Atomic Nuclei: Composition, Charge, and Mass

• Nuclei comprise protons and neutrons, except for ordinary hydrogen nuclei (single proton).
• Describing nuclei properties requires these quantities:
  • Atomic number (Z) = # of protons.
  • Neutron number (N) = # of neutrons.
  • Mass number (A)= # of nucleons (protons or neutrons).
• Representing nuclei uses X, where A is mass number, Z is atomic number, X is element symbol.
• Isotopes exist if same number of protons, but varying amounts of neutrons.
• Isotopes share a Z value, while N and A fluctuate.

Charge and Mass

• Protons carry a +e charge, electrons carry a -e charge, neutrons are neutral, where e = 1.602 177 33 × 10-19 C.
• Neutrons are hard to detect due to absence of charge.
• Protons are ~1836 times as massive as electrons whereby proton and neutron masses are practically equal.
• Unified mass unit (u) = standard unit for measuring atomic mass.
• One atom of carbon-12 isotope is exactly 12 u, 1 u= 1.660 559 × 10-27 kg
• Protons and neutrons have a mass of about 1 u, where electrons have a mass of a tiny fraction of a u.
• Rest energy of a particle is often expressed via Er = mc².
• The energy equivalent of one atomic mass unit is Er = mc² = (1.660 559 × 10-27 kg) (2.997 92 × 108 m/s)² = 1.492 431 × 10-10 J = 931.494 MeV.
• Nuclear physicists frequently use MeV/c² to express mass by 1 u = 931.494 MeV/c².

Size of Nuclei

• Rutherford's scattering experiments first investigated nuclei size/structure.
• Rutherford formulated how close an alpha particle gets to the targeted nucleus before reversal, using energy conservation.
• Kinetic energy converts completely to electrical potential energy in head-on collisions.
• Rutherford determined radius of gold nucleus was less than 3.2 × 10-14 m.
• Rutherford found positive charge concentrates in a small sphere, called the nucleus, with radius no greater than ~10-14 m.
• Small lengths are conveniently measured using femtometers (fm); 1 fm = 10-15 m.
• Nuclei are approximately spherical with an average radius: r = roA1/3, where ro = 1.2 × 10-15 m and A is the total number of nucleons
• Volume of a nucleus is directly proportional to total nucleon number, so density is almost the same across all nuclei.

Nuclear Stability

• Nucleons combine as tightly packed spheres when forming a nucleus.
• The nuclear force is an attractive force acting between nuclear particles, with short range.
• Protons attract via nuclear force, repelling each other via Coulomb force.
• The nuclear attractive force is stronger than the Coulomb repulsive force within nucleus.
• Nuclear force is practically independent of charge where forces are similar, apart from the additional repulsive Coulomb force for proton-proton relationships.
• N = Z for stable light nuclei, where N > Z for stable heavy nuclei
• As number of protons increases, more neutrons are needed to keep nucleus stable, diluting nuclear charge density.
• All elements with more than 83 protons lack stable nuclei and decay into other particles.
• Appendix B has masses and properties of isotopes.

Binding Energy

• The total mass of a nucleus is always less than its component nucleons.
• The total energy of a bound system is always less than the combined energy of the separated nucleons.
• Binding energy = the energy amount that has to be added to break a nucleus into separated neutrons and protons.
• Separating a deuteron into a proton and neutron means overcoming attractive nuclear force with at least 2.224 MeV of energy.
• Zero binding energy correlates to nucleus separation into component protons and neutrons without added energy.
• Plotting binding energy per nucleon against mass number for stable nuclei reveals that nuclei with mass numbers near 60 are most strongly bound.
• This fact is used for fission where energy is released.

Radioactivity

• Becquerel observed uranium salts emitting radiation that darkened photographic plates.
• Radioactivity = spontaneous emission of radiation.
• Alpha (α) particles: Emitted particles consist of 4/2He, or helium nuclei.
• Beta (β) particles: The particles emitted consist of either electron or a positron.
• Gamma (γ) rays: The emitted "rays" are high-energy photons.
• Electrons are designated e–, and positrons are desiganted e+.
• Radiation bends into 3 components using magnetic fields.
  • Gamma rays = no charge.
  • Alpha particles = positively charged
  • Beta particles = negatively charged.
• Radiation penetrating powers:
  • Alpha = barely penetrates paper.
  • Beta = penetrates a few millimeters of aluminum.
  • Gamma = penetrates lead (cm).

The Decay Constant and Half-Life

• Radioactive sample contain a number of N radioactive nuclei at some instant ΔN/Δt ∝ N, this is expressed mathematically as:ΔN = −λNΔt.

• λ = the decay constant (determines how quickly an isotope decays overtime).

• Activity R = the decay rate or number of decays per second (defined for a particular sample)

  • R = |ΔN/Δt|= λN

• Unit for activity is the curie(Ci), 1 Ci = 3.7 × 1010 decays/s

• SI unit for activity: the becquerel (Bq), 1 Bq = 1 decay/s

• number of nuclei at some time: N = N0e−λt

• N0 and N represents # of radioactive nuclei at time zero and time t respectively while, e =2.718..

• Useful parameters are half-life T1/2 =the time it takes for half an amount of radioactive nuclei to decay