Difference between revisions of "Estimating Atomic Transition Strengths"

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(Created page with '===Short Topical Videos=== * [http://youtu.be/gt0gTgozCbo Estimating the Strength of Atomic Transitions (Parsons, YouTube)] ===Reference Material=== * <latex> \documentclass[1…')
 
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<latex>
 
<latex>
 
\documentclass[11pt]{article}
 
\documentclass[11pt]{article}
\def\eboltz{e^{{-h\nu _0 \over kT}}}
+
\def\inv#1{{1 \over #1}}
 +
\def\ddt{{d \over dt}}
 +
\def\mean#1{\left\langle #1\right\rangle}
 
\def\sigot{\sigma_{12}}
 
\def\sigot{\sigma_{12}}
 
\def\sigto{\sigma_{21}}
 
\def\sigto{\sigma_{21}}
\def\wz{\omega_0}
+
\def\eval#1{\big|_{#1}}
 +
\def\tr{\nabla}
 
\def\dce{\vec\tr\times\vec E}
 
\def\dce{\vec\tr\times\vec E}
 
\def\dcb{\vec\tr\times\vec B}
 
\def\dcb{\vec\tr\times\vec B}
 +
\def\wz{\omega_0}
 +
\def\ef{\vec E}
 
\def\ato{{A_{21}}}
 
\def\ato{{A_{21}}}
 
\def\bto{{B_{21}}}
 
\def\bto{{B_{21}}}
 
\def\bot{{B_{12}}}
 
\def\bot{{B_{12}}}
\def\inv#1{\frac1{#1}}
 
\def\hf{\frac12}
 
 
\def\bfield{{\vec B}}
 
\def\bfield{{\vec B}}
\def\eval#1{\big|_{#1}}
+
\def\ap{{a^\prime}}
\def\tr{\nabla}
+
\def\xp{{x^{\prime}}}
\def\ef{\vec E}
+
\def\yp{{y^{\prime}}}
\def\Jbar{{\bar J}}
+
\def\zp{{z^{\prime}}}
 +
\def\tp{{t^{\prime}}}
 +
\def\upx{{u_x^\prime}}
 +
\def\upy{{u_y^\prime}}
 +
\def\e#1{\cdot10^{#1}}
 +
 
 
\usepackage{fullpage}
 
\usepackage{fullpage}
 
\usepackage{amsmath}
 
\usepackage{amsmath}
 
\usepackage{eufrak}
 
\usepackage{eufrak}
\usepackage{graphicx}
 
 
\begin{document}
 
\begin{document}
\title{ Estimating Atomic Transition Strengths (Einstein A's)}
+
\def\lya{Ly\alpha}
 +
\title{Estimating the Strength of Atomic/Molecular Lines}
 +
\section{ Einstein A's for $\lya$}
 +
 
 +
$\ato$ is a measure of the probability of decay per unit time, so
 +
$\ato^{-1}\sim lifetime$.  This should be about
 +
equal to the energy of the electron state divided by the average power radiated
 +
by an electron being accelerated:
 +
$$\ato^{-1}\sim{E\over P}\sim{\hbar\omega_0\over{2\over3}{e^2\ddot x^2
 +
\over c^3}}
 +
\sim{3\hbar\omega_0c^3\over2(e\ddot x)^2}$$
 +
Now $e\cdot\vec x=\vec d$ (the electric dipole moment) and
 +
$\ddot x\sim\omega_0^2x$ for a spring, so:
 +
$$\ato^{-1}\sim{3\hbar\wz c^3\over2d^2\wz^4}$$
 +
$$\boxed{\ato\sim{2d^2\wz^3\over3\hbar c^3}}$$
 +
For H: $d\sim ea_0$ and $\lambda_\lya=1216\AA$ so:
 +
$$\ato\sim5\e8s^{-1}$$
 +
 
 +
\section{ Magnetic Dipole for $\lya$}
 +
 
 +
The magnetic dipole of an electron is:
 +
$$\mu_e={e\hbar\over m_ec}$$
 +
Thus we can estimate the ratio of $\ato$ for magnetic dipole transitions to
 +
that of electric dipole transitions:
 +
$${\ato\eval{mag}\over\ato\eval{elec}}\sim\left({\mu_e\over d}\right)^2
 +
\sim\left({e^2\over\hbar c}\right)^2\sim\alpha^2$$
 +
This tells us that the magnetic dipole states (that is, fine and hyperfine
 +
states) are longer lived than electric dipole states by a factor of $\alpha^2$.
 +
$${\ato\eval{21cm}\over\ato\eval{Ly\alpha}}\sim\alpha^2
 +
\left({1216\AA\over21cm}\right)^3$$
 +
$$\ato\eval{21cm}\sim6\e{-15}s^{-1}$$
 +
The actual value is $2.876\e{-15}s^{-1}$.
 +
 
 +
\def\mfe{\mathfrak{E}}
 +
\section{ Electric Quadrupole}
 +
 
 +
If one is nearby a rotating quadrupole, one sees the $\mfe$  (electric)
 +
field rotating rigidly.
 +
However, from far away, there are kinks in the field, resulting in a retarded
 +
potential.  The radiation nearby goes as $r_{near}\sim\lambda$.  For a monopole,
 +
the electric field is $\mfe\sim{q\over r^2}$.  For a dipole, it is
 +
$\mfe={q\over r^2}
 +
{s\over r}$, where s is the charge separation.  For a quadrupole:
 +
$$\mfe={q\over r^2}\left({s\over r}\right)^2$$
 +
Since $P\propto \mfe^2$, the ratio of the powers emitted by a
 +
quadrupole vs. a dipole should be:
 +
$${P_{quad}\over P_{di}}\sim\left({s\over r}\right)^2
 +
\sim\left({s\over\lambda}\right)^2$$
 +
An acoustic analogy: a kettle whistle is a monopole, a
 +
guitar string is a dipole, and a tuning fork (with its two out-of-phase
 +
prongs) is a quadrupole.\\
 +
 
 +
Anyway, since $\ato\sim{P\over E}$,
 +
$$\boxed{{\ato\eval{quad}\over\ato\eval{di}}
 +
\sim\left({s\over\lambda}\right)^2}$$
 +
Thus $28\mu m$, the lowest quadrupole rotational transition of
 +
$H_2$, should have an $\ato$ of about:
 +
$$\ato\eval{28\mu m}\sim\ato\eval\lya\left({s\over\lambda_{H_2}}\right)^2
 +
\left({\lambda_\lya\over \lambda_{H_2}}\right)
 +
\sim\ato\eval\lya\left({a_0\over 28\mu m}\right)^2
 +
\left({1216\AA\over 28\mu m}\right)^3
 +
\sim7\e{-11}s^{-1}$$
 +
The actual value is $3\e{-11}s^{-1}$.
 +
 
 +
\section{ Radio Recombination Lines}
 +
 
 +
In HI, the $n=110\to n=109$ transition has a wavelength of 6 cm.  We can
 +
estimate its $\ato$:
 +
$$\ato\eval{6cm}\sim\ato\eval{Ly\alpha}
 +
\underbrace{\left({1216\AA\over6cm}\right)^3}_{change\atop in\ \lambda}
 +
\underbrace{\left({a_{110}\over a_0}\right)^2}_{change\atop in\ atom\ size}$$
 +
\def\bra#1{\langle #1|}
 +
\def\ket#1{|#1\rangle}
 +
This presents the question of which dipole moment to use.  It turns out
 +
we must use $\bra{i}\vec k\cdot\vec r\ket{f}$.
 +
 
 +
\section{ Back to $\sigma$}
  
 +
$$\sigot\eval{line\atop center}\sim{\lambda^2\over8\pi}{\ato\over\Delta\nu}$$
 +
Now $\Delta\nu\sim\nu$ for doppler broadening, and $\ato\sim\nu^3$, so for
 +
electric and magnetic dipole transitions:
 +
$$\sigot\sim\lambda^0$$
 +
So the cross-section for these transitions does not depend on wavelength.
  
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Revision as of 14:31, 24 September 2014

Short Topical Videos

Reference Material

Estimating the Strength of Atomic/Molecular Lines

1 Einstein A’s for

is a measure of the probability of decay per unit time, so . This should be about equal to the energy of the electron state divided by the average power radiated by an electron being accelerated:

Now (the electric dipole moment) and for a spring, so:

For H: and so:

2 Magnetic Dipole for

The magnetic dipole of an electron is:

Thus we can estimate the ratio of for magnetic dipole transitions to that of electric dipole transitions:

This tells us that the magnetic dipole states (that is, fine and hyperfine states) are longer lived than electric dipole states by a factor of .

The actual value is .

3 Electric Quadrupole

If one is nearby a rotating quadrupole, one sees the (electric) field rotating rigidly. However, from far away, there are kinks in the field, resulting in a retarded potential. The radiation nearby goes as . For a monopole, the electric field is . For a dipole, it is , where s is the charge separation. For a quadrupole:

Since , the ratio of the powers emitted by a quadrupole vs. a dipole should be:

An acoustic analogy: a kettle whistle is a monopole, a guitar string is a dipole, and a tuning fork (with its two out-of-phase prongs) is a quadrupole.\

Anyway, since ,

Thus , the lowest quadrupole rotational transition of , should have an of about:

The actual value is .

4 Radio Recombination Lines

In HI, the transition has a wavelength of 6 cm. We can estimate its :

This presents the question of which dipole moment to use. It turns out we must use .

5 Back to

Now for doppler broadening, and , so for electric and magnetic dipole transitions:

So the cross-section for these transitions does not depend on wavelength.