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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|^2} term, which is appreciable only for very intense fields, we now consider more fully the dominant term in the atom-field interaction,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A (r) = A\hat{z} e^{ikx}. }

Expanding the exponential, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | b \rangle} have the same parity, then the second term in the parentheses becomes important. Usually, it is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} times smaller. In particular, since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k r \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_z x = (p_z x - zp_x )/2 + (p_z x + zp_x )/2 . }

The first term of Eq. \ref{eq:hor4} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \hbar L_y/2} , and the matrix element becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{ieAk}{2 m} \langle b | \hbar L_y | a \rangle = - iAk \langle b | \mu_B L_y | a \rangle, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B = e\hbar /2 m} is the Bohr magneton. The magnetic field is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = - i k A \hat{y}} . Consequently, Eq.\ \ref{eq:hor5} can be written in the more familiar form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\vec{\mu} \cdot B} . (The orbital magnetic moment is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mu} = -\mu_B L} : the minus sign arises from our convention that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is positive.) We can readily generalize the matrix element to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int}(M1) = B \cdot \mu_B\langle b |L + g_sS| a\rangle, }

where we have added the spin dependent term from Eq. \ref{eq:hor_Hint}. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} indicates that the matrix element is for a magnetic dipole transition. The strength of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} transition is set by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B/c = \frac{1}{2}\frac{e\hbar}{mc}=\frac{1}{2}\frac{e^2}{\hbar c}\frac{\hbar^2}{e m} = \frac{1}{2}\alpha e a_0, }

so it is indeed a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [ r, H_0 ] = i\hbar p / m } , we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} (p_z x + z p_x) = \frac{m}{2i\hbar} ([z, H_0 ] x+ z[x, H_0 ]) = \frac{m}{2i\hbar} (- H_0 zx +zx H_0 ). }

So, the contribution of this term to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ba}} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ieA}{mc} \frac{km}{2\hbar i} \langle b | - H_0 zx + zx H_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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = i k A} . This is an electric quadrupole interaction, and we shall denote the matrix element by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle b |r:r|a\rangle} of a tensor product. It is straightforward to verify that the electric quadrupole interaction is also of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int}^{(2)} = H_{\rm int} (M1) + H_{\rm int} (E2). }

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int} (M1)} is real, whereas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm int} (E2)} is imaginary. Consequently,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |H_{ba} |^2} , the magnetic dipole and electric quadrupole rates are both smaller than the dipole rate by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -er}	-
Magnetic Dipole	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\mu_B(L+g_sS)}	+
Electric Quadrupole	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int}(T_{l,m}) = \langle n J M | T_{l,m} | n' J' M'\rangle, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{l,m}} is a spherical tensor operator of rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} . The operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{l,m}} transform under rotations like the spherical harmonics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle n J M | T_{l,m} | n' J' M'\rangle = \frac{\langle n J \| T_l \| n' J' \rangle}{\sqrt{2J+1}}\langle J' l, M', m| J M\rangle }

in terms of a reduced matrix element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle n J \| T_l \| n' J' \rangle} and a Clebsch-Gordan coefficient \linebreakFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J' l, M', m| J M\rangle} . In order for the latter to be nonzero, the triangle rule requires that \linebreakFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |J'-J| \leq l \leq |J'+J|} , while conservation of angular momentum requires $M=M'+m$. Since the operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle er} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B B} responsible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M1} transitions are both vectors, i.e. tensors of rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l=1} , these transitions are both governed by the dipole selection rules

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} |\Delta J| &= 0, 1;\\ |\Delta m| &= 0, 1. \end{align}}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bf r} is a polar vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bf L} is an axial vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E1} transitions are allowed only between states of opposite parity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} transitions are allowed only between states of the same parity. The operator responsible for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E2} transitions is a spherical tensor of rank 2. For example,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xz = (T_{2,-1}-T_{2,1})/4. }

Thus, electric quadrupole transitions are allowed only between states connected by tensors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2,m}(r)} , requiring:

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

In addition, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=0\rightarrow J'=0} transitions are forbidden in all of the cases considered above, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=J'=0} requires Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta L=0} (for any interaction that does not couple to spin) whereas absorption or emission of a photon implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\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.