Energy levels (Ch. 13) |
Nov. 13, 2017 |
(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.
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`{::}^2`2s`{::}^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: `{::}^{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. and
.
• 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.
.
Dye molecules are large and complex organic
material `->`
.
• Solid-state: .
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`{::}_2`O`{::}_3`, Ti Sapphire,
.
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.
.
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).
.