Artenum Innovative Solutions - Laboratory Framework

[./images/lab|{image:./images/lab|align=right|width=500|height=200}]
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T1 I - Introduction

The formalism of normalization of data arising from complete experiments has been performed for the first time by [../biblio/He_exp/Brauning98?action=download|Braunig et al (1998)] and extended by [../biblio/He_theo/Bouri06?action=download|Bouri et al (2006)]. We present here the principle of the technique in order to apply it to di-atomic molecules. Firstable, we need to introduce the laboratory frame, as it is shown in the figure above in spherical coordinates. It is usually defined with respect to the light and the spectrometer axis as follow:
- (Oz) : polarization axis 
- (Ox) : propagation axis
- (Oy) : spectrometer axis
Any (circular) permutation of the axis leave the system invariant (see below). 
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T1 II - Fully differential cross section
T1.1 1 - Solid Angles
In this space, the analysis of particles can be done within a solid angle (volume element), which can be expressed in energy scale by :[./images/eq1|{image:./images/eq1}] Although this representation defines an accurate volume element, it is more convenient to use the following "pseudo" volume element :[./images/eq1bis|{image:./images/eq1bis}]


We consider the situation where the molecular hydrogen is doubly ionized by a XUV photon (~~E=hv~~). The photo double ionization leads to the emission of __four particles__ imparting kinetic energy. The energy conservation can be expressed simply by:\\
[./images/eq2|{image:./images/eq2|align=center}]\\
where E1 is the energy of the electron labeled 1, E2 is the energy of the electron labeled 2, EN is the energy of one ion. At fixed photon energy (~~hv~~), the energy of one ion is determined by the __excess energy__ of the electrons (E1+E2). 

i. For the electrons, the description of all kinematics is given by the number of event counted in volume element of the ~~sixth order~~. It can be reduced to a volume element of the ~~fifth order~~ due to the invariance by rotation around the polarization axis (Oz) and only the difference of azimutal angles between particles is required : [./images/eq3|{image:./images/eq3}]
i. For di-atomic molecules only one ion is needed to know the alignement (homonuclear molecules) or the orientation (heteronuclear molecules). Since the excess energy is known, the solid angle with respect to the emission of electron 1 is simply given by: [./images/eq4|{image:./images/eq4}]

Finally the fully differential volume element of the ~~seven order~~ of the fixed in space molecule is: [./images/eq5|{image:./images/eq5|align=center}]

T1.1 2 - General formulation 
~~Th. Weber et al~~ has studied the photo double ionization of the hydrogen and introduced the fully differential cross section as: [./images/eq6|{image:./images/eq6|align=center}]\\
We will keep the same convention in the following. 

Theoritically, in the case of an infinite resolution, the number of event in volume element of the ~~seventh order~~, is directly proportional to the fully differential cross section. This relation is expressed with respect to the total amount of events (N) and the total photo-double ionization cross section by: 
[./images/eq7|{image:./images/eq7|align=center}]\\
Experimentally, the volume element has always a finite size (acceptance solid angles for each particle, experimental resolution...) and the previous expression has to be re-written:
[./images/eq8|{image:./images/eq8|align=center}]\\
As we can see the measure of the differential cross section cannot be extracted directly, but we can assume that the cross section is almost constant within a volume element "small enough". Thus, the differential cross section can be set outside the integrals, and only the volume element has to be calculated : 
[./images/eq9|{image:./images/eq9}]\\
It is straight forward to evaluate:\\
[./images/eq12|{image:./images/eq12|align=center}]\\
which can be approximated in the case of small acceptance angles (typically less then 10°) by:\\
[./images/eq13|{image:./images/eq13|align=center}]\\
Finally, the absolute experimental cross section can be evaluated by :\\
[./images/eq11|{image:./images/eq11|align=center} ]\\
This approach has been used with success for the photo double ionization of helium and good agreement with theory is achieved. Nevertheless one should keep in mind that :[./images/eq10|{image:./images/eq10}]
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