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Electric Quadrupole Transitions

According to Equation (8.195), the quantity that mediates spontaneous electric quadrupole transitions is

$\displaystyle \mbox{\boldmath$\epsilon$}$$\displaystyle \cdot{\bf d}_{if} =\frac{{\rm i}\,\omega_{if}}{2\,c}\,$$\displaystyle \mbox{\boldmath$\epsilon$}$$\displaystyle \cdot{\bf Q}_{if}\cdot{\bf n},$ (8.212)

where

$\displaystyle ({\bf Q}_{if})_{jk} = \langle i\vert\,x_j\,x_k - r^{\,2}\,\delta_{jk}/3\,\vert f\rangle.$ (8.213)

is the electric quadrupole matrix element. It follows, by analogy with with Equation (8.165), that the spontaneous emission rate associated with an electric quarrupole transition is

$\displaystyle \frac{dw_{i\rightarrow f}^{\,\rm spn}}{d{\mit\Omega}}= \frac{\alpha\,\omega_{if}^{\,5}}{8\pi\,\,c^{\,4}} \sum_{j=1,2}\vert$$\displaystyle \mbox{\boldmath$\epsilon$}$$\displaystyle _j\cdot {\bf Q}_{if}\cdot{\bf n}\vert^{\,2}.$ (8.214)

Here, $ d{\mit\Omega}$ is the solid angle associated with the direction of the emitted photon's normalized wavevector, $ {\bf n}$ . Moreover, $ \epsilon$ $ _{1,2}$ are the photon's two independent electric polarization vectors.

The selection rules for electric quadrupole transitions in a hydrogen-like atom are [23]

$\displaystyle {\mit\Delta} l$ $\displaystyle = 0,\pm 2$ (8.215)
$\displaystyle {\mit\Delta} m$ $\displaystyle = 0,\pm 1,\pm 2,$ (8.216)
$\displaystyle {\mit\Delta} m_s$ $\displaystyle =0,$ (8.217)

where $ l$ is the azimuthal quantum number, $ m$ the magnetic quantum number, and $ m_s$ the spin quantum number. A more general form of these selection rules is

$\displaystyle {\mit\Delta} j$ $\displaystyle = 0,\pm 1,\pm 2,$ (8.218)
$\displaystyle {\mit\Delta} m_j$ $\displaystyle = 0,\pm 1,\pm 2,$ (8.219)

where $ j$ and $ m_j$ are the standard quantum numbers associated with the total angular momentum of the system. Note, however, that electric quadrupole transitions between two $ j=0$ states, or between two $ j=1/2$ states, or between a $ j=1$ state and a $ j=0$ state, are forbidden [23].

Let us estimate the typical spontaneous emission rate for an electric quadrupole transition in a hydrogen atom. We expect the matrix element $ {\bf Q}_{if}$ , defined in Equation (8.216), to be of order $ a_0^{\,2}$ , where $ a_0$ is the Bohr radius. We also expect $ \omega_{if}$ to be of order $ \vert E_0\vert/\hbar$ , where $ E_0$ is the hydrogen ground-state energy. It thus follows from Equation (8.217) that

$\displaystyle \frac{dw_{i\rightarrow f}^{\,\rm spn}}{d{\mit\Omega}} \sim \alpha^{\,5}\,\omega_{if}\sim \alpha^{\,7}\,\frac{m_e\,c^{\,2}}{\hbar},$ (8.220)

which is of order $ \alpha^{\,2}$ times smaller than a typical electric dipole transition rate. [See Equations (8.180).]


next up previous
Next: Photo-Ionization Up: Time-Dependent Perturbation Theory Previous: Magnetic Dipole Transitions
Richard Fitzpatrick 2016-01-22