Difference between revisions of "Classical Bohr Atom"

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(Created page with '===Short Topical Videos=== * [http://youtu.be/424QV3tD4PE Radiation and Matter, to Order of Magnitude (Aaron Parsons)] ===Reference Material=== * [http://en.wikipedia.org/wiki/B…')
 
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<latex>
 
<latex>
 
\documentclass[11pt]{article}
 
\documentclass[11pt]{article}
 +
\def\sigot{\sigma_{12}}
 +
\def\sigto{\sigma_{21}}
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\def\wz{\omega_0}
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\def\dce{\vec\tr\times\vec E}
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\def\dcb{\vec\tr\times\vec B}
 +
\def\ato{{A_{21}}}
 +
\def\bto{{B_{21}}}
 +
\def\bot{{B_{12}}}
 
\def\inv#1{\frac1{#1}}
 
\def\inv#1{\frac1{#1}}
 +
\def\hf{\frac12}
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\def\bfield{{\vec B}}
 
\def\eval#1{\big|_{#1}}
 
\def\eval#1{\big|_{#1}}
 +
\def\tr{\nabla}
 +
\def\ef{\vec E}
 
\usepackage{fullpage}
 
\usepackage{fullpage}
 
\usepackage{amsmath}
 
\usepackage{amsmath}
 
\usepackage{eufrak}
 
\usepackage{eufrak}
 
\begin{document}
 
\begin{document}
\section*{ Radiation and Matter (to Order of Magnitude) }
 
  
 +
\section*{ Order of Magnitude Interaction of Radiation and Matter}
 +
 +
\subsection*{Energy Levels}
 +
\begin{itemize}
 +
\item Electronic Transitions:\par
 +
We'll start with the Bohr atom.  We begin by quantizing angular momentum:
 +
$$m_ev_ea_0=n\hbar$$
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If we balance the force required to keep the $e^-$ in a circular orbit
 +
with the electric force:
 +
$${m_ev_e^2\over a_0}={Ze^2\over a_0^2}$$
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$$a_0={\hbar^2\over m_ee^2Z}\approx0.52{\AA\over Z}$$
 +
A {\it Rydberg} is the energy required to ionize an H atom from the ground
 +
state.  It is $\sim13.6eV$.  We can estimate it by integrating the electric
 +
force from $r=a_0$ to $r=\infty$, but in reality, there is another factor of
 +
2:
 +
$$\boxed{Rydberg={Ze^2\over2a_0}={Z^2e^4m_e\over2\hbar^2}=13.6\cdot Z^2eV}$$
 +
\item Fine Structure:\par
 +
Fine Structure comes from the interaction of the magnetic moment of the $e^-$
 +
with the a $\bfield$ caused by the
 +
Lorentz-transformed Coulomb field of the proton (generated by the $e^-$'s
 +
motion).  The
 +
energy of a dipole interaction is $E=\vec\mu\cdot\bfield$, so we'd expect
 +
that:
 +
$$\Delta E\sim\mu B$$
 +
To estimate B: $B\sim{Ze\over a_0^2}{v\over c}$, and ${v\over c}\sim
 +
{e^2\over\hbar c}=\alpha=\inv{137}$ ($\alpha$ is the fine structure constant):
 +
$$B\sim{Ze^3\over\hbar ca_0^2}$$
 +
Estimating $\mu$: The $\bfield$ of a dipole goes as $B_{di}\sim{\mu\over r^3}$,
 +
so $\mu\sim B_e\eval{r_e}r_e^3$, where $r_e$ is the classical $e^-$ radius.
 +
Estimating that the rest mass energy of the $e^-$ should be about the
 +
electrostatic potential energy left in the $e^-$, $m_ec^2\sim{e\over r_e}$:
 +
$$r_e\sim{e^2\over m_ec^2}$$
 +
We can estimate $B_e$ by reverting to Maxwell's equation:
 +
$$\dcb={4\pi J\over c}$$
 +
$${B_e\over2\pi r_e}={4\pi\over c}{I\over4\pi r_e^2}$$
 +
Again, we estimate that $I\sim{e\over t_{spin}}$, and because the electron
 +
spin is quantized, $\hbar=m_er_e^2{2\pi\over t_{spin}}$.
 +
After some algebra, we get that:
 +
$$\mu_e={e\hbar\over2m_ec}$$
 +
\centerline{(Bohr magneton for $e^-$)}
 +
$$\mu_p={Ze\hbar\over2m_pc}$$
 +
\centerline{(Bohr magneton for nucleus)}
 +
Getting back to the energy:
 +
$$\begin{aligned}\Delta E&\sim{e\hbar\over2m_ec}{Ze\over a_0^2}\alpha\\
 +
&\sim Z^4\alpha^2\cdot Ryd\\ \end{aligned}$$
 +
 +
\item Hyperfine Structure:\par
 +
Instead of using the Bohr magneton, we use the intrinsic magnetic moment
 +
(spin) of the nucleus:
 +
$$B\eval{p}\sim{\mu_p\over a_0^3}\sim B_{fine}{m_e\over m_p}$$
 +
Thus:
 +
$$\Delta E\sim\Delta E_{fine}{m_e\over m_p}$$
 +
 +
Note that Fine and Hyperfine are magnetic dipole transitions.  $e^-$ level
 +
transitions are {\it electric} dipole transitions.  Magnetic dipole transitions
 +
are generally weaker.
 +
 +
\item  Vibrational Transitions in Molecules:\par
 +
Our general technique with vibrational transitions is to model them as
 +
harmonic oscillators.  Thus, they should have the characteristic harmonic
 +
energy series:
 +
$$E_n=(n+\hf)\hbar\wz$$
 +
For a harmonic oscillator, $\wz=\sqrt{k\over m}$.  We estimate that since
 +
the force for a spring is $k\cdot x$, and that force should be about the
 +
Coulomb force on $e^-$'s.  If we say that atoms stretch with respect to
 +
each other about a Bohr radius:
 +
$$ka_0\sim{e^2\over a_0^2}$$
 +
$$\Delta E\eval{vib\atop trans}\sim Ryd\cdot\sqrt{m_e\over A\cdot m_p}$$
 +
where A is the atomic mass \# of our atoms.
 +
 +
\item  Rotational Transitions in Molecules:\par
 +
The thing to remember is that angular momentum comes in units of $\hbar$.
 +
\end{itemize}
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Revision as of 17:11, 18 September 2013

Short Topical Videos

Reference Material

Order of Magnitude Interaction of Radiation and Matter

Energy Levels

  • Electronic Transitions:

We’ll start with the Bohr atom. We begin by quantizing angular momentum:

If we balance the force required to keep the in a circular orbit with the electric force:

A Rydberg is the energy required to ionize an H atom from the ground state. It is . We can estimate it by integrating the electric force from to , but in reality, there is another factor of 2:

  • Fine Structure:

Fine Structure comes from the interaction of the magnetic moment of the with the a caused by the Lorentz-transformed Coulomb field of the proton (generated by the ’s motion). The energy of a dipole interaction is , so we’d expect that:

To estimate B: , and ( is the fine structure constant):

Estimating : The of a dipole goes as , so , where is the classical radius. Estimating that the rest mass energy of the should be about the electrostatic potential energy left in the , :

We can estimate by reverting to Maxwell’s equation:

Again, we estimate that , and because the electron spin is quantized, . After some algebra, we get that:

(Bohr magneton for )

(Bohr magneton for nucleus) Getting back to the energy:

  • Hyperfine Structure:

Instead of using the Bohr magneton, we use the intrinsic magnetic moment (spin) of the nucleus:

Thus:

Note that Fine and Hyperfine are magnetic dipole transitions. level transitions are electric dipole transitions. Magnetic dipole transitions are generally weaker.

  • Vibrational Transitions in Molecules:

Our general technique with vibrational transitions is to model them as harmonic oscillators. Thus, they should have the characteristic harmonic energy series:

For a harmonic oscillator, . We estimate that since the force for a spring is , and that force should be about the Coulomb force on ’s. If we say that atoms stretch with respect to each other about a Bohr radius:

where A is the atomic mass # of our atoms.

  • Rotational Transitions in Molecules:

The thing to remember is that angular momentum comes in units of .