By C. Lu, A.W. Czanderna
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Extra resources for Applications of Piezoelectric Quartz Crystal Microbalances
In Miller and BolefTs model for the QCM  , the resonant frequency of a composite resonator was shown to depend upon the elastic properties of the deposited material [see Eq. (21)]. This dependence vanishes for the small mass approximation. In order to extend the range of mass load on a QCM, the role of elastic properties of the deposited material requires a more careful examination. Unfortunately, Eq. (21) as originally derived by Miller and Bolef is rather complicated and cannot be easily analyzed.
F * [ l - 2 (M f /M q ) + 3 ( M f / M q ) 2 . . ]. (18) By taking only the first three terms on the right-hand side of Eq. (18), it reduces to (fq-fc)/fq-(fq-fc)/2f2 - Mf/Mq-(3/2)(Mf/Mq). (19) If one neglects the second order term, Eq. (19) becomes (fq"fc)/fq = Mf/Mq, which is the same as Eq. (10). (20) 33 THEORY AND PRACTICE Stockbridge T s interpretation can be best justified for very thin films if one considers the discontinuous nature of film structure in the early thin film growth stages or the surface roughness of a typical quartz crystal plate, which is in the order of microns, in comparison to the film thickness.
It consists of a film characterized by a thickness tf, shear modulus yf, and densitypf; and a quartz crystal characterized by the corresponding parameters t q , y q , a n d p q . In Fig. 8, the coordinate is chosen so that the film-quartz interface is at x = 0. If a shear wave along the thickness direction is generated in this system, and assuming no losses in both media, the waves traveling in both the quartz and in the film can be described by the general wave equations: uQ (x,t) = A exp [ia)(p Q /y Q ) 1 / 2 x] + B exp [- ia)(p a /y Q ) 1 / 2 x] (31) and u f ( x , t ) = C exp [ia)(p f /uf) 1 / i x] + D exp [- io)(p f /y f ) 1 / z x] (32) respectively, where u q and Uf denote the time dependent particle displacements in the two media and GO is the angular frequency.