Frequency condition

In this section we will discuss more conventional spectroscopies absorption, emission and resonance Raman scattering. These spectroscopies are generally measured under single frequency conditions, and therefore our... 

The selection niles are derived tlnough time-dependent perturbation theory [1, 2]. Two points will be made in the following material. First, the Bolu frequency condition states that the photon energy of absorption or emission is equal... 

Boln- frequency condition. The second important feaUire is that i- 21 nonzero for an allowed... 

Atomic and Molecular Energy Levels. Absorption and emission of electromagnetic radiation can occur by any of several mechanisms. Those important in spectroscopy are resonant interactions in which the photon energy matches the energy difference between discrete stationary energy states (eigenstates) of an atomic or molecular system = hv. This is known as the Bohr frequency condition. Transitions between... 

This relation is called the Bohr frequency condition. If the energies on the right of this expression are each proportional to h ln2, then we have accounted for Rydberg s formula. We still have to explain why the energies have this form, but we have made progress. 

Bohr frequency condition The relation between the change in energy of an atom or molecule and the frequency of radiation emitted or absorbed ... 

Bohr frequency condition, 13 Bohr radius, 23 boiling, 314 boiling point, 314 alkanes, 737 anomalous, 184 boiling point prediction, 180 boiling point trends, 183 boiling-point elevation, 332 Boltzmann, L., 276 Boltzmann formula, 276 bomb calorimeter, 224... 

Energy loss under low frequency conditions is often used as a key factor for evaluating elastomer compositions. In this respect, the rolling resistance and skidding properties are usually evaluated by tan delta (tan 8) at 60°C and 0°C, where the 60°C value correlates with low frequency loss and 0°C correlates with skid resistance (high frequency loss). 

Extensive experiments were in fact needed before optimal test and acquisition conditions were eventually set (for details, see ). In any fixed strain and frequency conditions, data acquisition is made in order to record 10,240 points at the rate of 512 pt/s. Twenty cycles are consequently recorded at each strain step, with the immediate requirement that the instrument is set in order to apply a sufficient number of cycles (for instance, 40 cycles at 1.0 Hz, 20 cycles at 0.5 Hz the stability condition with the RPA) for the steady harmonic regime to be reached. Data acquisition is activated as the set strain is reached and stable. 

Fig. 5.8 Rate of Au(III) reduction as a function of ultrasound frequency. Conditions Au(III) 0.2 mM, 1 -propanol 20 mM, atmosphere Ar, ultrasound intensity 0.1 W mlA1 [33]...

In order to understand these observations it is necessary to resort to quantum mechanics, based on Planck s postulate that energy is quantized in units of E = hv and the Bohr frequency condition that requires an exact match between level spacings and the frequency of emitted radiation, hv = Eupper — Ei0wer. The mathematical models are comparatively simple and in all cases appropriate energy levels can be obtained from one-dimensional wave equations. 

According to the Bohr frequency condition the emitted radiation should have the exact frequency to excite a second Fe atom from its ground state, in the reverse process... 

Assumptions 1 to 3 as in model 8. Low-frequency test input and high-frequency conditioning input are positively correlated. After the in-phase inputs are introduced 14 times only, the test input is reintroduced and the changes in Xi Wi and A were investigated. 

Assumptions 1 to 3 as in model 9. Low-frequency test input and high-frequency conditioning input are anticorrelated. 

Transition probabilities. The interaction of quantum systems with light may be studied with the help of Schrodinger s time-dependent perturbation theory. A molecular complex may be in an initial state i), an eigenstate of the unperturbed Hamiltonian, Jfo I ) = E 10- If the system is irradiated by electromagnetic radiation of frequency v = co/2nc, transitions to other quantum states /) of the complex occur if the frequency is sufficiently close to Bohr s frequency condition,... 

The probability is a function of the incident energy per unit time, per unit area, I (co) Aco of the incident radiation in the frequency interval between co and co + Acu. We will also refer to /(co) as the spectral intensity of the incident radiation. The matrix element represents the expectation value of the dipole moment operator between initial and final state, hcofi = Ef—Ei is Bohr s frequency condition it is related to the energies of the initial and final states, i), /), and n designates the refractive index. 

The integration over / replaces the <5 function of frequency by h and fixes Ef at the value given by Bohr s frequency condition, Eq. 2.80. The integration over , must be done numerically. For each fixed value /,-, the summation over //is over only two terms (selection rules),... 

In this equation, the energies , and / of the initial and final states, i) and I/), and the dipole moment all refer to a pair of diatomic molecules hcvij = Ef — Ei is Bohr s frequency condition. With isotropic interaction, rotation and translation may be assumed to be independent so that the rotational and translational wavefuntions, population factors, etc., factorize. Furthermore, we express the position coordinates of the pair in terms of center-of-mass and relative coordinates as this was done in Chapter 5. 

This relation is called the Bohr frequency condition. Each spectral line arises from a specific transition (Fig. 1.18). 

Here is an empirical constant now known as the Rydberg constant its value is 3.29 X 101J Hz. This empirical formula for the lines, together with the Bohr frequency condition, strongly suggests that the energy levels themselves are proportional to rg[n1. 

The observation of discrete spectral lines suggests that an electron in an atom can have only certain energies. Transitions between these energy levels generate or absorb photons in accord with the Bohr frequency condition. 

Bohr frequency condition The relation between the change in energy of an atom or molecule and the frequency of radiation emitted or absorbed AE = hv. Bohr radius a0 In an early model of the hydrogen atom, the radius of the lowest energy orbit now a specific combination of fundamental constants (aG =... 

The Bohr frequency condition relates the characteristic frequencies of an atom to a set of characteristic energies... 

Having examined the leading interpretations of the quantum formalism, a more general theory of atomic structure, consistent with all points of view, could conceivably now be recognized. The first aspect, never emphasized in chemical theory, but fundamental to matrix mechanics, is that the observed frequencies that determine the stationary energy states of an atom, always depend on two states and not on individual electronic orbits. The same conclusion is reached in wave mechanics, without assumption. It means that an electronic transition within atoms requires the interaction between emitter and receptor states and the frequency condition AE(An) = hu, for all pairs in n. This condition by itself offers no rationale for the occurrence of the... 

If a molecule interacts with an electromagnetic field, a transfer of energy from the field to the molecule can occur only when Bohr s frequency condition is satisfied. Namely,... 

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