Frame rotating

As before, we note that the resonance frequency of a nucleus at position r is directly proportional to the combined applied static and gradient fields at that location. In a gradient G=G u, orthogonal to the slice selection gradient, the nuclei precess (in the usual frame rotating at coq) at a frequency ciD=y The observed signal therefore contains a component at this frequency witli an amplitude proportional to the local spin density. The total signal is of the fomi... 

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. 

Fig. 1.2 Behavior of the magnetization in a simple echo experiment. Top a free induction decay (FID) follows the first 90° pulse x denotes the phase of the pulse, i.e., the axis about which the magnetization is effectively rotated. The 180° pulse is applied with the same phase the echo appears at twice the separation between the two pulses and its phase is inverted to that of the initial FID. Bottom the magnetization vector at five stages of the sequence drawn in a coordinate frame rotating at Wo about the z axis. Before the 90° pulse, the magnetization is in equilibrium, i.e., parallel to the magnetic field (z) immediately aftertbe 90° pulse, it has been rotated (by90° ) into the transverse (x,y) plane as it is com-...

The so-called HORROR experiment by Nielsen and coworkers [26] introduced continuous rf irradiation recoupling to homonuclear spin-pairs and initiated the later very widely used concept of /-encoded recoupling. Using a irreducible spherical approach as described above, the HORROR experiment (Fig. 2d) is readily described as starting out with the dipolar coupling Hamiltonian in (10) and x-phase rf irradiation in the form Hrf = ncor(Ix +SX), also here without initial constraint on n. The dipolar coupling Hamiltonian transforms into tilted frame (rotation n/2 around Iy + Sy)... 

In a reference frame rotating about z with frequency go, the rf field Hi has two components (1) a dc component Hiy and (2) a component oscillating with frequency 2go. The 2G0-component has very little effect on and can be discarded. If the rf field is "on resonance" (go = goq), then, in the RRF, Hq disappear completely, and we are left with only the dc component of Hi,... 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

Eq. (6.26) is the TDSE in the Schrodinger picture. In general, it proves more convenient to discuss the time evolution of the driven system in a rotating frame, such as the frame rotating with the laser carrier frequency q- After transformation into the carrier frequency picture and application of the rotating wave approximation (RWA), the TDSE takes the form [92]... 

In order to describe strong-field interaction of the five-state system in Figure 6.9 with intense shaped femtosecond laser pulses, the theoretical formalism prepared in Section 6.3.2.1 is readily extended. The RWA Hamiltonian H f) for the five-state system in Figure 6.9 in the frame rotating with the carrier frequency reads... 

If the frame rotates at the Larmor frequency rotational field term of eq. (1.28) reaches co0jy. Since the effective field is zero, the Larmor equation (1.8 a) is obtained ... 

If we transform the problem to a frame rotating with the microwave field, it is static and cannot induce transitions. The transformation to the rotating frame, often used to describe two level magnetic resonance experiments, is discussed by Salwen37 and Rabi et al.3H. 

For a one-dimensional (ID) image of a static object like the projection in Figure 7.2a, bottom, the signal measured in a frame rotating at frequency G)0, is given by ... 

To simplify visualization of the problem, the static Cartesian frame is replaced by a frame rotating around the z axis at the carrier frequency (< ,). The situation is similar to that of an observer who is rotating with the frame as though riding a carousel. 

The (inertial) I-frame is an element of ordinary 3D (real) space it is origin localizable and can be rotated with respect to a second frame (e.g., Euler angles). Abstract quantum states are necessarily invariable to translations and rotations of such frames. Projected quantum states must be invariant to origin translations and I-frame rotations linear and angular momentum... 

Figure 1 (A) Semiclassical Bloch picture of a vibrational echo in a frame rotating...

FIGURE 2.8 Formation of the effective rf field Brii in a frame rotating at o> radians/second. Beff is the vector sum of applied rf field B, along x and the residual field along z resulting from B0 and the fictitious field that represents the effect of the rotating frame. 

Figure 9.11 shows the precession of two such vectors for the A nuclei in an AX system precessing in a frame rotating at the chemical shift of A. If a pulse has placed the magnetizations (of magnitude M ) along the y axis at time 0, the vectors precess in the xy plane as shown, according to the usual relations ... 

FIGURE 9.11 Precession of the two spin components of magnetization, Ma and in a frame rotating at the chemical shift frequency, as described in the text. 

So far, the equations in this chapter are based on the laboratory frame of reference. In Section 2.8 we saw that the description of magnetic resonance can often be simplified by using a frame rotating with angular frequency coabout the z axis, where pulse frequency (and reference frequency) used to observe the spin system. Now we want to express the density matrix in the rotating frame in order to facilitate our handling of time-dependent Hamiltonians that arise when radio frequency fields are applied. 

Each method employs a spin echo to refocus chemical shifts, so we can direct our attention to the precession of the coherences for the separate spin components in a frame rotating at the chemical shift frequency. The magnetizations for 13C in CH precess at J/2 in CH2 the central component remains fixed in the rotating frame, while the other components precess at J and CH3 has two components precessing at J/2 and two at 3//2. 

Forcing function, 143 periodic, 144 transient, 143 Fourier transform, 170 Fractional time, 29 Fractionation factor, 301 Fraction theorem, general partial, 85 Frame, rotating, 170 Franck-Condon principle, 435 Free energy, 211 transfer, 418... 

Inverting Eq. (A 1-18) to give n in terms of n, and carrying out the matrix multiplications, we obtain relationships between the components in the frame rotated to the angle x and those in the unrotated frame ... 

Transformed to a frame rotating at frequency to. the radiation field simplifies to... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

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