Oscillating atoms

The constant h, now called Planck s constant, has the value 6.626 X 10 i4 )-s. If the oscillating atom releases an energy E into the surroundings, then radiation of... 

Assignments for stretching frequencies can be approximated by the application of Hooke s law. In the application of the law, two atoms and their connecting bond are treated as a simple harmonic oscillator composed of two masses joined by a spring. The following equation, derived from Hooke s law, states the relationship between frequency of oscillation, atomic masses, and the force constant of the bond. 

In 1900, Max Planck (1858-1947) discovered a formula (now often called the Planck radiation law) that modeled curves like those shown in Figure 24-21 nearly perfectly. He followed this discoveiy by developing a theory that made two bold assumptions regarding the oscillating atoms or molecules in blackbody radiators. He assumed (1) that these species could have only discrete energies and (2) that they could absorb or emit energy in discrete units, or quanta. These assumptions, which are implicit in Equation 24-3, laid the foundation for the development of quantum theory and eventually won him the Nobel Prize in Physics in 1918. 

Kajiyama T, Tanaka K, Ohki I, Ge S-R, Yoon SJ, Takahara A (1994) Imaging of dynamic viscoelastic properties of a phase-separated polymer surface by forced oscillation atomic force microscopy. Macromolecules 27 7932-7934... 

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

The dependence of wave velocity on wavelength is called the dispersion of velocity. This phenomenon is related to displacements of oscillating atoms in antiphase. 

After switching off the excitation sources, the phase relations between the different oscillating atomic dipoles are altered by different relaxation processes, which perturb the atomic dipoles. We may classify these processes into two categories ... 

Another important task of the vibration spectroscopy is the determination of force constants if). For diatomic molecules, these can be obtained directly from the vibration frequencies (co) and the reduced mass (in) of oscillating atoms by the harmonic oscillator equation... 

The experimental values of force constants are presented in Tables SI 1.17 and S11.18, together with the xm calculated by Eqs 11.60 and 11.61. The force constants of MX crystalline compounds cannot be calculated by the harmonic oscillator formula from the lattice vibrations of crystals, because their calculations require taking into account crystal field effects and the polarizabilities and/or the effective charges of oscillating atoms. For this reason, the data on the optical force constants of crystalline compounds (especially the polyatomic ones) are incomplete and... 

The generic term van-der-Waals forces refers to any interaction between electrically neutral atoms and molecules which results from the non-uniform distribution of electrons around the atomic nuclei. This non-uniformity may be observable as a permanent polarisation of molecules (e.g. for water), yet it primarily exist on an instantaneous level due to the electron oscillations. Atoms and molecules are, therefore, fluctuating dipoles that interact via Coulombic forces, which eventually produces a net attraction. The strength of these forces diminishes very rapidly with the intermolecular distance r, ,- the corresponding energy potential uy obeys a power-law decay (Kralchevsky et al. 2002, Chap. 5.4.2) ... 

In the particular case of a nanoneedle at the end of a t3q>ical oscillating atomic force microscope cantilever (/ = 260 kHz), this evanescent length is of about 8 10 " m that can be larger than the... 

In the far infrared spectra of polymer as well as in the mid-infrared spectra, certain frequencies at which the resonance absorption of radiation by a system of oscillating atoms or molecules occurs are analyzed. However, in condensed matter, polymers in particular, at certain temperatures and frequences the processes are possible which also cause dissipation of energy of the electromagnetic field through configurational and other rearrangements. Various methods of dielectric or mechanical spectroscopy allow these processes to be detected in the form of the relaxation type absorption an a peak of losses immediately above Tg (the glass transition temperature) and the p, y and d peaks at temperatures below Tg [89,90]. 

The Fourier transform of the oscillation trains with random phase jumps yields again a Lorentzian line profile as derived in Sect.3.3. Summarizing, we can state that elastic and inelastic collisions which only perturb the phase or amplitude of an oscillating atom without changing its velocity cause homogeneous line broadening. 

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