Rotating vector diagrams

FIGURE 15.15 Quantities in dynamic testing (a) rotating vector diagram (b) stress and strain. 

Figure 10.2 Vector diagrams showing how a threefold rotation transforms v into vi, and how the latter is a linear combination of v and v36 specifically, vi = — h 3, +...

Figure 2.6 Diagram to describe the angular momentum associated with a rotating vector.

The amplitude of an RF pulse can be expressed in units of telsa (Bi). This corresponds to the magnitude (length) of the B vector in a rotating-frame vector diagram. Pulse amplitude is most commonly expressed in terms of the frequency of rotation of sample magnetization as it precesses around the B vector (for on-resonance pulses) during the pulse. 

Fig. 6.3.3 [Hou2] Vector diagrams of magnetization illustrating the principle of phase modulation in rotating-frame imaging for a selected volume element. Following an initial y pulse of length ti with variable flip angle 6, the magnetization is placed into the xz plane (left).

A connection with the magnetic parameters can be established with the aid of Equation (5) and the vector diagrams of Figure 4. To see the effects more clearly, projections of the vectors onto the xy plane are advantageous. As, furthermore, only differential precession effects intersystem crossing, that coordinate system is taken to rotate with the average frequency co, ... 

Equation (5.7) results in a match of the rotating-frame energies for H and and is called the Hartmann-Hahn condition. The match is produced when the applied carbon RF field (B ) is four times the strength of the applied proton RF field ( hX because yH/7c = 4. In Figure 5.5 the vector diagrams and pulse sequence are presented for this double rotating-frame experiment which results in CP. 

Figure 5.5 (a), (b), (c) Vector diagrams for a and C double rotating-frame CP experiment, (d) CP pulse sequence (Reprinted with permission from Jelinski [2].)... 

Fig. 8.1 - Phasor diagram for an alternating voltage E = A sin cor where A " is the maximum amplitude. The phasor is the rotating vector E.

In phasor terms, the rotating vectors are now separated on the polar diagram by the angle 0, Fig. 8.2. The response of simple circuit elements to the voltage E can be seen by applying Ohm s law, which for a pure resistance of value R is ... 

The Eqs. (24) and (25) are a pair of self-conjugate vector harmonics. The vector harmonics (V non) V (non)) in Eq. (24) are illustrated in Fig. 19. The 1 c component can be obtained by a rotation of the p" orbitals in Fig. 19 through 90° about the -I- z axis. A specific example illustrating how the vector harmonics are used to define the non-bonding component for a square pyramid based on the vector diagram is illustrated in Fig. 21. The important point to emphasise with this e set is that the n and it components contribute equally and the parity relationship interconverts the e components. This ensures the non-bonding character of this molecular orbital. The effect of the parity relationship is illustrated in Fig. 21. 

The situation in one particular form of the rotating frame experiment has points of similarity with the above arrangement, as shown in Fig. 2. A strong radiofrequency (r.f.) pulse is applied to the sample and after a time equivalent to a 7t/2 pulse, a njl phase shift is introduced into Hi. The njl pulse rotates Mq into the XY plane (first two vector diagrams) and the njl phase shift (third vector diagram) makes Hi and co-linear. Hi remaining constant during the phase... 

Figure 17 (A) Vector diagram illustrating the effect when a nominal 180° pulse is mis-set, resulting in only 170° rotation. (B) Illustration of how a composite 90° 180°, 90°, compensates for the effect of a mis-set pulse. Compensation is less complete for off-resonance signals.

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

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