The Larmor Frequency

Here Ti is the spin-lattice relaxation time due to the paramagnetic ion d is the ion-nucleus distance Z) is a constant related to the magnetic moments, i is the Larmor frequency of the observed nucleus and sis the Larmor frequency of the paramagnetic elechon and s its spin relaxation time. Paramagnetic relaxation techniques have been employed in investigations of the hydrocarbon chain... 

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

Chemical shift relates the Larmor frequency of a nuelear spin to its ehemieal environment The Larmor frequency is the preeession frequency Vg of a nuclear spin in a static magnetic field (Fig. 1.1). This frequency is proportional to the flux density Bg of the magnetic field vglBg = const.)... 

It is convenient to reference the chemical shift to a standard such as tetramethylsilane [TMS, (C//j)4Si] rather than to the proton fC. Thus, a frequency difference (Hz) is measured for a proton or a carbon-13 nucleus of a sample from the H or C resonance of TMS. This value is divided by the absolute value of the Larmor frequency of the standard (e.g. 400 MHz for the protons and 100 MHz for the carbon-13 nuclei of TMS when using a 400 MHz spectrometer), which itself is proportional to the strength Bg of the magnetic field. The chemical shift is therefore given in parts per million (ppm, 5 scale, Sh for protons, 5c for carbon-13 nuclei), because a frequency difference in Hz is divided by a frequency in MHz, these values being in a proportion of 1 1O. ... 

Figure 1.1. Nuclear precession nuclear charge and nuclear spin give rise to a magnetic moment of nuclei such as protons and carbon-13. The vector n of the magnetic moment precesses in a static magnetic field with the Larmor frequency vo about the direction of the magnetic flux density vector Bo...

We can use the angular frequency of TMS as the reference frequency of the rotating frame. Deducting this from the Larmor frequency of the signal will leave only the differential frequency (or, in other words, the chemical shift) associated with the magnetization vector of the signal. 

The amplitude represents a circular motion of the magnetization vector along the Larmor frequency of the nucleus is inherent to it and would therefore remain unchanged throughout the FID. 

Hence it is clear that if the two delay periods before and after the 180° pulses are kept identical, then refocusing will occur only when a selective 180° pulse is applied. This can happen only in a heteronuclear spin system, since a 180° pulse applied at the Larmor frequency of protons, for instance, will not cause a spin flip of the C magnetization vectors. 

In order for relaxation to occur through Wj, the magnetic field fluctuations need to correspond to the Larmor precession frequency of the nuclei, while relaxation via requires field fluctuations at double the Larmor frequency. To produce such field fluctuations, the tumbling rate should be the reciprocal of the molecular correlation time, i.e., f), so most efficient relaxation occurs only when voT, approaches 1. In very small, rapidly tumbling molecules, such as methanol, the concentration of the fluctuating magnetic fields spectral density) at the Larmor frequency is very low, so the relaxation processes Wj and do not occur efficiently and the nuclei of such molecules can accordingly relax very slowly. Such molecules have... 

Five different types of peaks can occur in 3D spectra. These are illustrated in an ABC spin system, in which the Larmor frequencies of the three nuclei are v,, Vg, and v, -, and the coherence transfers are associated with two different mixing processes, M, and Mj ... 

The basic principle underlying the development of images is simple (Lauterber, 1973). Consider a body cavity containing two pools of water in different quantities. In a uniform magnetic field, the NMR spectrum will consist of a single peak, since all the water molecules will process at the same frequency, irrespective of their spatial location. If, however, a linear field gradient is applied in the x -direction, the Larmor frequency of the water will increase linearly across the sample as a function of the x -coordinate, thereby creating a one-dimensional profile, or spectrum, of the sample (Fig. 7.21). 

Precession A characteristic rotation of the nuclear magnetic moments about the axis of the applied magnetic field Bo at the Larmor frequencies. Preparation period The first segment of the pulse sequence, consisting of an equilibration delay. It is followed by one or more pulses applied at the beginning of the subsequent evolution period. 

The quantity of interest is the precession of the components perpendicular to B0 that are measured in the experiment by induced voltage in the coil, which is subsequently amplified and demodulated. We can write them either as individual components Mx, M, or by a vector M+, which combines both of them. In the static field, the precession about B0 occurs with the Larmor frequency w0 = /B0. If we neglect those processes which dampen the amplitude of the rotating transverse magnetization as precession proceeds, this already describes the frequency that we pick up with our receiver coil, and it is the third and perhaps the most important of our three fundamental equations of NMR ... 

One further point needs to be mentioned when probing the feasibility of a particular experiment. Apart from its dependence on temperature and concentration (for instance of ions, solutes, impurities, isotopes), relaxation times - in particular the longitudinal relaxation time Tj - depend on the field strength. This can be understood from the concept that energy exchange is most efficient if the timescale of molecular motion is equal to the Larmor frequency. Often, molecular motion takes place over a wide range of frequencies, so that the func-... 

The Larmor frequency rf reference signal is generated using a DDS (direct digital synthesizer) board (FSW01, DST Inc., Asaka, Japan). This board can generate coherent and spectrally pure rf signals from 5 to 200 MHz, of which the frequency is controlled via the ISA bus. This board has an essential role in the NMR lock process. 

Our approach is to use the two-dimensional relaxation and diffusion correlation experiments to further enhance the resolution of different components. It is important to note that the correlation experiment, e.g., the Ti-T2 experiment, is different from two experiments of and T2 separately. For instance, the separate Ti and T2 experiment, in general, cannot determine the T1(/T2 ratio for each component. On the contrary, a component with a particular Tj and T2 will appear as a peak in the 2D 7i-T2 and the Ti/T2 ratio can be obtained directly. For example, small molecules often exhibit rapid rotation and diffusion in a solution and Ti/T2 ratio tends to be close to 1. On the other hand, the rotational dynamics of larger molecules such as proteins can be significantly slow compared with the Larmor frequency and resulting in a Ti/T2 ratio significantly larger than 1. 

In one example, the Tics of the non-crystalline methyl, methine and methylene carbons of iPP, 70% crystalline, were compared at room temperature with those of model atactic poly(propylene), hydrogenated poly(2-methyl-l,3-pentadiene) [163]. It was found that, within the experimental error, the Tic values of each of the carbons were the same in both polymers. The conclusion can then be reached that the fast segmental motion, at or near the Larmor frequency of... 

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