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论文范文
1.Introduction The first discovery of the piezoelectric effect was found in 1880 by Jacques and Pierre Curie. They discovered that a quartz crystal submitted to mechanical action will produce an electrical potential that is proportional to the force applied. This effect is called direct piezoelectric effect. In the next year, Gabriel Lippmann described an analytical method for the reverse piezoelectric effect and the Curie brothers demonstrated this theory in the laboratory. Since its discovery, the piezoelectricity has been used in different fields such as defence, medical diagnosis, particle detection, ultrasonics, motors, and echolocation [1, 2]. Due to the different vibrations modes in a piezoelectric material, the applications of this phenomenon can be different in each field. The first piezoelectric material used was the quartz; however, this material is not very good to transform the electrical energy into mechanic energy and its sensitivity is small. Later, in the middle of XIX century, lead zirconate titanate crystals (PZT) were developed. This material has a better sensitivity and frequency response. This material is produced using a very intense electrical field that polarizes the ceramic in a specific direction. Piezoceramic transducers have usually a regular geometry and the knowledge of the behaviour of piezoceramic disks is important for transducer design and applications [3, 4]. With these transducers, it is possible to generate vibrations from a few to several hundreds of kHz, demonstrating its feasibility to be used as ultrasonic sensors and actuators in this frequency range. For example, recent applications use piezoelectric sensors in order to detect the acoustic signal emitted by the interaction of elementary particles in a fluid target. In this sense, the dark matter detectors PICO bubble chambers use a superheated fluid target filled in a glass vessel [5, 6]. Under some thermodynamic conditions, the particle interaction produces a bubble nucleation within the fluid and acoustic waves are emitted during the bubble growth. These sensors are glued to the external walls of the vessel that contains the metastable fluid. The acoustic discernment of the different acoustic events depends forcefully on the complete sensor properties and there are constraints as well in the little size and radiopurity of the piezoelectric ceramics [7]. These detectors use circular piezoceramics (both cylindrical and disk types) in the transducer design for a frequency bandwidth up to 150 kHz. These show the influence of bonding the ceramic to the vessel in its acoustic response. Among others, it is extracted that there is an increase of the sensors sensitivity, mainly in low frequencies. This is due to a better adaptation of acoustic impedances between the medium and the ceramic, through the glass of the vessel [7]. However, the first step in the design of the final transducer is the choice of the type and size of the ceramic taking into account its final use. For this, and since low radioactivity is a must, the amount of ceramic used in the transducer is a factor of great importance. This is because the ceramic material contains lead, which is usually accompanied with heavy radioisotopes that are alpha emitters and thus a source of background through alpha-neutron reactions. In this paper, we explain a proven methodology to study the optimization of this type of sensors. For this, several circular section (disks or cylinders) PIC255 piezoceramics with different width and height are studied with analytical and numerical methods and the results obtained are contrasted by experimental measurements. In previous works with circular PZT, some authors studied the natural vibration modes of axial symmetric piezoceramics using Finite Element Methods (FEM) [3] focusing on the efficiency and transducer design. Theoretical analysis contrasted with experimental measurements of resonant vibration with interferometry and laser Doppler vibrometer have been used [8]. Additional characterization methods and theoretical approach formulas can also be found. Here, the main parameters of piezoceramics will be compared: the resonance, , and antiresonance, , frequencies of the electrical impedance and its amplitude, the product of each frequency resonance and the length (thickness, , or diameter, ) associated with the mode of vibration , and the piezoelectric coupling factor, . Moreover, the ratio of coupling factors of the lowest piezoelectric modes gives us a quantified estimation of the energy distribution in this frequency range. The conclusions of these studies lead us to have design principles to select a specific piezoceramic circular geometry with a radio-clean piezoelectric material that can be used in the next generation of dark matter bubble chamber detectors (PICO 500 L) [9]. 
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