Nuclei Set-3

Q1. A neutral $^{16}$O nucleus has $Z=8$, $N=8$ and the atomic mass $m(^{16}\mathrm{O})=15.994915\ \mathrm{u}$. Given $m_p=1.007276\ \mathrm{u}$, $m_n=1.008665\ \mathrm{u}$, $m_e=0.000549\ \mathrm{u}$ and $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$, the binding energy per nucleon of $^{16}$O (in MeV) is closest to

Q2. The binding energy per nucleon values are: $^2$H: $1.11\ \mathrm{MeV}$, $^4$He: $7.07\ \mathrm{MeV}$, $^{56}$Fe: $8.79\ \mathrm{MeV}$ and $^{238}$U: $7.63\ \mathrm{MeV}$. Consider (I) fusion of two deuterons into one $^4$He and (II) symmetric fission of $^{238}$U into two equal fragments each with BE/A $=8.60\ \mathrm{MeV}$. Which process releases the larger energy per nucleon and approximately how much?

Q3. Assertion (A): Even–even nuclei (both $Z$ and $N$ even) are generally more stable (higher binding energy) than neighbouring odd–odd nuclei of the same mass number $A$.

Reason (R): In the semi-empirical mass formula, the pairing term is

which increases binding for even–even nuclei.

Q4. Using the semi‑empirical mass formula and ignoring the pairing term, the most stable proton number $Z$ for a given mass number $A$ approximately satisfies

For $A=125$ with $a_c=0.714\ \mathrm{MeV}$ and $a_a=23.0\ \mathrm{MeV}$, the integer value of $Z$ nearest to the most stable isobar is

Q5. A sample contains two radioactive isotopes X and Y with half‑lives $T_{1/2,X}=1\ \mathrm{h}$ and $T_{1/2,Y}=8\ \mathrm{h}$. At $t=0$ the total count rate is $1200\ \mathrm{s^{-1}}$. After $3\ \mathrm{h}$ the total count rate is $600\ \mathrm{s^{-1}}$. Assuming no other sources, what percentage of the initial activity (at $t=0$) was due to isotope X (shorter lived)?

Q6. A parent nucleus $P$ has atomic mass $M_P=223.0180\ \mathrm{u}$. Possible daughter atomic masses are: for $\beta^-$ decay $M_{D^-}=223.0140\ \mathrm{u}$, for $\beta^+$ decay $M_{D^+}=223.0170\ \mathrm{u}$, and for $\alpha$ decay daughter mass $M_{D_\alpha}=219.0100\ \mathrm{u}$. Take $m_e=0.0005486\ \mathrm{u}$ and atomic mass of $^4$He as $4.002603\ \mathrm{u}$. Which decay channels are energetically allowed (Q>0) based on atomic masses?

Q7. Approximate the Coulomb barrier for an $\alpha$ particle at the surface of a $^{238}$U nucleus by $V=\dfrac{1}{4\pi\epsilon_0}\dfrac{2Ze^2}{R}$ with $R=r_0A^{1/3}$, $r_0=1.2\ \mathrm{fm}$ and $e^2/(4\pi\epsilon_0)=1.44\ \mathrm{MeV\cdot fm}$. For $Z=92$, $A=238$ this barrier is about $36\ \mathrm{MeV}$ while the $\alpha$ decay $Q$‑value is $\sim4.2\ \mathrm{MeV}$. Which statement is correct?

Q8. Assertion (A): If $\beta^+$ emission from a nucleus is energetically forbidden, electron capture (EC) from an inner orbital can still occur for the same parent nucleus. Reason (R): $\beta^+$ emission requires creation of a positron and thus needs at least $2m_ec^2$ extra energy, whereas EC simply converts a proton to a neutron by capturing an orbital electron and does not require creating a positron.

Q9. A parent nuclide decays to a daughter (initially absent) with half‑life $T_{1/2,1}=2.00\ \mathrm{d}$; the daughter has half‑life $T_{1/2,2}=5.00\ \mathrm{d}$. If initially only the parent is present, the daughter activity $A_2(t)$ rises to a maximum at time $t_{\max}$. Using decay constants $\lambda_i=\ln2/T_{1/2,i}$, $t_{\max}$ (in days) is approximately

Q10. Estimate the average mass density $\rho$ of a nucleus with $A=56$ using $R=r_0A^{1/3}$ with $r_0=1.20\ \mathrm{fm}$ and nucleon mass $m_N=1.67\times10^{-27}\ \mathrm{kg}$. Which is closest to the result?