Binding energy
Mass defect: a nucleus is not as massive as the sum of its constituent neutrons and protons. The
difference, the so-called mass defect is related to the Binding Energy.
Binding energy:
BE = Δm c 2 = (Z mp + (A − Z )mn − m(AZ X )) c 2 = ΔE [MeV] = 931,49 ⋅ Δm [u]
Nuclear rest masses are typically not tabulated, but can be deduced from the following:
M(AZ X ) = m(AZ X ) + Z me ⟶ BE = (Z M(11H ) + (A − Z )mn − M(AZ X )) c 2
Stability
Nuclei with a high Binding Energy per nuclide are very stable. The amount of BE /A is due to:
• Strong nuclear force (short-ranged)
• Coulomb force (long-ranged, repulsive (because only positive charges in nucleus))
Nuclear reactions and energetics
Conserved quantities in nuclear reactions
• Number of nucleons (‘total A remains equal’)
• Charge (‘total Z remains equal’)
• Energy (including rest mass)
• Linear momentum
• Angular momentum (spin)
A
Nuclear reaction ZX + AZ x → AZY + AZ y or AZ X (x, y) AZY
Q-Value
Change in kinetic energy (increase) or the change in rest mass (decrease). Conservation of energy
results in two equations for the Q-value for a nuclear reaction.
∑[
T′i − Ti] =
∑[
Q= mi c 2 − m′i c 2 ]
i i
Always be very careful when the number of protons or neutrons changes and free electrons are
created! You should then only consider the nuclear rest mass (rather than atomic reset mass).
Q-value will be lower than expected by E* if a reaction product leaves an electron or electrons in
an excited state.
, Radioactivity
Unstable nuclei decay spontaneously into daughter nucleus and other particle.
α-decay: a nucleus releases an entire helium nucleus
A
ZX → A−4Y + 42 He
Z−2
γ-decay: a nucleus in excited state decays to lower energy state and releases a photon.
A
Z X* → AZ X + γ
β −-decay: a nucleus releases an electron (neutron → proton)
A
ZX → A Y + 0 e + ve
Z+1 −1
β +-decay: a nucleus releases a positron (proton → neutron)
A
ZX → A Y + 01e + ve
Z−1
Electron capture: a proton in the nucleus combines with an orbiting electron to form a neutron.
A
ZX + 0 e → A Y + ve
−1 Z−1
Decay Kinetics
Decay is a spontaneous and stochastic process. We can only make a prediction on large numbers
of particles N:
d N(t)
= − λ N(t) ⟶ N(t) = N0e −λt where λ is the decay constant [s-1]
dt
Activity
The activity A is the number of decay events per second. The SI-unit is Becquerel: [Bq] = [s-1]
d N(t)
A≡− = λ N(t) ⟶ N(t) = λ N0e −λt
dt
Half-life
The half-life is the time t1/2 at which halve of the isotope has decayed. t1/2 is related to λ.
ln 2
t1/2 =
λ
Mass defect: a nucleus is not as massive as the sum of its constituent neutrons and protons. The
difference, the so-called mass defect is related to the Binding Energy.
Binding energy:
BE = Δm c 2 = (Z mp + (A − Z )mn − m(AZ X )) c 2 = ΔE [MeV] = 931,49 ⋅ Δm [u]
Nuclear rest masses are typically not tabulated, but can be deduced from the following:
M(AZ X ) = m(AZ X ) + Z me ⟶ BE = (Z M(11H ) + (A − Z )mn − M(AZ X )) c 2
Stability
Nuclei with a high Binding Energy per nuclide are very stable. The amount of BE /A is due to:
• Strong nuclear force (short-ranged)
• Coulomb force (long-ranged, repulsive (because only positive charges in nucleus))
Nuclear reactions and energetics
Conserved quantities in nuclear reactions
• Number of nucleons (‘total A remains equal’)
• Charge (‘total Z remains equal’)
• Energy (including rest mass)
• Linear momentum
• Angular momentum (spin)
A
Nuclear reaction ZX + AZ x → AZY + AZ y or AZ X (x, y) AZY
Q-Value
Change in kinetic energy (increase) or the change in rest mass (decrease). Conservation of energy
results in two equations for the Q-value for a nuclear reaction.
∑[
T′i − Ti] =
∑[
Q= mi c 2 − m′i c 2 ]
i i
Always be very careful when the number of protons or neutrons changes and free electrons are
created! You should then only consider the nuclear rest mass (rather than atomic reset mass).
Q-value will be lower than expected by E* if a reaction product leaves an electron or electrons in
an excited state.
, Radioactivity
Unstable nuclei decay spontaneously into daughter nucleus and other particle.
α-decay: a nucleus releases an entire helium nucleus
A
ZX → A−4Y + 42 He
Z−2
γ-decay: a nucleus in excited state decays to lower energy state and releases a photon.
A
Z X* → AZ X + γ
β −-decay: a nucleus releases an electron (neutron → proton)
A
ZX → A Y + 0 e + ve
Z+1 −1
β +-decay: a nucleus releases a positron (proton → neutron)
A
ZX → A Y + 01e + ve
Z−1
Electron capture: a proton in the nucleus combines with an orbiting electron to form a neutron.
A
ZX + 0 e → A Y + ve
−1 Z−1
Decay Kinetics
Decay is a spontaneous and stochastic process. We can only make a prediction on large numbers
of particles N:
d N(t)
= − λ N(t) ⟶ N(t) = N0e −λt where λ is the decay constant [s-1]
dt
Activity
The activity A is the number of decay events per second. The SI-unit is Becquerel: [Bq] = [s-1]
d N(t)
A≡− = λ N(t) ⟶ N(t) = λ N0e −λt
dt
Half-life
The half-life is the time t1/2 at which halve of the isotope has decayed. t1/2 is related to λ.
ln 2
t1/2 =
λ
