Core-Hole Lifetimes = FWHMs
Core-Hole Lifetimes = FWHM
Lifetime of a Core Hole = Core-Hole Lifetime
Γ = core-hole lifetime
Γ = ~10-15 seconds
Core-Hole Lifetime
Γτi = hbar
Γ = hbar/τi = (6.58 x 10-16)/τi
(seconds)
Γ = Γ x-ray + Γ Auger + Γ Coster-Kronig + ΓvibrationsÂ
Total FWHM = (FWHMx-ray2 + FWHMhemi-sphere broadening2 + FWHMelement2Â )1/2
Heisenberg’s Uncertainty Principle
ΔEΔt ≥ hbar
Core-Hole Lifetime Widths (FWHMs)
for subshell orbitals in same shell
Γs < Γp < Γd < Γf Â
for same subshell orbital
XPS Peak FWHM
(lifetime widths)
sFWHMÂ Â > pFWHMÂ Â > dFWHMÂ Â > fFWHM
Narrow FWHM means the lifetime of the hole is long.
Broad FWHM means that the lifetime of the hole is short.
NOTE:Â Â FWHM is inverse to core-hole lifetimes.
s, p, d and f orbital FWHM from PbCO3
| for PbCO3 | for PbCO3 | ||
| Shortest Lifetime | 4s FWHM =Â | 9.0 eV | |
| 4p3 FWHM = | 6.0 eV | ||
| 4d5 FWHM = | 3.5 eV | ||
| Longest Lifetime | 4f7 FWHM = | 1.2 eV |
Hole Lifetime Widths for Gases (Line width, meV)
from C. Nicolas and C Miron, J Electron Spectrosc. and Phenomena, Vol 185, p267-232 (2012)
Example Gas Spectra – Argon and Carbon Dioxide
from C. Nicolas and C Miron, J Electron Spectrosc. and Phenomena, Vol 185, p267-232 (2012)
INTRODUCTION
In the mid-1920s the German physicist Werner Heisenberg established his famous uncertainty principle, which states that the position and the momentum of a particle cannot be determined
simultaneously with unlimited accuracy. In an equivalent representation, this principle also applies to energy and time, via the inequality ΔEΔt ≥ hbar, where is the reduced Planck constant. A direct consequence of this relationship is widely used in spectroscopy and consists in the possibility to experimentally determine the lifetimes of excited states in atoms and molecules by mean of a spectroscopic line width measurement.
The specificity of inner-shell excitation or ionization is to produce highly excited species. In the general case, the decay to lower-lying states occurs either via radiative (X-ray fluorescence) or non-radiative (Auger decay or autoionization) processes. The lifetime Ï„ of the excited state is then proportional to the inverse of the sum of all partial transition rates to various final states. In the particular case of the two first rows atoms, the dominant de-excitation processes are represented by the Auger decay, in which one valence electron fills the core hole and a second one is ejected into the continuum.
In molecules, due to the localization of the core orbitals, only the part of the valence electrons wave functions close to the excited center will contribute to the Coulomb
matrix element describing the partial decay cross section (the one center approximation). In other words, the partial Auger decay rate will strongly depend on the given valence electron density in the vicinity of the core hole. Therefore electronegative ligands, which will deplete the electron density from the core hole location, are expected to lower the Auger rate and thus increase the lifetime of the corresponding core-excited or ionized state.
Moreover, to obtain the total Auger rate a summation over all the possible final states has to be performed. Consequently, larger the number of accessible states via the de-excitation, shorter the lifetime of the core hole, and this is the main reason for the decreasing core hole lifetime when moving from a free atom to the same atom in a molecule. However, this is only a general trend that might be subject to exceptions, as those related to the existence of symmetry based propensity rules.
Typical core hole lifetimes for K-shells and L-shells of light elements are lying in the femtosecond range and are too short to be directly measured routinely . In first approximation, in photoelectron or Auger electron spectroscopy, the Lorentzian line width is linked to the lifetime τ by the equation Γτi = hbar. However, when increasing the excitation energy from the photoionization threshold, and until the kinetic energy of the photoelectron becomes larger than that of the Auger electron, the interaction between these two particles via post-collision interaction (PCI) will further distort the spectral profile. Moreover, other broadening causes, such as the instrumental broadening itself (usually assumed to be Gaussian), or the Doppler broadening due to the thermal motion of the gas phase species, will also affect the line shape. To account for these contributions, the spectral line has to be modeled by a convolution between the asymmetric PCI-distorted line profile with a Gaussian profile including the monochromator bandwidth, the electron spectrometer bandwidth and the Doppler broadenings.
As far as the inner-shells of two first rows elements are concerned, typical line widths range from a few tens to a few hundreds of meV. From this intrinsic property of the system, interesting coherent phenomena may follow in core excitation/de-excitation, leading to interference process. One example is the so-called Lifetime Vibrational Interference (LVI), occurring in a core excitation process whenever the energy spacing of the vibrational levels is comparable to their natural line widths. Then inevitably several vibrational levels will be coherently excited and this will lead to interference between the various channels when the decay takes place toward the same electronic final states with their manifolds of vibrational states.
A second example is the Cohen-Fano type of interferences, which occur when a photoelectron is ejected from a delocalized molecular orbital. However, in the case of homonuclear diatomic molecules such interferences are only visible if the energy splitting between the symmetric (σg) and antisymmetric (σu) representations of the molecular orbitals is larger than the lifetime broadening. This is the case for the N2 molecule but similar observations have been also made recently in polyatomic molecules such as acetylene.
from Spectral Line page in Wikipedia
Spectral Line broadening and shift[edit]
There are a number of effects which control spectral line shape. A spectral line extends over a range of frequencies, not a single frequency (i.e., it has a nonzero linewidth). In addition, its center may be shifted from its nominal central wavelength. There are several reasons for this broadening and shift. These reasons may be divided into two general categories – broadening due to local conditions and broadening due to extended conditions. Broadening due to local conditions is due to effects which hold in a small region around the emitting element, usually small enough to assure local thermodynamic equilibrium. Broadening due to extended conditions may result from changes to the spectral distribution of the radiation as it traverses its path to the observer. It also may result from the combining of radiation from a number of regions which are far from each other.
Broadening due to local effects[edit]
Natural broadening[edit]
The lifetime of excited states results in natural broadening, also known as lifetime broadening. The uncertainty principle relates the lifetime of an excited state (due to spontaneous radiative decay or the Auger process) with the uncertainty of its energy. A short lifetime will have a large energy uncertainty and a broad emission. This broadening effect results in an unshifted Lorentzian profile. The natural broadening can be experimentally altered only to the extent that decay rates can be artificially suppressed or enhanced.[3]
Thermal Doppler broadening[edit]
The atoms in a gas which are emitting radiation will have a distribution of velocities. Each photon emitted will be “red”- or “blue”-shifted by the Doppler effect depending on the velocity of the atom relative to the observer. The higher the temperature of the gas, the wider the distribution of velocities in the gas. Since the spectral line is a combination of all of the emitted radiation, the higher the temperature of the gas, the broader the spectral line emitted from that gas. This broadening effect is described by a Gaussian profile and there is no associated shift.
Pressure broadening[edit]
The presence of nearby particles will affect the radiation emitted by an individual particle. There are two limiting cases by which this occurs:
- Impact pressure broadening or collisional broadening: The collision of other particles with the light emitting particle interrupts the emission process, and by shortening the characteristic time for the process, increases the uncertainty in the energy emitted (as occurs in natural broadening).[4] The duration of the collision is much shorter than the lifetime of the emission process. This effect depends on both the density and the temperature of the gas. The broadening effect is described by a Lorentzian profile and there may be an associated shift.
- Quasistatic pressure broadening: The presence of other particles shifts the energy levels in the emitting particle,[clarification needed] thereby altering the frequency of the emitted radiation. The duration of the influence is much longer than the lifetime of the emission process. This effect depends on the density of the gas, but is rather insensitive to temperature. The form of the line profile is determined by the functional form of the perturbing force with respect to distance from the perturbing particle. There may also be a shift in the line center. The general expression for the lineshape resulting from quasistatic pressure broadening is a 4-parameter generalization of the Gaussian distribution known as a stable distribution.[5]
Pressure broadening may also be classified by the nature of the perturbing force as follows:
- Linear Stark broadening occurs via the linear Stark effect, which results from the interaction of an emitter with an electric field of a charged particle at a distance {\displaystyle r}, causing a shift in energy that is linear in the field strength. {\displaystyle (\Delta E\sim 1/r^{2})}
- Resonance broadening occurs when the perturbing particle is of the same type as the emitting particle, which introduces the possibility of an energy exchange process. {\displaystyle (\Delta E\sim 1/r^{3})}
- Quadratic Stark broadening occurs via the quadratic Stark effect, which results from the interaction of an emitter with an electric field, causing a shift in energy that is quadratic in the field strength. {\displaystyle (\Delta E\sim 1/r^{4})}
- Van der Waals broadening occurs when the emitting particle is being perturbed by Van der Waals forces. For the quasistatic case, a Van der Waals profile[note 1] is often useful in describing the profile. The energy shift as a function of distance[definition needed] is given in the wings by e.g. the Lennard-Jones potential. {\displaystyle (\Delta E\sim 1/r^{6})}
Inhomogeneous broadening[edit]
Inhomogeneous broadening is a general term for broadening because some emitting particles are in a different local environment from others, and therefore emit at a different frequency. This term is used especially for solids, where surfaces, grain boundaries, and stoichiometry variations can create a variety of local environments for a given atom to occupy. In liquids, the effects of inhomogeneous broadening is sometimes reduced by a process called motional narrowing.
Broadening due to non-local effects[edit]
Certain types of broadening are the result of conditions over a large region of space rather than simply upon conditions that are local to the emitting particle.
Opacity broadening[edit]
Electromagnetic radiation emitted at a particular point in space can be reabsorbed as it travels through space. This absorption depends on wavelength. The line is broadened because the photons at the line center have a greater reabsorption probability than the photons at the line wings. Indeed, the reabsorption near the line center may be so great as to cause a self reversal in which the intensity at the center of the line is less than in the wings. This process is also sometimes called self-absorption.
Macroscopic Doppler broadening[edit]
Radiation emitted by a moving source is subject to Doppler shift due to a finite line-of-sight velocity projection. If different parts of the emitting body have different velocities (along the line of sight), the resulting line will be broadened, with the line width proportional to the width of the velocity distribution. For example, radiation emitted from a distant rotating body, such as a star, will be broadened due to the line-of-sight variations in velocity on opposite sides of the star. The greater the rate of rotation, the broader the line. Another example is an imploding plasma shell in a Z-pinch.
Radiative broadening[edit]
Radiative broadening of the spectral absorption profile occurs because the on-resonance absorption in the center of the profile is saturated at much lower intensities than the off-resonant wings. Therefore, as intensity rises, absorption in the wings rises faster than absorption in the center, leading to a broadening of the profile. Radiative broadening occurs even at very low light intensities.
Combined effects[edit]
Each of these mechanisms can act in isolation or in combination with others. Assuming each effect is independent, the observed line profile is a convolution of the line profiles of each mechanism. For example, a combination of the thermal Doppler broadening and the impact pressure broadening yields a Voigt profile.
However, the different line broadening mechanisms are not always independent. For example, the collisional effects and the motional Doppler shifts can act in a coherent manner, resulting under some conditions even in a collisional narrowing, known as the Dicke effect.