RU2794874C1 - Two-frequency resonator for a block of high-frequency transitions in polarized hydrogen and deuterium atoms - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: invention relates to the equipment of charged particle accelerators and is intended for use in sources of polarized particles: protons and deuterons and targets of polarized hydrogen and deuterium atoms. The device is a microwave resonator, which implementsσ transitions between the states of the hyperfine structure (HFS) in hydrogen and deuterium atoms, for which the direction of the magnetic field of the excited microwave oscillation coincides with the direction of the applied external static magnetic field. The resonators provide excitation of microwave oscillations at two frequencies: a frequency of about 1480 MHz for the RF transition in hydrogen atoms and a frequency of about 380 MHz for RF transitions in deuterium atoms. The resonators comprise electrodynamic systems symmetrical in longitudinal planes.

EFFECT: providing RF transitions with a high efficiency of proton vector polarization close to +1, both in hydrogen atoms, and vector polarization close to +1, and tensor polarization -2 (in combination with other blocks of RF transitions) by one resonator for deuterons in deuterium atoms while providing a single plane of polarization for both protons and deuterons, and increasing the efficiency of particle polarization.

3 cl, 12 dwg

Description

The invention relates to the technique of charged particle accelerators and is intended for use in sources of polarized ions - protons and deuterons and targets from polarized hydrogen and deuterium atoms.

Accelerated beams of polarized particles - protons and deuterons and targets of polarized atoms are used to study the fundamental properties of matter in experiments on the study of the spin structure of nucleons, the conservation of symmetries in nuclear interactions, and the spin dependence of nuclear interactions. The efficiency of expensive experiments being carried out is essentially determined both by the degree of polarization of the particle beam and by its intensity.

Beams of polarized protons and deuterons with a high degree of polarization are obtained by the atomic beam method, using the spatial separation of atoms in different states of the hyperfine structure during the passage of magnets with a sharply inhomogeneous field (such as Stern-Gerlach magnets) and exciting transitions between the states of the hyperfine structure (HFS) of atoms during the passage blocks of high-frequency (HF) transitions [Haeberli W 1967 Ann. Rev. Nucl. sci. 17 373-426].

The sources of polarized ions use blocks of RF transitions, working on the method of adiabatic passage Abraham-Winter [Abragam A and Winter J M 1958 Phys. Rev. Lett. 1374].

In this method, atoms are passed through a static magnetic field that has a gradient along the axis of the atomic beam. A resonator is placed in a gradient static magnetic field, which creates an RF magnetic field with a frequency corresponding to the energy difference between the spin states. The RF magnetic field can be directed perpendicular or parallel to the static magnetic field. In the first case, the excited transitions are called n-transitions, and in the second case, π-transitions. The efficiency of adiabatic RF transitions reaches values close to unity [A. Airapetian, N. Akopov et. al., "The HERMES polarized hydrogen and deuterium gas target in the HERA electron storage ring", Nuclear Instruments and Methods in Physics Research A 540 (2005) 68-101].

The essence of the invention is illustrated by the attached drawings.

и

, где

и

По вертикальной оси - энергия состояний в единицах энергии сверхтонкого расщепления в нулевом внешнем магнитном поле. Для атомов водорода энергия сверхтонкого расщепления

соответствует частоте ВЧ колебаний 1420,4 МГц, для дейтерия

соответствует частоте 328,4 МГц. ВЧ переходы разделяются на переходы в слабом, промежуточном и сильном поле в зависимости от отношения величины статического магнитного поля в блоке ВЧ перехода по отношению к критическому полю для атомов водорода и дейтерия.The energy diagrams of the states of the HFS of hydrogen atoms and deuterium atoms in a magnetic field are shown in Fig.1. The splitting of energy levels is due to the Zeeman effect and occurs due to the presence of a magnetic moment of hydrogen and deuterium atoms in the ground state 1S _1/2 . The horizontal axis shows the magnetic field in relative units

And

, Where

And

On the vertical axis - the energy of states in units of energy of hyperfine splitting in a zero external magnetic field. For hydrogen atoms, the hyperfine splitting energy

corresponds to the frequency of HF oscillations 1420.4 MHz, for deuterium

corresponds to a frequency of 328.4 MHz. RF transitions are divided into transitions in a weak, intermediate, and strong field, depending on the ratio of the static magnetic field in the RF transition block with respect to the critical field for hydrogen and deuterium atoms.

(см. фиг.1), а для атомов дейтерия - состояния

В сильном магнитном поле

ядерная поляризация атомов в этих состояниях будет близка к нулю, так как, например, для водорода в состоянии

проекция спина протона

а в состоянии

проекция спина протона

Atoms in states with Ms=+1/2, when passing through magnets with a sharply inhomogeneous field (separating magnets), are focused, and atoms in states with Ms=-1/2 are defocused, so that after passing through the magnets in the beam, mainly atoms in states with Ms=+1/2; for hydrogen atoms, these are the states of the hyperfine structure

(see figure 1), and for deuterium atoms - state

In a strong magnetic field

the nuclear polarization of atoms in these states will be close to zero, since, for example, for hydrogen in the state

proton spin projection

but able

proton spin projection

To obtain beams of polarized protons and deuterons, polarized atoms are ionized using the ionization of atoms by various methods, but mainly in a strong magnetic field. Therefore, the separation of atoms by spin states in magnets with a sharply inhomogeneous field is insufficient to obtain beams of protons and deuterons with a high degree of polarization. An increase in nuclear polarization is achieved by using transitions between the spin states of atoms that have passed through separating magnets.

Next, a simplified designation of RF transitions between the states of the STS of hydrogen and deuterium atoms is used in accordance with the notation of the STS in Fig.1, for example, the transition between the states |1> and |3> is designated as 1-3.

для водорода в состоянии

проекция спина протона

а в состоянии

проекция спина протона M_I=+1/2 (см. фиг.1).For the hydrogen atom, transitions 1-3 in a weak field and 2-4 in a strong field are used. The HF transition 2-4 for the hydrogen atom refers to a transitions in which the direction of the HF magnetic field coincides with the direction of the static magnetic field. After passing through the separating magnet and the RF transition block 2-4, the hydrogen atoms find themselves in states 1 and 4, which leads to the vector polarization of protons +1 after the ionization of atoms in a strong magnetic field, since in a strong magnetic field

for hydrogen in the state

proton spin projection

but able

the projection of the proton spin M _I =+1/2 (see figure 1).

The transition is excited by a high-frequency magnetic field, the frequency of the field is determined by the energy difference between the spin states between which the transition is excited and, consequently, by the value of the static magnetic field in which the atoms are located. The frequency of the RF vibration corresponding to the transition is calculated by the formula:

, а частота выражена в мегагерцах. В статическом поле В=15 mT, которое используется для перехода 2-4 в атомах водорода, F₂₄=1480,148 МГц.Where

, and the frequency is expressed in megahertz. In the static field B=15 mT, which is used for the 2-4 transition in hydrogen atoms, F ₂₄ =1480.148 MHz.

и

Затем пропускают пучок атомов через блок ВЧ перехода 1-4 так что после прохождения этого блока в пучке атомы дейтерия оказываются в состояниях

Далее пучок проходит через второй разделительный магнит, после которого в пучке остаются атомы дейтерия в состояниях

так как атомы в состояниях

при прохождении разделительного магнита дефокусируются и выбывают из пучка. Затем пучок атомов дейтерия пропускается через блок ВЧ перехода 3-5, так что после прохождения этого ВЧ перехода атомы дейтерия в пучке оказываются в состояниях

в которых тензорная поляризация атомов дейтерия имеет максимальное теоретическое значение -2. Реально достижимое значение тензорной поляризации определяется эффективностью ВЧ переходов 1-4 и 3-5.To obtain polarized deuterons with an extremely high degree of tensor polarization, a more complex scheme is used. First, using separating magnets, a beam of deuterium atoms is obtained in the states

And

Then a beam of atoms is passed through the RF transition block 1-4 so that after passing through this block in the beam, the deuterium atoms find themselves in the states

Next, the beam passes through the second separating magnet, after which deuterium atoms remain in the beam in the states

because the atoms are in states

when passing through the separating magnet, they are defocused and drop out of the beam. Then the beam of deuterium atoms is passed through the RF transition block 3-5, so that after passing through this RF transition, the deuterium atoms in the beam find themselves in the states

in which the tensor polarization of deuterium atoms has a maximum theoretical value of -2. The actually achievable value of the tensor polarization is determined by the efficiency of RF transitions 1-4 and 3-5.

(после первого разделительного магнита) пропускают через блок ВЧ перехода с промежуточным полем, в котором возбуждается переход 3-4 так что после перехода в пучке атомы оказываются в состояниях

Атомы в состоянии

дефокусируются затем при прохождении второго разделительного магнита, так что в пучке остаются атомы в состояниях

Пучок затем проходит через блок ВЧ переходов 2-6, так что после блока в пучке атомы оказываются в состояниях

в которых в сильном поле векторная поляризация дейтронов равна +1. Переходы 3-5 и 2-6 показаны на фиг.3. Частота ВЧ колебаний для этих переходов выбирается одинаковой для того, чтобы можно было использовать один и тот же ВЧ резонатор для обоих переходов и равна обычно - 380 МГц. Необходимая для обеспечения высокой эффективности ВЧ перехода (близкой к единице) величина составляющей Вх равна (1-2)⋅10^-4 Т [A. Airapetian, N. Akopov et. al., "The HERMES polarized hydrogen and deuterium gas target in the HERA electron storage ring", Nuclear Instruments and Methods in Physics Research A 540 (2005) 68-101].The 2-6 transition in deuterium atoms is used to obtain a deuteron vector polarization close to +1. In this case, the beam of deuterium atoms in the states

(after the first separating magnet) is passed through an RF transition block with an intermediate field, in which the 3-4 transition is excited, so that after the transition in the beam, the atoms are in the states

Atoms in the state

are then defocused during the passage of the second separating magnet, so that atoms remain in the beam in the states

The beam then passes through a block of RF transitions 2-6, so that after the block in the beam, the atoms are in the states

in which the vector polarization of deuterons is equal to +1 in a strong field. Transitions 3-5 and 2-6 are shown in Fig.3. The frequency of the RF oscillations for these transitions is chosen the same so that the same RF resonator can be used for both transitions and is usually equal to 380 MHz. The value of the Bx component required to ensure high efficiency of the RF transition (close to unity) is equal to (1-2)⋅10 ^-4 T [A. Airapetian, N. Akopov et. al., "The HERMES polarized hydrogen and deuterium gas target in the HERA electron storage ring", Nuclear Instruments and Methods in Physics Research A 540 (2005) 68-101].

Both transitions 3-5 and 2-6 for deuterium atoms refer to a transitions in which the HF magnetic field is parallel to the external static one. It should be noted that the presence of the transverse component of the HF magnetic field leads to the appearance of transitions 2-5 and 3-6, which are related to % transitions. Simultaneous excitation of transitions 2-6 and 2-5 leads to a decrease in the vector polarization of deuterons, and simultaneous excitation of transitions 3-5 and 3-6 leads to a decrease in the tensor polarization of deuterons.

Polarized ion sources with an atomic beam can alternately generate beams of polarized protons and deuterons. To switch the source operation from polarized protons to polarized deuterons, it is necessary, in particular, to use transitions 2-6 and 3-5 in deuterium atoms instead of the HF transition 2-4 in hydrogen atoms

The high-field RF transition device for hydrogen or deuterium atoms is shown schematically in Fig.4.

The static magnetic field has a vertical direction and is created by a C-shaped magnet, the poles of which are profiled to provide a gradient in the magnitude of the magnetic field in the direction of movement of the hydrogen or deuterium atoms in the magnet (shown in Fig.4 by an arrow). A high frequency magnetic field is generated in a resonator placed between the poles of a static magnet.

A solution that makes it possible to implement a resonant system in a limited volume and with a transverse component of the magnetic field Bx is the use of a two-wire line segment inside the resonator, the conductors of which, being vibrators, are short-circuited at one end on the resonator body and have open second ends. In order to provide the required frequency of oscillations with a wavelength in free space of λ~92 cm with resonator dimensions in any of the coordinates << λ/4, it is necessary to provide a strong capacitive load at the open ends.

One such device is used in the source of polarized deuterons [V.V. Fimushkin, D. E. Donets, A.D. Kovalenko, L.V. Kutuzova, Yu.V. Prokofichev, V.B. Shutov (JINR), A.S. Belov, V.N. Zubets, A.V. Turbabin (INR RAS), "Polarized Ion Source for the JINR accelerator complex", report on XVII WORKSHOP ON HIGH ENERGY SPIN PHYSICS DSPIN2017, Dubna, September 15, 2017, IOP Conf. Series: Journal of Physics: Conf.Series 938(2017) 012017], the vibrators in it are made as segments of a flat strip line. In a system of two vibrators, there are two oscillations with in-phase or anti-phase direction of surface currents. The working one is an oscillation with an antiphase distribution of currents.

The resonator device of the high-frequency transition block in a strong field is shown in Fig. 5, where:

1 - input cable (HF power input);

2 - measuring loop;

3 - strip vibrator;

4 - tuning (movable) rod.

The input of high-frequency power is carried out through a galvanic connection of the input cable (1) with one of the two strip vibrators (3) of the resonator. In a system of two vibrators, the existence of two HF oscillations is possible. The worker is an antiphase RF oscillation, for which the magnetic field of the RF oscillation in the working area between the vibrators is maximum. The frequency of the first HF oscillation of a vibrator with length L grounded at one end is approximately equal to f1 = c/(4L), where c is the speed of light and is reduced by introducing a capacitive load at the free end of the vibrator. The capacitance value is changed by changing the gap value by moving the adjusting rod (4). The measuring loop (2) can be used in the feedback system to measure the oscillation amplitude and stabilize the resonator frequency.

In polarized ion sources, switching from operation with polarized protons to operation with polarized deuterons requires two high-field RF transition blocks: one for operation with polarized hydrogen atoms, and the second for operation with deuterium atoms. To increase the intensity of the polarized ion beam in the source, it is important to reduce the total length of RF transition blocks. As a rule, the same static field magnet is used for HF transitions in hydrogen atoms (2-4) and deuterium atoms (2-6 and 3-5), but different resonators tuned to different frequencies for hydrogen atoms (~ 1480 MHz ) and deuterium atoms (~380 MHz). When switching from operation with polarized protons to operation with polarized deuterons, it is necessary to replace the resonators, which is a rather lengthy procedure, as it is associated with the admission of air into the vacuum installation, then pumping out the installation to a high vacuum, setting up the RF transition unit for working with hydrogen or deuterium atoms , measurements of the polarization of protons or deuterons. This procedure also turns out to be expensive if the replacement of the cavities results in a downtime of the operating accelerator.

To simplify the transition procedure and reduce its duration, a resonator has been developed in which it is possible to excite RF oscillations at two operating frequencies [V. Carassiti, G. Ciullo, P. Lenisa, A. Nass, Dual H and D cavity for the PAX target polarimeter, Physics of Particles and Nuclei 45, 283-284 (2014)]. This resonator is a prototype of the present invention, shown in Fig.6.

In this resonator, instead of one pair of parallel vibrators in a single-frequency resonator (see Fig.5), there are two pairs of parallel vibrators in the form of rods, offset relative to the plane of symmetry of the rectangular resonator housing and mounted on the same wall of the resonator. Vibrators for operation at a frequency of 1480 MHz when working with atomic hydrogen end with disks that form a capacitor that shortens the length of the resonator. Vibrators for operation at a frequency of 380 MHz (when working with atomic deuterium) are short-circuited to the resonator housing by two lumped capacitors, one of which is tunable. The resonator is tuned to the required frequency by changing the capacitance of the tunable capacitor. The static magnetic field is directed perpendicular to the plane in which the removable cover of the resonator housing is located (see Fig.6). In this case, the direction of the HF field, as follows from the geometry of the installation of the vibrators, does not coincide with the direction of the static field. With respect to the static magnetic field, the HF field both at a frequency of 1480 MHz and at a frequency of 380 MHz in this resonator has both a component parallel to the static field and a transverse component. This is a fundamental property of this resonator. This property leads to the simultaneous excitation of transitions 2-6 and 2-5 and 3-5 and 3-6. Moreover, if the resonance curves for the corresponding transitions overlap, then this simultaneous excitation of transitions leads to a decrease in the maximum achievable polarization in deuterium atoms. To achieve the maximum polarization of deuterons when using transitions 2-6 and 3-5, it is necessary to ensure that the HF magnetic field is parallel to the static magnetic field.

The first version of the proposed resonator, which provides two excitation modes at frequencies corresponding to transitions in hydrogen atoms 2-4 and deuterium 2-6 and 3-5, as well as the parallelism of the RF magnetic field to the static magnetic field, is shown in Fig.7. Dual-frequency resonator for RF transition block 2-4 for hydrogen atoms, 2-6 and 3-5 for deuterium atoms with high RF field uniformity and RF magnetic field direction along the static magnetic field:

1 - resonator housing;

2 - strip vibrators of the low-frequency system;

3 - strip vibrators of the high-frequency system.

The resonator contains two oscillatory systems, each of which consists of two strip vibrators symmetrical with respect to the YZ and XZ planes. The atoms pass through the resonator in the direction of the Z axis. Pairs of vibrators of each system are fixed with their bases on opposite walls of the resonator. For each pair of vibrators, the working one is an oscillation with an antiphase distribution of surface currents. The symmetry of the vibrators of both pairs relative to the YZ plane provides a single direction of the transverse component of the magnetic field for both working oscillations. This direction coincides with the direction of the static magnetic field in the RF transition block, which is directed along the X axis. This resonator differs from the prototype, in which the RF fields for the 2-4 transition in hydrogen atoms, 2-6 and 3-5 in deuterium atoms are directed at an angle to each other close to 70°-90°, and therefore do not coincide with the direction of the static magnetic field. This property of the proposed resonator leads to an increase in the efficiency of RF transitions 2-6 and 3-5 and is an advantage of the considered resonator design.

To ensure the maximum value of the transverse component of the magnetic field, the electrical lengths in each pair of vibrators must be equal. For this purpose, an independent individual adjustment element will be provided in the technical design for each vibrator. The criterion for the correct setting of each pair of vibrators is the required value of the frequency of the working oscillation with a minimum difference in the frequencies of the working and in-phase oscillations.

The electrical length of the low-frequency system vibrators is much greater than the length of the high-frequency vibrators. Therefore, the vibrators of the low-frequency system are made by envelopes, and the vibrators of the high-frequency system with subsequent bends. For a low frequency system, the high frequency surface is grounded.

The required value Vx ~ 2⋅10 ^-4 T corresponds to a low maximum electric field strength and the distance ~ (2-3) mm between the vibrators of the systems is sufficient to ensure electrical strength.

The maximum value of the Bx component for each oscillation is achieved in the region of the bases of the corresponding pair of vibrators. Therefore, the maxima of the Bx distributions for the operating oscillations of the systems are shifted along the beam axis (longitudinal z axis).

To ensure maximum uniformity (minimum heterogeneity) Vx in transverse directions, at the bases, the strip vibrators of both systems are made in the form of a cylindrical surface of segments with an opening angle of (45+-2) degrees, Fig.8. Cross section of vibrators with elements of cylindrical surfaces in the axial region with an opening angle of 45+-2 degrees. The axis of the beam is directed perpendicular to the plane of the figure.

The distribution of the RF magnetic field in the resonators, the determination of the dimensions of the elements, and the determination of the operating frequencies were modeled using the COMSOL software package.

Figure 9 shows the distribution of the HF magnetic field amplitude obtained by modeling on the axis of the first variant of the resonator along the Z axis. Red color shows the distribution for oscillations with a frequency of 1480 MHz, blue color shows the distribution for oscillations of a frequency of 380 MHz. Distribution of the RF magnetic field amplitude along the resonator axis for two modes of the first version of the resonator. The red color shows the distribution for oscillations with a frequency of 380 MHz, the distribution for oscillations with a frequency of 1480 MHz is shown in blue. The horizontal axis shows the distance along the beam axis in mm, the vertical axis shows the amplitude of the HF magnetic field in relative units.

In the proposed resonator, it is possible to excite an HF magnetic field both at a frequency of 1480 MHz and at a frequency of 380 MHz to implement HF transitions 2-4 in hydrogen atoms and transitions 2-6 and 3-5 in deuterium atoms. The direction of the RF magnetic field coincides with the direction of the static magnetic field, which leads to an increase in the efficiency of the above RF transitions.

To ensure the small dimensions of the resonator, it is necessary to provide a strong capacitive load at the open ends of the vibrators of the low-frequency system, which leads to small distances between the surfaces of the vibrators and the resonator, which must be maintained during manufacture with high accuracy.

The introduction of plates of microwave dielectric, tightly adjacent to the surface of the resonator and open ends of the vibrators of the low-frequency system allows both to increase the distance between the vibrators and the surface of the resonator, and to reduce the length of the vibrators, Fig.9. The dimensions of the dielectric plates and vibrators are selected as a result of numerical calculations, taking into account the characteristics of the dielectric. Thus formed the second version of the two-frequency resonator is shown in Fig.10. Dual-frequency resonator for RF transition block 2-4 for hydrogen atoms and 2-6 and 3-5 for deuterium atoms with high RF field uniformity and RF magnetic field direction along the static magnetic field:

1 - resonator housing;

2 - vibrators of the low-frequency system;

3 - vibrators of the high-frequency system;

4 - adjacent dielectric plates;

5 - beam axis.

A two-frequency resonator is also implemented with one oscillatory system of two vibrators fixed on one wall of the resonator. The frequency of the second RF oscillation grounded at one end of the vibrator length L is approximately equal to f2=3c/(4L). The required operating frequencies f1=380 MHz and f2=1480 MHz are provided both by selecting the length of the vibrators and by introducing a capacitive load, together with the placement of adjacent microwave dielectric plates, at the open ends of the vibrators. The thus formed third version of the resonator is shown in Fig.11. The required dimensions of the dielectric plates and vibrators are selected as a result of numerical calculations, taking into account the characteristics of the dielectric. As a working oscillation with a frequency of 380 MHz, the first antiphase HF oscillation of the vibrators is used, and as a working oscillation with a frequency of 1480 MHz, the second antiphase microwave oscillation is used. Dual-frequency resonator for RF transition block 2-4 for hydrogen atoms and 2-6 and 3-5 for deuterium atoms with high RF field uniformity and RF magnetic field direction along the static magnetic field.

In this design, Fig.11, the area of interaction of particles with the field of working RF oscillations expands along the beam axis to the full length of the resonator. The calculated amplitude distributions of the RF magnetic field along the beam axis for two modes of the resonator shown in Fig. 11 are shown in Fig. 12, where: 1 - distribution for oscillations with a frequency of 380 MHz, 2 - distribution for oscillations with a frequency of 1480 MHz. The horizontal axis shows the distance along the beam axis in mm, the vertical axis shows the amplitude of the HF magnetic field in relative units.

Claims (3)

1. A two-frequency resonator for a block of adiabatic RF transitions 2-4 in hydrogen atoms, 2-6 and 3-5 in deuterium atoms, including a housing, two RF power inputs and two feedback loops, containing two oscillatory systems of two strip vibrators each, providing the excitation of two oscillations with frequencies of about 380 MHz and about 1480 MHz in systems with vibrators, characterized in that in order to ensure the parallelism of the HF magnetic field vector to the direction of the static magnetic field, its uniformity and thus to increase the efficiency of HF transitions for hydrogen and deuterium atoms and the degree of polarization of protons and deuterons in a two-frequency resonator, the vibrators of the low-frequency and high-frequency systems are grounded on opposite walls of the resonator, the surface of the vibrators adjacent to the beam passage channel is made in the form of a cylindrical surface with an opening angle of 45 ± 2°, and the vibrators are made symmetrical with respect to the longitudinal planes with respect to direction of motion of the particle beam. 2. A two-frequency resonator for a block of adiabatic RF transitions 2-4 in hydrogen atoms, 2-6 and 3-5 in deuterium atoms according to claim 1, characterized in that in order to increase the distance between the vibrators of the low-frequency system and thus reduce the accuracy requirements between the resonator case and the open ends of the vibrators of the low-frequency system, adjacent plates of microwave dielectric with a relative permittivity from 2 to 10, a minimum dielectric loss tangent are introduced, and the dimensions of the plates are selected to provide the required frequencies of operating oscillations. 3. A two-frequency resonator for a block of adiabatic RF transitions 2-4 in hydrogen atoms, 2-6 and 3-5 in deuterium atoms according to claim 2, characterized in that it has only one oscillatory system of two vibrators grounded on one wall resonator, the first antiphase microwave oscillation was used as a working oscillation with a frequency of about 380 MHz, and the second antiphase oscillation of vibrators was used as a working oscillation with a frequency of about 1480 MHz, the frequencies of which are chosen both by choosing the length of the vibrators and by the dimensions of the adjacent dielectric plates.

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