2. Nuclear Stability and the Liquid Drop Model Flashcards
Why does the nucleus have less mass than the sum of individual nucleons?
The attractive strong nuclear force
The binding energy is the energy _______ when constituent protons and neutrons are brought together. It can also be thought of as the _______ __________ to separate a nucleus into constituent protons and neutrons.
Released
Energy released
What is the mass defect?
The mass difference between the mass of individual nucleons and their mass within a nucleus.
The _______ the mass defect, the greater the energy release when constituent nucleons are brought together to form a nucleus.
Higher
Give the equation for the nuclear binding energy
B = nuclear binding energy
Z = atomic number
m_p = proton mass
N = number of neutrons
m_n = neutron mass
M_N = M_A - Zm_e = nuclear mass of the nucleus ᴬX
c =speed of light
What is the difference between the mass of an atom and the mass of the constituent nucleus and electrons?
The sum of the electronic binding energies
Give the equation for the relation between the nuclear mass energy and the atomic mass energy
c = speed of light
M_N = nuclear mass
M_A = atomic mass
Z = atomic number
m_e = electron mass
B_i = electronic binding energy of the i-th electron
What is the usual value of electronic binding energies?
~ 10-100 keV
What is the usual value of atomic mass energies?
~ 1000’s of MeV
Why are electronic binding energies usually ignored?
Because they are much smaller than atomic mass energies.
Write the nuclear binding energy equation in terms of the atomic mass
B = nuclear binding energy
Z = atomic number
m_p = proton mass
N = number of neutrons
m_n = neutron mass
M_A = atomic mass of the nucleus ᴬX
m_e = electron mass
c =speed of light
[term] = M_N of the nucleus ᴬX
The most stable elements have the _______ binding energy per nucleon.
Highest
Describe the shape of the binding energy v. mass number curve
Iron is at the peak of the curve so is the most stable.
Give the equation for the volume of a nucleon
v = nucleon volume
r0 = radius of nucleon
Give the equation for the volume of a nucleus with A nucleons
V = nucleus volume
A = number of nucleons
v = nucleon volume
State the equation that relates the volume of a nucleus to its radius
V = nucleus volume
r = nucleus radius
Give the equation that relates the radius of a nucleus to the radius of its constituent nucleons
r = nucleus radius
r0 = nucleon radius
A = number of nucleons
What is the experimental value of the radius of a nucleon?
What is the Semi-Empirical Binding Energy model?
The simplest binding energy model that combines theory and experimental data to fit the binding energy curve. It treats the nucleus as a drop of incompressible fluid.
What are the 5 terms that contribute to the energy of the droplet in the Semi-Empirical Binding Energy model?
- Volume: from short-range (nearest neighbour) interactions
- Surface: from some nuclei being on the surface
- Coulomb: from repulsion between charged protons
- Asymmetry: due to energy levels of nucleus
- Pairing: from preference of nucleus to have even N and even Z
What is the Semi-Empirical Binding Energy volume term?
An observation that for all but the lightest nuclei, the binding energy per nucleon (B/A) is approximately constant because the binding energy is proportional to the number of interactions between nuclei (mediated by the strong nuclear force). This means that B is proportional to A.
Give the equation for the volume contribution to the Semi-Empirical Binding Energy
B_v = volume contribution to binding energy
a1 = constant
A = mass number
The volume term causes an ________ in binding energy.
Increase
The strong nuclear force is a _____ _____ force so it only interacts with its _______ _________. It is also charge ___________.
Short-range
Nearest neighbours
Charge independent
Describe the B/A against A plot containing only the volume contribution
What is the Semi-Empirical Binding Energy surface term?
Some nuclei are on the surface of the droplet, resulting in a lower binding energy as there are fewer nearest neighbour interactions.
Give the equation for the surface contribution to the Semi-Empirical Binding Energy
B_s = surface contribution to binding energy
a2 = constant
A = mass number
The surface term causes a ________ in binding energy.
Decrease
Describe the B/A against A plot containing only the volume and surface contributions
What is the Semi-Empirical Binding Energy coulomb term?
Some nuclei are protons with repulsive Coulomb interaction (i.e. they repulse each other). There are Z(Z-1)/2 pairs of protons in a nucleus with an atomic number, Z. Each pair give a negative contribution to the binding energy.
Give the equation for the coulomb contribution to the Semi-Empirical Binding Energy
B_c = coulomb contribution to binding energy
a3 = constant
Z = atomic number
A = mass number
The coulomb term causes a ________ in binding energy.
Decrease
Describe the B/A against A plot containing only the volume, surface, and coulomb contributions
What is the Semi-Empirical Binding Energy asymmetry term?
The shell model tells us the spacing between energy levels scales as 1/A, meaning that states with N ~ Z are now favoured.
What spin do protons and neutrons have?
1/2
What spin states can protons and neutrons have?
Spin up and spin down
What are half-integer spin particles known as?
Fermions
State the Pauli exclusion principle
No two IDENTICAL fermions can be in the same quantum state.
The state with the _______ energy has an equal number of protons and neutrons.
Minimum
Give the equation for the energy excess due to a difference between the number of neutrons and the atomic number
∆E = energy excess
N = number of neutrons
Z = atomic number
ε = spacing between energy levels
Give the equation for the asymmetry contribution to the Semi-Empirical Binding Energy
B_a = asymmetric contribution to binding energy
a4 = constant
A = mass number
Z = atomic number
The asymmetry term causes a ________ in binding energy.
Decrease
Describe the B/A against A plot containing only the volume, surface, coulomb, and asymmetry contributions
What is the Semi-Empirical Binding Energy pairing term?
In nature, we observe that nuclei with pairs of protons and pairs of neutrons are the most stable. This ‘pairing effect’ is more important for lighter nuclei.
Give the equation for the pairing contribution to the Semi-Empirical Binding Energy
B_p = pairing contribution to binding energy
a5 = constant
A = mass number
+ for even Z, even N
0 for even Z, odd N (or vice versa)
- for odd Z, odd N
What does the contribution of the pairing term depend on in the Semi-Empirical Binding Energy?
The atomic number and the neutron number within the nucleus.
Increase in energy:
+ for even Z, even N
Same energy:
0 for even Z, odd N (or vice versa)
Decrease in energy:
- for odd Z, odd N
Describe the B/A against A plot containing the volume, surface, coulomb, asymmetry, and pairing contributions
What is SEMF?
The semi-empirical binding mass formula for nuclear mass. It is the mass-equivalent to the semi-empirical binding energy formula.
State the semi-empirical binding mass formula for nuclear mass
M_N = nuclear mass
Z = atomic number
m_p = proton mass
A = mass number
a-terms = -B = - binding mass
