Modern Physics: Atomic Structure, Nuclear Processes, and Energy from Mass

Quantized energy levels

The idea that electrons in atoms can only have specific allowed energies (discrete “rungs”), not any value in a continuous range; this helps explain atomic stability and line spectra.

Energy-level transition

A change of an electron from one allowed energy level to another; the photon energy involved equals the energy difference between the two levels (ΔE = Ehigh − Elow).

Line emission spectrum

A spectrum consisting of bright lines at specific wavelengths, produced because atoms emit photons only for discrete energy differences between quantized levels.

Absorption spectrum

A spectrum with dark lines at specific wavelengths where photons are absorbed to raise electrons to higher energy levels; line positions correspond to the same energy gaps as emission lines for that element.

Photon

A particle of light that carries a discrete amount of energy; emitted or absorbed when an electron changes energy levels.

Emission (atomic)

Process where an electron drops from a higher to a lower energy level and the atom releases a photon with energy equal to the energy drop.

Absorption (atomic)

Process where an electron jumps from a lower to a higher energy level by absorbing a photon with exactly the required energy difference.

Photon relationships

Equations connecting photon energy and wave properties: E = hf, c = λf, and therefore E = hc/λ (higher energy ↔ higher frequency ↔ shorter wavelength).

Planck’s constant (h)

Constant that links photon energy to frequency in E = hf; h = 6.63 × 10^−34 J·s.

Electron-volt (eV)

A convenient energy unit for atomic scales; 1 eV = 1.60 × 10^−19 J.

Hydrogen Bohr energy levels

Model for hydrogen’s electron energies: E_n = −13.6 eV / n^2 (n = 1, 2, 3, …); negative sign indicates the electron is bound to the nucleus.

Nucleons

The particles that make up the nucleus: protons and neutrons.

Strong nuclear force

A very strong attractive force between nucleons that acts over extremely short distances and can overcome electric repulsion between protons to bind the nucleus.

Isotope

Atoms of the same element (same number of protons) that have different numbers of neutrons, giving different mass numbers.

Nuclear notation (^{A}_{Z}X)

Standard way to label nuclei: X is the element symbol, Z is atomic number (protons), A is mass number (protons + neutrons), and neutrons = A − Z.

Alpha decay

A decay where the nucleus emits an alpha particle (a helium-4 nucleus): ^AZ X → ^(A−4)(Z−2) Y + ^4_2 α.

Beta minus decay (β−)

A decay where a neutron turns into a proton and an electron is emitted: ^AZ X → ^A(Z+1) Y + ^0_−1 e (A stays the same; Z increases by 1).

Beta plus decay (β+) / positron emission

A decay where a proton turns into a neutron and a positron is emitted: ^AZ X → ^A(Z−1) Y + ^0_+1 e (A stays the same; Z decreases by 1).

Gamma emission

Emission of a high-energy photon from an excited nucleus: ^AZ X* → ^AZ X + γ; does not change A or Z, only the nucleus’s energy state.

Half-life (T_1/2)

The time required for half of the undecayed nuclei in a sample to decay; used in N(t) = N0(1/2)^(t/T_1/2).

Decay constant (λ)

Parameter describing the probability per unit time that a nucleus decays; appears in N(t) = N0 e^(−λt) and relates to half-life by T_1/2 = (ln 2)/λ.

Mass defect

The difference between the sum of the masses of separated protons/neutrons and the mass of the bound nucleus; the “missing” mass corresponds to binding energy.

Binding energy

Energy required to completely separate a nucleus into free protons and neutrons; equals the mass defect converted to energy (E = Δm c^2).

Mass–energy equivalence

Principle that mass can be converted to energy; for nuclear processes the key form is ΔE = Δm c^2 (mass-energy is conserved even if mass alone changes).

Atomic mass unit (u)

A convenient unit for nuclear masses; 1 u = 1.66 × 10^−27 kg (often paired with 1 u·c^2 ≈ 931 MeV for energy estimates).