Second-order Stark effect

The error is a systematic error arising from ac Stark effect, second order Doppler shift, a frequency offset between the cw dye laser oscillator and the high-powered pulsed amplifier output and laser metrology. Substantial improvement in the accuracy of this measurement appears technically feasible. The current theoretical value is(6,15)... 

Our new coordinates q[, qz, q t though dependent on the direction of the electric field, will not yet quite correspond to the parabolic coordinates used in the usual theory of the Stark effect. In order to obtain these, it would be necessary to go further in the approximation, by supposing that even the second order terms of the electric pertiurbation dominate over the relativistic terms. 

A simple consideration involving a slow mechanical transformation shows that if the energy quantity corresponding to the second-order Stark effect of a system is... 

From a study of the microwave spectrum of 2-methylselenophene, the second-order Stark effect in the ground state was determined.11 The technique used was double radiofrequency-microwave resonance. For the identification by the double resonance method transitions of chiefly the A-state were chosen. From these observations the components of the dipole moment of 2-methylselenophene and the total dipole moment were determined. 

Stark effect measurements for determination of molecular orientation and second-order molecular hyperpolarizability... 

Evaluation of second-order molecular hyperpolarizability using the quadratic Stark effect... 

The ab initio SCF cluster wavefunction has been used to investigate the bonding of CO and CN- on Cu,0 (5,4,1), (5 surface layer, 4 second layer and 1 bottom layer atoms), and to calculate their field dependent vibrational frequency shifts in fields up to 5.2 x 107 V/cm(46). A schematic view of the Cu10 (5,4,l)CO cluster is shown in Figure 8. In order to assess the significance of Lambert s proposal, that the linear Stark effect is the dominant factor in the field dependent frequency shift, the effect of the field was calculated by three methods. One is by a fully variational approach (i.e., the adsorbate is allowed to relax under the influence of the applied field) in which the Hamiltonian for the cluster in a uniform electric field, F, is given by... 

By taking into account the matrix elements off diagonal in n, the second and higher order contributions to the Stark effect can be calculated. If the calculation is carried through second order, the energies are given by1... 

When a) l/n3, the field required for ionization is E = 1/9n4, and as a> approaches l/n3 it falls to E=0.04n. These observations can be explained qualitatively in the following way. At low n, so that a> 1/n3, the microwave field induces transitions between the Stark states of the same n and m by means of the second order Stark effect. With only a first order Stark shift a state always has the same dipole moment and wavefunction, as indicated by the constant slope dW/d of the energy level curve. Thus when the field reverses, — — , the Rydberg electron s orbit does not change. With a second order Stark shift as well, the slope dW/d is not the same at E and —E, and as a result the dipole moment and wavefunction are not the same. If the field is reversed suddenly a single Stark state in the field E is projected onto several Stark states of the same n and m when E — - E. Since all the Stark states of the same n make transitions among themselves they ionize once the field is adequate to ionize one of them, the red one, at E = 1/9n4 for m n. 

In the case of strong excitation, in addition to the effects considered in Section 5.4, under the action of the dynamic Stark effect, transition from alignment to orientation may also take place [36, 243], which means that polarization moments of odd rank may emerge. This manifests itself in the second order of the expansion. Thus, we have... 

If polar diatomic molecules are previously aligned in a beam (see Section 6.2) there is another possibility, proposed in [43], of producing angular momenta orientation using alignment-orientation conversion in a homogeneous electric field due to the second-order Stark effect (see Section 5.4). We will consider this method in more detail since it is a nice example of how to make use of handling the different approaches presented in Chapter 5 simultaneously. 

For an accurate data analysis, a detailed understanding of systematic effects is necessary. Although they are significantly reduced with the improved spectroscopy techniques described above, they still broaden the absorption line profile and shift the center frequency. In particular, the second order Doppler shift and the ac-Stark shift introduce a displacement of the line center. To correct for the second order Doppler shift, a theoretical line shape model has been developed which takes into account the geometry of the apparatus as well as parameters concerning the hydrogen atom flow. The model is described in more detail in Ref. [13]. 

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. 

Now for CsF in its X ground state the value of A is zero the second 3-j symbol in (8.278) is then non-zero only if 1 + J + J is even, so that. / =. / I is a requirement. In other words, there can be no first-order Stark effect in this case. Equation (8.278) tells us that each rotational level J is mixed by the electric field with the adjacent rotational levels. / 1, and the Stark behaviour may therefore be represented by the following 3x3 truncated matrix. 

In our discussion of the Stark effect for CsF, we pointed out that (8.310) vanishes unless 1 + J + J is even in the 3- j symbol with zero arguments in the lower row therefore J = J 1 of necessity, and the Stark effect is second order. We showed that the second-order Stark energy could be obtained from second-order perturbation theory, to give the well-known expression (8.279) which we repeat again ... 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

In the "nonrigid symmetric-top rotors" (such as NH ), the second-order Stark effect is observed under normal circumstances. Indeed, field strengths of the order of 1 600 000 [V/m] are required to bring the interaction into the first-order regime in this case [18]. In contrast, very weak interactions suffice to make the mixed-parity states and appropriate for the description of optically active systems. Parity-violating neutral currents have been proposed as the interaction missing from the molecular Hamiltonian [Eq.(1)] that is responsible for the existence of enantiomers [14,19]. At present, this hypothesis is still awaiting experimental verification. 

A Stark effect for adsorbed sulfate on Pt electrodes has been reported for the 1200 cm symmetric stretching mode of the adsorbed ion [165]. A quadratic dependence of the band center on the applied electric field is observed (Fig. 60). But this field dependence changes with the degree of coverage. The frequency values extrapolated to zero coverage (singleton frequency) present a linear dependence on the applied electric field (Fig. 61). So we conclude that the second-order Stark effect is induced when the ions are close together on the surface. 

---