)(1.66×10−27kg/u)(1fm10−15m)=2.3×1017kg/m3.

Significance

  1. The radius of the Fe-56 nucleus is found to be approximately 5 fm, so its diameter is about 10 fm, or 10−14m.10−14m. In previous discussions of Rutherford’s scattering experiments, a light nucleus was estimated to be 10−15m10−15m in diameter. Therefore, the result shown for a mid-sized nucleus is reasonable.
  2. The density found here may seem incredible. However, it is consistent with earlier comments about the nucleus containing nearly all of the mass of the atom in a tiny region of space. One cubic meter of nuclear matter has the same mass as a cube of water 61 km on each side.
Check Your Understanding 10.1

Nucleus X is two times larger than nucleus Y. What is the ratio of their atomic masses?

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