Applications and Metrology at Nanometer Scale 1. Abdelkhalak El Hami

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      Introduction

      The scientific study of measurement is known as metrology. Any measure is based on a universally accepted standard and any measuring process is prone to uncertainty. In engineering science, measurement concerns various types of parameters. Legal metrology is imposed by a regulatory framework that the manufactured product must respect. Technical or scientific metrology involves the methods used to measure the technical characteristics of the manufactured product. In engineering sciences, measurement concerns various types of parameters. In a more general context of a systemic approach, metrology should also be considered in connection with other indicators of the production system. These measures enable the follow-up and development of the processes implemented for ensuring and optimizing product quality or reducing failure so that it meets client expectations. The ability of a product to meet quality and reliability expectations can be addressed in the design stage, according to a RBDO (Reliability-Based Design Optimization) approach described in the “Reliability of Multiphysical Systems Set Book 2”, entitled Nanometer-scale Defect Detection Using Polarized Light. More generally, RBDO makes it possible to consider the uncertain parameters of manufacturing processes, measurement and operational conditions in order to optimize the manufacturing process, the design parameters and the overall quality of the product.

      Reliability of Multiphysical Systems Set Book 2 focused on three levels of design for manufacturing an industrial product:

      In sciences, there are various ways to measure a dimension. The measuring instruments or methods employed depend on the scale at which metrology is approached. In order to describe the issues at stake for measurement at a given scale, we present the methods employed for the measurement processes at two scales of interest for scientists, namely the infinitely small, which corresponds to the Planck length of 1.6 x 10^–35 m, and the infinitely large, which corresponds to the diameter of the Universe evaluated at 8.8 x 10²⁶ m. This is to help the reader understand that, even though becoming an expert in a scientific field or in a given subject is not the objective, it is necessary to understand some basic tenets in order to master the methods used for successful metrology at a given scale.

      In 1899, Planck determined a unit of length lP=√(Gh/2πc³)≈ 1.6 x 10^–35 m, referred to as Planck length, based on fundamental constants: G, gravitational constant (6.6 x 10^–11 Nm² Kg^–2), h, Planck’s constant (6.64 x 10^–34 Js) and c, the speed of light (2.99,729,458 x 10⁸ ms^–1). This length cannot be measured with the measurement technologies available on Earth. Indeed, the smallest length measurable at the LHC (Large Hadron Collider) of CERN, the particle accelerator in which two protons are made to frontally collide in a ring of 26,659 km, which led to the discovery in 2012 of the Higgs boson, is approximately 10^–16 m, which is 19 orders of magnitude higher than the Planck length. CMS and ATLAS detectors were used in the observation of the Higgs boson, the latest prediction of the standard model not yet observed. The measurement at the scale of 10^–16 m is made by compressing energy to reach an infinitely small spatial volume.

The principle of measurement at the scale of fundamental particles is mainly based on three relations: the de Broglie relation between the momentum p and the wavelength λ, p=h/λ, which introduces the wave– particle duality for matter; the relation that links the energy E of a particle to its wave frequency or wavelength λ, such as proposed by Einstein to explain the photoelectric effect E = hc/λ; and the relation that links the energy E of a particle of rest mass m to its rest mass energy and to its kinetic energy associated with its momentum p=mv, E²= m²c⁴ + p²c², as mentioned in Einstein’s special theory of relativity. In the above formulas, v is the speed of the particle of mass m and c is the speed of light. The energy E can also be expressed by the formula E= γmc², where γ is given by γ =1/√(1-v²/c²). The speed of a particle is therefore given by v/c=√(1-(mc²/E)².

      In the LHC, the energy of a proton is 7 TeV (1.2 10^–6 J), far higher (by a factor of 7,500) than its rest energy, mc², which is 938 MeV. The formula for speed can then be rewritten as v/c = (1-(m²c⁴/2E²)), which is equal to 1 to the nearest 10^–8. Using the relation E= hc/λ, the resulting value of the wavelength is of the order of 10^–16 m, which gives the dimensions that can be reached in the LHC. The mass measured during two experiments at CERN in the LHC (8 TeV in 2012 and 13 TeV in 2015) is confirmed to the value of 125 GeV.

      To detect the Higgs boson, a particle of mass 125 GeV associated with the Higgs field, while the mass of a proton is 938 MeV, the proton is accelerated and consequently its kinetic energy is increased so that its energy given СКАЧАТЬ