Energy levels (Ch. 13) Nov. 13, 2017

Reading for atomic energy level (Laser Fundamentals, William & Silvast)

• Schrodinger equation: Model particle behavior$\text{ }-\frac{{\text{ℏ}}^{2}}{2m}{\nabla }^{2}\psi \left(\stackrel{\to }{r},t\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}V\left(\stackrel{\to }{r},t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-j\text{ℏ}\partial \frac{\psi \left(\stackrel{\to }{r},t\right)}{\partial t}$ where$\text{ }V\left(\stackrel{\to }{r},t\right)$ is potential set up by the coulomb force from nucleus,$\text{ }\psi$ complex wave function, m mass of electron.
• Time independent Schrodinger equation (TISE): For particle in stationary state (stay in orbit),$\text{ }\psi \left(\stackrel{\to }{r},t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\psi \left(\stackrel{\to }{r}\right){e}^{jEt/\text{ℏ}}$ and the equation becomes similar to Helmholtz equation$\text{ }-\frac{{\text{ℏ}}^{2}}{2m}{\nabla }^{2}\psi \text{\hspace{0.17em}}+\text{\hspace{0.17em}}V\psi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}E\psi$
where E is the eigenvalue of the equation corresponding to the energy level.
Electron probability over a volume dV =$|\psi {|}^{2}dV$ and$\text{ }\int |\psi {|}^{2}dV\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$
• Atomic levels for$\text{ }{H}_{2}$ : Hydrogen is the simplest atom. Its energy level$\text{ }{E}_{n}$ can be found with Bohr’s model by equating potential energy from Coulomb force to Kinetic energy of centrifugal force from orbiting electron$\text{ }{E}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{−}{M}_{r}\frac{{e}^{4}}{\left(4\pi {\epsilon }_{o}{\right)}^{2}2{\text{ℏ}}^{2}}\text{\hspace{0.17em}}\frac{1}{{n}^{2}}$ where reduced mass$\text{ }{M}_{r}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{m{M}_{p}}{m\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{M}_{p}}$ ,$\text{ }{M}_{p}$ is the mass of proton, n is principal quantum number. Also radius of the first Bohr orbit (n=1)$\text{ }{a}_{H}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\epsilon }_{o}{h}^{2}}{\pi m{e}^{2}}$
Hydrogen-like ions: Ions with one outer electron, e.g.$\text{ }H{e}^{+}$ ,$\text{ }L{i}^{2+}$ , $\text{ }{E}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{−}{M}_{r}\frac{{e}^{4}}{\left(4\pi {\epsilon }_{o}{\right)}^{2}2{\text{ℏ}}^{2}}\text{\hspace{0.17em}}{\left(\frac{Z}{n}\right)}^{2}$ where Z is the number of protons in nucleus.
• Detailed energy state: Solve TISE with separation of variable$\text{ }{\psi }_{nlm}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{R}_{nl}\left(r\right){\Theta }_{lm}\left(\theta \right){\Phi }_{m}\left(\phi \right)$ where$\text{ }l\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0\left(s\right),\text{\hspace{0.17em}}1\left(p\right),\text{\hspace{0.17em}}...,n-1$ is the azimuthal quantum number and$\text{ }m\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}±1,±2,...,\text{\hspace{0.17em}}±l$ is the magnetic quantum number.

$\text{ }{R}_{nl}\left(r\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\rho }^{l}{e}^{-\rho /2}{L}_{n+l}^{2l+1}\left(\rho \right)$

where where$\text{ }{L}_{n+1}^{2l+1}\left(\rho \right)$ is the Laguerre polynomial with$\text{ }\rho \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{2r}{{a}_{H}}$$\text{ }{\Theta }_{lm}\left(\theta \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{P}_{l}^{|m|}\left(x\right)\frac{\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{x}^{2}{\right)}^{1/\left(2|m|\right)}}{2l!}\frac{{d}^{|m|+1}}{d{x}^{|m|+1}}\left({x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1{\right)}^{l}$ where$\text{ }{P}_{l}^{|m|}\left(x\right)$ is the Legendre polynomials with$\text{ }x\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{cos}\theta$

$\text{ }{\Phi }_{m}\left(\phi \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{jm\phi }$

Spin quantum number$\text{ }s\text{\hspace{0.17em}}=\text{\hspace{0.17em}}±1/2$
Notice that we show outer unfilled subshell in energy state diagram as a shorthand notation e.g. excited C with$1{s}^{2}{2}^{2}2p3s$ is referred C (3s).
• Multi-electron atoms: Need to consider interaction of electrons including 1) kinetic energy (orbiting), 2) potential energy (Coulomb force), 3) mutual electrostatic energy of electrons, 4) spin-spin correlations and 5) spin-orbit energy.
1) and 2) are included in Bohr’s model
3) and 4) are accounted for by LS coupling which is commonly used for lasing transition.
5) is considered in J-J coupling (not common).
• LS coupling:$\text{ }{\text{}}^{2S+1}{L}_{J}$ where$\text{ }\stackrel{\to }{L}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\stackrel{\to }{L}}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\stackrel{\to }{L}}_{2}$ ,$\text{ }L\text{\hspace{0.17em}}=\text{\hspace{0.17em}}|\stackrel{\to }{L}|$ ,$\text{ }\stackrel{\to }{S}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\stackrel{\to }{S}}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\stackrel{\to }{S}}_{2}$ ,$\text{ }S=\text{\hspace{0.17em}}|\stackrel{\to }{S}|$ and$\text{ }J\text{\hspace{0.17em}}=\text{\hspace{0.17em}}|\stackrel{\to }{L}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{+}\text{\hspace{0.17em}}\stackrel{\to }{S}|$ , e.g. He and He Ne.
• Molecules: Each molecules has multiple atoms, i.e. more energy states for rotation and vibration.
Diatomic rotation ($\text{ }{E}_{r}$ with rotational quantum number$\text{ }r\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}},2...$ ) and vibration ($\text{ }{E}_{v}$ with vibrational quantum number$\text{ }v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}},2...$ )
Triatomic molecule, e.g. $\text{ }C{O}_{2}$ with 3 types of vibration -- asymmetric stretch, symmetric stretch and bending.
Dye molecules are large and complex organic material$\text{ }\to$ more complicated manifolds with wide range of possible lasing freq.
• Solid-state: Atoms/molecules are closer and interact with lattice (a periodic structure). Low energy levels remain sharp while the outer levels smear and form bands as atoms getting closer.
Lower band is called valence band with energy ($\text{ }{E}_{1}$ or$\text{ }{E}_{v}$ ). Upper band is called conduction band with energy ($\text{ }{E}_{2}$ or$\text{ }{E}_{c}$ ). A forbidden band is in between with bandgap energy$\text{ }{E}_{g}\text{\hspace{0.17em}}={E}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{E}_{1}$ Three types of solid -- metal (partial occupied conduction band), semiconductor (empty conduction band at 0K) and insulator (like semiconductor but with large$\text{ }{E}_{g}$ ).
Solid-state laser materials -- doped dielectric media, a host medium (transparent dielectric) with dopant (transition-metal, lanthanide ions), e.g. Ruby$\text{ }C{r}^{3+}$ :$\text{ }A{l}_{2}{O}_{3}$ , Ti Sapphire, Neodymium:YAG$\text{ }N{d}^{3+}:{Y}_{3}A{l}_{5}{O}_{2}$ ($1.064\mu m$ ), Neodymium:glass ($1.053\mu m$ ).
Semiconductors -- Similar to other solids with conduction band, valence band and bandgap energy but have 2 types of charge carriers (electrons and holes).
Band is not flat (more like parabolic) in spatial freq domain owing to periodic structure of lattice.
Direct bandgap (e.g. GaAs) is more favorable as lasing media than indirect bandgap (e.g. Si).
Man-made structures -- building heterojunctions with semiconductors having matched lattice constant but different$\text{ }{E}_{g}$ quantum well for 1D electron confinement, quantum wire for 2D electron confinement and quantum dot for 3D electron confinement.
• Occupation of energy level: Thermal distribution of particle can be modeled by Boltzmann distribution.
Probability$\text{ }P\left({E}_{m}\right)$ of finding atom at energy level$\text{ }{E}_{m}$$\text{ }P\left({E}_{m}\right)\propto \text{exp}\left(-{E}_{m}/\left(kT\right)\right)$ where m = 1, 2, 3, ...
Notice that we require$\text{ }\underset{m}{\overset{\text{}}{\Sigma }}P\left({E}_{m}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ and is not an integration of continuous function.
For laser media,$\text{ }\frac{{N}_{2}}{{N}_{1}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{g}_{2}}{{g}_{1}}\text{exp}\left(-\frac{{E}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{E}_{1}}{kT}\right)$ where$\text{ }{g}_{1,2}$ are degeneracies of energy levels 1 and 2.
Semiconductors follow Fermi-Dirac distribution$\text{ }f\left(E\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{e}^{\left(E\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{E}_{f}\right)/kT}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}$ where$\text{ }{E}_{f}$ is the Fermi-level (half population point). If$\text{ }E\gg {E}_{f}$ , it can be approximated by Boltzmann distribution.