Nuclear Properties and Structure Questions and Answers

Answer: B

The minimum photon energy is \(B\left(1 + \dfrac{B}{2mc^{2}}\right)\).

The photon must supply the binding energy B and also the recoil kinetic energy of the products, because its momentum E/c must be conserved. At threshold the two nucleons move off together with total momentum E/c and total rest energy mc² + B. Energy conservation with the bound state at rest:

\[ \begin{aligned} E + mc^{2} &= \sqrt{(mc^{2} + B)^{2} + E^{2}} \\ E &= B + \frac{B^{2}}{2mc^{2}} = B\left(1 + \frac{B}{2mc^{2}}\right) \end{aligned} \]

B alone ignores the recoil. The minus sign would make the threshold smaller than the binding energy, which energy conservation forbids, and B - mc² is negative for any real nucleus.

Answer: B

The ratio of nucleon numbers is 8.

Nuclear radius follows \( R = r_{0}A^{1/3} \), so \( A \propto R^{3} \):

\[ \begin{aligned} \frac{A_{A}}{A_{Al}} &= \left(\frac{R_{A}}{R_{Al}}\right)^{3} \\ &= 2^{3} = 8 \end{aligned} \]

Nucleus A has 8 × 27 = 216 nucleons. 16 would assume \( A \propto R^{4} \), and 40 or 14 do not come from the cube law.

Answer: B

The wavelength is nearly 9 F.

A 10 MeV proton is non-relativistic (its rest energy is 938 MeV), so its momentum is \(p = \sqrt{2mK}\):

\[ pc = \sqrt{2 \times 938 \times 10} \approx 137\ \text{MeV} \]

\[ \begin{aligned} \lambda &= \frac{hc}{pc} = \frac{1240\ \text{MeV fm}}{137\ \text{MeV}} \\ &\approx 9.0\ \text{fm} \end{aligned} \]

Since \(\lambda \propto 1/\sqrt{K}\), the other values belong to other energies: 7 F to about 17 MeV, 11 F to about 7 MeV and 13 F to about 5 MeV.

Answer: B

The binding energy per nucleon is nearly 8.54 MeV/nucleon.

Silver-107 has Z = 47 protons and N = 107 - 47 = 60 neutrons. The mass defect, using the hydrogen-atom mass for the proton so that the electrons cancel:

\[ \begin{aligned} \Delta m &= 47(1.007825) + 60(1.008665) \\ &\quad - 106.905091 \\ &= 107.887675 - 106.905091 \\ &= 0.982584\ \text{amu} \\ B &= 0.982584 \times 931.5 = 915.3\ \text{MeV} \\ B/A &= 915.3/107 = 8.55\ \text{MeV} \end{aligned} \]

That is close to the 8.5 MeV typical of medium nuclei. 8.10, 8.95 and 9.25 MeV/nucleon do not follow from these masses.