The effects of a passive Bi-Polar Grid (BPG) on
Ion Back-Flow (IBF) and Resolution

A passive BPG with $(V_{g}\pm\Delta g)$ can provide $\vec{E}$-field ratios without creating fluctuations and would eliminate dead time by constantly being opaque to ions while exploiting an external magnetic $\vec{B}$-field to remain transparent to electrons.

$$m\frac{d\vec{v}}{dt}=q\vec{E}+q(\vec{v}\times\vec{B})-\kappa\vec{v}$$

(1.1)

Charges follow the Langevin equation and lighter electrons will experience a continuous $(\vec{v}\times\vec{B})$ “push" allowing them to pass between the wires. Electrons coming from above the wire will get their 1^st push along the length of the wire: $\hat{F_{1}}=\pm\hat{x}\times\hat{z}=\mp\hat{y}$. Then they get a 2^nd weaker push which allows them to get past the wires: $\hat{F_{2}}=\mp\hat{y}\times\hat{z}=\mp\hat{x}$. Fig. 1 shows these effects along with the distortion both pinched and shifted in their y-position. Acceptable electron transparency with 0% ion transparency has been demonstrated by previous groups [1].

The general equation for the charge velocity in a gas as a function of $\vec{E}$ and $\vec{B}$ fields is given by

$$\vec{v}=\frac{\mu}{1+\omega^{2}\tau^{2}}\left(\vec{E}+\frac{\vec{E}\times\vec{B}}{|\vec{B}|}\omega\tau+\dfrac{(\vec{E}\cdot\vec{B})\vec{B}}{B^{2}}\omega^{2}\tau^{2}\right)$$

(1.2)

where $\mu=e\tau/m$ is the charge mobility with $\tau$ as the average time between collisions. $\omega=eB/m$ is the cyclotron frequency. Defining angle $\alpha$ between $\vec{E}$ and the positive x-direction gives $\vec{E}=(E\cos{\alpha},0,E\sin{\alpha})$. Together with $\vec{B}=(0,0,B)$ leads to

$$v_{x}=c(E\cos{\alpha})\qquad v_{y}=c(E\omega\tau\cos{\alpha})\qquad v_{z}=\mu E\sin{\alpha}$$

(1.3)

If the BPG wires aren’t uniformly spaced along their tensioned direction an $E_{y}$ component will be present and the charge velocity equations become

$$v_{x}=c(E_{y}\omega\tau+E_{x})\qquad v_{y}=c(-E_{x}\omega\tau+E_{y})\qquad v_{z}=c(E_{y}\omega^{2}\tau^{2}+E_{z})$$

(1.4)

where $c=\dfrac{\mu}{1+\omega^{2}\tau^{2}}$. In both cases, the velocities of main interest for ion blocking and electron transparency $v_{x}$ and $v_{z}$ are strongly dependent on $\omega\tau$. For electrons $\omega\tau\gg 1$ and the velocities will follow the $\vec{B}$-field and ignore the local $\vec{E}$-field of the BPG. For ions the converse is true.

Ion and electron transparencies, as a function of $\vec{B}$-field and $\Delta V$ volts on a linear BPG, were measured at Weizmann. A detector with MWPC gain wires was constructed. A strong ⁵⁵Fe X-Ray source was used and picoAmmeters were implemented to measure the small changes in current as the grid was slowly closed to ions. A mesh was used to help create a uniform drift field, as seen in Fig. 2, and to create a region where the $\vec{E}$-field above-to-below ratios can be better exploited.

Field ratios were varied measuring the electron and ion transparencies from the BPG and mesh $T^{g,m}_{e,i}$. Increased grid ratios meant decreased mesh ratios pulling more ions through it. $T_{e}^{g}<100\%$ requires an increased gain, which creates more IBF. A Figure of Merit (FoM) was constructed to quantify the effectiveness of the BPG [3]. 1^st term comes from $\vec{E}$ ratios, where higher ratios can extract more ions from the gain region. 2^nd term compensates for loss of primaries creating more IBF ions. Fig. 3 shows an example for transparencies and FoM.

$$FoM(\omega,\Delta V)=\frac{T^{m}_{i}(\omega,0)}{T^{m}_{i}(1,0)}\cdot\frac{T^{g}_{i}(\omega,\Delta V)}{T^{g}_{e}(\omega,\Delta V}\qquad\omega=E_{t}/E_{d}$$

(2.1)

From the equations of motion, and as seen in Fig. 1, the electrons fall "out of plane", and alternating wires "pinch" them together, hurting resolution. The error is cyclical with the same periodicity as the wires. These recurring shifts are known as Differential Non-Linearity (DNL), and can be corrected. Segmented pads are used to spread out the electron cloud charge collection with finer detail in order to get a more precise track position measurement.

Zig-zag pads [4], which themselves have DNL-corrected error due to their shape, are used. From the grid’s symmetric shift, perhaps BPG’s distortions will average out over the full particle track, regardless of the wire’s position with respect to the radial zig-zags. However, if the grid is aligned with the pads then their independent DNLs might combine for better resolution. A BPG with 2 linear and 2 radial wire configurations is placed above the pad-plane and goes inside a prototype TPC. We’ve taken it to Argonne National Lab (ANL) to test this hypothesis (Fig. 4).

The passive BPG source test demonstrates ion blocking with high electron transparency. The ANL test showed excellent resolution results and a paper with the full test description and analysis will soon be published.