Stark energy

Yttrium aluminum borate, YAlj (603)4 (abbreviated to YAB), is a nonlinear crystal that is very attractive for laser applications when doped with rare earth ions (Jaque et al, 2003). Figure 7.9 shows the low-temperature emission spectrum of Sm + ions in this crystal. The use of the Dieke diagram (see Figure 6.1) allows to assign this spectrum to the " Gs/2 Hg/2 transitions. The polarization character of these emission bands, which can be clearly appreciated in Figure 7.9, is related to the D3 local symmetry of the Y + lattice ions, in which the Sm + ions are incorporated. The purpose of this example is to use group theory in order to determine the Stark energy-level structure responsible for this spectrum. 

Fig. 6.16 Hydrogenic Stark energy levels (- -) before considering the effect of the core perturbation Vd(r). The alkali levels after incorporating the diagonal and off diagonal...

Fig. 15.2 Stark energy level diagram of the Na mt = 0 states, relevant to the multiphoton-assisted collisions. The vertical lines indicate the colhsional transfer and are drawn at the fields where they occur. The thick arrows correspond to the emitted photons (from ref. 8).

At the microscopic level, the nonlinearity of a molecular structure is described by the electric dipole interaction of the radiation field with the molecules. The resulting induced dipole moment and the Stark energy are given as (1,3)... 

The sum-over-states method is based on the perturbation expansion of the Stark energy term in which nonlinearities are introduced as a result of mixing with excited states. For example, the expression for Y(—3u) o),a),a)) which will be responsible for third harmonic generation is given as (25)... 

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 ... 

Diagonalisation of the Stark matrices enables us to plot the Stark energies, given values of B and /M), and the results are shown in figure 8.27 for the first three rotational levels, J = 0, 1 and 2. The parameter X is defined by A.2 = iJ E jB. In figure 8.28 we show plots of the effective electric moment of the molecule in the different J, M states listed in figure 8.27. With the aid of both diagrams, we are able to understand the principles of electric state selection, and the electric resonance transitions. 

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 ... 

For a dipole moment of 1.826 526 D and an electric field of 1500 V cm-1 the Stark energies of the M = 0 and M = 1 levels are calculated to be 308.31 and -154.16 kHz respectively. These shifts are very small because of the large separation between the rotational levels. 

In an electric field of 1475 V cm 1 the Stark energies are +298.119 kHz for Mj = 0, and are —149.064 kHz for Mj = 1. Using these values and the molecular constants given above, we may construct the hyperfine energy level diagram shown in figure 8.38. We label the levels with the basis state labels used in the above... 

Using (8.432) and (8.433) the Stark energies for J = 2, S2 = 2 can be readily calculated and the results are presented in figure 8.50 the initial splitting of the /1-doublets was determined from the electric resonance study to be 7.351 MHz for the v = 0 level. In small electric fields the parities of the states are essentially preserved, and transitions between the /1-doublets have their full electric dipole intensities. At higher electric fields, however, the opposite parity states are mixed and the electric dipole intensity decreases. It follows that so far as the intensities of the electric resonance transitions are concerned, low electric fields are desirable. On the other hand, Stern, Gammon,... 

Figure 8.50. Stark energies of the A -doublet levels for Q =2, J = 2. On the left-hand side, in zero field, the wave functions are the parity-conserved combinations given in equation (8.432). On the right-hand side, in strong field, the wave functions are the simple combinations shown, with parity not conserved.

For a discussion of the Stark energy of polar molecules, see Townes, C.H. Schawlow, A.L. "Microwave Spectroscopy",... 

The general cross section for the rotating molecule is a sum of two terms in e - and e similar to a (14). At small e, therefore, symmetric top molecules should have cross sections that go over to the dependence. The coefficient, however, depends directly on AW the first-order Stark energy shift and molecules that do not have a first-order Stark effect are expected to continue the e - dependence even at small energies. Linear molecules do not have first-order energy shifts, for example. It is interesting to note that... 

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

H.-Y. Meng, Y.-X. Zhang, S. Kang, T.-Y. Shi, M.-S. Zhan, Theoretical complex Stark energies of lithium by a complex scaling plus the B-spfine approach, J. Phys. B 41 (2008) 155003. 

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