What is the binding energy?
The energy required to separate nucleons into individual protons and neutrons -> binding energy, E(B)
What is the equation for E(B)?
E(B) = (Z(m(p)+m(e)) + Nm(n) - M)*c^2
What is the nuclear force?
For binding protons and neutrons despite the electrostatic repulsion of protons - this is the strong interaction.
What are 4 characteristics of the strong interaction?
- 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.
What is the Liquid Drop model?
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.
What is the first step in deriving an expression for binding energy of a nucleus?
Since nuclear forces show saturation there is a term proportional to A, i.e. C1*A, where C1 is extracted from experimental data.
What is the second step in deriving an expression for binding energy of a nucleus?
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
What is the third step in deriving an expression for binding energy of a nucleus?
Each one of the protons repels the others. Electric interaction proportional to 1/R: term = -C3*z(z-1)/A^1/3
What is the fourth step in deriving an expression for binding energy of a nucleus?
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
What is the fifth step in deriving an expression for binding energy of a nucleus?
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
What is, therefore, the equation for the binding energy of a nucleus?
E(B) = C1A-C2A^2/3 - C3z(z-1)/A^1/3 - C4(A-2Z)^2/A +-C5*A^-4/3
What s the equation for the mass of a neutral atom?
M = Z(m(p)-m(e))+Nm(n)-E(B)/c^2
What is the shell model of protons and neutrons in a nucleus?
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
What does the graph of V against r look like for the shell model of protons and neutrons?
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.
What are the “magic numbers” of protons or neutrons? What does this mean?
2, 8, 20, 28, 50, 82, 126: these nuclei are very stable.
What number of protons must there be for the velocity of an electron not to exceed c?
Z < 137
What is the heaviest nucleus found to be so far?
Z = 118, but naturally occurring: Uranium Z = 92
What 2 eigenvalues does the spin operator s(zhat) have?
ћ/2, -ћ/2 for up and down spin respectively
How do we work out the eigenvalues and eigenfunctions for s(zhat)?
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
What do we do with the 2x2 matrix λ multiplied by matrix(a,b)?
Find values of a and b for λ = ћ/2 and λ = -ћ/2, find 2 eigenfunctions matrix(1,0) and matrix(0,1)
What do we do after finding the eigenfunctions for s(zhat)?
Use s(zhat)X(sms) = m(s)ћ*X(sms)
What do we find s(xhat) equals?
s(xhat) = ћ/2 *matrix(0, 1, 1, 0) = ћ/2 *σ(xhat)
What do we find s(yhat) equals?
s(xhat) = ћ/2 *matrix(0, -i, i, 0) = ћ/2 *σ(yhat)
What do we find s(zhat) equals?
s(xhat) = ћ/2 *matrix(1, 0, 0, -1) = ћ/2 *σ(zhat)
