PHY492 Spring 2014, Nuclear and Elementary Particle Physics Homework 2, Due date: Monday, February 3, 2014 Please clearly state your assumptions, number the equations, and indicate logical connections between the lines. 1. (2pt x 5) Answer Yes or No to the following statements. Clearly and fully explain your reasoning. (a) In the Penning trap, ions are confined only using a magnetic field. (b) The most unstable isobar has the lightest mass among other isobars. (c) Beta-­‐decay lifetime of a nucleus (the ground state of the nucleus) has a single value irrespective of its decay branches. (d) In the liquid drop model, conditions for spontaneous fission are related to nuclear deformation effects and given by the competition between the volume and surface terms in the semi-­‐empirical mass formula. (e) In scattering experiments at relativistic energies, most of particles will be emitted in very forward angles because of Lorentz boost (Lorentz transformation from the center-­‐of-­‐mass frame to the laboratory frame). 2. (4pt) 210
Bi (Z=83, mean life 7.2 days) decays by beta-­‐particle emission to 210Po (Z=84, mean life 200 days), which in turn decays by alpha-­‐particle emission to 206Pb (Z=82). If a source initially contains only pure 210Bi, after how long will the rate of alpha-­‐particle emission reach a maximum? Use the following equation ( Eq.(2.67) in the text book ) for the sequential decay. 3. (4pt) The figures below show a comparison between the 52MeV deuteron elastic scattering data on 48Ti (left figure) and the Fraunhofer-­‐like diffraction form (J1(qR)/qR)2 (right figure). We can see that the third minimum of the data at 40 degrees (left) corresponds to the diffraction pattern at qR=7 [hbar] (right). Deduce the radius of 48Ti from this relation, and compare the result with a simple radius formula 1.2 x A1/3 [fm]. Assume the mass of deuteron of 1876 MeV/c2 and use the relation q = 2psin(θ/2) ( p: deuteron momentum [MeV/c], θ:scattering angle[deg]). 4. (4pt) Using the following formula, compute the total binding energy (B) and the value of B/A for 7Be (Z=4), 12C (Z=6), 12N (Z=7), 56Fe (Z=26), and 208Pb (Z=82), and compare with the experimental values of B/A available at http://www.nndc.bnl.gov/amdc/masstables/Ame2003/mass.mas03 (Note: this table gives B/A in keV) B (MeV) = 15.56A -­‐ 17.23A2/3 -­‐ 0.697 Z2/A1/3 -­‐ 93.14 (Z-­‐N)2/(2A)2 -­‐ 12δ/A1/2 (δ= -­‐1 for even-­‐even nuclei, δ=0 for even-­‐odd/odd-­‐even nuclei, δ=+1 for odd-­‐odd nuclei) 5. (1pt+2pt+1pt+2pt+2pt) A relic from an Egyptian tomb contains 1 mol (about 12g) of carbon with a measured activity of 4 × 10-­‐12 Ci (1Ci = 3.7×1010 Bq, Bq = 1 decay/second) for the radioactive decay of 14C. By following the procedures given below, determine how old the relic is. Assume the ratio of 14C/12C nuclei in a live tree is 1.3 × 10-­‐12, and the half-­‐life (T1/2) of 14C is 5420 years. (a) Convert the unit of the measured activity of the 14C decay to decay/second. (b) The number of unstable 14C nuclei in the relic follows the exponential decay curve N(t) = N(0) exp(-­‐t·∙ln2/T1/2). Calculate the activity A(t) (intensity of decays as a function of time t) using the function N(t). (c) Calculate the number N(0) of 14C, when the relic was buried in the tomb. Assume the 14C/12C ratio was the same as that of a live tree at present. Avogardo’s number is 6.0 × 1023/mol. (d) Calculate the activity A(0) of the 14C decay. (e) Determine the age Tage of the relic by considering the ratio A(Tage)/A(0). Note that A(Tage) is the value obtained in (a).