14 March 2000, 10:15 – 12:15 Dr. R. Egerton

------------------------------------------------------------------------------------------------------------

electron charge = 1.60 ´ 10^-19 C electron rest mass = 9.11 ´ 10^-31 kg

Planck’s constant h = 6.63 x 10^-34 J s first Bohr radius a₀ = 0.529 x 10^-9 m

1. (a) Under what conditions can electrons tunnel from one metal to another?

2. Sketch a labeled diagram of a scanning tunneling microscope (STM).

(c) Name one advantage of the constant-current mode of operation of the STM.

(d) Name one advantage of the constant-height mode of operation.

2. (a) Calculate the lowest energy of an electron confined (at zero potential) within a three-dimensional cubic box (with impenetrable walls) whose sides are 1 nm in length.

(b) What changes must be made to the physical conditions in (a) to make them appropriate to solving the Schroedinger equation for a hydrogen atom ?

Sketch graphs showing the probability P(r) of finding an electron at a distance r from the center of the hydrogen atom: (c) for the 1s state and (d) for the 2s state.

5. What is the most probable value of r for the 1s state ?

3. (a) Given the radius of a ²³⁸U nucleus is 7.4 fm, calculate the radius of a ⁵⁶Fe nucleus.

(b) Write down a definition of the binding energy E_b of an atomic nucleus.

(c) Sketch a graph showing the binding energy per nucleon (E_b/A) versus the mass number A of an atomic nucleus.

(d) According to the liquid drop model, what are the four contributions to the nuclear binding energy (list them using words, not symbols).

4. (a) Name the four fundamental forces between material particles, according to the Standard Model of particle physics.

(b) Name the field particles responsible for these forces.

(c) Write down which of the following particles are truly fundamental (structureless): electron, proton, quark, K-meson, muon, neutron, neutrino.

5. (a) Sketch a labeled diagram showing all the essential features of a cyclotron, as seen from above and including a typical particle trajectory .

(b) What physical effect limits the kinetic energy of the particles that can be accelerated by a cyclotron?

(c) How is this limit overcome in the case of a synchrotron?

4. Name one advantage of the bubble chamber over a cloud chamber, for recording particle tracks.

6. Write a concise summary (about 1 page) of ONE of the following:

1. the evolution of the universe OR (b) cold fusion OR (c) carbon nanotubes.

Ph 312 Final Exam (14 March 2000) Answers (R. Egerton)

1. (a) An electron can tunnel when the metals are separated by an insulator (or vacuum) whose thickness is less than several times the penetration depth (typically 0.1 nm).

(b) The diagram should show a sharp metal tip capable of being moved in x-, y- and z-directions (by piezoelectric elements) and placed very close to a specimen. A small voltage is applied between specimen and tip and the resulting tunneling current is monitored by a control computer.

(c) Constant-current mode maintains the tunneling gap constant, so that the STM tip does not crash into the specimen (very useful for rough specimens or large scan distances).

(d) Constant-height operation allows faster scan (no need to vary tip height by servo control) as required for the highest spatial resolution or recording real-time "movies".

2. (a) E = (1/2m) [pi hbar / L]^2 [n1^2 + n2^2 + n3^2]

= (0.5/9.11E-31) [(pi)(6.63E-34)/(2 pi 1E-9)]^2 [1 + 1 + 1]

= (5.488E29)(3.313E-25)^2 (3) = 1.807E-19 J = 1.13 eV

(b) Geometry must be changed to make it spherically symmetric and the potential must be changed from square-well shape to a hyperbolic potential well (V proportional to 1/r).

2. P(r) should start from zero at r=0, attain a maximum and return asymptotically to zero.
3. P(r) should start from zero, go through a maximum and return to zero, then go through a larger maximum before returning asymptotically to zero.
4. The most probable value of r is equal to the first Bohr radius, a0 = 0.529E-10 m.

1. (a) Nuclear radius R is proportional to A^(1/3),

so R(Fe) = R(U) (56/239)^(1/3)= (7.4 fm) (0.617) = 4.57 f

(b) Eb is the energy needed to separate all the nucleons to an infinite distance from each other

(c) Eb/A rises rapidly (with some oscillation) from zero (at A=0), achieves a broad maximum and then slowly decreases.

4. Bulk binding-energy term, surface-energy term (negative), electrostatic-repulsion term and N/Z-symmetry term.

1. (a) strong force, electromagnetic force, weak (nuclear) force and gravitational force

(b) gluon, photon, W (or Z) boson, graviton

(c) electron, quark, muon, neutrino

2. (a) The diagram should show the particle spiraling outward from the centre (due to a magnetic field perpendicular to the diagram) and passing between two "dees".

(b) Relativistic increase in particle mass limits the kinetic energy attainable.

(c) In the synchrotron, the applied magnetic field is increased (in a way which allows for the mass increased to keep the radius of curvature of the particle trajectory constant.

4. The bubble chamber can be cleared of particle tracks in a very short time, allowing operation at a high repetition rate.

P312fi00.doc