Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Nuclear and particle physics → Atomic Structure ↓


Nuclear binding energy


The concept of nuclear binding energy is important in understanding the stability and structure of atomic nuclei in the field of nuclear and particle physics. Simply put, nuclear binding energy is the energy required to split the nucleus of an atom into its constituent protons and neutrons. Alternatively, it is the energy released during the formation of the nucleus from these nucleons.

Atomic nucleus: The basics

An atom consists of a nucleus and electrons that orbit around it. The nucleus, which contains almost all of the atom's mass, is composed of protons and neutrons. Protons are positively charged, while neutrons are neutral particles. The number of protons (atomic number) determines the type of element, and the combination of protons and neutrons (mass number) determines the isotope of an element.

Understanding binding energy

To understand why binding energy is so important, consider that protons, all positively charged, repel each other due to electrostatic forces. The fact that they still stay together within a nucleus means that a strong attractive force is at work, called the nuclear force.

The difference in energy between the bound state of the nucleons inside the nucleus and the free state (when they are separated and not bound) is the nuclear binding energy. A more bound (or tightly bound) nucleus corresponds to a higher nuclear binding energy.

Formula of binding energy

To calculate the nuclear binding energy, we use the mass-energy equivalence principle provided by Einstein's famous equation:

E = mc^2

Where:

  • E is the energy
  • m is the mass
  • c is the speed of light in vacuum

Nuclear binding energy can be described as follows:

BE = (Z * mp + N * mn - M_nucleus) * c^2

Where:

  • BE is the nuclear binding energy
  • Z is the number of protons
  • N is the number of neutrons
  • mp is the mass of a proton
  • mn is the mass of a neutron
  • M_nucleus is the observed mass of the nucleus

This equation represents the mass defect: the discrepancy between the sum of the masses of the individual nucleons and the actual mass of the nucleus. The "missing mass" is converted into energy, the binding energy that holds the nucleus together.

Importance of binding energy

Nuclear binding energy is an indicator of the stability of the nucleus. When comparing the binding energy per nucleon (the average energy per particle in the nucleus), we find that iron and nickel have some of the highest values. These elements are at the peak of the binding energy per nucleon curve, which means they are the most stable.

Mass number binding energy per nucleon Iron, Nickel

Energy release in nuclear reactions

The emission or absorption of energy during nuclear reactions such as fission and fusion is intricately linked to binding energy.

Nuclear fission

In nuclear fission, a heavy nucleus splits into two smaller nuclei with some neutrons and energy is released. The source of this energy release is the difference in binding energy between the original nucleus and the resulting products.

Nuclear fusion

In contrast, nuclear fusion combines lighter nuclei to form heavier nuclei. Fusion releases a great deal of energy because the binding energy per nucleon of the resulting nucleus is greater than the binding energy of the original nucleus.

Lesson example: Calculating binding energy

Let's calculate the binding energy per nucleon for a helium-4 nucleus ((^4_2He)). Helium-4 contains 2 protons and 2 neutrons. If the mass of the proton is about 1.00728 atomic mass units (amu) and the mass of the neutron is about 1.00866 amu, and the observed mass of the helium nucleus is about 4.00150 amu, then:

Total mass of nucleons = 2(1.00728) + 2(1.00866) = 4.03188 amu
Mass defect = 4.03188 amu - 4.00150 amu = 0.03038 amu
Energy equivalent (in MeV) = 0.03038 amu * 931.5 MeV/amu ≈ 28.3 MeV
Binding energy per nucleon = 28.3 MeV / 4 = 7.075 MeV

This calculation shows that helium-4 is relatively stable and its binding energy per nucleon is about 7.1 MeV.

Implications in physics and cosmology

Analyzing binding energy has profound implications for understanding stellar phenomena. For example, stars shine because nuclear fusion occurs in their cores, converting hydrogen into helium and releasing energy in the form of binding energy differences. This process is key to understanding the life cycle of stars and the creation of elements in the universe.

Understanding nuclear binding energy provides insight into the processes that drive powerful explosions such as supernovae and the synthesis of elements heavier than iron in the universe, a process known as nucleosynthesis.

Conclusion

Nuclear binding energy is a fundamental concept that holds clues about the structure and stability of atomic nuclei. It not only explains how nuclei remain bound together despite the electrostatic repulsion between protons, but also helps us understand energy transformations during nuclear reactions. Whether it's the energy source powering the Sun or the destructive potential of nuclear weapons, binding energy is at the heart of nuclear physics and plays a vital role in both scientific discovery and the technological advancements of our modern world.


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Nuclear binding energy

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