Zeeman and Stark Effects

The first step in quadrupole resonance studies is the detection of lines in new compounds and the interpretation of the observed frequencies in the light of the molecular and crystalline structure of the compound. In the second step, physical studies are developed which yield informations on the temperature and pressure effects, on the isotopic shift, and on the Zeeman and Stark effects. 

Whilst the most important examples of Zeeman and Stark effects in 1A states are found in molecular beam studies, they can also be important in conventional absorption microwave rotational spectroscopy, as we describe in chapter 10. The use of the Stark effect to determine molecular dipole moments is a very important example. 

The remainder of this section is devoted to a simplified two-level treatment of the Zeeman and Stark effects in the presence of zero-field Stark effect and field-dependent interactions between basis functions 1M) and 2M). In the presence of a static field directed along the space Z-axis, Mj remains a good quantum number. The Zeeman and Stark Hamiltonians involve the interaction between a magnetic field or electric dipole, /r, in the molecule-fixed axis system and the space-fixed magnetic or electric field, F, parallel to the laboratory direction K. The interaction can be expressed in terms of direction cosines... 

Apart from the energy-level structures discussed above, molecules exhibit both the Zeeman and Stark effects. Further, hyperfine structure and isotopic shifts also occur. The occurrence of isotopic shifts is particularly simple to understand considering the substantially altered values of the reduced mass found in the vibrational and rotational energy expressions. 

G. Belin, I. Lindgren, I. Holmgren, S. Svanberg Hyperfine interaction, Zeeman and Stark effects for excited states in potassium. Phys. Scr. 12, 287 (1975)... 

Indirect but important usage of Zeeman and Stark effects is found in fundamental physics researches for example measurements of violation of symmetry in physical laws under time and space inversion known as the T-violation and the parity violation, respectively. Such measurements would eventually give... 

Closely related to the Zeeman effect are two others, the Paschen-Back effect, produced by very strong magnetic fields, and the Back-Goudsmit effect, observed with the spectra of elements having a nuclear magnetic moment, such as bismuth. See also Chemical Elements Electron Theory Paschen-Back Effect and Stark Effect. 

We dealt with the effects of applied static fields on the electronic Hamiltonian in section 3.7. In this section we first give the relevant terms for the nuclear Zeeman and Stark Hamiltonians and then perform the same coordinate transformations that proved to be convenient for the field-free molecular Hamiltonian. 

Finally, there is a way to use fixed frequency lasers for spectroscopy if one can achieve the tuning on the side of the molecules. Species with a permanent magnetic or electric dipole moment can be tuned into resonance by the Zeeman - or Stark-effect respectively. Tunability is very limited and therefore a densely distributed series of fixed frequency laser transitions is necessary for complete coverage of the spectrum. 

The first label, r, indicates the parity of the state functions, as we have just introduced in this section. The second label, r], indicates whether the Hamiltonian is time-even or time-odd. Time-even interactions are typically interactions associated with the electrostatic potential, such as the Jahn-Teller and Stark effects. Time-odd interactions are electrodynamic in nature, the most common one being the Zeeman interaction. We shall now study a function space that is invariant under time reversal... 

See also EPR, Methods Fluorescence Microscopy, Applications Fluorescent Molecular Probes Hole Burning Spectroscopy, Methods Laser Magnetic Resonance Laser Applications in Electronic Spectroscopy Laser Spectroscopy Theory Light Sources and Optics Luminescence Theory Near-IR Spectrometers Raman Optical Activity, Applications Symmetry in Spectroscopy, Effects of UV-Visible Absorption and Fluorescence Spectrometers Zeeman and Stark Methods in Spectroscopy, Applications. 

Not only can electronic wavefiinctions tell us about the average values of all the physical properties for any particular state (i.e. above), but they also allow us to tell us how a specific perturbation (e.g. an electric field in the Stark effect, a magnetic field in the Zeeman effect and light s electromagnetic fields in spectroscopy) can alter the specific state of interest. For example, the perturbation arising from the electric field of a photon interacting with the electrons in a molecule is given within die so-called electric dipole approximation [12] by ... 

From accurate measurements of the Stark effect when electrostatic fields are applied, information regarding the electron distribution is obtained. Further Information on this point is obtained from nuclear quadrupole coupling effects and Zeeman effects (74PMH(6)53). 

The prerequisite for the creation of orientation in the aligned state can also be formulated in terms of the time reversal properties of a Hamiltonian operator which represents the perturbation. As is shown in [276, 277] the alignment-orientation conversion may only take place if the time invariant Hamiltonian is involved. For instance, the Hamiltonian operator of the linear Zeeman effect is odd under time reversal and is thus not able to effect the conversion, whilst the operator of the quadratic Stark effect is even under time reversal and, as a consequence, the quadratic Stark effect can produce alignment-orientation conversion. 

We have calculated exactly the Zeeman effect for the levels IS, 3S and 3P. Indeed it is necessary to know the shift for all the hyperfine levels very well. These calculations are very classical and we just present the results in a Zeeman diagram (see Fig. 5). The most important part in the diagram is the crossing between the 38 2 (F=l, mp=-l) and 3P1/2(F=1, mj =0) levels, because the quadratic Stark effect is proportional to the square of the induced electric field and inversely proportional to the difference of energy between the two considered levels. Moreover the selection rules for the quadratic Stark effect in our case (E perpendicular to B) impose Am.F= l. So it is near this crossing that the motional Stark shift is large enough to be measured. In our calculations the Stark effect is introduced by the formalism of the density matrix [4] where the width of the levels are taken into account. The result of the calculation presented on... 

Although the resonance Is shifted by Zeeman- and motional Stark-effects due to a residual magnetic field of about 75 G in the transition region, the results are promising and lead to the expectation of improved precision in future excited state experiments. 

Owing to the special form of the eigenwave functions for t] f=0, and in accordance with the absence of first-order Zeeman effect, it may be shown that the magnetic dipolar contribution to nitrogen resonance line width is very small33,34). Lines are consequently narrow for many of the compounds studied, a very convenient feature when weak effects, like the Stark effect, are to be studied35). 

As with the Zeeman interaction discussed earlier, (1.43) is usually contracted to the space-fixed p = 0 component. An extremely important difference, however, is that in contrast to the nuclear spin Zeeman effect, the Stark effect in a 1Z state is second-order, which means that the electric field mixes different rotational levels. This aspect is thoroughly discussed in the second half of chapter 8 the second-order Stark effect is the engine of molecular beam electric resonance studies, and the spectra, such as that of CsF discussed earlier, are usually recorded in the presence of an applied electric field. 

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