Natural uranium found in the earth's crust contains the isotopes {}^{235}_{92}\mathrm{U} and {}^{238}_{92}\mathrm{U} in the atomic ratio 7\cdot3\times10^{-3} to 1. Assuming that at the time of formation these two isotopes were produced equally, estimate the time since formation. Given that the mean lives of both the isotopes are 1.03\times10^{9} years and 6.49\times10^{9} years, respectively.
Write down the Weizsäcker mass formula for the nuclear binding energy. Give short justification for each term of the formula.
List the main reactions in the pp chain leading from hydrogen to helium during stellar nucleosynthesis. Also mention the net effect of the reactions.
Explain the origin of the nuclear magnetic moment. Deduce expression for the magnetic dipole moment with the help of the Schmidt single particle model.
For a system consisting of one proton and one neutron (not necessarily a deuteron), write down the various possible states specifying clearly its isospin, spin and orbital quantum numbers.
Why is it not possible to detect the parity violation in weak interaction by observing only the beta decay rate? Justify your answer.
Explain the various methods of finding the size of the nucleus. How will you determine the nuclear radius from the observation of beta rays resulting from nuclear transition when the initial and final nuclei are mirror nuclei?
Given that the single particle energy separation between 1d_{5/2} and 1d_{3/2} in {}^{17}\mathrm{O} is 5\,\mathrm{MeV}. Calculate the strength of spin-orbit interaction. It is observed that 1d_{5/2} level is lower than 1d_{3/2} level.
Stable light nuclei have equal number of protons and neutrons, whereas heavy nuclei have excess of neutrons. Explain why.
Assuming equal masses for up (u) and down (d) quarks, find the ratio (\mu_n/\mu_p) of the magnetic moments of neutron and proton.
Nuclear forces are mediated by exchange of \pi-mesons of rest mass 140\,\mathrm{MeV}. Estimate the range of nuclear forces.
The maximum energy of a positron (e^+) released in the decay of \,{}_{6}^{13}\mathrm{C} atom into a \,{}_{7}^{13}\mathrm{N} atom is 1.202\,\mathrm{MeV}. If the mass of the \,{}_{6}^{13}\mathrm{C} atom is 13.003354\,\mathrm{u}, calculate the mass of the \,{}_{7}^{13}\mathrm{N} atom.
Obtain an expression for the magnetic moment of a nucleus having one nucleon outside the closed core. Use this to calculate the magnetic moment of \,{}_{8}^{17}\mathrm{O} nucleus.
Assuming that the neutron-proton interaction has a square well form V(r)=-V_0\quad\text{for }r\leq b =0\quad\text{for }r>b, the ground state wave function of deuteron nucleus is given as \psi(r)=A\sin kr\quad\text{for }r\leq b =Ce^{-\gamma r}\quad\text{for }r>b where k=\sqrt{\frac{M}{\hbar^2}(V_0+W)} and \gamma=\sqrt{\frac{MW}{\hbar^2}}. Here M is the nucleon mass, W is the binding energy of deuteron and A and C are constants.
(i) Show that for a just bound state of deuteron V_0b^2=\frac{\pi^2\hbar^2}{4M}.
(ii) Explain why deuteron is a loosely bound extended structure.
A \pi^--meson at rest decays into a \mu^--meson: \pi^-\to\mu^-+\bar{\nu}_{\mu} Calculate the kinetic energy of the \mu^--meson emitted in the reaction.
Which of the following elementary particle reactions/decays are allowed under various conservation laws? If allowed, write down the type of interaction and the characteristic time by which it would proceed:
(i) p + n \to \Lambda^0 + \Sigma^+
(ii) \pi^+ + n \to \Lambda^0 + K^+
(iii) p+n\to K^{+}+\Sigma^{+}
(iv) \pi^{0}\to\gamma+
(v) \bar{n}\to\bar{p}+e^{+}+\nu_e
Write the semi-empirical mass formula for nuclei and on its basis draw mass parabolas for odd and even isobars. What would be the most stable isobar in each case?
Write down the quark constituents of each of the following:
(i) \pi^+
(ii) K^+
(iii) \Delta^{++}
(iv) \Sigma^0
(v) \Omega^-
How does one explain the approximate constancy of average binding energy per nucleon (\mathrm{BE}/A) of nuclei in the region 30 \leq A \leq 170 in the plot of \mathrm{BE}/A versus mass number A?
Write two properties of deuteron which support the existence of non-central tensor force.
