The rotating frame

Ernst and W. A. Anderson, Application of Fourier transform spectroscopy to magnetic resonance, Rev. Sci. Instrum. 37, 93-102 (1966). 

Lowe and R. E. Norberg, Free-induction decays in solids , Phys. Rev. 107, 46-61 (1957). 

After the introductory remarks of the last section, we now dive into the rotating frame concept and consider how various magnetic fields and the magnetization behave in that frame. 

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 

There are many reasons for detecting more than just the 

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Figure Al.6.4. FVH diagram, showing the eoneept of adiabatie following. The Bloeh veetor, f, preeesses m a narrow eone about the rotating frame torque veetor, i2. As the detuning. A, ehanges from negative to positive, the field veetor, J , beeomes inverted. If the ehange m j is adiabatie the Bloeh veetor follows the...

The excitation field is the only field seen by the magnetization m the rotating frame. The magnetization processes about it. Starting from equilibrium (Mq=MqM ), transverse components are created and develop according to... 

Deverell C, Morgan R E and Strange J FI 1970 Studies of ohemioal exohange by nuolear magnetio relaxation in the rotating frame Mol. Phys. 18 553-9... 

AppHcations of soHd-state nmr include measuring degrees of crystallinity, estimates of domain sizes and compatibiHty in mixed systems from relaxation time studies in the rotating frame, preferred orientation in Hquid crystalline domains, as weU as the opportunity to characterize samples for which suitable solvents are not available. This method is a primary tool in the study of high polymers, zeoHtes (see Molecular sieves), and other insoluble materials. 

The Coriolis veclor lies in the same plane as the velocity vector and is perpendicular to the rotation vector. If the rotation of the reference frame is anticlockwise, then the Coriolis acceleration is directed 90° clockwise from the velocity vector, and vice versa when the frame rotates clockwise. The Coriolis acceleration distorts the trajectory of the body as it moves rectilinearly in the rotating frame. 

The twisting rubber hoses or o ringed elbows which connect the rotating pan necks with pipe and the wear plate to the face of the stationary valve The condition of the high impact nozzles mounted on the cloth wash manifold and, for wet cake discharge, the sluicing manifold The support rollers which take the vertical load of the entire machine and the horizontal thrust rolls that maintain the rotating frame concentric The toothed rim and sprocket which drives the pan filter... 

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. 

Very slow exchange. Slow exchange means that the lifetime ta = tb in each site is very long. Thus, a nucleus in site A precesses many times, at frequency (vq i a) in the rotating frame, before it leaves site A, and similarly for a nucleus in site B. Thus, there is time for absorption of energy from the radio-frequency field ffi, and resonance peaks appear at Va nd Vb in the laboratory frame. 

Figure 4-9. (Ai Precessing moment vectors in field tfo creating steady-state magnetization vector Afo. with//i = 0. (B) Immediately following application of a 90° pulse along the x axis in the rotating frame. (C) Free induction decay of the induced magnetization showing relaxation back to the configuration in A.

Different solid-state NMR techniques CPMAS NMR, the second moment of the signal, the spin-lattice relaxation time in the rotating frame T p) were combined to reach the conclusion that in the case of por-phine H2P the double-proton transfer is followed by a 90° rotation within the crystal (see Scheme 2). 

The effect of pulses on magnetization vectors is much easier to understand in the rotating frame than in the fixed frame. How do we arrive at the chemical-shift frequencies in the rotating frame ... 

Figure 1.33 The underlying principle of the Redfield technique. Complex Fourier transformation and single-channel detection gives spectrum (a), which contains both positive and negative frequencies. These are shown separately in (b), corresponding to the positive and negative single-quantum coherences. The overlap disappears when the receiver rotates at a frequency that corresponds to half the sweep width (SW) in the rotating frame, as shown in (c). After a real Fourier transformation (involving folding about n = 0), the spectrum (d) obtained contains only the positive frequencies.

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 spin-echo experiment therefore leads to the refocusing not only of the individual nuclear resonances but also of the field inhomogeneity components lying in front or behind those resonances, a maximum negative amplitude being observed at time 2t after the initial 90° pulse. The frequency of rotation of each signal in the rotating frame will depend on its chemical shift and after the vector has been flipped by the 180° pulse, it... 

Moreover, precession under selective irradiation occurs in the longitudinal plane of the rotating frame, instead of rotation in the transverse plane, which occurs during the evolution of the FID. The magnitude of the vector undergoing precession about the axis of irradiation decreases due to relaxation and field inhomogeneity effects. 

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