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

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

• Schrodinger equation: Model particle behavior $- \frac {\hbar} {2 m} \nabla^2 \psi ( \vec r , t)+V \psi ( \vec r , t ) = -j \hbar \frac {\partial \psi ( \vec r , t )} {\partial t}$ where V ( vec r , t ) is potential set up by the coulomb force from nucleus, psi is complex wave function, m mass of electron.
• Time independent Schrodinger equation (TISE): For particle in stationary state (stay in orbit), $\psi ( \vec r , t )= \psi ( \vec r ) e^{j \frac E \hbar t}$ and the equation becomes similar to Helmholtz equation $- \frac {\hbar} {2 m} \nabla^2 \psi ( \vec r )+V \psi ( \vec r ) = E \psi ( \vec r )$
where E is the eigenvalue of the equation corresponding to the energy level.
Electron probability over a volume dV = |psi|^2 dV and int |psi|^2 dV = 1
• Atomic levels for H_2: Hydrogen is the simplest atom. Its energy level 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 $E_n = -M_r \frac {e^4} {(4 \pi \epsilon_o )^2 2 \hbar^2} \frac 1 {n^2}$ where reduced mass M_r = {m M_p} / {m+M_p}, M_p is the mass of proton, n is principal quantum number. Also radius of the first Bohr orbit (n=1) a_H={epsilon_o h^2}/{pi m e^2}.
Hydrogen-like ions: Ions with one outer electron, e.g. He{::}^+, Li{::}^{2+}, $E_n = -M_r \frac {e^4} {(4 \pi \epsilon_o )^2 2 \hbar^2} \left( \frac Z n \right)^2$ where Z is the number of protons in nucleus.
• Detailed energy state: Solve TISE with separation of variable psi_{nlm} = R_{nl} (r) Theta_{lm} (theta) Phi_m (phi) where l = 0(s), 1(p), ... , n-1 is the azimuthal quantum number and m = 0, pm 1, pm 2, ..., pm l is the magnetic quantum number.

R_{nl} (r) = rho^l e^{-rho/2} L_{n+1}^{2l+1) (rho)

where L_{n+1}^{2l+1) (rho) is the Laguerre polynomial with rho = {2r}/{a_H}

Theta_{lm} (theta)=P_l^{|m|}(x)(1-x^2)^{1/{2|m|}}/{2l!} {d^{|m|+1}}/{dx^{|m|+1}} (x^2-1)^l where P_l^{|m|}(x) is the Legendre polynomials with x=cos theta

Spin quantum number s = pm 1/2
Notice that we show outer unfilled subshell in energy state diagram as a shorthand notation e.g. excited C with 1s{::}^22s{::}^22p3s 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: {::}^{2S+1}L_J where vec L = vec L_1 + vec L_2, L=|vec L|, vec S = vec S_1 + vec S_2, and J = |vec L + vec S|, e.g. He and He Ne.
• Molecules: Each molecules has multiple atoms, i.e. more energy states for rotation and vibration.
Diatomic rotation (E_r with rotational quantum number r = 0, 1, 2, ...) and vibration (E_v with vibrational quantum number v = 0, 1, 2, ...)
Triatomic molecule, e.g. CO{::}_2 with 3 types of vibration -- asymmetric stretch, symmetric stretch and bending.
Dye molecules are large and complex organic material -> 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 (E_1 or E_v). Upper band is called conduction band with energy (E_2 or E_c). A forbidden band is in between with bandgap energy E_{g} = E_2 - E_1. Three types of solid -- metal (partial occupied conduction band), semiconductor (empty conduction band at 0K) and insulator (like semiconductor but with large E_g).
Solid-state laser materials -- doped dielectric media, a host medium (transparent dielectric) with dopant (transition-metal, lanthanide ions), e.g. Ruby Cr{::}^{3+}: Al{::}_2O{::}_3, Ti Sapphire, Neodymium:YAG Nd{::}^{3+}:Y{::}_3Al{::}_50{::}_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 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 P(E_m) of finding atom at energy level E_m and P(E_m) prop e^(-{E_m}/{kT}) where m = 1, 2, 3, ...
Notice that we require sum_m P(E_m) = 1 and is not an integration of a continuous function.
For laser media, {N_2}/{N_1} = {g_2}/{g_1} e^{-{E_2-E_1}/{kT}} where g_{1,2} are degeneracies of energy levels 1 and 2.
Semiconductors follow Fermi-Dirac distribution f(E) = 1/{e^{{E-E_f}/{kT}}+1} where E_f is the Fermi-level (half population point). If E\>\>E_f, it can be approximated by Boltzmann distribution.