Ramo's Theorem Research Articles

Derivation of Relationships of Currents in External Circuit and Parameters of Sampl

Several methods to derive relationships between currents in the external circuit and variations in parameters of samples that induce such currents are considered. The relationships generalize the Shockley–Ramo theorem and may serve as a development of Kirchhoff laws for electrical circuits.

Wide-gap CdTe strip detectors for high-resolution imaging in hard X-rays

We propose a new strip configuration for CdTe X-ray detectors, named “Wide-gap CdTe strip detector”, in which the gap between adjacent strips is much wider than the width of each strip. It has been known that the observed energies of an incoming photon in adjacent strips can be utilized to achieve a position resolution finer than the strip pitch, if and only if the charge cloud induced by an incoming X-ray photon is split into multiple strips and their energies are accurately measured. However, with existing CdTe strip detectors, the ratio of such charge-sharing events is limited. An idea for a potential breakthrough to greatly enhance the ratio of charge-sharing events is to widen the gaps between strips on the detector. To test the concept, we developed a wide-gap CdTe strip detector, which has 64 platinum strip electrodes on the cathode side with some variations in strip pitches from 60 μm (30 μm strip and 30 μm gap width) to 80 μm (30 μm strip and 50 μm gap width). We evaluated the performance depending on the strip pitches by irradiating X-rays from 241Am on the detector. The charge loss due to the wider gaps on the detector was found to be significant to the extent that the assumption that the energy of an incoming photon for a charge-sharing event was the simple sum of the energies detected in adjacent strips lead to a significant degradation in the energy resolution in the accumulated spectrum, compared with those obtained with its predecessor having standard gap-widths. We then developed a new energy-reconstruction method to compensate for the charge loss. Application of the method to the data yielded a spectrum with a comparable spectral resolution with that of the predecessor. The ratio of the charge-sharing events for 17.8 keV events was doubled from that of the predecessor, from 24.3 to 49.9 percent. We assessed the results with a theoretical simulation based on the Shockley–Ramo theorem and found that they were in agreement.

Role of Auxiliary Potentials and Fields in the Shockley–Ramo Theorem for Inhomogeneous Locally Anisotropic Samples with Polarization

Role of Auxiliary Potentials and Fields in the Shockley–Ramo Theorem for Inhomogeneous Locally Anisotropic Samples with Polarization

Fluctuations in the collected charge in integrating photoconductive detectors under small and large signals: the variance problem

Charge collection efficiency (CE) η 0 under small signal conditions, corresponding to a uniform field in the detector medium, has been widely used in evaluating the performance of photoconductive detectors. The present paper answers the question, ‘What is the variance of the collected charge in an integrating detector as a function of photoinjection level and what are the errors if we continue to use the small signal equations?’ The variance in η 0 under small signals has been theoretically derived in the literature and has been a key factor in the detective quantum efficiency modeling of integrating detectors based on various semiconductors. is a noise source and can degrade the detector performance under incomplete charge collection. The statistical variance in the CE η 0, under small signals and the variance in the CE ηr under an arbitrary injection level r (injected charge divided by charge on the electrodes) have been studied using the Monte Carlo simulation model developed in this work to evaluate the difference between and from small to large signals. Initial injection of electron and hole pairs and their subsequent transport and trapping in the presence of an electric field, which is calculated from the Poisson equation, is used to calculate the photocurrent. Each injected carrier is tracked as it moves in the semiconductor until it is either trapped or reaches the collection electrode. Trapped carriers do not contribute to the photocurrent but continue to contribute to the field through the Poisson equation. The instantaneous photocurrent i ph(t) is calculated from the drift of the free carriers through the Shockley–Ramo theorem. i ph(t) is integrated over the duration of the photocurrent to calculate the total collected charge and hence the CE ηr . The variance in ηr is found from multiple simulations of ηr . The ηr and have been generated over varying charge injection ratios r, the electron and hole ranges μτ, mean photoinjection depths δ and drift mobility ratios b. At full injection, the deviation of the CE variance from the uniform field case ( = may be as much as 40% larger or 20% lower than the small signal model prediction. This study provides the extent of errors involved in the variance of the CE in non-uniform fields and quantifies the increase in errors that can occur in high injection cases. In practice, typical injection ratios are less than 0.2, which means that the magnitude of percentage error is less than 5%.

Remote sensing array (RSA) for linac beam monitoring

The purpose of the present work is to evaluate the feasibility of a novel real-time beam monitoring device for medical linacs which remotely senses charge carriers produced in air by the beam without intersecting and attenuating the beamline. The primary goal is to elaborate a theoretical concept of a possible detector geometry and underlying physical model that allows for determination of clinically relevant beam data in real time, namely MLC leaf positions and dose rate. The detector consists of two opposing electrode arrays arranged in two possible orientations around the beamline. Detection of charge carriers is governed by electromagnetic principles described by Shockley–Ramo theorem. Ions produced by ionization of the air column upstream of patient move laterally in an external electric field. According to the method of images, mirror charges and mirror currents are formed in the strip electrodes. Determination of MU rate and MLC positions using the measured signal requires solution of an inverse problem. In the present work we adopted a Least-Square approach and characterized detector response and sensitivity to detection of beam properties for different electrode geometries and MLC shapes. Results were dependent on MLC field shape and the leaf position within the active volume. The accuracy of determination of leaf positions were in the sub-mm range (up to 0.25–1 mm). Additionally, detector sensitivity was quantified by simulating ions/pulse delivered with a radiation transport deterministic computation in 1D in CEPXS/ONEDANT. For a 6 MV linac pulse, signal amplitude per pulse was estimated to be in the lower pA to fA range. We computationally demonstrated feasibility of the remote sensing detector capable of measuring beam parameters such as MLC leaf positions and dose range for each pulse. Future work should focus on optimizing the electrode geometry to increase sensitivity and better reconstruction algorithms to provide more accurate solutions of the inverse problem.

Full-volume characterization of an AGATA segmented HPGe gamma-ray detector using a $$^{152}Eu$$ source

Scanning tables use collimated gamma-ray sources to perform full volume characterization of position sensitive detectors. One of such tables is hosted at IPHC Strasbourg. It was designed and built within the AGATA collaboration. It exploits the pulse shape comparison scanning (PSCS) technique to build databases of pulses used to characterize the response of high purity germanium (HPGe) detectors and perform R&D on such crystals. Ultimately, measured databases could be used by the pulse shape analysis (PSA) algorithms employed in AGATA experiments. The table can perform full volume scans of large volume detectors in short times, with a good spatial resolution and at different energies. Lately, the table was upgraded with a new \(^{152}Eu\) source, which emits gamma rays in cascades of different energies. A scan with such source is performed for the first time. It allows to build different energy databases in one single scan. The present work aims at testing the performances of the PSCS technique with a multi-energetic source and verifying some assumptions of the Shockley–Ramo theorem which are at the base of the PSA algorithms used for gamma-ray tracking arrays.

Impact of charge trapping on the energy resolution of Ge detectors for rare-event physics searches

Charge trapping degrades the energy resolution of germanium (Ge) detectors, which require increased experimental sensitivity in searching for dark matter and neutrinoless double-beta decay. We investigate the charge trapping processes utilizing nine planar detectors fabricated from USD-grown crystals with well-known net impurity levels. The charge collection efficiency as a function of charge trapping length is derived from the Shockley–Ramo theorem. Furthermore, we develop a model that correlates the energy resolution with the charge collection efficiency. This model is then applied to the experimental data. As a result, charge collection efficiency and charge trapping length are determined accordingly. Utilizing the Lax model (further developed by CDMS collaborators), the absolute impurity levels are determined for nine detectors. The knowledge of these parameters when combined with other traits such as the Fano factor serve as a reliable indicator of the intrinsic nature of charge trapping within the crystals. We demonstrate that electron trapping is more severe than hole trapping in a p-type detector and the charge collection efficiency depends on the absolute impurity level of the Ge crystal when an adequate bias voltage is applied to the detector. Negligible charge trapping is found when the absolute impurity level is less than 1.0 × 1011 cm−3 for collecting electrons and 2.0 × 1011 cm−3 for collecting holes.

Nature of Currents Induced by Changes to the Sample Parameters. Capacitance and Noncapacitance Currents

In developing the ideas of the Ramo theorem, paper considers the nature of currents in an external circuit arising when the sample parameters are changed. It is shown that, besides currents induced by charge movement in the sample and capacitance currents, other currents are possible. Formulas and an illustrative example are presented, in which there are currents in the external circuit when the structure parameters are changed, although there are no capacitance currents or currents induced by charge movement.

Study of induced current of 3D-open-shell-electrode detector

Interactions between neighboring cells often occur in a semiconductor silicon detector array. That is, there are high-energy particles incident on one cell of the detector that generate an electric signal for the cell, while weak signals are also generated in other neighboring cells. Therefore, isolation of the detector cells is an important factor in the performance of the detector, particularly for its energy resolution. To eliminate the dead space in the 3D-trench-electrode detector proposed by Brookhaven National Laboratory, a novel three-dimensional (3D) detector, the 3D-Open-Shell-Electrode Detector (3DOSED), was proposed in 2018, in which good isolation between cells was achieved. To verify this and determine the optimal structure design, a Silvaco technology computer-aided design simulation tool was used to simulate the electrical characteristics of the detector in 3D. The distributions of the electric field and the weighting field of 3DOSEDs were obtained with different opening gap angles. A single minimum ionizing particle was used that was incident vertically on the middle of the central collection electrode and peripheral trench electrode of the detector cell, according to the Ramo theorem of the principle of induced-current generation, and the mathematical software matlab was employed to calculate the induced current and carrier drift time. The interference current between the neighboring cells of a 3DOSED was obtained. Good isolation was found between neighboring cells in the 3DOSED.

Nuclear fission investigation with twin ionization chamber

In this paper, we report recent results obtained in the development of digital pulse processing mathematics for prompt fission neutron (PFN) investigations using a twin ionization chamber (TIC) along with a fast neutron time-of-flight detector (ND). Due to some ambiguities in the literature concerning a pulse induction on TIC electrodes by fission fragment (FF) ionization, we first presented a detailed mathematical analysis of FF signal formation on the TIC anode. The analysis was done using the Shockley–Ramo theorem, which gives the relation between charged particle motion between TIC electrodes and the so-called weighting potential. The weighting potential was calculated by direct numerical solution of the Laplace equation (neglecting space charge) for the TIC geometry and ionization caused by FFs. Formulae for GI correction and digital pulse processing algorithms for PFN time-of-flight measurements and pulse shape analysis are presented and used in experiments for PFN investigations of two reactions, [Formula: see text]U(n[Formula: see text],f) and [Formula: see text]Cf(sf). Results of the measurements were compared to literature data to demonstrate the feasibility of the new developed techniques. These results were necessary for the development of a new PFN investigation facility consisting of a position sensitive fission fragment detector combined with 32 liquid scintillation neutron detectors.

Demonstration of induced current produced by a transited electron beam in an ultraviolet detector

An ultraviolet (UV) detector with a transmission photocathode is used to measure ultrashort electron beams with energies ranging from 1.8 to 3.0 keV. Both of the photocathode and collector are formed with planar transmission line structures, which could provide fast response time. The induced current in the collector produced by the transited electron beam is simulated using the Ramo theorem. The full width at half maximum (FWHM) of the pulses induced on the photocathode and collector are measured. While the photocathode is applied with -3.0 kV bias voltage, they are approximately 190 ps and 600 ps, respectively. The time of flight of the electron pulse from the photocathode to the collector is also measured, while the photocathode is applied with different bias voltage.

Charge collection efficiency in photoconductive detectors under small to large signals

Hecht collection efficiency η0 and its formulations for exponential absorption have been widely used in modeling charge collection efficiency in photoconductive detectors. The basic assumption of the Hecht formulation is that the electric field in the device is uniform, i.e., the photoinjected carriers do not perturb the field. Here, we have used Monte Carlo simulations to model the initial injection of electron and hole pairs and their subsequent transport and trapping in the presence of an electric field, which is calculated from the Poisson equation. Each injected carrier is tracked as it moves in the semiconductor until it is either trapped or reaches the collection electrode. Trapped carriers do not contribute to the photocurrent but continue to contribute to the field through the Poisson equation. The instantaneous photocurrent iph(t) is calculated from the drift of the free carriers through the Shockley–Ramo theorem. iph(t) is integrated over the duration of the photocurrent to calculate the total collected charge and hence the collection efficiency ηr. ηr has been calculated as a function of the charge injection ratio r, the electron and hole ranges (drift mobility and lifetime products, μτ), mean photoinjection depth δ, and drift mobility ratio b. The deviation of the collection efficiency ηr from the uniform field case η0 can be as much as 30% smaller than the small signal model prediction. However, for a wide range of electron and hole schubwegs and photoinjection ratios, typical errors remained less than 10% at full injection, the worst case. The present study provides partial justification to the wide-spread use of the uniform-field collection efficiency η0 formula in various applications, even under high injection conditions.

Sensitivity of X-ray image detectors: Space charge effects

A model for calculating the sensitivity of direct conversion X-ray image detectors is developed incorporating the space charge effect. The space charge perturbs the applied electric field across the thickness of the photoconductor, and thus the electric field becomes nonuniform. First, Poisson's equation is solved to obtain the electric field distribution. Then, the continuity equations for both holes and electrons are solved considering the perturbed electric field. Finally, the collected charge is determined from the charge carrier distributions and electric field profile using the Shockley–Ramo theorem. The developed X-ray sensitivity model is applied to polycrystalline mercuric iodide based detectors. The electric field collapses for a critical amount of trapped charge if the ratio of trapped electrons to holes is high (∼0.5). The pinch in electric field slows down the drift of free electrons and hence reduces the sensitivity of the detector. The developed model is fitted with the published experimental data, and the fitted values of effective mobility-lifetime products of charge carriers agree well with the published results. It is found that the space charge reduces the sensitivity in the screen-printed sample more significantly than in the vacuum deposited sample.

Designs and electric properties studied of 3D trench electrode Si detector with adjustable central collection electrode

In order to improve the radiation resistance of semiconductor detector, 3D trench electrode Si detector structures have been proposed by the Brookhaven National Laboratory (BNL) in 2009. In this paper, we propose a 3D trench electrode Si detector with adjustable central collection electrode, which integrates advantages of cylindrical units and parallel units. For those combined units, we use 1D Poisson equation as the first order approximation to study the changing of depletion in different regions. These results have been compared to those obtained from the full 3D simulation using a 3D-TCAD tool. According to calculations, we design a 3D trench electrode Si detector with adjustable central collection electrode. Furthermore, electrical characteristics of this type of detectors with different lengths of central collection electrode (lp+) have been simulated using the 3D-TCAD. Regarded as the main factor of signal-noise ratio, the geometric capacitance obtained from Silvaco TCAD simulations is compared with which is calculated by physical models when detectors size varies. Finally, the physic model of charge collection is established using Ramo theorem. The CCE (Charge Collection Efficiency) of detector unit is studied with this model (after Φ=1016 neq/cm2, average CCE is 38%).

Origin of Low-Energy Spurious Peaks in Spectroscopic Measurements With Silicon Detectors

When an uncollimated radioactive X-ray source illuminates a silicon PIN sensor, some ionizing events are generated in the nonimplanted gap between the active area of the sensor and the guard rings (GRs). Carriers can be collected by floating electrodes, i.e., electron accumulation layers at the silicon/oxide interface, and floating GRs. The crosstalk signals generated by these events create spurious peaks, replicas of the main peaks at either lower amplitude or of opposite polarity. We explain this phenomenon as crosstalk caused by charge collected on these floating electrodes, which can be analyzed by means of an extension of Ramo theorem.

Simulation study of signal formation in position sensitive planar p-on-n silicon detectors after short range charge injection

Segmented silicon detectors (micropixel and microstrip) are the main type of detectors used in the inner trackers of Large Hadron Collider (LHC) experiments at CERN. Due to the high luminosity and eventual high fluence of energetic particles, detectors with fast response to fit the short shaping time of 20–25 ns and sufficient radiation hardness are required. Charge collection measurements carried out at the Ioffe Institute have shown a reversal of the pulse polarity in the detector response to short-range charge injection. Since the measured negative signal is about 30–60% of the peak positive signal, the effect strongly reduces the CCE even in non-irradiated detectors. For further investigation of the phenomenon the measurements have been reproduced by TCAD simulations. As for the measurements, the simulation study was applied for the p-on-n strip detectors similar in geometry to those developed for the ATLAS experiment and for the Ioffe Institute designed p-on-n strip detectors with each strip having a window in the metallization covering the p+ implant, allowing the generation of electron-hole pairs under the strip implant. Red laser scans across the strips and the interstrip gap with varying laser diameters and Si-SiO2 interface charge densities (Qf) were carried out. The results verify the experimentally observed negative response along the scan in the interstrip gap. When the laser spot is positioned on the strip p+ implant the negative response vanishes and the collected charge at the active strip increases respectively. The simulation results offer a further insight and understanding of the influence of the oxide charge density in the signal formation. The main result of the study is that a threshold value of Qf, that enables negligible losses of collected charges, is defined. The observed effects and details of the detector response for different charge injection positions are discussed in the context of Ramo's theorem.

Calculation of acousto-electronic devices based on original models of piezoelectric and magnetostriction currents

A method of equivalent currents and energy method allow significant simplification of the calculation of acousto-electronic devices in comparison with alternative methods based on the normal Auld modes, Shockley–Ramo theorem, reciprocity principle, and Green function. A common character of four original models of piezoelectric and magnetostriction currents that serve as sources of electromagnetic perturbations is demonstrated and proven.

Neural-based pile-up correction and ballistic deficit correction of X-ray semiconductor detectors using the Monte Carlo simulation and the Ramo theorem

ABSTRACTPile-up distortion is a common problem in many nuclear radiation detection systems, especially in high count rates. It can be solved by hardware-based pile-up rejections, but there is no complete pile-up elimination in this way. Additionally, the methods can lead to poor quantitative results. Generally, time characteristics of semiconductor detector pulses are different from Scintillator detector pulses due to ballistic deficit. Hence, pulse processing-based pile-up correction in the detectors should consider this specification. In this paper, the artificial neural network pile-up correction method is applied for silicon detector piled-up pulses. For this purpose, the interaction of photons with a silicon detector is simulated by the MCNP4c code and the pulse current is calculated by Ramo's theorem. In this approach, we use a sub-Nyquist frequency sampling. The results show that the proposed method is reliable for pile-up correction and ballistic deficit in semiconductor detectors. The technique is remarkable for commercial considerations and high-speed, real-time calculations.

The influence of electron track lengths on the γ-ray response of compound semiconductor detectors

The charge-trapping effect in compound semiconductor γ-ray detectors in the presence of a uniform electric field is commonly described by Hecht׳s relation. However, Hecht׳s relation ignores the geometrical spread of charge carriers caused by the finite range of primary and secondary electrons (δ-rays) in the detector. In this paper, a method based on the Shockley–Ramo theorem is developed to calculate γ-ray induced charge pulses by taking into account the charge-trapping effect associated with the geometrical spread of charge carriers. The method is then used to calculate the response of a planar CdTe detector to energetic γ-rays by which the influence of electron track lengths on the γ-ray response of the detectors is clearly shown.

Can an “impulse response” really be defined for a photoreceiver?

In this paper we examine the validity of the concept of impulse response employed to characterize the time response and the signal-to-noise ratio of p-i-n and similar photodetecting devices. We analyze critically the way in which the formalism of analog linear systems has been extrapolated, by employing results from macroscopic electromagnetic theory such as the Shockley–Ramo theorem or any equivalent approach, to the extreme case of a single-photon detection. We argue that the concept of “response to an optical impulse” is ill-defined in the customary terms it is envisioned in the literature, this is, as an output current pulse having a certain predictable, calculated temporal shape, in response to the detection of an optical “Dirac delta” impulse, conceived in turn as the absorption of a single photon.