Quartz crystal microbalance

395:

disk. QCM crystals with vibration frequencies around 5 or 6 MHz usually have a planoconvex shape; at the rim the crystal is too thin for a standing wave with the resonance frequency. Thus, in both cases the thickness-shear vibration amplitude is greatest at the center of the disk. This means that the mass-sensitivity is peaked at the center also, with this sensitivity declining smoothly to zero towards the rim (For high-frequency crystals, the amplitude vanishes already somewhat outside the perimeter of the smallest electrode.) The mass-sensitivity is therefore very non-uniform across the crystal surface, and this non-uniformity is a function of the mass-distribution of the metal electrodes (or in the case of non-planar resonators, the quartz crystal thickness itself).

4279:. It only means that the (swollen) film is much more rigid than the ambient liquid. QCM data taken on the wet sample alone do not allow inference of the degree of swelling. The amount of swelling can be inferred from the comparison of the wet and the dry thickness. The degree of swelling is also accessible by comparing the acoustic thickness (in the Sauerbrey sense) to the optical thickness as determined by, for example, surface plasmon resonance (SPR) spectroscopy or ellipsometry. Solvent contained in the film usually does contribute to the acoustic thickness (because it takes part in the movement), whereas it does not contribute to the optic thickness (because the electronic 214:, and therefore by operating at a high resonant frequency it can be made very sensitive to small changes in that inertia as material is added to (or removed from) its surface. The sensitivity of gravitational mass measurements is, by comparison, limited by the Earth's gravitational field strength. We normally think of a balance as a way of measuring (or comparing) gravitational mass, as measured by the force that the earth exerts on the body being weighed. A few experiments have demonstrated a direct link between QCM and the 28: 269:(the stress-to-speed ratio) of the adjacent medium. (Some acoustic sensors for temperature or pressure make use of the fact that the speed of sound inside the crystal depends on temperature, pressure, or bending. These sensors do not exploit surface effects.) In the context of surface-acoustic wave based sensing, the QCM is also termed "bulk acoustic wave resonator (BAW-resonator)" or "thickness-shear resonator". The displacement pattern of an unloaded BAW resonator is a standing shear wave with 1713:) is the (small) stress exerted by the external surface. The function σ(t) may or may not be harmonic. One can always test for nonlinear behavior by checking for a dependence of the resonance parameters on the driving voltage. If linear acoustics hold, there is no drive level-dependence. Note, however, that quartz crystals have an intrinsic drive level-dependence, which must not be confused with nonlinear interactions between the crystal and the sample. 3931: 3651: 2110: 3679: 3411: 2477: 1923: 1705: 2702: 3094: 1550: 3332:= λ/4. The agreement between experiment and theory is often poor close to the film resonance. Typically, the QCM only works well for film thicknesses much less than a quarter of the wavelength of sound (corresponding to a few micrometres, depending on the softness of the film and the overtone order). 2732:

in 1959 and correlates changes in the oscillation frequency of a piezoelectric crystal with mass deposited on it. He simultaneously developed a method for measuring the resonance frequency and its changes by using the crystal as the frequency-determining component of an oscillator circuit. His method

1400:

The definition of the load impedance implicitly assumes that stress and speed are proportional and that the ratio therefore is independent of speed. This assumption is justified when the crystal is operated in liquids and in air. The laws of linear acoustics then hold. However, when the crystal is in

584:

Stress-compensated (SC) crystals are available with a doubly rotated cut that minimizes the frequency changes due to temperature gradients when the system is operating at high temperatures, and reduces the reliance on water cooling. SC-cut crystals have an inflection point of ~92 °C. In addition

302:

The QCM consists of a thin piezoelectric plate with electrodes evaporated onto both sides. Due to the piezo-effect, an AC voltage across the electrodes induces a shear deformation and vice versa. The electromechanical coupling provides a simple way to detect an acoustic resonance by electrical means.

148:

The frequency of oscillation of the quartz crystal is partially dependent on the thickness of the crystal. During normal operation, all the other influencing variables remain constant; thus a change in thickness correlates directly to a change in frequency. As mass is deposited on the surface of the

3944:

The frequency shift depends on the acoustic impedance of the material; the latter in turn depends on the viscoelastic properties of the material. Therefore, in principle, one can derive the complex shear modulus (or equivalently, the complex viscosity). However, there are certain caveats to be kept

1026:

There is a pitfall with the application of the electro-mechanical analogy, which has to do with how networks are drawn. When a spring pulls onto a dashpot, one would usually draw the two elements in series. However, when applying the electro-mechanical analogy, the two elements have to be placed in

758:

description and to a description in terms of acoustic reflectivities. They provide for a graphical representation of the resonator's properties and their shifts upon loading. These representations are not just cartoons. They are tools to predict the shift of the resonance parameters in response to

6986:

Vuoriluoto, Maija; Orelma, Hannes; Johansson, Leena-Sisko; Zhu, Baolei; Poutanen, Mikko; Walther, Andreas; Laine, Janne; Rojas, Orlando J. (2015-12-10). "Effect of Molecular Architecture of PDMAEMA–POEGMA Random and Block Copolymers on Their Adsorption on Regenerated and Anionic Nanocelluloses and

774:". Note: Here, speed means the time derivative of a displacement, not the speed of sound. There also is an electro-acoustic analogy, within which stresses (rather than forces) are mapped onto voltages. In acoustics, forces are normalized to area. The ratio of stress and speed should not be called " 572:

disrupted by internal stresses caused by temperature gradients in these temperature extremes (relative to room temperature, ~25 °C). These internal stress points produce undesirable frequency shifts in the crystal, decreasing its accuracy. The relationship between temperature and frequency is

294:. A seed crystal serves as an anchoring point and template for crystal growth. Grown crystals are subsequently cut and polished into hair-thin discs which support thickness shear resonance in the 1-30 MHz range. The "AT" or "SC" oriented cuts (discussed below) are widely used in applications. 4299:

The equations concerning viscoelastic properties assume planar layer systems. A frequency shift is also induced when the crystal makes contact with discrete objects across small, load-bearing asperities. Such contacts are often encountered with rough surfaces. It is assumed that the stress–speed

1405:

of strain (and speed) because the stress is transmitted across a finite number of rather small load-bearing asperities. The stress at the points of contact is high, and phenomena like slip, partial slip, yield, etc. set in. These are part of non-linear acoustics. There is a generalization of the

398:

Energy trapping slightly distorts the otherwise planar wave fronts. The deviation from the plane thickness-shear mode entails flexural contribution to the displacement pattern. If the crystal is not operated in vacuum, flexural waves emit compressional waves into the adjacent medium, which is a

394:

For crystals with high frequencies (10 MHz and higher), the electrodes at the front and the back of the crystal usually are key-hole shaped, thereby making the resonator thicker in the center than at the rim. The mass of the electrodes confines the displacement field to the center of the crystal

87:

are investigated as well. QCM has also been used to investigate interactions between biomolecules. Frequency measurements are easily made to high precision (discussed below); hence, it is easy to measure mass densities down to a level of below 1 μg/cm. In addition to measuring the frequency, the

201:

and more general, viscoelastic properties, are of much importance as well. The "non-gravimetric" QCM is by no means an alternative to the conventional QCM. Many researchers, who use quartz resonators for purposes other than gravimetry, have continued to call the quartz crystal resonator "QCM".

193:

sensitivity is easily reached. However, when the film thickness increases, viscoelastic effects come into play. In the late 1980s, it was recognized that the QCM can also be operated in liquids, if proper measures are taken to overcome the consequences of the large damping. Again, viscoelastic

4303:

Often, the external object is so heavy that it does not take part in the MHz oscillation of the crystal due to inertia. It then rests in place in the laboratory frame. When the crystal surface is laterally displaced, the contact exerts a restoring force upon the crystal surface. The stress is

2481:

Importantly, the QCM only probes the region close to the crystal surface. The shear wave evanescently decays into the liquid. In water the penetration depth is about 250 nm at 5 MHz. Surface roughness, nano-bubbles at the surface, slip, and compressional waves can interfere with the

567:

The resonance frequency of acoustic resonators depends on temperature, pressure, and bending stress. Temperature-frequency coupling is minimized by employing special crystal cuts. A widely used temperature-compensated cut of quartz is the AT-cut. Careful control of temperature and stress is

415:

can be excited electrically because only these induce charges of opposite sign at the two crystal surfaces. Overtones are to be distinguished from anharmonic side bands (spurious modes), which have nodal planes perpendicular to the plane of the resonator. The best agreement between theory and

571:

AT-cut crystals are singularly rotated Y-axis cuts in which the top and bottom half of the crystal move in opposite directions (thickness shear vibration) during oscillation. The AT-cut crystal is easily manufactured. However, it has limitations at high and low temperature, as it is easily

1929: 140:, can be as high as 10. Such a narrow resonance leads to highly stable oscillators and a high accuracy in the determination of the resonance frequency. The QCM exploits this ease and precision for sensing. Common equipment allows resolution down to 1 Hz on crystals with a fundamental 3926:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-\omega m_{\mathrm {F} }}{\pi Z_{q}}}\left(1-{\frac {Z_{\mathrm {Liq} }^{2}}{Z_{\mathrm {F} }^{2}}}\right)={\frac {-\omega m_{\mathrm {F} }}{\pi Z_{q}}}\left(1-J_{\mathrm {F} }{\frac {Z_{\mathrm {Liq} }^{2}}{\rho _{\mathrm {F} }}}\right)} 2886: 4212: 585:

to their high temperature inflection point, they also have a smoother cubic relationship and are less affected by temperature deviations from the inflection point. However, due to the more difficult manufacturing process, they are more expensive and are not widely commercially available.

4239:

b) It is difficult to infer the viscoelastic correction factor from QCM data. However, if the correction factor differs significantly from unity, it may be expected that it affects the bandwidth Δ(w/2) and also that it depends on overtone order. If, conversely, such effects are absent

328:

would be radiated into the bulk and potentially be reflected back to the crystal from the opposing cell wall. Such reflections are avoided with transverse waves. The range of penetration of a 5 MHz-shear wave in water is 250 nm. This finite penetration depth renders the QCM

350:. By fitting a resonance curve to the conductance curve, one obtains the frequency and bandwidth of the resonance as fit parameters. In ring-down, one measures the voltage between the electrodes after the exciting voltage has suddenly been turned off. The resonator emits a decaying 2298: 3231: 6289:

Johannsmann, Diethelm; Reviakine, Ilya; Rojas, Elena; Gallego, Marta (2008-10-28). "Effect of sample heterogeneity on the interpretation of QCM data: comparison of combined quartz crystal microbalance/atomic force microscopy measurements with finite element method modeling".

958:. It is characterized by a parameter φ. While φ is dimensionless for usual transformers (the turns ratio), it has the dimension charge/length in the case of electromechanical coupling. The transformer acts as an impedance converter in the sense that a mechanical impedance, 127:

and mechanical deformation is well known; this allows probing an acoustic resonance by electrical means. Applying alternating current to the quartz crystal will induce oscillations. With an alternating current between the electrodes of a properly cut crystal, a standing

3646:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-Z_{\mathrm {F} }}{\pi Z_{q}}}{\frac {Z_{\mathrm {F} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)-iZ_{\mathrm {Liq} }}{Z_{\mathrm {F} }+iZ_{\mathrm {Liq} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)}}} 2482:

measurement of viscosity. Also, the viscosity determined at MHz frequencies sometimes differs from the low-frequency viscosity. In this respect, torsional resonators (with a frequency around 100 kHz) are closer to application than thickness-shear resonators.

337:

Economic ways of driving a QCM make use of oscillator circuits. Oscillator circuits are also widely employed in time and frequency control applications, where the oscillator serves as a clock. Other modes of operation are impedance analysis, QCM-I, and ring-down,

2304: 303:

Otherwise, it is of minor importance. However, electromechanical coupling can have a slight influence on the resonance frequency via piezoelectric stiffening. This effect can be used for sensing, but is usually avoided. It is essential to have the electric and

7024:

Mohan, Tamilselvan; Niegelhell, Katrin; Zarth, Cíntia Salomão Pinto; Kargl, Rupert; Köstler, Stefan; Ribitsch, Volker; Heinze, Thomas; Spirk, Stefan; Stana-Kleinschek, Karin (2014-11-10). "Triggering Protein Adsorption on Tailored Cationic Cellulose Surfaces".

399:

problem when operating the crystal in a liquid environment. Standing compressional waves form in the liquid between the crystals and the container walls (or the liquid surface); these waves modify both the frequency and the damping of the crystal resonator.

1758: 4233:" and the corresponding thickness "Sauerbrey thickness". Even though the Sauerbrey thickness can certainly serve to compare different experiments, it must not be naively identified with the geometric thickness. Worthwhile considerations are the following: 2724:

to show that it was derived by applying the Sauerbrey equation to the frequency shift. The shift in bandwidth is zero if the Sauerbrey equation holds. Checking for the bandwidth therefore amounts to checking the applicability of the Sauerbrey equation.

603:

QCM (EQCM) is particularly advanced. Using the EQCM, one determines the ratio of mass deposited at the electrode surface during an electrochemical reaction to the total charge passed through the electrode. This ratio is called the current efficiency.

4427: 6520:

Friedt, Jean-Michel; Choi, Kang-Hoon; Francis, Laurent A.; Campitelli, Andrew (2002-02-25) . "Simultaneous Atomic Force Microscope and Quartz Crystal Microbalance Measurements: Interactions and Displacement Field of a Quartz Crystal Microbalance".

5382:

Stokich, Theodore M.; Radtke, Douglas R.; White, Christopher C.; Schrag, John L. (1998-06-04) . "An instrument for precise measurement of viscoelastic properties of low viscosity dilute macromolecular solutions at frequencies from 20 to 500 kHz".

3935:

Apart from the term in brackets, this equation is equivalent to the Sauerbrey equation. The term in brackets is a viscoelastic correction, dealing with the fact that in liquids, soft layers lead to a smaller Sauerbrey thickness than rigid layers.

4064: 329:

surface-specific. Also, liquids and gases have a rather small shear-acoustic impedance and therefore only weakly damp the oscillation. The exceptionally high Q-factors of acoustic resonators are linked to their weak coupling to the environment.

5945:(NB. Possible mixup of sources? While all three authors and the journal exist (and they published elsewhere in this journal), the existence of this particular article needs to be verified as it could not be found in online repositories so far.) 1556: 2536: 2956: 1415: 149:

crystal, the thickness increases; consequently the frequency of oscillation decreases from the initial value. With some simplifying assumptions, this frequency change can be quantified and correlated precisely to the mass change using the

390:

To avoid dissipation of vibration energy (damping the oscillation) by the crystal holder, which touches the crystal at the rim, the vibration should be confined to the center of the crystal platelet. This is known as energy trapping.

144:

in the 4 – 6 MHz range. A typical setup for the QCM contains water cooling tubes, the retaining unit, frequency sensing equipment through a microdot feed-through, an oscillation source, and a measurement and recording device.

5611:

Arnau Vives, Antonio; Sogorb, Tomás; Jiménez, Yolanda (2002-06-21) . "Circuit for continuous motional series resonant frequency and motional resistance monitoring of quartz crystal resonators by parallel capacitance compensation".

4228:

The correct interpretation of the frequency shift from QCM experiments in liquids is a challenge. Practitioners often just apply the Sauerbrey equation to their data and term the resulting areal mass (mass per unit area) the

245:) are investigated as alternatives to quartz, mainly (but not only) for use at high temperatures. Such devices are also called "QCM", even though they are not made out of quartz (and may or may not be used for gravimetry). 2105:{\displaystyle ={\frac {1}{\pi Z_{q}}}\,{\frac {-1+i}{\sqrt {2}}}{\sqrt {\rho \omega \left(\eta ^{\prime }-i\eta ^{\prime \prime }\right)}}={\frac {i}{\pi Z_{q}}}{\sqrt {\rho \left(G^{\prime }+iG^{\prime \prime }\right)}}} 2736:

Because the film is treated as an extension of thickness, Sauerbrey’s equation only applies to systems in which (a) the deposited mass has the same acoustic properties as the crystal and (b) the frequency change is small

5986: 5251: 1381: 517: 2761: 4073: 1236: 307:

boundary conditions well under control. Grounding the front electrode (the electrode in contact with the sample) is one option. A π-network sometimes is employed for the same reason. A π-network is an arrangement of

580:

near room temperature. As a consequence the AT-cut quartz crystal is most effective when operating at or near room temperature. For applications which are above room temperature, water cooling is often helpful.

1738:

perpendicular to the surface normal (thickness-shear mode). There are neither compressional waves nor flexural contributions to the displacement pattern. There are no nodal lines in the plane of the resonator.

1726:

For a number of experimental configurations, there are explicit expressions relating the shifts of frequency and bandwidth to the sample properties. The assumptions underlying the equations are the following:

766:. In the same way as the current through a network of resistors can be predicted from their arrangement and the applied voltage, the displacement of a network of mechanical elements can be predicted from the 2170: 1027:

parallel. For two parallel electrical elements the currents are additive. Since the speeds (= currents) add when placing a spring behind a dashpot, this assembly has to be represented by a parallel network.

3108: 6919:

Du, Binyang; Johannsmann, Diethelm (2004). "Operation of the Quartz Crystal Microbalance in Liquids: Derivation of the Elastic Compliance of a Film from the Ratio of Bandwidth Shift and Frequency Shift".

206:

balance. At resonance, the force exerted upon the crystal by the sample is balanced by a force originating from the shear gradient inside the crystal. This is the essence of the small-load approximation.

1267:

The small-load approximation is central to the interpretation of QCM-data. It holds for arbitrary samples and can be applied in an average sense. Assume that the sample is a complex material, such as a

6327:

Johannsmann, Diethelm (2001-06-07) . "Derivation of the shear compliance of thin films on quartz resonators from comparison of the frequency shifts on different harmonics: A perturbation analysis".

4327:’’), where the imaginary part quantifies a withdrawal of energy from the crystal oscillation (for instance due to viscoelastic effects). For such a situation, the small-load approximation predicts 1023:

is the thickness of the plate. The transformer often is not explicitly depicted. Rather, the mechanical elements are directly depicted as electrical elements (capacitor replaces a spring, etc.).

4236:

a) The QCM always measures an areal mass density, never a geometric thickness. The conversion from areal mass density to thickness usually requires the physical density as an independent input.

2472:{\displaystyle \eta ^{\prime \prime }={\frac {1}{2}}{\frac {\pi Z_{q}^{2}}{\rho _{\mathrm {Liq} }\,f}}\,{\frac {\left(\left(\Delta \left(w/2\right)\right)^{2}-\Delta f^{2}\right)}{f_{f}^{2}}}} 1918:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {i}{\pi Z_{q}}}\,{\frac {\sigma }{\dot {u}}}={\frac {i}{\pi Z_{q}}}Z_{\mathrm {ac} }={\frac {i}{\pi Z_{q}}}{\sqrt {\rho i\omega \eta }}} 6409:

Bandey, Helen L.; Martin, Stephen J.; Cernosek, Richard W.; Hillman, A. Robert (1999-04-28). "Modeling the Responses of Thickness-Shear Mode Resonators under Various Loading Conditions".

189:. For thin films, the resonance frequency is usually inversely proportional to the total thickness of the plate. The latter increases when a film is deposited onto the crystal surface. 63:

is disturbed by the addition or removal of a small mass due to oxide growth/decay or film deposition at the surface of the acoustic resonator. The QCM can be used under vacuum, in gas

7112: 6446:

Lucklum, Rolf; Behling, Carsten; Hauptmann, Peter (1999-05-21). "Role of Mass Accumulation and Viscoelastic Film Properties for the Response of Acoustic-Wave-Based Chemical Sensors".

4470: 4284: 613: 339: 1155:

The BvD circuit predicts the resonance parameters. One can show that the following simple relation holds as long as the frequency shift is much smaller than the frequency itself:

6786:

Martin, Stephen J.; Granstaff, Victoria Edwards; Frye, Gregory C. (October 1991). "Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading".

1393:, and must be solved numerically. Approximate solutions also exist, which go beyond the small-load approximation. The small-load approximation is the first order solution of a 4332: 3952:

It is often difficult to disentangle effects of inertia and viscoelasticity. Unless the film thickness is known independently, it is difficult to obtain unique fitting results.

7122: 3986: 1700:{\displaystyle {\frac {\Delta (w/2)}{f_{f}}}={\frac {1}{\pi Z_{q}}}\,{\frac {2}{\omega u_{0}}}\left\langle \sigma \left(t\right)\sin \left(\omega t\right)\right\rangle _{t}} 1112:, cannot be determined from a single measurement. It is inferred from the comparison of the loaded and the unloaded state. Some authors use the BvD circuit without the load 2697:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}\approx {\frac {i}{\pi Z_{q}}}{\frac {-\omega ^{2}u_{0}m_{\mathrm {F} }}{i\omega u_{0}}}=-{\frac {2\,f}{Z_{q}}}m_{\mathrm {F} }} 5957: 253:

The QCM is a member of a wider class of sensing instruments based on acoustic waves at surfaces. Instruments sharing similar principles of operation are shear horizontal

6013:

Buttry, Daniel A.; Ward, Michael D. (September 1992). "Measurement of interfacial processes at electrode surfaces with the electrochemical quartz crystal microbalance".

3089:{\displaystyle \Delta f=-{\frac {f_{f}}{\pi }}\left(\arctan {\frac {Z_{\mathrm {F} }}{Z_{q}}}\tan \left({\frac {2\pi f}{Z_{\mathrm {F} }}}m_{\mathrm {F} }\right)\right)} 1545:{\displaystyle {\frac {\Delta f}{f_{f}}}={\frac {1}{\pi Z_{q}}}\,{\frac {2}{\omega u_{0}}}\left\langle \sigma \left(t\right)\cos \left(\omega t\right)\right\rangle _{t}} 5984:

Bruckenstein, Stanley; Shay, Michael (1985-06-25). "An in situ weighing study of the mechanism for the formation of the adsorbed oxygen monolayer at a gold electrode".

1283:) can be calculated in one way or another, a quantitative analysis of the QCM experiment is in reach. Otherwise, the interpretation will have to remain qualitative. 123:. The piezoelectric effect has found applications in high power sources, sensors, actuators, frequency standards, motors, etc., and the relationship between applied 1077:

The figure on the right shows the Butterworth-van Dyke (BvD) equivalent circuit. The acoustic properties of the crystal are represented by the motional inductance,

31:

Photograph of typical quartz crystal resonators as used for QCM, metallised with gold electrodes (left: front electrode, right: back electrode) by vapor deposition.

5525:

Eggers, Frieder "Frederico"; Funck, Theodor (1987). "Method for measurement of shear-wave impedance in the MHz region for liquid samples of approximately 1 ml".

770:

of the network and the applied force. The electro-mechanical analogy maps forces onto voltages and speeds onto currents. The ratio of force and speed is termed "

92:

factor (equivalent to the resonance bandwidth) is often measured to help analysis. The dissipation factor is the inverse quality factor of the resonance, Q = w/f

5955:

Schumacher, Rolf (April 1990). "The Quartz Microbalance: A Novel Approach to the In-Situ Investigation of Interfacial Phenomena at the Solid/Liquid Junction ".

5527: 5344: 6166:

Reed, Christopher "Chris" E.; Kanazawa, K. Keiji; Kaufmann, James H. (1990) . "Physical description of a viscoelastically loaded AT-cut quartz resonator".

594: 4631:

Lu, Chih-Shun; Czanderna, Alvin Warren, eds. (1984). "Introduction, History, and Overview of Applications of Piezoelectric Quartz Crystal Microbalances".

4268:

d) Complex samples often have fuzzy interfaces. A "fluffy" interface will often lead to a viscoelastic correction and, as a consequence, to a non-zero Δ(

6523: 6366: 4288: 166: 6815:

Domack, Arno; Prucker, Oswald; Rühe, Jürgen; Johannsmann, Diethelm (1997-07-01). "Swelling of a polymer brush probed with a quartz crystal resonator".

4300:

ratio may be replaced by an average stress–speed ratio, where the average stress just is the lateral force divided by the active area of the crystal.

627:, are available for analysis. The latter quantifies processes which withdraw energy from the oscillation. These may include damping by the holder and 3958:

For films in air, the small-load approximation must be replaced by the corresponding results from perturbation theory unless the films are very soft.

265:

resonators. Surface acoustic wave-based devices make use of the fact that the reflectivity of an acoustic wave at the crystal surface depends on the

4476: 2881:{\displaystyle \tan \left({\frac {\pi \Delta f}{f_{f}}}\right)={\frac {-Z_{\mathrm {F} }}{Z_{q}}}\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)} 7142: 5815:

Cumpson, Peter; Seah, Martin (1990). "The quartz crystal microbalance radial/polar dependence of mass sensitivity both on and off the electrodes".

5165:

Elam, Jeffrey W.; Pellin, Michael J. (2005-04-16). "GaPO4 Sensors for Gravimetric Monitoring during Atomic Layer Deposition at High Temperatures".

4927:

Lu, Chih-Shun; Lewis, Owen (November 1972). "Investigation of film-thickness determination by oscillating quartz resonators with large mass load".

4207:{\displaystyle \Delta \left(\omega /2\right)={\frac {8}{3\rho _{\mathrm {F} }Z_{q}}}f_{f}^{\,4}m_{\mathrm {F} }^{3}n^{3}\pi ^{2}J^{\prime \prime }} 3663:

denote the film and the liquid. Here, the reference state is the crystal immersed in liquid (but not covered with a film). For thin films, one can

1031: 5860: 1299: 441: 4272:/2) as well as an overtone-dependent Sauerbrey mass. In the absence of such effects, one may conclude that the outer interface of film is sharp. 727:

usually is not strictly proportional to the bandwidth (although it should be according to the BvD circuit; see below). Also, in absolute terms,

5046:

Johannsmann, Diethelm (2008). "Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance".

1160: 5342:

McSkimin, Herbert J. (1952). "Measurement of Dynamic Shear Viscosity and Stiffness of Viscous Liquids by Means of Traveling Torsional Waves".

7072: 6071: 4886: 4737: 4652: 4275:

e) When the viscoelastic correction, as discussed in (b), is insignificant, this does by no means imply that the film is not swollen by the

3398:. Unless the density of the film is known independently, the QCM can only measure mass per unit area, never the geometric thickness itself. 424:= 13. On low harmonics, energy trapping is insufficient, while on high harmonics, anharmonic side bands interfere with the main resonance. 67:("gas sensor", first use described by King) and more recently in liquid environments. It is useful for monitoring the rate of deposition in 6972: 6959: 6364:

Nakamoto, Takamichi; Moriizumi, Toyosaka (1990-03-17) . "A Theory of a Quartz Crystal Microbalance Based upon a Mason Equivalent Circuit".

4496:

Heterogeneous samples will, in general, lead to scattering of acoustic waves, which is not captured by just calculating the average stress.

2293:{\displaystyle \eta ^{\prime }=-{\frac {\pi Z_{q}^{2}}{\rho _{\mathrm {Liq} }\,f}}\,{\frac {\Delta f\Delta \left(w/2\right)}{f_{f}^{2}}}} 181:

The classical sensing application of quartz crystal resonators is microgravimetry. Many commercial instruments, some of which are called

6989: 5496:

Lin, Zuxuan; Ward, Michael D. (February 1995). "The Role of Longitudinal Waves in Quartz Crystal Microbalance Applications in Liquids".

5048: 6252:

Laschitsch, Alexander; Johannsmann, Diethelm (1999-03-22) . "High frequency tribological investigations on quartz resonator surfaces".

3335:

Note that the properties of a film as determined with the QCM are fully specified by two parameters, which are its acoustic impedance,

3226:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-1}{\pi Z_{q}}}Z_{\mathrm {F} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)} 5817: 5697:"A simple setup to simultaneously measure the resonant frequency and the absolute dissipation factor of a quartz crystal microbalance" 358: 370: 5799: 5742:; Bailey, Emerson D. (1954). "Method for Determining the Viscoelastic Properties of Dilute Polymer Solutions at Audio-Frequencies". 5584: 4713: 4689: 1286:

The limits of the small-load approximation are noticed either when the frequency shift is large or when the overtone-dependence of Δ

804:. Synonymous terms are "surface impedance" and "acoustic load." The load impedance is in general not equal to the material constant 4964:

Bruckenstein, Stanley; Shay, Michael (October 1985). "Experimental aspects of use of the quartz crystal microbalance in solution".

6132: 5701: 5614: 5093: 3309: 170: 2496:

The frequency shift induced by a thin sample which is rigidly coupled to the crystal (such as a thin film), is described by the

361:

Impedance analysis is based on electrical conductance curve. The central parameters of measurement are the resonance frequency f

324:

Most acoustic-wave-based sensors employ shear (transverse) waves. Shear waves decay rapidly in liquid and gaseous environments.

6788: 6448: 6411: 6292: 6110: 5498: 5167: 4519: 6864:(1999) . "Viscoelastic Acoustic Response of Layered Polymer Films at Fluid-Solid Interfaces: Continuum Mechanics Approach". 5211: 6671:

Kanazawa, K. Keiji; Gordon II, Joseph G. (1985). "The oscillation frequency of a quartz resonator in contact with liquid".

5452: 5313: 7179: 6643: 6575: 4606: 4283:

of a solvent molecule does not change when it is located inside a film). The difference in dry and wet mass is shown with

347: 215: 137: 27: 2733:

continues to be used as the primary tool in quartz crystal microbalance experiments for conversion of frequency to mass.

5447: 5308: 4993:

Ward, Michael D.; Buttry, Daniel A. (1990-08-31). "In Situ Interfacial Mass Detection with Piezoelectric Transducers".

4465: 182: 202:

Actually, the term "balance" makes sense even for non-gravimetric applications if it is understood in the sense of a

5424: 6751: 6485: 6329: 6254: 6168: 5885: 5744: 5576: 5445:

Zhang, Chao; Vetelino, John F. (2003-06-01). "Chemical sensors based on electrically sensitive quartz resonators".

4929: 3962:

For thin films in liquids, there is an approximate analytical result, relating the elastic compliance of the film,

2756:> 0.05, the Z-match method must be used to determine the change in mass. The formula for the Z-match method is: 1294:/2) is analyzed in detail in order to derive the viscoelastic properties of the sample. A more general relation is 556: 222: 5128:

Fritze, Holger; Tuller, Harry L. (2001-02-05) . "Langasite for high-temperature bulk acoustic wave applications".

6207:; Knoll, Wolfgang (1992-09-15) . "Viscoelastic properties of thin films probed with a quartz-crystal resonator". 5246: 4763:"Studying Soft Interfaces with Shear Waves: Principles and Applications of the Quartz Crystal Microbalance (QCM)" 4561: 158: 5203: 1276:, or a droplet. If the average stress-to-speed ratio of the sample at the crystal surface (the load impedance, 677:/2. The use of Γ is motivated by a complex formulation of the equations governing the motion of the crystal. A 6146: 286:

When the QCM was first developed, natural quartz was harvested, selected for its quality and then cut in the

5130: 6204: 7132: 6673: 5739: 5570: 5303: 4422:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {N_{S}}{\pi Z_{q}}}{\frac {\kappa _{S}^{*}}{\omega }}} 699:, Γ, is half the bandwidth at half maximum. Using a complex notation, one can treat shifts of frequency, Δ 343: 5204:"Pulse mode shear horizontal-surface acoustic wave (SH-SAW) system for liquid based sensing applications" 3949:

The viscoelastic parameters themselves usually depend on frequency (and therefore on the overtone order).

778:" (in analogy to the mechanical impedance) because this term is already in use for the material property 6957:

Johannsmann, Diethelm (1999-02-26). "Viscoelastic analysis of organic thin films on quartz resonators".

6599: 5927: 5791: 1247: 254: 5864: 4059:{\displaystyle {\frac {\Delta \left(\omega /2\right)}{-\Delta f}}\approx \eta \omega J_{F}^{\,\prime }} 2531:

is the (average) mass per unit area. Inserting this result into the small-load-approximation one finds

6885: 6826: 6760: 6686: 6532: 6494: 6375: 6338: 6263: 6218: 6177: 6054:. Springer Series on Chemical Sensors and Biosensors. Vol. 5 (1 ed.). Berlin / Heidelberg: 5826: 5753: 5710: 5623: 5536: 5394: 5353: 5202:

Martin, Fabrice; Newton, Michael I.; McHale, Glen; Melzak, Kathryn A.; Gizeli, Electra (2004-01-15).

5139: 5102: 5057: 5004: 4938: 4776: 4717: 4574: 771: 266: 68: 797:

the speed of sound). The ratio of stress and speed at the crystal surface is called load impedance,

6747:"Characterization of a thickness-shear mode quartz resonator with multiple nonpiezoelectric layers" 5779: 5385: 4966: 4549: 2729: 2721: 1394: 755: 747: 262: 5664:(November 1988). "Impedance Analysis of Quartz Oscillators, Contacted on One Side with a Liquid". 354:, where the resonance parameters are extracted from the period of oscillation and the decay rate. 7063: 6922: 6901: 6875: 6842: 6633: 6548: 6391: 6114: 5842: 5639: 5284: 5028: 4598: 4445: 4440: 2497: 2491: 775: 186: 150: 141: 7152: 5091:

Mueller, R; White, W (1968). "Direct Gravimetric Calibration of a Quartz Crystal Microbalance".

373:

Ring-down yields the equivalent information in time-domain measurements. The dissipation factor

6151: 7091: 7044: 7027: 7006: 6939: 6817: 6646: 6587: 6579: 6465: 6428: 6309: 6234: 6209: 6128: 6085: 6077: 6067: 5795: 5590: 5580: 5276: 5268: 5228: 5184: 5073: 5020: 4995: 4900: 4892: 4882: 4841: 4804: 4743: 4733: 4685: 4658: 4648: 4590: 4431:

The QCM allows for non-destructive testing of the shear stiffness of multi-asperity contacts.

1386: 1273: 325: 238: 72: 6043: 411:, typically indexed by the number of nodal planes parallel to the crystal surfaces. Only odd 7133:

Quartz crystal microbalance for vacuum applications (HV and UHV) to monitor thin film growth

7081: 7068:"Strongly Stretched Protein Resistant Poly(ethylene glycol) Brushes Prepared by Grafting-To" 7036: 6998: 6968: 6931: 6893: 6834: 6797: 6768: 6717:"Measurement of the viscosity of media by means of shear vibration of plane piezoresonators" 6694: 6540: 6502: 6457: 6420: 6383: 6346: 6301: 6271: 6226: 6185: 6059: 6024: 6015: 5995: 5966: 5834: 5761: 5718: 5673: 5631: 5544: 5507: 5463: 5402: 5361: 5324: 5260: 5220: 5176: 5147: 5110: 5065: 5012: 4975: 4946: 4874: 4833: 4824: 4794: 4784: 4725: 4640: 4582: 4528: 4460: 1402: 1030: 849: 600: 577: 552: 120: 79:, in particular) to surfaces functionalized with recognition sites. Larger entities such as 5661: 5306:; Stevenson, Adrian C. (January 1992). "A Love plate biosensor utilising a polymer layer". 559:

usually are proportional to each other. The QCM operates in the range of linear acoustics.

416:

experiment is reached with planar, optically polished crystals for overtone orders between

221:

Crystalline α–quartz is by far the most important material for thickness-shear resonators.

7126: 6866: 6612: 6124: 6055: 5940: 4870: 4681: 4570: 4450: 2150: 864: 101: 6047: 4860: 2165:. For viscoelastic liquids (η’ = η(ω), η’’≠ 0), the complex viscosity can be obtained as 631:

losses inside the electrode or the crystal. In the literature some parameters other than

7164: 6889: 6830: 6764: 6690: 6536: 6498: 6379: 6342: 6267: 6222: 6181: 5830: 5757: 5714: 5627: 5540: 5398: 5357: 5143: 5106: 5061: 5008: 4942: 4780: 4721: 4578: 316:

the two electrodes. This makes the device less susceptible to electrical perturbations.

5784: 4799: 4762: 4644: 4455: 4280: 4230: 2119: 868: 696: 678: 573: 270: 64: 56: 6897: 6698: 6120:

Mechanics of Solids - Waves in Elastic and Viscoelastic Solids (Theory and Experiment)

5999: 5467: 5224: 547:

the amplitude of electrical driving. The piezoelectric strain coefficient is given as

7173: 7146: 7136: 7117: 6716: 6682: 6637: 6395: 5846: 5643: 5566: 5562: 5548: 5328: 4979: 4602: 3664: 2146: 836:

The electro-mechanical analogy provides for mechanical equivalents of a resistor, an

551: = 3.1·10 m/V for AT-cut quartz crystals. Due to the small amplitude, 313: 97: 6905: 6846: 6552: 5838: 5288: 5032: 4710:

The Quartz Crystal Microbalance in Soft Matter Research - Fundamentals and Modeling

4621:(NB. This was partially presented at Physikertagung in Heidelberg in October 1957.) 3981:

is independent of film thickness. It is an intrinsic property of the film. One has

1269: 291: 273:

at the crystal surface. This makes the analysis particularly easy and transparent.

162: 154: 17: 7153:

The Principles of QCM-I with impedance analysis and dissipation monitoring (QCM-D)

5016: 2161:/2) are equal and opposite. They scale as the square root of the overtone order, 734:– being an electrical quantity and not a frequency – is more severely affected by 720:

is an output parameter of some instruments based on advanced oscillator circuits.

628: 6861: 6118: 5692: 4864: 4291:

for instance in protein adsorption on nanocellulose and in other soft materials.

635:

itself are used to quantify bandwidth. The Q-factor (quality factor) is given by

218:

by comparing traceable (gravitational mass) weighings with QCM measurements.

6483:

Benes, Ewald (February 1984). "Improved quartz crystal microbalance technique".

1735: 1140:

then change their value in the presence of the load (they do not if the element

955: 841: 735: 89: 4869:. Springer Series on chemical sensors and biosensors. Vol. 5. Heidelberg: 4554:"Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung" 4553: 6973:

10.1002/(SICI)1521-3935(19990301)200:3<501::AID-MACP501>3.0.CO;2-W

4729: 2706:

If the density of the film is known, one can convert from mass per unit area,

837: 436:

of lateral displacement rarely exceeds a nanometer. More specifically one has

304: 287: 129: 7002: 6650: 6583: 6230: 6081: 5272: 4896: 4747: 4662: 4594: 6838: 5594: 4822:

Grate, Jay W. (2000). "Acoustic Wave Microsensor Arrays for Vapor Sensing".

2115: 1410:), is periodic in time and synchronous with the crystal oscillation one has 1376:{\displaystyle Z_{L}=-iZ_{q}\tan \left(\pi {\frac {\Delta f}{f_{f}}}\right)} 512:{\displaystyle u_{0}={\frac {4}{\left(n\pi \right)^{2}}}dQU_{\mathrm {el} }} 433: 351: 258: 198: 190: 60: 52: 7095: 7086: 7067: 7048: 7010: 6943: 6591: 6469: 6432: 6313: 6238: 5970: 5893: 5892:. International Crystal Manufacturing Co., Inc. (ICM). 2007. Archived from 5677: 5280: 5232: 5188: 5077: 5024: 4845: 4808: 1734:

The distortion of the crystal is given by a transverse plane wave with the

357: 71:

systems under vacuum. In liquid, it is highly effective at determining the

4248:, Sauerbrey thickness same on all overtone orders) one may assume that (1- 1731:

The resonator and all cover layers are laterally homogeneous and infinite.

1055:

is the motional capacitance (inversely proportional to the stiffness) and

369: 51:(QCN)) measures a mass variation per unit area by measuring the change in 6880: 6544: 5420: 4636: 1231:{\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {i}{\pi Z_{q}}}Z_{L}} 767: 412: 408: 309: 133: 116: 6801: 6387: 6028: 5511: 4532: 4517:

King, Jr., William H. (August 1964). "Piezoelectric sorption detector".

197:

Today, microweighing is one of several uses of the QCM. Measurements of

5987:

Journal of Electroanalytical Chemistry and Interfacial Electrochemistry

5252:

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control

4767: 4586: 4276: 2501: 1257:

is the acoustic impedance of material. For AT-cut quartz, its value is

845: 790: 763: 751: 599:

The QCM can be combined with other surface-analytical instruments. The

124: 84: 76: 7040: 6935: 6461: 6424: 6350: 6305: 5914:

Ballato, Arthur; Lukaszek, Theodore J.; Eernisse, Errol Peter (1979).

5765: 5723: 5696: 5635: 5365: 5264: 5180: 5151: 5114: 4950: 4837: 4789: 1741:

All stresses are proportional to strain. Linear viscoelasticity holds.

153:. Other techniques for measuring the properties of thin films include 6772: 6506: 6275: 6189: 6063: 5406: 5069: 4761:

Johannsmann, Diethelm; Langhoff, Arne; Leppin, Christian (May 2021).

112: 6746: 4878: 3406:

For a film immersed in a liquid environment, the frequency shift is

1119:. This circuit is also called “four element network”. The values of 7062:

Emilsson, Gustav; Schoch, Rafael L.; Feuz, Laurent; Höök, Fredrik;

6089: 4904: 3325:= π/2) define the film resonances. At the film resonance, one has 1029: 203: 80: 5249:(July 1998). "Review of shear surface acoustic waves in solids". 1406:

small-load equation dealing with this problem. If the stress, σ(

703:, and bandwidth, ΔΓ, within the same set of (complex) equations. 1041:

is the electrical (parallel) capacitance across the electrodes.

211: 346:

as a function of driving frequency is determined by means of a

4635:. Methods and Phenomena. Vol. 7 (1 ed.). Amsterdam: 2137:ρ) is the acoustic impedance of the medium. ρ is the density, 910:) the speed). For a point mass undergoing oscillatory motion 6642:. Bell Telephone Laboratories series (1 ed.). New York: 1062:

is the motional resistance (quantifying dissipative losses).

746:

Modeling of acoustic resonators often occurs with equivalent

6639:

Piezoelectric Crystals and Their Applications to Ultrasonics

4199: 4196: 4051: 3973:. The shear compliance is the inverse of the shear modulus, 2316: 2313: 2179: 2090: 2087: 2071: 2019: 2016: 2000: 6566:

Borovikov, V. V.; Dialnyan, R. A.; Shmyt’ko, I. M. (1987).

993:. For planar piezoelectric crystals, φ takes the value φ = 4633:

Applications of Piezoelectric Quartz Crystal Microbalances

6745:

Granstaff, Victoria Edwards; Martin, Stephen J. (1994) .

2748:

If the change in frequency is greater than 5%, that is, Δ

1401:

contact with a rough surface, stress can easily become a

194:

effects contribute strongly to the resonance properties.

5666:

Berichte der Bunsengesellschaft für Physikalische Chemie

4068:

For thin films in air an analogous analytical result is

7113:

Quartz Crystal Microbalance with Dissipation monitoring

5863:. Epson Toyocom Corporation. 2007-03-22. Archived from 4712:. Soft and Biological Matter (1 ed.). Heidelberg: 4471:

Quartz crystal microbalance with dissipation monitoring

1048:

is the motional inductance (proportional to the mass).

819:ρ). Only for propagating plane waves are the values of 4265:

c) Complex samples are often laterally heterogeneous.

4335: 4076: 3989: 3682: 3414: 3111: 2959: 2764: 2539: 2307: 2173: 1932: 1761: 1559: 1418: 1302: 1163: 444: 6109:

Thurston, Robert N. (1984-07-02) . "Chapter 36". In

4304:

proportional to the number density of the contacts,

3977:. In the thin-film limit, the ratio of Δ(w/2) and –Δ 290:. However, most of today's crystals are grown using 6860:Voinova, Marina V.; Rodahl, Michael; Jonson, Mats; 6123:. Vol. IV (new revised ed.). Heidelberg: 762:Equivalent circuits build on the electromechanical 5958:Angewandte Chemie International Edition in English 5783: 4421: 4206: 4058: 3925: 3645: 3225: 3088: 2880: 2696: 2471: 2292: 2104: 1917: 1699: 1544: 1375: 1272:, a sand pile, a froth, an assembly of spheres or 1230: 875:). For a dashpot, the impedance by definition is 511: 407:Planar resonators can be operated at a number of 6710: 6708: 4544: 4542: 3103:For a viscoelastic film, the frequency shift is 2720:. The thickness thus derived is also called the 954:/(iω). Piezoelectric coupling is depicted as a 673:. The half-band-half-width, Γ, is Γ = 5421:"Basic Technology of Quartz Crystal Resonators" 1034:Butterworth-van-Dyke (BvD) equivalent circuit. 7143:Tutorial on modelling the behavior of the QCM 6618: 6145:(NB. Originally published as volume VIa/4 of 1709:Angular brackets denote a time average and σ( 100:in the system and is related to the sample's 8: 5606: 5604: 5528:Journal of Physics E: Scientific Instruments 5345:Journal of the Acoustical Society of America 2728:The Sauerbrey equation was first derived by 713:, is also used as a measure of dissipation. 5916:IEEE Transactions on Sonics and Ultrasonics 1105:is the load impedance. Note that the load, 856:), the point mass (quantified by the mass, 595:Electrochemical quartz crystal microbalance 185:, are available. These devices exploit the 6987:Evidence of Interfacial Water Expulsion". 6524:Japanese Journal of Applied Physics Part 1 6367:Japanese Journal of Applied Physics Part 1 6203:Johannsmann, Diethelm; Mathauer, Klemens; 6046:. In Steinem, Claudia; Janshoff, Andreas; 5786:Introduction to Quartz Crystal Unit Design 5377: 5375: 4678:Piezoelectric Transducers and Applications 706:The motional resistance of the resonator, 167:Multi-Parametric Surface Plasmon Resonance 7085: 6879: 6058:(published 2006-09-08). pp. 49–109. 6044:"Studies of Viscoelasticity with the QCM" 5722: 4798: 4788: 4703: 4701: 4408: 4403: 4397: 4388: 4374: 4368: 4357: 4346: 4336: 4334: 4224:Interpretation of the Sauerbrey thickness 4195: 4185: 4175: 4165: 4159: 4158: 4148: 4147: 4142: 4129: 4118: 4117: 4104: 4088: 4075: 4050: 4049: 4044: 4004: 3990: 3988: 3909: 3908: 3898: 3886: 3885: 3879: 3872: 3871: 3847: 3831: 3830: 3817: 3801: 3795: 3794: 3784: 3772: 3771: 3765: 3745: 3729: 3728: 3715: 3704: 3693: 3683: 3681: 3628: 3627: 3616: 3615: 3587: 3586: 3569: 3568: 3549: 3548: 3526: 3525: 3514: 3513: 3491: 3490: 3483: 3474: 3458: 3457: 3447: 3436: 3425: 3415: 3413: 3211: 3210: 3199: 3198: 3176: 3175: 3162: 3144: 3133: 3122: 3112: 3110: 3069: 3068: 3055: 3054: 3037: 3018: 3007: 3006: 3000: 2978: 2972: 2958: 2866: 2865: 2854: 2853: 2830: 2818: 2817: 2807: 2792: 2775: 2763: 2687: 2686: 2674: 2665: 2659: 2644: 2625: 2624: 2614: 2604: 2594: 2585: 2572: 2561: 2550: 2540: 2538: 2461: 2456: 2441: 2425: 2407: 2383: 2382: 2375: 2362: 2361: 2349: 2344: 2334: 2324: 2312: 2306: 2282: 2277: 2259: 2239: 2238: 2231: 2218: 2217: 2205: 2200: 2190: 2178: 2172: 2086: 2070: 2056: 2047: 2034: 2015: 1999: 1982: 1959: 1958: 1949: 1936: 1931: 1899: 1890: 1877: 1864: 1863: 1850: 1837: 1822: 1817: 1816: 1807: 1794: 1783: 1772: 1762: 1760: 1691: 1635: 1622: 1621: 1612: 1599: 1588: 1572: 1560: 1558: 1536: 1480: 1467: 1466: 1457: 1444: 1433: 1419: 1417: 1360: 1346: 1326: 1307: 1301: 1222: 1209: 1196: 1185: 1174: 1164: 1162: 1066:is the effective area of the crystal and 1012:is the piezoelectric stress coefficient ( 499: 498: 480: 458: 449: 443: 4922: 4920: 4477:Tapered element oscillating microbalance 3301:is the film's viscoelastic compliance, ρ 1744:Piezoelectric stiffening may be ignored. 368: 356: 26: 6721:Instruments and Experimental Techniques 5655: 5653: 5302:Gizeli, Electra; Goddard, Nicholas J.; 4509: 4489: 4315:. The spring constant may be complex (κ 3955:Electrode effects can be of importance. 3242:is the acoustic impedance of the film ( 2896:is the wave vector inside the film and 608:Quantification of dissipative processes 568:essential in the operation of the QCM. 528:the amplitude of lateral displacement, 7073:ACS Applied Materials & Interfaces 6608: 6597: 5936: 5925: 5491: 5489: 5479: 5477: 4311:, and their average spring constant, κ 3384:is not algebraically independent from 965:, appears as an electrical impedance, 612:For advanced QCMs, such as QCM-I and 536:the piezoelectric strain coefficient, 320:Shear waves decay in liquids and gases 136:, which is the ratio of frequency and 4859:Steinem, Claudia; Janshoff, Andreas; 3667:the above equation to first order in 2486:Inertial loading (Sauerbrey equation) 1016:= 9.65·10 C/m for AT-cut quartz) and 7: 6960:Macromolecular Chemistry and Physics 5886:"What do you mean by a crystal cut?" 4220:’’ is the viscous shear compliance. 3940:Derivation of viscoelastic constants 2524:is the amplitude of oscillation and 1753:For a semi-infinite medium, one has 7066:; Dahlin, Andreas B. (2015-04-15). 6990:The Journal of Physical Chemistry B 6617:(NB. V. V. Borovikov translates to 5049:Physical Chemistry Chemical Physics 249:Surface acoustic wave-based sensors 177:Gravimetric and non-gravimetric QCM 5818:Measurement Science and Technology 4676:Arnau Vives, Antonio, ed. (2004). 4645:10.1016/B978-0-444-42277-4.50007-7 4339: 4160: 4119: 4077: 4022: 3993: 3910: 3893: 3890: 3887: 3873: 3832: 3796: 3779: 3776: 3773: 3730: 3686: 3629: 3617: 3594: 3591: 3588: 3570: 3556: 3553: 3550: 3527: 3515: 3492: 3459: 3418: 3212: 3200: 3177: 3115: 3070: 3056: 3008: 2960: 2867: 2855: 2819: 2781: 2688: 2626: 2543: 2434: 2396: 2369: 2366: 2363: 2248: 2242: 2225: 2222: 2219: 1868: 1865: 1765: 1563: 1422: 1349: 1167: 681:resonance frequency is defined as 654:, is the inverse of the Q-factor: 503: 500: 326:Compressional (longitudinal) waves 25: 7118:What is QCM and how does it work? 6898:10.1238/Physica.Regular.059a00391 6715:Borovikov, A. P. (January 1976). 5448:Sensors and Actuators B: Chemical 5309:Sensors and Actuators B: Chemical 4714:Springer International Publishing 1749:Semi-infinite viscoelastic medium 563:Effects of temperature and stress 7139: (archived February 3, 2013) 6568:Soviet Physics-Technical Physics 5702:Review of Scientific Instruments 5615:Review of Scientific Instruments 5423:. Fortiming Corporation. 2008 . 5094:Review of Scientific Instruments 2122:of the viscosity, respectively. 616:, both the resonance frequency, 576:. The cubic relationship has an 171:dual polarisation interferometry 7149: (archived January 6, 2013) 7129: (archived August 14, 2009) 6111:Truesdell III, Clifford Ambrose 5427:from the original on 2018-08-27 5304:Lowe, Christopher "Chris" Robin 5212:Biosensors & Bioelectronics 4708:Johannsmann, Diethelm (2015) . 4612:from the original on 2019-02-26 3969:’ to the ratio of Δ(w/2); and Δ 972:, across the electrical ports. 96:(see below); it quantifies the 6042:Johannsmann, Diethelm (2007). 1580: 1566: 1091:, and the motional resistance 1: 6699:10.1016/S0003-2670(00)82721-X 6644:D. Van Nostrand Company, Inc. 6576:American Institute of Physics 6000:10.1016/S0022-0728(85)80057-7 5660:Beck, Ralf; Pittermann, Udo; 5468:10.1016/S0925-4005(03)00094-7 5225:10.1016/S0956-5663(03)00257-4 5017:10.1126/science.249.4972.1000 738:problems than the bandwidth. 342:. In impedance analysis, the 115:is one member of a family of 5861:"Who needs Crystal Devices?" 5329:10.1016/0925-4005(92)80044-X 4980:10.1016/0013-4686(85)85005-2 3352:and its mass per unit area, 2500:. The stress is governed by 1084:, the motional capacitance, 650:. The “dissipation factor”, 7123:Quartz crystal microbalance 4466:Thin-film thickness monitor 3402:Viscoelastic film in liquid 695: + iΓ, where the 37:quartz crystal microbalance 7196: 6752:Journal of Applied Physics 6486:Journal of Applied Physics 6330:Journal of Applied Physics 6255:Journal of Applied Physics 6169:Journal of Applied Physics 5745:Journal of Applied Physics 5577:Cambridge University Press 5549:10.1088/0022-3735/20/5/011 4930:Journal of Applied Physics 4680:(1 ed.). Heidelberg: 2489: 2141:, the speed of sound, and 759:the addition of the load. 750:. Equivalent circuits are 592: 298:Electromechanical coupling 49:quartz crystal nanobalance 6619: 5839:10.1088/0957-0233/1/7/002 5247:Gulyaev, Yuri Vasilyevich 4730:10.1007/978-3-319-07836-6 2903:its thickness. Inserting 1147:is explicitly included). 159:surface plasmon resonance 47:(QMB), sometimes also as 7003:10.1021/acs.jpcb.5b07628 6231:10.1103/PhysRevB.46.7808 5575:(2 ed.). New York: 3294:is the film thickness. 2153:(η’ = const, η’’ = 0), Δ 1246:is the frequency of the 1151:Small-load approximation 540:the quality factor, and 6839:10.1103/PhysRevE.56.680 6147:Encyclopedia of Physics 5740:Rouse, Jr., Prince Earl 5131:Applied Physics Letters 3287:is the wave vector and 1008:is the effective area, 7087:10.1021/acsami.5b01590 6674:Analytica Chimica Acta 6653:. ark:/13960/t4xh07b19 5971:10.1002/anie.199003293 5678:10.1002/bbpc.198800327 5572:The Art of Electronics 4562:Zeitschrift für Physik 4550:Sauerbrey, Günter Hans 4423: 4208: 4060: 3927: 3647: 3227: 3090: 2882: 2698: 2504:, which implies σ = -ω 2473: 2294: 2106: 1919: 1701: 1546: 1377: 1232: 1074: 1073:is the load impedance. 513: 382: 366: 239:gallium-orthophosphate 32: 6862:Kasemo, Bengt Herbert 6052:Piezoelectric Sensors 5792:Van Nostrand Reinhold 5693:Kasemo, Bengt Herbert 4866:Piezoelectric Sensors 4424: 4209: 4061: 3928: 3648: 3228: 3091: 2883: 2699: 2474: 2295: 2107: 1920: 1717:Viscoelastic modeling 1702: 1547: 1395:perturbation analysis 1378: 1233: 1033: 623:, and the bandwidth, 514: 372: 360: 255:surface acoustic wave 30: 7180:Weighing instruments 6789:Analytical Chemistry 6545:10.1143/JJAP.41.3974 6449:Analytical Chemistry 6412:Analytical Chemistry 6293:Analytical Chemistry 5780:Bottom, Virgil Eldon 5499:Analytical Chemistry 5483:IEC standard 60444-1 5168:Analytical Chemistry 4520:Analytical Chemistry 4333: 4074: 3987: 3680: 3412: 3109: 2957: 2762: 2537: 2305: 2171: 1930: 1759: 1557: 1416: 1300: 1161: 772:mechanical impedance 532:the overtone order, 442: 365:and the bandwidth w. 344:electric conductance 121:piezoelectric effect 119:that experience the 69:thin-film deposition 7064:Lim, Roderick Y. H. 6997:(49): 15275–15286. 6890:1999PhyS...59..391V 6831:1997PhRvE..56..680D 6802:10.1021/ac00020a015 6765:1994JAP....75.1319G 6691:1985AcAC..175...99K 6634:Mason, Warren Perry 6537:2002JaJAP..41.3974F 6499:1984JAP....56..608B 6388:10.1143/JJAP.29.963 6380:1990JaJAP..29..963N 6343:2001JAP....89.6356J 6268:1999JAP....85.3759L 6223:1992PhRvB..46.7808J 6182:1990JAP....68.1993R 6029:10.1021/cr00014a006 5890:Quartz Crystal FAQs 5831:1990MeScT...1..544C 5758:1954JAP....25.1312S 5715:1996RScI...67.3238R 5628:2002RScI...73.2724A 5541:1987JPhE...20..523E 5512:10.1021/ac00100a001 5399:1994JRheo..38.1195S 5386:Journal of Rheology 5358:1952ASAJ...24..355M 5144:2001ApPhL..78..976F 5107:1968RScI...39..291M 5062:2008PCCP...10.4516J 5009:1990Sci...249.1000W 5003:(4972): 1000–1007. 4967:Electrochimica Acta 4943:1972JAP....43.4385L 4781:2021Senso..21.3490J 4722:2015qcms.book.....J 4579:1959ZPhy..155..206S 4533:10.1021/ac60215a012 4413: 4170: 4153: 4055: 3903: 3806: 3789: 2722:Sauerbrey thickness 2466: 2354: 2287: 2210: 2145:= i ωη is the 2114:η’ and η’’ are the 943:. The spring obeys 867:(quantified by the 848:(quantified by the 756:continuum mechanics 748:electrical circuits 742:Equivalent circuits 589:Electrochemical QCM 428:Amplitude of motion 132:is generated. The 45:quartz microbalance 18:Sauerbrey thickness 6115:Bell III, James F. 5662:Weil, Konrad Georg 4639:. pp. 1–393. 4587:10.1007/BF01337937 4446:Sauerbrey constant 4441:Sauerbrey equation 4419: 4399: 4204: 4154: 4138: 4056: 4040: 3923: 3881: 3790: 3767: 3643: 3370:. The wave number 3223: 3086: 2878: 2694: 2498:Sauerbrey equation 2492:Sauerbrey equation 2469: 2452: 2340: 2290: 2273: 2196: 2102: 1915: 1697: 1542: 1403:nonlinear function 1373: 1228: 1075: 776:acoustic impedance 754:equivalent to the 509: 383: 367: 333:Modes of operation 282:Resonator crystals 257:(SH-SAW) devices, 187:Sauerbrey relation 183:thickness monitors 151:Sauerbrey equation 142:resonant frequency 33: 7080:(14): 7505–7515. 7041:10.1021/bm500997s 7035:(11): 3931–3941. 7028:Biomacromolecules 6936:10.1021/la035965l 6818:Physical Review E 6796:(20): 2272–2281. 6607:Missing or empty 6531:(6A): 3974–3977. 6462:10.1021/ac981245l 6456:(13): 2488–2496. 6425:10.1021/ac981272b 6419:(11): 2205–2214. 6351:10.1063/1.1358317 6306:10.1021/ac8013115 6300:(23): 8891–8899. 6217:(12): 7808–7815. 6210:Physical Review B 6127:. pp. 257–. 6073:978-3-540-36567-9 6048:Wolfbeis, Otto S. 5935:Missing or empty 5766:10.1063/1.1721552 5752:(10): 1312–1320. 5724:10.1063/1.1147494 5691:Rodahl, Michael; 5672:(11): 1363–1368. 5636:10.1063/1.1484254 5366:10.1121/1.1906904 5265:10.1109/58.710563 5181:10.1021/ac050349a 5175:(11): 3531–3535. 5152:10.1063/1.1345797 5115:10.1063/1.1683352 5056:(31): 4516–4534. 4974:(10): 1295–1300. 4951:10.1063/1.1660931 4937:(11): 4385–4390. 4888:978-3-540-36567-9 4861:Wolfbeis, Otto S. 4838:10.1021/cr980094j 4790:10.3390/s21103490 4775:(10): 3490-3573. 4739:978-3-319-07835-9 4654:978-0-444-42277-4 4417: 4395: 4363: 4136: 4029: 3916: 3854: 3807: 3752: 3710: 3641: 3481: 3442: 3308:The poles of the 3169: 3139: 3099:Viscoelastic film 3062: 3024: 2987: 2836: 2798: 2680: 2651: 2592: 2567: 2467: 2380: 2332: 2288: 2236: 2151:Newtonian liquids 2100: 2054: 2029: 1980: 1979: 1956: 1913: 1897: 1857: 1832: 1830: 1814: 1789: 1642: 1619: 1594: 1487: 1464: 1439: 1385:This equation is 1366: 1264:= 8.8·10 kg m s. 1216: 1191: 486: 210:The QCM measures 43:) (also known as 16:(Redirected from 7187: 7100: 7099: 7089: 7059: 7053: 7052: 7021: 7015: 7014: 6983: 6977: 6976: 6954: 6948: 6947: 6930:(7): 2809–2812. 6916: 6910: 6909: 6883: 6881:cond-mat/9805266 6857: 6851: 6850: 6812: 6806: 6805: 6783: 6777: 6776: 6773:10.1063/1.356410 6742: 6736: 6735: 6733: 6732: 6712: 6703: 6702: 6668: 6662: 6661: 6659: 6658: 6630: 6624: 6622: 6621: 6616: 6610: 6605: 6603: 6595: 6563: 6557: 6556: 6517: 6511: 6510: 6507:10.1063/1.333990 6480: 6474: 6473: 6443: 6437: 6436: 6406: 6400: 6399: 6361: 6355: 6354: 6324: 6318: 6317: 6286: 6280: 6279: 6276:10.1063/1.369745 6249: 6243: 6242: 6200: 6194: 6193: 6190:10.1063/1.346548 6163: 6157: 6155: 6144: 6142: 6141: 6106: 6100: 6099: 6097: 6096: 6064:10.1007/5346_024 6039: 6033: 6032: 6023:(6): 1335–1379. 6016:Chemical Reviews 6010: 6004: 6003: 5994:(1–2): 131–136. 5981: 5975: 5974: 5952: 5946: 5944: 5938: 5933: 5931: 5923: 5911: 5905: 5904: 5902: 5901: 5882: 5876: 5875: 5873: 5872: 5857: 5851: 5850: 5812: 5806: 5805: 5789: 5776: 5770: 5769: 5735: 5729: 5728: 5726: 5709:(9): 3238–3241. 5688: 5682: 5681: 5657: 5648: 5647: 5608: 5599: 5598: 5559: 5553: 5552: 5522: 5516: 5515: 5493: 5484: 5481: 5472: 5471: 5462:(1–3): 320–325. 5456: 5442: 5436: 5435: 5433: 5432: 5417: 5411: 5410: 5407:10.1122/1.550608 5379: 5370: 5369: 5339: 5333: 5332: 5323:(1–3): 131–137. 5317: 5299: 5293: 5292: 5243: 5237: 5236: 5208: 5199: 5193: 5192: 5162: 5156: 5155: 5125: 5119: 5118: 5088: 5082: 5081: 5070:10.1039/b803960g 5043: 5037: 5036: 4990: 4984: 4983: 4961: 4955: 4954: 4924: 4915: 4914: 4912: 4911: 4856: 4850: 4849: 4825:Chemical Reviews 4819: 4813: 4812: 4802: 4792: 4758: 4752: 4751: 4705: 4696: 4695: 4673: 4667: 4666: 4628: 4622: 4620: 4618: 4617: 4611: 4558: 4546: 4537: 4536: 4527:(9): 1735–1739. 4514: 4497: 4494: 4461:Piezoelectricity 4428: 4426: 4425: 4420: 4418: 4412: 4407: 4398: 4396: 4394: 4393: 4392: 4379: 4378: 4369: 4364: 4362: 4361: 4352: 4351: 4350: 4337: 4213: 4211: 4210: 4205: 4203: 4202: 4190: 4189: 4180: 4179: 4169: 4164: 4163: 4152: 4146: 4137: 4135: 4134: 4133: 4124: 4123: 4122: 4105: 4100: 4096: 4092: 4065: 4063: 4062: 4057: 4054: 4048: 4030: 4028: 4017: 4016: 4012: 4008: 3991: 3932: 3930: 3929: 3924: 3922: 3918: 3917: 3915: 3914: 3913: 3902: 3897: 3896: 3880: 3878: 3877: 3876: 3855: 3853: 3852: 3851: 3838: 3837: 3836: 3835: 3818: 3813: 3809: 3808: 3805: 3800: 3799: 3788: 3783: 3782: 3766: 3753: 3751: 3750: 3749: 3736: 3735: 3734: 3733: 3716: 3711: 3709: 3708: 3699: 3698: 3697: 3684: 3652: 3650: 3649: 3644: 3642: 3640: 3639: 3635: 3634: 3633: 3632: 3622: 3621: 3620: 3599: 3598: 3597: 3575: 3574: 3573: 3562: 3561: 3560: 3559: 3537: 3533: 3532: 3531: 3530: 3520: 3519: 3518: 3497: 3496: 3495: 3484: 3482: 3480: 3479: 3478: 3465: 3464: 3463: 3462: 3448: 3443: 3441: 3440: 3431: 3430: 3429: 3416: 3305:is the density. 3232: 3230: 3229: 3224: 3222: 3218: 3217: 3216: 3215: 3205: 3204: 3203: 3182: 3181: 3180: 3170: 3168: 3167: 3166: 3153: 3145: 3140: 3138: 3137: 3128: 3127: 3126: 3113: 3095: 3093: 3092: 3087: 3085: 3081: 3080: 3076: 3075: 3074: 3073: 3063: 3061: 3060: 3059: 3049: 3038: 3025: 3023: 3022: 3013: 3012: 3011: 3001: 2988: 2983: 2982: 2973: 2887: 2885: 2884: 2879: 2877: 2873: 2872: 2871: 2870: 2860: 2859: 2858: 2837: 2835: 2834: 2825: 2824: 2823: 2822: 2808: 2803: 2799: 2797: 2796: 2787: 2776: 2730:Günter Sauerbrey 2713:, to thickness, 2703: 2701: 2700: 2695: 2693: 2692: 2691: 2681: 2679: 2678: 2669: 2660: 2652: 2650: 2649: 2648: 2632: 2631: 2630: 2629: 2619: 2618: 2609: 2608: 2595: 2593: 2591: 2590: 2589: 2573: 2568: 2566: 2565: 2556: 2555: 2554: 2541: 2478: 2476: 2475: 2470: 2468: 2465: 2460: 2451: 2447: 2446: 2445: 2430: 2429: 2424: 2420: 2419: 2415: 2411: 2384: 2381: 2379: 2374: 2373: 2372: 2355: 2353: 2348: 2335: 2333: 2325: 2320: 2319: 2299: 2297: 2296: 2291: 2289: 2286: 2281: 2272: 2271: 2267: 2263: 2240: 2237: 2235: 2230: 2229: 2228: 2211: 2209: 2204: 2191: 2183: 2182: 2111: 2109: 2108: 2103: 2101: 2099: 2095: 2094: 2093: 2075: 2074: 2057: 2055: 2053: 2052: 2051: 2035: 2030: 2028: 2024: 2023: 2022: 2004: 2003: 1983: 1981: 1975: 1974: 1960: 1957: 1955: 1954: 1953: 1937: 1924: 1922: 1921: 1916: 1914: 1900: 1898: 1896: 1895: 1894: 1878: 1873: 1872: 1871: 1858: 1856: 1855: 1854: 1838: 1833: 1831: 1823: 1818: 1815: 1813: 1812: 1811: 1795: 1790: 1788: 1787: 1778: 1777: 1776: 1763: 1706: 1704: 1703: 1698: 1696: 1695: 1690: 1686: 1685: 1681: 1663: 1643: 1641: 1640: 1639: 1623: 1620: 1618: 1617: 1616: 1600: 1595: 1593: 1592: 1583: 1576: 1561: 1551: 1549: 1548: 1543: 1541: 1540: 1535: 1531: 1530: 1526: 1508: 1488: 1486: 1485: 1484: 1468: 1465: 1463: 1462: 1461: 1445: 1440: 1438: 1437: 1428: 1420: 1382: 1380: 1379: 1374: 1372: 1368: 1367: 1365: 1364: 1355: 1347: 1331: 1330: 1312: 1311: 1237: 1235: 1234: 1229: 1227: 1226: 1217: 1215: 1214: 1213: 1197: 1192: 1190: 1189: 1180: 1179: 1178: 1165: 902:the force and (d 850:drag coefficient 844:, which are the 578:inflection point 518: 516: 515: 510: 508: 507: 506: 487: 485: 484: 479: 475: 459: 454: 453: 348:network analyzer 21: 7195: 7194: 7190: 7189: 7188: 7186: 7185: 7184: 7170: 7169: 7161: 7127:Wayback Machine 7109: 7107:Further reading 7104: 7103: 7061: 7060: 7056: 7023: 7022: 7018: 6985: 6984: 6980: 6956: 6955: 6951: 6918: 6917: 6913: 6867:Physica Scripta 6859: 6858: 6854: 6814: 6813: 6809: 6785: 6784: 6780: 6744: 6743: 6739: 6730: 6728: 6714: 6713: 6706: 6670: 6669: 6665: 6656: 6654: 6632: 6631: 6627: 6620:В. В. Боровиков 6606: 6596: 6565: 6564: 6560: 6519: 6518: 6514: 6482: 6481: 6477: 6445: 6444: 6440: 6408: 6407: 6403: 6363: 6362: 6358: 6326: 6325: 6321: 6288: 6287: 6283: 6251: 6250: 6246: 6205:Wegner, Gerhard 6202: 6201: 6197: 6165: 6164: 6160: 6149: 6139: 6137: 6135: 6125:Springer-Verlag 6108: 6107: 6103: 6094: 6092: 6074: 6056:Springer-Verlag 6041: 6040: 6036: 6012: 6011: 6007: 5983: 5982: 5978: 5954: 5953: 5949: 5934: 5924: 5913: 5912: 5908: 5899: 5897: 5884: 5883: 5879: 5870: 5868: 5859: 5858: 5854: 5814: 5813: 5809: 5802: 5778: 5777: 5773: 5737: 5736: 5732: 5695:(1998-06-04) . 5690: 5689: 5685: 5659: 5658: 5651: 5610: 5609: 5602: 5587: 5561: 5560: 5556: 5524: 5523: 5519: 5495: 5494: 5487: 5482: 5475: 5450: 5444: 5443: 5439: 5430: 5428: 5419: 5418: 5414: 5381: 5380: 5373: 5341: 5340: 5336: 5311: 5301: 5300: 5296: 5245: 5244: 5240: 5206: 5201: 5200: 5196: 5164: 5163: 5159: 5127: 5126: 5122: 5090: 5089: 5085: 5045: 5044: 5040: 4992: 4991: 4987: 4963: 4962: 4958: 4926: 4925: 4918: 4909: 4907: 4889: 4879:10.1007/b100347 4871:Springer-Verlag 4863:, eds. (2007). 4858: 4857: 4853: 4821: 4820: 4816: 4760: 4759: 4755: 4740: 4707: 4706: 4699: 4692: 4682:Springer-Verlag 4675: 4674: 4670: 4655: 4630: 4629: 4625: 4615: 4613: 4609: 4571:Springer-Verlag 4556: 4552:(April 1959) . 4548: 4547: 4540: 4516: 4515: 4511: 4506: 4501: 4500: 4495: 4491: 4486: 4451:Sauerbrey layer 4437: 4384: 4380: 4370: 4353: 4342: 4338: 4331: 4330: 4326: 4322: 4318: 4314: 4310: 4297: 4261: 4254: 4226: 4191: 4181: 4171: 4125: 4113: 4109: 4084: 4080: 4072: 4071: 4018: 4000: 3996: 3992: 3985: 3984: 3968: 3942: 3904: 3867: 3860: 3856: 3843: 3839: 3826: 3819: 3758: 3754: 3741: 3737: 3724: 3717: 3700: 3689: 3685: 3678: 3677: 3673: 3623: 3611: 3610: 3606: 3582: 3564: 3563: 3544: 3521: 3509: 3508: 3504: 3486: 3485: 3470: 3466: 3453: 3449: 3432: 3421: 3417: 3410: 3409: 3404: 3397: 3390: 3383: 3376: 3369: 3365: 3358: 3351: 3345: 3341: 3331: 3324: 3318: 3304: 3300: 3293: 3286: 3279: 3272: 3268: 3262: 3258: 3252: 3248: 3241: 3206: 3194: 3193: 3189: 3171: 3158: 3154: 3146: 3129: 3118: 3114: 3107: 3106: 3101: 3064: 3050: 3039: 3036: 3032: 3014: 3002: 2993: 2989: 2974: 2955: 2954: 2950: 2946: 2939: 2932: 2925: 2917: 2909: 2902: 2895: 2861: 2849: 2848: 2844: 2826: 2813: 2809: 2788: 2777: 2771: 2760: 2759: 2719: 2712: 2682: 2670: 2661: 2640: 2633: 2620: 2610: 2600: 2596: 2581: 2577: 2557: 2546: 2542: 2535: 2534: 2530: 2523: 2516: 2510: 2494: 2488: 2437: 2403: 2399: 2395: 2391: 2390: 2389: 2385: 2357: 2356: 2336: 2308: 2303: 2302: 2255: 2251: 2241: 2213: 2212: 2192: 2174: 2169: 2168: 2128: 2082: 2066: 2065: 2061: 2043: 2039: 2011: 1995: 1994: 1990: 1961: 1945: 1941: 1928: 1927: 1886: 1882: 1859: 1846: 1842: 1803: 1799: 1779: 1768: 1764: 1757: 1756: 1751: 1724: 1719: 1674: 1670: 1653: 1649: 1645: 1644: 1631: 1627: 1608: 1604: 1584: 1562: 1555: 1554: 1519: 1515: 1498: 1494: 1490: 1489: 1476: 1472: 1453: 1449: 1429: 1421: 1414: 1413: 1356: 1348: 1342: 1338: 1322: 1303: 1298: 1297: 1282: 1263: 1256: 1245: 1218: 1205: 1201: 1181: 1170: 1166: 1159: 1158: 1153: 1146: 1139: 1132: 1125: 1118: 1111: 1104: 1097: 1090: 1083: 1072: 1061: 1054: 1047: 1040: 1022: 1003: 992: 985: 978: 971: 964: 953: 949: 942: 935: 924: 897: 881: 874: 869:spring constant 862: 855: 832: 825: 810: 803: 784: 744: 733: 726: 719: 712: 694: 687: 672: 645: 622: 610: 601:electrochemical 597: 591: 565: 546: 527: 494: 468: 464: 463: 445: 440: 439: 430: 405: 388: 386:Energy trapping 364: 335: 322: 312:, which almost 300: 284: 279: 251: 244: 236: 232: 228: 179: 110: 95: 59:resonator. The 23: 22: 15: 12: 11: 5: 7193: 7191: 7183: 7182: 7172: 7171: 7168: 7167: 7160: 7159:External links 7157: 7156: 7155: 7150: 7140: 7130: 7120: 7115: 7108: 7105: 7102: 7101: 7054: 7016: 6978: 6949: 6911: 6852: 6807: 6778: 6737: 6704: 6663: 6625: 6558: 6512: 6475: 6438: 6401: 6374:(5): 963–969. 6356: 6319: 6281: 6244: 6195: 6158: 6133: 6101: 6072: 6034: 6005: 5976: 5947: 5906: 5877: 5852: 5825:(7): 544–555. 5807: 5800: 5771: 5738:Sittel, Karl; 5730: 5683: 5649: 5600: 5585: 5567:Hill, Winfield 5563:Horowitz, Paul 5554: 5517: 5506:(4): 685–693. 5485: 5473: 5437: 5412: 5371: 5334: 5294: 5259:(4): 935–938. 5238: 5219:(6): 627–632. 5194: 5157: 5120: 5101:(3): 291–295. 5083: 5038: 4985: 4956: 4916: 4887: 4851: 4832:(7): 627–648. 4814: 4753: 4738: 4697: 4690: 4668: 4653: 4623: 4538: 4508: 4507: 4505: 4502: 4499: 4498: 4488: 4487: 4485: 4482: 4481: 4480: 4474: 4468: 4463: 4458: 4456:Weighing scale 4453: 4448: 4443: 4436: 4433: 4416: 4411: 4406: 4402: 4391: 4387: 4383: 4377: 4373: 4367: 4360: 4356: 4349: 4345: 4341: 4324: 4320: 4316: 4312: 4308: 4296: 4295:Point contacts 4293: 4281:polarizability 4259: 4252: 4231:Sauerbrey mass 4225: 4222: 4201: 4198: 4194: 4188: 4184: 4178: 4174: 4168: 4162: 4157: 4151: 4145: 4141: 4132: 4128: 4121: 4116: 4112: 4108: 4103: 4099: 4095: 4091: 4087: 4083: 4079: 4053: 4047: 4043: 4039: 4036: 4033: 4027: 4024: 4021: 4015: 4011: 4007: 4003: 3999: 3995: 3966: 3960: 3959: 3956: 3953: 3950: 3941: 3938: 3921: 3912: 3907: 3901: 3895: 3892: 3889: 3884: 3875: 3870: 3866: 3863: 3859: 3850: 3846: 3842: 3834: 3829: 3825: 3822: 3816: 3812: 3804: 3798: 3793: 3787: 3781: 3778: 3775: 3770: 3764: 3761: 3757: 3748: 3744: 3740: 3732: 3727: 3723: 3720: 3714: 3707: 3703: 3696: 3692: 3688: 3671: 3638: 3631: 3626: 3619: 3614: 3609: 3605: 3602: 3596: 3593: 3590: 3585: 3581: 3578: 3572: 3567: 3558: 3555: 3552: 3547: 3543: 3540: 3536: 3529: 3524: 3517: 3512: 3507: 3503: 3500: 3494: 3489: 3477: 3473: 3469: 3461: 3456: 3452: 3446: 3439: 3435: 3428: 3424: 3420: 3403: 3400: 3395: 3388: 3381: 3374: 3367: 3363: 3356: 3349: 3343: 3339: 3329: 3322: 3316: 3302: 3298: 3291: 3284: 3277: 3270: 3266: 3260: 3256: 3250: 3246: 3239: 3221: 3214: 3209: 3202: 3197: 3192: 3188: 3185: 3179: 3174: 3165: 3161: 3157: 3152: 3149: 3143: 3136: 3132: 3125: 3121: 3117: 3100: 3097: 3084: 3079: 3072: 3067: 3058: 3053: 3048: 3045: 3042: 3035: 3031: 3028: 3021: 3017: 3010: 3005: 2999: 2996: 2992: 2986: 2981: 2977: 2971: 2968: 2965: 2962: 2948: 2944: 2937: 2930: 2923: 2915: 2907: 2900: 2893: 2876: 2869: 2864: 2857: 2852: 2847: 2843: 2840: 2833: 2829: 2821: 2816: 2812: 2806: 2802: 2795: 2791: 2786: 2783: 2780: 2774: 2770: 2767: 2717: 2710: 2690: 2685: 2677: 2673: 2668: 2664: 2658: 2655: 2647: 2643: 2639: 2636: 2628: 2623: 2617: 2613: 2607: 2603: 2599: 2588: 2584: 2580: 2576: 2571: 2564: 2560: 2553: 2549: 2545: 2528: 2521: 2514: 2508: 2490:Main article: 2487: 2484: 2464: 2459: 2455: 2450: 2444: 2440: 2436: 2433: 2428: 2423: 2418: 2414: 2410: 2406: 2402: 2398: 2394: 2388: 2378: 2371: 2368: 2365: 2360: 2352: 2347: 2343: 2339: 2331: 2328: 2323: 2318: 2315: 2311: 2285: 2280: 2276: 2270: 2266: 2262: 2258: 2254: 2250: 2247: 2244: 2234: 2227: 2224: 2221: 2216: 2208: 2203: 2199: 2195: 2189: 2186: 2181: 2177: 2126: 2120:imaginary part 2098: 2092: 2089: 2085: 2081: 2078: 2073: 2069: 2064: 2060: 2050: 2046: 2042: 2038: 2033: 2027: 2021: 2018: 2014: 2010: 2007: 2002: 1998: 1993: 1989: 1986: 1978: 1973: 1970: 1967: 1964: 1952: 1948: 1944: 1940: 1935: 1912: 1909: 1906: 1903: 1893: 1889: 1885: 1881: 1876: 1870: 1867: 1862: 1853: 1849: 1845: 1841: 1836: 1829: 1826: 1821: 1810: 1806: 1802: 1798: 1793: 1786: 1782: 1775: 1771: 1767: 1750: 1747: 1746: 1745: 1742: 1739: 1732: 1723: 1720: 1718: 1715: 1694: 1689: 1684: 1680: 1677: 1673: 1669: 1666: 1662: 1659: 1656: 1652: 1648: 1638: 1634: 1630: 1626: 1615: 1611: 1607: 1603: 1598: 1591: 1587: 1582: 1579: 1575: 1571: 1568: 1565: 1539: 1534: 1529: 1525: 1522: 1518: 1514: 1511: 1507: 1504: 1501: 1497: 1493: 1483: 1479: 1475: 1471: 1460: 1456: 1452: 1448: 1443: 1436: 1432: 1427: 1424: 1371: 1363: 1359: 1354: 1351: 1345: 1341: 1337: 1334: 1329: 1325: 1321: 1318: 1315: 1310: 1306: 1280: 1261: 1254: 1243: 1225: 1221: 1212: 1208: 1204: 1200: 1195: 1188: 1184: 1177: 1173: 1169: 1152: 1149: 1144: 1137: 1130: 1123: 1116: 1109: 1102: 1095: 1088: 1081: 1070: 1059: 1052: 1045: 1038: 1020: 1001: 990: 983: 976: 969: 962: 951: 947: 940: 933: 922: 895: 879: 872: 860: 853: 830: 823: 808: 801: 782: 743: 740: 731: 724: 717: 710: 697:imaginary part 692: 685: 670: 643: 620: 609: 606: 593:Main article: 590: 587: 564: 561: 544: 525: 505: 502: 497: 493: 490: 483: 478: 474: 471: 467: 462: 457: 452: 448: 429: 426: 404: 401: 387: 384: 362: 334: 331: 321: 318: 299: 296: 283: 280: 278: 275: 250: 247: 242: 234: 230: 226: 178: 175: 109: 106: 93: 75:of molecules ( 57:quartz crystal 24: 14: 13: 10: 9: 6: 4: 3: 2: 7192: 7181: 7178: 7177: 7175: 7166: 7163: 7162: 7158: 7154: 7151: 7148: 7147:archive.today 7144: 7141: 7138: 7137:archive.today 7134: 7131: 7128: 7124: 7121: 7119: 7116: 7114: 7111: 7110: 7106: 7097: 7093: 7088: 7083: 7079: 7075: 7074: 7069: 7065: 7058: 7055: 7050: 7046: 7042: 7038: 7034: 7030: 7029: 7020: 7017: 7012: 7008: 7004: 7000: 6996: 6992: 6991: 6982: 6979: 6974: 6970: 6966: 6962: 6961: 6953: 6950: 6945: 6941: 6937: 6933: 6929: 6925: 6924: 6915: 6912: 6907: 6903: 6899: 6895: 6891: 6887: 6882: 6877: 6873: 6869: 6868: 6863: 6856: 6853: 6848: 6844: 6840: 6836: 6832: 6828: 6824: 6820: 6819: 6811: 6808: 6803: 6799: 6795: 6791: 6790: 6782: 6779: 6774: 6770: 6766: 6762: 6758: 6754: 6753: 6748: 6741: 6738: 6726: 6722: 6718: 6711: 6709: 6705: 6700: 6696: 6692: 6688: 6684: 6683:Elsevier B.V. 6680: 6676: 6675: 6667: 6664: 6652: 6648: 6645: 6641: 6640: 6635: 6629: 6626: 6623:in Cyrillic.) 6614: 6601: 6593: 6589: 6585: 6581: 6577: 6573: 6569: 6562: 6559: 6554: 6550: 6546: 6542: 6538: 6534: 6530: 6526: 6525: 6516: 6513: 6508: 6504: 6500: 6496: 6492: 6488: 6487: 6479: 6476: 6471: 6467: 6463: 6459: 6455: 6451: 6450: 6442: 6439: 6434: 6430: 6426: 6422: 6418: 6414: 6413: 6405: 6402: 6397: 6393: 6389: 6385: 6381: 6377: 6373: 6369: 6368: 6360: 6357: 6352: 6348: 6344: 6340: 6337:(11): 6356–. 6336: 6332: 6331: 6323: 6320: 6315: 6311: 6307: 6303: 6299: 6295: 6294: 6285: 6282: 6277: 6273: 6269: 6265: 6261: 6257: 6256: 6248: 6245: 6240: 6236: 6232: 6228: 6224: 6220: 6216: 6212: 6211: 6206: 6199: 6196: 6191: 6187: 6183: 6179: 6175: 6171: 6170: 6162: 6159: 6153: 6148: 6136: 6130: 6126: 6122: 6121: 6116: 6112: 6105: 6102: 6091: 6087: 6083: 6079: 6075: 6069: 6065: 6061: 6057: 6053: 6049: 6045: 6038: 6035: 6030: 6026: 6022: 6018: 6017: 6009: 6006: 6001: 5997: 5993: 5989: 5988: 5980: 5977: 5972: 5968: 5964: 5960: 5959: 5951: 5948: 5942: 5929: 5921: 5917: 5910: 5907: 5896:on 2016-03-03 5895: 5891: 5887: 5881: 5878: 5867:on 2007-07-18 5866: 5862: 5856: 5853: 5848: 5844: 5840: 5836: 5832: 5828: 5824: 5820: 5819: 5811: 5808: 5803: 5801:9780442262013 5797: 5793: 5788: 5787: 5781: 5775: 5772: 5767: 5763: 5759: 5755: 5751: 5747: 5746: 5741: 5734: 5731: 5725: 5720: 5716: 5712: 5708: 5704: 5703: 5698: 5694: 5687: 5684: 5679: 5675: 5671: 5667: 5663: 5656: 5654: 5650: 5645: 5641: 5637: 5633: 5629: 5625: 5621: 5617: 5616: 5607: 5605: 5601: 5596: 5592: 5588: 5586:0-521-37095-7 5582: 5578: 5574: 5573: 5568: 5564: 5558: 5555: 5550: 5546: 5542: 5538: 5534: 5530: 5529: 5521: 5518: 5513: 5509: 5505: 5501: 5500: 5492: 5490: 5486: 5480: 5478: 5474: 5469: 5465: 5461: 5457: 5454: 5449: 5441: 5438: 5426: 5422: 5416: 5413: 5408: 5404: 5400: 5396: 5392: 5388: 5387: 5378: 5376: 5372: 5367: 5363: 5359: 5355: 5351: 5347: 5346: 5338: 5335: 5330: 5326: 5322: 5318: 5315: 5310: 5305: 5298: 5295: 5290: 5286: 5282: 5278: 5274: 5270: 5266: 5262: 5258: 5254: 5253: 5248: 5242: 5239: 5234: 5230: 5226: 5222: 5218: 5214: 5213: 5205: 5198: 5195: 5190: 5186: 5182: 5178: 5174: 5170: 5169: 5161: 5158: 5153: 5149: 5145: 5141: 5137: 5133: 5132: 5124: 5121: 5116: 5112: 5108: 5104: 5100: 5096: 5095: 5087: 5084: 5079: 5075: 5071: 5067: 5063: 5059: 5055: 5051: 5050: 5042: 5039: 5034: 5030: 5026: 5022: 5018: 5014: 5010: 5006: 5002: 4998: 4997: 4989: 4986: 4981: 4977: 4973: 4969: 4968: 4960: 4957: 4952: 4948: 4944: 4940: 4936: 4932: 4931: 4923: 4921: 4917: 4906: 4902: 4898: 4894: 4890: 4884: 4880: 4876: 4872: 4868: 4867: 4862: 4855: 4852: 4847: 4843: 4839: 4835: 4831: 4827: 4826: 4818: 4815: 4810: 4806: 4801: 4796: 4791: 4786: 4782: 4778: 4774: 4770: 4769: 4764: 4757: 4754: 4749: 4745: 4741: 4735: 4731: 4727: 4723: 4719: 4715: 4711: 4704: 4702: 4698: 4693: 4691:3-540-20998-0 4687: 4683: 4679: 4672: 4669: 4664: 4660: 4656: 4650: 4646: 4642: 4638: 4634: 4627: 4624: 4608: 4604: 4600: 4596: 4592: 4588: 4584: 4580: 4576: 4572: 4568: 4565:(in German). 4564: 4563: 4555: 4551: 4545: 4543: 4539: 4534: 4530: 4526: 4522: 4521: 4513: 4510: 4503: 4493: 4490: 4483: 4478: 4475: 4472: 4469: 4467: 4464: 4462: 4459: 4457: 4454: 4452: 4449: 4447: 4444: 4442: 4439: 4438: 4434: 4432: 4429: 4414: 4409: 4404: 4400: 4389: 4385: 4381: 4375: 4371: 4365: 4358: 4354: 4347: 4343: 4328: 4307: 4301: 4294: 4292: 4290: 4286: 4282: 4278: 4273: 4271: 4266: 4263: 4258: 4251: 4247: 4243: 4237: 4234: 4232: 4223: 4221: 4219: 4214: 4192: 4186: 4182: 4176: 4172: 4166: 4155: 4149: 4143: 4139: 4130: 4126: 4114: 4110: 4106: 4101: 4097: 4093: 4089: 4085: 4081: 4069: 4066: 4045: 4041: 4037: 4034: 4031: 4025: 4019: 4013: 4009: 4005: 4001: 3997: 3982: 3980: 3976: 3972: 3965: 3957: 3954: 3951: 3948: 3947: 3946: 3939: 3937: 3933: 3919: 3905: 3899: 3882: 3868: 3864: 3861: 3857: 3848: 3844: 3840: 3827: 3823: 3820: 3814: 3810: 3802: 3791: 3785: 3768: 3762: 3759: 3755: 3746: 3742: 3738: 3725: 3721: 3718: 3712: 3705: 3701: 3694: 3690: 3675: 3670: 3666: 3665:Taylor-expand 3662: 3658: 3653: 3636: 3624: 3612: 3607: 3603: 3600: 3583: 3579: 3576: 3565: 3545: 3541: 3538: 3534: 3522: 3510: 3505: 3501: 3498: 3487: 3475: 3471: 3467: 3454: 3450: 3444: 3437: 3433: 3426: 3422: 3407: 3401: 3399: 3394: 3387: 3380: 3373: 3362: 3355: 3348: 3338: 3333: 3328: 3321: 3315: 3311: 3306: 3297: 3290: 3283: 3276: 3265: 3255: 3245: 3238: 3233: 3219: 3207: 3195: 3190: 3186: 3183: 3172: 3163: 3159: 3155: 3150: 3147: 3141: 3134: 3130: 3123: 3119: 3104: 3098: 3096: 3082: 3077: 3065: 3051: 3046: 3043: 3040: 3033: 3029: 3026: 3019: 3015: 3003: 2997: 2994: 2990: 2984: 2979: 2975: 2969: 2966: 2963: 2952: 2943: 2936: 2929: 2921: 2913: 2906: 2899: 2892: 2888: 2874: 2862: 2850: 2845: 2841: 2838: 2831: 2827: 2814: 2810: 2804: 2800: 2793: 2789: 2784: 2778: 2772: 2768: 2765: 2757: 2755: 2751: 2746: 2744: 2740: 2734: 2731: 2726: 2723: 2716: 2709: 2704: 2683: 2675: 2671: 2666: 2662: 2656: 2653: 2645: 2641: 2637: 2634: 2621: 2615: 2611: 2605: 2601: 2597: 2586: 2582: 2578: 2574: 2569: 2562: 2558: 2551: 2547: 2532: 2527: 2520: 2513: 2507: 2503: 2499: 2493: 2485: 2483: 2479: 2462: 2457: 2453: 2448: 2442: 2438: 2431: 2426: 2421: 2416: 2412: 2408: 2404: 2400: 2392: 2386: 2376: 2358: 2350: 2345: 2341: 2337: 2329: 2326: 2321: 2309: 2300: 2283: 2278: 2274: 2268: 2264: 2260: 2256: 2252: 2245: 2232: 2214: 2206: 2201: 2197: 2193: 2187: 2184: 2175: 2166: 2164: 2160: 2156: 2152: 2148: 2147:shear modulus 2144: 2140: 2136: 2132: 2125: 2121: 2117: 2112: 2096: 2083: 2079: 2076: 2067: 2062: 2058: 2048: 2044: 2040: 2036: 2031: 2025: 2012: 2008: 2005: 1996: 1991: 1987: 1984: 1976: 1971: 1968: 1965: 1962: 1950: 1946: 1942: 1938: 1933: 1925: 1910: 1907: 1904: 1901: 1891: 1887: 1883: 1879: 1874: 1860: 1851: 1847: 1843: 1839: 1834: 1827: 1824: 1819: 1808: 1804: 1800: 1796: 1791: 1784: 1780: 1773: 1769: 1754: 1748: 1743: 1740: 1737: 1733: 1730: 1729: 1728: 1721: 1716: 1714: 1712: 1707: 1692: 1687: 1682: 1678: 1675: 1671: 1667: 1664: 1660: 1657: 1654: 1650: 1646: 1636: 1632: 1628: 1624: 1613: 1609: 1605: 1601: 1596: 1589: 1585: 1577: 1573: 1569: 1552: 1537: 1532: 1527: 1523: 1520: 1516: 1512: 1509: 1505: 1502: 1499: 1495: 1491: 1481: 1477: 1473: 1469: 1458: 1454: 1450: 1446: 1441: 1434: 1430: 1425: 1411: 1409: 1404: 1398: 1396: 1392: 1388: 1383: 1369: 1361: 1357: 1352: 1343: 1339: 1335: 1332: 1327: 1323: 1319: 1316: 1313: 1308: 1304: 1295: 1293: 1289: 1284: 1279: 1275: 1271: 1265: 1260: 1253: 1249: 1242: 1238: 1223: 1219: 1210: 1206: 1202: 1198: 1193: 1186: 1182: 1175: 1171: 1156: 1150: 1148: 1143: 1136: 1129: 1122: 1115: 1108: 1101: 1094: 1087: 1080: 1069: 1065: 1058: 1051: 1044: 1037: 1032: 1028: 1024: 1019: 1015: 1011: 1007: 1000: 996: 989: 982: 975: 968: 961: 957: 946: 939: 932: 928: 921: 917: 913: 909: 905: 901: 893: 889: 885: 878: 870: 866: 859: 851: 847: 843: 839: 834: 829: 822: 818: 814: 807: 800: 796: 792: 788: 781: 777: 773: 769: 765: 760: 757: 753: 752:algebraically 749: 741: 739: 737: 730: 723: 716: 709: 704: 702: 698: 691: 688: = 684: 680: 676: 669: 665: 662: = 661: 658: = 657: 653: 649: 642: 639: = 638: 634: 630: 626: 619: 615: 607: 605: 602: 596: 588: 586: 582: 579: 575: 569: 562: 560: 558: 554: 550: 543: 539: 535: 531: 524: 519: 495: 491: 488: 481: 476: 472: 469: 465: 460: 455: 450: 446: 437: 435: 427: 425: 423: 419: 414: 410: 402: 400: 396: 392: 385: 380: 376: 371: 359: 355: 353: 349: 345: 341: 332: 330: 327: 319: 317: 315: 314:short-circuit 311: 306: 297: 295: 293: 292:seed crystals 289: 281: 276: 274: 272: 268: 264: 260: 256: 248: 246: 240: 237:, "LGS") and 224: 219: 217: 213: 212:inertial mass 208: 205: 200: 195: 192: 188: 184: 176: 174: 172: 168: 164: 160: 156: 152: 146: 143: 139: 135: 131: 126: 122: 118: 114: 107: 105: 103: 99: 91: 86: 82: 78: 74: 70: 66: 62: 58: 54: 50: 46: 42: 38: 29: 19: 7165:QCM mini-FAQ 7077: 7071: 7057: 7032: 7026: 7019: 6994: 6988: 6981: 6964: 6958: 6952: 6927: 6921: 6914: 6871: 6865: 6855: 6822: 6816: 6810: 6793: 6787: 6781: 6759:(3): 1319–. 6756: 6750: 6740: 6729:. Retrieved 6727:(1): 223–224 6724: 6720: 6678: 6672: 6666: 6655:. Retrieved 6638: 6628: 6609:|title= 6600:cite journal 6571: 6567: 6561: 6528: 6522: 6515: 6490: 6484: 6478: 6453: 6447: 6441: 6416: 6410: 6404: 6371: 6365: 6359: 6334: 6328: 6322: 6297: 6291: 6284: 6262:(7): 3759–. 6259: 6253: 6247: 6214: 6208: 6198: 6176:(5): 1993–. 6173: 6167: 6161: 6138:. Retrieved 6134:0-38713163-9 6119: 6104: 6093:. Retrieved 6051: 6037: 6020: 6014: 6008: 5991: 5985: 5979: 5962: 5956: 5950: 5937:|title= 5928:cite journal 5919: 5915: 5909: 5898:. Retrieved 5894:the original 5889: 5880: 5869:. Retrieved 5865:the original 5855: 5822: 5816: 5810: 5790:. New York: 5785: 5774: 5749: 5743: 5733: 5706: 5700: 5686: 5669: 5665: 5622:(7): 2724–. 5619: 5613: 5571: 5557: 5532: 5526: 5520: 5503: 5497: 5459: 5446: 5440: 5429:. Retrieved 5415: 5393:(4): 1195–. 5390: 5384: 5349: 5343: 5337: 5320: 5307: 5297: 5256: 5250: 5241: 5216: 5210: 5197: 5172: 5166: 5160: 5135: 5129: 5123: 5098: 5092: 5086: 5053: 5047: 5041: 5000: 4994: 4988: 4971: 4965: 4959: 4934: 4928: 4908:. Retrieved 4865: 4854: 4829: 4823: 4817: 4772: 4766: 4756: 4709: 4677: 4671: 4632: 4626: 4614:. Retrieved 4566: 4560: 4524: 4518: 4512: 4492: 4430: 4329: 4305: 4302: 4298: 4274: 4269: 4267: 4264: 4256: 4249: 4245: 4244:/2) « Δ 4241: 4238: 4235: 4227: 4217: 4215: 4070: 4067: 3983: 3978: 3974: 3970: 3963: 3961: 3943: 3934: 3676: 3668: 3660: 3656: 3655:The indices 3654: 3408: 3405: 3392: 3385: 3378: 3371: 3360: 3353: 3346: 3336: 3334: 3326: 3319: 3313: 3307: 3295: 3288: 3281: 3274: 3263: 3253: 3243: 3236: 3234: 3105: 3102: 2953: 2941: 2934: 2927: 2919: 2911: 2904: 2897: 2890: 2889: 2758: 2753: 2749: 2747: 2745:< 0.05). 2742: 2738: 2735: 2727: 2714: 2707: 2705: 2533: 2525: 2518: 2511: 2505: 2495: 2480: 2301: 2167: 2162: 2158: 2154: 2142: 2138: 2134: 2130: 2123: 2113: 1926: 1755: 1752: 1725: 1710: 1708: 1553: 1412: 1407: 1399: 1390: 1384: 1296: 1291: 1287: 1285: 1277: 1270:cell culture 1266: 1258: 1251: 1240: 1239: 1157: 1154: 1141: 1134: 1127: 1120: 1113: 1106: 1099: 1092: 1085: 1078: 1076: 1067: 1063: 1056: 1049: 1042: 1035: 1025: 1017: 1013: 1009: 1005: 998: 994: 987: 980: 979:is given by 973: 966: 959: 944: 937: 930: 926: 919: 915: 911: 907: 903: 899: 891: 887: 883: 876: 857: 835: 827: 820: 816: 812: 805: 798: 794: 786: 779: 761: 745: 728: 721: 714: 707: 705: 700: 689: 682: 674: 667: 663: 659: 655: 651: 647: 640: 636: 632: 624: 617: 611: 598: 583: 570: 566: 548: 541: 537: 533: 529: 522: 520: 438: 431: 421: 417: 406: 397: 393: 389: 378: 377:is equal to 374: 336: 323: 301: 285: 277:Instrumental 261:devices and 252: 220: 209: 196: 180: 163:spectroscopy 155:ellipsometry 147: 111: 104:properties. 102:viscoelastic 48: 44: 40: 36: 34: 6967:(3): 501–. 6874:(5): 391–. 6825:(1): 680–. 6493:(3): 608–. 6150: [ 5965:(4): 329–. 5535:(5): 523–. 5451: [ 5352:(4): 355–. 5312: [ 5138:(7): 976–. 4573:: 206–222. 3674:, yielding 2933:as well as 1736:wave-vector 1722:Assumptions 1248:fundamental 956:transformer 863:), and the 842:capacitance 789:with ρ the 736:calibration 90:dissipation 6731:2019-02-28 6685:: 99–105. 6657:2019-03-01 6140:2019-03-01 6095:2019-03-01 6090:2006935375 5961:(Reviww). 5900:2007-05-30 5871:2007-05-30 5431:2019-03-03 4910:2019-03-01 4905:2006935375 4616:2019-02-26 4504:References 929:) we have 838:inductance 833:the same. 305:dielectric 271:anti-nodes 130:shear wave 6651:608479473 6636:(1950) . 6584:0038-5662 6396:122368118 6082:1612-7617 5847:250845734 5644:109041806 5273:0885-3010 4897:1612-7617 4748:2213-1736 4663:0377-9025 4603:122855173 4595:0044-3328 4415:ω 4410:∗ 4401:κ 4382:π 4348:∗ 4340:Δ 4200:′ 4197:′ 4183:π 4115:ρ 4086:ω 4078:Δ 4052:′ 4038:ω 4035:η 4032:≈ 4023:Δ 4020:− 4002:ω 3994:Δ 3945:in mind: 3906:ρ 3865:− 3841:π 3824:ω 3821:− 3763:− 3739:π 3722:ω 3719:− 3695:∗ 3687:Δ 3604:⁡ 3539:− 3502:⁡ 3468:π 3451:− 3427:∗ 3419:Δ 3187:⁡ 3156:π 3148:− 3124:∗ 3116:Δ 3044:π 3030:⁡ 2998:⁡ 2985:π 2970:− 2961:Δ 2842:⁡ 2811:− 2782:Δ 2779:π 2769:⁡ 2657:− 2638:ω 2602:ω 2598:− 2579:π 2570:≈ 2552:∗ 2544:Δ 2435:Δ 2432:− 2397:Δ 2359:ρ 2338:π 2317:′ 2314:′ 2310:η 2249:Δ 2243:Δ 2215:ρ 2194:π 2188:− 2180:′ 2176:η 2091:′ 2088:′ 2072:′ 2059:ρ 2041:π 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