Electric Quadrupole Transitions

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

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

where

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

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

$${\mit\Delta} l = 0,\pm 2$$ (8.215)

$${\mit\Delta} m = 0,\pm 1,\pm 2,$$ (8.216)

$${\mit\Delta} m_s =0,$$ (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

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

$${\mit\Delta} m_j = 0,\pm 1,\pm 2,$$ (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

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

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