Rotating frame density matrix

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

The steady-state density matrix equation in the rotating frame, in operator form, is given in equation 12,... 

We know that rf pulses rotate magnetization vectors and alter the state of the spin system. To examine such effects quantitatively in terms of alterations in the density matrix, it will be helpful to express the density matrix in the rotating frame of reference. 

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. 

We now look at the evolution of the density matrix for the one spin system as the magnetization precesses in the rotating frame. Once more, we apply Eq. 11.16 in the same manner we did in Section 11.2 to take into account the effect of the rotating frame. In this case we obtain... 

The computational procedure follows closely the steps of an actual m.p. experiment see Fig. 1. The spin system, which is initially in thermal equilibrium, is hit by a preparation pulse Pp. Thereafter, one component of the transverse nuclear magnetization created by Pp, say My, is measured and the measurement is repeated at intervals of the cycle time The resulting time series My(qtJ,q = 0,...,(2 " - 1), if Fourier transformed. For simulations we accordingly first specify the initial condition of the spin system, that is, the initial value of the spin density matrix g(t) in the rotating frame. Our standard choice Pp, = P implies p(0) fy == the sum running over k = We then follow the evolution... 

To gain some insight into the theory of multi-pulse NMR, the WAHUHA sequence (t, -Fx, t, —y, T, T, -Fy, t, —x, t) is applied in the rotating coordinate frame to a pair of spins coupled by the dipole-dipole interaction [Sch9]. The pulses are considered to be infinitely short. Their flip angle is 90°, and they are applied in -Fx, —y, -Fy, and —x directions of the rotating coordinate frame. The density matrix p to- -6x) after completion of a WAHUHA cycle is obtained from the density matrix p(ta) before application of the cycle by transformation with the evolution operator U(i) of the cycle. 

Spin-spin relaxation is handled by second order perturbation theory [ Abr 1 ] of the density matrix equation of motion (2.2.62) in the rotating frame (RCF) [Abrl],... 

Evaluation of the density matrix is facilitated by transformation of the spin Hamiltonian into the rotating frame, the z -axis of which is tilted at the magic angle tilted rotating frame),... 

We find the density matrix elements from the master equation of the system. In the frame rotating with the laser frequency a>L and within a secular approximation, in which we ignore all terms oscillating with (colrf - a>L) and ((]>2d — the master equation for the density operator of the system is given by... 

The BPP expression was derived under the assumptions that the dipolar interactions formed a perturbation on the Zeeman levels and that the time dependent part of the dipolar interaction could be treated by time dependent perturbation theory or equivalently the density matrix approach to determine the relaxation expression. This does not rule out a BPP type expression for relaxation in the rotating frame. Specifically, a BPP type approach can be used to derive the following expression for T due to rotational motion under basically... 

Upon transforming to an interaction representation (i.e., a reference frame rotating at frequency w) in which the density matrix is defined by Equation 7, and invoking the rotating wave approximation which consists of dropping all high-frequency motions with respect to u . Equation 25 becomes... 

Since the signal field is detected in the same direction as that of the incoming field, it is proportional to the expectation value of the dipole moment operator, p q), at time t associated with the density matrix element containing the phase factor e (i.e. the 01 coherence). After transforming back to the non-rotating frame, the QCL expectation value of p(q) leads to the following expression for the linear ORF ... 

We can now use these explicit forms to calculate the effect of RF pulses on the deviation matrix densities for spin 1 /2 nuclei. In the case of a r/2 pulse with the magnetic field aligned with the x-direction of the rotating frame, we have ... 

We can summarize the results of this section saying that the effects of RF pulses can be described in the rotating frame in terms of the rotation operators (usually around the X, y, —x,and —y axes) applied to the deviation density matrix starting from thermal equilibrium. The evolution of the system after or between the RF pulses is described as a free precession aroimd the z-axis, with a frequency that depends on the frequency offset and is therefore different for nuclei experiencing distinct local fields (due to chemical shifts, for example). 

For an ensemble of nuclei with spin / > 1/2 experiencing no quadrupolar interaction, such as in an isotropic liquid or a crystal with cubic symmetry, the equations describing the time evolution of the density matrix under action of static and RF magnetic fields are a natural extension of the / = 1/2 case. The Hamiltonian contains only the Zeeman and RF terms the effects of RF pulses are described by rotations of the spin operators around the transverse axes in the rotating frame, whereas free evolution corresponds to rotations around the z-axis. 

When intermediate regime) the evolution of the density matrix in the rotating-frame exhibits a much more complex behavior as function of the pulse length and this is the basis of the method known as nutation NMR spectroscopy [10,19]. The distinction between selective (ft>i -C coq), non-selective ([Pg.71]

The purpose of this paper is to present the density matrix formalism of angular momentum with half- and integer spin quantum numbers using the spectrum dissolving theorem. The density matrix formalism was developed for the laboratory and rotating frame in order to obtain a complete analytical representation for the spin excitation and response scheme. The density matrix contained in the rotation operator will beeome elear, as oppossed to the approximate treatment, and thereby from the theoretieal process of off-resonance to that of just-resonance through the state of near-resonance will be visualized continuously. 

The density matrix is used to investigate the effects of radiofrequency pulses on spin systems. In the rotating frame, the r.f. Hamiltonian in angular frequency units is time-independent and has a form given by Eq. (2.28).,... 

Next we need to define the form of the time evolution operator (Liouvillian) for the density matrix described by the SLE. The molecule being partitioned in two fragments, as described above, we have (i) two local frames respectively fixed on the pahnitate chain (CF) and on the tempo probe (PB) these are chosen with their respective z axes directed along the rotating bond, for convenience (ii) the molecular frame (MF), fixed on the pahnitate chain this is the frame which diagonalizes the... 

The excited state density matrix. In order to solve equation (16.43) it is necessary to express the excitation term in the laboratory frame of reference. Using the two rotation operators we have... 

The density matrix in the laboratory frame is obtained by inverting the sequence of the rotations, giving... 

We complete our illustration of the power of the density matrix treatment by seeking the stationary solutions of equation (17.55) in the rotating frame. By setting all time... 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

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