In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(−t/τ) (exponential decay).
In simple linear systems
Mechanics: Damped unforced oscillator
Let the homogeneous differential equation:
m d 2 y d t 2 + γ d y d t + k y = 0 {\displaystyle m{\frac {d^{2}y}{dt^{2}}}+\gamma {\frac {dy}{dt}}+ky=0}
model damped unforced oscillations of a weight on a spring. The displacement will then be of the form y ( t ) = A e − t / T cos ( μ t − δ ) {\displaystyle y(t)=Ae^{-t/T}\cos(\mu t-\delta )} . The constant T ( = 2 m / γ {\displaystyle =2m/\gamma } ) is called the relaxation time of the system and the constant μ is the quasi-frequency.
Electronics: RC circuit In an RC circuit containing a charged capacitor and a resistor, the voltage decays exponentially:
V ( t ) = V 0 e − t R C , {\displaystyle V(t)=V_{0}e^{-{\frac {t}{RC}}}\ ,}
The constant τ = R C {\displaystyle \tau =RC\ } is called the relaxation time or RC time constant of the circuit. A nonlinear oscillator circuit which generates a repeating waveform by the repetitive discharge of a capacitor through a resistance is called a relaxation oscillator.
In condensed matter physics In condensed matter physics, relaxation is usually studied as a linear response to a small external perturbation. Since the underlying microscopic processes are active even in the absence of external perturbations, one can also study "relaxation in equilibrium" instead of the usual "relaxation into equilibrium" (see fluctuation-dissipation theorem).
Stress relaxation In continuum mechanics, stress relaxation is the gradual disappearance of stresses from a viscoelastic medium after it has been deformed.
Dielectric relaxation time In dielectric materials, the dielectric polarization P depends on the electric field E. If E changes, P(t) reacts: the polarization relaxes towards a new equilibrium, i.e., the surface charges equalize. It is important in dielectric spectroscopy. Very long relaxation times are responsible for dielectric absorption. The dielectric relaxation time is closely related to the electrical conductivity. In a semiconductor it is a measure of how long it takes to become neutralized by conduction process. This relaxation time is small in metals and can be large in semiconductors and insulators.
Liquids and amorphous solids An amorphous solid such as amorphous indomethacin displays a temperature dependence of molecular motion, which can be quantified as the average relaxation time for the solid in a metastable supercooled liquid or glass to approach the molecular motion characteristic of a crystal. Differential scanning calorimetry can be used to quantify enthalpy change due to molecular structural relaxation. The term "structural relaxation" was introduced in the scientific literature in 1947/48 without any explanation, applied to NMR, and meaning the same as "thermal relaxation".
Spin relaxation in NMR
In nuclear magnetic resonance (NMR), various relaxations are the properties that it measures.
Chemical relaxation methods
In chemical kinetics, relaxation methods are used for the measurement of very fast reaction rates. A system initially at equilibrium is perturbed by a rapid change in a parameter such as the temperature (most commonly), the pressure, the electric field or the pH of the solvent. The return to equilibrium is then observed, usually by spectroscopic means, and the relaxation time measured. In combination with the chemical equilibrium constant of the system, this enables the determination of the rate constants for the forward and reverse reactions.
Monomolecular first-order reversible reaction A monomolecular, first order reversible reaction which is close to equilibrium can be visualized by the following symbolic structure:
A → k B → k ′ A {\displaystyle {\ce {A}}~{\overset {k}{\rightarrow }}~{\ce {B}}~{\overset {k'}{\rightarrow }}~{\ce {A}}}
A ↽ − − ⇀ B {\displaystyle {\ce {A <=> B}}}
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