Ewald dimension

Pc- (c) Dipole density p. (d) Water contribution to the surface potential x calculated from the charge density Pc by means of Eq. (1). All data are taken from a 150 ps simulation of 252 water molecules between two mercury phases with (111) surface structure using Ewald summation in two dimensions for the long-range interactions. 

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

Variational electrostatic projection method. In some instances, the calculation of PMF profiles in multiple dimensions for complex chemical reactions might not be feasible using full periodic simulation with explicit waters and ions even with the linear-scaling QM/MM-Ewald method [67], To remedy this, we have developed a variational electrostatic projection (VEP) method [75] to use as a generalized solvent boundary potential in QM/MM simulations with stochastic boundaries. The method is similar in spirit to that of Roux and co-workers [76-78], which has been recently... 

Figure 5.3 Comparison of Ewald and bare electrostatic potential in one dimension. See Eq. (5.32).

In the Ewald construction (Figure 3.17), a circle with a radius proportional to 1/A and centered at C, called the Ewald circle, is drawn. In three dimensions it is referred to as the Ewald sphere or the sphere of reflection. The reciprocal lattice, drawn on the same scale as that of the Ewald sphere, is then placed with its origin centered at 0. The crystal, centered at C, can be physically oriented so that the required reciprocal lattice point can be made to intersect the surface of the Ewald sphere. 

FIGURE 3.17. The construction of an Ewald sphere of reflection, illustrated in two dimensions (the Ewald circle), (a) Bragg s Law and the formation of a Bragg reflection hkl. The crystal lattice planes hkl are shown, (b) Construction of an Ewald circle, radius 1/A, with the crystal at the center C and Q-C-0 as the incident beam direction. 

Ewald sphere, sphere of reflection A geometrical construction used for predicting conditions for diffraction by a crystal in terms of its reciprocal lattice rather than its crystal lattice. It is a sphere, of radius 1/A (for a reciprocal lattice with dimensions d = X/d). The diameter of this Ewald sphere lies in the direction of the incident beam. The reciprocal lattice is placed with its origin at the point where the incident beam emerges from the sphere. Whenever a reciprocal lattice point touches the surface of the Ewald sphere, a Bragg reflection with the indices of that reciprocal lattice point will result. Thus, if we know the orientation of the crystal, and hence of its reciprocal lattice, with respect to the incident beam, it is possible to predict which reciprocal lattice points are in the surface of this sphere and hence which planes in the crystal are in a reflecting position. 

However, not all systems require periodic boundary conditions in all spatial directions.For membranes, for example, only two dimensions are periodic, while the third one is finite. In that case, the Ewald method is computationally highly inefficient and wouid not aiiow to treat more than a few hundred charged particles. We present two alternative approaches, the MMM2D and ELC methods, which allow for computational efficiency similar to the bulk case. It is also simple to adapt the MMM2D method for systems with only one periodic dimension. 

The Ewald method can also be formulated for partially periodic boundary conditions, i.e. systems where only one or two of the three spatial dimensions are periodic. These geometries are applied for example in simulations of membranes or nano-pores, where some dimensions are supposed to have finite extend. In this case, however, the method scales like 0 N ), which only allows the treatment of moderately large systems with a few hundred charges. 

Changing the orientation of the crystal reorients the reciprocal lattice bringing different reciprocal lattice points on to the surface of the Ewald sphere. An ideal powder contains individual crystallites in all possible orientations with equal probability. In the Ewald construction, every reciprocal lattice point is smeared out onto the surface of a sphere centered on the origin of reciprocal space. This is illustrated in Figure 1.9. The orientation of the vector is lost and the three-dimensional vector space is reduced to one dimension of the modulus of the vector A ti-... 

In three dimensions, the circular intersection of the smeared reciprocal lattice with the Ewald sphere results in the diffracted X-rays of the reflection hkl forming coaxial cones, the so-called Debye-Scherrer cones (Figure 1.11). 

Fig. 3 Ewald construction. The white half-circle indicates the Ewald sphere in two dimensions. The points of intersection between the reciprocal lattice rods and the Ewald sphere form the set of reciprocal lattice points (bright) which obey Bragg s law and appear as diffraction spots in the diffraction pattern. Zero-, first- and second-order Laue zone are indicated. Eor electron diffraction in TEM, the ratio between the radius of the Ewald sphere and the reciprocal lattice unit is larger than visualized in the figure. (View this art in color at www.dekker. com.)...

In practice, most applications of the slab-adapted Ewald smn in three dimensions employ vacuum spaces that are three to five times thicker than the original substrate separation [252-257]. The energies obtained with the approximate Ewald sum then coincide almost perfectly with those of the rigorous method discussed in Section 6.3.1. For a systematic discussion of the errors involved in the. slab-adapted version, we refer the interested reader to Ref. 256. 

These size effects can easily be pictured in terms of the reciprocal lattice. Ideally, each reciprocal lattice point is sharp and diffraction will only occur when the Ewald sphere intersects a particular point. The effect of finite crystal dimensions is to modify the shape of each reciprocal lattice point by drawing the point out in reciprocal space in a direction normal to the small dimension in real space. The following descriptions are a first approximation only. (More accurate depictions of the change in shape of a reciprocal lattice reflection can be determined by applying diffraction theory.) A square crystal of side w will have each reciprocal lattice point modified to a cross with each arm of approximate length 1 /w, extended along a direction perpendicular to the square faces a cubic crystal will be similar, with each arm pulled out along directions normal to the cube faces. A spherical crystal of diameter w will have each reciprocal lattice point broadened into a sphere of approximately l/w diameter a needle of diameter w... 

Before we can measure the intensity of a Bragg reflection, we need to determine where and from what direction to orient the X-ray detector. A geometrical description of diffraction, the Ewald sphere, allows us to calculate which Bragg reflections will be formed if we know the orientation of the crystal with respect to the incidentX-ray beam. In the Ewald construction (shown in two dimensions in Fig. 11), a sphere of radius 1/X is drawn with the crystal at its center and the reciprocal lattice on its surface. A Bragg reflection is produced when a reciprocal lattice point touches the surface of the Ewald sphere. As the orientation of the crystal is changed, so is the orientation of its reciprocal lattice. 

As an example for results obtained by the constrained MD method, we briefly discuss some of the differences between the free energy profiles of Li+, F , and I" ions on the mercury surface. In this study [192], only the water-metal interactions are described by the SCF interaction energies [40]. The ions interact with the surface exclusively by means of image interactions. Ewald summation in two dimensions is used to properly describe the long range polarization effects near the interface. 

By making a small modification to the Ewald construction, we obtain a more informative representation of the Laue method. By multiplying all the dimensions of the construction by the wavelength A, we obtain a lattice -h fcb -h /c ) and a sphere of radius 1. Thus, for polychromatic radiation, we obtain a superposition of lattices of variable dimensions intersected by a single... 

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