Larmor relation

The Larmor relation, Eq. 2.45, is the basis for determining the nuclear precession frequency. In many instances we are concerned only with the magnitude of the... 

In this and subsequent expressions we drop Planck s constant h in order to convert to the more useful unit of hertz, rather than joules. To further simplify the notation, we substitute the Larmor relation, Eq. 2.14, into Eq. 6.3 to get... 

The normal method of scanning NMR spectra (or, for that matter, other spectra) consists of altering the wavelength of the electromagnetic energy supplied and observing absorption whenever the Larmor relation for a particular nucleus is met. This is known as continuous-wave (CW) spectroscopy. 

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... 

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.)... 

The relation between matter and ether was rendered clearer by Lord Kelvin s vortex-atom theory, which assumed that material atoms are vortex rings in the ether. The properties of electrical and magnetic systems have been included by regarding the atom as a structure of electrons, and an electron as a nucleus of permanent strain in the ether— a place at which the continuity of the medium has been broken and cemented together again without fitting the parts, so that there is a residual strain all round the place (Larmor). 

The quantum mechanics treatment of diamagnetism has not been published. It seems probable, however, that Larmor s theorem will be retained essentially, in view of the marked similarity between the results of the quantum mechanics and those of the classical theory in related problems, such as the polarisation due to permanent electric dipoles and the paramagnetic susceptibility. f Thus we are led to use equation (30), introducing for rK2 the quantum mechanics value... 

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

Let us now derive the equations that relate the spatial information to the signal behavior. As we have seen previously, a spin at position r possesses a Larmor frequency co(r) = y B(r) = v( Bo + g r). It is convenient to subtract the reference value, given by the average field, oi0 = v B0. so that we obtain the frequency difference relative to an (arbitrarily chosen) position r= 0 ... 

Larmor, Joseph. 1900. Aether and Matter A Development of the Dynamical Relations of the Aether to Material Systems on the Basis of the Atomic Constitution of Matter. Cambridge Cambridge University Press. 

Let us suppose that there exists a linear combination of the Larmor frequencies such that a1oj1 -j- a2a>2 Aco where the a< are integers close to one and where Aco is the line width. In this case the F(,) terms of the perturbation induce an exchange of quanta between the two Zeeman subsystems, the energy balance being taken up by the dipole-dipole subsystem. One of the quasi-invariants is thus destroyed but the combination Mjax — M2ja.2 remains constant. As in chemical thermodynamics,19 it is useful here to introduce a reaction coordinate f to characterize the state of the system we then have the relations ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

The torque is simply the rate of change of angular momentum and since magnetic moment is related to angular momentum by Equation (1), one may solve Equation (6) to find the motion of the spin vector. The spin precesses about H and the angular frequency of this precession, known as the Larmor precession, is yH. This situation is illustrated in Fig. 2. 

The issues with respect to obtaining chemical information within an imaging experiment are considered next. The description of image acquisition given in Section II.A.l was based on the assumption that the Larmor frequency of a nuclear spin is directly related to its location in the sample, as determined by the applied magnetic field gradient. As discussed by Callaghan (13), this is precisely true only... 

This general and important relationship, irrespective of the value of /, is called Larmor s equation. It relates the intensity of the magnetic field in which the nuclei are located to the electromagnetic radiation frequency that induces resonance hence, a signal in the spectrum (see Table 9.1 and Fig. 9.1). 

Using the effective magnetic field Bc[t for a nucleus with I — 1/2, Larmor s relation becomes ... 

According to the Larmor equation (1.8), chemical shifts can be related to field differences zlBs, measurable in millitesla. 

In order to rotate the magnetization vectors of all nuclear spins within the range of Larmor frequencies to be observed, the pulse must not only be adjusted for 90", so that yB1 tp = 71/2 (eq. 2.2)), but must also be very strong, so that y B, 2 7i A (eq. (2.3)). These requirements give the relation between pulse width and spectral width ... 

NMR spectrum is obtained by Fourier transformation as the distribution of frequency components that are present in an FID or an echo signal sampled over a period in time. Therefore, a rapidly decaying signal (short T2) has a broad function over frequency because of a wide spread of Larmor frequencies and vice-versa. The linewidth in Hz at half height, Avj/2, is related to T2 relaxation time by the equation,... 

The nuclear Overhauser effect (NOE) is a consequence of the modulation of the dipole-dipole interactions (through space) between different nuclei and is correlated with the inverse sixth power of the internuclear distance. Experimentally, the NOE is the fractional change in intensity of one resonance when another resonance is irradiated in a double-irradiation experiment. The NOE phenomenon is intimately related to spin relaxation. The NOE varies as a function of the product of the Larmor frequency, co0, and the rotational correlation time, tc. In small molecules, tc is short relative to uo"1. In this extreme motional narrowing situation, the frequency... 

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