Transformation to the Rotating Frame

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 wrf/27r is usually chosen to be the 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. 

The transformation can be made by applying Eq. 11.16, with a Hamiltonian appropriate to a fictitious magnetic field (see Section 2.8) that would cause procession at a frequency of — wrf (equivalent to the frame moving at + r  

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The evolution of the density operator can be calculated in the Eigenframe first and the result is then transformed to the rotating frame. For simplicity, a pulse sequence with two PIPs (refer to Fig. 20) is taken into account first. The calculation is then extended to a general case with any number of PIPs. 

The above density operator needs to be transformed to the rotating frame before calculating the evolution of the density operator by the PIP2. It can be done by a frame transformation of... 

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. 

As in the classical case the transformation to the rotating frame has eliminated the time dependence of the r.f. perturbation 3(t) and has introduced the effective static magnetic field... 

The field phasor is a continuously rotating phasor in the space, whose angular position keeps changing with the position of the rotor with respect to the stationary stator. Let the rotor field displacement under the stationary condition with respect to the stator be denoted by angle /3as shown in Figure 6.11, This displacement will continue to change and will rotate the rotor (field frame). All the phasor quantities of the stator arc now expressed in terms ol the field frame. Figure 6.11 shows these two equivalent stator side phasors transformed to the rotor frame. 

A transformation into the rotating frame of reference of the nuclear spin system and integrating over all positions then allows us to rewrite eqn (5) as... 

All the effective RF fields created by a PIP are shifted from the carrier frequency fa. To calculate the rath excitation band for example, a transformation from the rotating frame to a new one (or the second rotating frame) is often needed, where the rath field plays the role of a new carrier. This transformation can be achieved by a unitary operator of1... 

The average propagator is introduced to describe the echo attenuation in the displacement experiments of Fig. 5.4.4 in terms of the particle displacement R along the gradient direction during the gradient-pulse delay A. To this end (5.4.26) is transformed to the rotated coordinate frame of Fig. 5.4.5 with axes defined in (5.4.21). By the same derivation used for (5.4.22) one arrives at... 

The reason why these anti-phase terms are preserved can best be seen by transforming to a tilted co-ordinate system whose z-axis is aligned with the effective field seen by each spin. For the case of a strong fi, field placed close to resonance the effective field seen by each spin is along x, and so the operators are transformed to the tilted frame simply by rotating them by -90° about y... 

As is well known, the 3x3 matrix Oy can be diagonalized by an appropriate orthogonal coordinate transformation (rotational transformation), provided it is a symmetric matrix generally it is considered to be symmetric because of its physical meaning. If the principal-axes frame of o, where o is expressed by a diagonal matrix, is transformed to the laboratory frame by a rotational transformation R(o, /3, y) which is defined by three Eulerian angles a, /3 and y, then the representations of o in both frames are related to each other by the equation = (5)... 

The QCLE with the field-matter interaction term in Eq. (34) may be solved by making the following transformation to a rotating frame ... 

As described previously, the analysis of the effect of RE pulses is best conducted by using the concept of the rotating frame. The transformation of the density operator from the laboratory to the rotating frame (which rotates around the z-axis with frequency = -72k) is accomplished with the use of the rotation operator [4] ... 

The effect of a RF pulse can be obtained by transforming the complete Hamiltonian Hz + Hrf from the laboratory to the rotating-frame [4], which is equivalent to expressing the effective Hamiltonian associated with the rotating-frame effective field given in Equation (2.3.6) ... 

In transforming Eq. [4] to the rotating frame of reference we make use of the exponential operator, exp(--iVy ). Its properties can best be seen from a series expansion of the exponential, viz. 

To describe the motion of spins under the influence of a resonant rf field, Hj, it is conventional in magnetic resonance to transform to a rotating frame representation. In the rotating fr me, the spins are subject to an effective field,... 

Phasor rotator to transform the field frame coordinates to the stator frame coordinates. 

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