Higher-Order Radiation Processes

▼ Category:8.421

► Category:Atoms in electric fields

  Atoms

  Atoms in electric fields

  Atoms in Magnetic Fields

  Coherence

  Effects of the Nucleus on Atomic Structure

  Fine Structure and Lamb Shift

  Higher-Order Radiation Processes

  Interaction of an atom with an electromagnetic field

  Old Coherence

  Resonances

  Two-Photon Excitation

Beyond the dipole approximation

Recall that the interaction Hamiltonian for an atom in an electromagnetic field is given by

$H_{\mathrm {int} }=-{\frac {e}{mc}}p\cdot A+{\frac {e^{2}}{2mc^{2}}}|A|^{2}+g_{s}\mu _{B}S\cdot ({\bf {\nabla }}\times A),$

where the last term we have so far considered only for static magnetic fields. Neglecting, as before, the $|A|^{2}$ term, which is appreciable only for very intense fields, we now consider more fully the dominant term in the atom-field interaction,

$H_{ba}={\frac {e}{\rm {mc}}}\langle b|p\cdot A(r)|a\rangle .$

For concreteness, we shall take A(r) to be a plane wave of the form

$A(r)=A{\hat {z}}e^{ikx}.$

Expanding the exponential, we have

$H_{ba}={\frac {eA}{\rm {mc}}}\langle b|p_{z}(1+ikx+(ikx)^{2}/2+\dots )|a\rangle .$

Thus far in the course, we have considered only the first term, the dipole term. If dipole radiation is forbidden, for instance if $|a\rangle$ and $|b\rangle$ have the same parity, then the second term in the parentheses becomes important. Usually, it is $\alpha$ times smaller. In particular, since

$kr\approx {\frac {\hbar \omega }{\hbar c}}a_{0}\approx {\frac {e^{2}/a_{0}}{\hbar c}}a_{0}\approx {\frac {e^{2}}{\hbar c}}=\alpha ,$

the expansion in \ref{eq:hor3} is effectively an expansion in $\alpha$. We can rewrite the second term as follows:

$p_{z}x=(p_{z}x-zp_{x})/2+(p_{z}x+zp_{x})/2.$

The first term of Eq. \ref{eq:hor4} is $-\hbar L_{y}/2$, and the matrix element becomes

$-{\frac {ieAk}{2m}}\langle b|\hbar L_{y}|a\rangle =-iAk\langle b|\mu _{B}L_{y}|a\rangle ,$

where $\mu _{B}=e\hbar /2m$ is the Bohr magneton. The magnetic field is $B=-ikA{\hat {y}}$. Consequently, Eq.\ \ref{eq:hor5} can be written in the more familiar form $-{\vec {\mu }}\cdot B$. (The orbital magnetic moment is ${\vec {\mu }}=-\mu _{B}L$: the minus sign arises from our convention that $e$ is positive.) We can readily generalize the matrix element to

$H_{\rm {int}}(M1)=B\cdot \mu _{B}\langle b|L+g_{s}S|a\rangle ,$

where we have added the spin dependent term from Eq. \ref{eq:hor_Hint}. $M1$ indicates that the matrix element is for a magnetic dipole transition. The strength of the $M1$ transition is set by

$\mu _{B}/c={\frac {1}{2}}{\frac {e\hbar }{mc}}={\frac {1}{2}}{\frac {e^{2}}{\hbar c}}{\frac {\hbar ^{2}}{em}}={\frac {1}{2}}\alpha ea_{0},$

so it is indeed a factor of $\alpha$ weaker than a dipole transition, as we argued above.

The second term in Eq.\ \ref{eq:hor4} involves $(p_{z}x+zp_{x})/2$. Making use of the commutator relation $[r,H_{0}]=i\hbar p/m$, we have

${\frac {1}{2}}(p_{z}x+zp_{x})={\frac {m}{2i\hbar }}([z,H_{0}]x+z[x,H_{0}])={\frac {m}{2i\hbar }}(-H_{0}zx+zxH_{0}).$

So, the contribution of this term to $H_{ba}$ is

${\frac {ieA}{mc}}{\frac {km}{2\hbar i}}\langle b|-H_{0}zx+zxH_{0}|a\rangle =-{\frac {eAk}{2c}}{\frac {E_{b}-E_{a}}{\hbar }}\langle b|zx|a\rangle ={\frac {ieE\omega }{2c}}\langle b|zx|a\rangle ,$

where we have taken $E=ikA$. This is an electric quadrupole interaction, and we shall denote the matrix element by

$H_{\rm {int}}(E2)^{\prime }={\frac {ie\omega }{2c}}\langle b|zx|a\rangle E.$

The prime indicates that we are considering only one component of a more general expression involving the matrix element $\langle b|r:r|a\rangle$ of a tensor product. It is straightforward to verify that the electric quadrupole interaction is also of order $\alpha$.

The total matrix element of the second term in the expansion of Eq.\ \ref{eq:hor3} can be written

$H_{\rm {int}}^{(2)}=H_{\rm {int}}(M1)+H_{\rm {int}}(E2).$

Note that $H_{\rm {int}}(M1)$ is real, whereas $H_{\rm {int}}(E2)$ is imaginary. Consequently,

$|H_{\rm {int}}^{(2)}|^{2}=|H_{\rm {int}}(M1)|^{2}+|H_{\rm {int}}(E2)|^{2}.$

The magnetic dipole and electric quadrupole terms do not interfere.

Because transition rates depend on $|H_{ba}|^{2}$, the magnetic dipole and electric quadrupole rates are both smaller than the dipole rate by $\alpha ^{2}\sim 5\times 10^{-5}$. For this reason they are generally referred to as {\it forbidden} processes. However, the term is used somewhat loosely, for there are transitions which are much more strongly suppressed due to other selection rules, as for instance triplet to singlet transitions in helium. \begin{table}

Transition		Operator	Parity
Electric Dipole	$E1$	$-er$	-
Magnetic Dipole	$M1$	$-\mu _{B}(L+g_{s}S)$	+
Electric Quadrupole	$E2$	$-er:r$	+

\caption{Summary of dipole and higher-order radiation processes.} \end{table}

Selection rules

A forbidden transition, then, is one that is weaker than an electric dipole-allowed transition by $\alpha ^{n}$ and only appears in some higher-order approximation. Examples of such higher-order effects are the magnetic dipole and electric quadrupole terms described above, multiphoton processes, the relativistic effects which allow singlet to triplet transitions in helium, and hyperfine interactions within the nucleus. To derive selection rules for the transitions we have discussed above, it is useful to express the matrix elements in terms of spherical tensor operators:

$H_{int}(T_{l,m})=\langle nJM|T_{l,m}|n'J'M'\rangle ,$

where $T_{l,m}$ is a spherical tensor operator of rank $l$. The operators $T_{l,m}$ transform under rotations like the spherical harmonics $Y_{l,m}$, and any operator can be written as a linear combination of these spherical tensors. By the Wigner-Eckart Theorem, we can express the matrix element

$\langle nJM|T_{l,m}|n'J'M'\rangle ={\frac {\langle nJ\|T_{l}\|n'J'\rangle }{\sqrt {2J+1}}}\langle J'l,M',m|JM\rangle$

in terms of a reduced matrix element $\langle nJ\|T_{l}\|n'J'\rangle$ and a Clebsch-Gordan coefficient \linebreak$\langle J'l,M',m|JM\rangle$. In order for the latter to be nonzero, the triangle rule requires that \linebreak$|J'-J|\leq l\leq |J'+J|$, while conservation of angular momentum requires $M=M'+m$. Since the operators $er$ and $\mu _{B}B$ responsible for $E1$ and $M1$ transitions are both vectors, i.e. tensors of rank $l=1$, these transitions are both governed by the dipole selection rules

${\begin{aligned}|\Delta J|&=0,1;\\|\Delta m|&=0,1.\end{aligned}}$

Since ${\bf {r}}$ is a polar vector and ${\bf {L}}$ is an axial vector, $E1$ transitions are allowed only between states of opposite parity and $M$ transitions are allowed only between states of the same parity. The operator responsible for $E2$ transitions is a spherical tensor of rank 2. For example,

$xz=(T_{2,-1}-T_{2,1})/4.$

Thus, electric quadrupole transitions are allowed only between states connected by tensors $T_{2,m}(r)$, requiring:

${\begin{aligned}|\Delta J|&=0,1,2;\\|\Delta m|&=0,1,2.\end{aligned}}$

In addition, $J=0\rightarrow J'=0$ transitions are forbidden in all of the cases considered above, since $J=J'=0$ requires $\Delta L=0$ (for any interaction that does not couple to spin) whereas absorption or emission of a photon implies $|\Delta L|=1$.

 This discussion of matrix elements, selection rules, and radiative
 processes barely skims the subject.  For an authoritative
 treatment, the books by Shore and Menzel and by Sobelman are
 recommended, as well as "Angular Momentum" by D.M. Brink and G.R. Satchler.