## 2 Plasma Emission and Multi-Element Detection  
For a laser-induced plasma that is optically thin and in local thermodynamic equilibrium (LTE), the number of photons \(\Delta\)n\({}_{\text{ki}}\) emitted within the time interval \(\Delta\)t due to transitions between two atomic quantum states k and i is given by [Equation (1)]:  
\[\Delta\text{n}_{\text{ki}}/\Delta\text{t}=\text{N}\ \text{A}_{\text{ki}}\ \text{g}_{\text{k}}\ \text{exp}[-\text{E}_{\text{k}}/\text{k}_{\text{B}}\ \text{T}_{\text{e}}]/Z(\text{T}_{\text{e}}), \tag{1}\]  
where N is the number of neutral atoms of the respective chemical element in the plasma, A\({}_{\text{ki}}\) the transition probability, g\({}_{\text{k}}\) the degeneracy factor of state k, E\({}_{\text{k}}\) the energy of the upper level of transition, k\({}_{\text{B}}\) the Boltzmann constant, Z(T\({}_{\text{e}}\)) the partition function, and T\({}_{\text{e}}\) the electron temperature (= plasma temperature). The number of detected photons n\({}_{\text{det}}\) depends strongly on the measurement setup and the instrumentation used [Equation (2)]:  
\[\text{n}_{\text{det}}=\Delta\text{n}_{\text{ki}}/\Delta\text{t}\times\left( \Delta\Omega/2\ \pi\right)\text{\tau}_{\text{g}}\ \text{Y}_{\text{det}}. \tag{2}\]  
The solid angle covered by the collection optics is \(\Delta\Omega\), the measurement time (gate width) is \(\text{\tau}_{\text{g}}\), and the total detection efficiency is \(\text{\tau}_{\text{det}}\) (which includes efficiencies of the light guiding system with mirrors, lenses, and fibers, of the optical imaging, and of spectrometer and electronic detector). As an example, we discuss the ablation of an iron sample containing 10 ppm of carbon impurity and the optical emission of carbon in the laser-induced plasma. With nanosecond lasers, the ablated mass per pulse is typically around 300 ng, corresponding to 3.2 \(\times\) 10\({}^{15}\) Fe atoms and N = 3.2 \(\times\) 10\({}^{10}\) C atoms (ignoring ions in the following). The most intense emission line of C is in the UV range at wavelength \(\lambda_{\text{ki}}\) = 193.09 nm and the corresponding spectroscopic parameters are \(\text{A}_{\text{ki}}\) = 3.7\(\times\)10\({}^{8}\) s\({}^{-1}\), g\({}_{\text{k}}\) = 3, and E\({}_{\text{k}}\) = 7.7 eV. For a typical plasma temperature of T\({}_{\text{e}}\) = 9000 K the partition function of neutral C is \(Z\) = 9.8. The C atoms are emitting \(\Delta\text{n}_{\text{ki}}/\Delta\text{t}\approx\) 1.8 \(\times\) 10\({}^{14}\) photons per second. For a typical LIBS setup equipped with an Echelle spectrometer and ICCdetector and with \(\Delta\Omega/2\pi\approx 10^{-2}\), \(\tau_{\rm g}\approx 10\) u\(\rm s\), and \(\gamma_{\rm det}\approx 1.25\times 10^{-5}\), the total detected emission signal for carbon is \(n_{\rm det}\approx 430\) photons. As a consequence of such rather low signals, the limits of detection (LOD) and of quantification (LOQ) in LIBS are typically in the ppm range (or higher, depending on the element considered). With double-pulse LIBS the emission of the laser-induced plasma can be enhanced and lower LOD values are achieved [1; 6]. The measured signals depend also on the type of sample material (matrix effect) and very different limits for the same analyte may be obtained for different matrices. The measured LOD and LOQ values for many analyte elements in different matrices can be found in the literature [50]. The LOD values for solid materials are summarized in the table of elements in Figure A3 (Appendix A). The compositional analysis of industrial materials requires multi-element detection capability. Modern electronic components used in mobile phones, computers, and cars are comprised of up to 60 chemical elements, for example. In LIBS, many different elements can be measured at the same time, making this technique especially suited for the analysis of materials with complex composition.