exam 2 3145

A (mass number)

number of nucleons (protons + neutrons) in the nucleus

Z (atomic number)

number of protons; Z alone decides the element

N (neutron number)

N = A − Z

n in a reaction

neutron, ¹₀n (A=1, Z=0)

p in a reaction

proton, ¹₁H (A=1, Z=1)

d in a reaction

deuteron (deuterium nucleus), ²₁H (A=2, Z=1)

t in a reaction

triton (tritium nucleus), ³₁H (A=3, Z=1)

α

alpha particle, ⁴₂He (A=4, Z=2)

γ

gamma ray (photon), A=0, Z=0

Reaction shorthand X(a,b)Y

X + a → Y + b (target, incoming particle, outgoing particle, product)

How to find the product of a reaction

balance total A and total Z on both sides, then look up the element from Z

β⁻ decay

emits ⁰₋₁e; A unchanged, Z goes UP by 1 (decay particles always go on the RIGHT side)

β⁺ decay

emits ⁰₊₁e; A unchanged, Z goes DOWN by 1

Electron capture (EC)

nucleus absorbs an electron; A unchanged, Z goes DOWN by 1

α decay

A goes down by 4, Z goes down by 2

Four conservation laws in nuclear reactions

total energy, nucleons, charge, momentum

1 amu energy equivalent

931.5 MeV (c² = 931.5 MeV/amu)

MeV to joules

multiply by 1.602×10⁻¹³ J/MeV

Rest-mass energy

E [MeV] = m [amu] × 931.5 MeV/amu

mc² with m in grams and c in cm/s gives

ergs; multiply by 10⁻⁷ to get joules

Q-value from masses

Q = (Σm_before − Σm_after) × 931.5 MeV/amu (BEFORE minus AFTER)

Q > 0 means

exothermic: energy is released

Q < 0 means

endothermic: the incoming particle must bring at least |Q| of kinetic energy

Mass of p in any Q, BE, or Sn calculation

neutral ¹H = 1.007825 amu (NOT 1.007276)

When to use m_p = 1.007276 amu

only for a bare proton by itself, e.g. its own rest-mass energy

Decimal places for amu in Q-value calculations

always 6

If amu does not cancel in an energy calculation

you forgot × 931.5 MeV/amu; the answer is still a mass

Mass excess Δ

Δ = (M − A) × 931.5 MeV/amu

Q from mass excesses

Q = ΣΔ_before − ΣΔ_after (BEFORE minus AFTER)

Atomic mass from Δ

M = A + Δ/931.5

Δ of C-12

exactly 0

Δ sanity check

|Δ| is under ~100 MeV; if you get ~10⁵ MeV you forgot to subtract A

Binding energy BE

BE = [Z·m(¹H) + N·mₙ − M(ᴬZ)] × 931.5 MeV/amu

BE/A

average binding energy per nucleon = BE / A (MeV/nucleon)

BE/A sanity check

between ~1 (²H) and ~8.8 (peak near Fe-56) MeV/nucleon; U-235 ≈ 7.6

Separation energy of the last neutron

Sₙ = [M(ᴬ⁻¹Z) + mₙ − M(ᴬZ)] × 931.5 MeV/amu; typically 5-8 MeV

Q from binding energies

Q = ΣBE_after − ΣBE_before (AFTER minus BEFORE, opposite of the mass method)

BE of a single neutron or proton

0

Why both fusion and fission release energy

both move nuclei toward higher BE per nucleon (peak at Fe)

Recoil: parent nucleus at rest means

the two products have equal and opposite momentum

Momentum of a particle with mass

p = mv = √(2mE)

Momentum of a photon

p = E/c

Daughter recoil energy given E_α

E_D = (m_α / m_D) × E_α

E_α from Q (alpha decay)

E_α = Q × m_D / (m_D + m_α)

E_D from Q (alpha decay)

E_D = Q × m_α / (m_D + m_α)

Gamma recoil energy

E_N = E_γ² / (2 × M[amu] × 931.5 MeV/amu)

Recoil sanity check

α-decay recoil is tens to ~100 keV; γ recoil is eV to a few keV

Recoil method the instructor prefers

conservation of momentum ("always the most accurate")

Atom density N

number of atoms (or molecules) in 1 cm³

N for a single substance

N = ρ N_A / M

Avogadro's number

N_A = 0.6022×10²⁴ atoms/mol

Atoms of one element in a molecule

N_element = (atoms per molecule) × N_molecule, where N_molecule = ρ N_A / M_molecule

ρ and M in N = ρN_A/M must describe

the SAME thing (molecule density with molecule weight)

w/o

weight percent: fraction of the mixture's MASS

a/o

atom percent: fraction of the ATOMS

Percent vs decimal rule

convert to a decimal once on the givens line, then never divide by 100 again

N_i for a weight-percent mixture

N_i = w_i ρ N_A / M_i (the isotope's own M)

Average atomic weight from w/o

1/M_mix = Σ w_i / M_i

Average atomic weight from a/o

M_mix = Σ γ_i M_i

N_i for an atom-percent mixture

N_i = γ_i × (ρ N_A / M_mix)

M_mix sanity check

must lie between the lightest and heaviest isotope masses

"Uranium enriched to x w/o in U-235" means

x% of the URANIUM is U-235, not x% of the UO₂

Weight fraction of uranium in UO₂

w_U = M_U / (M_U + 2×15.9994)

U-235 atom density in UO₂

N₂₅ = w₂₅ × w_U × ρ_UO₂ × N_A / M₂₅

Oxygen atom density in UO₂

N_O = 2 × ρ_UO₂ × N_A / M_UO₂

UO₂ sanity check

N_O = 2(N₂₅ + N₂₈)

Solid atom density sanity check

10²² to 10²³ atoms/cm³ (0.01 to 0.1 in ×10²⁴ units)

Six neutron interactions

elastic (n,n); inelastic (n,n'); radiative capture (n,γ); charged-particle (n,p),(n,α); neutron-producing (n,2n),(n,3n); fission (n,f)

Elastic scattering (n,n)

kinetic energy conserved; nucleus left in ground state

Inelastic scattering (n,n')

nucleus left in an excited state, then emits a γ; endothermic

Radiative capture (n,γ)

neutron absorbed, nucleus emits a γ; exothermic

Charged-particle reaction (n,p) or (n,α)

neutron absorbed, proton or alpha emitted; can be endo- or exothermic

Neutron-producing reaction (n,2n)

high-energy neutron knocks out one or more extra neutrons; endothermic

Fission (n,f)

neutron absorbed by a fissile nucleus, which splits apart

Microscopic cross-section σ

effective cross-sectional area a nucleus presents to a neutron for a given reaction; barns or cm²

1 barn

10⁻²⁴ cm²

Macroscopic cross-section Σ (instructor's exam definition)

relative probability per unit path length that a neutron will undergo a collision as it moves in the medium; Σ = Nσ; cm⁻¹

Total cross-section

σ_t = σ_s + σ_a

Scattering cross-section

σ_s = σ_e + σ_i

Absorption cross-section

σ_a = σ_γ + σ_f + σ_p + σ_α + σ_2n + ...

Mean free path

λ = 1/Σ_t; average distance a neutron travels before an interaction (cm)