Second-order stark energy

The energies of the levels in an electric field can be calculated by numerical diagonalisation of the above matrix for different values of the electric field and the J, M quantum numbers. However, perturbation theory has also often been used and we may readily derive an expression for the second-order Stark energy using the above matrix elements. The result is as follows ... 

Figure 8.27. Second-order Stark energies for the first three rotational levels of a heteronuclear diatomic molecule in a 1 state [48]. The parameter X is defined by X2 = //q 1 /. 5. Note that the states with M= 1 or 2 remain rigorously degenerate, irrespective of field strength.

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 ... 

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... 

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... 

We have thus far treated the K (n + 2)s states as having no Stark shifts. While they have no first order Stark shift, they do have a second order shift due to their dipole interaction with the p states, which are removed from the s states by energies large compared to the microwave frequency. The microwave field does not produce appreciable sidebands of the s state since it has no first order Stark shift. However, it does induce a Stark shift to lower energy. Not surprisingly the Stark shift produced by a low frequency microwave field of amplitude E is the same as the second order Stark shift produced by a static field Es/j2, they have the same value of (E2). Careful inspection of Fig. 10.9 reveals that the resonances observed with high microwave powers shift with power. 

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 our picture of microwave ionization the n dependence of the ionization fields comes from the rate limiting step between the bluest n and reddest + 1 Stark states. It would be most desirable to study this two level system in detail, but in Na this pair of Stark levels is almost hopelessly enmeshed in all the other levels. In K, however, there is an analogous pair of levels which is experimentally much more attractive [17,18]. The K energy levels are shown in Fig. 6. All are m = 0 levels, and we are interested in the 18s level and the Stark level labelled (16,3). We label the Stark states as (n, k) where n is the principal quantum number and k is the zero field state to which the Stark state is adiabatically connected. As shown in Fig. 6, the (16, k) Stark states have very nearly linear Stark shifts and the 18s state has only a very small second order Stark shift, which is barely visible on the scale of Fig. 6. The 18s and (16,3) states have an avoided crossing at a field of 753 V/cm due to the coupling produced by the finite size of the K-" core [19]. 

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

Type of Rotational Energy in Stark Energy Second-Order Stark ... 

It is instructive to compare these results with the behavior of the Is state. In the first place, the effect of a uniform electric field on the Is level is zero, to first order, because (ls i/ ls) vanishes for reasons of symmetry. Only when first-order corrections are made to the 1 s wavefunction are energy effects seen, and these occur in the second-order energy terms. Therefore, the Is state gives a second-order Stark effect, but no first-order effect. The Is state gives no first-order effect because the spherically symmetric zeroth-order wavefunction has no electric dipole to interact with the field. But the proper zeroth-order wave functions for some of the = 2 states, given by Eqs. (12-81) and (12-82), do provide electric dipoles in opposite directions that interact with the field to produce first-order energies of opposite signs. 

FIGURE 3 Effect of the square-wave Stark-modulation voltage on the appearance of the J = 2 3 transition of DCS. Note that as the Stark lobes labeled by M are displaced from the field-off line, the intensity of the line increases. The Stark effect is second order. The energy level diagram for a linear molecule in an electric field is shown in the inset. The allowed AM = 0 transitions are also depicted. 

That part of the black body spectrum coincident with the atomic transition frequencies leads to the transitions which redistribute the population. In contrast, all the energy of the black body radiation contributes to the shift of the energy levels. The energy shift is a second order ac Stark shift, and for state n the shift AWb is given by10... 

It appears likely that the statistical uncertainty will eventually be reduced to around 100 kHz, so we consider sources of systematic error which may be expected to enter at this level. The uncertainty in the second-order Doppler shift (450kHz/eV) will be reduced to 100 kHz by a 5% measurement of the beam energy. The AC Stark shift of the 2S-3S transition will be around 70 kHz for the present laser intensity, and can be extrapolated to zero intensity by varying the UV power. Finally, as mentioned above, the systematic uncertainties will be quite different from those in the microwave and quench anisotropy measurements. 

In most problems it is either unnecessary or impracticable to carry the approximation further, but in some cases the second-order calculation can be carried out and is large enough to be important. This is especially true in cases in which the first-order energy W is zero, as it is for the Stark effect for a free rotator, a problem which is important in the theory of the measurement of dipole moments (Sec. 49/). 

In order to show that the translational Zeeman effect is negligible, we take an average translational velocity, Fq =yA TjM, which at T = — 60 °C leads to Fo=201 m/sec for ethylene oxide. At 25 kGauss this corresponds to a cross field Fts of 5 V/cm. With b = l-88 Debye for ethylene oxide, tlus leads to an estimate of the Stark effect energy, < ts > = lyMb TsI =4.7 MHz. We now take the most critical case, the Stark effect perturbation of the loi and lio rotational levels respectively, where the rotational energy difference in the denominator is smallest (11.4 GHz), and we get a second order perturbation in the order of 2 kHz which may come into the range of the experimental accuracy of... 

The term AWi (= Ajr ) is the first-order energy shift, and AW, (= Cjr ) is the second-order shift. Equation 16 has the form of (2), and in the region of validity of the rotational Stark effect the simple Langevin theory applies. However, the effective angular momentum is determined by the first two terms of (16), and the effective polarizability is determined by the last two terms. 

Equation (14) clearly indicates that molecules whose linear spectra show intense (i.e., large /tge) low-energy (i.e., low ge = hujgg) CT (i.e., large A/ige, either positive or negative, as indicated by solvatochromism or by Stark spectroscopy) transitions are good candidates for second-order NLO investigations. 

The shift in the energy of a quantum-mechanical system caused by an applied electric field is called the Stark effect. The first-order (or linear) Stark effect is given by (14.16), and from (14.17) it vanishes for a system with no permanent electric dipole moment. The second-order (or quadratic) Stark effect is given by the energy correction and is proportional to the square of the applied field. 

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