Quartz vibrational modes

The fact that the order parameter vanishes above does not mean that Nature does not have an inkling of things to come well below (or above) T. Such indicators are indeed found in many instances in terms of the behaviour of certain vibrational modes. As early as 1940, Raman and Nedungadi discovered that the a-) transition of quartz was accompanied by a decrease in the frequency of a totally symmetric optic mode as the temperature approached the phase transition temperature from below. Historically, this is the first observation of a soft mode. Operationally, a soft mode is a collective excitation whose frequency decreases anomalously as the transition point is reached. In Fig. 4.4, we show the temperature dependence of the soft-mode frequency. While in a second-order transition the soft-mode frequency goes to zero at T, in a first-order transition the change of phase occurs before the mode frequency is able to go to zero. 

The single crystal of gas hydrate prepared from H2 + CO2 and H2 + CO2 + THF mixtures was analyzed by in situ Raman spectroscopy using a laser Raman microprobe spectrophotometer with multichannel CCD detector. In the present study, the single crystal was defined as the gas hydrate crystal for which the Raman peak of the intermolecular 0-0 vibration mode can be detected. The argon ion laser beam (wavelength 514.5 nm, power 100 mW) or He-Ne laser beam (wavelength 632.8 nm, power 35 mW) condensed to 2 pm in spot diameter was irradiated to the sample through the upper sapphire (or quartz) window. The backscatter of the opposite direction was taken in with the same lens. The spectral resolution was about 1 cm ... 

Example calculation for quartz. We show, as an example, the calculation of a partition function ratio for quartz at 25°C (Table 2). The input data are taken from the compilation of frequencies and frequency shifts factors (ra /ra) given in Polyakov and Kharlashina (1994), with minor modifications (see below). The shift factors in this compilation are mostly from Sato and McMillan (1987), who directly measured the spectrum of quartz. The degeneracy column in the table gives the number of vibrational modes with a particular frequency, and therefore the number of times that mode must be counted in the calculations. Including degeneracies, there are 27 vibrational modes. 

AT and BT plates are made to vibrate primarily in the thickness shear mode. However, it is important to realize that quartz crystals can be made to vibrate in any one or combination of these modes. The vibrational modes can be induced electrically, acoustically, thermally or by some combination of all of these factors. The thickness shear frequency response of an AT plate can be described in terms of the fundamental frequency. For a rectangular AT cut plate, the equation for calculating the approximate frequency of vibration is ... 

Fig. 11. (a) Equivalent series resistance of quartz and langasite single crystals as a function of resonator vibration modes, (b) electromechanical coupling factor of langasite series as a function of piezoelectric constant. 

This is the correct name for most popular mass sensors, although they are better known as Quartz Crystal Microbalances (QCMs). A piezoelectric crystal vibrating in its resonance mode is a harmonic oscillator. For microgravimetric applications, it is necessary to develop quantitative relationships between the relative shift of the resonant frequency and the added mass. In the following derivation, the added mass is treated as added thickness of the oscillator, which makes the derivation more intuitively accessible. 

Figure 4.3 shows the shear-mode vibration of a quartz crystal of mass M and thickness t. At resonance, the wavelength X is... 

The crystal cut determines the mode of oscillations. Shear vibrations are generated if one large crystal face moves parallel with respect to the underlying planes as in QCMs with AT-cut a-quartz crystals. This crystal wafer is prepared by cutting the quartz at approximately 35.17° from its Z-axis. A typical crystal plate is a cylindrical disk of a diameter 10 mm and thickness about 0.7 to 0.1 mm for resonant operation in the 2 to 15 MHz frequency range. This type of crystals shows weak dependence of the resonant frequency on the temperature and stress for room temperature operation. 

The addition of mass provides the means of transduction for many chemical sensors, including surface acoustic wave (SAW) devices, quartz crystal microbalances (QCM), and microcantilevers. In all these devices, the mass addition either perturbs the vibration, oscillations, or deflection within the transducer. The mode of transduction in an optical interferometer can also be linked to mass addition the sensor s response is altered by refractive index changes in the material being monitored. It is possible that this change can be elicited solely from refractive index changes without the addition of mass, although in sensing a particular... 

The lattice vibrations for the simple tetrahedral lattice wore studied in Section 9-A. The state of the distortion of the lattice was specified by giving the displacement (5r, of each atom. We then made a transformation to normal coordinates u., each corresponding to a normal mode frequency w(k), and these were plotted as a function of k in Fig. 9-2. There were three curves for each atom in the primitive cell. We see immediately that there will be difficulties in complex structures in quartz there arc 27 sets of modes, and oven in the simple molecular lattice there are 9. This complexity suggests that one should proceed by computer. One such approach was taken by Bell, Bird, and Dean (1968). They took a large cluster... 

The typical and much discussed effect of phase transitions is a so-called soft mode, A soft mode is a vibration, the frequency of which nears zero as the physical parameter (mostly the temperature but sometimes also the pressure or the external electric field) approaches its critical point. One of the fir.st soft modes was observed by Raman et al. in the a / quartz transition (Raman and Nedungadi, 1940). The theory of these modes was proposed by Cochran (Cochran, 1960, 1961). It turns out that the soft mode is simply the vibration that, due to its form, allows the transformation from one phase to the other. At the transition point, the restoring forces disappear and the frequency approaches zero. Extensive reviews of the application of spectroscopy in connection with the investigation of phase transitions have been provided by Rao and Iqbal (Rao and Rao, 1978 Iqbal, 1986). 

---