Measurements of mechanical dissipation in high sound velocity materials: implications for resonant-mass gravitational radiation detectors

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PHYSICS LETTERS A

29 July 1991

Measurements of mechanical dissipation in high sound velocity materials: implications for resonant-mass gravitational radiation detectors En-Ke Hu Department of Physics and Gravitational Physics Institute, Zhongshan University, Guangzhou 510275, China

C. Zhou, L. Mann, P.F. Michelson Department of Physics and High Energy Physics Laboratory, Stanford University, Stanford, CA 94305, USA

and

J.C. Price Department of Physics, Universityof Colorado, Boulder, CO 80302, USA Received 3 April 1991; accepted for publication 10 April 199 l Communicated by J.P. Vigier The sensitivity of resonant-mass gravitational radiation detectors depends on both the antenna cross-sectionand the detector noise. The cross-sectionis determined by the sound velocity vsand density p of the antenna material, while the principal detector noise sources are thermal Nyquist noise and noise due to the readout electromechanicalamplifier. The thermal noise is proportional to T/Q, where T is the temperature and Q is the antenna's mechanical quality factor. For a given frequency and antenna geometry, the cross-sectionis proportional to pvs. Thus the speed of sound and Q are important figures-of-meritin selectingthe antenna material. Materials with high vsare available that in principle could provide about a hundred-fold increase in the crosssection of resonant-mass gravity wave detectors as compared to current generation detectors. In this Letter we report the results of measurements of the temperature-dependent mechanical losses associated with excitation of the fundamental longitudinal acoustic mode in several potentially suitable materials. We also discuss the impact that these materials could have on the sensitivity of resonant-massdetectors~

1. Introduction Cryogenic, resonant-mass gravitational radiation detectors have been u n d e r development in several laboratories for more than a decade [ l ]. Recently, several of these detectors were operated in coincidence for the first time [2 ]. Most of these detectors consists of a massive solid right cylinder made from a l u m i n u m , with a f u n d a m e n t a l longitudinal resonance near 1 kHz. A gravitational wave pulse, incident with suitable polarization a n d direction, will induce a d y n a m i c strain in any odd-order longitudinal mode of the a n t e n n a provided that the gravity wave pulse has nonzero energy spectral density at the

resonant frequency of the mode. The principle of operation of these detectors is based on the detection of a sudden change in the vibrational state of such a mode against the background detector noise sources that include thermal Nyquist noise and the noise due to the readout transducer and amplifier and other detection electronics. It is conventional to characterize the sensitivity of a resonant-mass detector to an impulsive source in terms of two quantities: the pulse detection noise temperature Td a n d the integrated a n t e n n a cross section 27. These quantities are related to the energy spectral density of a gravitational wave pulse O(co)

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that can be detected with unity signal-to-noise ratio (SNR) by

O( oJa)~,f( O, ¢)=ka Td,

(1)

where oga is the antenna resonant frequency and f ( 0 , 0) ~<1 is an orientation factor that depends on the direction and polarization of the incoming wave [ 3 ]. For a cylindrical antenna the frequency integral of the cross-section, averaged over polarization and direction, is given by

32GM (vs~ 2 S= ~15~cn ---5 \c] '

(2)

where M is the antenna mass, G is the gravitational constant, c is the speed of light, and n = 1, 3, 5, 7 .... is the mode number. Ta, usually referred to as the detector noise temperature, depends on the detector noise sources. If the readout amplifier noise dominates (as it should if the quantity T/Q is sufficiently small), then Td = T~, where T~ is the amplifier noise temperature. Note that Td is much smaller than the temperature T. During the past decade the main focus of technical development in resonant-mass detectors has been on the electromechanical transducer and amplifier with the goal of minimizing Td. With second-generation low-temperature detectors, operating at 4 K and utilizing low-noise superconductive instrumentation, sensitivities of order Td ~ 10 mK have been achieved [1,2]. For a 5000 kg antenna, this detector noise temperature corresponds to h ~ 10- ~s at the detector noise level, h is the dimensionless gravitational wave amplitude. In the near future it is likely that resonant-mass detectors will achieve h ~ 10 -20 by operating at T = 50 mK and by utilizing improved readout instrumentation [ 1,4,5 ]. When this sensitivity is achieved the so-called linear amplifier quantum limit Table 1 Properties of selected high sound speed materials. (The percentages in the first column are the sample purities. )

AI 6063 A120a (94%) AI203 (99.5%) Be SiC

210

p (g/cm 3)

vl (km/s)

_r/_r~

2.7 3.62 3.89 1.85 3.2

5 9 10 13 11.8

1 25 46 113 87

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(corresponding to h = 3 × 10 -21 for a 5000 kg antenna operating at 1 kHz) will then be within reach through further improvements in readout intrumentation. One approach to improving detector sensitivity beyond h ~ 10- 2o or 3 × 10- 2~is to consider antenna materials with higher sound speeds than aluminum. The potential gain in sensitivity available can be seen from eq. (2). The cross-section for the fundamental mode is proportional to My 2. The antenna mass is M=pnREL, where R is the radius of the antenna. The length of the antenna L is determined by the frequency of the antennaf~ and the speed of sound, and is approximately proportional to vs/f~. If we consider antennas with fixed aspect ratio (geometry), then the cross-section is proportional to pv~. Table 1 lists the density, sound speed and the frequency integrated cross-section of resonant-mass antennas for several materials with high sound speeds.

2. Q requirements for a gravitational radiation antenna

In order to utilize any of the materials listed in table 1, two requirements must be met. First, a method of fabricating a large mass antenna should be available and, second, the mechanical quality factor of the antenna must be high enough that the amplifier noise will be the dominant noise source. Because, as we discuss below, the antenna Q requirements are relatively modest if the detector is operated at very low temperature, we do not believe that it is necessary to obtain Q values as high as those obtained in single crystal materials [ 6 ]. Therefore, we have focussed our investigations on amorphous and polycrystalline materials such as those listed in table 1. This has the additional benefit that relatively straightforward fabrication techniques can be used to make a massive antenna. The antenna Q requirement is established by comparing the antenna thermal noise sources with the amplifier noise sources. If the condition

TL Q> Ta Af

(3)

is met, then the detector noise temperature will satisfy Td<~2Ta, where Ta is the amplifier noise tem-

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perature [ 1,7 ]. If Q is much larger than this value then To ~ Ta. The detection bandwidth Af is determined by the transducer and amplifier noise sources. These noise sources can be characterized by the mechanical amplifier noise temperature 7", and the noise resistance r , = (Sf/Su)1/2. Sf and Su are the spectral densities of the force noise and velocity noise, respectively [ 1 ]. Because r, is usually compared to the mechanical impedance of the antenna, its is useful to define a dimensionless noise resistance by Rn=r./ f,M. For a mechanical amplifier directly connected to a high-Q antenna, the detection bandwidth isf, R,. Since typically R , ~ 10 -5, the above Q requirement can be very stringent. Fortunately a method, first identified by Paik [7 ], exists for substantially increasing the detection bandwidth by introducing a mechanical resonator as a matching device between the antenna and the amplifier. This technique has been used on a detector at Stanford University and a bandwidth of ~ 10 Hz has been achieved [8,9]. Richard [ 10] has pointed out that additional stages of resonant coupling can further improve the noise matching. Price [ 11 ] has done the most complete analysis of multimode detector systems and shown that for a given value of R, and a given number of modes, there is an optimum mass for each matching resonator element that makes the system most tolerant to mechanical losses. In the limit that many modes are used, the fractional detector bandwidth Af/fa can approach unity and the Q requirement on the antenna (and the matching resonators) becomes relatively easy to satisfy: Q > T/Ta. Given the present state of development of resonant transducer technology, a fractional bandwidth of order 0.1 appears feasible [ l, 12 ]. We will adopt this number for the estimates of Q requirements that follow. As previously discussed, the next generation of resonant-mass detectors, utilizing aluminum antennas and operating at T ~ 50 mK, will likely achieve Td ~ l p.K corresponding to h ~ 10-2°. Assuming that an antenna fabricated from a high sound speed material has a cross-section 100 times that of an aluminum antenna, to reach the same sensitivity with such an antenna, also cooled to 50 mK, would require Q ~ 104. To reach sensitivities h ~ l0 -21 will require Q>~ 10 6. Note that the linear amplifier quantum limit for this antenna corresponds to

h = 3 × 10-22 Q ~ 2 x 107.

29 July 1991 and

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3. Measurements We have measured the Q values of several small resonators fabricated from the materials listed in table 1. The Be and SiC resonators were cylindrical bars and the samples of A1203 were rectangular bars. A small piezoelectric transducer ( P Z T ) was glued to each end of a sample, one transducer being used for excitation of the longitudinal mode of interest and the other for signal detection. All of the samples were suspended by means of a simple wire loop suspension inside of a vacuum can in which the pressure was maintained below 10 - 6 Torr during the measurements. The samples were cooled relatively rapidly to 2.5 K. The resonant frequency and Q of each sample were measured as the resonator was slowly warmed to 300 K. The cylindrical beryllium sample had a resonant frequency of 42.805 kHz at 4 K. The behaviour of Q as a function of temperature was typical of many metals, increasing as the temperature decreased (fig. lb). The variation of resonant frequency with temperature is shown in fig. I a. We also measured two A1203 samples with different purities. AD995 is a sample with 99.5% purity and AD94 a sample with 94% purity. The resonant frequency of AD995 was about 21.8 kHz and that of AD94 was 20.1 kHz. The temperature dependence of the resonant frequency is shown in fig. 2a. The temperature dependence of Q is shown in fig. 2b. Both samples had a flat plateau of Q in the low temperature regime. AD995 also had a pronounced absorption peak a t about 100 K. Although both samples exhibited similar behaviour, except for the absorption peak, the Q value of the higher purity sample was an order of magnitude higher than that of the lower purity one. Note that the impurity level of AD995 was about ten times lower than that of AD94. This indicates that purity is an essential factor in determining Q. Finally, we investigated two reaction-bonded (RB) silicon carbide samples (resonant frequencies were about 20 kHz) manufactured by the Carborundum 211

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the t e m p e r a t u r e d e p e n d e n c e o f Q a n d the resonant frequency o b t a i n e d for both samples. F o r temperatures lower than 10 K, Q is almost i n d e p e n d e n t o f temperature. N e a r 40 K, Q increases sharply as temperature increases a n d reaches the m a x i m u m value o f about 2.3 × 106 near 54 K. To investigate the pos-

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sibility that the behavior of Q, in particular the presence of narrow absorption peaks, depended on external causes, we repeated the measurements on each sample several times, varying the size of the piezoelectric excitation and readout transducers and the

length of the suspension wire loops. We obtained reproducible results, indicating that the observed dependence of the Q values on temperature is an: intrinsic property of the samples.

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4. Discussion

Table 2 is the summary of measured Q values of all the samples discussed above. Although none of the high sound speed materials that we have thus far investigated has the required Q ( I> 10 6 at 4 K or lower temperature), the measured Q value (4.35 x l05) for SiC at 4 K is encouraging. Silicon carbide is the most promising material identified so far because it not only had the highest Q values among the high sound speed materials we studied but also the structure of 214

the SiC samples we measured was very imperfect, and so it may be possible to greatly improve the Q values. We have not yet investigated the ultralow temperature properties of this material. The SiC samples measured actually are not pure SiC but a mixture of SiC and free silicon. Fig. 4 is a micrograph of a polished and etched SiC sample. By means of microprobe measurements we determined that the dark areas are SiC and the light areas are free silicon. The SiC grains are surrounded by free silicon. There are two phases present in the SiC

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Table 2 Measured Q of high sound speed materials. Temperature (K)

4.2 77

Q Be

A1203 (99.5%)

AI203 (94%)

SiC

2.4× 105 2.6x 104

2.5× 105 2.1 X l05

1.9x l04 1.5x 104

4.35× l05 1.06× l06

Fig. 4. Micrograph of polished and etched SiC sample ( 100 times magnification). White areas are free Si. There are two phases of SiC present. The core of the SiC grain is a-SiC and the surrounding shell is ~SiC. grains. By powder X-ray diffraction analysis, we determined that the two phases are 12H a-SiC and 3C ~SiC. The grain cores are probably a-SiC (hexagonal and rhombohedral polytypes) and the reaction product around the core is probably ~SiC (cubic polytype). The measured density p of the sample is 2.93 g / c m a. From the relation between the measured density and the weight fractions, we determined that the weight fraction of free silicon is 24%. It has been reported in the literature [ 13 ] that the existence of free silicon will greatly change the mechanical properties of SiC. For example, the Young

modulus of SiC containing 25% free silicon is only 78% of that of pure SiC and the fracture toughness is reduced by 26%. The large fraction of free Si may also limit the mechanical quality factor. In this regard we note that silicon carbide, fabricated by reaction-sintering [ 13 ], can be made with less than 10% free silicon content. We also note that heat treatment can cause phase transformation. Annealing SiC at 2700°C can result in all of the ~ S i C being transformed into a-SiC [ 14] so that only one phase exists. It is possible that such

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a single phase sample would have lower mechanical dissipation,

5. Conclusion The technology o f resonant-mass gravitational radiation detectors has developed to the p o i n t that detectors have been operated with burst sensitivities at the detector's noise level o f h = 1 0 - ~8. These detectors operate at 4 K. A new generation o f detectors, designed to operate at ~ 50 mK, will likely achieve h ~ 10 -2°. One a p p r o a c h to surpassing this goal is to consider a n t e n n a materials with higher sound speeds than a l u m i n u m , the material o f choice to date. A two order o f magnitude increase in a n t e n n a cross-section is then possible. In this p a p e r we have investigated several such materials. Silicon carbide has been identified as particularly promising.

Acknowledgement This work was s u p p o r t e d by the N a t i o n a l Science F o u n d a t i o n o f C h i n a a n d by the U.S. N a t i o n a l Science F o u n d a t i o n under grants INT80-06403 a n d

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PHY89-14914. We thank R. Penny and W. Duffy for useful discussions.

References [I]P.F. Michelson, J.C. Price and R.C. Taber, Science 237 (1987) 150. [2 ] E. Amaldi et al., Astron. Astrophys. 216 ( 1989 ) 325. [3] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation ( Freeman, San Francisco, 1973 ). [4 ] P.F. Michelson et al., in: Experimental gravitational physics, eds. P.F. Michelson, Hu En-ke, G. Pizzella (World Scientific, Singapore, 1988 ). [5 ] P.F. Michelson et al., in: Prec. Elizabeth and Frederik White Research Conference, Australian Academy of Sciences, Canberra, October 1990, to be published. [6] V.B. Braginsky, V.B. Mitrofanov and V.I. Panov, Systems with small dissipation (University of Chicago Press, Chicago, 1985 ). [7] H.J. Paik, J. Appl. Phys. 47 (1976) 1168. [8] S.P. Boughn et al., Rev. Sci. Instrum. 61 (1990) 1. [9] S.P. Boughn et al., Astrophys. J. Lett. 261 (1982) LI9. [ 10] J.-P. Richard, Phys. Rev. Lett. 52 (1984) 165. [ 11 ] J.C. Price, Phys~Rev. D 36 (1987) 3555. [12] P.F. Michelson and R.C. Taber, Phys. Rev. D 29 (1984) 2149. [13] P. Kennedy, in: Non-oxide technical and engineering ceramics, ed. S. Hampshire (Elsevier, Amsterdam, 1986 ). [ 14 ] M.M. Dobson, Silicon carbide alloys (Partheon, Carnforth, Lancashire, 1986).