Atomic Physics & Spectroscopy – Comprehensive Bullet-Point Notes

Historical Evolution of the Atom Concept

• Ancient Greek philosophies

  • Atomism (matter made of discrete, indivisible units)
  • Continuum theory (matter infinitely divisible; championed by Aristotle → gained wider acceptance)

• Dalton’s “Billiard-Ball” model (early 1800s)

  • Atom = tiny, indivisible, indestructible solid sphere
  • No internal structure accounted for

• J. J. Thomson’s “Plum-Pudding” model (1897)

  • Atom = positively-charged, diffuse sphere with embedded electrons
  • First inclusion of sub-atomic particle (electron) but still no true nucleus

• Rutherford nuclear model (1911)

  • Gold-foil/α-particle scattering → tiny, dense, positively-charged nucleus contains almost all mass; electrons orbit in empty space
  • Problems under classical E-M theory
  • Charged particle in circular motion should radiate continuously → electron spirals into nucleus & emits continuous spectrum (not observed)

Classical One-Electron (Planetary) Model

• Simplifying assumptions (adiabatic approximation)

  • Nucleus charge $q_N = +Ze$, mass ≫ electron mass → nucleus assumed fixed
  • Electron of charge $-e$ moves in circular orbit of radius $r$

• Mechanics

  • Coulomb potential: $U(r) = - \dfrac{kZe^2}{r}$
  • Force balance (centripetal): $\dfrac{mv^2}{r}=\dfrac{kZe^2}{r^2} \;\Rightarrow\; mv^2 = \dfrac{kZe^2}{r}$

• Model correctly gives qualitative stability but fails to reconcile with radiation losses predicted by Maxwell equations.

Bohr Atom (1913) – Postulates & Derivations

• Extension of Rutherford by imposing quantum conditions

  1. Electron moves only in certain allowed circular orbits around nucleus under Coulomb force.
  2. Light emitted/absorbed only when electron jumps between allowed states; photon energy $E= h\nu = \hbar \omega$.
  3. Quantisation of angular momentum: $L = n\hbar = n\dfrac{h}{2\pi}, \; n = 1,2,3,\dots$

• From $L = mvr = n\hbar$ and Coulomb force balance:
  $r<em>n = \dfrac{n^2\hbar^2}{mk</em>e e^2Z} \quad\Longrightarrow\quad r<em>1 (\text{H}) \equiv a</em>0 = 0.05297\,\text{nm}$

• Electron speed
  $v<em>n = \dfrac{k</em>e e^2Z}{\hbar} \; \dfrac{1}{n}$

• Energies (kinetic + potential)
  $E<em>n = -\dfrac{mk</em>e^2 e^4 Z^2}{2\hbar^2}\;\dfrac{1}{n^2} = -\dfrac{13.6\,\text{eV}\,Z^2}{n^2}$

  • Ionisation potential (ground → $n=\infty$): $E_i = 13.6\,Z^2\,\text{eV}$ (e.g.
  • H : $13.6\,\text{eV}$
  • He$^{+}$ : $54.4\,\text{eV}$)

• Rydberg formula for spectral lines
  $\frac{1}{\lambda}=R\left(\frac{1}{n<em>l^2}-\frac{1}{n</em>u^2}\right)$

  • $R = 1.0967758\times10^{7}\,\text{m}^{-1}$
  • Series:
  • Lyman $n_l = 1$ (UV)
  • Balmer $n_l = 2$ (visible)
  • Paschen $n_l = 3$ (IR)
  • Brackett $n_l = 4$ (far-IR)

• Example – first Lyman line (n=2→1)

  • Energy gap $10.2\,\text{eV}$ → $\lambda \approx 122\,\text{nm}$ (UV)

• Limitations

  • Works only for one-electron (hydrogenic) atoms
  • Circular-orbit assumption ad-hoc; no intensities/transition rates
  • Cannot treat multi-electron, unbound, or non-circular behaviour

Quantum-Mechanical Rescue – Old Quantum Mechanics

• Heisenberg (1924-25) formulated matrix mechanics → overcame Bohr flaws using non-commuting algebra.
• Introduction of wave-particle duality (Davisson & Germer electron diffraction 1925).
• Led to probabilistic interpretation and Uncertainty Principle.

Heisenberg Uncertainty Principle (HUP)

• Statement: simultaneous measurement precisions obey $\Delta x\,\Delta p_x \ge \dfrac{\hbar}{2}$.
• Thought experiment: electron through narrow slit (Young’s double-slit)
  • Narrower slit → smaller $\Delta x$ but diffraction increases $\Delta p_x$.
• Illustrative calculations
  • Electron $v = 1.10\times10^7\,\text{m/s}$ with $0.10\%$ precision → $\Delta x_{min} \approx 110\,\text{nm}$.
  • Baseball (150 g, $42\,\text{m/s}\pm1\,\text{m/s}$) → $\Delta x \sim 7\times10^{-34}\,\text{m}$ (macroscopic objects essentially unaffected).

Zeeman Effect

• Spectral-line splitting in external static magnetic field $\mathbf B$ due to interaction between $\mathbf B$ and magnetic dipole from orbital (and spin) angular momentum.
• Types
  • Normal Zeeman (spin-zero configurations) → equally spaced triplet, selection rule $\Delta m_l=0,\pm1$.
  • Anomalous Zeeman (spin involved) → more complex pattern.
• Energetics: $\Delta E = m<em>l \mu</em>B B$ (simple case), $\mu<em>B = \dfrac{e\hbar}{2m</em>e}$.
• Applications: NMR, ESR, MRI, Mössbauer spectroscopy; astrophysical $B$-field measurement.

Quantum Numbers & Atomic Orbitals

• Four quantum numbers describe electron states
  1. Principal $n = 1,2,3,\dots$ → size & energy
  2. Orbital (azimuthal) $\ell = 0\dots(n-1)$ → shape; labels $s(0),p(1),d(2),f(3),g(4)…$
  3. Magnetic $m_\ell = -\ell,\dots,0,\dots,\ell$ → orientation (2\ell+1 values)
  4. Spin $m_s=\pm\tfrac12$ (introduced later)
• Example $n=3$
  • \ell = 0 → $m_\ell=0$ (1 state)
  • \ell =1 → $m_\ell=-1,0,1$ (3 states)
  • \ell =2 → $m_\ell=-2,-1,0,1,2$ (5 states)
  • Total spatial states = 9 (each doubles with spin)
• For hydrogenic atoms energy depends only on $n$: $E_n=-13.6/n^2\,\text{eV}$.

Electron Spin Multiplicities

• Multiplicity $2S+1$, where $S$ = total spin.
  • Singlet: $S=0$, paired spins (↑↓)
  • Doublet: one unpaired electron (for radicals)
  • Triplet: $S=1$, two unpaired electrons (↑↑)

Chemical Bonding Overview

• Ionic bonding – electron transfer (e.g.
  NaCl)
• Covalent bonding – electron sharing (C–C in diamond, polyethylene)
• Van der Waals forces – weak dipole-dipole attractions; instantaneous or permanent dipoles (e.g.
  H₂O interactions)

Introduction to Spectroscopy

• Spectroscopy: study of interaction between matter and electromagnetic (EM) radiation.
• Interaction type depends on photon energy (wavelength $\lambda$ or frequency $\nu$):
  • $\gamma$, X-ray – inner-shell electronic transitions
  • UV/Vis – outer-electron excitations
  • IR – molecular vibrations/rotations
  • Microwave – rotations / NMR (nuclear spin) at radio frequencies

Electromagnetic Spectrum Quick Numbers

• $\lambda$ ranges (approx.)
  • Gamma < $0.01\,\text{nm}$
  • X-rays $0.01 – 10\,\text{nm}$
  • UV $10 – 380\,\text{nm}$
  • Visible $380 – 780\,\text{nm}$ (ROYGBV)
  • IR $0.78 – 1000\,\mu\text{m}$
  • Microwave cm to m; Radio > $1\,\text{m}$

UV/Visible Spectroscopy

• Chromophores absorb due to $\pi\rightarrow\pi^<em>$ or $n\rightarrow \pi^</em>$ transitions.
• Example data (in hexane):
  • Ethene $\lambda_{max}=171\,\text{nm}, \varepsilon\approx1.5\times10^{4}$
  • Ethanal $n\rightarrow\pi^*$ at $290\,\text{nm}$ (low intensity)
• Solvent effects: identical solvent required for comparison owing to solvatochromic shifts.

IR Spectroscopy

• Typical mid-IR range $4000-400\,\text{cm}^{-1}$ (2.5–25 µm).
• Absorption bands correspond to vibrational modes.

Scattering Phenomena

• Rayleigh scattering – elastic (initial & final quantum states identical) → responsible for blue sky.
• Raman scattering – inelastic; shift in energy equals vibrational quantum.
  • Stokes (scattered lower energy), Anti-Stokes (higher energy).
  • Raman active modes complement IR; water is weakly IR-opaque but Raman-transparent → aqueous studies viable.
  • Instrumentation requires intense monochromatic source (lasers) and sensitive detectors (Raman lines ~0.001 % of incident intensity).
• Surface-Enhanced Raman Scattering (SERS)
  • Enhancement factors $\sim10^{6}$ near rough Au/Ag/Cu surfaces.
  • Mechanisms: (1) local electromagnetic field via surface plasmon resonance, (2) charge-transfer complexes.
  • Enables ultra-sensitive surface/adsorbate analysis.

Key Formulae & Constants (Quick Reference)

• Coulomb constant ke = \dfrac{1}{4\pi\varepsilon0} = 8.988\times10^9\,\text{N·m}^2\text{/C}^2
• Bohr radius $a<em>0 = \dfrac{\hbar^2}{mk</em>e e^2} = 0.529\,Å$
• Rydberg constant $R = 1.097\times10^{7}\,\text{m}^{-1}$
• Planck constant h = 6.626\times10^{-34}\,\text{J·s}, \; \hbar = h/2\pi
• Electron mass $m_e = 9.109\times10^{-31}\,\text{kg}$
• Bohr magneton $\mu<em>B = \dfrac{e\hbar}{2m</em>e}=9.274\times10^{-24}\,\text{J/T}$

Conceptual Connections & Implications

• Transition from deterministic classical orbits (Rutherford) to quantised, probabilistic description (Bohr → QM) illustrates paradigm shift in physics.
• HUP sets fundamental measurement limits, directly impacting nanotechnology, electron microscopy & quantum computing.
• Zeeman splitting underpins modern magnetic-resonance imaging and astrophysical magnetometry.
• Spectroscopic techniques (UV/Vis, IR, Raman, SERS) form analytical backbone across chemistry, biology, materials science (e.g.
  drug analysis, forensic trace detection, semiconductor process monitoring).
• Ethical/practical implications: MRI diagnostic power vs.
  cost; laser safety in Raman; quantum measurement limits in encryption.

Worked-Example Sheet (selected)

1. Hydronic energy: $E_{n,z} = -\dfrac{13.6\,\text{eV}\,Z^2}{n^2}$
   • Compute $E_3$ for He$^{+}$: $E=-54.4/9 = -6.04\,\text{eV}$
2. Wavelength of Balmer transition $n=6\to2$:
   $\frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{6^2}\right)=R\left(\frac{1}{4}-\frac{1}{36}\right)=R\frac{8}{36}= \frac{2R}{9}$
   $\lambda \approx \frac{9}{2R}= \frac{9}{2\times1.097\times10^{7}}\,\text{m}=410\,\text{nm}$ (violet region).
3. Electron position uncertainty (speed 5×10³ m/s ±0.003 %)
   $p=mv;\;\Delta p=3\times10^{-5}p\Rightarrow\Delta x\ge\dfrac{\hbar}{2\Delta p}\approx0.77\,\text{mm}$.

End of compiled study notes.