Frequency mass dependence

Use Equation VIII-1 to determine the effective mass of the cantilever if the cantilever has a spring constant C = 20 N/m, the minimum detectable force gradient is hF/dz = 4 X 10 N/m, and the frequency shift is 200 kHz. How does the frequency shift depend on distance from the surface if the force has a 1/z distance dependence ... 

Mass resonant analyzer. A mass analyzer for mass-dependent resonant-energy transfer and measurement of the resonance frequency, power, or ion current of the resonant ions. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Molecules vibrate at characteristic frequencies, which depend both on the difficulty of the motion (the so-called force constant) and on the masses of the atoms involved. The more difficult the motion and the lighter the atomic masses, the higher the vibrational frequency. For a diatomic molecule the vibrational frequency is proportional to ... 

In many of the normal modes of vibration of a molecule the main participants in the vibration will be two atoms held together by a chemical bond. These vibrations have frequencies which depend primarily on the masses of the two vibrating atoms and on the force constant of the bond between them. The frequencies are also slightly affected by other atoms attached to the two atoms concerned. These vibrational modes are characteristic of the groups in the molecule and are useful in the identification of a compound, particularly in establishing the structure of an unknown substance. 

A nonlinear molecule consisting of N atoms can vibrate in 3N — 6 different ways, and a linear molecule can vibrate in 3N — 5 different ways. The number of ways in which a molecule can vibrate increases rapidly with the number of atoms a water molecule, with N = 3, can vibrate in 3 ways, but a benzene molecule, with N = 12, can vibrate in 30 different ways. Some of the vibrations of benzene correspond to expansion and contraction of the ring, others to its elongation, and still others to flexing and bending. Each way in which a molecule can vibrate is called a normal mode, and so we say that benzene has 30 normal modes of vibration. Each normal mode has a frequency that depends in a complicated way on the masses of the atoms that move during the vibration and the force constants associated with the motions involved (Fig. 2). 

The quartz balance is a tool for detecting the increase (or decrease) of the film mass deposited onto the surface of a quartz resonator, connected to the driving circuit, and registering the shift in a frequency. The dependence is expressed by the Sauerbray equation (Sauerbray 1964) ... 

Image current detection is (currently) the only nondestructive detection method in MS. The two mass analyzers that employ image current detection are the FTICR and the orbi-trap. In the FTICR ions are trapped in a magnetic field and move in a circular motion with a frequency that depends on their m/z. Correspondingly, in the orbitrap ions move in harmonic oscillations in the z-direction with a frequency that is m/z dependent but independent of the energy and spatial spread of the ions. For detection ions are made... 

In a quadrupole mass spectrometer, the ions pass into a path between four rods that are attached to an electric circuit that applies a range of frequencies to the rods. Ions resonate in the quadrupole until a certain frequency, which depends on their mass and charge, is reached and then the ions exit the quadrupole and are measured. A diagram of a quadrupole mass spectrometer is given in Section 13.2.3, Figure 13.5. 

In Equation 5.34 to is the harmonic frequency, v the vibrational quantum number, and xe and ye the first and second anharmonicity constants (mass dependent, co x /(coxe) = X /X = il/il, l, and i are vibrational reduced masses). The ZPE(v = 0) contribution to RPFR through first order is thus... 

Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation here is that the mass dependence of all corrections of order a." Za.Y obtained above is exact, as was proved in [1, 2], and there is no additional mass dependence beyond the one already present in (3.7)-(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Za) m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the exact mass dependence, and the only mass scale in the low frequency part, which depends also on multiphoton exchanges, is the reduced mass. 

The F and G matrices may now be combined to give two two-dimensional equations, one for the A, modes and one for the E modes. To illustrate what these modes actually look like in a real case, they are depicted for ND3 in Figure 10.10. These drawings are based on calculations (cf. Wilson, Decius, and Cross for details) from experimentally observed frequencies. The molecule ND3 is used rather than NH3 because the inverse mass dependence of the amplitudes would make the vectors on the nitrogen atom of NH3 im-practically small compared to those on the hydrogen atoms. 

The vibrational frequency depends on the reduced mass and the force constant. Often the individual bonds in polyatomic molecules generate vibrational frequencies which depend only slightly on the rest of the molecule. For example, carbon-oxygen double bonds are found in a wide variety of organic molecules (such as acetone, (CH3)2C=0). The C=0 stretch is excited at v = 1750 cm-1 in virtually any such molecule, and is often used to confirm the existence of a C=0 group in an unknown... 

We shall now examine the relationship between bond stretching and frequency in more detail. Hooke s law told us to expect frequency to depend on both mass and bond strengths, and we can illustrate this double dependence with a series of bonds of various elements to carbon. 

When an atom is displaced from its equilibrium position in a molecule, it is subject to a restoring force which increases with the displacement. A spring follows the same law (Hooke s law) a chemical bond is therefore formally similar to a spring that has weights (atoms) attached to its two ends. A mechanical system of this kind possesses a natural vibrational frequency which depends on the masses of the weights and the stiffness of the spring. 

The operation of AW devices in liquids has been reported and models have been developed to interpret observed behavior for the TSM [37-41], FPW [20] and SH-APM devices [16]. For sensing in liquids, the effective surface mass depends on the thickness of the liquid/coating layer that is coupled to die pre agating AW. The diickness of this layer depends on the density and viscosity of the contacting liquid as well as cqierating frequency. For diin-frlm, acoustically... 

Practical problems associated with infrared dichroism measurements include the requirement of a band absorbance lower than 0.7 in the general case, in order to use the Beer-Lambert law in addition infrared bands should be sufficently well assigned and free of overlap with other bands. The specificity of infrared absorption bands to particular chemical functional groups makes infrared dichroism especially attractive for a detailed study of submolecular orientations of materials such as polymers. For instance, information on the orientation of both crystalline and amorphous phases in semicrystalline polymers may be obtained if absorption bands specific of each phase can be found. Polarized infrared spectroscopy can also yield detailed information on the orientational behavior of each component of a pol3mier blend or of the different chemical sequences of a copoljnner. Infrar dichroism studies do not require any chain labelling but owing to the mass dependence of the vibrational frequency, pronounced shifts result upon isotopic substitution. It is therefore possible to study binary mixtures of deuterated and normal polymers as well as isotopically-labelled block copolymers and thus obtain information simultaneously on the two t3q>es of units. 

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