Chapter 01Basic sciencesLast reviewed: · What's new
Atomic and Nuclear Physics
This chapter's English text is a draft translation (prepared with Claude and checked against a terminology glossary and automatic checks); specialist review is in progress. Please report errors with the feedback button.
Learning objectives and modules2 modules · about 1 h 30 min
Learning objectives
After this chapter you will be able to:
Distinguish the contributions of the Thomson, Rutherford, Bohr and quantum mechanical models of the atom.
Define isotope, isobar, isotone and isomer with examples used in nuclear medicine.
Relate mass defect and binding energy to mass–energy equivalence and interpret the binding energy per nucleon curve.
Predict nuclear stability and the expected direction of decay from the neutron-to-proton ratio.
Explain characteristic X-ray and Auger electron emission and how they vary with atomic number.
Modules
The chapter is divided into study modules without changing the order of the text: read, watch the animations, then answer the questions linked to those headings.
The idea of the atom as the smallest indivisible unit of matter was put forward by Democritus in ancient Greece. Modern atomic physics took shape with experimental discoveries at the end of the 19th century.
Dalton Model (1803)
Atoms are indivisible, indestructible spherical particles. Atoms of the same element are identical. Chemical reactions occur by rearrangement of atoms. The model lost its validity with the discovery of subatomic particles.
Thomson Model – "Plum Pudding" (1897)
J.J. Thomson discovered the electron with his cathode ray experiment. In this model the atom consists of negative electrons embedded in a positively charged sphere. The net charge of the atom is zero. Thomson also determined the charge-to-mass ratio of the electron (e/m).
Rutherford Model – "Nuclear Atom" (1911)
The Geiger–Marsden alpha scattering experiment proved that the atom has a small, dense, positively charged nucleus at its centre. Most alpha particles passed through the gold foil, but a small fraction was scattered through large angles, showing that the mass is concentrated in the nucleus. Electrons orbit the nucleus. However, the model cannot explain why, according to classical electromagnetic theory, orbiting electrons do not continuously radiate energy and spiral into the nucleus.
Bohr Model (1913)
Niels Bohr introduced the concept of quantised orbits for the hydrogen atom:
•
Electrons can exist only at certain energy levels (n = 1, 2, 3...)
•
An electron moving between orbits emits or absorbs a photon equal to the energy difference: ΔE = hν,
–
h (Planck constant):A universal constant of approximately 6.626 × 10⁻³⁴ Js.
–
ν (nu / frequency): Frequency of the light or electromagnetic wave (Hz or s⁻¹).
•
Ground state: n = 1, the lowest energy level
•
Excited state: n > 1
•
The electron with the highest binding energy is in the K shell (n = 1)
Although the Bohr model was successful for hydrogen, it could not explain atoms with many electrons.
BOARD TIP
▸
The Bohr model is frequently asked in board exams. Key point: electrons can exist only at certain energy levels, and the energy of the photon emitted in a transition is calculated as ΔE = hν. Characteristic X-rays are based on this principle!
Figure 1.1: Development of atomic models from Dalton to Bohr (schematic).
Did this figure help you understand the topic?
Quantum Mechanical Model (1926)
With the Schrödinger wave equation, probability clouds (orbitals) replaced exact electron orbits. Four quantum numbers uniquely describe each electron in an atom:
•
Principal quantum number (n): Energy level (shell). n = 1, 2, 3... → K, L, M...
•
Angular momentum quantum number (l): Shape of the subshell. l = 0, 1, 2... (n-1) → s, p, d, f
•
Magnetic quantum number (mₗ): Determines the orientation of the orbital in space. Takes integer values from −l to +l (including 0).
•
Spin quantum number (mₛ): Orientation of the electron's intrinsic angular momentum (spin). Takes only the values +½ or −½.
Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. This principle determines electron configuration and the periodic table.
Figure 1.2: Shell capacities (2n²): subshells, orbitals and the Pauli exclusion principle (schematic).
Did this figure help you understand the topic?
Shell Capacities:
Each shell can hold at most 2n² electrons:
•
K shell (n=1): 2 electrons
•
L shell (n=2): 8 electrons
•
M shell (n=3): 18 electrons
•
N shell (n=4): 32 electrons
Atomic Structure Terminology
•
Atomic number (Z): Number of protons → determines the identity of the element
•
Mass number (A): Number of protons + neutrons (A = Z + N)
•
Isotope: Same Z, different N (e.g. I-123, I-125, I-131 → all Z=53)
•
Isobar: Same A, different Z (e.g. Mo-99 and Tc-99 → A=99)
•
Isotone: Same N, different Z
•
Isomer: Same Z and A, different energy state (e.g. Tc-99m and Tc-99)
BOARD TIP
▸
Distinguishing isotope/isobar/isotone/isomer is a classic board question. Remember: isotope → same protons (Z), isobar → same A (mass), isotone → same neutrons (N), isomer → everything the same but a different energy (metastable). Tc-99m is an isomer of Tc-99!
Figure 1.3: Isotope, isotone, isobar and isomer relationships on the chart of nuclides (around Tc-99).
Did this figure help you understand the topic?
Characteristic X-rays and Auger Electrons
When an electron is removed from an inner shell (photoelectric effect, electron capture, etc.), an electron from an outer shell fills the vacancy. This transition has two possible outcomes:
1. Characteristic X-ray: The energy difference is emitted as a photon. The photon energy equals the energy difference between the two shells. It is named Kα, Kβ, etc. (transition to the K shell).
2. Auger electron: The transition energy is transferred, without a photon, to another electron of the same atom, which is ejected from the atom. This electron is the Auger electron.
•
Energy calculation: The energy of the Kα X-ray equals the difference between the binding energies of the K and L shells (EK − EL). For a KLL Auger electron, the binding energy of the ejected L electron is subtracted once more: E ≈ EK − 2EL.
WHY IT MATTERS · Fluorescence yield and Coster-Kronig transitions
What it is: An inner-shell vacancy is filled either by a characteristic X-ray or by an Auger electron; the probability of X-ray emission is the fluorescence yield (ω). A Coster-Kronig transition is a type of Auger transition in which the vacancy moves between subshells of the same shell (e.g. L1 → L3) and a very low-energy electron is ejected (the Figure 1.4 animation shows both routes for a K vacancy).
•
In numbers: The K-shell fluorescence yield rises with Z: oxygen 0.008, calcium 0.16, technetium 0.78, iodine 0.88. For the L and M shells it stays low even in heavy atoms (iodine L subshells 0.04-0.08), so each vacancy turns into an electron cascade. Auger and Coster-Kronig electrons per decay: Tc-99m ~4, In-111 ~15, I-125 ~25 (depending on the calculation method).
•
Why it matters clinically: Most of these electrons have energies from a few eV to a few keV and ranges of nanometres to micrometres: an atom decaying close to DNA causes dense, high-LET-like damage, while outside the cell the effect is negligible. This is why Auger therapy (I-125, In-111, Tb-161) must deliver the agent into the cell nucleus.
•
In the exam: "A K vacancy gives mostly X-rays at high Z, so why does I-125 emit many Auger electrons?" → The outer shells have low fluorescence yields; the vacancy cascade produces many low-energy Auger and Coster-Kronig electrons.
•
Sources: Krause, J Phys Chem Ref Data 1979; Howell (AAPM TG-6), Med Phys 1992; Ku et al., EJNMMI Radiopharm Chem 2019.
BOARD TIP
▸
Auger electron emission dominates in low-Z atoms, whereas characteristic X-ray emission dominates in high-Z atoms. This is key to understanding the biological effect of Auger emitters such as I-125 → Auger electrons have a very short range and, close to DNA, act like high-LET radiation.
Electron capture by I-125 leaves a vacancy in the K shell, filled with emission of a characteristic X-ray or an Auger electron; fluorescence yield and the Auger cascade at DNA scale are shown.
Continuous stream · no seeking
Key point: in light atoms the vacancy is mostly filled with an Auger electron, in heavy atoms with an X-ray; I-125 releases ~25 Auger electrons per decay and can cause a double-strand break if close to DNA.
Source: binding energies Bearden and Burr 1967, Kα₁ Bearden 1967, ωK Krause 1979; Auger electron yield Howell 1992; DSB Krisch et al. 1976, Lobachevsky and Martin 2000, damage range Martin and Haseltine 1981. Shell radii not to scale. Cascade shell energies approximated from binding energies (Bearden & Burr 1967).
How to use: one click pauses and resumes; in the fluorescence yield step choose the element with ↑ ↓ or by tapping a row; in the Auger cascade tap a vacancy to see the energy of the electron emitted from that shell; ← → change steps.
Speed
Did this animation help you understand the topic?
Figure 1.4: Filling of an inner-shell vacancy: characteristic X-ray and Auger electron (schematic).
1.2 Nuclear Structure and Nuclear Forces8 questions
Components of the Nucleus
The nucleus consists of protons and neutrons (collectively called "nucleons"):
•
Proton: Positively charged (+1e), mass ≈ 1.007276 amu (938.3 MeV/c²)
•
Neutron: Uncharged, mass ≈ 1.008665 amu (939.6 MeV/c²) → slightly heavier than the proton
•
A free neutron is unstable and turns into a proton by beta decay with a half-life of ~10.2 minutes
The nuclear radius is estimated empirically by the formula:
r ≈ r₀ × A1/3, where r₀ ≈ 1.2-1.4 fm (femtometre, 10⁻¹⁵ m)
Nuclear Forces
There are four fundamental forces in nature, and understanding them is important in nuclear medicine physics:
1. Strong nuclear force: The force that holds nucleons together in the nucleus. Its range is very short (~1-3 fm), but within this range it is ~100 times stronger than the electromagnetic force. It is charge-independent (equal between proton–proton, proton–neutron and neutron–neutron). It is the main reason for nuclear stability.
2. Electromagnetic force: The repulsive force between protons (Coulomb force). Its range is infinite, but it is weaker than the strong force. In large nuclei (high Z), Coulomb repulsion increases with the number of protons → instability.
3. Weak nuclear force: The force responsible for beta decay. It converts a neutron into a proton (or a proton into a neutron). Its range is very short (~10⁻¹⁸ m).
4. Gravitational force: Negligibly weak on the nuclear scale.
BOARD TIP
▸
If the board exam asks "which force holds the nucleus together?" → the strong nuclear force.
▸
"Which force is responsible for beta decay?"
→ The weak nuclear force. Do not confuse the two!
Nuclear Models
1. Liquid Drop Model:
–
Basis of the Bethe–Weizsäcker semi-empirical mass formula
–
Likens the nucleus to a drop of incompressible liquid
–
Explains binding energy with five terms: volume, surface, Coulomb, asymmetry and pairing terms
–
Successfully explains the fission process
2. Shell Model:
–
Proposes that nucleons, like electrons, occupy energy levels (shells)
–
Magic numbers: 2, 8, 20, 28, 50, 82, 126 → nuclei with these numbers of protons or neutrons are especially stable
–
Even–even nuclei (even Z and N) are the most stable, odd–odd nuclei the least stable
–
Only 4 stable odd–odd nuclei occur in nature: ²H, ⁶Li, ¹⁰B, ¹⁴N
BOARD TIP
▸
Magic numbers (2, 8, 20, 28, 50, 82, 126) may be asked in board exams. Pb-208 is a doubly magic nucleus (Z=82, N=126) → one of the most stable heavy nuclei in nature. Tc-99, the decay product of Tc-99m, is itself unstable (β⁻, T½ ≈ 2.1×10⁵ years); technetium has no stable isotope at all (Mattauch isobar rule).
Mass–energy equivalence, set out by Albert Einstein in 1905 within the special theory of relativity, is a cornerstone of nuclear physics:
E = mc²
•
E: Energy (joule or eV)
•
m: Mass (kg or amu)
•
c: Speed of light (3 × 10⁸ m/s)
The equation states that mass can be converted into energy and energy into mass.
Unit Conversions
Unit conversions frequently used in nuclear medicine:
•
1 amu (atomic mass unit) = 931.5 MeV/c²
•
1 eV = 1.602 × 10⁻¹⁹ J
•
1 amu = 1.66054 × 10⁻²⁷ kg
•
Electron mass = 0.000549 amu = 0.511 MeV/c²
•
Proton mass = 1.007276 amu = 938.3 MeV/c²
•
Neutron mass = 1.008665 amu = 939.6 MeV/c²
BOARD TIP
▸
1 amu = 931.5 MeV and electron mass = 0.511 MeV are the conversion values most often asked in board exams. In positron–electron annihilation the total energy = 2 × 0.511 = 1.022 MeV → two photons of 511 keV each are emitted in opposite directions at 180°. This is the physical basis of PET imaging!
Mass–Energy Relationship in Nuclear Reactions
In nuclear reactions total mass–energy is conserved. Mass alone, however, is not:
•
Exothermic reaction (Q > 0): The total mass of the products is less than that of the reactants → the mass difference is converted into energy (fission, radioactive decay)
•
Endothermic reaction (Q <0): The total mass of the products is greater than that of the reactants → energy must be supplied from outside for the reaction to occur (some nuclear reactions)
The mass of a nucleus is always less than the sum of the masses of the free nucleons that make it up. This difference is called the "mass defect" (Δm):
Δm = [Z × mₚ + N × mₙ] − Mnucleus
•
Z: Number of protons
•
N: Number of neutrons
•
mₚ: Proton mass
•
mₙ: Neutron mass
•
M(nucleus): Measured mass of the nucleus
This "missing mass" has not disappeared; according to Einstein's E = mc² it has been converted into binding energy.
Binding Energy
Binding energy (BE) is the minimum energy required to separate the nucleons of a nucleus completely:
BE = Δm × 931.5 MeV/amu
The higher the binding energy, the more stable the nucleus.
Figure 1.5: Calculation of mass defect and binding energy for He-4.
Did this figure help you understand the topic?
Binding Energy per Nucleon (BE/A)
The binding energy per nucleon (BE/A) is used to compare the stability of nuclei:
•
Light nuclei (A <20): BE/A is low and rises rapidly
•
Medium-mass nuclei: BE/A reaches its maximum
•
Fe-56 (iron): BE/A ≈ 8.8 MeV → highest binding energy per nucleon → most stable nucleus
•
Heavy nuclei (A > 60): BE/A decreases slowly (because of Coulomb repulsion)
This curve explains two important mechanisms of energy production:
•
Fusion: Light nuclei combine → BE/A increases → energy is released (the Sun's energy source)
•
Fission: Heavy nuclei split → BE/A increases → energy is released (nuclear reactors, Mo-99 production)
BOARD TIP
▸
That Fe-56 has the highest BE/A value, meaning "the most stable nucleus", is a classic board question. You should also be able to explain from the BE/A curve why fission and fusion release energy.
Figure 1.6: Binding energy per nucleon (BE/A) curve; energy release by fusion and fission.
Did this figure help you understand the topic?
Nucleon Separation Energy
The energy needed to remove a single nucleon (proton or neutron) from a nucleus is called the "nucleon separation energy". It differs from the total binding energy and varies with the shell structure. Separation energy is particularly high in nuclei with magic numbers.
Alpha decay: a ⁴He nucleus is emitted → Z decreases by 2, A decreases by 4
–
Example: Ra-223, Ra-226, Rn-222
4. Excited nuclei:
–
Isomeric transition (IT): A gamma ray is emitted; Z and A do not change
–
Internal conversion (IC): The gamma energy is transferred to an orbital electron
–
Example: Tc-99m → Tc-99 (IT, 140 keV gamma)
BOARD TIP
▸
Look at the N/Z ratio to decide the type of decay: neutron excess → β⁻, proton excess → β⁺ or EC, heavy nucleus → α. If the board exam asks "which decay does F-18 undergo?" → β⁺ (positron emission) → the basis of PET.
The real chart of nuclides shows stable nuclei and decay modes in colour; neutron excess decays by β⁻, proton excess by β⁺/EC and heavy nuclei by α towards stability.
Continuous stream · no seeking
Key point: as Z rises, stable nuclei need a higher neutron-to-proton ratio; nuclei outside the band decay in the direction that brings them towards it.
Source: IAEA LiveChart of Nuclides (ENSDF/NUBASE), 3383 ground states.
How to use: drag the map, zoom with the wheel or two fingers, tap a nucleus to see its decay path walk briefly to a stable nucleus, after which the walkthrough resumes; in the free step ← → change the neutron number (N) and ↑ ↓ the proton number (Z); Space or ⏸ pauses; in the other steps ← → change steps.
Speed
Did this animation help you understand the topic?
Figure 1.7: The band of stability and the decay directions of nuclei outside it (schematic).
Even–Odd Rule and Stability
The stability of nuclei depends on whether their numbers of protons and neutrons are even or odd:
•
Even–even (Z even, N even): Most stable → ~166 stable nuclei in nature
•
Even–odd or odd–even: Intermediate stability → ~110 stable nuclei in nature
•
Odd–odd (Z odd, N odd): Least stable → only 4 stable nuclei in nature (²H, ⁶Li, ¹⁰B, ¹⁴N)
Natural Radioactive Series
There are three main radioactive decay series in nature:
•
Uranium series: U-238 → ... → Pb-206 (stable)
•
Actinium series: U-235 → ... → Pb-207 (stable)
•
Thorium series: Th-232 → ... → Pb-208 (stable)
All three series end with stable lead isotopes. Alpha and beta decays occur in succession along these series.
Key Points for the Board
Atomic Models
•
Thomson → discovered the electron (cathode rays, 1897)
•
Rutherford → discovered the nucleus (alpha scattering experiment, 1911)
•
Bohr → quantised energy levels (1913), ΔE = hν
•
Quantum mechanical model → probability clouds (orbitals), 4 quantum numbers, Pauli exclusion principle
Basic Definitions – Do Not Confuse
•
Isotope → same Z (protons), different N → Example: I-123, I-125, I-131
•
Isobar → same A (mass), different Z → Example: Mo-99, Tc-99
•
Isotone → same N (neutrons), different Z
•
Isomer → same Z and A, different energy state → Example: Tc-99m / Tc-99
Critical Numerical Values
•
1 amu = 931.5 MeV
•
Electron/positron mass = 0.511 MeV
•
Annihilation photon energy = 511 keV (each)
•
β⁺ decay threshold = 1.022 MeV
•
Tc-99m gamma energy = 140 keV
•
I-131 gamma energy = 364 keV
Nuclear Forces
•
Strong nuclear force → holds the nucleus together
•
Weak nuclear force → responsible for beta decay
•
Electromagnetic force → repulsion between protons (Coulomb)
•
Gravitational force → negligible on the nuclear scale
Stability Rules
•
Magic numbers: 2, 8, 20, 28, 50, 82, 126
•
Example of a doubly magic nucleus: Pb-208 (Z=82, N=126)
•
Even–even nuclei are the most stable, odd–odd the least stable
•
Only 4 stable odd–odd nuclei in nature: ²H, ⁶Li, ¹⁰B, ¹⁴N
When an inner-shell electron leaves, a vacancy remains. Here the I-125 nucleus captures a K electron (electron capture), becomes Te-125 and emits a neutrino. The same vacancy can arise from the photoelectric effect or internal conversion. The atom fills it with an electron from a higher shell.
Step 1/5
The band of stability
Each square on the chart is a nucleus: neutrons (N) increase to the right, protons (Z) upward. White squares are stable nuclei and form a narrow band. For light nuclei the band lies on the N = Z line; as Z rises more neutrons are needed to offset Coulomb repulsion between protons and the band bends toward the neutron side (N/Z = 1.54 for Pb-208). There are no stable nuclei above Z = 83. Tap a square to see its properties.
The band of stabilityStableβ⁻β⁺ / ECαOther (p, n, SF)