Cross-section
From Online Dictionary of Crystallography
Section efficace (Fr). Wirkungsquerschnitt (Ge). Sezione d'urto (It). 反応断面積 (Ja). Эффективное поперечное сечение (Ru). Sección eficaz (Sp).
Definition
Cross-section is a measure of the probability of interaction between the incident photons with the material via photoabsorption or scattering processes. It is the effective area that will yield a transition process for a perpendicularly incident flux of one particle per unit area. The total interaction cross-section [math]\sigma_{tot}[/math] is usually represented as a sum over the individual photon interaction cross-sections (per atom):
[math]\sigma_{tot} = \sigma_{coh} + \sigma_{incoh} +\tau_{PE} + \kappa_n + \kappa_e + \sigma_{p.n.}[/math]
[following standard tabulations] or
[math]\sigma_{tot} = \sigma_{coh} + \sigma_{incoh} +\sigma_{PE} + \sigma_n + \sigma_e + \sigma_{p.n.}[/math]
[following IUCr International Tables nomenclature].
The components are, in order, the coherent scattering cross-section, the incoherent scattering cross-section, the photoelectric absorption cross-section, the nuclear (coherent) pair-production cross-section, the nuclear incoherent (or triplet) cross-section and the photonuclear cross-section. These cross-sections are conventionally given in barns/atom (1 barn = [math]10^{−28}[/math] m2 = 100 fm2). For most crystallographic applications the energy range of interest is [math]3\ \mathrm{keV} \lt E \lt 200\ \mathrm{keV}[/math] where the photoelectric, coherent and incoherent cross-sections are dominant. The three corresponding nuclear cross-sections are minor in this energy regime but become dominant from 1 MeV – 10 MeV. It should be noted that some of the cross-sections can be in phase. Where there is significant coherence, the scattering amplitudes add vectorially, and this can contribute more than the summation of the cross-sections. When coherences are high, partitioning into separate cross-sections is no longer appropriate.
The atomic form factor can be represented as the sum of the angle-dependent component, and the anomalous real and imaginary energy-dependent components (the latter two are also referred to as the dispersive or resonant contributions):
[math]f = f_0(\theta) + f^\prime(E) + if^{\prime\prime}(E)[/math].
The imaginary component of the atomic form factor [math]\mathrm{Im}(f) = f^{\prime\prime}[/math] (in electrons per atom) [math]= f_2[/math] (used by Henke et al.) is directly related to the atomic photoabsorption cross-section given as [math]\sigma_{PE}[/math] or [math]\mu_{PE}[/math] in different references:
[math]\mathrm{Im}(f) = f^{\prime\prime}(E) = f_2(E) = {E\sigma_{PE}(E)\over2hcr_e}[/math]
where [math]r_e[/math] is the Bohr electron radius and other symbols have their usual meanings (Chantler, 2000). The Kramers–Kronig relation expresses causality and determines the real component of the (atomic) form factor in terms of the imaginary component:
[math]f^\prime(E,Z) = f^\prime(\infty, Z) - P\int_0^\infty {{\varepsilon^\prime f^{\prime\prime}(\varepsilon^\prime)\over (\hbar\omega)^2 - (\varepsilon^\prime)^2} d\varepsilon^\prime}[/math]
where [math]E = \hbar\omega[/math] is the photon energy, [math]\varepsilon^\prime[/math] is the energy above the electron binding energy of the intermediate (bound or continuum) state, and [math]P[/math] represents the Cauchy principal value. Fundamental constants and conversion factors are given (for example) by Chantler (1995).
In measurements as in XAFS, the flux is often measured before [math]I(0)[/math] and after [math]I(t)[/math] the sample of thickness [math]t[/math] (in cm). The Beer–Lambert law suggests
[math]\Big[{\mu\over\rho}\Big] = {{-\ln[I(t)/I(0)]}\over{\rho t}}[/math].
In practice fluxes must be corrected for background signal, normalization and scattering, following
[math]\Big[{\mu\over\rho}\Big] = {{-\ln \Big[\overline{\big((I(t)_d - D_d) \big/ ((I(0)_u - D_u)\big)_s}\Big/\overline{\big((I(t)_d - D_d) \big/ (I(0)_u - D_u)\big)_b} \Big]}\over{\rho t}}[/math]
where the subscript [math]s[/math] refers to the intensity measured with a sample in the path of the beam and the subscript [math]b[/math] refers to the intensity measured without a sample in the path of the beam (correcting, for example, for air attenuation and window effects). The subscript [math]u[/math] refers to the upstream detector or monitor, while the subscript [math]d[/math] refers to the downstream detector. [math]D[/math] refers to the dark current or the electronic noise for either the upstream or downstream detectors. This formula does not explicitly correct for the scattering components, and is valid for the absorption cross-section [e.g. Chantler et al. (2001), de Jonge et al. (2005)].
To derive the mass absorption coefficient (or equivalent linear coefficients) including back-scattering and correcting for harmonics, further corrections are sometimes important. For central X-ray energies, the photoelectric absorption is often dominant and hence this can be a useful approximation. However, many measurements of XAFS use fluorescence, and the normalization is further complicated by self-absorption, and in particular there is no clean [math]I(t)[/math] measurement.
References
- Chantler, C. T. (1995). J. Phys. Chem. Ref. Data 24, 71–643
- Chantler, C. T. (2000). J. Phys. Chem. Ref. Data 29(4), 597–1056
- Chantler, C. T., Tran, C. Q., Barnea, Z., Paterson, D., Cookson, D. J. and Balaic, D. X. (2001). Phys. Rev. A, 64, 062506
- de Jonge, M. D., Tran, C. Q., Chantler, C. T., Barnea, Z., Dhal, B. B., Cookson, D. J., Lee, W.-K. and Mashayekhi, A. (2005). Phys. Rev. A, 71, 032702