Nuclear Binding Energy and Nuclear Radiation

Nuclear Binding Energy and Mass Deficit

• Experiments show the total mass of a nucleus is less than the sum of the masses of its constituent nucleons.

• Mass Defect:

  • Defined as the difference between the measured mass of a nucleus and the total mass of its constituents.

  • Calculated using the formula:

  • \Delta m=Zmp+(A-Z)mn-mtotal
    

    • Where:

    • Z = proton number

    • A = nucleon number

    • mp = mass of a proton (kg)

    • mn = mass of a neutron (kg)

    • m{total} = measured mass of the nucleus (kg)

Mass-Energy Equivalence

• A system of separated nucleons has a greater mass than that of bound nucleons.

• Binding Energy:

  • The energy required to separate a nucleus into its individual protons and neutrons.

  • Formation of a nucleus from isolated nucleons releases energy, indicating it's an exothermic reaction.

  • Calculated using:

  • E = \Delta m \cdot c^2

    • Where:

    • E = energy (J)

    • m = mass (kg)

    • c = speed of light ($3.00 \times 10^8 \ m/s$)

• Mass-Energy Equivalence:

  • Proposed by Einstein: Matter can become energy and vice versa.

• Examples of mass-energy equivalence include:

  • Fusion of hydrogen into helium in stars.

  • Fission of uranium in nuclear power plants.

  • Nuclear explosions.

  • High-energy collisions in particle accelerators.

Atomic Mass Unit (a.m.u)

• Unified atomic mass unit (u or a.m.u) is approximately the mass of one proton or neutron:

  • 1{ u}=1.66\times10^{-27}\text{ kg}

• Defined as one-twelfth of a carbon-12 atom, which has a mass of exactly 12 u.

• Since mass and energy are interchangeable:

  • 1{ u}=931.5ext{ MeV}

• Practically used in nuclear physics for expressing the mass of subatomic particles.

Binding Energy per Nucleon

• Defined as the total binding energy of a nucleus divided by the number of nucleons.

• A higher binding energy per nucleon implies greater stability:

  • Iron (A = 56) has the highest binding energy per nucleon, making it the most stable element.

• Key Features on Binding Energy Graph:

  • Low A values (light elements):

  • Lower binding energy per nucleon.

  • More likely to undergo fusion (e.g., Helium-4, Carbon-12).

  • Helium-4 is particularly stable.

  • High A values (heavy elements):

  • High initial binding energy per nucleon that gradually decreases with increasing A.

  • Heavier elements are typically less stable and likely to undergo fission.

Nuclear Fusion

• Defined as the process where small nuclei combine to form larger nuclei, releasing energy.

• Primarily occurs with low mass nuclei (like hydrogen and helium).

  • Example: Fusion of deuterium and tritium forms helium and releases energy.

Conditions for Nuclear Fusion

• High kinetic energy required to overcome the electrostatic repulsion between protons:

  • Achievable in environments with temperatures around 100 \times 10^6 Kelvin.

• Requires very high density.

Fusion Products

• When two protons fuse, deuterium is produced.

• In stellar cores, deuterium fuses with a tritium nucleus to create a helium nucleus and release energy, sustaining stellar processes.