oscillator strength

An atom or a molecule can absorb light and undergo a transition from

one quantum state to another.

The oscillator strength $f\_\{12\}$ of a transition from a lower state

$|1\backslash rangle$ to an upper state $|2\backslash rangle$ may be defined by

:$$

f_{12} = \frac{2 }{3}\frac{m_e}{\hbar^2}(E_2 - E_1) \sum_{\alpha=x,y,z}

| \langle 1 m_1 | R_\alpha | 2 m_2 \rangle |^2,

where $m\_e$ is the mass of an electron and $\backslash hbar$ is

the reduced Planck constant. The quantum states $|n\backslash rangle,\; n=$ 1,2, are assumed to have several

degenerate sub-states, which are labeled by $m\_n$. "Degenerate" means

that they all have the same energy $E\_n$.

The operator $R\_x$ is the sum of the x-coordinates $r\_\{i,x\}$

of all $N$ electrons in the system, i.e.

:$$

R_\alpha = \sum_{i=1}^N r_{i,\alpha}.

The oscillator strength is the same for each sub-state $|n\; m\_n\backslash rangle$.

The definition can be recast by inserting the Rydberg energy $\backslash text\{Ry\}$ and Bohr radius $a\_0$

:$$

f_{12} = \frac{E_2 - E_1}{3\, \text{Ry}} \frac{\sum_{\alpha=x,y,z}

| \langle 1 m_1 | R_\alpha | 2 m_2 \rangle |^2}{a_0^2}.

In case the matrix elements of $R\_x,\; R\_y,\; R\_z$ are the same, we can get rid of the sum and of the 1/3 factor

:$$

f_{12} = 2\frac{m_e}{\hbar^2}(E_2 - E_1) \, | \langle 1 m_1 | R_x | 2 m_2 \rangle |^2.

To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum $\backslash boldsymbol\{p\}$. In absence of magnetic field, the Hamiltonian can be written as $H=\backslash frac\{1\}\{2m\}\backslash boldsymbol\{p\}^2+V(\backslash boldsymbol\{r\})$, and calculating a commutator $[H,x]$ in the basis of eigenfunctions of $H$ results in the relation between matrix elements

:$$

x_{nk}=-\frac{i\hbar/m}{E_n-E_k}(p_x)_{nk}.

.

Next, calculating matrix elements of a commutator $[p\_x,x]$ in the same basis and eliminating matrix elements of $x$, we arrive at

:$$

\langle n|[p_x,x]|n\rangle=\frac{2i\hbar}{m}\sum_{k\neq n} \frac{|\langle n|p_x|k\rangle|^2}{E_n-E_k}.

Because $[p\_x,x]=-i\backslash hbar$, the above expression results in a sum rule

:$$

\sum_{k\neq n}f_{nk}=1,\,\,\,\,\,f_{nk}=-\frac{2}{m}\frac{|\langle n|p_x|k\rangle|^2}{E_n-E_k},

where $f\_\{nk\}$ are oscillator strengths for quantum transitions between the states $n$ and $k$. This is the Thomas-Reiche-Kuhn sum rule, and the term with $k=n$ has been omitted because in confined systems such as atoms or molecules the diagonal matrix element $\backslash langle\; n|p\_x|n\backslash rangle=0$ due to the time inversion symmetry of the Hamiltonian $H$. Excluding this term eliminates divergency because of the vanishing denominator.{{cite book|author1=Edward Uhler Condon|author2=G. H. Shortley|title=The Theory of Atomic Spectra|url=https://books.google.com/books?id=hPyD-Nc_YmgC|accessdate=26 July 2013|year=1951|publisher=Cambridge University Press|isbn=978-0-521-09209-8|page=108}}

In crystals, the electronic energy spectrum has a band structure $E\_n(\backslash boldsymbol\{p\})$. Near the minimum of an isotropic energy band, electron energy can be expanded in powers of $\backslash boldsymbol\{p\}$ as $E\_n(\backslash boldsymbol\{p\})=\backslash boldsymbol\{p\}^2/2m^*$ where $m^*$ is the electron effective mass. It can be shown{{cite journal |doi=10.1103/PhysRev.97.869|title=Motion of Electrons and Holes in Perturbed Periodic Fields|journal=Physical Review|volume=97|issue=4|pages=869|year=1955|last1=Luttinger|first1=J. M.|last2=Kohn|first2=W.|bibcode=1955PhRv...97..869L}} that it satisfies the equation

:$$

\frac{2}{m}\sum_{k\neq n}\frac{|\langle n|p_x|k\rangle|^2}{E_k-E_n}+\frac{m}{m^*}=1.

Here the sum runs over all bands with $k\backslash neq\; n$. Therefore, the ratio $m/m^*$ of the free electron mass $m$ to its effective mass $m^*$ in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the $n$ band into the same state.{{cite book|publisher=Springer|location=Berlin | doi=10.1007/978-3-642-91116-3_3| chapter=Elektronentheorie der Metalle| title=Aufbau Der Zusammenhängenden Materie| pages=333| year=1933| last1=Sommerfeld| first1=A.| last2=Bethe| first2=H.| isbn=978-3-642-89260-8}}

• Atomic spectral line
• Sum rule in quantum mechanics
• Electronic band structure
• Einstein coefficients

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Category:Spectroscopy

Category:Atoms

Category:Crystals