Faraday cup
A Faraday cup is a metal (conductive) cup designed to catch charged particles in vacuum. The resulting current can be measured and used to determine the number of ions or electrons hitting the cup. The Faraday cup was named after Michael Faraday who first theorized ions around 1830. Examples of devices which use Faraday cups include space probes (Voyager 1, & 2, Parker Solar Probe, etc.) and mass spectrometers. Principle of operation. When a beam or packet of ions hits the metal, it gains a small net charge while the ions are neutralized. The metal can then be discharged  to measure a small current proportional to the number of impinging ions. The Faraday cup is essentially part of a circuit where ions are the charge carriers in vacuum and it is the interface to the solid metal where electrons act as the charge carriers (as in most circuits). By measuring the electric current (the number of electrons flowing through the circuit per second) in the metal part of the circuit, the number of charges being carried by the ions in the vacuum part of the circuit can be determined. For a continuous beam of ions (each with a single  charge), the total number of ions hitting the cup per unit time is where N is the number of ions observed in a time t (in seconds), I is the measured current (in amperes) and e is the elementary charge (about 1.60 × 10 C). Thus, a measured current of one nanoamp (10 A) corresponds to about 6 billion ions striking the Faraday cup each second. Similarly, a Faraday cup can act as a collector for electrons in a vacuum (e.g. from an electron beam). In this case, electrons simply hit the metal plate/cup and a current is produced . Faraday cups are not as sensitive as electron multiplier detectors, but are highly regarded for accuracy because of the direct relation between the measured current and number of ions. In plasma diagnostics. The Faraday cup utilizes a physical principle according to which the electrical charges delivered to the inner surface of a hollow conductor are redistributed around its outer surface due to mutual self-repelling of charges of the same sign – a phenomenon discovered by Faraday. The conventional Faraday cup is applied for measurements of ion (or electron) flows from plasma boundaries and comprises a metallic cylindrical receiver-cup – 1  (Fig. 1) closed with, and insulated from, a washer-type metallic electron-suppressor lid – 2 provided with the round axial through enter-hollow of an aperture with a surface area formula_2. Both the receiver cup and the electron-suppressor lid are enveloped in, and insulated from, a grounded cylindrical shield – 3 having an axial round hole coinciding with the hole in the electron-suppressor lid – 2. The electron-suppressor lid is connected by 50 Ω RF cable with the source formula_3 of variable DC voltage formula_4. The receiver-cup is connected by 50 Ω RF cable through the load resistor  formula_5 with a sweep generator producing saw-type pulses formula_6. Electric capacity formula_7 is formed of the capacity of the receiver-cup – 1 to the grounded shield – 3 and the capacity of the RF cable. The signal from formula_5 enables an observer to acquire an I-V characteristic of the Faraday cup by oscilloscope. Proper operating conditions: formula_9 (due to possible potential sag) and formula_10, where formula_11 is the ion free path. Signal from formula_5 is the Faraday cup I-V characteristic which can be observed and memorized by oscilloscope In Fig. 1: 1 – cup-receiver, metal (stainless steel). 2  – electron-suppressor lid, metal (stainless steel). 3 – grounded shield, metal (stainless steel). 4 – insulator (teflon, ceramic). formula_7 – capacity of Faraday cup. formula_5 – load resistor. Thus we measure the sum formula_15 of the electric currents through the load resistor formula_5: formula_17 (Faraday cup current) plus the current formula_18 induced through the capacitor formula_7 by the saw-type voltage formula_20of the sweep-generator: The current component formula_21 can be measured at the absence of the ion flow and can be subtracted further from the total current formula_22 measured with plasma to obtain the actual Faraday cup I-V characteristic formula_23 for processing . All of the Faraday cup elements and their assembly that interact with plasma are fabricated usually of temperature-resistant materials (often these are stainless steel and teflon or ceramic for insulators). For processing of the Faraday cup I-V characteristic, we are going to assume that the Faraday cup is installed far enough away from an investigated plasma source where the flow of ions could be considered as the flow of particles with parallel velocities directed exactly along the Faraday cup axis. In this case, the elementary particle current formula_24 corresponding to the ion density differential formula_25 in the range  of velocities between formula_26 and formula_27 of ions flowing in through operating aperture formula_28 of the electron-suppressor can be written in the form where formula_29 is elementary charge, formula_30 is the ion charge state, and formula_31 is the one-dimensional ion velocity distribution function. Therefore, the ion current at the ion-decelerating voltage formula_20 of the Faraday cup can be calculated by integrating Eq. () after substituting Eq. (), where the lower integration limit is defined from the equation formula_33 where formula_34 is the velocity of the ion stopped by the decelerating potential formula_20, and formula_36 is the ion mass. Thus  Eq. () represents the I-V characteristic of the Faraday cup. Differentiating Eq. () with respect to formula_20, one can obtain the relation where the value formula_38 is an invariable constant for each measurement. Therefore, the average velocity formula_39 of ions arriving into the Faraday cup and their average energy formula_40 can be calculated (under the assumption that we operate with a single type of ion) by the expressions where formula_41 is the ion mass in atomic units. The ion concentration formula_42 in the ion flow at the Faraday cup vicinity can be calculated by the formula which follows from Eq. () at  formula_43, and from the conventional condition for distribution function normalizing Fig. 2 illustrates the I-V characteristic formula_44 and its first derivative formula_45 of the Faraday cup with formula_46 installed at output of the Inductively coupled plasma source powered with RF 13.56 MHz and operating at 6 mTorr of H2. The value of the electron-suppressor voltage (accelerating the ions) was set experimentally at formula_47, near the point of suppression of the secondary electron emission from the inner surface of the Faraday cup. Error sources. The counting of charges collected per unit time is impacted by two error sources: 1 ) the emission of low-energy secondary electrons from the surface struck by the incident charge and 2) backscattering (~180 degree scattering) of the incident particle, which causes it to leave the collecting surface, at least temporarily. Especially with electrons, it is fundamentally impossible to distinguish between a fresh new incident electron and one that has been backscattered or even a fast secondary electron.