Zero-field energy

Figure 5.12. But even for systems with finite rhombicity the zero-field energy matrix can be reduced to 2 x 2 dimensionality...

Again, in our example of I) = 2 cm1 and EID = 0.025, even a small a-value of a = E/10 gives A2 = 0.064 cnr1. The complete zero-field energy manifold is schematically depicted in Figure 7.2. 

The label IAmsl = 2 does not mean that two quanta hv are absorbed it is simply a somewhat unfortunate but widely divulged notation to indicate a transition (A E = hv) between two levels that we happen to have labeled as 1+1) and 1-1). Strictly speaking, these labels should only apply to the strong-field situation of (S S) B S) as we discussed in Chapter 4. In the present example of Figure 11.1 we are in the weak-to-intermediate field regime (S S > B S), which means that the actual wavefunctions are linear combinations of the ones in Equation 11.5. In particular, a rhombic E-term mixes the 1+1) and 1-1) levels as can be seen from its appearance in nondiagonal positions in the zero-field energy matrix... 

And the combination of Heisenberg superexchange plus Zener double exchange results in zero-field energy levels in terms of the system spin S... 

Values for the Zero-Field Energies, Dipole Moments (p), Polarizabilities (a), HyperpolarizabiUties (fi), and Second Hyperpolarizabilities (y) for the Non-BO H2 Isotopomer Series"... 

Expectation Values for Zero-Field Energies, Virial Coefficients (r ), Dipole Moments (p), and Static Polarizabilites (a) for Non-BO LiH/D for Various Expansion Lengths (m) ... 

In adiabatic ionization what is important is the ordering of the zero field energy levels, not their precise zero field energies. 

As shown by Fig. 14.15, the resonances occur near zero field, and it is easy to calculate the small Stark shifts with an accuracy greater than the linewidths of the collisional resonances. As a result it is straightforward to use the locations of the collisional resonances to determine the zero field energies of the p states relative to the energies of the s and d states. Since the energies of the ns and nd states have been measured by Doppler free, two photon spectroscopy,22 these resonant collision measurements for n = 27, 28, and 29 allow the same precision to be transferred to the np states. If we write the quantum defect dp of the K np states as... 

Most of the high precision spectroscopy of He Rydberg states has been done by microwave resonance, which is probably the best way of obtaining the zero field energies. Wing et a/.8-12 used a 30-1000pA/cm2 electron beam to bombard He gas at 10-5-10-2 Torr. As electron bombardment favors the production of low states, it is possible to detect A transitions driven by microwaves. The microwave power was square wave modulated at 40 Hz, and the optical emission from a specific Rydberg state was monitored. When microwave transitions occurred to or... 

Table 13 Secular equation and zero-field energies for T-type terms under symmetry lowering... 

Fig. 7 Zero-field energy levels for reduction of T-terms of Oh to multiplets of D 4 systems with Aax > 0...

Case 2 On increasing the CF strength 10Dq (which reduces A0) the biquadratic spin-spin interaction takes on significance. The zero-field energy levels lie at (/7) = -2a (doublet) and e(rs) = +a (quartet). The ZFS is S = 3a. 

On small tetragonal distortion the /s(O0 multiplet starts to split into two multiplets I (D ) (ground) and separated by the zero-field energy... 

For a tetragonal bipyramid the zero-field energy gap adopts moderate values, and on the extreme negative tetragonality it reaches a limit of D/hc = - 30 cm-1, whereas for the very positive tetragonality it is D/hc <... 

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... 

For the zero-field problem F remains a good quantum number, but J is not because of the hyperfine mixing. The spin spin, spin orbit and spin rotation energies have already been listed in table 8.6. The complete zero-field energy matrix, including the hyperfine terms, is as follows. 

Figure 8.39. Behaviour of the energy levels of HF in the J = 1 level in magnetic fields from 0 to 8 kG. The electric field was 2952 V cm-1. The vertical arrows indicate the five transitions measured by de Leeuw and Dymanus [89]. For the zero field energy level pattern, see figure 8.38.

Santos-Lemus and Yartsev (1995) analyzed earlier data on NIPC doped PC by Santos-Lemus (1983) and Santos-Lemus and Hirsch (1986). The temperature dependence was analyzed by a Poole-Frenkel model. From plots of the activation energy versus E /2, zero-field energies were determined for different NIPC concentrations. The values were in the range of 0.60 to 0.73 eV. From a plot of the zero-field activation energy versus p K a value of 0.98 eV was derived for the potential barrier of an isolated NIPC molecule. 

Ulanski et al. (1990) described hole mobilities of poly(paracyclophane) (PDE) and PDE doped with 4% tetracyanoethylene (TCNE). The field dependencies were described as log i < E. The temperature dependencies were described by an Arrhenius relationship with a field-dependent activation energy. For PDE, the zero-field energy was 0.80 eV. The incorporation of TCNE resulted in an increase in mobility of approximately an order of magnitude. The authors suggested that this is caused by a decrease in the plane-to-plane distance in a cyclophane unit due to complexing with TCNE. 

Following the procedures outlined in the calculations presented above, substitution of the Zeeman coefficients and zero-field energies into the Van Vleck equation yields the following expression for the magnetic susceptibility ... 

Fig. 10. Zero-field energy ladder for deuterated rhodoso-chloride, [Cr4(ND3),2 (ODlelClg, calculated from best-fit parameters in Giidel and Hauser (68).

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