Fine-structure states

The branching ratios for the fine structure states of the 0(3Pj) were measured by placing a 2-inch diameter active area photomultiplier tube... 

X-ray investigation of inorganic pigments yields information on the structure, fine structure, state of stress, and lattice defects of the smallest coherent regions that are capable of existence (i.e., crystallites) and on their size. This information cannot be obtained in any other way. Crystallite size need not be identical with particle size as measured by the electron microscope, and can, for example, be closely related to the magnetic properties of the pigment. 

The fine structure intervals of the alkali atoms often fall in the 1-10 MHz range, in which case the transition between spin orbit and uncoupled states can be made either diabatically or adiabatically. Jeys et al.16 have observed the transition from an adiabatic to a diabatic passage from the coupled fine structure states to the uncoupled states. With a pulsed laser, they excited Na atoms from the 3p1/2 state to the 34d3/2 state with o polarized light, which leads to 25% my = 1/2 atoms and... 

While the fine structure transitions are inherently magnetic dipole transitions, it is in fact easier to take advantage of the large A = 1 electric dipole matrix elements and drive the transitions by the electric resonance technique, commonly used to study transitions in polar molecules.37 In the presence of a small static field of 1 V/cm in the z direction the Na ndy fine structure states acquire a small amount of nf character, and it is possible to drive electric dipole transitions between them at a Rabi frequency of 1 MHz with an additional rf field of 1 V/cm. 

When the fine structure frequencies fall below 100 MHz they can also be measured by quantum beat spectroscopy. The basic principle of quantum beat spectroscopy is straightforward. Using a polarized pulsed laser, a coherent superposition of the two fine structure states is excited in a time short compared to the inverse of the fine structure interval. After excitation, the wavefunctions of the two fine structure levels evolve at different rates due to their different energies. For example if the nd3/2 and nd5/2 mf = 3/2 states are coherently excited from the 3p3/2 state at time t = 0, the nd wavefunction at a later time t can be written as40... 

The final expression for the population of OH in a particular rotational state j (which, as a consequence of the electronic spin, is half-integer in this case) and in one of the four possible electronic fine-structure states, 2n1/2(A, A") and 2n3/2( 4/, A"), which we designate by the index l = 1-4, is given by... 

Incidentally we note that the structures seen in Figures 11.8 and 11.9 are the remnants of the oscillations due to the sin2(j7e) term in Equation (10.7). The inclusion of the electronic fine-structure states tends to damp them. Since the structures are out-of-phase for the two A-doublet states, averaging over 2II A ) and 2Yl A") smears them out and leads to relatively smooth rotational distributions. 

Figure 1. Optogalvanic signal for stepwise excitation of sodium (3s - 3p — nd, ns) in an H,-air flame. Each transition is split into two components by the fast mixing of the fine structure states, 3p,/t — 3pi/t. The data are not normalized for the variation of laser power with wavelength. At this level of sensitivity the one-photon signal (3s —> 3p) is undetectable.

If we put gs = 2, we find that for the lowest rotational level of the 2 n 3/2 state, J = 3/2, the g-factor is 4/5. For any rotational level of the 2 n 1/2 state, however, (1.67) predicts a g-factor of zero. For a perfect case (a) molecule, therefore, we cannot use magnetic resonance methods to study 2ni/2 states. Fortunately perhaps, most molecules are intermediate between case (a) and case (b) so that both fine-structure states are magnetic to some extent. The other point to notice from (1.67) is that the g-factor decreases rapidly as J increases. 

The spin-orbit coupling therefore removes the degeneracy of the so-called fine-structure states listed above. For example, simply by substituting the appropriate values of J, L and S, the three fine-structure components arising from the L = 1, S= 1 configuration are calculated to have the following first-order spin orbit energies ... 

The q = 1 terms link the fine structure states the importance of this mixing is maximised when the separation between the fine-structure states is small (i.e. small spin-orbit splitting), and when the rotational constant B is large. Consequently molecules like CIO, BrO and 10 are good case (a)2 n systems, whereas CH obeys case (b) coupling, even in its lowest rotational levels. The OH radical approximates more closely to case (a) in its lowest rotational levels, but goes over to case (b) as the rotational quantum number increases. As we mentioned above, the transition from case (a) to case (b) is discussed in detail in chapter 9 it is not, of course, confined to 2n molecules. 

We are now in a position to examine the details of the Zeeman effect in the para-H2, TV = 2 level, and thereby to understand Lichten s magnetic resonance studies. For each Mj component we may set up an energy matrix, using equations (8.180) and (8.181) which describe the Zeeman interactions, and equations (8.201), (8.206) and (8.214) which give the zero-field energies. Since Mj = 3 components exist only for J = 3, diagonalisation in this case is not required. For Mj = 2 the J = 2 and 3 states are involved. For Mj = 0 and I, however, the matrices involve all three fine-structure states and take the form shown below in table 8.7. Note that /. is equal to a0 + 3 63-2/4 and the spin-rotation terms have been omitted. The diagonal Zeeman matrix elements are... 

Figure 8.19. First-order Zeeman splitting ofthe N = 2, J = 1, 2, 3 fine-structure states for the c3nu state of H2.

Comparison of these matrices shows at once that the /I-doublet splitting in the 2nj/2 fine-structure state is determined primarily by the diagonal elements, and will... 

Complete matrices for the parity-conserved 2 If fine-structure states (exclusive of nuclear spin terms) may now be constructed by combining the A-doubling matrices given above with the spin-orbit and rigid body rotation matrices given in our discussion of the LiO spectrum. The matrix representation is block diagonal for each value of J and each parity. The results for the positive and negative parity states are as follows. 

In addition to the extensive data on the yl-doublet transitions of OH already described, Meerts and Dymanus [142] also measured similar spectra of the species OD, SH and SD. The SH and SD radicals were produced by reacting H atoms with either H2S or D2S in the molecular beam source. In all cases hyperfine components of the A -doublet transitions were measured for a number of rotational levels in both fine structure states. The theoretical analysis of the spectra was similar to that already described for OH, with the addition of deuterium quadrupole interactions in OD and SD. In table 8.26 we fist the A-doubling and nuclear hyperfine constants determined for the four species. Note that the parameters fisted above for OH differ slightly from those given subsequently by Brown, Kaise, Kerr and Milton [115]. 

This is, in essence, the result needed to construct figure 8.47. There is more to be done, however, because it is necessary to use (8.430) to derive matrix elements for the parity-conserved functions, and then to take note of the rotational distortion which mixes the fine-structure states. This mixing can be represented by an effective 12 value, which is designated 12 eff in table 8.28, where the results of the Stark experiments are listed. 

Figure 8.48. Lower rotational levels and fine structure states of CO in its a 311 state.

Figure 9.31. FIR laser magnetic resonance spectrum of CO in the a 3n state, observed using the 393.6 pm line from formic acid [62]. This spectrum arises from the J = 7 — 6 rotational transition in the Q = 2 fine-structure state, and the transitions obey the selection rule A Mj = +1. The lower Mj states are indicated in the diagram.

This spectrum is a spectroscopist s dream because apart from its beauty, its assignment is straightforward. The Zeeman effect is essentially linear, and the g-factor identifies both the fine-structure state (3 second-order Zeeman components. The hyperfrne and A-doublet splittings are readily apparent. Many resonances arising from transitions within the two lowest fine-structure states were observed and assigned. 

Figure 10.53. Lowest rotational levels in the fine-structure states of NO.

Klisch, Belov, Schieder, Winnewisser and Herbst [157] have combined all of the data for NO to produce a current best set of molecular constants for three isotopomers, presented in table 10.15. The data used, apart from their own terahertz studies, included the A-doubling of Meerts and Dymanus [156, 158], the sub-millimetre transitions of 15N160 and 14NlsO, and Fourier transform data from Salek, Winnewisser and Yamada [159]. These last authors were able to study the magnetic dipole transitions between the two fine-structure states. The values of the spin orbit constant A for the less common isotopomers come from Amiot, Bacis and Guelachvih [160]. 

J = 3/2, 5/2 and 7/2 levels of both fine-structure states. Also shown are the /l-doublet transitions observed, first by Dousmanis, Sanders and Townes [4], and subsequently by ter Meulen and Dymanus [165] andMeertsandDymanus [166]. The later studies [166] used molecular beam electric resonance methods which were described in chapter 8, and the most accurate laboratory measurements of transitions within the lowest rotational level were those of ter Meulen and Dymanus [165] using a beam maser spectrometer, also described in chapter 8. In the years following these field-free experiments, attention... 

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