Selection rules

The selection rules in spectroscopy are the restrictions and the prohibition of transitions between the levels of a quantum-mechanical system with absorption or emission of a photon imposed by conservation laws and symmetry.

Content

Dipole and multipole transitions

Optical transitions between the levels of a quantum-mechanical system are classified according to multipolarity: dipole transitions, quadrupole transitions, octupole transitions, etc. These are the so-called electrical transitions. In addition, there are magneto-dipole transitions, and, accordingly, magneto-quadrupole transitions, etc. Usually, dipole transitions in intensity follow before quadrupole, quadrupole before octupole - the higher the multipole, the weaker the quantum mechanical system interacts with light. But if the matrix element of the dipole transition is equal to zero, transitions of higher multipolarity are also observed. Magnetic dipole transitions are less intense than electric dipole transitions, but more intense than electric quadrupole transitions. Accordingly, electric quadrupole ones are more intense than magneto-quadrupole transitions, and those, in turn, are electric octupole transitions, etc.

Conditional spectroscopic designations of transitions are as follows: E1 is an electric dipole transition, E2 is an electric quadrupole transition, E3 is an octupole transition, etc .; M1 is the magnetic dipole transition, M2 is the magnetic quadrupole transition, etc.

The matrix element of the dipole transition is defined as $-\langle f|e\mathbf {r} |i\rangle$ ${\ displaystyle - \ langle f | e \ mathbf {r} | i \ rangle}$ where $|i\rangle$ ${\ displaystyle | i \ rangle}$ the wave function of the initial state of the system, and $|f\rangle$ ${\ displaystyle | f \ rangle}$ Is the wave function of the final state of the system in the notation of sconce and ket vectors, e is the electron charge, and $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ - radius vector. By analogy, the matrix element of the magnetic dipole transition is determined, namely $-\langle f|\mu (2\mathbf {S} +\mathbf {L} )|i\rangle$ ${\ displaystyle - \ langle f | \ mu (2 \ mathbf {S} + \ mathbf {L}) | i \ rangle}$ where $\mathbf {S}$ ${\ displaystyle \ mathbf {S}}$ - spin operator, $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ - operator of the orbital momentum.

Transitions between levels are called allowed transitions if the matrix element of the dipole transition is nonzero. In this case, the spectral lines are intense. Transitions between levels are called forbidden transitions if the matrix element of the dipole transition is zero. Despite the name, forbidden transitions can occur due to higher multipole or in the presence of third bodies. Their spectral intensity is less.

Harmonic Oscillator

The allowed transitions of the harmonic oscillator satisfy the selection rule:

n_{f}=n_{i}\pm 1

{\ displaystyle n_ {f} = n_ {i} \ pm 1}

,

where n _f and n _i are the quantum numbers of the final and initial state, respectively. That is, transitions can only occur between neighboring states. Considering that the states of a harmonic oscillator are equidistant, this leads to the existence of a single line in the emission or absorption spectrum.

Magnetic quantum number

For magnetic quantum number

\Delta m=0,\,\pm 1

{\ displaystyle \ Delta m = 0, \, \ pm 1}

.

The light that is emitted when switching from $\Delta m=0$ ${\ displaystyle \ Delta m = 0}$ linearly polarized . When going from $\Delta m=\pm 1$ ${\ displaystyle \ Delta m = \ pm 1}$ circularly polarized light is emitted.

Total moment quantum number

For the quantum number of the total moment of a multielectron system

\Delta J=0,\pm 1

{\ displaystyle \ Delta J = 0, \ pm 1}

.

In addition, transitions between states in which both quantum numbers of the total moment are equal to zero are forbidden.

Orbital quantum number

For orbital quantum number

\Delta L=\pm 1

{\ displaystyle \ Delta L = \ pm 1}

.

If we talk about multi-electron systems in atoms, then the following selection rules should be considered:
1. Transitions between terms of different multiplicity are prohibited.
2. Magnetic-dipole transitions are forbidden if the radial quantum number changes.
3. EL transitions have parity (-1) ^L , transitions ML - (-1) ^{L + 1} .
4. For the transitions EL and ML, the inequality $\ |J^{1}-J^{2}|\leq \Delta L\leq J^{1}+J^{2}$ ${\ displaystyle \ | J ^ {1} -J ^ {2} | \ leq \ Delta L \ leq J ^ {1} + J ^ {2}}$ ,Where $\Delta L$ ${\ displaystyle \ Delta L}$ - change in the orbital quantum number, $\ J^{1}$ ${\ displaystyle \ J ^ {1}}$ and $\ J^{2}$ ${\ displaystyle \ J ^ {2}}$ - start and end full moment.

Literature

Bili M.U. Atomic physics. - Kiev: Vishka school, 1973.
Serbo V.G., Khriplovich I. B. Quantum mechanics. NSU, 2008 .-- 274 p.