Nuclear Binding and Structure

Question 1

What is the binding energy?

Answer 1

The energy required to separate nucleons into individual protons and neutrons -> binding energy, E(B)

Question 2

What is the equation for E(B)?

Answer 2

E(B) = (Z(m(p)+m(e)) + Nm(n) - M)*c^2

Question 3

What is the nuclear force?

Answer 3

For binding protons and neutrons despite the electrostatic repulsion of protons - this is the strong interaction.

Question 4

What are 4 characteristics of the strong interaction?

Answer 4

• Independent of charge
• Short-ranged ~10^-15 m
• Nuclear matter nearly const density which suggest that each nucleon interacts with the other nucleons only in its immediate vicinity
• Nuclear force favours binding of pairs of protons (up and down) or neutrons (up and down) with opposite spins.

Question 5

What is the Liquid Drop model?

Answer 5

Based on observation that nearly all nuclei have same density, individual nucleons are analagous to molecules in a liquid held together by Van der Waals interactions and/or hydrogen binding and surface tension effects.

Question 6

What is the first step in deriving an expression for binding energy of a nucleus?

Answer 6

Since nuclear forces show saturation there is a term proportional to A, i.e. C1*A, where C1 is extracted from experimental data.

Question 7

What is the second step in deriving an expression for binding energy of a nucleus?

Answer 7

Nucleons on surface of a nucleus are less tightly bound than those in interior of nucleus…negative term, proportional to 4πR^2 -> -C2*A^2/3

Question 8

What is the third step in deriving an expression for binding energy of a nucleus?

Answer 8

Each one of the protons repels the others. Electric interaction proportional to 1/R: term = -C3*z(z-1)/A^1/3

Question 9

What is the fourth step in deriving an expression for binding energy of a nucleus?

Answer 9

From experiments, nuclei appear to need a balance between energies associated with neutrons and protons so that N~Z for small A and N slightly greater than Z for lare nuclei: -C4*(A-2Z)^2/A

Question 10

What is the fifth step in deriving an expression for binding energy of a nucleus?

Answer 10

Nuclear force favours pairing of protons and neutrons. Positive term if Z and N are even, negative if both are odd, zero otherwise: +-C5*A^-4/3

Question 11

What is, therefore, the equation for the binding energy of a nucleus?

Answer 11

E(B) = C1A-C2A^2/3 - C3z(z-1)/A^1/3 - C4(A-2Z)^2/A +-C5*A^-4/3

Question 12

What s the equation for the mass of a neutral atom?

Answer 12

M = Z(m(p)-m(e))+Nm(n)-E(B)/c^2

Question 13

What is the shell model of protons and neutrons in a nucleus?

Answer 13

For protons, there is an additional potential energy associated with the Coulomb repulsion -> each proton considered to interact with a sphere of uniform charge density of radius R and total charge (Z-1)e

Question 14

What does the graph of V against r look like for the shell model of protons and neutrons?

Answer 14

Vcoulomb at top is curve curving down to x-axis, Vnuc is bottom curving up, so Vtot is combination of these so a bit higher than Vnuc.

Question 15

What are the “magic numbers” of protons or neutrons? What does this mean?

Answer 15

2, 8, 20, 28, 50, 82, 126: these nuclei are very stable.

Question 16

What number of protons must there be for the velocity of an electron not to exceed c?

Answer 16

Z < 137

Question 17

What is the heaviest nucleus found to be so far?

Answer 17

Z = 118, but naturally occurring: Uranium Z = 92

Question 18

What 2 eigenvalues does the spin operator s(zhat) have?

Answer 18

ћ/2, -ћ/2 for up and down spin respectively

Question 19

How do we work out the eigenvalues and eigenfunctions for s(zhat)?

Answer 19

Choose 2x2 matrix, so s(zhat) = ћ/2*matrix(1, 0, 0, -1), and then multiply this by matrix (a,b), where the 2x2 matrix = λ

Question 20

What do we do with the 2x2 matrix λ multiplied by matrix(a,b)?

Answer 20

Find values of a and b for λ = ћ/2 and λ = -ћ/2, find 2 eigenfunctions matrix(1,0) and matrix(0,1)

Question 21

What do we do after finding the eigenfunctions for s(zhat)?

Answer 21

Use s(zhat)X(sms) = m(s)ћ*X(sms)

Question 22

What do we find s(xhat) equals?

Answer 22

s(xhat) = ћ/2 *matrix(0, 1, 1, 0) = ћ/2 *σ(xhat)

Question 23

What do we find s(yhat) equals?

Answer 23

s(xhat) = ћ/2 *matrix(0, -i, i, 0) = ћ/2 *σ(yhat)

Question 24

What do we find s(zhat) equals?

Answer 24

s(xhat) = ћ/2 *matrix(1, 0, 0, -1) = ћ/2 *σ(zhat)

Question 25

What property of the spin matrices σ(xhat), σ(yhat) and σ(zhat) do we find?

Answer 25

σ(xhat)^2 = σ(yhat)^2 = σ(zhat)^2 = matrix(1, 0, 0, 1) = 1, and s(hat)^2 = s(xhat)^2 + s(yhat)^2 + s(zhat)^2 = 3ћ^2/2 *matrix(1, 0, 0, 1)

Question 26

What is the weak force responsible for?

Answer 26

Responsible for beta decay of some isotopes and nuclear fusion processes.

Question 27

What are the 4 fundamental particles in QM?

Answer 27

electrons, electron neutrinos, protons and neutrons.

Question 28

Which forces do quarks encounter?

Answer 28

strong, EM and weak

Question 29

Which forces do leptons encounter?

Answer 29

charged electrons, muons, tauons etc experience EM and weak but not strong, neutrinos experience only weak

Question 30

What are interactions in QFT mediated by?

Answer 30

Exchange of ‘virtual’ photons.

Question 31

What is a gauge boson?

Answer 31

Each of the 3 forces described by QFT corresponding to exchange of spin-1 force carrying particle known as a gauge boson.

Question 32

What is the gauge boson for the EM force?

Answer 32

The photon -> massless.

Question 33

What is the gauge boson for the strong force?

Answer 33

The gluon -> massless.

Question 34

What is the gauge boson for the weak force?

Answer 34

Mediated by charged W+ and W- bosons which are about 80 times more massive than protons.

Question 35

What are Feynman diagrams?

Answer 35

Used to describe some amplitude for a process on the way to summing over all appropriate processes to complete a calculation.

Question 36

What is one example of a Feynman diagram?

Answer 36

1 electron emits a photon γ which another electron absorbs. Interaction points are called g1, and g1 is proportional to charge, α proportional to g1^2, and measures the strength of the coupling.

Question 37

What is the equation for the dimensionless measure α?

Answer 37

α = e^2/4πε0ћc ~ 1/137

Question 38

What is another example of a Feynman diagram?

Answer 38

Muon decay: a muon emits W- and turns imto a muon neutrino. The W- boson decays into an electron and an anti-neutrino.

Question 39

What is the Higgs boson for?

Answer 39

Used to provide the mechanism by which all other particles acquire mass.