Linewidths transition

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... 

Physical background. MAS will narrow the inliomogeneously broadened satellite transitions to give a series of sharp sidebands whose intensity envelopes closely follow the static powder pattern so that the quadnipole interaction can be deduced. The work of Samoson [25] gave real impetus to satellite transition spectroscopy by showing that both the second-order quadnipolar linewidths and isotropic shifts are fiinctions of / and Some combinations of / and produce smaller second-order quadnipolar effects on the satellite lines than... 

Once the basic work has been done, the observed spectrum can be calculated in several different ways. If the problem is solved in tlie time domain, then the solution provides a list of transitions. Each transition is defined by four quantities the mtegrated intensity, the frequency at which it appears, the linewidth (or decay rate in the time domain) and the phase. From this list of parameters, either a spectrum or a time-domain FID can be calculated easily. The spectrum has the advantage that it can be directly compared to the experimental result. An FID can be subjected to some sort of apodization before Fourier transfomiation to the spectrum this allows additional line broadening to be added to the spectrum independent of the sumilation. 

Figure C3.5.9. Contributions to the homogeneous linewidth of a C=0 stretching transition of W(CO)g (Q 2000...

In other work, the impact of thermal processing on linewidth variation was examined and interpreted in terms of how the resist s varying viscoelastic properties influence acid diffusion (105). The authors observed two distinct behaviors, above and below the resist film s glass transition. For example, a plot of the rate of deprotection as a function of post-exposure processing temperature show a change in slope very close to the T of the resist. Process latitude was improved and linewidth variation was naininiized when the temperature of post-exposure processing was below the film s T. 

Finally, instmmental broadening results from resolution limitations of the equipment. Resolution is often expressed as resolving power, v/Av, where Av is the probe linewidth or instmmental bandpass at frequency V. Unless Av is significantly smaller than the spectral width of the transition, the observed line is broadened, and its shape is the convolution of the instrumental line shape (apparatus function) and the tme transition profile. 

SWCNTs have been produced by carbon arc discharge and laser ablation of graphite rods. In each case, a small amount of transition metals is added to the carbon target as a catalyst. Therefore the ferromagnetic catalysts resided in the sample. The residual catalyst particles are responsible for a very broad ESR line near g=2 with a linewidth about 400 G, which obscures the expected conduction electron response from SWCNTs. 

The effect of a pressure of 80 and 150 MPa on the spin-state transition has been also studied [169], a series of spectra obtained at 150 MPa being shown in Fig. 32. The speetra show relaxation effects as line broadening and linewidth asymmetry. Calculated spectra were obtained in the same way as at ambient pressure. Rate constants for a number of temperatures are listed in Table 12, the parameter values resulting from an Arrhenius plot of the rate constants being listed in Table 13. In Fig. 33, the quantity 5g of Eq. (36) has been plotted as a... 

The second two terms in (5) are called T-terms or two-photon terms , which have (Qe g — 2hco) in the denominator corresponding to the excitation of an electron into the higher-lying excited state e. If we consider a resonance condition where hco = hcoeig/2 and assume that the transition linewidth is narrow, rt>s, < < h coeg imaginary part of the T-term can then be expressed in SI units as ... 

Reducing the linewidth of the lowest energy one-photon transition. Minimizing T increases d2PA(ft)), which allows for photons to closely approach the 1PA edge without one-photon losses... 

Transitions have a natural linewidth associated with their lifetime (via the uncertainty principle) but this is usually small. 

The natural linewidth comes from the lifetime, r, of the upper state of a spontaneous transition, which is related to the Einstein A coefficient so that r = A l faster transitions have shorter lifetimes and vice versa, and similarly an allowed transition will have a short lifetime for the upper state whereas forbidden transitions will have a long lifetime. The lifetime consideration is very important in the laboratory where transitions have to occur on the timescale of the experiment, otherwise they are not observed. Hence in the laboratory allowed transitions are observed and in general (but not specifically) forbidden transitions are not seen. For astronomy this does not matter. So what if a forbidden transition has a lifetime of 30 million years - the Universe is 15 billion years old - if you wait long enough it will happen. The rules of spectroscopy need to be understood but in space anything goes ... 

The natural linewidth is the smallest contribution to the line profile of a transition and is only rarely seen as limiting within the laboratory. For an electronic transition with a lifetime of 10000 ps the linewidth is of order 0.00053 cm-1 but for a rotational transition the lifetime linewidth 5.3 x 10-15 cm-1. The best microwave spectra recorded in the laboratory have a linewidth of a few Hz or 10-12 cm-1, which is close (but not very) to the natural linewidth limit. 

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