Oscillator mass

More recently methods have also been developed to measure the adsorbed amount on single surfaces and not onto powders. Adsorption to isolated surfaces can, for instance, be measured with a quartz crystal microbalance (QCM) [383]. The quartz crystal microbalance consists of a thin quartz crystal that is plated with electrodes on the top and bottom (Fig. 9.11). Since quartz is a piezoelectric material, the crystal can be deformed by an external voltage. By applying an AC voltage across the electrodes, the crystal can be excited to oscillate in a transverse shear mode at its resonance frequency. This resonance frequency is highly sensitive to the total oscillating mass. For an adsorption measurement, the surface is mounted on such a quartz crystal microbalance. Upon adsorption, the mass increases, which lowers the resonance frequency. This reduction of the resonance frequency is measured and the mass increase is calculated [384-387],... 

Mass absorbed in the presence of oscillations Mass absorbed in the absence of oscillations... 

We next assume that the oscillator is harmonic, so that En = Bm0(n + ). It is also well-known that qmi, oc 5m n i, and that qn n j =, /nh/2/.irate constants vanish except between neighboring pairs of levels, and kn n i = nko, where ko is given by... 

Positions of some bands in IR absorption spectrum of the pigment/Ag nanoparticles (Fig. 1) are shifted in comparison to their position in the IR spectrum of pure ultramarine. It means silver to participate in the reaction with ultramarine. Silver gets the bond with Si-0 and Al-O tetrahedra to form Si-O-Ag and Al-O-Ag bonds. Hence, the oscillator masses increase and the oscillation frequencies decrease. So, adding silver particles results in the vibration band shift to the long-wave range. Besides, the intensity of absorption decreases because of the bond energy redistribution, ultramarine... 

Here P and P2 are the momenta of the two oscillating masses the harmonic frequency of the uncoupled oscillators at equilibrium positions iA o is ( ox. The coupling potential proportional to jx depends only on the distance between equilibrium positions of the two oscillating masses at .X Q = Xq/ + 47. ). 

Hn is the Hermite polynomial of degree n, m is the oscillator mass, and Uqx is the mean-square amplitude in the ground state. The scattering process, which is very important for the studies by the neutron techniques, is characterized by the scattering function... 

Eq. (4) and all consequences can be generalized to a set of harmonic oscillators. Then, combination bands can be observed (see below Sec. 3). Both MSAs and effective oscillator masses highlight the very nature of normal modes. 

Fitting procedures give information on wave functions via mean-square displacements (ufj for each vibration and effective oscillator masses. It transpires that proton dynamics for bending modes correspond very closely to isolated harmonic oscillators with a mass of 1 amu [Ikeda 2002], They are largely de-... 

As a conclusion for this section, INS studies of KHCOs single-crystals provide the most detailed, and hopefully the most tutorial, view of proton dynamics ever obtained. The limitation of optical techniques to establishing an unambiguous representation of proton dynamics is emphasized. Effective oscillator masses of 1 amu are determined for each normal mode. Then arises a new fundamental question which mechanisms can account for the decoupling of proton dynamics from the lattice ... 

The choice of normal coordinates is arbitrary [Wilson 1964] and definitions at variance from (11) can be found in text books, see for example [Cohen-Tan-noudji 1977], However, the under determination holds only in the classical regime. The effective oscillator mass for coupled proton oscillators is clearly 1 amu according to the scattering function (see above Sec. 3). Only normal coordinates defined in (11) have a physical reality in the quantum regime. At the present time, there is no obvious justification, apart from experiments, for this choice. 

Neutron scattering experiments shed a new light on proton dynamics in the extended arrays of hydrogen bonded centrosymmetric dimer entities of the KHCO3 crystal. Proton dynamics are decoupled from the lattice. Measurements of effective oscillator masses (namely, 1 amu) contribute to full determination of normal coordinates. 

Equations (13.26) and (13.29b) now provide an exact result, within the bilinear coupling model and the weak coupling theory that leads to the golden rule rate expression, for the vibrational energy relaxation rate. This result is expressed in terms of the oscillator mass m and frequency ca and in tenns of properties of the bath and the molecule-bath coupling expressed by the coupling density A ((a)g (a) at the oscillator frequency... 

Notice that, is unitless p is given in atomic mass units, 1 amu.) If the atomic displacement of the scattering atom in the molecule, u, is exclusively derived from one vibration then the mass, fi, given by Eq. (5.9) is the oscillator mass for that unique mode, fXy. Otherwise, //, represents only a parameterisation in terms of an effective mass,... 

The ratio of the intensity of an overtone to its fundamental is given by the effective mass, men, which increases as extra vibrational contributions increase the total displacements of the scattering atoms. Ultimately, the effective mass, Weff, may be quite different from the oscillator mass, /Xy. This is related to, but subtly different from, effects observed on direct geometry spectrometers ( 5.3.2). 

The oscillator mass Hv is 3.2 amu. This must be compared to the mass of the hydrogen atoms in the librational motion of the ammonium ion, 4 amu. Of importance here is not the precise value of the oscillator mass but that it is light, no assignment of this band to heavy atom motions is tenable. 

Single particle dynamics occurs where there are no interactions between hydrogen atoms in the metal so the oscillator mass, p, is close to unity and the INS spectra extend over several harmonics ( 5.2.1.2). 

This vibration involves almost exclusively the motion of the hydrogen atom against the undeforming molecule. It has an oscillator mass close to unity, which gives it its strong INS intensity. 

This sensor is also based on the principle shown in Fig. 7.2.1. The oscillating masses are manufactured from a wafer bulk material by an etching process (bulk micromachining). The masses are suspended from the frame by U-shaped structures, representing springs, which serve also as coupling of the masses to each other. 

Fig. 7.2.8 Sil icon micromachined yaw-rate sensor with electromagnetic drive 1, coupling spring 2, magnet 3, oscillating direction 4, oscillating mass 5, acceleration sensor 6, direction of Coriolis acceleration 7, spring to...

The functional principle is again in line with Fig. 7.2.1. Oscillating masses and acceleration sensors are produced in the same plane and consist (depending on the micromachining technique) for example of 2-10 pm thick epitactical deposited polycrystalline silicon. 

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