A point source emitting alpha particles is placed at a distance of 1m from a counter which records any alpha particle falling on its 1 cm² window. If the source contains 6.0 × 10¹⁶ active nuclei and the counter records a rate of 50000 counts/second, find the decay constant. Assume that the source emits alpha particles uniformly in all directions and the alpha particles fall nearly normally on the window.

A. A radioactive nucleus ‘A’ undergoes a series of decays as given below :

The mass number and atomic number of are 176 and 71 respectively. Determine the mass and atomic numbers of and A.

B. Write the basic nuclear processes underlying and decays.

Nuclei with magic no. of proton Z = 2, 8, 20, 28, 50, 52 and magic no. of neutrons N = 2, 8, 20, 28, 50, 82 and 126 are found to be very stable.

(i) Verify this by calculating the proton separation energy

S_p for ¹²⁰Sn (Z = 50) and ¹²¹Sb = (Z = 51).

The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by

Sp = (M_{Z–1, N} + M_H – M_Z,N) c².

Given ¹¹⁹In = 118.9058u, ¹²⁰Sn = 119.902199u,

¹²¹Sb = 120.903824u, ¹H = 1.0078252u.

(ii) What does the existence of magic number indicate?

The activity R of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

(i) Plot the graph of R versus t and calculate half-life from the graph.

(ii) Plot the graph of versus t and obtain the value of half-life from the graph.

Define the term ‘decay constant’ of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegrations/s after 20 hr. and 30 hr. respectively from start. Calculate the half life and initial number of nuclei at t = 0.

(a) Write the relation between half-life and average life of a radioactive nucleus.

(b) In a given sample two isotopes A and B are initially present in the ratio of 1:2. Their half-lives are 60 years and 30 years respectively. How long will it take so that the sample has these isotopes in the ratio of 2:1?

A charged capacitor of capacitance C is discharged through a resistance R. A radioactive sample decays with an average-life τ. Find the value of R for which the ratio of the electrostatic field energy stored in the capacitor to the activity of the radioactive sample remains constant in time.

Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is:

Assume that we start with 1000 ³⁸S nuclei at time t = 0. The number of ³⁸Cl is of count zero at t = 0 and will again be zero at t = ∞. At what value of t, would the number of counts be a maximum?

Which sample, A or B shown in Fig. 13.2 has shorter mean-life?