First Order Frequencies

For a nucleus with I = 1, the first order ENDOR spectrum consists of four transitions at frequencies 

According to (3.3), 4 I ENDOR transitions will be observed for a nucleus with arbitrary spin I and with an unresolved hf structure in the EPR display. If the hf structure is resolved, however, each mrstate can be saturated individually and either a four-line ENDOR spectrum (EPR observer -1 mi I) or only a two-line ENDOR spectrum (EPR observer m, = I) will be observed. 

For Bo oriented parallel to one of the principal axes of coaxial g, A and Q tensors, the first order transition frequencies in (3.3) reduce to 

A further contribution to the first order ENDOR frequencies arises from the nuclear dipole-dipole interaction = IDK between the two nuclei I and K. The shifts of the ENDOR lines of nucleus I due to %CD are described by mKD33(ms), with D33(m ) = R3CI(ms)DCK(ms)R3/cI(ms)cK(ms). In transition metal complexes this interaction is 

4 Throughout this paper the principal values of magnetic coupling tensors are denoted by lower indices x, y, z if the corresponding principal axes coincide with the g tensor axes. In all other cases, indices 1, 2, 3 are used 

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Fig. lOa-c. Higher order splittings in symmetry planes Single crystal nitrogen ENDOR spectrum of Cu(TPP) diluted into (H20)Zn(TPP) with Bo normal to the porphyrin plane B0 = 327.7 mT. a) Observed spectrum. (Adapted from Ref. 66) b) Transition frequencies obtained by numerical diagonalization of the full spin Hamiltonian matrix (Four nitrogen nuclei). (Ref. 68) c) First order frequencies, (Eq. (3.10))... 

The first-order frequency shift vanishes for m = 3, which means that the central transition for noninteger spin nuclei is not affected to first order by quadrupolar interactions (see Fig. 2). It is clearly advantageous to work with these, as the — 1 <- 0 and 0 <-> 1 transitions for integer spin nuclei are always shifted. 

Figure 3.39. First-order (frequency dependent) phase errors arise from a dephasing of magnetisation vectors during the pre-acquisition delay which follows the excitation pulse. When data collection begins, vectors with different frequencies have developed a significant phase difference which varies across the spectrum.

TABLE 12. Taft a constants, activation energies and first-order frequency factors (A) for the depolymerization of substituted polysiloxanes. Major depolymerization products formed in the thermal degradation of substituted polysiloxanes are D 3 and D 4 (from Reference 156 reproduced by permission of John Wiley Sons, Inc)... 

No-pair energies (eV) of the 2s — 2p3/2 transition in Li-like uranium. Bq and are first-order frequency-independent and frequency-dependent Breit energies, respectively. Bo X Bn and x B are corresponding higher-order Breit energies, respectively. 

AH) = heat of reaction, Btu/mole of A converted k = zero order rate constant, moles A/ft -sec k = first order frequency factor, moles A/atm-ft -sec A[ = initial moles of A per unit mass of feed o = total moles of feed per unit mass of feed n, = total moles of reacting system per unit mass of feed xa = moles of A converted per unit mass of feed P = pressure, atm... 

We observed the phase-dependent quantum interference in the double A system realized with cold Rb atoms coupled by four laser fields. The coherently coupled four-level double A-type system realized with the laser coupling scheme for the Rb Di transitions is shown in Fig. 8 and the simplified experimental set up is depicted in Fig. 4(b). An extended-cavity diode laser with a beam diameter 3 mm and output power 50 mW is used as the coupling laser. The driving electric current to the diode laser is modulated at 5=181 MHz with a modulation index -0.5, which produces two first-order frequency sidebands separated by 362 MHz. The two sidebands are tuned to the Rb Di F=3—>F =2 and F=3 F =3 transitions respectively and serve as the two coupling fields due to a tt phase difference between the two sidebands). Another... 

Much of the important theoretical development involving compensation cuts for quartz crystals has been done by EerNisse (1975,1976). EerNisse has been able to show that the frequency-stress coefficient is determined by the O cut angle, and the first-order frequency-temperature coefficient is determined almost entirely by the 6>-cut angle. Several standard cut types, such as the thermal-transient-compensated (TTC-) cut (Klusterset al, 1977),TS-cut (Holland, 1964), and SC-cut all refer to double rotated cuts with 9 34° and O 22°. A similar cut is the IT-cut by Bottom and Ives (1951). The IT-cut has 9 = 34°17 and O = 19° 06. The frequency-temperature coefficient is zero for the IT-cut. 

The first-order frequency shift is zero for m = i, so that the central transition for non-integer spins (such as Al with I = j) is not affected by quadrupolar interactions to first order. It is thus advantageous to work with such nuclei, especially since the central transition is normally the only one which is observed other transitions are so broadened and shifted as to be unobservable. 

Friedman s isoconversional method [14] involves an Arrhenius analysis at constant levels of conversion, and we determined the apparent first-order frequency factor and activation energy at 1 % intervals using both LLNL and AKTS kinetics analysis programs. 

Figure Bl.3.8. A WMEL diagram for die three-colour fifth order qiiasi-Ramaii echo . As usual, the first pair of field actions creates the Raman coherence which is allowed both to dephase and walk off with time. This is followed by a second pair of field actions, which creates a different but oppositely phased Raman coherence (now hf) to the first. Its frequency is at oi - = e y Provided that frequencies are identified...

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

A simple method for predicting electronic state crossing transitions is Fermi s golden rule. It is based on the electromagnetic interaction between states and is derived from perturbation theory. Fermi s golden rule states that the reaction rate can be computed from the first-order transition matrix and the density of states at the transition frequency p as follows ... 

HyperChem can calculate transition structures with either semi-empirical quantum mechanics methods or the ab initio quantum mechanics method. A transition state search finds the maximum energy along a reaction coordinate on a potential energy surface. It locates the first-order saddle point that is, the structure with only one imaginary frequency, having one negative eigenvalue. 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

First-order spectra (mulliplels) are observed when the eoupling constant is small compared with the frequency difference of chemical shifts between the coupling nuclei This is referred to as an A n spin system, where nucleus A has the smaller and nucleus X has the considerably larger chemical shift. An AX system (Fig. 1.4) consists of an T doublet and an X doublet with the common coupling constant J x The chemical shifts are measured from the centres of eaeh doublet to the reference resonance. 

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

Frequency response characteristics of first-order systems... 

Fig. 6.4 Graphical display of frequency domain data for a first-order system.

As can be seen from equation (6.34), each time the frequency doubles (an increase of one octave) the modulus halves, or falls by 6dB. Or alternatively, each time the frequency increases by a factor of 10 (decade), the modulus falls by 10, or 20 dB. Hence the HF asymptote for a first-order system has a slope which can be expressed as —6 dB per octave, or —20 dB per decade. 

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