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Fluorescence

When X-ray photoabsorption ejects an inner-shell electron (see attenuation & transport), it leaves a vacancy in a deep level. The atom relaxes by dropping an outer electron into the hole, and the released energy comes out either as a characteristic X-ray photon (fluorescence) or by ejecting a second electron (the Auger process). The Fluorescence tab previews the characteristic-photon channel; it is X-ray-only and hidden for electron and neutron beams.

Fluorescence (X-ray)

Characteristic lines

Because the shell energies are sharply defined, the emitted photon energy is the difference of two binding energies,

\[E_\gamma = E_B(\text{inner shell}) - E_B(\text{outer shell}),\]

and is therefore characteristic of the element:

  • K lines — vacancy in the \(K\) shell filled from \(L\) (\(K\alpha\)) or \(M\) (\(K\beta\)).
  • L lines — vacancy in the \(L\) shell filled from \(M\)/\(N\) (\(L\alpha\), \(L\beta\), …).

Only transitions allowed by the dipole selection rules appear, which is why the spectrum is a few discrete lines (K\(\alpha_1\), K\(\alpha_2\), K\(\beta_1\), L\(\alpha_1\), …) rather than a continuum. Their energies follow Moseley's law; in the screened-hydrogenic approximation,

\[E_{n_2\to n_1} \approx R_\infty hc\,(Z-\sigma)^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \qquad \text{so}\qquad \sqrt{E} \propto (Z-\sigma),\]

with \(\sigma\) a screening constant. For \(K\alpha\) (\(n_2{=}2\to n_1{=}1\), \(\sigma\approx1\)) this reduces to \(E_{K\alpha}\approx R_\infty hc\,(Z-1)^2\left(1-\tfrac14\right)\). This monotonic, electron-count-driven \(Z\) dependence is the basis of elemental identification (EDX/WDX).

Fluorescence yield

The competition between radiative and Auger relaxation is captured by the fluorescence yield

\[\omega = \frac{\Gamma_r}{\Gamma_r + \Gamma_a},\]

the probability that a given vacancy decays by emitting a photon rather than an Auger electron (\(\Gamma_r\), \(\Gamma_a\) are the radiative and Auger rates).

  • For light elements the Auger channel dominates, so \(\omega_K\) is small (well below 1% for C, N, O) — light elements fluoresce weakly, which is why they are hard to detect by EDX.
  • For heavy elements the radiative channel wins and \(\omega_K \to\) near 1.

The complementary Auger yield \(a\) takes the rest, so

\[\omega + a = 1 ,\]

and a small \(\omega\) means most vacancies decay by Auger emission. Within the radiative channel, the share of one particular line \(\ell\) (e.g. \(K\alpha_1\) vs \(K\beta_1\)) is its branching ratio

\[p_{\ell\mid X} = \frac{\Gamma_\ell}{\sum_{\ell'\in X}\Gamma_{\ell'}},\]

the relative radiative rate within shell \(X\). ReciPro lists \(\omega_K\) for each element and the strongest line in the spectrum.

What the preview does and does not model

The EDX emission lines plot draws each characteristic line as a stick at its photon energy with height proportional to

\[\text{(atomic fraction)} \times \text{(radiative rate)} \times \omega.\]

This is a qualitative preview of where the lines fall and their rough relative heights. It deliberately omits the factors that a real, quantitative EDX/XRF spectrum requires:

  • whether the incident energy is actually above the absorption edge needed to create the vacancy — a line is drawn even if it cannot be excited at the current energy;
  • the excitation cross section (how efficiently the incident beam creates the vacancy at the chosen energy);
  • self-absorption of the emitted photons within the sample (matrix effects);
  • detector efficiency and resolution.

So the preview is for line identification and relative-position reasoning, not for quantitative composition.

From preview to quantification

A real EDX/XRF analysis converts line intensities into concentrations through a matrix (ZAF) correction — for atomic number (\(Z\)), absorption (\(A\)) of the emitted photons on their way out of the sample, and secondary fluorescence (\(F\)) excited by other lines — combined with the excitation cross section and detector response noted above. In full form the measured intensity of line \(\ell\) from element \(i\) is

\[I_\ell \;\propto\; C_i\,\Phi_0\,\sigma_{\text{ion},X,i}(E_0)\,\omega_{X,i}\,p_{\ell\mid X}\,\epsilon(E_\ell)\,A_\text{matrix}(E_0,E_\ell),\]

with \(C_i\) the concentration, \(\Phi_0\) the incident flux, \(\sigma_\text{ion}\) the ionisation cross section, \(\omega\) the fluorescence yield, \(p_{\ell\mid X}\) the branching ratio, \(\epsilon\) the detector efficiency, and \(A_\text{matrix}\) the absorption / secondary-fluorescence correction. ReciPro's preview keeps only the \(C_i\,p_{\ell\mid X}\,\omega\) part (atomic fraction × radiative rate × yield) and drops the rest, so it places the lines and gives their intrinsic relative strengths so that they can be recognised in a measured spectrum.

See also