Self-Assessment Quiz — Chapter 1

Test your understanding of the core concepts before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) In the Geiger-Marsden experiment, approximately what fraction of alpha particles scattered at angles greater than 90 degrees?

(a) 1 in 10 (b) 1 in 100 (c) 1 in 8,000 (d) 1 in 1,000,000

Q2. (Multiple Choice) The Rutherford scattering cross section is proportional to:

(a) $\sin^{-2}(\theta/2)$ (b) $\sin^{-4}(\theta/2)$ (c) $\cos^{-4}(\theta/2)$ (d) $\theta^{-4}$

Q3. (True/False) The Rutherford cross section diverges as $\theta \to 0$ because the Coulomb force has infinite range.

Q4. (Short Answer) What is the relationship between the impact parameter $b$ and the scattering angle $\theta$ in Rutherford scattering? Define all symbols.

Q5. (Multiple Choice) Chadwick identified the neutron by analyzing:

(a) Gamma ray spectra from beryllium (b) Recoil energies of different target nuclei struck by neutral radiation from beryllium (c) Tracks in a cloud chamber from beryllium bombardment (d) Magnetic deflection of radiation from beryllium

Q6. (True/False) The neutron is slightly lighter than the proton.

Q7. (Short Answer) Define: isotopes, isotones, and isobars. Give one example of each.

Q8. (Multiple Choice) Approximately how many nuclides have been experimentally observed?

(a) ~250 (b) ~1,000 (c) ~3,300 (d) ~10,000

Q9. (Short Answer) Why does the valley of stability curve toward $N > Z$ for heavy nuclei?

Q10. (Multiple Choice) The nuclear magic numbers are:

(a) 2, 6, 14, 28, 50, 82, 126 (b) 2, 8, 20, 28, 50, 82, 126 (c) 2, 8, 18, 32, 50, 72, 126 (d) 2, 10, 28, 50, 82, 114, 126

Q11. (True/False) ${}^{208}\text{Pb}$ is doubly magic, with $Z = 82$ and $N = 126$.

Q12. (Short Answer) State the empirical formula for the nuclear radius as a function of mass number. What does the $A^{1/3}$ dependence imply about nuclear density?

Q13. (Multiple Choice) Nuclear density is approximately:

(a) $0.016\,\text{nucleons/fm}^3$ (b) $0.16\,\text{nucleons/fm}^3$ (c) $1.6\,\text{nucleons/fm}^3$ (d) $16\,\text{nucleons/fm}^3$

Q14. (Short Answer) Define "binding energy." Is the binding energy of a stable nucleus positive or negative? Why?

Q15. (True/False) A nucleus with a larger binding energy per nucleon ($B/A$) contains more energy than one with a smaller $B/A$.

Q16. (Multiple Choice) The nucleus with the highest binding energy per nucleon is:

(a) ${}^{4}\text{He}$ (b) ${}^{56}\text{Fe}$ (c) ${}^{62}\text{Ni}$ (d) ${}^{208}\text{Pb}$

Q17. (Short Answer) Explain in two or three sentences why both nuclear fusion and nuclear fission can release energy, using the binding energy per nucleon curve.

Q18. (Short Answer) The neutron separation energy $S_n$ for ${}^{208}\text{Pb}$ is 7.37 MeV, while for ${}^{209}\text{Pb}$ it is only 3.82 MeV. What does this dramatic change tell you about $N = 126$?

Solutions

A1. (c) 1 in 8,000. This was the remarkable observation — the plum pudding model predicted negligible large-angle scattering.

A2. (b) $\sin^{-4}(\theta/2)$. This is the signature angular dependence of Coulomb scattering from a point charge, derived from $b = a\cot(\theta/2)$.

A3. True. The long-range $1/r$ Coulomb potential means that even very distant encounters produce (very small) deflections. The cross section integrates to infinity because there is no natural cutoff to the impact parameter. In practice, electron screening provides this cutoff.

A4. $b = a\cot(\theta/2)$, where $b$ is the impact parameter (perpendicular distance from the projectile's initial trajectory to the target nucleus), $\theta$ is the scattering angle, and $a = kz_1z_2e^2/(2T)$ is half the distance of closest approach for a head-on collision, with $z_1, z_2$ the projectile and target charges, $T$ the kinetic energy, and $k = 1/(4\pi\epsilon_0)$.

A5. (b) Recoil energies of different target nuclei. Chadwick measured the maximum recoil energies of protons and nitrogen nuclei, using the ratio to deduce the mass of the neutral particle.

A6. False. The neutron ($m_n = 939.565\,\text{MeV}/c^2$) is slightly heavier than the proton ($m_p = 938.272\,\text{MeV}/c^2$) by about $1.293\,\text{MeV}/c^2$. This is why free neutrons are unstable and undergo beta decay ($n \to p + e^- + \bar{\nu}_e$) with a half-life of about 10.2 minutes.

A7. - Isotopes: Same $Z$, different $N$. Example: ${}^{12}\text{C}$ and ${}^{14}\text{C}$ (both $Z = 6$). - Isotones: Same $N$, different $Z$. Example: ${}^{13}\text{C}$ and ${}^{14}\text{N}$ (both $N = 7$). - Isobars: Same $A$, different $Z$. Example: ${}^{14}\text{C}$ and ${}^{14}\text{N}$ (both $A = 14$).

A8. (c) ~3,300. Of these, only about 252 are truly stable. The vast majority are radioactive.

A9. The nuclear (strong) force acts between all nucleon pairs ($pp$, $nn$, $pn$) and is short-ranged. The Coulomb force acts only between proton pairs and is long-ranged, growing as $Z^2$. For heavy nuclei, the Coulomb repulsion among the many protons would destabilize the nucleus unless extra neutrons are added. The additional neutrons contribute to the nuclear attraction (diluting the Coulomb effect per nucleon) without adding Coulomb repulsion, so $N > Z$ is required for stability.

A10. (b) 2, 8, 20, 28, 50, 82, 126. These arise from the nuclear shell model (Chapter 6) and are analogous to the noble gas electron numbers in atomic physics.

A11. True. ${}^{208}\text{Pb}$ has $Z = 82$ (magic) and $N = 208 - 82 = 126$ (magic), making it doubly magic and the heaviest stable doubly magic nucleus.

A12. $R = r_0 A^{1/3}$ with $r_0 \approx 1.21\,\text{fm}$. Since $V \propto R^3 \propto A$, the density $\rho = A/V$ is constant (independent of $A$). This is nuclear density saturation — all nuclei have approximately the same interior density, about $0.16\,\text{nucleons/fm}^3$.

A13. (b) $0.16\,\text{nucleons/fm}^3$. This is nuclear saturation density, equivalent to about $2.3 \times 10^{17}\,\text{kg/m}^3$.

A14. The binding energy $B(A,Z) = [Zm_p + Nm_n - M(A,Z)]c^2$ is the energy required to completely disassemble the nucleus into its constituent free protons and neutrons. For any stable (or bound) nucleus, $B > 0$ — the nucleus has less mass than its parts, and you must add energy to break it apart. A negative binding energy would mean the nucleus is unbound.

A15. False. This is a common and important misconception. A nucleus with larger $B/A$ has less mass per nucleon — it is more tightly bound. The binding energy represents the energy you would need to add to disassemble it, not energy "stored" inside. Moving to a higher $B/A$ releases energy (the products have less total mass).

A16. (c) ${}^{62}\text{Ni}$, with $B/A = 8.795\,\text{MeV/nucleon}$. ${}^{56}\text{Fe}$ ($B/A = 8.790$) is very close but slightly lower. The common claim that iron-56 is the most tightly bound is a widespread misconception.

A17. The binding energy per nucleon curve rises steeply for light nuclei, peaks near $A \approx 62$, and decreases slowly for heavy nuclei. Fusion of light nuclei moves up the curve toward the peak, producing more tightly bound products and releasing the mass-energy difference. Fission of heavy nuclei also moves up the curve (from the other side), producing medium-mass fragments that are more tightly bound per nucleon. In both cases, the products have higher $B/A$ and therefore less mass, with the mass difference appearing as kinetic energy.

A18. The sharp drop in $S_n$ from 7.37 MeV to 3.82 MeV when going from ${}^{208}\text{Pb}$ ($N = 126$) to ${}^{209}\text{Pb}$ ($N = 127$) indicates that $N = 126$ is a magic number — a closed neutron shell. The 126th neutron completes a shell and is tightly bound; the 127th neutron begins a new shell and is much less tightly bound, analogous to the drop in ionization energy when moving from a noble gas to an alkali metal in atomic physics.