Bohr's model and physics of the atom

• Energy in the nth state: $E_n = -\frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$

• For hydrogen (Z=1): $E_n = \frac{E_1}{n^2} = -\frac{13.6\text{ eV}}{n^2}$

• Derivation steps:

  1. Kinetic energy: $K = \frac{1}{2}mv^2 = \frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$
  2. Potential energy: $V = -\frac{Ze^2}{4\pi\varepsilon_0r} = -\frac{mZ^2e^4}{4\varepsilon_0^2h^2n^2}$
  3. Total energy: $E = K + V = -\frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$

• Energy scales with $Z^2$ and inversely with $n^2$

• All energy levels are negative (bound states)

• E = 0 corresponds to ionization (electron completely removed)

• About 1 in 8000 alpha particles was deflected by more than 90° (turned back)
• This revealed:
  • The existence of a very heavy, positively charged nucleus
  • The nucleus contains virtually all mass of the atom
  • The nucleus occupies only about 10^-15 of the atom's volume
  • The atom must contain mostly empty space
• This observation led to the nuclear model of the atom, replacing Thomson's model

• Ground state wave function: $\psi(r) = \psi_{100} = \sqrt{\frac{Z^3}{\pi a_0^3}}e^{-r/a_0}$

• Linear probability density: $P(r) = \frac{4}{a_0^3}r^2e^{-2r/a_0}$

• Comparison with Bohr's model:

  • Bohr: electron exists only at radius r = a₀
  • QM: electron has maximum probability at r = a₀, but exists with decreasing probability at all distances
  • QM probability density peaks exactly at Bohr radius
  • Wave function never becomes zero (electron can be found anywhere)

• This spherically symmetric probability distribution (s-orbital) replaces the concept of a fixed circular orbit

For Li²⁺ (Z=3):

1. First Bohr orbit radius: $r_1 = \frac{a_0}{Z} = \frac{0.053\text{ nm}}{3} = 0.0177\text{ nm}$

2. Ground state energy: $E_1 = -Z^2 \times 13.6\text{ eV} = -3^2 \times 13.6\text{ eV} = -122.4\text{ eV}$

3. Wavelength from n=2 to n=1 transition: $\frac{1}{\lambda} = RZ^2(\frac{1}{1^2} - \frac{1}{2^2}) = R \times 3^2 \times \frac{3}{4} = 9R \times \frac{3}{4} = 6.75R$

   $\lambda = \frac{1}{6.75R} = \frac{1}{6.75 \times 1.097 \times 10^7 m^{-1}} = 13.5\text{ nm}$

This is in the far ultraviolet region of the spectrum.

Need for electron motion:

• Stationary electrons would fall into the nucleus due to electrostatic attraction
• Circular orbital motion creates centripetal force that counterbalances the attractive Coulomb force
• This maintains stable electron-nucleus separation

Fundamental problem created:

• According to Maxwell's electromagnetic theory, accelerating charged particles must radiate energy
• Orbiting electrons are constantly accelerating (changing direction)
• This radiation would cause continuous energy loss
• The electron would spiral inward and eventually collapse into the nucleus
• The atom should be unstable according to classical physics
• Observed atomic stability contradicted this theoretical prediction

This contradiction required quantum mechanics (later Bohr's model) to resolve by introducing quantized orbits where electrons don't radiate energy.

Hydrogen spectral series demonstrate energy quantization through:

1. Discrete line patterns: Each series shows sharply defined spectral lines at specific wavelengths rather than continuous emission, proving energy states are quantized.

2. Mathematical expression: The wavelengths fit the Rydberg formula: $\frac{1}{\lambda} = R\left(\frac{1}{n^2} - \frac{1}{m^2}\right)$ Where:

   • R ≈ 1.097 × 10⁷ m⁻¹ (Rydberg constant)
   • n = lower energy level (1 for Lyman, 2 for Balmer, 3 for Paschen)
   • m = higher energy level (m > n)

3. Energy transitions: Each spectral line represents a discrete energy change when an electron jumps from a higher to lower orbital: $E_m - E_n = hc/\lambda$

4. Series convergence: Each series approaches a limit wavelength as m approaches infinity, corresponding to the minimum energy needed to ionize from the respective lower state.

In Bohr's atomic model:

• Orbit radius is proportional to n²: rₙ = n²·r₁
• For hydrogen atom:
  • First orbit (n=1): r₁ = 53 pm (Bohr radius or a₀)
  • Second orbit (n=2): r₂ = 4r₁ = 212 pm
  • Third orbit (n=3): r₃ = 9r₁ = 477 pm
  • Fourth orbit (n=4): r₄ = 16r₁ = 848 pm

For hydrogen-like ions with atomic number Z:

• rₙ = n²·r₁/Z

Note: As the orbit radius increases, the electron is less tightly bound to the nucleus, corresponding to higher energy states (less negative energy values).

Contradictions between quantum mechanics and Bohr's model:

1. Electron position:

   • Bohr model: Electron exists at fixed radius (r = a₀ for n=1, r = 4a₀ for n=2)
   • Quantum mechanics: Electron has a probability distribution across all distances

2. For ground state (n=1, l=0):

   • Maximum probability at r = a₀ (coincidentally matching Bohr radius)
   • But significant probability of finding electron at r < a₀ and r > a₀

3. For n=2, l=0 state:

   • Bohr model predicts electron at exactly r = 4a₀
   • Quantum mechanics shows two probability maxima (near r = a₀ and r = 5.4a₀)
   • Has a node where probability is zero (impossible in Bohr model)

This demonstrates that electrons don't orbit in fixed circular paths but exist as three-dimensional probability clouds (orbitals).

In a He-Ne laser, pumping occurs through:

1. Electric discharge ionizes gas atoms
2. Free electrons accelerate in the electric field
3. High-energy electrons collide with helium atoms, exciting them to metastable state (E₃ = 20.61 eV)
4. Excited helium atoms collide with neon atoms
5. Energy transfer occurs: He* + Ne → He + Ne* (excited neon at E₂ = 20.66 eV)

Helium is included because:

• Its metastable state (20.61 eV) closely matches neon's upper lasing level (20.66 eV)
• This energy resonance enables efficient energy transfer
• The metastable helium state has long lifetime, acting as an energy reservoir
• Without helium, direct electrical excitation of neon would be inefficient for population inversion

This indirect pumping scheme solves the challenge of selectively populating the upper laser level of neon.

Q-switching and mode-locking are techniques for generating intense laser pulses:

Q-switching:

• Artificially increases cavity loss (lowers "Q factor") during pumping
• Population inversion builds to much higher levels than normal
• Suddenly restoring low loss (high Q) produces giant pulse
• Energy stored across entire gain medium releases almost simultaneously
• Produces pulses of nanosecond duration
• Peak power millions of times higher than continuous operation
• Methods: rotating mirrors, electro-optic shutters, saturable absorbers

Mode-locking:

• Forces multiple longitudinal modes to maintain fixed phase relationship
• Creates constructive interference at regular intervals
• Pulse duration: Δt ≈ 1/Δν (inverse of spectral bandwidth)
• Produces ultrashort pulses (picoseconds to femtoseconds)
• Repetition rate = c/2L (time to complete one round trip in cavity)
• Methods: active (modulators) or passive (saturable absorbers)

Both techniques convert continuous pumping energy into intense, short bursts of light for applications requiring extremely high peak powers.