Energy levels

The interaction of light with an absorption center involves a change in energy. The absorption of energy by an absorption center will cause it to move from a 

When energy is lost from an absorption center, it moves from the excited state back to the ground state. The energy can be released directly by passing from Ex to Eq (Fig. 9.14a), with an energy output given by 

For many optical processes, the number of atoms in each energy state is of critical importance. Under thermal equilibrium, the relative population of two energy levels, E() and E, is given by the Boltzmann law  

The simplest example is a particle confined to a box (Fig. 9.1). As the box is made smaller or the particles less massive the energy levels become further apart. For a cubic box the energy levels may be expressed as 

Polyatomic molecules can also have rotational and vibrational energy and we can write (to a good approximation) 

The rotational and vibrational energies are also quantized. The energy levels 

We see that even in its lowest energy level when v = 0 a vibrator has energy hv. This is called its zero-point energy. Its existence is due to the uncertainty principle which is a fundamental consequence of quantum mechanics. It tells us that we cannot simultaneously specify exactly both the momentum and position of a particle. As particles are confined to a limited region of space (as will occur in vibrational motion) their linear momentum cannot be exactly zero and so the particles energy also cannot be zero. Hence an oscillator can never lose all its energy. 

Note 28toi and are those temperatures at which AT becomes equal to the energy corresponding to the first rotational and vibrational transitions respectively. 

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Briliouin zones Electronic theory of metals divides the electronic slates of a metal into a series of broad energy levels known as Briliouin zones. 

Racah parameters The parameters used to express quantitatively the inter-electronic repulsion between the various energy levels of an atom. Generally expressed as B and C. The ratios between B in a compound and B in the free ion give a measure of the nephelauxetic effect. ... 

When subjected to an electron bombardment whose energy level is much higher than that of hydrocarbon covalent bonds (about 10 eV), a molecule of mass A/loses an electron and forms the molecular ion, the bonds break and produce an entirely new series of ions or fragments . Taken together, the fragments relative intensities constitute a constant for the molecule and can serve to identify it this is the basis of qualitative analysis. 

Nuclear magnetic resctnance involves the transitions between energy levels of the fourth quantum number, the spin quantum number, and only certain nuclei whose spin is not zero can be studied by this technique. Atoms having both an even number of protons and neutrons have a zero spin for example, carbon 12, oxygen 16 and silicon 28. 

Fig. V-14. Energy level diagram and energy scales for an n-type semiconductor pho-toelectrochemical cell Eg, band gap E, electron affinity work function Vb, band bending Vh, Helmholtz layer potential drop 0ei. electrolyte work function U/b, flat-band potential. (See Section V-9 for discussion of some of these quantities. (From Ref. 181.)...

XES, Soft x-ray emission An x-ray or electron beam Energy levels and chemical... 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

Calculate the value of the first three energy levels according to the wave mechanical picture of a particle in a one-dimensional box. Take the case of nitrogen... 

Bolir fiirther postidated that quantum jumps between the different allowed energy levels are always accompanied by absorption or emission of a photon, as required by energy conservation, viz. 

The energy level spectrum of the hamionic oscillator is completely regular. The ground state energy is given... 

It should be mentioned that the single-particle Flamiltonians in general have an infinite number of solutions, so that an uncountable number of wavefiinctions [/ can be generated from them. Very often, interest is focused on the ground state of many-particle systems. Within the independent-particle approximation, this state can be represented by simply assigning each particle to the lowest-lying energy level. If a calculation is... 

It is not possible to solve this equation analytically, and two different calculations based on the linear variational principle are used here to obtain the approximate energy levels for this system. In the first,... 

The variation of tlie frequency can be approximated by a series in the number of quanta, so the energy levels are given by... 

Often, it is a fair approximation to tnmcate the series at the quadratic tenn with The energy levels are then approximated as... 

Wliat does one actually observe in the experunental spectrum, when the levels are characterized by the set of quantum numbers n. Mj ) for the nonnal modes The most obvious spectral observation is simply the set of energies of the levels another important observable quantity is the intensities. The latter depend very sensitively on the type of probe of the molecule used to obtain the spectmm for example, the intensities in absorption spectroscopy are in general far different from those in Raman spectroscopy. From now on we will focus on the energy levels of the spectmm, although the intensities most certainly carry much additional infonnation about the molecule, and are extremely interesting from the point of view of theoretical dynamics. 

Figure Al.2.8. Typical energy level pattern of a sequence of levels with quantum numbers nj for the number of quanta in the symmetric and antisymmetric stretch. The bend quantum number is neglected and may be taken as fixed for the sequence. The total number of quanta (n + n = 6) is the polyad number, which...

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