See also: [[Compton Imaging]] # Shockley-Ramo Theorem To read the authoritative paper, see [[@heReviewShockleyRamo2001]]. ## Thesis-Like Writing The process of measuring radiation with room-temperature semiconductors involves measuring the total current induced upon an electrode as charge carriers diffuse through a detector material. In theory, it is possible to deduce the total charge induced upon an electrode of interest from the movement of a point like charge by integrating the electric field over a surface which encloses the electrode. $\oint_S \varepsilon \vec{E} \cdot d \vec{S} $ However, in practice, given the non-uniformity of the electric field in the detector material as the charge carrier drifts, this integration can prove to be computationally difficult or even impossible. The Shockley-Ramo Theorem is a mathematical theorem that is the backbone of modern pixelated radiation detector design. The theorem states that the induced charge $Q$ upon a conducting electrode of interest is proportional to the movement of a point-like charge $q$ through a charge-sensitive device between an initial position and a final position with unitless "weighting potentials" $\phi_0(\vec{x_i})$ , and $\phi_0(\vec{x_f})$ respectively. This is written more rigorously as $Q = -q \left[ \phi_0 (x_f) - \phi_0 (x_i)\right] $ or more simply $Q = -q \Delta \phi_0 (x)$ One may take the first-order time derivative of this equation to determine the instantaneous current on the electrode of interest: $i = -q \vec{v} \cdot \vec{E}_0 (\vec{x}) $ where $\vec{E}_0(\vec{x})$ is the instantaneous "weighting electric field" at $\vec{x}$ while $\vec{v}$ is the velocity of the charge carrier. The "weighting potential" $\phi_0(\vec{x})$ and "weighting electric field", $\vec{E}_0 (\vec{x})$ are not the true parameters found in the detector bulk, but are instead theoretical parameters that depend on the electrode geometry and material's dielectric constant. These terms can be found by solving the Poisson Equation $\nabla^2 \phi_0 = 0 $ with the boundary conditions: $ \begin{cases} \phi|_{S_i} = 1 & i = k \\ \phi|_{S_i} = 0 & i \neq k \\ \end{cases} $ where $S_i$ is the $i$-th surface in the system, and $k$ is the surface of interest in the system. In the context of radiation detectors, these values are defined by the conditions met when the electrode of interest is set to 1V, all other electrodes are set to 0V (are grounded) and all other charges (space charge) are removed. It incorrect, however, to assume that the weighting potential is the same as the true electric potential field in the detector itself. However, in the use case of radiation detection, it is, presently, only necessary to know the final signal amplitude. Given that electric fields are conservative (i.e. path independent), it is sufficient to determine the difference in potential between the point charge's starting position and ending position. It is also unrealistic to assume that there will only be a single charge traveling through the detector bulk during a given photon measurement. Rather, as is more likely the case, electron "clouds" of varying shapes and sizes will be generated and will travel through the detector material. Luckily, the Shockley-Ramo theorem can be extended to consider many simultaneous charges in a detector at the same time, through the linearity of Poisson's Equation and the Linear Superposition Principle. Since electron clouds are simply made up of many individual point-like charge carriers, it is possible to apply the Shockley-Ramo Theorem to each charge carrier that comprises the entire cloud to generate an output signal. It is worth recognizing that the shape of the cloud and the velocity at which it travels both have an impact on the waveform shape, which can be analyzed in post-processing. Between the linearity of the mathematical operations and the relative uniformity of the weighting potentials, the Shockley-Ramo Theorem is an exceptionally useful tool which sets the foundation for any and all measurements with pixelated radiation detectors. ## Summary The Shockley-Ramo theorem is a mathematical shortcut that allows us to determine an induced charge $Q$ from the movement of a charge $q$ based on its starting potential $\phi_0(\vec{x_i})$ and its ending potential $\phi_0(\vec{x_f})$. As the foundation of modern [[CdZnTe (CZT) Detectors]], the Shockley-Ramo Theorem allows for intentional electrode design to capture information about ionizing radiation from the generation and transport of electric charges. The theorem itself is fundamentally based on the conservation of energy and is proven in [[@heReviewShockleyRamo2001]]. ## The Mathematical Definition $Q = -q \Delta \phi_0 (x)$ $ i = q\vec{v} \cdot \vec{E}_0(x) $ ### Mathematical Term Definitions - $q \equiv$ the charge, - $\vec{v} \equiv$ the instantaneous velocity of the charged particle - $\phi_0 \equiv$ the weighting potential - $\vec{E}_0 \equiv$ the weighting field The Shockley-Ramo Theorem allows us to easily measure the charge incident on an electrode when all others are grounded, this is possible based on **CONSERVATION OF ENERGY** in an electric field. Deposited Energy -> Moving charge $q$ -> Induced Charge $Q$ -> Output Signal -- ## Other Relevant Notes: - [[Three Fundamental Basics to Understand when Considering the Shockley-Ramo Theorem]] - [[Proof of the Shockley-Ramo Theorem]] - [[Weighting Potentials Must Obey Boundary Conditions from the Shockley-Ramo Theorem]] - [[Jim's Two Key Shockley-Ramo Factors]]