Difficulty: Introductory to Intermediate | Prerequisites: Basic understanding of atoms (protons, neutrons, electrons), energy units (eV, MeV), and electric charge. If you are not comfortable with atomic structure from chemistry, review that first.
This topic covers the building blocks of the atomic nucleus and the forces that hold it together. It sits near the end of PHYS 116, bridging classical physics with modern physics. You need a working understanding of atoms, electric charge, and basic energy concepts. The chapter answers three questions: what is the nucleus made of, why does it stay together, and what happens when it falls apart? If you have been away for a few weeks, make sure you are comfortable with Coulomb's law and conservation of energy before diving in.
Atoms have nuclei made of protons and neutrons, held together by the strong nuclear force. The number of protons sets the element; the number of neutrons sets the isotope. The mass of a nucleus is always slightly less than the mass of its parts added up, and that missing mass (the "mass defect") is what keeps everything glued together, expressed as binding energy.
Nucleon
A proton or neutron, either of the two particles found in the atomic nucleus. Think of it as the generic name for "stuff in the nucleus."
Atomic number (Z)
The number of protons in a nucleus. This is what determines the element. Change Z and you change the element entirely.
Mass number (A)
The total number of nucleons (protons + neutrons) in a nucleus. In simple terms, A = Z + N.
Isotope
Atoms of the same element (same Z) but with different numbers of neutrons, giving them different mass numbers. Think of isotopes as siblings: same family (element), different weight.
Mass defect (Δm)
The difference between the total mass of the individual nucleons and the actual mass of the assembled nucleus. In simple terms, the nucleus weighs less than its parts, and that "missing" mass has been converted into binding energy.
Binding energy (B)
The energy required to completely disassemble a nucleus into separate protons and neutrons. The higher the binding energy per nucleon, the more stable the nucleus. This is the nuclear equivalent of "how hard is it to pull this apart."
Strong nuclear force
The short-range attractive force between nucleons that holds the nucleus together, overpowering the electrostatic repulsion between protons. Think of it as nuclear glue: incredibly strong but only works when nucleons are very close together (about 1 fm).
Alpha particle (α)
A helium-4 nucleus: 2 protons and 2 neutrons bundled together. In simple terms, it is the packet a heavy nucleus throws out during alpha decay.
A nucleus is written as ᴬ_Z X, where X is the element symbol, A is the mass number (top), and Z is the atomic number (bottom).
Example: ²³⁸₉₂U means uranium with 92 protons and 238 total nucleons. The neutron count is N = A − Z = 238 − 92 = 146.
Isotopes share the same atomic number (same element) but have different mass numbers.
Carbon has three common isotopes: ¹²C, ¹³C, and ¹⁴C. All have 6 protons; they differ only in neutron count (6, 7, and 8 neutrons respectively).
The number of neutrons determines which isotope you are dealing with. The number of protons determines which element.
In alpha decay, a parent nucleus ᴬ_Z X emits an alpha particle (⁴₂He) and becomes a daughter nucleus.
The daughter has 2 fewer protons (Z − 2) and 4 fewer nucleons (A − 4).
The element changes because Z changes. This is a nuclear transmutation.
When nucleons combine into a nucleus, the resulting atom is lighter than the sum of its parts. The difference is the mass defect Δm.
Example: a helium-4 atom has a mass of 4.00260 u, but 2 hydrogen atoms + 2 neutrons have a combined mass of 4.03298 u. The mass defect is 0.03038 u.
Binding energy B = (Z × m_H + N × m_n − m_atom) × 931.49 MeV/u
m_H = 1.007825 u (hydrogen atom mass), m_n = 1.008665 u (neutron mass)
The factor 931.49 MeV/u converts atomic mass units to energy via E = mc².
Iron-56 has Z = 26 protons and N = 30 neutrons. Atomic mass = 55.934940 u.
B = (26 × 1.007825 + 30 × 1.008665 − 55.934940) × 931.49
B ≈ 492 MeV total binding energy.
⁵⁶Fe has one of the highest binding energies per nucleon of any nucleus, which is why iron sits at the peak of the binding energy curve and is exceptionally stable.
Divide total binding energy by A to get binding energy per nucleon. Higher values mean greater stability.
Helium-4 example: B = 28.30 MeV for 4 nucleons, so B/A ≈ 7.08 MeV per nucleon.
Nuclei near ⁵⁶Fe have the highest B/A. Lighter nuclei can gain stability by fusion; heavier nuclei can gain stability by fission. Both processes move toward the iron peak.
The strong force is attractive and acts between all nucleon pairs (proton-proton, proton-neutron, neutron-neutron) equally.
It is extremely short-range. It is negligible when nucleons are more than a few femtometres (fm) apart.
At about 1 fm (nucleons "touching"), the force is zero, in equilibrium.
Below 1 fm, the strong force becomes repulsive, preventing nucleons from collapsing onto each other.
Two protons experience both the attractive strong force and a repulsive electrostatic (Coulomb) force.
A proton-neutron pair or a neutron-neutron pair experiences only the strong force (neutrons carry no charge).
The strong force is much stronger than the Coulomb repulsion at nuclear distances, which is why protons can coexist in the nucleus despite their positive charges.
In large nuclei, protons on opposite sides are too far apart for the strong force to act between them, but the long-range Coulomb repulsion still reaches. This is why very heavy nuclei tend to be unstable.
Making a neutron-rich nucleus more stable can involve converting a neutron into a proton (beta-minus decay). For example, ¹²B has 5 protons and 7 neutrons; converting one neutron into a proton brings it to ¹²C (6 protons, 6 neutrons), which is more balanced and stable.
Neutron count
N = A − Z
Binding energy
B = (Z × m_H + N × m_n − m_atom) × 931.49 MeV/u
where m_H = 1.007825 u, m_n = 1.008665 u
Alpha decay
ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂He + energy
Constants to memorise: 931.49 MeV/u is the conversion factor from atomic mass units to energy. m_H and m_n should be on your formula sheet, but know which is which.
Binding energy per nucleon explains why the sun runs on fusion (combining light nuclei toward the iron peak) and why nuclear power plants use fission (splitting heavy nuclei toward the iron peak). Both processes release energy because the products are more tightly bound.
The strong force's short range is the reason there is a practical upper limit on element size. Beyond about uranium, nuclei become too large for the strong force to hold together reliably, which is why superheavy elements are all unstable and short-lived.
Students often think that the strong force acts over long distances like gravity or the Coulomb force. It does not. It is effectively zero beyond a few femtometres.
Students sometimes confuse atomic number (Z, protons) with mass number (A, total nucleons). Z determines the element. A determines the isotope.
A common error is thinking the mass defect means mass has disappeared. Mass has not disappeared; it has been converted into binding energy via E = mc². The "missing" mass is the energy cost of pulling the nucleus apart.
Students often assume more nucleons always means more stable. Stability depends on the binding energy per nucleon, not the total number of nucleons. ⁵⁶Fe is more stable than ²³⁸U despite having far fewer nucleons.
⚠️ You will almost certainly be asked to calculate binding energy using the formula. Make sure you can plug in Z, N, m_H, m_n, and the atomic mass without mixing up the values.
⚠️ Know the difference between total binding energy and binding energy per nucleon. Exam questions often ask which nucleus is more stable, and the answer depends on B/A, not B alone.
⚠️ Alpha decay notation comes up frequently. Be able to write the daughter nucleus given a parent, reducing Z by 2 and A by 4.
⚠️ Understand why the strong force is short-range and how this explains the instability of heavy nuclei. This is a common conceptual question.
True or False: Isotopes of the same element have different numbers of protons.
False. Isotopes have the same number of protons (same Z) but different numbers of neutrons.
Fill in the blank: The binding energy of a nucleus is calculated using the formula B = (Z × ___ + N × ___ − m_atom) × 931.49.
m_H (hydrogen atom mass) and m_n (neutron mass).
True or False: The strong nuclear force acts over distances comparable to the size of an atom.
False. The strong force is effective only at nuclear distances (a few femtometres), far smaller than atomic size.
Fill in the blank: In alpha decay, the daughter nucleus has ___ fewer protons and ___ fewer total nucleons than the parent.
2 fewer protons and 4 fewer nucleons.
Q: Uranium-238 has 92 protons. How many neutrons does it have?
A: N = A − Z = 238 − 92 = 146 neutrons.
Q: Calculate the binding energy of helium-4 given that its atomic mass is 4.00260 u, m_H = 1.007825 u, and m_n = 1.008665 u.
A: B = (2 × 1.007825 + 2 × 1.008665 − 4.00260) × 931.49 = 0.03038 × 931.49 ≈ 28.30 MeV.
Q: Write the alpha decay equation for ²³⁸₉₂U.
A: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He + energy. The daughter is thorium-234.
Q: Why does the strong force not prevent very heavy nuclei from being unstable?
A: The strong force is short-range (a few fm). In a very large nucleus, protons on opposite sides are too far apart for the strong force to act between them, but the Coulomb repulsion, being long-range, still pushes them apart. The result is instability.
Q: Which has a higher binding energy per nucleon: ⁵⁶Fe or ²³⁸U?
A: ⁵⁶Fe. Iron-56 sits at the peak of the binding energy per nucleon curve, making it the most tightly bound (and therefore most stable) nucleus.
This material connects directly to nuclear decay (Ch. 30.4-30.5), which covers what happens when unstable nuclei break apart. The binding energy curve also underpins nuclear medicine (Ch. 30.6) and energy generation, both fission reactors and stellar fusion.
The strong force introduced here is one of the four fundamental forces in physics. If you continue to modern physics or particle physics, you will encounter it again in the context of quarks and gluons.
Coulomb's law from electrostatics (Ch. 20-21) is essential background. The competition between Coulomb repulsion and the strong force is the central tension in nuclear stability.
nuclear structure, nuclear stability, isotope, isotopes, atomic number, mass number, nucleon, proton, neutron, binding energy, binding energy per nucleon, mass defect, strong nuclear force, strong force, alpha decay, alpha particle, helium-4 nucleus, nuclear notation, ⁵⁶Fe iron-56, ²³⁸U uranium-238, E = mc², mass-energy equivalence, nuclear transmutation, PHYS 116, general physics, University of Washington