Brownian thermal noise is the phase noise caused at nonzero temperature by random motions of the reflecting faces of mirrors in a GW interferometer. A reflecting face can move either because it is displaced by its suspension system or because it undergoes internal stresses. At finite temperature the two effects are possible. We address here the internal stresses. Consider a massive body at temperature . If , the atoms constituting the body are excited and have random motions around their equilibrium position. The fact that they are strongly coupled to neighboring atoms makes possible the propagation of elastic waves of various types, reflecting on the faces, and the onset of stationary waves. One can show that, for a finite body (e.g., a cylinder of silica), there is a discrete infinity of such stationary waves, each corresponding to a particular elastic normal mode. At thermal equilibrium, the state of the body can be represented by a linear superposition of all the modes, with random relative phases, and, due to the energy equipartition theorem, the same energy ( is the Boltzmann constant). The motion of atoms near a limiting surface of the body will modify its shape slightly, and, if we consider the reflecting face of a mirror, a surface distortion is a possible cause of phase change in the reflected beam, in other words, of a noise. Estimation of the resulting spectral density of phase noise is the internal thermal noise problem in massive mirrors.

8.1 The Fluctuation-Dissipation theorem and Levin’s generalized
coordinate method

8.2 Infinite mirrors noise in the substrate

8.3 Infinite mirrors, noise in coating

8.3.1 Coating Brownian thermal noise: LG modes

8.3.2 Coating Brownian thermal noise: Flat modes

8.4 Finite mirrors

8.4.1 Equilibrium equations

8.4.2 Boundary conditions

8.4.3 Strain energy

8.4.4 Explicit displacement and strain tensor

8.5 Coating Brownian thermal noise: finite mirrors

8.2 Infinite mirrors noise in the substrate

8.3 Infinite mirrors, noise in coating

8.3.1 Coating Brownian thermal noise: LG modes

8.3.2 Coating Brownian thermal noise: Flat modes

8.4 Finite mirrors

8.4.1 Equilibrium equations

8.4.2 Boundary conditions

8.4.3 Strain energy

8.4.4 Explicit displacement and strain tensor

8.5 Coating Brownian thermal noise: finite mirrors

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