Precession, nuclear

When exposed to a static magnetic field B0, a spinning nucleus behaves like a gyroscope in a gravitational field. As illustrated by Fig. 1.2, the spin axis - which coincides with the magnetic moment vector p - precesses about B0. The frequency of precession, v0, is known as the Larmor frequency of the observed nucleus. 

In a stationary external magnetic field, Hq, a nucleus of spin / has 21 +1 quantitized energy levels. This means that there is only one possible energy transition for a nucleus / = 1/2, a vastly simpler situation compared to energy transition of electrons 

The relationship between atomic number, atomic mass and nuclear spin number 

Mass number Atomic number Spin number, / 

Since the nucleus is spinning, the nucleus also possesses angular momentum, L, whose vector is co-linear with and linearly proportional to fi (the spinning motion being common to both nuclear charge and mass), i.e. 

These equations indicate that at any instant, changes in fi are perpendicular to both ]1 

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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...

Chemical shift The difference between the nuclear precession frequency and the carrier frequency. 

Larmor frequency The nuclear precession frequency about the direction of Bo. Its magnitude is given by yBo/27T. 

Modulation The variation in amplitude and/or phase of an oscillatory signal by another function, e.g., modulation of the nuclear precession frequency of one nucleus by the nuclear precession frequency of a correlated nucleus in COSY spectra. 

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... 

The SIN defined by Equation 7.6 for a given NMR resonance is proportional to the square of the nuclear precession frequency (mo, rad/s), the magnitude of the transverse magnetic field (Bi) induced in the RE coil per unit current (/), the number of spins per unit volume (Ns), the sample volume (Vs), and a scaling constant that accounts for magnetic field inhomogeneities. The SIN is inversely proportional to the noise generated in the RE receiver and by the sample (Vnoise) as defined by the Nyquist theorem,... 

In pulsed NMR, the magnetic field is turned on for the time necessary to rotate the magnetization vector into a plane called the 90° rotation or 90° pulse. The field is turned off and the magnetization vector rotates at a nuclear precession frequency relative to the coil. This induces an NMR signal that decays with time as the system returns to equilibrium. This signal is called the free induction decay (FID). 

Fig. 2.1. Preferred nuclear precession about the direction of the magnetic field (a) and the effect of a 90°. radio frequency pulse (b).

In Mossbauer spectroscopy, we encounter two types of expectation values for the electronic spin4 6 that we illustrate briefly for an iron site with S = 1/2 and g 2, taking the applied field along z. If the spin relaxation rate (spin flips between the Ms= + 1/2 and Ms= —1/2 sublevels) is slow compared to the nuclear precession frequency (which is typically 10—30 MHz Larmor precession around Bint or quadrupole precession), the nucleus senses the Fe atom in either the Ms= + 1/2 or Ms =1/2 state during the absorption process. In this case, we have (Sz) = + 1/2 for spin up and (Sz) = —1/2 for spin down. Each electronic level produces a Mossbauer spectrum, and these two spectra are weighted by the probability (given by the... 

In this expression r is the inter-chain hopping time and ts is the phonon scattering time along a chain. The quantity s = (d2/a2) is the ratio of the anisotropic to isotropic contribution of the hyperfine interaction and /(cd) is the spectral density of the interaction, with coe and con being electron and nuclear precession frequencies respectively,... 

The exact value of v0 is designed to be slightly offset from the range of nuclear precession frequencies to be examined (the spectral width, SW, in hertz). Therefore, the SW can be no greater (and preferably less) than Av, leading to the relationship... 

The simplest theoretical approach to exchange is via the Bloch equations, to which terms are added to reflect the rate phenomena. The spectra shown in Fig. 2.14 are obtained from such a treatment. It is apparent that the line shapes depend on the ratio R/ vA — vB), where the exchange rate R = 1/t. Thus fast and slow are measured with respect to differences in the nuclear precession frequencies in the two sites. Exchange rates can be measured by analysis of line shapes and by certain pulse experiments, as described in later chapters. 

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... 

Plots of J o>) versus oj for different values of Tc are shown in Fig. 8.1. The values of rc should be compared with the reciprocal of the nuclear Larmor frequency w0. For either a very short or very long rc the value of J 0 is relatively small. J(to) reaches its maximum when rc = l/o)0, that is, when the average molecular tumbling frequency is equal to the nuclear precession... 

Figure 2. Nuclear precession about the magnetic field axis. The nucleus is in the ground state.

It was shown in the previous section that the emitted rf signal from excited nuclear spins (the FID) is detected as a time dependent oscillating voltage which steadily decays as a result of spin relaxation. In this form the data is of little use to us because it is a time domain representation of the nuclear precession frequencies within the sample. What we actually want is a display of the frequency components that make up the FID as it is these we relate to transition energies and ultimately chemical environments. In other words, we need to transfer our time domain data into the frequency domain. 

In an adiabatic fast passage experiment, the sweep rate is chosen so that no appreciable relaxation takes place during the sweep (the "fast condition), and the nuclear precession around He is always rapid compared with the rotation of He (the "adiabatic" condition). The first condition means that M remains constant in length during the sweep while the second condition ensures that whatever initial relationship the magtiz-ation had with respect to Hg remains throughout the sweep. In particular, if the nuclei were in thermal equilibrium with... 

Low-spin iron (III) ions have an electron hole in the izg orbitals. Therefore, these centers have 5 = 1/2 and spin-orbit interaction contributes considerably to the magnetic hyperfine field. Low-spin ironflll) compounds in solution always show a rather complicated magnetic Mossbauer pattern at temperatures around 4.2 K and low external fields, which means that the relaxation rate of these centers is lower than the nuclear precession rate of 10 s. Sometimes a magnetic splitting is observed even at 77 K. Therefore, in order to pin down 5 and AEq, it is advisory to measure between 100 and... 

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