The geometry of MICROMEGAS detector is then, reproduced by form of cuboids Figure 2(b). The coordinate system used is showing in Figure 2(a), where:

where p_g is the Micro-Mesh pitch p_strip is the strip pitch.

The knowledge of the shape of the electric field lines close to the micromesh is a key issue for an optimal operation of the detector and especially for an efficient transfer of electrons to the amplification gap. The electric field is homogeneous in both the conversion and the amplification gap. It exhibits a funnel like shape around the openings of the micro-grid: field lines are highly compressed towards the middle of the openings, into a small pathway equal to a few microns in diameter. The compression factor is directly proportional to the ratio of the electric fields between the two gaps. Figure 3 displays details of the field lines near the grid used in the present test.

The electrons liberated in the conversion gap by the ionizing radiation follow these lines and are focused into the multiplication gap where amplification process takes place. The ratio between the Electric Field in the amplification gap and that in the conversion gap must be set at large values (>5) to permit a full electron transmission, and to reduce a part of ion cloud, produced in the avalanche, to escape into the conversion gap.

The mathematic term of potential is often calculated analytically using conformal transformations or Schwartz- Chritoffel transformation [7] . Our work is based on the use of Green functions [8] . The configuration that we want to study is presented in Figure 4.

After all calculus, considering all processes, we established the following form of Potential:

where is the charge density in (y, z), and G_D is the Dirichlet Function (Green Function) the weighting potential is presented in Figure 5.

The weighing field is deduced from Equation (2)

Equations (2) and (3) have been implemented in a MATLAB program; we treat different cases related to the geometric shapes of a track to determine the optimal width and pitch of the tracks. The results of the 3D potential and field distributions are shown in Figure 6 and Figure 7, respectively.

Note that the estimate of the potential distribution is insufficient to determine the optimal width of the tracks to get a good configuration MICROMEGAS detector. But it seems that the configurations (w1 = 280 microns, w2 = 317 microns and D) are satisfactory because the distributions are well presented. By against the corresponding distributions of the widths w = 100 µm tracks, have signal distortion (deformation) which causes instabilities and system disturbances.

The distributions of different configurations of weighting field are shown in the figure above. Note that the distributions of peaks increase with the growth of D. Moreover, the peaks of the first column of the field distribution (w = 100 microns) are very close, which produces overlaps (spark between the wire of the tracks). The remaining columns have peaks with a larger spacing. It is concluded that sufficient choice of track width minimizes system instability, as an example of the third column (w = 317 microns), which has only one peak band which further improves the stability; have good performance (better resolution) to MICROMEGAS.

Table 1 shows some input output parameters used in the MICROMEGAS detector modeling.

From Equation (4) [9] , [10] and Table 2 shown below [11] , we can estimate the charge collected Q_i(t) (I = 1 to 5) in MICROMEGAS for different proportions of each gas mixture; using X-rays (5.9 keV) as a radiation source, the same program remains what we need to change the values of the coefficients A_i and B_i [11] - [13] . These parameters are adjusted by experience [14] .

where T_w is the first Townsend Coefficient calculated in [12] , n₀ is the electron number in Micro-Mesh.

The results are shown in Figure 9 and Figure 10.

First, we note that the temporal evolution of the total charge for both gas mixtures keeps the same evolution qualitatively. On the other hand, the charge decreases with the increase in the proportion of the quencher, and increases with the increase the electric field, in addition, the variation for Ne-isobutane is greater than that of Ne-DME. It is interesting to note that the total charge is simply the gain multiplied by the primary charge, and that in fact the amplified electrons do not contribute much to the final detected charge. The main contribution is clearly due to ions [15] . The proportion between the two contributions relates directly to the gain of the chamber. The fraction f_e of the signal due to electrons is given by:

Similarly, we can calculate the proportion of charge induced by ions:

Q_e is the charge induced by electron.

Table 2 and the Equations (4a) and (4b) allowed us dressed a Table 3, the results of charge fractions for two electric field configurations are presented in Figure 11.

From this figure, it is found that the fraction of charge induced by the electron for each gas mixing follows an exponential law; it decreases with increasing the amplification field and the proportion of the quencher. It tends to a minimum value (saturation) until the field is very intense. However, the proportion of charge induced by ions increases with increasing the field and the quencher, it tends to a limit value in higher field. The boundary value was: