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Fig. 7. Shown is the bulk modulus of ZrW2O8 as a function of temperature meas- ured using the synchronous-digitization RUS system. Throughout this temperature range, the thermal expansion coefficient is negative.
Fig. 6. The time of flight computed using the correlated signals as a function of fre- quency for several choices of which cycles to overlap.
while all further processing is done mathematically. The first process uses the correlation function
to convolute the specimen echo h(t) with the buffer rod echo g(t) to obtain a symmetric “pulse” that preserves the original timing information. The result is shown in Fig. 2 where the sample echo, Fig. 2b is correlated with both itself and the buffer rod echo, Fig. 2c. Next, simple fitting procedures are used to determine the time delay between the two compo- nents of Fig. 2c, but with different choices for the exact cycle of the correlated buffer rod signal (Fig. 2c left) to be used to measure the time delay to a particular cycle of the correlated S signal (Fig. 2c right). Frequency is then varied, and the computation is repeated. The results, shown in Fig. 6 show that with the correct choice for the cycles to be used to deter- mine time of flight, the delay time is independent of fre- quency, and requires no user input, such as a visual guess as to what the correct overlap should be.
Thus by using a minimal electronics system without many of the complications needed for analog determination of time delays, and recording only the minimally-processed raw data digitally, it is possible to implement mathematical processes to determine unambiguously the time of flight of an acoustic pulse. The all-digital measurement of pulse-echo time of flight has the same absolute accuracy as the full inherent precision of the very best pulse-echo-overlap systems, and is simpler and cheaper than an analog-based system.
An example
Let us examine a bit of physics that begs for the use of
both RUS and PE ultrasound techniques to find the answer to
an interesting question. ZrW2O8 is a cubic material with a
volume thermal expansion coefficient that is negative from
liquid helium temperatures to well above room tempera-
10
That is, as it warms up it contracts as if pressure were applied. This suggests an obvious measurement, that of the elastic stiffness as a function of temperature. Most materials expand on warming, accompanied by a softening of the elas-
ture.
tic stiffness. But if a material contracts on warming, should it not become stiffer? Using RUS we measured all the elastic moduli in one pass on a beautiful monocrystal and found that ZrW2O8 softens on warming11 even as its volume shrinks (Fig. 7). This surprise then suggests that if one applies pres- sure at constant temperature, unlike most other materials, ZrW2O8 should get softer. But this cannot be done with RUS, and so we resorted to PE in a SiC anvil-type pressure cell,12 bouncing sound through the anvils into the monocrystal specimen and out again. Sure enough, compressing this solid makes it softer (Fig. 8). The explanation for this is related to
12
The root of it is that even at a microscopic level, the angles and bonds among rigid substructures of the unit cell are such that heat leads to vibrations that make ZrW2O8 shrink like a rubber band when warmed, but under compression the same bonds begin to bend in a highly non-linear way, introducing a route
to compression that is soft.AT
the famous Euler column instability problem.
Fig. 8. The response of the longitudinal modulus and the shear modulus to pressure of ZrW2O8 measured in a SiC-anvil pressure cell using the digital PE system described here. Note that the material softens under compression, opposite what most materials do, but consistent with the softening that occurs on warming as the material contracts.
Digital Ultrasonics 21