Atomic and Nuclear Physics

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Thomson's Atomic Model An atom is a solid sphere with uniform positive charge and mass, with electrons embedded like seeds in a watermelon. Successfully explained thermionic emission, photoelectric emission, and ionization. Failed to explain $\alpha$-particle scattering and origin of spectral lines. Rutherford's $\alpha$-Scattering Experiment Most $\alpha$-particles pass undeflected. Some deflected at small angles. A few (1 in 1000) deflected $> 90^\circ$. Very few returned ($180^\circ$). Number of scattered particles: $N \propto \frac{1}{\sin^4(\theta/2)}$ Distance of Closest Approach ($r_0$) Minimum distance where initial kinetic energy is converted to potential energy: $$ \frac{1}{2}mv^2 = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(2e)}{r_0} \implies r_0 = \frac{Ze^2}{\pi\epsilon_0 mv^2} = \frac{4kZe^2}{mv^2} $$ Impact Parameter ($b$) Perpendicular distance of the velocity vector of the $\alpha$-particle from the center of the nucleus: $$ b = \frac{Ze^2 \cot(\theta/2)}{4\pi\epsilon_0 \frac{1}{2}mv^2} $$ Large $b \implies$ undeflected $\alpha$-particles. Small $b \implies$ large scattering of $\alpha$-particles. Rutherford's Atomic Model Most mass ($>99.95\%$) and all positive charge concentrated in a small region called the atomic nucleus. Nucleus is positively charged, size $\approx 10^{-15}$ m (1 Fermi). Atom is mostly empty space; electrons revolve around the nucleus. Failure of Rutherford's Model Stability of Atom: Accelerated charged particles should radiate energy, causing electrons to spiral into the nucleus. Spectrum: Predicted continuous spectrum, but observed is line spectrum. Did not explain electron distribution outside the nucleus. Bohr's Atomic Model Proposed for hydrogen-like atoms (single electron, charge $Ze$). Postulates Electrons move in stable circular orbits without radiating energy. Angular momentum is quantized: $L = mvr_n = n\frac{h}{2\pi}$ Energy radiation occurs only during electron transitions between permitted orbits ($E_2 - E_1 = h\nu$). Bohr's Orbits for Hydrogen and H-like Atoms Radius of orbit: $r_n = \frac{n^2h^2\epsilon_0}{\pi m Ze^2} = 0.53 \frac{n^2}{Z} \text{ Å}$ Speed of electron: $v_n = \frac{Ze^2}{2\epsilon_0 nh} = \frac{cZ}{137n} \approx 2.2 \times 10^6 \frac{Z}{n} \text{ m/s}$ Quantity Formula Dependency on $n$ and $Z$ Angular speed ($\omega_n$) $\frac{\pi m Z^2 e^4}{2\epsilon_0^2 n^3 h^3}$ $\omega_n \propto \frac{Z^2}{n^3}$ Frequency ($f_n$) $\frac{m Z^2 e^4}{4\epsilon_0^2 n^3 h^3}$ $f_n \propto \frac{Z^2}{n^3}$ Time period ($T_n$) $\frac{4\epsilon_0^2 n^3 h^3}{m Z^2 e^4}$ $T_n \propto \frac{n^3}{Z^2}$ Angular momentum ($L_n$) $n\frac{h}{2\pi}$ $L_n \propto n$ Magnetic moment ($\mu_n$) $i_n A_n = \frac{e h}{4\pi m_e} n$ (Bohr magneton) $\mu_n \propto n$ Drawbacks of Bohr's Atomic Model Valid only for one-electron atoms (H, He$^+$, Li$^{2+}$). Assumed circular orbits, but Sommerfeld proposed elliptical. Could not explain spectral line intensities. Assumed stationary nucleus, but it rotates. Could not explain fine structure of spectral lines. Could not explain Zeeman effect (magnetic field splitting) or Stark effect (electric field splitting). Did not explain doublets in spectra (e.g., Sodium). Energy of Electron in $n^{th}$ Orbit Potential Energy ($U$): $U = -\frac{kZe^2}{r_n}$ Kinetic Energy ($K$): $K = \frac{kZe^2}{2r_n} = \frac{|U|}{2}$ Total Energy ($E$): $E = K+U = -\frac{kZe^2}{2r_n} = -\frac{me^4 Z^2}{8\epsilon_0^2 n^2 h^2} = -R_y \frac{Z^2}{n^2} = -13.6 \frac{Z^2}{n^2} \text{ eV}$ Ionization Energy: Energy to remove electron from an orbit to infinity ($E_\text{ionization} = -E_n = 13.6 \frac{Z^2}{n^2} \text{ eV}$). Excitation Energy: Energy to move electron from lower to higher energy level ($E_\text{excitation} = E_\text{final} - E_\text{initial}$). Binding Energy (B.E.): Energy released when constituents form a system (or needed to separate them). For H atom, B.E. = 13.6 eV. Hydrogen Spectrum and Spectral Series When an excited hydrogen atom returns to a lower state, it emits photons of specific wavelengths, forming spectral series. Energy of emitted radiation: $\Delta E = E_2 - E_1 = R_y c h Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) = 13.6 Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV}$ Frequency of emitted radiation: $\nu = \frac{\Delta E}{h} = R_y c Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ Wave number/Wavelength: $\frac{1}{\lambda} = \frac{\nu}{c} = R_y Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ Number of spectral lines: For transition from $n_2$ to $n_1$: $N_E = \frac{(n_2-n_1+1)(n_2-n_1)}{2}$. For transition to ground state ($n_1=1$): $N_E = \frac{n(n-1)}{2}$. Spectral Series Transition $\lambda_\text{max}$ $\lambda_\text{min}$ Region Lyman $n_2 = 2,3,4...\infty \to n_1 = 1$ $\frac{4}{3R_y}$ $\frac{1}{R_y}$ Ultraviolet Balmer $n_2 = 3,4,5...\infty \to n_1 = 2$ $\frac{36}{5R_y}$ $\frac{4}{R_y}$ Visible Paschen $n_2 = 4,5,6...\infty \to n_1 = 3$ $\frac{144}{7R_y}$ $\frac{9}{R_y}$ Infrared Brackett $n_2 = 5,6,7...\infty \to n_1 = 4$ $\frac{400}{9R_y}$ $\frac{16}{R_y}$ Infrared Pfund $n_2 = 6,7,8...\infty \to n_1 = 5$ $\frac{900}{11R_y}$ $\frac{25}{R_y}$ Infrared Line Wavelengths First line: $n_2 = n_1+1 \implies \lambda_\text{max} = \frac{n_1^2(n_1+1)^2}{(2n_1+1)R_y}$ Series limit: $n_2 = \infty \implies \lambda_\text{min} = \frac{n_1^2}{R_y}$ Quantum Numbers Describe the state of an electron in an atom. Principal Quantum Number ($n$): Determines main energy level/shell (K, L, M, N...). $n = 1, 2, 3, \ldots$ Electron energy $E_n \propto -1/n^2$, radius $r_n \propto n^2$. Orbital (Azimuthal) Quantum Number ($l$): Determines subshell (s, p, d, f) and shape of orbital. $l = 0, 1, 2, \ldots, (n-1)$. Orbital angular momentum: $L = \sqrt{l(l+1)}\frac{h}{2\pi}$. Magnetic Quantum Number ($m_l$): Determines orientation of orbital in space. $m_l = -l, \ldots, 0, \ldots, +l$. Number of orientations for a given $l$ is $(2l+1)$. Spin (Magnetic) Quantum Number ($m_s$): Describes electron's intrinsic angular momentum (spin). $m_s = +\frac{1}{2}$ (spin up) or $-\frac{1}{2}$ (spin down). Electronic Configurations of Atoms Pauli's Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers. Aufbau Principle: Electrons fill orbitals of lowest energy first. Hund's Rule: When filling degenerate orbitals, electrons occupy them singly with parallel spins before pairing up. Nucleus Rutherford's experiment established the nucleus as a small, dense, positively charged center. Stability determined by balance between attractive nuclear force (protons & neutrons) and repulsive electrical force (protons). Neutron Fundamental particle, constituent of all nuclei (except H). Discovered by Chadwick. Free neutron is unstable: $n \to p + e^- + \bar{\nu}$. Charge: Neutral. Mass: $1.6750 \times 10^{-27}$ kg. Spin angular momentum: $\frac{1}{2}\frac{h}{2\pi}$. Half-life: 12 minutes. Penetration power: High. Types: Slow (thermal) and fast. Thermal neutrons have energy $\approx 0.025$ eV. Types of Nuclei Isotopes: Same atomic number ($Z$), different mass number ($A$). E.g., $^1_1H, ^2_1H, ^3_1H$. Isobars: Same mass number ($A$), different atomic number ($Z$). E.g., $^3_1H, ^3_2He$. Isotones: Same number of neutrons ($A-Z$). E.g., $^4_2He, ^5_3Li$. Mirror Nuclei: Same mass number ($A$), proton and neutron numbers interchanged (or $Z$ differs by 1). E.g., $^3_1H, ^3_2He$. Size of Nucleus Nuclear radius ($R$): $R \propto A^{1/3} \implies R = R_0 A^{1/3}$, where $R_0 \approx 1.2 \times 10^{-15}$ m (1 Fermi). Nuclear volume ($V$): $V \propto R^3 \propto A$. Nuclear density ($\rho$): $\rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} \approx 2.38 \times 10^{17} \text{ kg/m}^3$. Is nearly constant for all nuclei. Nuclear Force Forces that bind nucleons together. Exchange forces: Result from exchange of mesons (Yukawa's theory). $\pi$-mesons ($\pi^+, \pi^-, \pi^0$) mediate the force. Short range ($\approx 10^{-15}$ m). Strongest forces in nature. Attractive. Charge independent. Non-central. Atomic Mass Unit (amu) Defined as $\frac{1}{12}$th mass of a carbon-12 atom. 1 amu $\approx 1.66054 \times 10^{-27}$ kg. Energy equivalent of 1 amu: $931 \text{ MeV}$. Pair Production and Annihilation Pair Production: Energetic $\gamma$-ray photon ($h\nu$) absorbed by nucleus, creating an electron ($e^-$) and a positron ($e^+$). $$ h\nu \to e^+ + e^- $$ Minimum energy for $\gamma$-photon: $2 m_e c^2 \approx 1.02 \text{ MeV}$. Pair Annihilation: Electron and positron combine, producing two $\gamma$-photons. $$ e^+ + e^- \to h\nu + h\nu $$ Nuclear Stability Neutron-proton ratio ($N/Z$): Lighter nuclei: $N/Z \approx 1$ for stability. Heavy nuclei: $N/Z > 1$ for stability (more neutrons needed to counteract proton repulsion). Even/Odd numbers of $Z$ or $N$: Even $Z$, Even $N$ (even-even) are most stable ($\approx 60\%$ of stable nuclides). Odd $Z$, Odd $N$ (odd-odd) are least stable (only 5 known). Binding Energy per Nucleon: Higher B.E./nucleon $\implies$ more stable nucleus. Mass Defect and Binding Energy Mass Defect ($\Delta m$): $\Delta m = (Z m_p + (A-Z) m_n) - M_\text{nucleus}$. ($M_\text{nucleus}$ is actual mass of nucleus). Binding Energy (B.E.): Energy equivalent to mass defect. B.E. $= \Delta m c^2$. If $\Delta m$ is in amu, B.E. $= \Delta m \times 931 \text{ MeV}$. Packing Fraction: $\frac{\Delta m}{A}$. Smaller value means greater stability. B.E. per Nucleon: $\frac{\text{B.E.}}{A}$. Binding Energy Curve Graph of B.E./nucleon vs. mass number ($A$). Peaks at $A=4, 8, 12, 16, 20$ (e.g., $^4_2He, ^8_4Be, ^{12}_6C, ^{16}_8O, ^{20}_{10}Ne$) indicating higher stability. Maximum B.E./nucleon for $A=56$ ($^{56}_{26}Fe$), value is $8.8 \text{ MeV/nucleon}$. For $A > 56$, B.E./nucleon gradually decreases (e.g., for $U^{238}$, it's $7.5 \text{ MeV}$). Nuclear Reactions Process where identity of a nucleus changes due to bombardment by energetic particles. $X(a,b)Y \implies X + a \to Y + b + Q$ Q-value: Energy absorbed (endothermic, $Q 0$). $Q = (\text{Mass of reactants} - \text{Mass of products})c^2$. Conservation Laws: Mass number ($A$) conserved. Charge number ($Z$) conserved. Momentum conserved. Total energy conserved. Common Reactions: $(p,n), (p,\alpha), (p,\gamma), (n,p), (\gamma,n)$. Nuclear Fission Splitting of a heavy nucleus into two lighter nuclei with energy liberation. Discovered by Otto Hahn and Fritz Strassmann. Example: $^1_0n + ^{235}_{92}U \to ^{236}_{92}U^* \to ^{141}_{56}Ba + ^{92}_{36}Kr + 3^1_0n + Q$. Energy released: $\approx 200 \text{ MeV}$ per fission. Neutrons released are fast ($2 \text{ MeV}$), need to be slowed down for chain reaction. Chain Reaction Self-sustaining sequence of nuclear fissions. Neutron Reproduction Factor ($k$): $\frac{\text{Rate of neutron production}}{\text{Rate of neutron loss}}$. $k=1$: Steady (critical size/mass). $k>1$: Accelerates (supercritical, atom bomb). $k Controlled Chain Reaction Uncontrolled Chain Reaction Controlled artificially No control One neutron used for fission More than one neutron used Slow rate Fast rate $k=1$ $k>1$ Energy Large explosive energy Nuclear reactors Atom bomb Nuclear Reactor Device for controlled chain reactions to generate energy or produce isotopes. Fissionable Material (Fuel): $^{235}U, ^{232}Th, ^{239}Pu$. Moderator: Slows down fast neutrons (graphite, heavy water $D_2O$). Control Material: Absorbs neutrons to control reaction rate (cadmium rods). Coolant: Removes heat (water, $CO_2$). Protective Shield: Concrete wall to protect from radiation. Uses: Electric power, radioisotope production, $^{239}Pu$ manufacturing. Nuclear Fusion Two or more light nuclei combine to form a heavier nucleus with energy liberation. Requires high pressure ($\approx 10^6 \text{ atm}$) and high temperature ($\approx 10^7 - 10^8 \text{ K}$). Example: Proton-proton chain in stars. $4^1_1H \to ^4_2He + 2e^+ + 2\nu_e + 2\gamma + 26.73 \text{ MeV}$. Energy released is much larger than fission for same fuel mass. Plasma: Ionized gas at high temperatures ($\approx 10^8 \text{ K}$). Atom Bomb Hydrogen Bomb Fission process ($^{235}U$) Fusion process (deuterium + tritium) Critical size important No limit to critical size Explosion at normal T, P High T, P required Less energy More energy (more dangerous) Radioactivity Spontaneous emission of radiation by heavy elements. Discovered by Becquerel (Uranium salt, 1896). Pierre and Marie Curie discovered Radium. Unaffected by physical/chemical changes. All elements with $Z > 82$ are naturally radioactive. Artificial/induced radioactivity: Lighter elements made radioactive by bombardment. Nuclear Radiations Rutherford's experiment showed $\alpha, \beta, \gamma$ rays. Features $\alpha$-particles $\beta$-particles $\gamma$-rays Identity Helium nucleus ($^4_2He^{2+}$) Fast electron ($e^-$) EM waves (photons) Charge $+2e$ $-e$ Zero Mass $4 m_p$ $m_e$ Massless Speed $\approx 10^7 \text{ m/s}$ $1\% \text{ to } 99\% \text{ of } c$ $c$ Kinetic Energy $4-9 \text{ MeV}$ Varies up to $1.2 \text{ MeV}$ Varies up to $2.23 \text{ MeV}$ Penetration Power 1 (stopped by paper) 100 (100x $\alpha$) 10000 (100x $\beta$) Ionization Power 10000 100 1 E/M Field Effect Deflected Deflected Not deflected Energy Spectrum Line and discrete Continuous Line and discrete Interaction with Matter Produces heat Produces heat Photoelectric effect, Compton effect, pair production Types of Decay $\alpha$-decay: Emission of $\alpha$-particle ($^4_2He$). $^{A}_{Z}X \to ^{A-4}_{Z-2}Y + ^4_2He$. $N$ and $Z$ decrease by 2, $A$ decreases by 4. $\beta$-decay: $\beta^-$-decay: Emission of electron ($e^-$) and antineutrino ($\bar{\nu}$). $^{A}_{Z}X \to ^{A}_{Z+1}Y + e^- + \bar{\nu}$. Neutron transforms to proton ($n \to p + e^- + \bar{\nu}$). $N$ decreases by 1, $Z$ increases by 1, $A$ unchanged. $\beta^+$-decay (Positron Emission): Emission of positron ($e^+$) and neutrino ($\nu$). $^{A}_{Z}X \to ^{A}_{Z-1}Y + e^+ + \nu$. Proton transforms to neutron ($p \to n + e^+ + \nu$). $N$ increases by 1, $Z$ decreases by 1, $A$ unchanged. Electron Capture: Nucleus captures an orbital electron, converting a proton to a neutron. $^{A}_{Z}X + e^- \to ^{A}_{Z-1}Y + \nu$. $N$ increases by 1, $Z$ decreases by 1, $A$ unchanged. $\gamma$-decay: Emission of $\gamma$-ray photon. Excited nucleus ($^{A}_{Z}X^*$) transitions to lower energy state ($^{A}_{Z}X$) by emitting $\gamma$-photon. No change in $A, Z, N$. Intensity of $\gamma$-decay: $I = I_0 e^{-\mu x}$. Radioactive Disintegration Law of Radioactive Disintegration: Rate of decay is proportional to the number of atoms present. $$ -\frac{dN}{dt} = \lambda N $$ Where $N$ is number of atoms, $\lambda$ is decay constant. $$ N(t) = N_0 e^{-\lambda t} $$ $$ M(t) = M_0 e^{-\lambda t} $$ Activity ($A$): Rate of disintegration. $A = -\frac{dN}{dt} = \lambda N = A_0 e^{-\lambda t}$. Units: Becquerel (Bq = 1 disintegration/sec), Curie (Ci = $3.7 \times 10^{10}$ Bq), Rutherford (Rd = $10^6$ Bq). Half-life ($T_{1/2}$): Time for half of the substance to decay. $$ T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda} $$ Mean life ($\tau$): Average lifetime of a radioactive atom. $$ \tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693} \approx 1.44 T_{1/2} $$ Radioactive Series Sequence of decays until a stable nuclide is reached. Series Parent Stable End Product 4n (Thorium) $^{232}_{90}Th$ $^{208}_{82}Pb$ 4n+1 (Neptunium) $^{237}_{93}Np$ $^{209}_{83}Bi$ 4n+2 (Uranium) $^{238}_{92}U$ $^{206}_{82}Pb$ 4n+3 (Actinium) $^{227}_{89}Ac$ $^{207}_{82}Pb$ Successive Disintegration and Radioactive Equilibrium If $A \to B \to C$, then rate of disintegration of $A$ and $B$ are $\lambda_1 N_1$ and $\lambda_2 N_2$. Net rate of formation of $B = \lambda_1 N_1 - \lambda_2 N_2$. Radioactive Equilibrium: $\lambda_1 N_1 = \lambda_2 N_2 \implies \frac{N_1}{N_2} = \frac{\lambda_2}{\lambda_1} = \frac{T_{1/2,1}}{T_{1/2,2}}$. Uses of Radioactive Isotopes Medicine: Diagnostics (Cr-51, Na-24, Hg-203, I-131), cancer treatment (Co-60), skin diseases (P-31). Archaeology: Carbon dating (C-14), age of meteorites (K-40), age of Earth. Agriculture: Pest control (Co-60), fertilizers (P-32). Tracers: Studying biochemical reactions. Industries: Leak detection, age of planets.