Nuclear Binding Energy and Reactions

• Nuclei are subatomic particles.
• The nucleus of an atom holds most of the atom's mass and can have multiple electric charges.
• Most properties of atoms and molecules result from atomic electrons, not the nuclei.
• The existence of elements results from different charges within the nuclei.
• Energy involved in natural processes traces back to nuclear reactions.
• Nuclear reaction energies can be 10^6-10^10 times larger than chemical energies of reactions.
• Nuclear energy has applications in reactors, weapons, medical imaging (NMR), and radioactive dating.

Nuclear Composition

• Nucleons consist of neutrons and protons.
• Atomic nuclei of the same element have the same number of protons but can have different numbers of neutrons.
• Atomic structure was known before nuclear structure because nuclear forces are stronger than electric forces.
• It's harder to break a nucleus to see what's inside.
• Charges in nuclear structure are in MeV range, while charges in electron structure of atom are in eV range.
• Proton has a charge of +e and a mass 1836 times that of an electron.
• Neutron is uncharged and has a mass slightly greater than a proton.
• Atomic number is the number of protons in the nucleus, which equals the number of electrons in a neutral atom.

Isotopes

• Isotopes are atoms of the same element can have different numbers of neutrons.
• For a given element, isotopes exist.
• Approximately 99.9% of hydrogen nuclei are single protons.
• Deuterium contains one neutron.
• Tritium contains two neutrons.
• Deuterium is stable, while tritium is radioactive and changes into an isotope of helium.
• At any given time, there are only 2 kg of tritium of natural origin.
• Heavy water is water where deuterium atoms are combined with oxygen atoms instead of hydrogen atoms.

Nuclide Notation

• A is the mass number.
• Z is the atomic number.
• X is the chemical symbol of element.
• N = A - Z
• A is the number of protons in nucleus and number of electrons in neutral atom.
• N is the number of neutrons in nucleus.
• Hydrogen (¹H), deuterium (²H), and chlorine (³⁵Cl and ³⁷Cl) are examples.

Atomic Mass

• It refers to the mass of neutral atoms (nucleus + electrons).
• Mass units (u) define atomic mass
• The mass is relative to the ¹²C atom.
• It is defined as 12 u".
• One atomic mass unit converts to 1 u = 1.66054 x 10⁻²⁷ kg.
• One atomic mass unit is energy equivalent is 931.49 MeV/u.
• Proton mass is approximately 1.6726 x 10⁻²⁷ kg
• Proton mass is approximately 1.007276 u
• Neutron mass is approximately 1.6750 x 10⁻²⁷ kg
• Neutron mass is approximately 1.008665 u
• Mass of ¹H atom is approximately 1.007825 u

Nuclear Properties (Size)

• The first estimate of nuclear size came from Rutherford's scattering experiment.
• Alpha particles deflected by a target nucleus in a thin foil.
• Findings were consistent with Coulomb's Law if alpha particles didn't get too close the nucleus.
• To consider point masses and charges, distances must exceed 10⁻¹⁴ m for smaller particle.

Determining Nuclear Radius

• Use particle scattering with fast electrons and neutrons.
• Electrons interact with nucleus via electric forces, and scattering reveals charge distribution.
• Neutrons interact with nucleus via nuclear forces, and scattering reveals nuclear matter distribution.
• Analysis requires the de Broglie wavelength of scattered particles to be smaller than the nucleus radius.
• Volume of nucleus is directly proportional to number of nucleons it contains (mass number A).
• The study yields density of nucleons is nearly the same in interior of different nuclei.
• Nuclear radius is R = R₀A¹/³, where R₀ ≈ 1.2 x 10⁻¹⁵ m = 1.2 fm (femtometer or fermi).
• R₀ is approximate because nuclei do not have sharp boundaries.
• R₀ is a little less when deduced from electron scattering and more from neutron scattering.
• Nuclear mass and nuclear charge are not identically distributed through a nucleus.

Nuclear Density in Carbon and Gold

• The calculated radius R for ¹²C nucleus is (1.2)(¹²)^(⅓) fm ≈ 2.7 fm.
• The diagram illustrates nuclear density in ⁵⁹Co and ¹⁹⁷Au versus radial distance from center.
• R₀ ≈ 4.5 fm
• R_Au ≈ 7 fm

Finding Density of ¹²C Nucleus

• Atomic mass of ¹²C is 12 u.
• Nuclear density is given as ρ = mass/volume.
• ρ = (12 u) / (⁴⁄₃πR³) = (12 u) * (1.66 x 10⁻²⁷ kg/u) / (⁴⁄₃π (2.7 x 10⁻¹⁵)^(3) m³)
• ρ ≈ 2.4 x 10¹⁷ kg/m³.
• The figure is essentially the same for all nuclei since R scales with mass number A.
• There is a direct proportionality to mass number of nucleus.

Nuclear Spin and Magnetic Moment

• Nuclei with odd numbers of protons/neutrons are fermions with spin quantum number s = ½.
• The magnitude of spin angular momentum is given by |S| = √(s(s+1))ħ = (√3 / 2) ħ.
• Associated with spin angular momentum is the spin magnetic moment.
• It is described by the magnetic quantum number ms = ±½.
• In nuclear physics, magnetic moments are expressed in nuclear magnetons (µ_N).
• A nuclear magneton is given by µ_N = eħ / (2m_p) ≈ 5.051 x 10⁻²⁷ J/T ≈ 3.152 x 10⁻⁸ eV/T, where mp is the proton mass.
• The nuclear magneton is smaller than the Bohr magneton (µ_B = eħ / (2m_e) ≈ 9.274 x 10⁻²⁴ J/T).

Spin Magnetic Moments

• Spin magnetic moments of P and n have components in any z-direction.
• Proton: µ_pz = ±2.793 µ_N
• Neutron: µ_nz = ±1.913 µ_N

Parity

• The two possibilities for signs are from -½, +½ and ± for µ_pz.
• Because µ_pz is in same direction as spin S, then for µ_nz it is opposite to S.
• P and n spin moments and spin angular momentum are parallel in both cases.
• An example is the single proton hydrogen nucleus which yields the total angular momentum S = ½ ħ.
• Nuclei with more nucleons may have orbital angular momentum due to motion inside the nucleus.
• Total angular momentum for such a nucleus is the vector sum of spin and orbital angular momenta of its nucleons

Nuclear Binding Energy

• Energy holds a nucleus together.
• Deuterium, with a neutron and a proton, should have a mass equal to a free proton and neutron.
• Mass of ¹H is 1.007825 u; mass of a neutron is 1.008665 u for an expected mass of ²H of 2.016490u.
• The measured mass of ²H is 2.014102 u, or about 0.002388 u less than the above configuration.
• The "missing mass" corresponds to energy that is released when the nucleons bind together.
• Energy equivalent is ΔE = (0.002388 u)(931.49 MeV/u) = 2.224 MeV
• Experiments that break up a deuterium nucleus into constituent particles confirm the energy. It requires around 2.224 MeV.
• If less energy than 2.224 MeV is given to a ²H nucleus, then the nucleus stays together.
• If more is given, any extra energy becomes kinetic energy of product (n, p).
• Greater binding energy correlates to larger energy need to break the molecule.
• E_b = [Z m(¹H) + N m(n) − m(²X)] (931.49 MeV/u)

Mass Defect Equation Terms

• The mass defect E_b represents the binding energy in MeV of a nucleus (²X).
• N=A-Z represents the number of neutrons.
• The atomic mass number of the nucleus, and N is neutron, so
• m(¹H) is the atomic mass of hydrogen,
• m(n) is the neutron mass, and m(²X) is the atomic mass of the nucleus (²X).
• All terms are measured in atomic mass units (u).

Case Examples of Eb

• E_b for ²H (deuterium) is 2.224 MeV
• E_b for ²⁰⁹Bi (isotope of Bismuth) is 1640 MeV.
• Typical binding energy is 8 x 10¹⁰ kJ/kg
• By comparison, the heat of vaporization of water is 2.26 x 10³ kJ/kg
• Comparison continued (Heat given off by gasoline is 47x10³kJ/kg. (1.7 million times smaller)).

Binding Energy Per Nucleon

• Binding energy tells about the relative stability of nucleus.
• Divide entire binding energy by number of nucleons.
• For ²H E_b/nucleon is 2.224 MeV.
• This yields 1.112 MeV/nucleon
• The binding energy for ²⁰⁹Bi it is 1640 MeV/209 or 7.8 MeV/nucleon.

Mass and Nucleon Data

• Beryllium: 208.980 u.
• Plots show higher energy per nuleon leads to more stable nucleus.
• The greatest is the B⋅E/nucleon and the more stable is core.
• Highest max is 8.8 mev/nucleon:
• Fe56; it exhibits the greatest stable nucleus.
• First main element of this site is the following: If a massive nucleus is divided to generate secondary nuclear forms, greater power transfer will be realized.
• The formula that derives energy will need less original nucleus.
• Then, if an origin is broken to make second nucleus, over flow of power will resutl.
• If energy goes under 0 mev, there will beno further flow of poweer.
• Energy/Nucleon (235 Nucleon) = 188
• Total energy can range up t 0. 8MEV. *8 million megavatts.

Nuclear Fission and Fusion

• Nuclear fission releases energy for a single atomic event.
• Ordinary chemical reactions releases ev/atom powers for the nucleus.
• Nuclear fission creates for nucleus up to millions as much as an atom.
• Combines two nucleis to a new one.
• Fission nucleus increases beam,
• Light nucleus to new high power (helium) is known as fusion.
• From origin sunlights , this has all energy for stars also. Total BE is complex than single hydro proton can offer to continue.
• Stable core explains main energy factor to light nucleus for new core.
• Atomic BE is 160,7065.
• Atomit proton to all parts BE(2H) , then for new BE: