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Joseph Weber used a massive aluminum cylinder and measured its deformation, in his attempt to detect gravitational waves. Why did he choose aluminium? Would a solid diamond deform as much as solid rubber of the same size, weight and shape? (It takes more force for a diamond ball to be compressed than for a soft rubber ball to be deformed by the Earth's gravity.)

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    $\begingroup$ An interesting question; perhaps you could back it up with some of your prior research (so we don't repeat stuff you've already read) So if you have read the Wikipedia pages or perhaps the articles in online magazines, please link to those. $\endgroup$
    – James K
    Oct 14, 2023 at 18:40
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    $\begingroup$ Getting a large diamond is a whole problem unto itself… $\endgroup$
    – Jon Custer
    Oct 14, 2023 at 22:03
  • $\begingroup$ Well, the question was the choice of material and the desired characteristics of the chosen material. The comparison of diamond and rubber was just a theoretical example. $\endgroup$
    – Peter
    Oct 15, 2023 at 8:16

2 Answers 2

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The concept of the "resonant bar" detector for gravitational waves is that you get your cylinder of material to oscillate at its resonant frequency when pinged by a gravitational wave source. You then have some fancy electronics to measure the displacement of the ends of the bar at this resonant frequency.

The big problem is that the bar will be oscillating over a broad range of frequencies all the time just because of thermal motion.

In order to isolate the gravitational wave signal you want the resonance to be as sharp as possible so that you have the best chance of detecting the gravitational wave signal above a thermal noise signal that will then be collected from just a very narrow bandwidth around the sharp resonant frequency.

To clarify - the gravitational wave source "pings" the detector (thinking here of an impulsive signal from something like a supernova), but the cylinder then resonates for a time inversely proportional to the width of the resonance (i.e. like an almost undamped oscillator). This then allows you to integrate for a long time (perhaps hundreds of seconds) and thus compete with noise in only a very narrow range of frequencies.

At the same time, the amplitude response of the detector will be proportional to its mass and you want the resonant frequency to be at around the frequency where you expect to get gravitational waves, where the resonant frequency is proportional to the sound speed divided by the length of the bar.

So this is a classic design problem where you have multiple, conflicting constraints that you try and optimise. You want a bar made of a material that is dense, but has a high Q (quality) factor to give a sharp resonance and of a size that has a resonance at $\sim 1$ kHz (what was thought to be the right sort of signal frequency for a core-collapse supernova for example) and which is also not so massive that you need immensely expensive infrastructure to isolate from the Earth. Then there is also the question of money and how exotic a material you are prepared to purchase.

For all these reasons Weber settled on a bar made of aluminium, around 2-m in length, 1-m in diameter and weighing about 4000 kg. This has a characteristic resonant frequency at $f_r = \sim v/2L$. With the sound speed of aluminium $v\simeq 5000$ m/s and bar length $L=2$ m, $f_r \simeq 1.25$ kHz. It is also small enough to fit in a laboratory and not too massive to suspend from a modest frame. But the key thing appears to be a high Q-factor, which for such an aluminium cylinder at room temperature is around $\sim 10^6$ (i.e. the frequency width of the resonance is about a million times narrower than the resonance frequency) but 20 times higher at lower temperatures (e.g., Coccia and Niinikowski 1984).

Aluminium still appears to be the material of choice for these detectors although there is at least one example (the "Niobe" detector), that used pure Niobium at cryogenic temperature (Aguiar 2010), but which I imagine is very costly.

Diamond and Rubber

The Q-factor for single crystal diamond diamond could reach a few $10^5$ (so lower than Aluminium) and the sound speed is about 12000 m/s. A 1 kHz resonant detector would therefore need a pure "diamond cylinder" of length 1-m and would not work as well as aluminium even if such an object could be made.

Rubber has a very low Q factor. That is why is it used in damping systems.

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Weber never had a chance to discover a GW. The first reason is that the detector is only sensitive within an extremely small bandwidth $\Delta f=f_{resonance}/Q=1250/10^6$ Hz due to the extremely high Q factor $(Q\approx10^6)$ . How likely is it to find a GW source that produces a continuous signal in the narrow frequency range $1249.9988~Hz<f<1250.0013$ Hz?

But Weber tried to discover short-term coincidences $\Delta t<1$ second between two distant bars. With a high-Q resonator you can only detect continuous GW because it takes a very long time for the resonator to accumulate enough energy for the amplitude to become measurable. From Küpfmüller's uncertainty principle $\Delta t\Delta f>0.5$ follows $\Delta t>400$ seconds. What Weber measured was just noise.

A diamond resonator would be worse because the Q factor is larger.

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    $\begingroup$ The Q factor is lower for diamond unless the whole thing is held at cryogenic temperatures. The source does not have to produce GWs only in that frequency band, it could produce them over a broader range (as of course is predicted for supernovae), some of which would excite the resonance. The point of the resonant detector is that a short (possibly broadband) signal excites the resonance, which then "rings" for a long time and it is this that allows a longer integration time and gives you the chance to detect it over the noise. $\endgroup$
    – ProfRob
    Nov 16, 2023 at 0:57

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