Offset rotating-frame frequency

Note that we use A v to refer to the rotating-frame frequency (sometimes called the resonance offset). This is the difference between the Larmor frequency and the reference frequency v0 - vr. The above equation shows that the same physical law expressed in the equation on the left-hand side (precession rate is proportional to y and to B0) is operating in the equation on the right-hand side (resonance offset is proportional to y and to fires) in the rotating frame of reference, as long as we introduce the pseudofield. In the NMR spectrum, A v is the distance from the center of the spectral window to the NMR peak (Fig. 6.2), also represented as 2 in units of radians per second. If the peak is in the downfield half (left half) of the spectrum, the Larmor frequency is greater than the reference frequency ( v0 > vr) and we have a positive resonance offset (A v > 0). This corresponds to the motion of the net magnetization... 

From the discussion so far we can see that when an RF field is being applied there are two magnetic fields in the rotating frame. First, there is the RF field (or B field) of magnitude By, we will make this field static by choosing the rotating frame frequency to be equal to —n>Rp. Second, there is the reduced field, AB, given by (- 2/y). Since 2 = (co0 - comL fram.) and comL fram. = — urf it follows that the offset is... 

Fig. 18. The relationship between the excitations in the rotating frame by a RF pulse and in the Eigenframe by a PIP. The offset in the Eigenframe S is measured from the effective carrier fr( = A/+/rr, where the frequency shift is A/= Atp/lnAr. Reprinted from Ref. 49 with permission from Elsevier.

If the magnetic field gradient is applied for a short time period (i.e., a pulse ), as opposed to continuously during which time data are acquired, instead of imposing a time-independent modified resonance frequency on a nucleus as determined by its spatial position, the nuclear spin is given a phase offset (say fi) after application of the pulse characteristic of its spatial position when the pulse was applied. In the rotating frame of the spin system, this phase offset, (j)i is equal to yg i, where <5 is the duration of the applied gradient, Zi the position of the spin. 

Transport measurements performed using pulsed magnetic field gradients are most clearly understood in the context of a more mathematical framework. It follows from Eq. (2) that the phase shift (i.e., the instantaneous phase offset in resonance frequency) (f) t) acquired (in the rotating frame) following application of... 

These simple product operators precess in the x -y plane of the rotating frame at a frequency corresponding to the chemical shift in hertz relative to the center of the spectral window (the resonance offset Av = v0 — vr). The chemical shift frequency Av can also be represented as the angular velocity 2 in units of rad/s ( 2 = 2ttAv). Using 2 allows us to skip all the 2tt terms. 

In the rotating frame of reference, this corresponds to the trajectory of z magnetization of a spin under the action of a rf field along the x axis. In this case, corresponds to the offset of the spin and corresponds to the Rabi frequency I f = - yB,/(277), which is proportional to the amplitude of the rf field. In the zero-quantum frame, which is spanned by the operators (ZQ), (ZQ)y, and (ZQ), the frequency corresponds to the... 

In heteronuclear Hartmann-Hahn experiments, a rf sequence is irradiated simultaneously at two frequencies vf and Vg. These experiments are conveniently analyzed in the corresponding doubly rotating frame (Ernst et al., 1987). In this frame, the free-evolution Hamiltonian contains offset terms and for / and 5 spins, isotropic homonuclear /-/ and 5-5 coupling terms and, and truncated heteronuclear coupling terms... 

If we let (p Acu i, then eqn (9) corresponds to a calculation of the effect of a frequency offset term —Awl in the rotating frame Hamiltonian. Thus a modulation by exp (iqAcDt is caused by the offset, shifting the spectrum to Aco as seen in fig. 2. 

Applying the rf pulse at a frequency 00 near the Larmor frequency, only the central transition, which is unaffected by the first-order quadrupolar interaction, is effectively irradiated. The satelHte transitions, which are moved away from the Larmor frequency by oo due to the first-order quadrupolar interaction, are well off-resonance. Thus, in general, an additional term accounting for this offset must be considered. In the rotating frame, the Hamiltonian during the pulse is... 

Fig. 2.2.7 Magnetic fields in the rotating frame. Depending on the offset S2o of the NMR frequency from the rotation frequency a>rt of the rotating frame, a fictitious magnetic field of magnitude = —Qaly acts along the z-direction. Along the y-direction the rf magnetic field of amplitude B = —a> ly is applied. The vector sum of both forms the effective field B ts< around which the magnetization is rotating with frequency o>eff = —...

Taking the effective field into consideration is, particularly, important for pulses of weak amplitude co. Such pulses are selective pulses, because the rotation angle depends on the offset S2q of the NMR frequency from the rf frequency, which serves as a reference for the rotating frame. The angle 6 by which the effective field is tilted from the z-direction (Fig. 2.2.7) is given by... 

Fig. 9.2.7 Illustration of the fast adiabatic passage through resonance. The magnetization M follows the direction of the effective field The effective field in the rotating frame is the vector sum of the fictitious field Sbc and rf excitation field B. Both fields are applied in orthogonal directions. Because the fictitious field is proportional to the resonance offset S2, the magnitude of the fictitious field and thus the direction of the effective field can be changed by adjusting the resonance offset frequency S2.

The jump-and-retum (JR) pulse sequence consists of two hard tt/2 pulses that are opposite in phase, at the solvent frequency. The first -nil pulse brings, in the rotating frame, all of the spins to the +y direction, then during a subsequent delay t the water protons remain directed along +y whilst the other spins rotate in the plane by m, where w is the frequency offset from the water (i.e. pulse) frequency. The second ttH pulse is then applied to bring all of the spins into the z-x plane, thus the amplitude response is sin wt. t is chosen so that... 

Figure 2.12. Chemical shifts in the rotating frame. Vectors evolve according to their offsets from the reference (transmitter) frequency vq. Here this is on-resonance for spins A (vo = va) whilst spins X move ahead at a rate of -l-v Hz (=vx - Vo).

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