Fundamental resonance frequency

The fundamental resonant frequency of a pipe open at both ends ... 

If an electric held of the proper frequency is applied across the quartz crystal, the crystal wiU oscillate in a mechanically resonant mode. These condihons correspond to the creation of a standing acoustic shear wave that has a node midpoint between the two faces of the crystal and two antinodes at both faces of the disk. This is depicted schematically in Eig. 21.20b. In an EQCM experiment the crystals are operated at the fundamental resonant frequency that is a function of the thickness of the crystal. A crystal with a thickness of 330pm has a resonant frequency of 5 MHz. Crystals with these characteristics are commercially available. In an EQCM experiment, an alternating electric field of 5 MHz is applied to excite the quartz crystal into... 

Table 29.3 lists the new cavity dimensions that resulted in well-organized oscillations. The fundamental resonance frequencies for these cavities ranged between 20 and 40 kHz for the coupled convective-acoustic mode and 6 and 9 kHz for the Helmholtz mode. As before, the dominant frequencies included not only the fundamental mode frequencies, but also many higher harmonics and overtones. 

Fundamental resonant frequency of the unperturbed quartz resonator... 

The chemical shift is measured in parts per million (ppm) and is designated by the Greek letter delta (5). The resonant frequency for a particular nucleus at a specific position within a molecule is then equal to the fundamental resonant frequency of that isotope (e.g., 50.000 MHz for 13C) times a factor that is slightly greater than 1.0 due to the chemical shift ... 

A graph of the resonant frequencies over a very narrow range of frequencies centered on the fundamental resonant frequency of the nucleus of interest (e.g., 13C at 50.000 MHz) is called a spectrum, and each peak in the spectrum represents a unique chemical environment within the molecule being studied. For example, cycloheptanone has four peaks due to the four unique carbon positions in the molecule (Fig. 1.3). Note that symmetry in a molecule can make the number of unique positions less than the total number of carbons. 

The fundamental resonant frequency (/0) shifts when a thin film is deposited on the surface of the quartz crystal. Under the assumption that the density and the shear modulus of the film are the same as those of quartz and that the film is uniform (constant density and thickness) and covering the acoustically active area of the whole... 

The utility of NMR spectroscopy stems from the fact that all nuclei of a particular isotope do not have the same fundamental resonant frequency as required by Eq. 3. In fact a spectrum of resonance absorptions may be observed for a particular nuclide in a given compound. These may be related directly to the chemical structure of the compound under investigation. 

The quartz crystal resonator is a useful device for the study of thin-layer and interfacial phenomena. The crystals commonly employed have a fundamental resonance frequency of 5 -10 MHz and a resolution of the order of 0.1 -0.5 Hz. This high resolution makes the device sensitive to a myriad of physical phenomena, some of which are interrelated and some quite independent of each other. It cannot be overemphasized that the quartz crystal resonator acts as a true microbalance (more appropriately a nanobalance) only if in the course of the process being studied, the nature of the interface (its roughness, sHp-page, the density and viscosity of the solution adjacent to it, and the structure of the solvent in contact with it) is maintained constant. 

The quartz resonator with a fundamental resonance frequency of 5 MHz forms the bottom plate of a measuring chamber that holds approximately... 

Microcantilevers have a fundamental resonance frequency of vibration, V, that is dependent on its spring constant, k, and effective mass, m, as shown in equation (1). 

When the resonant condition of a thickness-shear oscillation is satisfied for AT-cut piezoelectric crystals, a shear wave propagates through the bulk of the material, perpendicular to the faces of the crystals. The fundamental resonance frequency, fo is given by ... 

If severe tray vibration is expected, beams should be designed for the fundamental resonant frequency of the tray panel. Trays may be tied together by rod arrangements to raise their resonant frequencies. 

For torsional waves, the relationship between p and the fundamental resonant frequency/is given by... 

For flexural vibrations, the equations for the fundamental resonant frequencies can be written for cylindrical and rectangular specimens respectively as... 

From all applications of the EQCM, in situ microgravimetry is the most popular. Thus, the EQCM technique allows in situ mtmitoring of the mass changes on the surface of the working electrode, which usually is an Au electrode deposited on a quartz crystal. The sensitivity with which the mass is detected depends on the resonance frequency of the quartz crystal. Thus, if the resonance frequency of the quartz is high, lower masses changes of order ng/cm can be detected. T3q)ical fundamental resonance frequencies are 5, 9, and 10 MHz, but resonance frequencies from kilohertz until gigahertz were already used in different sensors [36]. 

Supported Edge JormrxmmTTTTTior Fundamental Resonance Frequency 30 Hz... 

Fig. 15. Relation between (overtone mode resonant frequency/fundamental resonant frequency) ratio and (diameter/thickness) ratio.

Usually, typical frequencies for the nonresonance methods are in the range from 0.01 Hz to 100 Hz, which should be below the fundamental resonance frequency. At higher frequencies, the dynamic mechanical properties can also be measured using wave propagation methods [1,4]. 

We used a set of silicon and silicon nitride V-shaped cantilevers with nominal spring constants, kn, from 0.06 to 47 N/m, stiffer cantilevers were applied to harder materials. Actual values of spring constants were determined the fundamental resonant frequency calibration proposed earlier and cross-checked by both cantilever-against-cantilever technique and added mass technique (see review of different approaches in Ref. 6). Tip radii were measured using mixed gold nanoparticles. Measured radii were in the range of 20 - 60 nm. Shape of the tip ends at a nanometer scale was virtually symmetrical and for most tips, spherical approximation could be used. 

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