Imaginary spectrum

The real and imaginary spectra obtained by Fourier transformation of FID signals are usually mixtures of the absorption and dispersion modes as shown in Fig. 2.13 (a). These phase errors mainly arise from frequency-independent maladjustments of the phase sensitive detector and from frequency-dependent factors such as the finite length of rf pulses, delays in the start of data acquisition, and phase shifts induced by filtering frequencies outside the spectral width A. 

The angle 0 can be thought of as a rotation of the two mutually perpendicular vectors representing the real and imaginary spectra. The problem of phase correction boils down to finding the correct phase rotation angle 0. Well, actually it is a little more complicated because the phase correction 0 is usually a linear function of the chemical shift (8). Defining the line... 

Following the Fourier transform, two data sets are generated representing the real and imaginary spectra (Section 3.3) so the real part with which one usually deals contains half the data points of the original FID (in the absence of further manipulation), and its data size, SI, is therefore TD/2. Digital resolution is then ... 

Adjust zero time until the sum of squares of the imaginary spectrum is zero (i.e., the FID is then centred about zero). 

The actual Fourier transform is a digital calculation, so not all frequencies are tested. In fact, the number of frequencies tested is exactly equal to the number of time values sampled in the FID. If we start with 16,384 complex data points in our FID (16,384 real data points and 16,384 imaginary data points), we will end up with 16,384 data points in the real spectrum (the imaginary spectrum is discarded). Another difference from the above description is that the actual Fourier transform algorithm used by computers is much more efficient than the tedious process of multiplying test functions, one by one, and calculating the area under the curve of the product function. This fast Fourier transform (FFT) algorithm makes the whole process vastly more efficient and in fact makes Fourier transform NMR possible. 

Recall that the raw NMR data (FID) consists of two numbers for each data point one real value and one imaginary value. After the Fourier transform, there are also two numbers for each frequency point one real and one imaginary. In a perfect world, the real spectrum would be in pure absorptive mode (normal peak shape) and the imaginary spectrum would be in pure dispersive (up/down) mode. In reality, each spectrum is a mixture of absorptive and dispersive modes, and the proportions of each can vary with chemical shift (usually in a linear... 

Absorptive spectrum = (real spectrum) x cos(0) + (imaginary spectrum) x sin(0)... 

In die Fourier transform of a real time series, die peakshapes in the real and imaginary halves of die spectrum differ. Ideally, the real spectrum corresponds to an absorption lineshape, and die imaginary spectrum to a dispersion lineshape, as illustrated in Figure 3.20. The absorption lineshape is equivalent to a pure peakshape such as a Lorentzian or Gaussian, whereas die dispersion lineshape is a little like a derivative. 

Typically, both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye , and the first-order correction is then used to adjust the phase of a signal far away from the first in a similar manner. Ideally, the two chosen resonances should be as far apart in the spectrum as possible to maximise the frequency-dependent effect. Experimentally, this process of phase correction involves mixing of the real and imaginary parts of the spectra produced by the FT process such that the final displayed real spectrum is in pure absorption mode whereas the usually unseen imaginary spectrum is pure dispersion. 

Because the harmonic coefficients are complex, we can plot the spectrum in either Cartesian or polar form. The Cartesian form has a real spectrum and imaginary spectrum as shown... 

Rgure 10.8 Spectra for a square wave with delay (a) real spectrum, (b) imaginary spectrum,... 

It is possible to understand the fine structure in the reflectivity spectrum by examining the contributions to the imaginary part of the dielectric fiinction. If one considers transitions from two bands (v c), equation A1.3.87 can be written as... 

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

The observable NMR signal is the imaginary part of the sum of the two steady-state magnetizations, and Mg. The steady state implies that the time derivatives are zero and a little fiirther calculation (and neglect of T2 tenns) gives the NMR spectrum of an exchanging system as equation (B2.4.6)). 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

For each EA spectrum, the transmission T was measured with the mechanical chopper in place and the electric field off. The differential transmission AT was subsequently measured without the chopper, with the electric field on, and with the lock-in amplifier set to detect signals at twice the electric-field modulation frequency. The 2/ dependency of the EA signal is due to the quadratic nature of EA in materials with definite parity. AT was then normalized to AT/T, which was free of the spectral response function. To a good approximation [18], the EA signal is related to the imaginary part of the optical third-order susceptibility ... 

The half-width (at half-height) and the shift of any vibrational-rotational line in the resolved spectrum is determined by the real and imaginary parts of the related diagonal element TFor linear molecules the blocks of the impact operator at k = 0,2 correspond to Raman scattering and that at k = 1 to IR absorption. The off-diagonal elements in each block T K, perform interference between correspond-... 

---