Covariance mapping

Feldman, A. Antoine, M. Lin, J. Demirev, P. Covariance mapping in matrix-assisted laser desorption/ionization time-of-flight mass spectrometry. Rapid Comm. Mass Spectrom. 2003,17, 991-995. 

Figure 17.20. Phosphorus covariance map and fluorescence P-NEXAFS spectra collected from a coastal marine sediment (Brandes et al., 2007). Covariance map is color-coded Green represents P, blue represents Si, and red represents Na fluorescence signals. Regions 1,3, and 5 have spectra consistent with organic phosphorus or polyphosphate, while regions 2 and 4 closely match calcium phosphate, specifically the mineral apatite. See color insert.

In order to correlate the different ion species ejected from the same Coulomb explosion pathway, the covariance mapping technique [17] was introduced, in which correlations among the fragment ions are extracted from the fluctuation in the ion signals of a large number of TOF mass spectra recorded at respective laser shots. When the fragment ions are produced through the same explosion pathway, their covariance becomes positive. 

From the covariance map, in which the covariance coefficients [17] are plotted on a two-dimensional plane, we can identify the respective explosion pathways and show how the momentum components along the TOF axis of the different ion species are correlated. The characteristic patterns appearing in the double covariance map for the ion pairs (0+, 0+), (0+, C+) and (0+, 02+) ejected from C02 at the field intensity of 1015 W/cm2 were interpreted in terms of a large amplitude bending motion [18], where the widths of the bond angle distribution were found to be constant different charge states (z = 3-6). 

Since the covariance map is constructed based on one-dimensional TOF mass spectra, the angular distribution of fragment ions with respect to the laser polarization direction is only indirectly incorporated into the map through the momentum distribution along the TOF axis (see [19] for the extension to two-dimensional measurements). In order to derive three-dimensional momentum vector distributions of the fragment ions together... 

Fig. 3. A cross section of the ion-ion correlation analyser. The upper drift tube is used for covariance mapping, the lower one for high-resolution conventional TOF spectra...

Fig. 4. Covariance map of N2, with the conventional time-averaged TOF spectrum placed along the X and y axes. The correlated features 1-4 are discussed in the text. The inset shows the strong feature 4 using a larger range of grey scale...

In 1989, Frasinski et al. [16] introduced a new triple coincidence technique called covariance mapping. A brief explanation of the technique can be given by considering again the process [N2 ]-+Nb and denoting by ti and... 

The power of covariance mapping will be illustrated by one example. When Lavancier et al. [21] studied the multiple ionization of CO at 305 nm, analysis of their conventional TOP spectra suggested that the process [CO ]-+ + O... 

Fig. 5a, b. Covariance map of CO with the laser E field a parallel to b perpendicular to the drift tube axis. The features labelled 1-6 are discussed in the text... 

Before leaving the subject of ion-ion covariance mapping, we comment on experiments measuring ion angular distributions. The very first experiment on the MEDI of N2 noted the strong directionality of the fragment ion emission relative to the laser E field [8], but no quantitative measurements were made. Subsequently isoelectronic CO was studied at 600 nm [25] and an example of the covariance maps produced when the laser E field is (a) parallel to and (b) perpendicular to the drift tube axis is shown in Fig. 5. 

We have seen, in the case of diatomic molecules, that two-dimensional covariance maps allow correlation of the ion pairs. The situation with regard to the multiple ionization of triatomic molecules is more complex. Similar two-dimensional maps serve to indicate correlations between pairs of ions but at high laser intensities it is probable that three ions will be produced. Although one might infer that three ions have been created simultaneously, the only sure way of confirming the creation and subsequent fragmentation of a triple ion is to use three-dimensional covariance mapping. 

Fig. 6. Covariance map of N O, with the conventional ion TOF spectrum placed along the x and y axes...

What is surprising is the lack of peaks in the TOF spectrum (and structure on the covariance map) associated with single ions. The total energy release (about 40 eV) suggests that, as the three atoms move apart and as the laser E field increases, there comes a point when the molecule is suddenly six-times ionized. One can speculate on the mechanism involved, but this interesting phenomenon deserves further study both experimentally and theoretically. 

Pigure 9 shows an electron-ion covariance map obtained at the same time as Pig. 8b. Por convenience the time-averaged electron and ion TOP spectra are shown alongside the x and y axes. The electron TOP is virtually structureless, the slight forward-backward asymmetry and the peak at 292 ns being due to a small inhomogeneity in the extraction field. The forward electrons are unaffected and a kinetic energy scale is shown for both electrons and protons. In order to ensure that this lack of structure was not the result of poor instrumental resolution, an electron TOP spectrum of Xe was taken under identical conditions. A series of ATI peaks was observed in both the forward and backward direction, separated by about 2 eV. 

Perhaps the most interesting application of electron-electron covariance mapping relates to the question of the major mode of multiple ionization of atoms and molecules. Luk et al. [34] studied the multiple ionization process in Xe using a laser of 193 nm wavelength and 10 ps pulse length and conventional ion TOP spectroscopy they suggested that it was direct (a collective, instantaneous emission of many electrons). Lambropoulos [35] pointed out that, with a laser of such modest rise time, the ionization must proceed sequentially. In fact L Huillier et al. [36] had also studied Xe at 532 nm and observed a knee in the curve of log (ion counts) vs log (laser intensity) for Xe that they attributed to direct double ionization. 

Much insight in the dissociation patterns of molecules can be obtained by using the so called covariance mapping technique [48], Terawatt laser excitation allowed such experiments for CO2 to be performed in a non-explored power density regime [49]. 

Shen, P Hauri, D. Ross, J. Oefner, P. J. Analysis of glycolysis metabolites by capillary zone electrophoresis with indirect UV detection.7. Capillary Electrophor. 1996,3,155-163. Card, D. A. Fohner, D. E. Sato, S. Buzza, S. A. Castleman, A. W., Jr. Covariance mapping of ammonia clusters evidence of the coimectiveness of clusters with coulombic explosion. J. Phys. Chem. 1997,101, 3417-3423. 

Covariance NMR mostly refers to any NMR experiment whose resulting data are subjected at some point to covariance analysis, covariance transformation or covariance treatment. Covariance NMR processing describes the steps that compute the covariance from a matrix of NMR data and yields the covariance map. The covariance map is equivalent to a NMR spectrum obtained after Fourier transformation, if the covariance was calculated obeying certain mathematical constraints, cf further below. In other words. 

Assuming that x and y of Eq. (5.1) are spectroscopic data series and that they are arranged such that S is a spectrum of Nj data points, the elements Cy of the covariance matrix C, also called the covariance map, are computed according to Eq. (5.2)... 

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