Fourier transformation pulse shapes

Fig. 7. (a) Spin-echo (90-T-180 pulse sequence) for "B NMR at 61 MHz in a sample that contains 10% B. (b) Fourier transformed line shape. [Reprinted with permission from Solid State Communications, Vol. 43, S. G. Greenbaum, W. E. Carlos, and P. C. Taylor, The coordination of boron in a-Si (B,H). Copyright 1982 Peigamon Press, Ltd.]... 

Relaxation times are peculiarly important in FT NMR work because they determine the rate of pulsing and accumulation thus fast pulsing is possible for quadrupolar nuclei, and can make up for low receptivity. Conversely, the FT method is peculiarly informative on relaxation rates, because of the signal intensities and frequencies in the FID which are recovered in the Fourier transformed line shape. 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

Fig. 1.10 Soft rf pulses (left) in the shape of a sine (sin x/x) function, and their Fourier transforms (right), being equivalent to the excited slice in the presence of a constant magnetic field gradient. The well defined sine function (top) produces an excitation that is a slice...

Fig. 7. The excitation profiles (n= — 2 to 2) by a periodic pulse of / Jt sin(jr//7) /Jt, where the solid lines are computer simulated results, solid circles are calculated with the effective RF fields, and open circles are obtained from the Fourier transformation of the RF shape. In the computer simulation, the pulse is composed of 4001 steps and has a pulsewidth r = 5 ms. Reprinted from Ref. 27 with permission from Elsevier.

Similarly to non-selective experiments, the first operation needed to perform experiments involving selective pulses is the transformation of longitudinal order (Zeeman polarization 1 ) into transverse magnetization or ly). This can be achieved by a selective excitation pulse. The first successful shaped pulse described in the literature is the Gaussian 90° pulse [1]. This analytical function has been chosen because its Fourier transform is also a Gaussian. In a first order approximation, the Fourier transform of a time-domain envelope can be considered to describe the frequency response of the shaped pulse. This amounts to say that the response of the spin system to a radio-frequency (rf) pulse is linear. An exact description of the... 

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

To introduce the application of ultrashort laser sources in microscopy, we want to review some properties of femtosecond pulses first for a comprehensive introduction the reader may refer to one of the established textbooks on femtosecond lasers (Diels and Rudolph 2006). The most important notion is the Fourier transform relation between the temporal shape of a pulse and the spectrum necessary to create it. This leads to the well-known time-bandwidth product for the pulse temporal width (measured as full width at half maximum, FWHM) At and the pulse spectral width Av. 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

Fourier Transform-limited 100 fs, 800 nm, 1015 W cm 2 laser pulse and (b) the optimum result obtained by means of an 80-parameter unrestricted optimisation (dashed line) and a restricted 3-parameter optimisation (full line). The inset in (b) shows the evolution of the fitness value for the 80 parameter optimisation (full squares maximum fitness, open squares average fitness), (c) Autocorrelation trace of the optimal pulse corresponding to the 80 parameters optimisation. The pulse shapes consists of two pulses of 120 fs of equal amplitude separated by 500 fs. 

The results of two different optimisations of the production of charged states >11+ are presented in Fig. 2b. The dashed curve is the TOF distribution obtained when optimising 80 independent phases across the spectrum. By contrast with the Fourier Transform-limited pulse, ions up to 25+ are present in the TOF distribution The corresponding pulse shape (as determined from the autocorrelation in Fig. 2c) is a sequence of two pulses of equal amplitude and separated by 500 fs. To test the importance of the time delay between the two pulses, we performed restricted optimisations where a periodic phase was applied across the spectrum along with a quadratic term. In this case the period and amplitude of the oscillatory part... 

Transverse relaxation is caused by the distribution and fluctuation of the resonance frequency of the A spins. The distribution-induced relaxation is called free induction decay. The free induction decay curve is the Fourier transform of the spectral shape of the A spins. This spectral shape depends on the intensity and the pulse width of the incident microwave, when the total width of ESR spectrum is large as is the case for radical species in solids. Therefore, the analysis of the free induction decay curve gives no information on the nature of radical species in solids unless the pulse width is narrow enough to cover the entire ESR spectrum. 

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