Rotational displacements

Perhaps the best-characterized lesion in DNA associated with uv inactivation and mutagenesis is that involving the intrastrand photodimerization of adjacent thymine residues this lesion is almost wholly repaired by photodissociation of the dimers at shorter wavelengths in the photoreactivation process. Production of the chh dimer in this case, promoted by the configuration of adjacent molecules on the same sugar-phosphate strand, must however involve a rotational displacement of 36°, following the reduction of 0.6 A in molecular separation. 

The characters of P are related to those of P. Consider first the effect on, say Rxt of a C(6) rotation about some axis, not necessarily the x axis. This rotation will move the rotation displacement vectors in such a manner as to transform Rm into a vector R mr where H is the vector obtained by applying C(6) directly to Rx. Thus, for proper rotations, the matrices describing how A, Ry, and R, transform are exactly those matrices that describe how ordinary position vectors along the xt y, and z axes transform the matrix in the representation P corresponding to any C(6) is therefore the same as the matrix in P that corresponds to C(0) and the characters for proper rotations are the same for P as for P. 

Thus, because of the space-group symmetry, the displacement at (NK) is equal to Ru(nK), the rotated displacement from the equivalent site (nk). 

The wave vector q in the transformed displacement (E i)ua(nn) appears in the scalar product with a in the eigenvalue of the function operator ( a/), and is unaffected by a pure translation, as we should expect. Under the pure rotation (/ 0), the rotated displacement... 

In the incommensurate phase T < 7] the magnetic structure forms a helix along the c-axis. For the theoretical analysis of the magnetic properties of copper metaborate based on a phenomenological thermodynamic potential, it is essential that the crystal symmetry has no center of inversion. The inversion operation enters only in combination with rotational displacement around the c-axis by 90° A and 433. Therefore, in the thermodynamic... 

One MC pass is defined as N attempted MC steps, once for each of the N molecules. A single MC step consists of an attempted random translational displacement of the com, a random rotational displacement around a randomly chosen axis and a random intramolecular rotation around the central C-C bond joining the two rings. In the simulation of the liquid beiphenyl, prior to equilibration at 400 K, the liquid was kept at 900 K for a few hundred MC passes. For all the condensed phases, equilibration was carried out for about 10000 MC passes and the averages were calculated over an equivalent number of MC passes. 

Reproducing the in and out of plane scattering restricted the solution to the Ni-terminated octopolar model uniquely, plus the following symmetry-related atomic relaxations in each atomic layer p (p=0 for the apex layer of atoms). Note that 3 of the 4 atoms in layer p of the unit cell have symmetry-related vertical displacements s, whereas the independent atom has vertical displacement (see Fig. 10). Likewise, 5pj defines a radial displacement of symmetry-related atoms away from the in-plane position of the apex atom (see Fig. 10). The Pp,s terms define the possible rotational displacements. Detailed fitting reveals that all 5pj and pp terms are negligible except 6i=0.117 0.015A. 

For the formulation of the lattice dynamics in a molecular crystal, only six coupled equations of motion are necessary three for translation and three for rotation of a single molecule, if all the molecules in the unit cell are connected by symmetry operations. (This assumption does not hold e.g. for dipolar disorder in dimethyl-naphthalene or dimethyl-anthracene crystals (see Sect. 5.7).) The translational-rotational displacements u(lk) of the kth molecule in the Ith unit cell are taken to be given by propagating plane waves u(lfe) = U(fk)e F Here, S2(K)... 

The solid [2.2]-p-cyclophane was sublimated at 383 K, and the vapors were passed through a pyrolitic quartz tube at 873 K. This compound converts into PX without formation of detectable side products under these conditions [29, 30]. As stated in reference 22, solid PX most probably contains polymer chain precursors in the form of monomer stacks transforming into polymer chains as a result of the rather small rotational displacement of molecules without their translation movement. The assumed pattern of the solid-state PX polymerization is presented in Figure 2.1. 

The nature of the phase transitions and proton-deuteron dynamics in the crystals of 9-hydroxyphenalenone derivatives have been investigated by means of solid state NMR spectroscopy. It was found that the antiferroelectric ordering of deuterons along the hydrogen bonds in 5-methyl-9-deuterioxyphenalenone and 5-bromo-9-deuterioxyphenalenone occurs below the phase transition temperatures at 44 K and 34 K, respectively. The microscopic origin of the incommensurate modulation in 5-bromo-9-deuterioxyphenalenone below 34 K was attributed to slight rotational displacements, which are fixed at commensurate angles below the lock-in transition at 22 K. 

At room temperature, these HBs can break and re-form quite rapidly, leading to the rearrangement of the network. At the microscopic level, we need to understand the lifetime and bond-breaking mechanism of individual HBs between two neighboring water molecules. In the case of a water molecule, such bond-breaking events lead to the spatial/rotational displacements that in turn lead to the translational and rotational diffusion of water molecules. Therefore, the basic mechanism of the microscopic dynamics in water is expected to be different from a simple liquid made of nearly spherical units, Uke in the case of argon. 

Strain is a parameter which describes the relative displacement of particles within a deforming body. In the general case of a body subjected to stress, normal (or linear) and shear strains will occur together with translational and rotational displacements of elements within the body. 

Choudhury and Nimbalkar (2007) extended the work of Zeng and Steedman (2000) for the retaining wall under passive condition considering vertical seismic excitation and presented the results of rotational displacements of rigid retaining walls using planar failure mechanism. Basha... 

Rotational displacement of the wall is expected to take place only when angular acceleration (fl) exceeds the threshold acceleration for rotation Once rotation commences, it is assumed to continue at a constant acceleration equal to until the relative velocity between the... 

The total rotational displacement is the summation of individual rotations during the entire earthquake motion. 

The influence of seismic parameters on the magnitude of seismic passive resistance, sliding and rotational displacements is discussed in the following sections. 

For example, it can be found in Figure 8 that the assumption of planar failure surface overestimates the critical acceleration values (k g ) to the extent of 11.52% for = 30°. In addition, for a constant value of k = 0.46 and

planar failure mechanism predicts the magnitude of sliding displacement (-9) value as 10 cm, whereas composite failure mechanism predicts S value as 70 cm, the difference reaches as high as 85.71%. It may be noted in Figure 9 that the assumption of planar failure surface overestimates the value of k and difference reaches as high as 34.48% for 4> = 30°, and further for a constant value of = 0.40, planar failure mechanism predicts a rotational displacement (6) of 2.0°, whereas compos-... 

Figure 9. Comparison of rotational displacements (0 ) estimated using planar and composite failure mechanisms for different and (f> values.

For = 30° and 6 = 15°, the assumption of planar failure mechanism overestimates the critical seismic acceleration values for sUd-ing and rotation by 11.52 and 34.48% respectively, and further it underestimates the sliding and rotational displacements to the extent of 85.71 and 76.47% respectively. Accordingly the error in the estimation can be magnified as magnitude of 6 approaches 4> hence the planar failure mechanism is notthe appropriate one to predict the passive... 

Choudhury, D., Nimbalkar, S. (2007). Seismic rotational displacement of gravity walls by pseudodynamic method passive case. Soil Dynamics and Earthquake Engineering, 27, 242-249. doi 10.1016/j.soildyn.2006.06.009... 

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