Covariance mapping: Difference between revisions

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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 Thomas, M Larsson, L Foucar, J Ullrich, K Motomura, S Mondal, K Ueda, T Osipov, L Fang, B F Murphy, N Berrah, C Bostedt, J D Bozek, S Schorb, M Messerschmidt, J M Glownia, J P Cryan, R Coffee, O Takahashi, S Wada, M N Piancastelli, R Richter, K C Prince, and R Feifel "Dynamics of Hollow Atom Formation in Intense X-ray Pulses Probed by Partial Covariance Mapping" ''Phys. Rev. Lett.'' '''111''' 073002 (2013), [http://hdl.handle.net/10044/1/11746 open access]</ref>. X-ray pulses are focused on neon atoms and ionize them. The kinetic energy spectra of the photoelectrons ejected from neon are recorded at each laser shot using a suitable spectrometer (here a [[Time-of-flight mass spectrometry|time-of-flight spectrometer]]). The single-shot spectra are sent to a computer, which calculates and displays the covariance map.

'''Figure 1: Schematics of thea covariance mapping experiment.''' The experiment was performed at the [[LCLS#LCLS|LCLS FEL]] at [[Stanford University]].''' <ref name="LJF13"/>

===The need to correlate photoelectrons===

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===The principle===

Consider a random function <math>X_i = X_iX_n(E)</math>, where index <math>in</math> labels a particular instance of the function and <math>E</math> is the independent variable. In the context of the FEL experiment, <math>X_iX_n(E)</math> is a digitized electron energy spectrum produced by laser shot <math>in</math>. As the electron energy <math>E</math> takes a range of discrete values, the spectra <math>X_i(E)E_i</math>, the spectra can be regarded as row vectors of experimental data in need of analysis.:

:<math> \mathbf{X}_n = [X_n(E_1), X_n(E_2), X_n(E_3), ... ] </math>.

The simplest way to analyse the data is to average the spectra over <math>N</math> laser shots:

:<math> \langle \mathbf{X} \rangle = \langle X(E)\rangle =frac{1}{N} \sum^{N}_{in=1} X_i(E)\mathbf{X}_n </math>.

... covariance formula