Exercises — Chapter 1

Rutherford Scattering

Problem 1.1 ⭐ Alpha particles with kinetic energy $T = 7.7\,\text{MeV}$ are scattered from a thin aluminum foil ($Z = 13$).

(a) Calculate the distance of closest approach $d_0$ for a head-on collision ($b = 0$).

(b) Compare $d_0$ to the nuclear radius of aluminum ($R \approx 1.21 \times 27^{1/3}\,\text{fm}$). Is the Rutherford formula valid at this energy?

(c) Calculate the impact parameter $b$ for scattering at $\theta = 90°$.

Problem 1.2 ⭐ Show that the Rutherford scattering formula can be written as:

$$\frac{d\sigma}{d\Omega} = \left(\frac{z_1 z_2 e^2}{4E_{\text{cm}}}\right)^2 \frac{1}{\sin^4(\theta/2)}$$

where $E_{\text{cm}}$ is the kinetic energy in the center-of-mass frame. Express $E_{\text{cm}}$ in terms of the lab kinetic energy $T_{\text{lab}}$ and the masses $m_1$ (projectile) and $m_2$ (target).

Problem 1.3 ⭐⭐ The original Geiger-Marsden experiment used 5.5 MeV alpha particles on gold ($Z = 79$, $A = 197$).

(a) Calculate the Rutherford cross section $d\sigma/d\Omega$ at $\theta = 45°$, $90°$, and $135°$. Express your answers in barns/sr.

(b) Verify the $\sin^{-4}(\theta/2)$ dependence by computing the ratios of cross sections at these angles.

(c) If the gold foil is $0.4\,\mu\text{m}$ thick ($\rho_{\text{Au}} = 19.3\,\text{g/cm}^3$, $A_{\text{Au}} = 197$), calculate the number of gold atoms per unit area $n_t$, and hence the probability that an alpha particle scatters into a $1\,\text{cm}^2$ detector at $\theta = 90°$ located $10\,\text{cm}$ from the foil.

Problem 1.4 ⭐⭐ The total Rutherford cross section is infinite (because $d\sigma/d\Omega \to \infty$ as $\theta \to 0$). In practice, atomic electron screening limits the maximum impact parameter.

(a) Estimate the maximum impact parameter as the Thomas-Fermi screening radius: $b_{\max} \approx a_0 Z^{-1/3}$, where $a_0 = 0.529 \times 10^{-10}\,\text{m}$ is the Bohr radius. For gold, compute $b_{\max}$.

(b) What minimum scattering angle $\theta_{\min}$ corresponds to $b_{\max}$? Use 5.5 MeV alphas on gold.

(c) Compute the approximate "screened" total cross section by integrating from $\theta_{\min}$ to $\pi$.

Problem 1.5 ⭐⭐⭐ Derivation: Center-of-mass correction. In the Rutherford derivation, we assumed the target nucleus was infinitely heavy. For a target of finite mass $M$:

(a) Show that the scattering in the center-of-mass frame gives the same formula but with $T$ replaced by $E_{\text{cm}} = T \cdot M/(m+M)$ where $m$ is the projectile mass.

(b) For alpha particles on helium ($m = M$), calculate the ratio of the corrected cross section to the uncorrected (infinite-mass target) cross section at $\theta_{\text{lab}} = 60°$.

(c) Show that for $\theta_{\text{cm}} = 180°$ (head-on), the lab scattering angle is $\theta_{\text{lab}} = 180°$ only if $m < M$.

Problem 1.6 ⭐⭐⭐ Deviations from Rutherford scattering. At sufficiently high energies (or small angles of approach), the projectile reaches the nuclear surface and the scattering deviates from pure Coulomb.

(a) Estimate the energy at which 7.5 MeV alpha particles on ${}^{208}\text{Pb}$ would begin to show deviations from Rutherford scattering at $\theta = 180°$. (Hint: Set $d_0 = R_\alpha + R_{\text{Pb}}$.)

(b) The Coulomb barrier for the alpha-Pb system is approximately $V_C = kz_1z_2e^2/(R_\alpha + R_{\text{Pb}})$. Calculate $V_C$ and compare to part (a).

Chadwick's Neutron Discovery

Problem 1.7 ⭐ In Chadwick's experiment, neutrons from the ${}^{9}\text{Be}(\alpha, n){}^{12}\text{C}$ reaction were observed to give: - Maximum proton recoil energy: $T_p = 5.7\,\text{MeV}$ - Maximum nitrogen recoil energy: $T_N = 1.4\,\text{MeV}$

(a) Assuming elastic collisions with a neutral particle of mass $m_n$, show that the maximum recoil energy of a target of mass $M$ is $T_{\max} = 4m_n M T_n / (m_n + M)^2$.

(b) Eliminate $T_n$ and solve for $m_n$. Compare your answer to the modern neutron mass.

Problem 1.8 ⭐⭐ The Joliot-Curies initially interpreted the beryllium radiation as gamma rays.

(a) If gamma rays of energy $E_\gamma$ scatter elastically off a proton at rest, show using relativistic kinematics that the maximum proton recoil energy is:

$$T_p^{\max} = \frac{2E_\gamma^2}{m_p c^2 + 2E_\gamma}$$

(b) For $T_p^{\max} = 5.7\,\text{MeV}$, what gamma ray energy is required?

(c) The Q-value of ${}^{9}\text{Be}(\alpha, \gamma){}^{13}\text{C}$ is $10.65\,\text{MeV}$. With 5.3 MeV alpha particles from polonium, what is the maximum available gamma energy (assuming the ${}^{13}\text{C}$ recoils)? Show this is far less than needed, disproving the gamma-ray hypothesis.

Nuclear Notation and the Chart of Nuclides

Problem 1.9 ⭐ For each of the following nuclides, state the values of $Z$, $N$, and $A$:

(a) ${}^{40}\text{K}$ (b) ${}^{14}\text{C}$ (c) ${}^{99}\text{Tc}$ (d) ${}^{235}\text{U}$ (e) ${}^{3}\text{He}$ (f) ${}^{209}\text{Bi}$

Problem 1.10 ⭐ Classify each pair as isotopes, isotones, isobars, or none of these:

(a) ${}^{14}\text{C}$ and ${}^{14}\text{N}$

(b) ${}^{16}\text{O}$ and ${}^{18}\text{O}$

(c) ${}^{13}\text{C}$ and ${}^{14}\text{N}$

(d) ${}^{40}\text{Ar}$ and ${}^{40}\text{K}$ and ${}^{40}\text{Ca}$

(e) ${}^{3}\text{H}$ and ${}^{3}\text{He}$

Problem 1.11 ⭐⭐ Using the data in a chart of nuclides (or the NNDC website at www.nndc.bnl.gov):

(a) List all stable isotopes of tin ($Z = 50$). How many are there? Why is tin special?

(b) Identify the stable isobars of $A = 96$. (Remember that a stable nucleus cannot undergo beta decay.)

(c) How many stable nuclei have $N = 50$? List them.

Problem 1.12 ⭐ On a chart of nuclides ($N$ horizontal, $Z$ vertical), sketch the approximate locations of:

(a) The $N = Z$ line

(b) The valley of stability

(c) The magic number lines $N = 20, 28, 50, 82, 126$ and $Z = 20, 28, 50, 82$

(d) Label the regions of $\beta^-$ decay and $\beta^+$/EC decay relative to the valley of stability.

Nuclear Sizes

Problem 1.13 ⭐ Calculate the nuclear radius (using $R = r_0 A^{1/3}$, $r_0 = 1.21\,\text{fm}$) for:

(a) ${}^{4}\text{He}$ (b) ${}^{27}\text{Al}$ (c) ${}^{56}\text{Fe}$ (d) ${}^{208}\text{Pb}$ (e) ${}^{238}\text{U}$

Problem 1.14 ⭐⭐ The nuclear density can be expressed in several equivalent ways.

(a) Calculate $\rho_{\text{nuc}}$ in nucleons/fm$^3$ using $r_0 = 1.21\,\text{fm}$.

(b) Convert to kg/m$^3$. (Use $1\,\text{u} = 1.66054 \times 10^{-27}\,\text{kg}$, $1\,\text{fm} = 10^{-15}\,\text{m}$.)

(c) Calculate the mass of a sphere of nuclear matter with radius $1\,\text{cm}$. Express in tonnes.

(d) A neutron star has a mass of $\sim 1.4 M_\odot$ ($M_\odot = 2 \times 10^{30}\,\text{kg}$) and a radius of $\sim 10\,\text{km}$. Estimate its average density and compare to nuclear density.

Problem 1.15 ⭐⭐ The Fermi distribution $\rho(r) = \rho_0 / [1 + \exp((r - R)/a)]$ describes the nuclear charge density.

(a) Show that $\rho(R) = \rho_0 / 2$ (justifying calling $R$ the "half-density radius").

(b) Define the "10–90 thickness" $t$ as the distance over which $\rho$ drops from $0.9\rho_0$ to $0.1\rho_0$. Show that $t = 4a\ln 3 \approx 4.39a$.

(c) For $a = 0.54\,\text{fm}$, calculate $t$. Is the nuclear surface sharp or diffuse?

Problem 1.16 ⭐⭐⭐ Electron scattering form factor. In Born approximation, the elastic electron-nucleus scattering cross section is:

$$\frac{d\sigma}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_{\text{Mott}} |F(q)|^2$$

where $F(q) = \frac{1}{Ze}\int \rho_{\text{ch}}(\mathbf{r}) e^{i\mathbf{q}\cdot\mathbf{r}} d^3r$ is the form factor and $q = 2k\sin(\theta/2)$ is the momentum transfer.

(a) For a uniform sphere of charge with radius $R$, show that:

$$F(q) = \frac{3[\sin(qR) - qR\cos(qR)]}{(qR)^3}$$

(b) The first diffraction minimum occurs where $F(q) = 0$. Show this happens at $qR \approx 4.49$ (the first zero of the numerator).

(c) For ${}^{208}\text{Pb}$, the first diffraction minimum in $500\,\text{MeV}$ electron scattering occurs at $\theta \approx 18°$. Extract the nuclear radius $R$ from this measurement.

Binding Energy

Problem 1.17 ⭐ Calculate the binding energy $B$ and binding energy per nucleon $B/A$ for:

(a) ${}^{2}\text{H}$ (deuterium): $M = 2.014102\,\text{u}$

(b) ${}^{3}\text{He}$: $M = 3.016029\,\text{u}$

(c) ${}^{3}\text{H}$ (tritium): $M = 3.016049\,\text{u}$

(d) ${}^{16}\text{O}$: $M = 15.994915\,\text{u}$

(e) ${}^{62}\text{Ni}$: $M = 61.928345\,\text{u}$

Problem 1.18 ⭐ The binding energy of ${}^{4}\text{He}$ is 28.296 MeV while that of ${}^{3}\text{He}$ is 7.718 MeV and that of ${}^{3}\text{H}$ is 8.482 MeV.

(a) Which is more tightly bound per nucleon, ${}^{3}\text{He}$ or ${}^{3}\text{H}$?

(b) Account for the difference in terms of nuclear forces and Coulomb repulsion. (Hint: ${}^{3}\text{H}$ and ${}^{3}\text{He}$ are mirror nuclei.)

(c) How much more tightly bound per nucleon is ${}^{4}\text{He}$ compared to the $A = 3$ nuclei? Why is ${}^{4}\text{He}$ so exceptionally stable?

Problem 1.19 ⭐⭐ Calculate the energy released in the following reactions using binding energies or atomic masses:

(a) ${}^{2}\text{H} + {}^{3}\text{H} \to {}^{4}\text{He} + n$ (DT fusion)

(b) ${}^{235}\text{U} + n \to {}^{141}\text{Ba} + {}^{92}\text{Kr} + 3n$ (one possible fission channel)

Use: $B({}^{2}\text{H}) = 2.224\,\text{MeV}$, $B({}^{3}\text{H}) = 8.482\,\text{MeV}$, $B({}^{4}\text{He}) = 28.296\,\text{MeV}$, $B({}^{235}\text{U}) = 1783.87\,\text{MeV}$, $B({}^{141}\text{Ba}) = 1174.20\,\text{MeV}$, $B({}^{92}\text{Kr}) = 783.18\,\text{MeV}$.

Problem 1.20 ⭐⭐ Separation energies and magic numbers. Calculate the neutron separation energy $S_n$ for the following nuclides and identify where discontinuities reveal magic numbers:

(a) Calculate $S_n$ for ${}^{16}\text{O}$ and ${}^{17}\text{O}$.

(b) Calculate $S_n$ for ${}^{40}\text{Ca}$ and ${}^{41}\text{Ca}$.

(c) In each case, interpret the jump in $S_n$ in terms of magic numbers.

Problem 1.21 ⭐⭐ The iron-nickel peak. It is commonly stated that ${}^{56}\text{Fe}$ is the "most stable nucleus." This is imprecise. Using the following atomic masses:

(a) Calculate $B/A$ for each.

(b) Which has the highest $B/A$?

(c) Explain the distinction between "most tightly bound per nucleon" (${}^{62}\text{Ni}$), "most bound per unit mass" (also ${}^{62}\text{Ni}$), and "produced in greatest abundance in stars" (${}^{56}\text{Fe}$, because of nuclear statistical equilibrium at $Z = N$ and favoring even-even nuclei).

Problem 1.22 ⭐⭐⭐ Energy from fusion: the pp chain. In the Sun, four protons are converted to one ${}^{4}\text{He}$ nucleus through a series of reactions (the pp chain), with two positrons and two neutrinos emitted.

(a) Write the net reaction: $4 {}^{1}\text{H} \to {}^{4}\text{He} + 2e^+ + 2\nu_e$. Calculate the total energy released using atomic masses (careful: the positrons annihilate with electrons).

(b) The Sun's luminosity is $L_\odot = 3.846 \times 10^{26}\,\text{W}$. How many pp chain reactions occur per second?

(c) How many tonnes of hydrogen are consumed per second?

(d) Approximately $2\%$ of the energy is carried away by neutrinos and is not deposited in the Sun. What is the neutrino luminosity in watts?

Problem 1.23 ⭐⭐⭐ Energy from fission. A nuclear reactor operates at $1\,\text{GW}$ (thermal) using ${}^{235}\text{U}$ fuel. Each fission releases approximately $200\,\text{MeV}$.

(a) How many fission events occur per second?

(b) How many kg of ${}^{235}\text{U}$ are consumed per day?

(c) A coal-fired plant of the same power output burns coal with an energy content of $30\,\text{MJ/kg}$. How many tonnes of coal are needed per day? Compare to the uranium consumption.

Computational Problems

Problem 1.24 💻 ⭐⭐ Write a Python script that computes and plots the Rutherford scattering cross section $d\sigma/d\Omega$ (in barns/sr) versus scattering angle $\theta$ (from $5°$ to $175°$) for:

(a) 5.5 MeV alphas on gold

(b) 5.5 MeV alphas on aluminum

(c) 10 MeV alphas on gold

Plot all three on the same log-scale plot and verify the $Z^2$ and $T^{-2}$ dependences.

Problem 1.25 💻 ⭐⭐ Using the binding energy data from the chapter (or a more complete dataset from the NNDC):

(a) Reproduce the $B/A$ vs. $A$ curve.

(b) Identify the five nuclides with the highest $B/A$.

(c) Add a plot of neutron separation energy $S_n$ vs. $N$ for the calcium isotopes ($Z = 20$, $N = 18$ to $32$). Identify the magic number $N = 20$ and $N = 28$.

Problem 1.26 💻 ⭐⭐⭐ Simulating Rutherford scattering. Write a Python simulation that:

(a) Generates random impact parameters $b$ uniformly distributed in $[0, b_{\max}]$

(b) Computes the scattering angle $\theta = 2\arctan(a/b)$ for each

(c) Histograms the resulting angular distribution

(d) Overlays the theoretical Rutherford cross section (properly normalized)

Use 100,000 projectiles and choose $b_{\max}$ large enough that the minimum angle $\theta_{\min} \lesssim 5°$.

Problem 1.27 ⭐⭐⭐ 🔬 Nuclear charge radii systematics. The rms charge radius of a nucleus can be parameterized as $\langle r^2 \rangle^{1/2} = r_0 A^{1/3}$ with different values of $r_0$ for different mass regions.

(a) Using the tabulated rms charge radii from Angeli and Marinova (Atomic Data and Nuclear Data Tables, 2013), extract $r_0$ for light ($A < 40$), medium ($40 < A < 120$), and heavy ($A > 120$) nuclei.

(b) Plot $\langle r^2 \rangle^{1/2}$ vs. $A^{1/3}$ and assess the linearity.

(c) Identify nuclides that deviate significantly from the trend. Are they associated with magic numbers or deformation?

Problem 1.28 🔬 Beyond Rutherford: nuclear rainbow scattering. At energies above the Coulomb barrier, elastic scattering shows oscillatory patterns in the cross section due to nuclear refraction and diffraction — analogous to atmospheric rainbows.

(a) Read the review by Khoa et al. (Journal of Physics G, 2007, vol. 34, p. R111) on nuclear rainbow scattering.

(b) For ${}^{16}\text{O} + {}^{16}\text{O}$ elastic scattering at $E_{\text{lab}} = 350\,\text{MeV}$, estimate the Coulomb barrier and the grazing angular momentum.

(c) Explain qualitatively why nuclear rainbow scattering probes the nuclear optical potential in the interior of the nucleus, complementing elastic electron scattering which probes only the charge distribution.