In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of
absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
or
emission
Emission may refer to:
Chemical products
* Emission of air pollutants, notably:
**Flue gas, gas exiting to the atmosphere via a flue
** Exhaust gas, flue gas generated by fuel combustion
** Emission of greenhouse gases, which absorb and emit radi ...
of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
in transitions between
energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...
s of an atom or molecule.
For example, if an emissive state has a small oscillator strength,
nonradiative decay will outpace
radiative decay. Conversely, "bright" transitions will have large oscillator strengths. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition.
Theory
An atom or a molecule can absorb light and undergo a transition from
one quantum state to another.
The oscillator strength
of a transition from a lower state
to an upper state
may be defined by
:
where
is the mass of an electron and
is
the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. The
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s
1,2, are assumed to have several
degenerate sub-states, which are labeled by
. "Degenerate" means
that they all have the same energy
.
The operator
is the sum of the x-coordinates
of all
electrons in the system, etc.:
:
The oscillator strength is the same for each sub-state
.
The definition can be recast by inserting the
Rydberg energy
In spectroscopy, the Rydberg constant, symbol R_\infty for
heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
and
Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
:
In case the matrix elements of
are the same, we can get rid of the sum and of the 1/3 factor
:
Thomas–Reiche–Kuhn sum rule
To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum
. In absence of magnetic field, the Hamiltonian can be written as
, and calculating a commutator