Inharmonic oscillator

Fig. 4.3 Scheme of electronic excitation of a diatomic molecule, also called potential energy curve (inharmonic oscillator). A ground state, B first excited state, 1 direct dissociation from exited ground state, 2 direct dissociation from ground state, 3 radiation transfer from exited state, 4 excitation from ground state. 

A complex molecule can be considered as a system of coupled inharmonic oscillators. Empir-... 

Linear Synthesis. The most popular method of synthesis is so-called Additive Synthesis , where the output is a sum of oscillators. While it is commonly assumed that the oscillators produce sinusoids (Fourier synthesis), in fact, they can be any waveform. Furthermore, with static additive synthesis, a pre-mixed combination of harmonics was stored in the lookup table. Unfortunately, this doesn t permit inharmonic partials. Dynamic Fourier synthesis allows the amplitudes and frequencies of the partials to be varied relative to each other. Computationally, it is important to recognize the that updating oscillator coefficients for large numbers of oscillators can be expensive. 

The challenging task of high frequency resonators arises from inharmonic modes, often called spurious modes or spurs. These are characterized by movement of particles in several regions of the quartz disc in antiphase. If the frequency separation between the harmonic and inharmonic modes is not sufficiently large, modes can efficiently be coupled. Frequency jumps of quartz oscillators are an unwanted effect that impedes a reliable frequency measurement. Spurious modes, especially those which lie close to the resonance frequency, enhance the challenges of oscillator design. 

At very low temperatures, these lattice oscillations comprise almost all of the heat capacity. At higher temperatures, a correction for the inharmonicity of the lattice oscillations must be considered. In addition, contributions from group oscillations and rotations about main-chain bonds must also be added at higher temperatures. Finally, a contribution from defects may also be needed. 

Spurious modes are inharmonic modes that may also appear in the vibrational spectrum of the quartz resonator. The spurious modes usually occur in high-resistance modes, and they generally occur at frequencies that are higher than the desired oscillation frequency. The occurrence of spurious modes is increased for crystals operating in an overtone mode as shown in Fig. 3.3. 

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