Vector flipping

Most often the hypothesis H concerns the value of a continuous parameter, which is denoted 0. The data D are also usually observed values of some physical quantity (temperature, mass, dihedral angle, etc.) denoted y, usually a vector, y may be a continuous variable, but quite often it may be a discrete integer variable representing the counts of some event occurring, such as the number of heads in a sequence of coin flips. The expression for the posterior distribution for the parameter 0 given the data y is now given as... 

The spin-echo experiment therefore leads to the refocusing not only of the individual nuclear resonances but also of the field inhomogeneity components lying in front or behind those resonances, a maximum negative amplitude being observed at time 2t after the initial 90° pulse. The frequency of rotation of each signal in the rotating frame will depend on its chemical shift and after the vector has been flipped by the 180° pulse, it... 

Figure 2.3 Spin-echo experiment. The behavior of nucleus X in an AX spin system is shown. (A) Application of the second 180° pulse to nucleus X in the AX hetero-nuclear system results in a spin-flip of the two X vectors across the x -axis. But the direction of rotation of the two X vectors does not change, and the two vectors therefore refocus along the —y axis. The spin-echo at the end of the t period along the -y axis results in a negative signal. (B) When the 180° pulse is applied to nucleus A in the AX heteronuclear system, the spin-flip of the X vectors...

In the first case, the 180° pulse is applied to nucleus X, causing the two vectors of nucleus X to flip across the x -axis. But no change occurs in the direction of their rotation during the second delay period, so at the end of this period the vectors are refocused along the —y -axis, producing a negative signal. 

Hence it is clear that if the two delay periods before and after the 180° pulses are kept identical, then refocusing will occur only when a selective 180° pulse is applied. This can happen only in a heteronuclear spin system, since a 180° pulse applied at the Larmor frequency of protons, for instance, will not cause a spin flip of the C magnetization vectors. 

Figure 5.10 (A) Selective spin-flip pulse sequence for recording heteronuclear 2D / resolved spectra. (B) Its effect on magnetization vectors. The selective 180° pulse in the middle of the evolution period eliminates the large one-bond coupling constants, /< ...

Flip angle The angle by which a vector is rotated by a pulse. 

Why, then, is the magnetisation density used The answer is that the magnetisation density is important for certain approximations which are usually made in analysing neutron scattering experiments. In the standard polarised neutron diffraction (PND) experiment [5], only one parameter is measured - the so-called flipping ratio . It is impossible to determine a vector quantity like the magnetisation density from a single number, unless some assumptions are made. The assumptions usually made are ... 

Another process responsible for a fluctuation of the local magnetic field is Neel relaxation. It corresponds to the flip of the crystal magnetization vector from one easy direction of anisotropy to another. The correlation time of this... 

The chain configuration is made to vary in time by allowing the beads to move one at a time. When an interior bead i (i = 1, 2,... N — 1) moves (or flips ) the vectors [Pg.306]

Fig, 1. The basic stochastic process. Position of bead i is indicated by a filled circle before it has flipped and an open circle after it has flipped. Solid lines represent the bond vectors hdashed lines indicate bond vectors ba, and boi+, after the flip. 

A radio-frequency (rf) alternating field initiates any NMR experiment. At resonance, the field vector B, rotates with Larmor frequency (v, = v0) perpendicularly to the static field vector B0, as shown in Fig. 2.1. In this situation, the nuclear magnetic moments will precess about both fields B0 and B,. Provided B1 extends along the x axis at a certain instant, the double cone of precession will rotate about the x axis in the yz plane (Fig. 2.1 (a) -> (b)). The flip angle 0 relative to the z axis at a given field strength B, depends on the pulse width t of the rf field, in the range of some ps, so that... 

The action of S on I na np> gives thus a term proportional to I na np> plus a sum over occupation number vectors, where the spin functions of an alpha orbital and a beta orbital have been flipped. Only permutations of the singly occupied orbitals are included. Inanp> is thus in general not an... 

Figure 8. Oscillogram, reflecting growth and turn of the magnetization vector of the MnF2 sample during sublattice flipping near Ho 92 kOe. The arrow shows the turn direction in increasing field ( ip = 40 ) [1,2].

Figure 9. Diagram of the antiferromagnetism vector L (left-hand) and the magnetization M (right-hand) turn at sublattice flipping in tilted field (ip < ipc)- The arrows show the turn directions in the I and II phases. The dashed line encloses the angles, corresponding to the regions of phase instability and realization of domain boundaries [1,2].

A further interesting result concerns spin component conservation. Each rotational level of this 2S state has two spin components, one (termed F ) with the spin and rotational angular momentum vectors parallel, and the other (F2) having them antiparallel. Excitation of an F component results in transfer primarily to F- components of other rotational levels, and similarly for F2 excitation. In a collision, the OH would as soon exchange several hundred cm" of energy as flip its spin around at no energy cost. Similar results were also observed in the fluorescence scans of the NH molecule (16). 

Fig. 9. Schematic illustration of how the strength of 13C-13C dipolar coupling between neighbouring 13C spin pairs in polyethylene changes by virtue of the change in orientation of the 13C-,3C vector (described by 0 and Oj) with respect to the applied field, Bo, when the polyethylene chain undergoes a 180° flip about the molecular chain axis.

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