Lattice line

Kochendorfer addresses superposition of symmetrical lattice line ghosts originating from the dilation - although it is readily shown that dilatation superposition (Mellin convolution) always causes the observed profile to become asymmetrical [125,126]. 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

Fig. 11. Single layers of MoS on pyrrhotite crystallite (49). Pyrrhotite lattice lines are 5.8. A apart.

Make a cut along one lattice line and then deform the sheet so as to accommodate an additional 60° sector. 

Inputting solid particles at fixed positions, of different sizes simulates a solid phase in the fluid lattice (Fig. 4). The number of fluid particles per node and their interaction law (collisions) affect the physical properties of real fluid such as viscosity. Particle movements are divided into the so called propagation step (spatial shift) and collisions. Not all particles take part in the collisions. It strongly depends on their current positions on the lattice in a certain LGA time step. In order to avoid an additional spurious conservation law [13], a minimum of two- and three-body collisions (FHP1 rule) is necessary to conserve mass and momentum along each lattice line. Collision rules FHP2 (22 collisions) and FHP5 (12 collisions) have been used for most of the previous analyses [1],[2],[14], since the reproduction of moisture flow in capillaries, in comparison to the results from NMR tests [3], is then the most realistic. 

The relation of an individual image or diffraction pattern to the 3D structure factor set is important to understand. The 3D data for a single-layer crystal falls on a set of reciprocal lattice lines, where one line passes through each of the diffraction spots in the pattern for an untilted crystal such as shown in Fig. 2. The data from a single crystal falls on a central section of the 3D data, that is a plane which passes through the origin and whose orientation corresponds to the orientation of the crystal. Thus, a crystal tilted to any orientation contributes one data point to each lattice line within the resolution limit for that particular orientation. 

Fig. 2 a Representation of the 3D diffraction data from the monolayer tubulin crystals. Amplitudes (shown here) and phases vary continuously along reciprocal lattice lines that pass through each of the diffraction spots in the diffraction pattern of an untilted crystal, shown in the horizontal plane. The two planes perpendicular to this plane show amplitudes along some of the lattice lines, b Three lattice line curves. The vertical axis represents intensity in an arbitrary unit, while the horizontal axis is z, the height along the lattice line in reciprocal space. Note the odd numbered curve (17,6) although the intensities are weak, the peaks are well defined because of the large number of measurements... 

Figure 7.7 Fast but anisotropic segmental motion results in a solid-like contribution to the NMR signal. This contribution is expressed in terms of a fractional contribution q of the second moment M2 of the rigid lattice line of a single chain or residual dipolar interactions between protons. The line splitting caused by the dipole-dipole interaction depends on the orientation angle q of the internuclear vector of the coupling protons in the magnetic field B(). The distribution of orientation angles changes with the...

The most interesting case arises for simple shear flows (A = 0). It can be proven mathematically that time-periodic motion is possible only when the direction of the flow is parallel to either a lattice plane or a lattice line. These are respectively termed slide and tube flows the origin of such terminology is evident from Fig. 4. Very similar configurations have been experimentally observed by Ackerson and Clark (1983). 

The study of the electronic structure of impurities and defects in solids has a long tradition, both because of its own intrinsic theoretical interest and because of the technological importance in improving the performance of solid state devices. Lattice defects can be point defects (such as substitutional or interstital foreign atoms, vacancies, antisite defects in composite lattices), line defects (such as dislocations), planar defects (such as boundaries, adatom surfaces, stacking faults corresponding to misplaced planes of atoms), and so on. 

This construction also shows why the diffracted beams from planes of a zone are arranged on a cone in the Laue method. All reciprocal-lattice lines representing the planes of one zone lie on a plane passing through the origin of the reciprocal lattice. This plane cuts the reflection sphere in a circle, and all the diffracted beam vectors S must end on this circle, thus producing a conical array of diffracted beams, the axis of the cone coinciding with the zone axis. 

Figure 4.4. Representation of a reciprocal lattice line s projection when producing a diffraction pattern with a polycrystalline sample...

The required number of points is approximately equal to the volume of the spherical octant just defined. We may easily convince ourselves of this with the aid of fig. 6 (though this, of course, is only two-dimensional). Since the co-ordinates of the lattice-points are whole numbers, the small squares formed by the lattice-lines have the... 

A straight line that passes through two, and hence, an infinite number of lattice points is a lattice line. A simple translation vector T = [7a + Kb + Wc (U,V,W being coprime integers) defines the direction of a set of parallel lattice lines equivalent by translation. It is easy to see that the greater the separation between the lines, the smaller is the norm of the translation T. ... 

The analogy between the equations representing the edges and faces of a crystal on the one hand, with lattice lines and lattice planes on the other is the foundation of the theory of the periodic nature of crystal structures. This interpretation of the law of rational indices was formulated by the French abbe Auguste Bravais (1811-1863) as follows ... 

The faces of a crystal are parallel to lattice planes mth a high density of lattice points the edges are parallel to lattice lines generated by short translations. 

In the rotating crystal method, a single crystal is used with dimensions of the order of 0.1 to 0.5 mm (smaller than the diameter of the primary beam). The importance of undesired phenomena such as absorption of the beam and extinction (Section 3.3.2) increases with the size of the crystal. The crystal executes a rotation about a lattice line [(77 IT] of the translation lattice. Hence, the crystal must be precisely aligned. The planes of the reciprocal lattice whose lattice points hkl satisfy the equation... 

Fig. 7 Schematic representation of the possible modes of epitaxy. Overlayer lattice points are depicted as small filled circles and substrate lattice points are depicted as larger shaded circles and the overlayer primitive cells are indicated by solid lines, (a) In a commensurate overiayer, every overlayer lattice point coincides with a substrate lattice point and certain overlayer [h, fc] lines in this case [1,4] overlayer lattice lines, coincide with primitive [0,1] substrate lines. Although one matrix element is zero in this example, this is not required for commensurism.

Commensurism (POP) p,q,r, and are all integers. All the overlayer lattice points lie simultaneously on two primitive substrate lattice lines and coincide with symmetry-equivalent substrate points (Fig. 7a). This can be described alternatively as POP coincidence. Each primitive overlayer lattice vector is an integral multiple of an identically oriented (primitive or nonprimitive) substrate lattice vector. This condition is generally regarded the energetically most favorable one with respect to the overlayer-substrate interface because the surface potentials of the two opposing lattices, which have periodicities that conform to the lattice dimensions, are phase coherent. This... 

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