Atomic-Molecular-Optical Physics
Table of Contents
- 1. Absorption and Emission
- 2. Reflection
- 3. Refraction
- 4. Incandescence
- 5. Luminescence
- 6. Transition Dipole Moment
- 7. Radiation
- 8. Scattering
- 9. X-Ray Emission
- 10. Auger Effect
- 11. References
1. Absorption and Emission
1.1. Emission Spectrum
Rydberg formula (1888) The spectral lines of hydrogen follows: \[ \nu_{jk} = cR \left( \frac{1}{k^2} - \frac{1}{j^2} \right) \] where \( R \) is called the Rydberg constant.
Rydberg-Ritz Combination Principle (1908) \[ \nu_{jk} = \nu_{jm} + \nu_{mk}. \]
1.2. Emission Coefficient
Emission coefficient is the energy emitted by a unit volume per unit time into a unit solid angle: \[ \varepsilon := \frac{\dd{E}}{\dd{V}\dd{t}\dd{\Omega}}. \]
1.3. Einstein Coefficients
- published in 1917
The (spontaneous) emission coefficient \( \varepsilon \) is proportional to the number density via the A coefficient: \[ \varepsilon = \sum_{j\to k}\frac{h\nu_{jk}}{4\pi} n_j A_{jk} \]
1.4. Einstein's Relations
The relations between Einstein A coefficients and B coefficients.
1.5. Franck-Condon Principle
The vibronic transitions happens in a way that minimizes the change in nuclear configuration.
The potential landscape is drawn as a function of normal coordinates \(q_i\).
2. Reflection
3. Refraction
3.1. Refractive Index
3.1.1. History
Cauchy (1836) \[ n^2 = a + \frac{b}{\lambda^2} + \frac{d}{\lambda^4} + \frac{f}{\lambda^6} + \cdots. \]
Sellmeier (1872) Formulated after the discovery of anomalous dispersion by Christiansen and Kundt, to account for the anomalies. \[ n^2 - 1 = \sum_k \frac{a_k\lambda^2}{\lambda^2 - \lambda_k^2}. \]
Helmholtz (1875) Helmholtz hypothesized the damping of the ether. \[ n^2 - 1 = \sum_k \frac{a_k\lambda^2}{\lambda^2 - \lambda_k^2 + b_k\lambda^2}. \] Although the original motivation is now proven to be wrong, the term itself remains to be important.
Lorentz and Drude The molecules within matter is modelled as harmonic oscillators.
As a plane electromagnetic wave \( E_s(z,t) = E_0\exp(i\omega (t - z/c)) \) passees through a thin dispersive medium, the retarded wave
\begin{align*} E'(z,t) &= E_s(z,t - \Delta t) \\ &= \exp(-i\omega\Delta t) E_s(z,t) \\ &\approx (1 - i\omega \Delta t) E_s(z,t) \\ & = \left( 1 - \frac{i\omega(n-1) \Delta z}{c} \right)E_s(z,t) \\ \end{align*}is due to the superposition with the wave \( E_q(z,t) \) generated by the oscillating molecules
\begin{align*} E'(z,t) &= E_s(z,t) + E_q(z,t) \\ &= E_s(z,t) - \frac{i\omega\eta q x_0}{2\varepsilon_0 c} \exp(i\omega(t - z/c)). \end{align*}Thus, we find that the refractive index is \[ n - 1 = \frac{qx_0N}{2\varepsilon_0E_0}. \]
Let us model the molecules with a forced damped oscillator \[ m \ddot{x} = - \omega_0^2x - \gamma \dot{x} + qE_0e^{i\omega t} \] the solution would have the amplitude \[ x_0 = \frac{qE_0}{m}\frac{1}{\omega_0^2 - \omega^2 + i\gamma\omega}. \]
Plugging \( x_0 \) into the equality we drived earlier, the formula for the index of refraction is obtained: \[ n-1= \frac{e^2N}{2\varepsilon_0m}\frac{1}{\omega_0^2 - \omega^2 + i\gamma\omega}. \]
The imaginary part in this formula constitute the decay of the electromagnetic field.
Lorenz-Lorentz Equation \[ \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi k_e}{3}N\chi \] where \( \chi \) is the polarizability.
Ladenburg and Reiche (1923) \[ \mathbf{P}_k = \frac{c^3 \mathbf{E}}{32 \pi^4 k_e} \sum_{j> k} \frac{A_{jk}}{\nu_{jk}^2(\nu_{jk}^2 -\nu^2)} \]
Kramers (1924) \[ \mathbf{P}_k = \frac{c^3 \mathbf{E}}{32 \pi^4 k_e} \left[ \sum_{j> k} \frac{A_{jk}}{\nu_{jk}^2(\nu_{jk}^2 -\nu^2)} - \sum_{j < k} \frac{A_{kj}}{\nu_{kj}^2(\nu_{kj}^2 -\nu^2)} \right] \]
4. Incandescence
- Thermal Radiation
5. Luminescence
5.1. Jablonski Diagram
Energy levels of electronic states and vibrational levels of a molecule are arranged vertically, grouped by spin-multiplicity horizontally. Ground vibrational states are indicated with thick lines.
Radiative transitions is drawn with straight arrow, and nonradiative transitions are drawn with squiggly arrows.
5.2. Grotrian Diagram
- Term Diagram
Show allowed electronic transitions between energy levels of atoms.
5.3. Franck-Condon Principle
Vibronic transition occurs in a way that minimizes the change in nuclear configuration.
5.3.1. Franck-Condon Diagram
- Franck-Condon Energy Diagram
Potential energy diagram along a nuclear configuration.
5.4. Photoluminescence
- PL
Light emitted by matter excited by light.
5.4.1. Fluorescence
- 형광
The electron decays from an excited singlet state to singlet state.
5.4.2. Phosphorescence
- 야광
Delayed light emission after excitation
The electron is excited to a singlet state and moves to a triplet state via intersystem crossing. The decay to ground singlet state is forbidden and much slower. It was first observed in phosphorus, hence the name.
5.4.3. Radiophotoluminescence
Radiation exposure cause photoluminescence
5.5. Thermoluminescence
Light emitted when heated.
5.6. Electroluminescence
- EL
Light emitted by matter excited by electric field or current.
5.7. Triboluminescence
Quartz emits light when rubbed against each other. This is due to electrons jump over voltage gap created by Triboelectric effect.
5.8. Chemiluminescence
Excited by chemical reactions
5.9. Cathodoluminescence
Excited by electron beams
5.10. Bioluminescence
Excited by biochemical reactions in organisms
6. Transition Dipole Moment
6.1. Definition
The transition from state to state has a dipole moment defined by \[ \mathbf{d}_{a\to b} = \langle \psi_b|q\mathbf{r} |\psi_a\rangle. \]
The transition happens only when the dipole moment can interact with the electromagnetic wave.
7. Radiation
8. Scattering
8.1. Reyleigh Scattering
Elastic scattering by electron
8.2. Raman Scattering
Inelastic scattering due to the interaction with the material's vibration, rotation state, or phonon in crystal.
Stokes Raman the molecule gains energy Anti-Sotkes Raman the molecule loses energy
\[ I \propto \left|\mathbf e_s^{\,*}\cdot \boldsymbol{\alpha}_R\cdot \mathbf e_i\right|^2, \] where \(\mathbf e_i\) and \(\mathbf e_s\) are the incident and detected polarization vectors, and \(\boldsymbol{\alpha}_R\) is the Raman tensor.
8.3. Compton Scattering
Inelastic scattering due to the interaction with valence or free electron, with recoiling electron.
\[ \lambda' - \lambda = \frac{h}{m_e c}(1-\cos\theta), \] where \(\theta\) is the photon scattering angle
9. X-Ray Emission
Core electron is knocked away by external impact usually from electron. Core electron is filled by deexcitation from higher energy states other than the ones forbidden by the orbital selection rules, and X-ray radiation is emitted.
- 2s → 1s forbidden
- \(\mathrm{K}_{\alpha}\) 2p → 1s
- \(\mathrm{K}_{\alpha1}\colon 2p_{3/2} \to 1s_{1/2}\) or L₃ → K
- \(\mathrm{K}_{\alpha2}\colon 2p_{1/2} \to 1s_{1/2}\) or L₂ → K
- \(\rm K_{\beta}\) 3p → 1s
- \(\rm L_{\alpha}\) 3d → 2p
Spin orbit coupling splits each of these lines into two, for example \(\rm K_{\alpha1}\) and \(K_{\alpha2}\).
9.1. Spectroscopic Subshell Notation
where the subscript is the total angular momentum \(\mathbf{J}\) of the electron under spin-orbit coupling. They are in order of increasing energy.
10. Auger Effect
French [o.ʒe]
When electron deexcites, other electron can take up the energy and get emitted.
The binding energy of the secondary electron is subtracted from the emitted energy from deexcitation.
For example, the L₁-K transition, can cause L₂ shell electron to be emitted with the energy of \(B(K) - B(L₁) - B(L₂)\). Notice that the L₁-K transition is allowed because no photon is involved here.
10.1. Fluorescent Yield
The fraction of characteristic x-ray emission to the number of vacancies.
The fluorescent yield can be defined for the vacancies in specific shell, like \(\omega_{\rm K}, \omega_{\rm L}\).
Fluorescent yield increases as Z increases.