Relaxation and resonance

When the ball is set in motion by a force, F, it oscillates under the Hooke s Law extension and compression of the spring, but its motion is retarded by the viscous drag of the surrounding medium. In infrared vibrational spectroscopy, the ball is an atom set in excited motion by the irradiation and there is no surrounding viscous medium. In dynamic mechanical relaxation, the ball is part of a molecule set in motion by an applied stress and other chains form the surrounding medium. 

Then the total Newtonian equation of motion of the ball has the exciting force, F, the acceleration and deceleration of the vibration, the drag of the viscous medium and the Hooke s Law restoring force of the spring. This gives for the force and the restraints acting in opposite directions  

this is a second-order linear differential equation containing inertial, viscous and spring force terms. Unfortunately, such a differential equation does not have an explicit solution, so we have to look at limiting cases. 

when the ball is moving slowly we note that this represents low vibration frequencies. Under these circumstances the inertial term, a(d x /,  

This is the first-order linear differential relaxation equation. So we can say that at low frequencies the movement is relaxation. 

---

Yaouanc, A. and de Reotier, P.D. (2010) Muon Spin Rotation, Relaxation, and Resonance Applications to Condensed Matter, Oxford University Press, Oxford. 

R. Kubo, in Fluctuations, Relaxation and Resonance in Magnetic Systems, D. Ter Haar, ed., Oliver and Boyd, Edinburgh, 1961, p. 23. 

Kubo R. A stochastic theory of line-shape and relaxation. In Haar DT, eds. Fluctutation, Relaxation and Resonance in Magnetic Systems. Edinburgh Oliver and Boyd, 1961. 

Ye, X. Demidov, A. Rosea, F. Wang, W. Kumar, A. lonascu, D. Zhu, L. Barrick, D. Wharton, D. Champion, R M., Investigations of heme protein absorption line shapes, vibrational relaxation, and resonance Raman scattering on ultrafast time scales. J. Phys. Chem. A 2003, 107, 8156-8165. 

The two features are attributable to the same type of paraelectric center and correspond to its relaxation and resonance interactions with the alternating electric field. Pirc and Dick first gave a unified theoretical treatment of both peaks. Their treatment applied primarily to crystalline media and is qualitatively applicable to glasses if paraelectric centers are assumed to have a distribution of energy splittings and relaxation times instead of fixed values. In the case of glasses, the theory of low-energy excitations was initially formulated to explain their low-temperature thermal properties Pure vitreous silica apparently contains... 

Gygax, F.N., A. Amato, D. Andieica, M. Pinkpank and A. Schenck, 1999, Book of abstracts, 8th Int. Conf. on Muon Spin Rotation, Relaxation and Resonance, Les Diablerets, Switzerland, August/ September 1999, unpublished. 

In our permittivity units, the complex response of the system looks Hke both relaxation and resonance (Figure 12.16). 

It is important to emphasize the physical diflerence between relaxational and resonance motions because confusion frequently arises due to the use of similar words to describe physically very dissimilar phenomena. 

Kubo R (1962) In ter Haar D (ed) Fluctuations relaxation and resonance in magnetic systems. Plenum, N-Y... 

In comparisons of muons with protons and of muonium with hydrogen atoms, pronounced quantum effects occur whenever dynamics are involved. In this way, muons have been utilized to probe a large variety of properties and materials insulators, semiconductors, metals, superconductors, insulators, gases, liquids, crystalline and amorphous solids, static and dynamic magnetic properties of all kinds, electron mobility, quantum diffusion, chemical reactivity and molecular structure and dynamics. The term adopted for the broad field of muon spin spectroscopy techniques, fiSR, emphasizes the analogy with other types of magnetic resonance for example EPR. juS represents muon spin , and R in a more general sense stands simultaneously for rotation , relaxation and resonance . 

---