Lorentzian resonances

The normalization is that giving an area of 2, since for a large the lineshape consists of two separated Lorentzian resonances, as shown in Fig. 3. 

Figure 3.9. The Fourier transform of spectra of Fig. 3.8 Projection PK(a>) of the pure excitonic state on the eigenstates of the coupled system of an exciton K and the effective photon continuum in a 2D lattice, for various values of the wave vector K (KXd). The vertical peak represents a discrete state, whose weight is represented by a rectangle (a-d). For an exciton with K < K0, the continuum band (matter-contaminated photons) dominates the spectrum, with a quasi-lorentzian resonance (a,/). For K > K0, the discrete state dominates (d,e). In the intermediate region (b, c) the spectrum reflects the complicated behavior of its Fourier transform cf. Fig. 3.8. 

A second type of eigenstates, illustrated in Fig. 3.10, is the band of states formed by the exciton-contaminated photon continuum. Far from the critical area cK co0, this band presents a lorentzian resonance (cf. Fig. 3.9), whose temporal instability (cf. Fig. 3.8) is described by an exponential decay. Thus, the exact solution leads back to that of second-order perturbation theory, obtained in Section III.A.2.b above. 

Figure 13 Calculated absorption-type spectrum for DCO. Energy normalization is such that E = 0 corresponds to D-I-CO with CO at equilibrium. The dotted line marks the quantum mechanical threshold and the numbers indicate the pure CO stretching states (0, U2,0). There are 29 bound states. Because of the logarithmic scale, the Lorentzian resonance profiles have an unusual shape. Reproduced, with permission of the American Institute of Physics, from Ref. 15.

For a sharp, Lorentzian resonance peak (as shown schematically in Fig. 2) the composite oscillator s internal friction, Qq, is... 

For frequencies between the resonator resonances, the losses are high and the threshold will not be reached. In the case of a Lorentzian resonance profile, for instance, the loss factor has increased to about ten times (yo) at frequencies that are 3Ayr away from the resonance center yq. 

The transition probability shows a series of Lorentzian resonance peaks. Those appear when the energy of the incoming electron in the x direction, E — is close to the real part of the resonance energy, e. Due to the symmetry of the problem, tunneling is allowed only through even resonance states. 

As an example of this method, we present a simulation of this technique using experimentally realistic values (/). We assume that the laser source is an ultrabroadband titanium-sapphire laser producing transform-limited pulses of uniform power spectral density between 700-1000 nm. The bandwidth from 800-1000 nm is reserved for stimulating CARS, while the bandwidth from 700-800 nm is used as the reference pulse. In this simulation, we desire to simulate the measurement of the relative amounts of DNA (deoxyribonucleic acid) which has a resonance at 1094 cm" and RNA (ribonucleic acid) which has a resonance at 1101 cm A hypothetical Raman spectrum was created which has Lorentzian resonances at both frequencies. A pulse is designed such that... 

In isotropic liquid systems, this component obeys an exponential relaxation, corresponding when Fourier transformed to a unique Lorentzian resonance line (line width a few H ). When molecular motions are anisotropic (in the range 10 to 10 Hz) the resulting average interaction leads to a structure in the resonance spectrum, reflecting the distribution of anisotropies in the system . This non-zero average interaction may be checked clearly by the presence of a pseudo-solid echo after a convenient pulse sequence . The following points may be emphasized respectively in the absence presence of an external constraint imposed on the network. 

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

Raman gain coefficient, whose maximum occurs at exact resonance, - oig = For a Lorentzian lineshape, the maximum gain coefficient is given by... 

The bracketed term in Eq. (4-60b) describes a Lorentzian line shape for the NMR absorption band. The maximum in the band occurs at the resonance frequency, wq. Expressed in units of X0W0T2/2, the maximum value of x" s 1 at one-half this maximum peak height we find, by substitution, that (wq — w) = IIT. Using w = 2 ttv to convert to frequency (in Hz) gives (vq — v) = 3-7 T 2. However, the peak width is twice this, or... 

In a Mdssbauer transmission experiment, the absorber containing the stable Mdssbauer isotope is placed between the source and the detector (cf. Fig. 2.6). For the absorber, we assume the same mean energy q between nuclear excited and ground states as for the source, but with an additional intrinsic shift A due to chemical influence. The absorption Une, or resonant absorption cross-section cr( ), has the same Lorentzian shape as the emission line and if we assume also the same half width , cr( ) can be expressed as ([1] in Chap. 1)... 

In the case of resonance absorption of synchrotron radiation by an Fe nucleus in a polycrystalline sample, the frequency dependence of the electric field of the forward scattered radiation, R(oj), takes a Lorentzian lineshape. In order to gain information about the time dependence of the transmitted radiation, the expression for R(oj) has to be Fourier-transformed into R(t) [6]. 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

Thus a single Lorentzian line is obtained that is centered at a weighted average resonant frequency and has a width proportional to a weighted average T2x plus a term proportional to the average lifetime and the square of the separation of the slow exchange resonances. 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

We have seen in Chapter 2 that in EPR spectroscopy one usually varies the magnetic held instead of the frequency, because the use of a mechanically rigid micro-wave resonator dictates the frequency to be constant. For this reason, the Lorentzian distribution in Equation 4.5 is frequently rewritten as a distribution in resonance fields as... 

---