## 2 Design of Detector Parameters
### Crosstalk Effect of the Inter-Electrode Capacitance  
The spatial resolution of the XS anode is primarily determined by the electronic noise, which is mainly induced by the charge-sensitive amplifier (CSA) and increases with input capacitance adding [16]. The anode is a periodic structure, and the circuits of each channel are independent of each other. The capacitance of one channel is:  
\[C_{in}=C_{i}+C_{ie}+C_{a}+(1+A)\cdot C_{f} \tag{3}\]  
where \(C_{i}\) is the strip electrode capacitance, \(C_{ie}\) is the inter-electrode capacitance of the anode strip, \(C_{a}\) is the amplifier input capacitance, \(C_{f}\) is the feedback capacitance, and \(A\) is the open loop gain of the integrated amplifier. Crosstalk directly affects the signals on the strip electrodes because of the inter-electrode capacitance \(C_{ie}\) of the XS anode. Optimization of the inter-electrode capacitance is conductive to reduce the electronic noise.  
For an ideal XS anode, the charges on the strips could be expressed as:  
\[Q=C_{mutual}V\text{Where, }Q=\begin{bmatrix}Q_{1}\\ \vdots\\ Q_{n}\end{bmatrix},\ C_{mutual}=\begin{bmatrix}C_{11}&&\\ &\ddots&\\ &&C_{mn}\end{bmatrix},\ V=\begin{bmatrix}V_{1}\\ \vdots\\ V_{n}\end{bmatrix} \tag{4}\]  
However, the crosstalk effect cannot be neglected in practice. The expression of charges on strips can be written in Maxwell's capacitance matrix:  
Figure 5: The variation of charge ratio with insulation thickness for different area ratio Ra. The four different size XS anodes have the same period of 690 \(\upmu\)m and top strip width w1 of 253 \(\upmu\)m, and the bottom strip width w2 is 350 \(\upmu\)m, 400 \(\upmu\)m, 450 \(\upmu\)m, and 500 \(\upmu\)m, respectively.  
\[Q^{\prime}=C_{Maxwell}V\text{ Where, }Q^{\prime}=\begin{bmatrix}Q^{ \prime}_{1}\\ \vdots\\ Q^{\prime}_{n}\end{bmatrix},\ C_{Maxwell}=\begin{bmatrix}\sum\limits_{i=1}^{n}C_{1i}&-C_{12}& \cdots&-C_{1n}\\ -C_{21}&\ddots&&\vdots\\ \vdots&&&-C_{(n-1)n}\\ -C_{n1}&\cdots&-C_{n(n-1)}&\sum\limits_{i=1}^{n}C_{ni}\end{bmatrix} \tag{5}\]  
where \(C_{ii}\) is the capacitance of strip \(i\) to the vacuum chamber, \(C_{ij}\) is the capacitance between the strip \(i\) and \(j\), \(C_{ij}=C_{ji}\). Substituting Equation (5) into Equation (4), the relationship between the ideal charge and the actual charge can be obtained, and the charge can be corrected to remove crosstalk.  
\[Q^{\prime}=C_{Maxwell}C_{matual}^{-1}Q \tag{6}\]  
The XS anode model established by COMSOL mentioned earlier is also used to optimize inter-electrode capacitance. Taking one top strip and one bottom strip as examples, the variation of total inter-electrode capacitance (\(C_{ie}=\sum\limits_{i\
eq j}C_{ij}\)) with insulation thickness for different strip widths is depicted in Figure 6. It can be seen that, for the same strip width, \(C_{ie}\) decreases with the increase in insulation thickness. For the same insulation thickness, \(C_{ie}\) increases with the increase in strip width. The top and bottom strips can be regarded as parallel plate capacitors. The width of the strip and insulation thickness corresponds to the area of the capacitor plate and the dielectric thickness, respectively. This explains the variation of inter-electrode capacitance with different parameters.  
To sum up, reducing the inter electrode capacitance requires increasing the thickness of the insulation layer and reducing the width and area of the strip. However, this will lead to the imbalance of charge ratio and the reduction in power utilization. Therefore, when designing anode parameters, it is necessary to comprehensively consider the system requirements and make trade-offs.