Selection rules

Definition

Selection rules refer to the conditions under which the quantum mechanical transition matrix elements for a process are different from zero (and hence the process is allowed), due to constraints derived from the symmetry properties of the states involved and those of the transition operator. Since the set of symmetries for a quantum system form a group, one can classify the transformation properties of the states (eigenfunctions) and the transition operator according to the irreducible representations of the group itself. In such a case the Wigner–Eckart theorem dictates the conditions for the process to occur (see any standard text on group theory).

In the most common example for transition theory, if we classify the system states (eigenfunctions) according to the full rotation group and expand the transition operator in multipoles as

[math]\mathbf{p}e^{i\mathbf{k \cdot r}} = \sum\limits_l f_l(r)\mathbf{Y}_l(\hat{\mathbf{r}})[/math]

then the angular part of the matrix element will be proportional to [math]\langle J_f|Y_l|J_i\rangle[/math] where [math]J=L+S \quad (j=l+s)[/math] in a Russell–Saunders coupling scheme. Therefore, from addition of angular momenta and the fact that the transition operator does not affect the spin, we derive

[math]|J_f - J_i| \le l \le J_f + J_i[/math]

[math]|L_f - L_i| \le l \le L_f + L_i[/math]

[math]\Delta S=0; \Delta M_S = 0; \Delta M_j = m_i,[/math]

which provide a set of selection rules in this case. [math]J_f=0[/math] to [math]J_i=0[/math] transitions are forbidden by the above rule, since the lowest multipole operator (dipole radiation) is a vector ([math]l=1; m_l =\pm 1,0[/math]). Conditions on symmetries include those relating to parity, orbital angular momentum quantum number, spin quantum number, (multi-)polarity of the photon field causing the transition, polarization of the photon field causing the transition, etc. For summation of angular momenta, such as for molecular levels; hyperfine structure including nuclear angular momenta; or coupling within Russell–Saunders, j–j or mixed schemes, a vector triangle summation must generally be followed as above, which provides most selection rules. Different polarizations have different selection rules, so an edge or XAFS spectrum using (polarized) synchrotron radiation will have a different shape and structure depending upon whether the incident X-ray field is linearly polarized, circularly polarized or partially polarized.

Electric dipole transition is only the dominant lowest-order transition coupling, tending to be dominant for low energies or low-Z elements (even in compounds). However, for such elements as transition metals, higher-order terms including electric quadrupole radiation and magnetic dipole radiation become stronger and have complementary selection rules for atomic, molecular and condensed matter quantum systems. Higher-order radiation is crucial for the interpretation of [math]K\alpha[/math] spectra satellites, for continuum photoionization amplitudes and XAFS and absorption edges, and for pre-edge features.

History

In widespread tabulations of photoeffect for atomic systems (viz. Creagh in International Tables for X-ray Crystallography, Volume C, FFAST in the USA and XCOM/Hubbell), some authors have clarified the significance in the computations of dipole, higher-order or 'all-order’ computations.