MIDSUMMER EXAMINATIONS 2001 PHYSICS, PHYSICS WITH ASTROPHYSICS PHYSICS WITH SPACE SCIENCE & TECHNOLOGY PHYSICS WITH MEDICAL PHYSICS

Transcription

1 No. of Pages: 6 No. of Questions: 10 MIDSUMMER EXAMINATIONS 2001 Subject PHYSICS, PHYSICS WITH ASTROPHYSICS PHYSICS WITH SPACE SCIENCE & TECHNOLOGY PHYSICS WITH MEDICAL PHYSICS Title of Paper MODULE PA266 opt.322 ATOMS AND NUCLEI Time Allowed Three hours Instructions to candidates Answer ALL questions in SECTION A. Allow one hour. Section A carries 40 marks. Answer THREE questions ONLY in SECTION B. The marking scheme for each question in SECTION B is indicated by the numbers in square brackets in the right-hand margin. A separate answer book MUST be used for Sections A & B. A LIST OF PHYSICAL CONSTANTS IS ATTACHED SECTION A A1. a) State the Pauli Exclusion Principle. b) What set of quantum numbers best defines a spin-orbit coupled electron state and what physical quantities do these quantum numbers represent? c) Under what circumstances might a different set of quantum numbers be preferred when describing the quantum state of an electron in an atom? d) Suggest an alternative set of quantum numbers appropriate to (c). A2. a) How many fine structure components contribute to the n = 3 2 (Hα) emission line of hydrogen? b) Draw a Grotrian diagram to illustrate the relevant transitions and label each energy level with an appropriate term symbol. c) Specify all the selection rules that apply in the above case.

2 2. A3. a) Calculate the Q value of the process Po Bi + p, Hence decide whether this process can occur spontaneously. What is the relevance of this calculation to the decay modes of heavy nuclei? b) Sketch a graph to show how the potential energy of an α particle in the vicinity of a nucleus depends on its distance from the centre of the nucleus and indicate the nature of the forces experienced by the α particle. By considering the form of the graph, explain why the Rutherford scattering law breaks down when the α particle energy is sufficiently high. 209 Atomic masses in u : 84Po Bi H A4. a) i) What is meant by the activation energy for nuclear fission? ii) What is used to supply the activation energy for fission of U in a natural uranium reactor? iii) Why does fission of U also occur in a natural uranium reactor? iv) What is the function of the moderator in a natural uranium reactor? b) A solar neutrino detector consists of a tank of GaCl 3 solution on the Earth's surface. Solar neutrinos cause inverse β decay of 71 31Ga and the decay events 2 are counted by particle detectors. The tank has a cross-sectional area of 10 m and a height of 3 m. The solar neutrino flux is m 2 s 1. Estimate the number of events counted per year. number density of 71 31Ga nuclei in solution = 5 10 cross-section for inverse β decay = m 2 27 m 3

3 3. SECTION B B1. Describe the Stern-Gerlach experiment and discuss its importance in establishing the fact that electrons carry spin angular momentum. [10] In a Stern-Gerlach experiment performed with hydrogen atoms, deflections of ±1 o are observed for atoms of average kinetic energy J subject to a magnetic field gradient of 400 T m 1 over a distance of 9.4 cm. Use this information to obtain an estimate of the gyromagnetic ratio associated with the electron spin. [8] Compare the derived value for the gyromagnetic ratio of an electron with that expected for a classical charged spinning particle. [2] B2. Describe the main features of the spectrum of atomic helium and explain how these features may be interpreted in terms of the LS coupling scheme. [12] Assuming LS coupling determine the possible states of an atom with two optical electrons in the following configurations: (a) 1s 1 3d 1 ; (b) 2p 1 3p 1 ; (c) 4s 2. [8] Note: LS coupling is also known as Russell-Saunders coupling. B3. The sodium resonance line consists of a doublet with components at 5890 and 5896 Å. Identify the electronic transitions which produce this doublet structure. [2] When the sodium resonance line is subjected to an applied magnetic field, it exhibits further line splitting. Describe the line splitting pattern which is observed as the magnetic field is increased gradually from zero to a very high value. [6] What is the origin of this line splitting? [6] In a particular study, the spectrum of the sodium resonance line is recorded when a magnetic field of 50 T is applied (NB. This is an exceptionally strong magnetic field). At what wavelengths will spectral line components be observed? [6]

4 4. B4. Define the binding energy of a nucleus. [2] An approximate expression for the binding energy of an odd-even nucleus is B(Z, A) =? a 1 A? a 2 A 2/3 Z(Z -1) (2Z - A) 2? a 3? a A 1/3 4, A where Z is the atomic number, A is the mass number and a 1, a 2, a 3 and a 4 are constants. Give the missing signs and explain the physical meaning of each term. [8] State the additional term that appears in the expressions for the binding energy of odd-odd and even-even nuclei and explain its physical origin. [2] Find the energy released in the process At Bi +α. Hence, decide whether this process can occur spontaneously. [8] a 1 = 15.5 MeV, a 2 = 16.8 MeV, a 3 = 0.72 MeV, a 4 = 23 MeV. Binding energy of α particle = 28.3 MeV. B5. Sketch the energy spectrum of the particles emitted in α and β decay and label the main features of the spectrum in each case. [4] State the physical origin of the features in the α and β spectra and explain why the spectra have a very different form. [4] Define the Q value of an α decay reaction and derive a relation between the Q value and the kinetic energy of the α particle. [6] The highest α particle energy in α decay of 227 Th to 223 Ra is MeV and the next highest energy is MeV. Calculate the energy of the γ ray that is emitted after the MeV α particle. [6] Note: For a heavy nucleus of mass m and mass number A, the α to nuclear mass ratio m α /m is approximately 4/A.

5 5. B6. Name the conservation laws that govern any nuclear reaction and define the Q value of the reaction. Express Q in terms of (a) the initial kinetic energies of the reactants and the final kinetic energies of the products and (b) the masses of the reactants and products. [4] Consider the reaction a + X Y + b where a and b are particles, X and Y are nuclei and X is stationary. Use the conservation laws to show that the relation between the kinetic energies T a and T b of the particles is (m Y + m b )T 1/2 b =(T a m a m b ) 1/2 cosθ ± { T a m a m b cos 2 θ + [ Qm T + T a (m Y m a )](m Y + m b )} 1/2, where m a, m b, m Y are masses and θ the angle between the momenta of a and b. [8] Use this result to show that a negative Q reaction cannot proceed unless T a is greater than a certain threshold energy and obtain an expression for this energy. [3] Calculate the Q values of the reaction 11 5B + p 11 6 C + n and estimate the threshold energy. [2] It is desired to use this reaction to produce 11 6C. What steps could be taken to maximise the production rate? [3] Atomic masses in u : 11 5B H C n

6 6. PHYSICAL CONSTANTS elementary charge e = C electron rest mass me = kg proton rest mass mp = kg neutron rest mass mn = kg Planck constant h = 2π h = J s speed of light in vacuum c = m s 1 Boltzmann constant k = J K 1 Bohr magneton µ B = J T 1 Stefan constant σ = W m 2 K 4 radiation constant a = 4σ / c = J m 3 K 4 Avogadro number N A = mol 1 gas constant R = J mol 1 K 1 gravitational constant G = N m 2 kg 2 permittivity of vacuum εo = F m 1 permeability of vacuum µo = 4π 10 7 H m 1 1 atomic mass unit u = kg MeV 1eV = J ASTRONOMICAL CONSTANTS mass of Sun M O = kg radius of Sun R O = km luminosity of Sun L O = J s-1 one parsec pc = m mean Earth-Sun distance 1 a.u. = m mass of Earth M O + = kg mean radius of Earth R O + = 6370 km