In fact, virtually all oscillators become anharmonic when their pump amplitude increases beyond some threshold, and as a result it is necessary to use nonlinear equations of motion to describe their behavior.
32.
Pulsed lasers are most often used because TPA is a third-order nonlinear optical process, and therefore is most efficient at very high anharmonic oscillator model for depicting vibrational behavior of molecules.
33.
At the higher end of possible heat capacities, heat capacity may exceed " R " by modest amounts, due to contributions from anharmonic vibrations in solids, and sometimes a modest contribution from conduction electrons in metals.
34.
An anharmonic oscillator ( in contrast to a simple harmonic oscillator ) is one in which the potential energy is not quadratic in the extension " q " ( the generalized position which measures the deviation of the system from equilibrium ).
35.
It has been extended by P . N . Keating to include anharmonic effects ( and calculate third-order elastic constants ), and many other researchers have extended it to include forces between the covalent bonds, and augment it in other ways.
36.
In the paper, Heisenberg tried to explain the energy levels of a one-dimensional anharmonic oscillator, avoiding the concrete but unobservable representations of electron orbits by using observable parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states.
37.
Further examples of anharmonic oscillators include the large-angle pendulum; nonequilibrium semiconductors that possess a large hot carrier population, which exhibit nonlinear behaviors of various types related to the effective mass of the carriers; and ionospheric plasmas, which also exhibit nonlinear behavior based on the anharmonicity of the plasma.
38.
The speed of sound is also slightly sensitive, being subject to a second-order anharmonic effect, to the sound amplitude, which means that there are non-linear propagation effects, such as the production of harmonics and mixed tones not present in the original sound ( see parametric array ).
39.
In his work a simple example of an anharmonic oscillator evolving under the force of the form as a superposition of incoherent sinusoidal oscillations with continuous spectrum was used to show that depending on a specific approximation time scale the evolution of the system can be either deterministic, or a stochastic process satisfying Fokker-Planck equation, or even a process which is neither deterministic nor stochastic.
40.
Where \ mathrm { H _ o } is the ground-state Hamiltonian, the linear term vanishes ( as the ground state is found by minimizing the energy in terms of the ionic position ), and \ Gamma _ { ij, \ alpha \ beta } and \ Psi _ { ijk, \ alpha \ beta \ gamma } are the second-and third-order phonon dispersion, while the anharmonic ( third order and higher ) terms are responsible for thermal expansion as well as the phonon up-( multiple low-energy optical phonons combine to form a high-energy phonon ) and downconversion ( a high-energy phonon splits into multiple low-energy phonons ).
