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.
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