• R ab —the distance between absorbing atom a and backscattering atom b. Since the
phase of the backscattered wave depends on the distance travelled R ab , absent
other terms, the EXAFS would follow sin(2kR ab ). The amplitude of the EXAFS
diminishes as
1
R
2
ab
.
The EXAFS effect also depends on the type of neighbor. This is because the
strength of the scattering depends on the atomic number of the neighbor, through:
• |f b (π,k)|—the electron-atom backscattering amplitude
The type of neighbor also affects the phase of the scattered wave through:
• β b (k)—the electron-atom backscattering phase shift
The type of absorbing atom also contributes a phase shift through:
• α a (k)—the central atom phase shift
Of course, if there is more than one neighbor at the same distance, then the effect
will be multiplied by:
• N b —the number of scattering atoms at a particular distance.
If the interatomic distances are slightly different, then the sine waves from
different species in the sample will be slightly out of phase. For a Gaussian
distribution of such distances with an rms deviation σ ab , this reduces the EXAFS
by:
• exp(À2σ ab
2 k
2 )—the Debye-Waller factor.
Finally, there are inelastic effects that can rob EXAFS intensity, including a
damped exponential dependence of the photoelectron mean free path λ:
• exp(À2R ab /λ)
and an additional reduction by:
• S 0
2 (k)—a factor that includes losses due to multi-electron excitations.
We now look in detail at the contributions of each of thse terms.
6.4.1 Scattering Functions
In the above equation, which describes EXAFS as a sum of damped sine waves, to
first order, there are only three terms that affect the frequency and phase of the
oscillations. The most important term is of course the information that we want—the
absorber-backscatterer distance R ab . The other two terms are so-called phase shifts
that prevent EXAFS analysis from being a slam-dunk procedure. These phase shifts
arise as the electron wave propagates across the potentials of the absorbing atom and
the backscattering atom. Although one can find published values for phase shifts for
different atoms [225,226], nowadays, they are usually calculated as needed within
larger software packages such as FEFF [227,228].
6.4 Single Scattering EXAFS Equation
143
phase of the backscattered wave depends on the distance travelled R ab , absent
other terms, the EXAFS would follow sin(2kR ab ). The amplitude of the EXAFS
diminishes as
1
R
2
ab
.
The EXAFS effect also depends on the type of neighbor. This is because the
strength of the scattering depends on the atomic number of the neighbor, through:
• |f b (π,k)|—the electron-atom backscattering amplitude
The type of neighbor also affects the phase of the scattered wave through:
• β b (k)—the electron-atom backscattering phase shift
The type of absorbing atom also contributes a phase shift through:
• α a (k)—the central atom phase shift
Of course, if there is more than one neighbor at the same distance, then the effect
will be multiplied by:
• N b —the number of scattering atoms at a particular distance.
If the interatomic distances are slightly different, then the sine waves from
different species in the sample will be slightly out of phase. For a Gaussian
distribution of such distances with an rms deviation σ ab , this reduces the EXAFS
by:
• exp(À2σ ab
2 k
2 )—the Debye-Waller factor.
Finally, there are inelastic effects that can rob EXAFS intensity, including a
damped exponential dependence of the photoelectron mean free path λ:
• exp(À2R ab /λ)
and an additional reduction by:
• S 0
2 (k)—a factor that includes losses due to multi-electron excitations.
We now look in detail at the contributions of each of thse terms.
6.4.1 Scattering Functions
In the above equation, which describes EXAFS as a sum of damped sine waves, to
first order, there are only three terms that affect the frequency and phase of the
oscillations. The most important term is of course the information that we want—the
absorber-backscatterer distance R ab . The other two terms are so-called phase shifts
that prevent EXAFS analysis from being a slam-dunk procedure. These phase shifts
arise as the electron wave propagates across the potentials of the absorbing atom and
the backscattering atom. Although one can find published values for phase shifts for
different atoms [225,226], nowadays, they are usually calculated as needed within
larger software packages such as FEFF [227,228].
6.4 Single Scattering EXAFS Equation
143