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[[Image:Schematics of LCLS covariance mapping experiment.png|thumb|400px|'''Figure 1: Schematics of a covariance mapping experiment.''' The experiment was performed at the [[LCLS#LCLS|LCLS FEL]] at [[Stanford University]].<ref name="LJF13"/>]]
Covariance mapping is particularly well suited to [[free-electron laser]] (FEL) research, where the x-ray intensity is so high that the large number of photoelectron and photoions produced at each pulse overwhelms simpler [[Photoelectron photoion coincidence spectroscopy|coincidence techniques]]. Figure 1 shows a typical experiment.<ref name="LJF13">L J Frasinski, V Zhaunerchyk, M Mucke, R J Squibb, M Siano, J H D Eland, P Linusson, P v.d. Meulen, P Salén, R D
===The need for correlations===
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===How to read the map===
[[Image:Full covariance map of neon.png|thumb|450px|'''Figure 3: A covariance map revealing correlations between electrons emitted from neon (and from some N<sub>2</sub> and water vapour contamination).''' The map is constructed shot by shot from electron energy spectra recorded at the photon energy of 1062 eV, which are shown along the ''x'' and ''y'' axes after averaging over 480 000 FEL shots. Volumes of the features on the map give relative probabilities of various ionisation sequences, which can be
The covariance map obtained in the FEL experiment<ref name="LJF13"/> is shown in Fig. 3. Along the ''x'' and ''y'' axes the averaged spectra <math>\langle\mathbf{X}\rangle</math> and <math>\langle\mathbf{Y}\rangle</math> are shown. These spectra are resolved on the map into pairwise correlations between energies of electrons coming from the same process. For example, if the process is the first process depicted in Fig. 2 (PP), then two low-energy electrons are ejected from the Ne core giving a positive island in the bottom-left corner of the map (one of the white ones). The island is positive because if one of the electrons is detected, there is higher than average probability of also detecting the other electron and the covariance of the signals at the two energies takes a positive value.
The volumes of the islands are directly proportional to the relative probabilities of the ionisation processes.<ref name="LJF89"/> This useful quality of the map follows from a property of the [[Poisson distribution]], which governs the number of neon atoms in the focal volume and the number of electrons produced at a particular energy, <math>X_n(E_i)</math>. The property employed here is that the [[variance]] of a Poisson distribution is equal to its [[mean]] and this property is also inherited by covariance. Therefore, the covariance plotted on the map is proportional to the number of neon atoms that produce pairs of electrons of particular energies. This makes covariance much more suitable for particle counting experiments than other bivariate estimators, such as [[Pearson's correlation coefficient]].
On the diagonal of the map there is an autocorrelation line. It is present there because the same spectra are used for the ''x'' and ''y'' axes. Thus, if an electron pulse is present at a particular energy on one axis, it is also present on the other axis giving the variance signal along the <math>E_x = E_y</math> line, which is usually stronger than the neighbouring covariance islands. The mirror symmetry of the map with respect to this line has the same origin. The autocorrelation line and the mirror symmetry are not present if two different detectors are used for the ''x'' and ''y'' signals, for example where one detector is used to detect ions and another to detect electrons.<ref name="LJF92">L J Frasinski, M Stankiewicz, P A Hatherly, G M Cross and K Codling “Molecular H<sub>2</sub> in intense laser fields probed by electron-electron electron-ion and ion-ion covariance techniques” ''Phys. Rev. A'' '''46''' R6789–R6792 (1992), [http://hdl.handle.net/10044/1/11612 open access]</ref>
[[Image:Parts of covariance map of neon.png|thumb|450px|'''Figure 4:
Much more information is present on the map than on the averaged, 1D spectrum. The single, often broad and indistinct peaks on the 1D spectrum are resolved into several islands on the map. Fig. 4 shows magnified core-core and core-valence regions with several ionisation sequences identified unambiguously. In the D<sub>KV</sub> process the two electrons ejected share arbitrarily the energy available from a single proton producing a conspicuous line ''E<sub>x</sub>'' + ''E<sub>y</sub>'' = const in the left panel of Fig. 4. Impurities, such as water vapour or nitrogen, give islands usually away from the islands of the species studied (see Fig. 3b,e,f).
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==References==
{{reflist}}
* L J Frasinski "Covariance mapping techniques" ''J. Phys. B: At. Mol. Opt. Phys.'' '''49''' 152004 (2016), [http://iopscience.iop.org/article/10.1088/0953-4075/49/15/152004 open access] (review article)▼
[[Category:Covariance and correlation]]
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[[Category:Quantum chemistry]]
[[Category:Photochemistry]]
▲==Other reading==
▲* L J Frasinski "Covariance mapping techniques" ''J. Phys. B: At. Mol. Opt. Phys.'' '''49''' 152004 (2016), [http://iopscience.iop.org/article/10.1088/0953-4075/49/15/152004 open access] (review article)
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