Upper states

The population in the upper state as a flinction of time is shown in figure A1.6.2. There are several important things to note. At early times, resonant and non-resonant excitation produce the same population in the upper state because, for short times, the population in the upper state is independent of the Rabi frequency ... 

One should also notice that resonant excitation completely cycles the population between the lower and upper state with a period of 2ji/0. Non-resonant excitation also cycles population between the states but never completely depopulates the lower state. Finally, one should notice that non-resonant excitation cycles population between the two states at a faster rate than resonant excitation. 

Figure Al.6.2. The population in the upper state as a fiinction of time for resonant excitation (frill curve) and for non-resonant excitation (dashed curve).

At 321 mn there is a vibronic origin marked This has one quantum of v, the antisynnnetric C-H stretching mode, in the upper state. Its intensity is induced by a distortion along This state has B2 vibrational symmetry. The direct product of B2 and A2 is B, so it has B vibronic syimnetry and absorbs x-polarized light. One can also see a 4 6,, vibronic origin which has the same syimnetry and intensity induced by... 

A very weak peak at 348 mn is the 4 origin. Since the upper state here has two quanta of v, its vibrational syimnetry is A and the vibronic syimnetry is so it is forbidden by electric dipole selection rules. It is actually observed here due to a magnetic dipole transition [21]. By magnetic dipole selection rules the A2- A, electronic transition is allowed for light with its magnetic field polarized in the z direction. It is seen here as having about 1 % of the intensity of the syimnetry-forbidden electric dipole transition made allowed by... 

In spectroscopy it is common for transitions to be observed as absorptive lines because the Boltzmaim distribution, at equilibrium, ensures a higher population of the lower state than the upper state. Examples where emission is observed, which are by definition non-equilibrium situations, are usually cases where excess population is created in the higher level by infiising energy into the system from an external source. 

The path length is set by the experimental configuration while a is known for each transition (such as OO O J—> OO l, J 1 or OO l J—> 00 2, J 1). Thus, a measurement of zi///provides the partial pressure P of molecules produced in probed states such as OO O Jor 00 1 J. (Strictly, optical probing measures the difference in the partial pressures between the upper and lower states of the probed transition however, in practice, the lower state population is always much larger than the upper state population so that the probe senses only the lower state population in the experiment.)... 

Figure 6. Bending potential curves for the X Ai, A B electronic system of BH2 [33,34], Full hotizontal lines K —Q vibronic levels dashed lines /f — I levels dash-dotted lines K — 2 levels dotted lines K — 3 levels. Vibronic levels of the lower electronic state are assigned in benf notation, those of the upper state in linear notation (see text). Zero on the energy scale corresponds to the energy of the lowest vibronic level.

The use of U — L to indicate a transition between an upper state U and a lower state L is general in spectroscopy and will apply throughout the book. 

If the radiofrequency spectmm is due to emission of radiation between pairs of states - for example nuclear spin states in NMR spectroscopy - the width of a line is a consequence of the lifetime, t, of the upper, emitting state. The lifetime and the energy spread, AE, of the upper state are related through the uncertainty principle (see Equation 1.16) by... 

So far as rule 2 is concerned, since AJ is conventionally taken to refer to J -J", where J is the quantum number of the upper state and J" that of the lower state of the transition, AJ = — 1 has no physical meaning (although it emerges from the quantum mechanics). It is commonly, but incorrectly, thought that AJ = +1 and AJ = — 1 refer to absorption and emission, respectively in fact AJ = +1 applies to both. Transition wavenumbers or frequencies are given by... 

The transition moment (Equation 2.13) for a transition between lower and upper states with vibrational wave functions and j/[ respectively is given by... 

Note that the upper state quantum number of a transition is given first and the lower state quantum number second. 

A close look at Figure 6.8 reveals that the band is not quite symmetrical but shows a convergence in the R branch and a divergence in the P branch. This behaviour is due principally to the inequality of Bq and Bi and there is sufficient information in the band to be able to determine these two quantities separately. The method used is called the method of combination differences which employs a principle quite common in spectroscopy. The principle is that, if we wish to derive information about a series of lower states and a series of upper states, between which transitions are occurring, then differences in wavenumber between transitions with a common upper state are dependent on properties of the lower states only. Similarly, differences in wavenumber between transitions with a common lower state are dependent on properties of the upper states only. 

For the first line, J" = 2 (lower state) and J = Q (upper state)... 

More accurately, we can use the method of combination differences, while still neglecting centrifugal distortion, to obtain B" and B. Transitions having wavenumbers v[5(J — 2)] and v[0 J + 2)] have a common upper state so that the corresponding combination difference A4F(J) is a function of B" only ... 

In the case of H2O it is easy to see from the form of the normal modes, shown in Figure 4.15, that all the vibrations Vj, V2 and V3 involve a change of dipole moment and are infrared active, that is w=l-0 transitions in each vibration are allowed. The transitions may be labelled Ig, 2q and 3q according to a useful, but not universal, convention for polyatomic molecules in which N, refers to a transition with lower and upper state vibrational quantum numbers v" and v, respectively, in vibration N. 

If the upper state is a combination or overtone level, may not be a single symmetry... 

The symbols and indicate allowed and forbidden transitions, respectively, whichever is the upper state. 

Figure 7.20(b) illustrates the case where r c r". An example of such a transition is the Mulliken band system of C2 (see Table 7.6 and Figure 7.17). The value of is 1.2380 A in the D state and 1.2425 A in the X state. Here the most probable transition is from A to B with no vibrational energy in the upper state. The transition from A to C...

In Figure 7.25 are shown stacks of rotational levels associated with two electronic states between which a transition is allowed by the -F -F and, if it is a homonuclear diatomic, g u selection rules of Equations (7.70) and (7.71). The sets of levels would be similar if both were states or if the upper state were g and the lower state u The rotational term values for any X state are given by the expression encountered first in Equation (5.23), namely... 

If the spectrum is observed in emission it is the rotational populations in the upper state which determine relative intensities. They may or may not be equilibrium Boltzmann populations, depending on the conditions under which the molecule got into the upper state. 

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