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 → Radioactivity ↓


Half life


Introduction to radioactivity

Radioactivity is a fundamental concept in nuclear and particle physics, where certain types of atoms or isotopes undergo spontaneous transformation. This transformation involves the emission of ionizing particles and radiation. Studying these processes helps us understand the structure of atomic nuclei and their underlying forces.

Atoms are made up of protons, neutrons, and electrons. The nucleus of an atom contains protons and neutrons, while electrons orbit the nucleus. Radioactive elements have unstable nuclei that disintegrate by emitting radiation, eventually turning into a different element. This process is known as radioactive decay.

Types of radioactive decay

There are several types of radioactive decay, each of which emits different particles:

  • Alpha decay: An alpha particle (two protons and two neutrons) is released from the nucleus.
  • Beta decay: When a neutron is converted into a proton or vice versa, a beta particle (electron or positron) is emitted.
  • Gamma decay: The nucleus releases energy in the form of gamma rays, which are high-energy photons.

What is half-life?

The concept of half-life is important in understanding the process of radioactive decay. The half-life of a radioactive substance is defined as the time it takes for half of the radioactive nuclei in a sample to decay. It is an exponential process, meaning that a proportion of the substance decreases by half during each half-life period. This is a typical property of decay processes.

Mathematical description of half-life

The decay of radioactive substances obeys the exponential decay law. If N(t) is the number of undisintegrated nuclei at time t, then the decay can be described by the equation:

N(t) = N_0 * e^(-λt)

Where:

  • N_0 is the initial number of nuclei.
  • λ (lambda) is the decay constant, which is unique for each radioactive substance.
  • e is the base of the natural logarithm, which is approximately equal to 2.71828.

The half-life T_{1/2} is related to the decay constant by the following formula:

T_{1/2} = ln(2) / λ

where ln is the natural logarithm.

Visual example: Exponential decay

Imagine a sample containing 1000 radioactive nuclei. If the half-life is 5 years, then after 5 years, about 500 nuclei will remain. After another 5 years (10 years total), about 250 will remain, and so on. Every 5 years the number of undisintegrated nuclei is halved.

Applications of half-life

Radiocarbon dating

One of the best-known applications of half-life is radiocarbon dating. This method is used to estimate the age of organic material, such as fossils, by measuring the amount of carbon-14, which is a radioactive isotope of carbon. The half-life of carbon-14 is about 5730 years. By comparing the ratio of carbon-14 and carbon-12 in a sample, scientists can determine how long it has been since the organism died.

Medical uses

In medicine, certain isotopes are used in diagnostic imaging, such as PET scans. These isotopes are chosen based on their half-life to ensure they can decay quickly enough to minimize radiation exposure to patients. For example, technetium-99m, which is widely used in medical imaging, has a half-life of about 6 hours, making it ideal for short diagnostic tests.

Nuclear power

In nuclear power plants, the control of radioactive isotopes is critical for safe operations. Understanding half-lives helps manage radioactive waste. Isotopes with long half-lives, such as plutonium-239, require careful long-term storage, while isotopes with short half-lives decay quickly to safe levels.

Lesson example: Calculating the remaining nuclei

Suppose a radioactive isotope has a half-life of 10 years, and you start with a sample containing 8000 radioactive atoms. You can calculate the number of atoms remaining after a certain period of time using the concept of half-life.

After one half-life (10 years), the number of atoms remaining is:

N(10) = 8000 * (1/2) = 4000

The number remaining after two half-lives (20 years) is:

N(20) = 4000 * (1/2) = 2000

As we proceed through these calculations, it is noticeable that with each half-life period the amount of undissociated atoms is halved.

Importance and limitations of half-life

Half-life is an essential aspect of radioactivity, providing a consistent measure for understanding and predicting the behavior of radioactive materials over time. However, it is important to remember that half-life is a statistical concept that applies well to large numbers of atoms due to the random nature of individual nuclear decay.

In practice, it is often applied in areas where accurate predictions of decay over time are required, while compensating for its probabilistic nature. The accuracy of using the half-life as a predictive measure is reduced in systems with a very small number of atoms.

Conclusion

Understanding the concept of half-life provides a deeper insight into nuclear processes and the behavior of radioactive minerals or elements. Whether it is in scientific research, medical applications, archaeological studies or energy production, understanding how isotopes decay over time allows us to harness and manage the benefits and challenges posed by radioactivity.


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Half life

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