PX282 - C9b - bound-bound absorption

bound-bound absorption

discrete wavelengths would be absorbed, so a delta function is expected in a $κ_{λ}$ - $λ$ plot
in reality, there is a 'broadening' in the line:

κ_{λ}^{b b} \propto g_{m} N_{n} ψ (λ)

where,
$g_{m} =$ statistical weight,
$N_{m} =$ number density in state,
$ψ (λ) =$ line profile

the energy of the line:

E_{m} - E_{n} = Δ E = h ν ⟹ λ = \frac{h c}{Δ E}

line broadening can occur due to several reasons

natural broadening

it arises from the uncertainty principle:

\begin{matrix} (1) & Δ E Δ t \geq ℏ \end{matrix}

the electron is in an excited state for a finite time, so the energy can only have an uncertain value:

\begin{matrix} Δ E = \frac{h c}{λ^{2}} Δ λ = \frac{ℏ}{Δ t} [from (1)] \\ \frac{h c}{λ} - \frac{h c}{λ + Δ λ} = \frac{h}{2 π Δ t} \\ Δ λ = \frac{λ^{2} + λ Δ λ}{2 π c Δ t} \end{matrix}

since $Δ λ ≪ λ :$

Δ λ \approx \frac{λ^{2}}{2 π c Δ t} = \frac{λ^{2}}{2 π c} (\frac{1}{Δ t_{n}} + \frac{1}{Δ t_{m}})

eg: $λ = 656 n m$ , for a $H_{α}$ line:
$n = 2$ , $m = 3$
$Δ t_{2} \approx Δ t_{3} = 10^{- 8} s$
$⟹ Δ λ = 4.6 \times 10^{- 5} n m$

doppler broadening

it arises from the motion of the absorbing atom
assuming atoms follow the maxwell-boltzmann distribution
the most probable velocity:

v = \sqrt{\frac{2 k T}{m}}

the doppler shift:

\frac{Δ λ}{λ} = \pm \frac{v}{c}

since it could be in either direction, there is a factor of $2 :$

Δ λ = \frac{2 λ}{c} \sqrt{\frac{2 k T}{m}}

eg: for $H_{α}$ line in the sun, $T = 5700 K$ , $m_{H} = 1.67 \times 10^{-} 27 k g :$

∴ Δ λ = 0.043 n m

pressure broadening

it arises from the uncertainty principle again
collisions shorten the lifetime of electron in a state
from the uncertainty principle:

Δ E \sim \frac{ℏ}{Δ t}

if $Δ t ↓: Δ E ↑ ⟹ Δ λ ↑$
estimate for $Δ t :$

Δ t \sim \frac{l}{v} = \frac{1 / n σ}{v}

where, $l = 1 / n σ =$ mean free path, $v =$ velocity of atom, $n =$ number density, $σ =$ collision cross-section

Δ λ \sim \frac{λ^{2}}{π c Δ t} = \frac{λ^{2}}{π c} n σ \sqrt{\frac{2 k T}{m}}

here, $Δ λ \propto n \sqrt{T} ⟹$ higher $p$ , higher $Δ λ$
eg:
- $n_{⊙} \sim 1.5 \times 10^{23} m^{- 3}$
- $σ \approx 2 π r^{2} = 2 π a_{0}^{2} = 1.8 \times 10^{- 20} m^{2}$ , where, $a_{0} =$ bohr radius

∴ Δ λ \approx 2 \times 10^{- 5} n m

image: K. N. Liou, An Introduction to Atmospheric Radiation

natural and pressure broadening form the lorentzian in the $κ - λ$ plot

stellar rotation

there will be constant doppler shifts from a rotating star:

\frac{Δ λ}{λ} = \frac{v_{r o t}}{c}

eg: for the sun, $P_{r o t} = 27 d a y s$ , $R_{⊙} = 7 \times 10^{8} m$ , $v_{r o t} \sim 2 \times 10^{3} m s^{- 1}$
$∴ Δ λ \approx 0.004 n m$