#zettel #research/imaging #research/radiography/event-distinguishing # Subpixel Imaging in Pixelated 3D-CZT Detectors Pixelated 3D-CZT detectors are capable of providing information about photon interactions within the crystal bulk. Prior to the development of subpixel sensing techniques developed by [[@zhuSubPixelPositionSensing2011]], the position resolution of 3D-CZT was limited to its pixel pitch. From works that demonstrated that the movement of charge carries induced transient signals on non-collecting neighbor pixels, Zhu demonstrated that it was possible to determine the centroid of an electron cloud over the collecting anode pixel. These methods to deduce the subpixel position of the electron cloud are detailed below: ## How This Works The [[Shockley-Ramo Theorem]] determines an imaginary weighting potential for each electrode, regardless of whether the pixel is the collecting pixel, a neighboring pixel or otherwise. As electrons diffuse through the crystal bulk toward the anode pixel array, they will induce charges on multiple pixels due to the nature of the weighting potential shape. this is visualized by a figure from [[@narita_anode_2004]] that shows thow the weighing potential from a non-collecting pixel can stil induce a transient signal. ![[neighbor_weighting_potential.png]] For the trajectory of a charge "near" the neighbor pixel, the signal amplitude will be much large than one far from the neighbor pixel. Because the pixels are all the same shape and size and biased at the same voltage, this symmetry can be used to determine a lateral position with more precision than just the pitch of the pixels. ## Opposing Neighbor Ratio Consider a 3-by-3 shown below that corresponds to a subset of pixels. | 11 / UL | 12 / UC | 13 / UR | | ------- | ------- | ------- | | 21 / ML | 22 / MC | 23 / MR | | 31 / LL | 32 / LC | 33 / LR | In this case the 22 / MC pixel is the collecting pixel while the surrounding pixels are neighbor pixels. The tessellation symmetry can be exploited to determine the subpixel position using a concept known as the "opposing neighbor ratio". In this method, the ratios of the amplitudes of the neighboring pixels $s_{ij} (x,y,z)$ can be used to map a position in the $x$ and $y$ direction of the subpixel position. In general the transient signal is depth-independent unless it occurs near the anode. Hence, for simplicity, these amplitudes are reported solely based on their horizontal and vertical position, $s_{ij}(x,y)$, $ Rn_x(x,y) = \frac{s_{21}(x,y) - s_{23}(x,y)}{s_{21}(x,y) + s_{23}(x,y)} $ $ Rn_y(x,y) = \frac{s_{12}(x,y) - s_{32}(x,y)}{s_{12}(x,y) + s_{32}(x,y)} $ This method will generally work for a Trigger + 4 Readout system on the ASIC. Should a more precise measure be desired, a Trigger + 8 readout system can be used, along with a ratio that joins the left, and right signals into a single value respectively to determine the x-position. $ s_l(x) ≈ s_l(x, y) =s_{11}(x, y) + s_{21}(x, y) + s_{31}(x, y) $ $ s_r(x) ≈ s_r(x, y) =s_{13}(x, y) + s_{23}(x, y) + s_{33}(x, y) $ $\Rightarrow R_x(x) = \frac{s_l(x) - s_r(x)}{s_l(x) + s_r(x)} $ This is also done for the top and bottom signals to determine the y-position $ s_t(y) ≈ s_t(x, y) =s_{11}(x, y) + s_{12}(x, y) + s_{13}(x, y) $ $ s_b(y) ≈ s_b(x, y) =s_{31}(x, y) + s_{32}(x, y) + s_{33}(x, y) $ $ \Rightarrow R_y(y) = \frac{s_t(y) - s_b(y)}{s_t(y) + s_b(y)} $ ## Other Methods That Are Less Effective 1. the ratio between center pixel amplitude and neighbor pixels (neighbor-center ratio) 2. the ratio between the two corner neighbors and the center pixel signal (corner-neighbor ratio). these are omitted here but are listed in Yuefeng's thesis. ## Some Assumptions - This algorithm determines the charge-weighted centroid of the electron cloud, and assumes that this centroid is relatively close to the photon interaction positions that incites the electron cloud. - this is a reasonable assumption at lower energies. But as [[@zhangEVENTSRECONSTRUCTION3D2005]] demonstrated, the size of the electron clouds can become multiple millimeters in diameter. Because of this, we will need a reason to correct for this. ## Related Thoughts: - This is only function for electron clouds that are generally from energy depositions below 1 MeV. At higher energies, the centroid of the electron cloud is significantly displaced from its inciting position where the gamma ray interacted with an electron in the crystal lattice. - There are ways that we can do subpixel sensing for charge sharing events over two pixels as shown by [[@zhuSubpixelSensingChargeSharing2022]]