Ensemble of spins

Fig. 4 (a) Pulse scheme for signal enhancement of the CT and schematic representations of the population distribution for an ensemble of spin-3/2 nuclei under (b) thermal equilibrium, (c) satellite saturation, and (d) satellite inversion... 

The symbols Rauto and Rcross within the relaxation matrix are the auto- and cross-relaxation rates, respectively. and (l2Z) are the longitudinal magnetizations of spin 1 and 2, respectively, and the brackets indicate averaging over the whole ensemble of spins. Rcross in terms of the spectral densities is given by... 

In an external magnetic field, Bq, the magnetization, Mq, of an ensemble of spins I is given by Curie s equation (10-13) ... 

The pulse sequence starts with a preparation period P, which usually allows the ensemble of spins - still partially perturbed by the pulses applied in the preceding scan -to return back to the equilibrium state. The preparation period may also be used to force this ensemble of spins to a defined non-equilibrium state according to the operators needs. 

In a real system, there is not just one isolated nucleus. In reality, there are many nuclei, and all them can occupy a particular spin state. After a certain time after the application of the external magnetic field, the spin system will reach the state of thermal equilibrium with a thermostat. This means that we should consider an ensemble of spins consequently, applying the canonical ensemble methodology, it is easy to calculate the ratio of the populations of the two spin energy states [45] ... 

Now that the concept of coherence has been introduced, let us make our model of the ensemble of spins a little more accurate. Instead of lining up the spins in a row, we move their magnetic vectors to the same origin, with the South pole of each vector placed at the same point in space (Fig. 5.3(a)). Furthermore, we need to consider both quantum states, the up cone (a or lower energy state) and the down cone (/3 or higher energy state). 

Now consider the effect of a 180° pulse on the ensemble of spins represented in Fig. 5.3. The RF pulse is actually a rotation, and we will see in Chapter 6 that this rotation is exactly analogous to the precession of magnetic vectors around the B0 field. The pulse itself can be viewed as a magnetic field (the Bi field) oriented in the x-y plane, perpendicular to the B0 field, and for the short period when it is turned on it exerts a torque on the individual nuclear magnets that makes them precess counterclockwise around the B field. This is shown in Fig. 5.4. Each magnetic vector is rotated by 180°, so the entire structure of two cones is turned upside down, with the upper cone and all its magnetic vectors turned down to become the lower cone, and the lower cone turned up to become the upper cone. This... 

The symbols we have been using to represent spin states (I, S-y, 2IySz, etc.) of the entire ensemble of spins are actually operators they can operate on a spin state (of a single spin pair in our Ha, Hb system) and spit out another spin state. We already saw this with the raising and lowering operators ... 

These vectors are just a column of numbers representing the coefficients of the four pure spin states ci, C2, C3 and c4. They do not describe the whole ensemble of spins, just one Ha - Hb pair. 

A complete description of the quantum mechanical wavefunction for any given spin system in an ensemble of spin systems is both infeasible and unnecessary, given that in NMR spectroscopy it is the properties of the nuclear spins that are of primary interest. And as NMR experiments deal with a large ensemble of spin systems the basic element required for a complete description of the system is... 

In an ensemble of spins eqn (3.1.6) has to be summed over all coupling pairs. The coupling tensor is denoted by D. Its trace is zero,... 

The spectral density is a measure of the amplitude of the M-quantum component of the nuclear spin interaction oscillating at frequency Mo as a result of molecular motion. Of course, we should also recognize that since H(t) varies randomly in time, otherwise identical spin systems will have different H(t) at any given time t. Thus, we need to perform an average over the ensembles of spin systems making up the total sample. We denote this ensemble average by a bar, and thus we replace CM in Eq. (11) with... 

We generally observe the result of a measurement on an ensemble of spins, so we first ask what do we directly observe in an NMR experiment The answer is always a voltage induced in an inductor by the component of a magnetic moment perpendicular to the static field which is parallel to the z axis V t) This is to say that our signal is proportional to a compo-... 

In order to begin to understand the ideas behind, and information available from experiments probing MQCOH in polymers, let us remind ourselves of the meaning of phase coherence in quantum mechanics. We start with the simplest case, a noninteracting ensemble of spin j systems, and with spin basis functions that are eigen functions of the largest interaction present, the Zeeman Hamiltonian These are I, m) =, ) = a), and, - ) = )3). A spin 2 system will have single particle wavefunction... 

In an ensemble of spin noninteracting systems represented by such a phase-coherent superposition, the many-particle wavefunction is a product of the single-particle states,... 

Following excitation, the net magnetization vector M will almost always have a component precessing in the xy plane this component returns to its equilibrium position through a process called relaxation. Relaxation occurs when an ensemble of spins are distributed among their available allowed spin states contrary to the Boltzmann equation (Equation 1.5). Relaxation occurs through a number of different relaxation pathways and is itself a very demanding and rich subdiscipline of NMR. The two... 

Relaxation. The return of an ensemble of spins to the equilibrium distribution of spin state populations. 

Lobo and Ramanathan have combined adiabatic and Hartmann-Hahn cross-polarization for sensitivity enhancement in solid-state separated local field 2D NMR experiments of partially ordered systems The magnetization in double- and zero-quantum reservoirs of an ensemble of spin-1/2 nuclei has been examined and their role in determining the sensitivity of a class of separated local field NMR experiments based on Hartmann-Hahn cross-polarization has been described. Lobo and Ramanathan report that for the liquid crystal system studied, a large dilute spin-polarization, obtained initially by the use of adiabatic cross-polarization, can enhance the sensitivity of the above experiment. The signal enhancement factors, however, are found to vary and depend on the local dynamics. The experimental results have been utilized to obtain the local order-parameters of the system. 

The spin-lattice relaxation rate of a particular set of equivalent nuclei is the first-order time constant of the energy exchange process for those nuclei and we now shall use the rotating reference frame model to illustrate one of several possible methods whereby these time constants may be measured. Consider our ensemble of spins. When they are at thermal equilibrium with the lattice, their net magnetisation can be represented as a vector directed along the +z-axis (see... 

Fig. 1. (a) Energy levels diagram and allowed transitions of an ensemble of spins with I = 1/2 in the absence and the presence of Bo- The small arrows indicate the orientation of nuclear spins relative to Bo- (b) Energy diagram and allowed single-quantum transitions for spins with / = 1. 

NMR relaxation The process by which an ensemble of spins returns to their equilibrium state. 

This is the Curie expression for the magnetization of an ensemble of spin 1/2 nuclei. 

Conventional NMR deals with a large ensemble of spins. It means that the state of the system is in a statistical mixture, which is obviously inadequate for QIP. However, the NMR ability for manipulating spins states worked out by Cory et al. [24] and Chuang et al. [23] resulted in elegant methods for creating the so called effectively pure or pseudo-pure states. Behind the idea of the pseudo-pure states is the fact that NMR experiments are only sensitive to the traceless deviation density matrix. Thus, we might search for transformations that, applied to the thermal equilibrium density matrix, produce a deviation density matrix with the same form as a pure state density matrix. Once such state is created, all remaining unitary transformations will act only on such a deviation density matrix, which will transform as a true pure state. 

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