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\rangle to an upper state |2\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 \hbar is the reduced Planck constant. The quantum states |n\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\rangle. The definition can be recast by inserting the Rydberg energy \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. == Thomas–Reiche–Kuhn sum rule ==

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 \boldsymbol{p}. In absence of magnetic field, the Hamiltonian can be written as H=\frac{1}{2m}\boldsymbol{p}^2+V(\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\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 \langle n|p_x|n\rangle=0 due to the time inversion symmetry of the Hamiltonian H. Excluding this term eliminates divergency because of the vanishing denominator. == Sum rule and electron effective mass in crystals ==

In crystals, the electronic energy spectrum has a band structure E_n(\boldsymbol{p}). Near the minimum of an isotropic energy band, electron energy can be expanded in powers of \boldsymbol{p} as E_n(\boldsymbol{p})=\boldsymbol{p}^2/2m^* where m^* is the electron effective mass. It can be shown 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\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. == See also ==