Chapters 1-5 (Nuclei)

Question 1

Nuclide

Answer 1

Nucleus w/ specific A and Z values

Question 2

Isotone

Answer 2

Nuclei w/ same no of neutrons (i.e. fixed (A-Z) value) but diff no of protons

Question 3

Isobar

Answer 3

Nuclei w/ same A but diff Z

Question 4

Mirror nuclei

Answer 4

2 nuclei w/ odd A where no of p+s in one nuclei = no of neutrons in other and vice-versa

Question 5

Binding energy and why is it -ve

Answer 5

energy required to free nucleons -> -ve as need to put energy IN

Question 6

What force is binding energy due to?

Answer 6

Strong force

Question 7

Why is the binding energy term (over c^2) known as the “mass defect”?

Answer 7

As the strong force that causes the binding energy causes reduction in total mass of nucleus from just sum of mass of constituent nucleons.

Question 8

Describe relationship b/w A and binding energy

Answer 8

Binding energies per nucleon increases sharply as A increases
Peaks at iron then slowly decreases for more massive nuclei.

Question 9

The Mass of a nucleus contains the binding energy, which can be given by a semi-empirical formula. This formula assumes that the nucleus can be thought of as what? Why was this eventually changed and what was it changed to?

Answer 9

Assumes nucleus is like a liquid drop, where the nucleons are all on the surface of this drop. (might be wrong)

However, this model underestimated binding energies of “magic nuclei” , whose number of p+s or ns equal to one of the numbers in the list: 2,8,20,28,50,82,126. (Doubly magic if both p+s and ns equal numbers in list)

This however was explained by “Shell Model”, where each nucleon is moving in some potential and their energy levels are classified in terms of the quantum numbers n, l and j. [like multi-layered liquid drop I think?]

Question 10

Note 5 features of the Shell model

Answer 10

1. For spherically symmetric potential, the wavefunction of any nucleon is: psi_nlm = R_nl (r) * Y^m _l (theta, phi) [same as wavefunction of electron in H]
2. The energy eigenvalues will depend on n and l but are degenerate in m.
3. Energy levels come in “bunches” called “shells” w/ large energy gaps b/w each shell.
4. In ground state, nucleons fill available energy levels from bottom up, w/ 2 in each energy level.
5. Potential follows Saxon-Woods model.

Question 11

Why would using a Simple Harmonic Potential to model the potential of nucleons be unsuccessful?

Answer 11

Would yield equally spaced energy levels and would not see the shell structure and hence the magic no.s

Question 12

What quantum numbers does the energy level depend on?

Answer 12

s, l and j

Question 13

One of the magic numbers was predicted to be 40 but turned out to be 50. What caused this increase? Write a general equation for its contribution.

Answer 13

Spin-Orbit coupling contribution to potential.

V(r) -> V(r) + W(r)L.S [L and S are vectors]

Question 14

What values does total angular momentum “j” take?

Answer 14

j = l +/- 1/2

Question 15

Equation for energy shift caused by spin-orbit coupling. [Hint: J_Lower(J_Lower + 1) - J_Higher(J_Higher + 1)]

Answer 15

delta E proportional to j(j+1) - l(l+1) - s(s+1)

Question 16

Do states with higher j have higher or lower energy?

Answer 16

lower

Question 17

The spin-orbit effect is v. large. What implications does this have?

Answer 17

Leads to crossing over of energy levels into diff shells. Hence, there are now more energy levels in shell with magic number 50 than previously thought, giving the reason for why magic no increased from 40 to 50.

Question 18

Spin of a pair of nucleons

Answer 18

0

Question 19

Parity of a pair of nucleons

Answer 19

1 (“even parity”)

Question 20

If Z and A are even, what is the spin and parity?

Answer 20

0 spin
1 (even parity)

Question 21

If A or Z is odd, there will be an unpaired nucleon. What is the spin and parity now?

Answer 21

spin = j value of unpaired nucleon
parity = (-1)^l where l is the orb ang mom of unpaired nucleon.

Question 22

If Z and A are odd, there will be an unpaired p+ w/ j1 and unpaired neutron w/ j2.

Answer 22

spin: takes values b/w mod(j1-j2) and mod(j1+j2) in integer steps.

parity = (-1)^{l1 + l2}

Question 23

Why do nuclei w/ and odd no. of p+s and/or ns have a magnetic dipole moment?

Answer 23

Because they have a non-zero intrinsic spin

Question 24

5 Assumptions made in Rutherford experiment

Answer 24

1. Nuclear charge of atom = Ze and that mass of proton and neutron are the same.
2. Nucleus is seen by alpha particle as a point particle because alpha particle has low energy, therefore has large wavelength, so nucleus appears as a point to it.
3. m_nucleus&raquo_space; m_alpha, therefore can neglect nuclear recoil
4. Laws of CM and Em apply and there are no other forces present
5. ELASTIC collision

Question 25

If alpha particle does not hit the nucleus head-on but by a distance given by the impact parameter “b”, does the scattering angle get smaller or larger?

Answer 25

Smaller, therefore there is not a full 180 degree rebound like there is in a head-on collision.

Question 26

What is the relationship between scattering angle and impact parameter (equation)

Answer 26

tan(theta/2) = D/2b

Question 27

What results were obtained from the Rutherford Experiment and what do they imply?

Answer 27

High probability that the alpha particle isn’t scattered much and low probability of backscattering (scattering angle > 90 degrees)

-> Suggests atom is mostly empty