transformations of matter at the nuclear level and laws acting at near-light speeds and high energies
Formula for quantization of orbital momentum: m·vₙ·rₙ = h / 2π · n — the electron's orbital angular momentum can only take discrete values, multiples of Planck's constant
According to Bohr, an electron can only move in specific orbits where its orbital angular momentum is a multiple of **h / 2π**. This condition explains the existence of stable energy levels without radiation.
Formula: ν = (E₂ − E₁) / h — emission frequency is determined by the energy difference between levels divided by Planck's constant
When transitioning from level **E₁** to **E₂**, an atom emits or absorbs a photon with frequency **ν**, corresponding to the energy difference between the levels. This formula is a direct consequence of the quantized nature of spectra.
Rydberg formula: νₘᵢₙ = R·(1/n² − 1/m²) — emission frequency during transitions between atomic energy levels, where R — Rydberg constant, n and m — quantum numbers, m > n
The Rydberg formula describes the spectral lines of hydrogen: emission frequency depends on the initial (m) and final (n) levels. The constant **R** is universal for hydrogen-like systems.
Formula: ΔE = Δm·c² = Δm·931.5 MeV — nuclear binding energy is proportional to the mass defect, where c — speed of light, Δm — difference between nucleon mass and nuclear mass
Binding energy is the energy required to break a nucleus into individual nucleons. Mass defect (**Δm**) is the difference between the sum of the masses of the nucleons and the mass of the nucleus. Mass-energy equivalence is used: **E = mc²**, often converting mass to MeV using the factor 931.5.
Formula: Δm = Σm_nucleons − m_nucleus — mass defect is the difference between the sum of the masses of individual nucleons and the mass of the nucleus
To calculate the mass defect, the difference between the total mass of the constituent particles before the reaction and the actual mass of the formed nucleus is used. This effect reflects energy stability: the larger **Δm**, the stronger the nucleus.
Formula: N = N₀·2^(−t / T₁/₂) — the number of undecayed nuclei decreases exponentially over time, where N₀ — initial number of nuclei, t — time, T₁/₂ — half-life
This formula describes the exponential decrease in the number of radioactive nuclei **N** over time. **N₀** is the initial number of atoms, **t** is the elapsed time, and **T1/2** is the half-life, at which half of the nuclei remain. This is fundamental for calculating activity, lifespan, and material safety.
Formula: l = l₀·√(1 − v² / c²) — the length of a moving body decreases along the direction of motion, where l₀ — proper length, v — body's velocity, c — speed of light
The length of an object moving at velocity **v** relative to an observer appears shorter than its proper length **l₀** (length in the rest frame). The effect becomes significant at speeds close to the speed of light **c**.
Formula: t = t₀ / √(1 − v² / c²) — time in a moving reference frame increases compared to proper time, where t₀ — proper time, v — velocity, c — speed of light
Time measured in a moving reference frame **t** runs slower than the proper time **t₀** in a stationary frame. This means that for a moving object, time slows down compared to a stationary one.
Formula: v = (v₁ + v₂) / (1 + v₁·v₂ / c²) — when adding velocities in relativistic mechanics, the speed of light limit is taken into account, where v₁ and v₂ — velocities to be added, c — speed of light
In special relativity, velocities do not simply add arithmetically. This formula shows how two velocities **v₁** and **v₂** are combined to yield a resultant velocity **v**, where the resultant velocity will never exceed the speed of light **c**.
Formula: p = m₀·v / √(1 − v² / c²) — momentum of a particle with non-zero rest mass moving at near-light speed, taking into account relativistic effects
A particle's momentum increases with its speed, especially as it approaches the speed of light. **m₀** is the particle's rest mass, **v** is its velocity. This effect is a consequence of relativistic mass increase.
Formula: m = m₀ / √(1 − v² / c²) — the mass of a body increases with increasing speed, where m₀ — rest mass, v — body's velocity, c — speed of light
The mass of an object increases as its speed increases. **m₀** is the object's rest mass. At speeds close to the speed of light, mass tends towards infinity.
Formula: ΔE = Δm·c² — energy arising from a change in mass is proportional to the mass defect and the square of the speed of light
Einstein's famous formula showing that mass and energy are equivalent. A change in mass **Δm** corresponds to a change in energy **ΔE**. This is a fundamental principle of nuclear physics and SRT.
Formula for total energy: E = mc² — the energy of a body is proportional to its mass and the square of the speed of light, reflecting the equivalence of mass and energy
The total energy of a particle includes both its rest energy (associated with its rest mass) and its kinetic energy, associated with its motion. Here, **m** is the relativistic mass.
Formula for relativistic kinetic energy: W = m₀·c²·(1 / √(1 − v² / c²) − 1) — the energy of motion of a body with non-zero rest mass at near-light speed
Kinetic energy in SRT differs from the classical formula. It accounts for the relativistic increase in mass and approaches infinity as the object's speed approaches the speed of light.