where
is Boltmann's constant,
T
is the temperature,
m
is the mass and
is the loss angle or loss factor of the oscillator of angular
resonant frequency
. This loss factor is the phase lag angle between the
displacement of the mass and any force applied to the mass at a
frequency well below
. In the case of a mass on a spring the loss factor is a measure
of the mechanical loss associated with the material of the
spring. For a pendulum, most of the energy is stored in the
lossless gravitational field. Thus the loss factor is lower than
that of the material which is used for the wires or fibres used
to suspend the pendulum. Indeed following Saulson [87
] it can be shown that for a pendulum of mass
m, suspended on four wires or fibres of length
l, the loss factor of the pendulum is related to the loss factor
of the material by
where
I
is the moment of the cross section of each wire, and
T
is the tension in each wire whose material has a Young's modulus
E
. In general for most materials it appears that the intrinsic
loss factor is essentially independent of frequency over the
range of interest for gravitational wave detectors (although care
has to be taken with some materials in that a form of damping
known as thermo-elastic damping can become important for wires of
small cross-section [73] and for some bulk crystalline materials [17
]). In order to estimate the internal thermal noise of a test
mass, each resonant mode of the mass can be regarded as a
harmonic oscillator. When the detector operating range is well
below the resonances of the masses, following Saulson [87], the effective spectral density of thermal displacement of the
front face of each mass can be expressed as:
In this formula
m
is the mass of the test mass,
is an angular frequency in the operating range of the detector,
is the resonant angular frequency of the fundamental mode,
is the intrinsic material loss, and
is a correction factor to include the effect of summation of the
motion over the higher order modes of the test mass (taking into
account the effect of a finite optical beam size and correction
for the effective masses of the modes). Typically, as calculated
by Gillespie and Raab [41],
is a number less than 10. A different and more general treatment
of internal thermal noise using evaluation of the relevant
mechanical impedance has been carried out by Bondu et al. [15]. This was based on work of Yuri Levin [62] and gives good agreement with the results of Gillespie and
Raab.
In order to keep thermal noise as low as possible the
mechanical loss factors of the masses and pendulum resonances
should be as low as possible. Further the test masses must have a
shape such that the frequencies of the internal resonances are
kept as high as possible, must be large enough to accommodate the
laser beam spot without excess diffraction losses, and must be
massive enough to keep the fluctuations due to radiation pressure
at an acceptable level. Test masses range in mass from 6 kg
for GEO 600 to 30 kg for the first proposed upgrade to
LIGO. To approach the best levels of sensitivity discussed
earlier the loss factors of the test masses must be
or lower, and the loss factor of the pendulum resonances should
be smaller than
. Discussions relevant to this are given in [82
,
80]. Obtaining these values puts significant constraints on the
choice of material for the test masses and their suspending
fibres. One viable solution which should allow detector
sensitivities to approach the level desired for upgraded
detectors is to use fused silica masses hung by fused silica
fibres [19,
84], as the intrinsic loss factors in samples of synthetic fused
silica have been measured at around
[65,
95]. Still, the use of other materials such as sapphire is being
seriously considered for future detectors as mentioned in
section
6
[18,
57,
82]. The technique of hydroxy-catalysis bonding provides a method
of jointing oxide materials in a suitably low loss way to allow
`monolithic' suspension systems to be constructed [83]. A picture of such a prototype fused silica mass suspended by
two fused silica fibres, which has been constructed in Glasgow
and is being tested at the University of Perugia, is shown in
Fig.
5
.
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Gravitational Wave Detection by Interferometry (Ground
and Space)
Sheila Rowan and Jim Hough http://www.livingreviews.org/lrr-2000-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |