The Bohr Frequency Condition

This modification of the Planck equation was suggested by Bohr (1922 Nobel Prize for Physics) and is sometimes referred to as the Bohr frequency condition. 

It applies to a large range of transitions between energy states of nuclei, electrons in atoms, rotational, vibrational and electronic changes in ions and molecules, and the transitions responsible for the optical properties of metals and semiconducting materials. 

Line spectra are so-called because of the way in which they are observed, the emitted radiation passing through a slit onto a photographic plate and registering as a line . 

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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. 

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... 

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. 

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

Fig. 5.4.1b). If the nucleus is supplied with sufficient energy to fulfill the Bohr frequency condition or resonance condition, AE = hv0, a transition becomes possible. This condition can be achieved by applying an alternating radio frequency field (rf), B, perpendicular to B0 and rotating in the xy plane with a frequency, v, equal to v0. In this situation, v, = v = ( >/2re)B0 [Eq. (3)] defines the resonance condition. As the nucleus absorbs energy, its magnetic moment fi rotates away from B0 toward the xy plane (Fig. 5.4.1a). 

If the perturbation frequency co is approximately in resonance with co o, thus satisfying the Bohr frequency condition hco = E — Eq, then Eq (4.88) reduces to... 

This is called the Bohr frequency condition. We now understand that the atotnic transition energy AE is equal to the energy of a photon, as proposed earlier by Planck and Einstein. 

The electronic transitions in atoms and molecules (and in solids) are associated with two electronic states of the system by the Bohr frequency condition... 

We should expect that such a molecule, possessing an electric moment, would radiate according to the quantum theory the quantum frequencies will, however, differ from the classical ones. The energies of the stationary states are given by (1). Since only one Fourier term occurs, in the motion the quantum number can change by +1 or —1 only, and the Bohr frequency condition gives therefore for the emission (m+l)->-m ... 

This is not surprising, for the principles used are not really consistent on the one hand the classical differential relation is replaced by a difference relation, in the shape of the Bohr frequency condition,... 

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

The equation AE hv is called the Bohr frequency condition and is the basic assumption that as the electron falls from one level to another, the energy evolved is given off as a photon of energy E hv. The energies and the energy differences are quantized, and so are the frequencies. The isolated atom emits hght of only certain definite frequencies, as observed in the bright line spectrum. ... 

This equation shows that only the Fourier component B((o ) is involved in the excitation to the state m, in agreement with the Bohr frequency condition. At absorption, the field loses the energy cOmn. corresponding to a photon. 

The relationship of electromagnetic radiation to changes of energy in nuclei, atoms, molecules and metals the Bohr frequency condition... 

The Bohr frequency condition was introduced, which relates the difference in energy between any two energy levels to the energy of a photon that is either absorbed or emitted in a radiative transition. 

In conventional photokinetic studies, radiation effects an electronic transition from the ground state of the molecule to some electronically excited state. Coupled with the electronic transition there are changes in the vibrational and rotational state of the molecule. For a transition to occur the Bohr frequency condition, Ae = hv = HcqIX, must be obeyed. Absorption at a specific wavelength is determined by the molar absorption coefficient which, for photochemically interesting problems, is proportional to the square of the electric dipole transition moment... 

This is the Bohr frequency condition, if you recall from Chapter 9. 

Equation 14.1 is written in the original form of the Bohr frequency condition The difference in energy of the two quantum states equals the energy of the photon, which equals hv. 

These assumptions are brought together in the Bohr frequency condition, which relates the frequency v (nu) of radiation to the difference in energy AE between two states of an atom or molecule ... 

We can relate this energy difference to the properties of the light that can bring about the transition. From the Bohr frequency condition (eqn 9.1), this energy separation corresponds to a frequency of... 

The energy of a photon emitted or absorbed, and therefore the frequency, v (nu), of the radiation emitted or absorbed, is given by the Bohr frequency condition (Section 9.1) ... 

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