Modern Nuclear Physics: Comprehensive Studies on Nuclei, Radioactivity, and Detectors

Historical Foundations and the Atomic Nucleus

• J. J. Thomson's Model (1907):

  • Proposed that the atom is a sphere of uniform positive charge with an equal number of negative charges (electrons) embedded within it.

  • The atomic radius was estimated at the order of $10^{-10}\,\text{m}$.

• Rutherford's Alpha-Scattering Experiment (1911):

  • Procedure: Rutherford directed a beam of high-velocity alpha particles ($+2e$ charge, $4\,\text{a.m.u.}$ mass) at a thin gold foil (thickness approximately $4 \times 10^{-6}\,\text{m}$). A fluorescent zinc sulphide ($ZnS$) screen was used to detect deflections.

  • Observations: Most alpha particles passed through undeflected; some were deflected at small angles; a very few were reflected back (large-angle scattering).

  • Conclusions:

    1. The atom consists mostly of empty space.

    2. A central, massive, positively charged core exists, called the Nucleus.

    3. The nucleus contains almost the entire mass of the atom.

    4. Electrons revolve around the nucleus in circular orbits.

    5. The radius of the nucleus is approximately $10^{-14}\,\text{m}$ to $10^{-15}\,\text{m}$, while the atom's radius is $10^{-10}\,\text{m}$.

• Distance of Nearest Approach ($r_0$):

  • The size of the nucleus is estimated by calculating the point at which an alpha particle's kinetic energy is entirely converted into electrostatic potential energy due to Coulombic repulsion.

  • $\text{K.E.} = \text{P.E.}$

  • $E = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(2e)}{r_0}$

  • $r_0 = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{E}$

  • Substituting $E = \frac{1}{2} m u^2$:

  • $r_0 = \frac{Ze^2}{\pi\epsilon_0 m u^2}$

Constituents of Nuclei: Theories and Failures

• Proton-Electron Theory (Pre-1932):

  • Hypothesis: Nuclei were thought to consist of protons and electrons. For example, Nitrogen ($A=14$, $Z=7$) was thought to have 14 protons and 7 electrons inside the nucleus, plus 7 electrons outside.

  • Reasons for Failure:

    1. Heisenberg Uncertainty Principle: If an electron ($\Delta x \approx 10^{-14}\,\text{m}$) were confined in the nucleus, its energy $E \approx pc$ would be roughly $20\,\text{MeV}$. However, observed beta-decay electrons only have energies of $3\text{--}4\,\text{MeV}$.

    2. Nuclear Spin: Nitrogen ($A=14$) has an integral spin ($I=1$). A 21-particle system (14p + 7e) would require a half-integral spin.

    3. Magnetic Moment: Electrons have magnetic moments roughly 2000 times larger than protons. If electrons were in the nucleus, nuclear magnetic moments would be much larger than experimentally observed.

• Proton-Neutron Theory (Post-1932):

  • Discovery of Neutron: James Chadwick (1932) discovered the neutron by bombarding Beryllium with alpha particles. The neutron ($n^0$) is a neutral particle with mass $1.675 \times 10^{-27}\,\text{kg}$ ($1.00866\,\text{a.m.u.}$).

  • Current Model: A nucleus consists of $Z$ protons and $N$ neutrons, where $A = Z + N$.

  • Support: Uncertainty principle energy for nucleons is roughly $0.1\,\text{MeV}$ to $52\,\text{KeV}$, which is consistent with nuclear energy scales. Resultant spins for even $A$ are integral and for odd $A$ are half-integral, matching observation.

Fundamental Properties of the Nucleus

• Nuclear Size and Radius:

  • The volume of the nucleus is proportional to the mass number $A$.

  • $\frac{4}{3}\pi R^3 \propto A$

  • $R = R_0 A^{1/3}$

  • Where $R_0$ is a constant typically valued between $1.2\,\text{fm}$ and $1.6\,\text{fm}$ ($1\,\text{fermi} = 10^{-15}\,\text{m}$).

• Nuclear Density ($\rho$):

  • $\rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} = \frac{1.66 \times 10^{-27}\,\text{kg} \times A}{\frac{4}{3}\pi (R_0 A^{1/3})^3}$

  • The density is roughly $2.9 \times 10^{17}\,\text{kg/m}^3$ and is nearly constant for all nuclei.

• Nuclear Charge:

  • Total charge is $Ze$, where $e = 1.6 \times 10^{-19}\,\text{C}$. Protons are the sole charge carriers.

• Nuclear Parity ($P$):

  • Refers to the behavior of the wave function $\Psi(x, y, z)$ under coordinate inversion.

  • If $\Psi(-x, -y, -z) = +\Psi(x, y, z)$, parity is even ($P=+1$).

  • If $\Psi(-x, -y, -z) = -\Psi(x, y, z)$, parity is odd ($P=-1$).

  • Parity is related to orbital quantum number $l$: $P = (-1)^l$.

• Nuclear Statistics:

  • Nuclei with odd $A$ follow Fermi-Dirac statistics (Fermions).

  • Nuclei with even $A$ follow Bose-Einstein statistics (Bosons).

• Nuclear Magnetic Moment ($\mu$):

  • Measured in Nuclear Magnetons ($\mu_N$).

  • $\mu_N = \frac{e\hbar}{2m_p} = 5.05 \times 10^{-27}\,\text{J/T}$.

  • Proton magnetic moment: $+2.792\,\mu_N$.

  • Neutron magnetic moment: $-1.913\,\mu_N$.

• Electric Quadrupole Moment ($Q$):

  • Measures the deviation from spherical symmetry.

  • $Q = \frac{1}{e} \int (3z^2 - r^2) \rho \, d\tau$

  • $Q = 0$: Spherical.

  • Q > 0: Prolate (elongated).

  • Q < 0: Oblate (flattened).

Mass Stability and Binding Energy

• Atomic Mass Unit (a.m.u.):

  • Defined as $1/12\text{th}$ of the mass of a Carbon-12 atom.

  • $1\,\text{a.m.u.} = 1.66 \times 10^{-27}\,\text{kg} = 931.25\,\text{MeV}$.

• Mass Defect ($\Delta m$):

  • The difference between the sum of the masses of individual nucleons and the actual mass of the nucleus.

  • $\Delta m = ZM_p + (A - Z)M_n - M_{\text{nucleus}}$

• Packing Fraction ($f$):

  • Defined as mass defect per nucleon: $f = \frac{M - A}{A}$.

  • Smaller packing fraction correlates to higher stability.

• Binding Energy ($B$):

  • The energy required to break the nucleus into its constituent protons and neutrons.

  • $B = \Delta m c^2 = [ZM_H + (A - Z)M_n - M_{\text{atom}}]c^2$

• Binding Energy per Nucleon ($B/A$):

  • Average value is approximately $8\,\text{MeV}$ for most nuclei.

  • Maximum peak at $\text{Fe}^{56}$ ($8.8\,\text{MeV}$).

  • Lower for light nuclei (A < 20) and very heavy nuclei (A > 240).

  • Drops to $7.6\,\text{MeV}$ for $U^{238}$.

  • Cyclic peaks at $A = 4, 8, 12, 16, 20$ suggest stable alpha-particle-like structures.

Radioactivity and Decay Laws

• Definition: Spontaneous emission of radiations ($\alpha, \beta, \gamma$) from unstable heavy nuclei to achieve a stable state.

• Radioactive Decay Law:

  • The rate of disintegration is proportional to the number of undecayed atoms present.

  • $\frac{dN}{dt} = -\lambda N$

  • $N = N_0 e^{-\lambda t}$

  • Where $\lambda$ is the disintegration constant.

• Half-Life ($T_{1/2}$):

  • Time for half of the initial nuclei to decay.

  • $T_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693}{\lambda}$

• Average Life ($T_a$):

  • The reciprocal of the decay constant.

  • $T_a = \frac{1}{\lambda} = 1.44 \, T_{1/2}$

• Units of Activity:

  • Curie (Ci): $3.7 \times 10^{10}\,\text{disintegrations/sec}$.

  • Rutherford (rd): $10^6\,\text{disintegrations/sec}$.

  • Becquerel (Bq): $1\,\text{disintegration/sec}$.

Decay Modes and Nuclear Transmutation

• Alpha ($\alpha$) Decay:

  • Emission of a Helium nucleus (${}_2He^4$).

  • ${}_Z X^A \rightarrow {}_{Z-2} Y^{A-4} + {}_2 He^4 + Q$

  • Daughter element shifts two places lower in the periodic table.

  • Geiger-Nuttall Law: $\log(\lambda) = A \log(R) + B$. Longer range implies shorter half-life.

• Beta ($\beta$) Decay:

  • $\beta^-$ Emission: Neutron converts to proton: $n^0 \rightarrow p^+ + e^- + \bar{\nu}$.

  • $\beta^+$ Emission: Proton converts to neutron: $p^+ \rightarrow n^0 + e^+ + \nu$.

  • Electron Capture: Nucleus absorbs an inner shell electron: $p^+ + e^- \rightarrow n^0 + \nu$.

  • Neutrino Hypothesis (Pauli, 1931): To conserve energy, momentum, and spin in beta decay, a neutral, massless particle called a neutrino ($\nu$) or antineutrino ($\bar{\nu}$) must be emitted.

• Gamma ($\gamma$) Decay:

  • De-excitation of a nucleus from an excited to a ground state with the emission of a high-energy photon.

  • Internal Conversion: Instead of emitting a gamma ray, the excitation energy is transferred directly to an orbital (usually K-shell) electron, which is ejected.

Nuclear Models

• Liquid Drop Model (Bohr/Wheeler):

  • Analogies: Constant density, saturation of forces, and spherical shape due to surface tension.

  • Semi-Empirical Mass Formula:

  • $B = a_v A - a_s A^{2/3} - a_c \frac{Z(Z-1)}{A^{1/3}} - a_a \frac{(A-2Z)^2}{A} \pm \delta$

  • Terms: Volume ($a_v$), Surface ($a_s$), Coulomb ($a_c$), Asymmetry ($a_a$), and Pairing energy ($\delta$).

  • Successfully explains nuclear fission.

• Shell Model (Mayer/Jensen):

  • Based on Magic Numbers: 2, 8, 20, 28, 50, 82, 126.

  • Nuclei with these nucleon counts are exceptionally stable.

  • Assumes nucleons move independently in a central potential well with strong spin-orbit coupling ($l\text{-}s$ coupling).

  • Predicts nuclear spin, parity, and magnetic moments accurately.

Nuclear Forces

• Yukawa's Meson Theory (1935):

  • Strong nuclear forces arise from the exchange of virtual particles called pi-mesons ($\pi$, mass $\approx 270\,m_e$).

  • Types: $\pi^+$, $\pi^-$, $\pi^0$.

• Properties of Nuclear Forces:

  1. Strongest forces in nature: Roughly 100 times stronger than electromagnetism.

  2. Short Range: Effective only within roughly $2\,\text{fm}$. At distancing below $0.5\,\text{fm}$, the force becomes strongly repulsive.

  3. Saturation: A nucleon only binds to its immediate neighbors.

  4. Charge Independent: $n\text{-}n$ force $\approx p\text{-}p$ force $\approx n\text{-}p$ force (if Coulomb effects are removed).

  5. Spin Dependent: Stronger when spins are parallel ($I=1$, triplet) than antiparallel ($I=0$, singlet).

  6. Non-central: Forces depend on the orientation of the spin relative to the radius vector.

Interaction of Radiation with Matter

• Heavy Charged Particles (Protons, Alpha):

  • Lose energy primarily through ionization and excitation of orbital electrons.

  • Bohr's Stopping Power Formula: $-\frac{dE}{dx} \propto \frac{Z^2 n}{v^2}$.

• Light Charged Particles (Electrons):

  • Lose energy via ionization and Bremsstrahlung (braking radiation) emitted when decelerating near a nucleus.

  • Radiation loss becomes dominant at high energies and in heavy (high-$Z$) materials: $\frac{(dE/dx)_{\text{rad}}}{(dE/dx)_{\text{ion}}} \approx \frac{EZ}{800}$.

• Gamma Ray Interactions:

  1. Photoelectric Effect: Gamma photon transfers its full energy to an orbital electron, which is ejected ($h\nu = w_0 + K.E.$). Dominant at low energies (< 0.5\,\text{MeV}).

  2. Compton Effect: Elastic scattering of a photon by a free electron. The wavelength increases: $\Delta\lambda = \lambda' - \lambda = \frac{h}{m_0 c}(1 - \cos(\phi))$. Dominant at intermediate energies.

  3. Pair Production: A photon transforms into an electron-positron pair in the field of a nucleus. Minimum energy required: $2m_0 c^2 = 1.02\,\text{MeV}$. Dominant at high energies (> 10\,\text{MeV}).

Detectors and Accelerators

• Detectors:

  • Ionization Chamber: Operates in the saturation region; pulse height is proportional to primary ionization.

  • Proportional Counter: Employs gas multiplication (Townsend avalanche); pulse height is proportional to energy loss.

  • Geiger-Muller (G.M.) Counter: High-voltage operation; produces large, uniform pulses regardless of energy. Requires quenching (filling with Argon + Methane/Halogens) to prevent continuous discharge.

  • Scintillation Counter: Uses fluorescent crystals (e.g., $NaI(Tl)$, $ZnS$) and a Photomultiplier Tube (PMT).

• Accelerators:

  • Linear Accelerator (LINAC): Accelerates ions in a straight line using alternating R.F. voltages through drift tubes of increasing length: $L_n = \frac{T}{2} \sqrt{\frac{2nqV}{m}}$.

  • Cyclotron: Uses a magnetic field and two D-shaped electrodes. Frequency: $f = \frac{Bq}{2\pi m}$. Energy is limited by relativistic mass increase.

  • Betatron: Accelerates electrons using magnetic induction in a constant-radius orbit. Betatron condition: $\frac{d\Phi}{dt} = 2\pi r^2 \frac{dB}{dt}$.

  • Synchrotron: Synchronizes magnetic field and R.F. frequency to maintain a constant orbit as the particle becomes relativistic.

Nuclear Reactors

• Components:

  1. Fuel: Fissionable material ($U^{235}$, $Pu^{239}$).

  2. Moderator: Slows down fast neutrons to thermal energies (<1\,\text{eV}) to increase fission probability. (e.g., Graphite, $D_2O$, $Be$).

  3. Control Rods: Absorb neutrons to maintain a controlled chain reaction (e.g., Cadmium, Boron).

  4. Coolant: Removes fission heat (e.g., $CO_2$, liquid sodium, heavy water).

  5. Shielding: Thick concrete wall ($\approx 2.8\,\text{m}$) to protect against radiation.

• Safety Processes: Scramming refers to the automatic emergency insertion of control rods to shut down the reactor.