Three-dimensional waves

The Fourier series that the crystallographer seeks is p(x,y,z), the three-dimensional electron density of the molecules under study. This function is a wave equation or periodic function because it repeats itself in every unit cell. The waves described in the preceeding equations are one-dimensional they represent a numerical value/(x) that varies in one direction, along the x-axis. How do we write the equations of two-dimensional and three-dimensional waves First, what do the graphs of such waves look like  

When you graph a function, you must use one more dimension than specified by the function. You use the additional dimension to represent the numerical value of the function. For example, in graphing f (x), you use the y-axis to show the numerical value off (x). [In Fig. 2.13, they-axes are used to represent/ ), the height of each wave at point x.] Graphing a two-dimensional function f(x,y) requires the third dimension to represent the numerical value of of the function. 

For example, imagine a weather map with mountains whose height at location (x,y) represents the temperature at that location. Such a map graphs a two-dimensional function t(x,y), which gives the temperature t at all locations (x,y) on the plane represented by the map. If we must avoid using the third dimension, for instance in order to print a flat map, the best we can do is to draw a contour map on the plane map (Fig. 5.1), with continuous lines (contours, in this case called isotherms) representing locations having the same temperature. 

Graphing the three-dimensional function p(x,y,z) in the same manner would require four dimensions, one for each of the spatial dimensions x, y, and z, and a fourth one for representing the value of p. Here a contour map is the only choice. In three dimensions, contours are continuous surfaces (rather than lines) on which the function has a constant numerical value. A contour 

The blue netlike surface in Plate 2 is also a contour map of a three-dimensional function. It represents a surface on which the electron density p(x,y,z) of adipocyte lipid binding protein (ALBP) is constant. Imagine that the net encloses 98% (or some specified value) of the protein s electron density, and so the net is in essence an image of the protein s surface. 

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Loettgers A, Untch A, Keller H-M, Schinke R, Werner H-J, Bauer C and Rosmus P 1997 Ab initio study of the photodlssoclatlon of HCO In the first absorption band three-dimensional wave packet... 

To picture the spatial distribution of an electron around a nucleus, we must try to visualize a three-dimensional wave. Scientists have coined a name for these three-dimensional waves that characterize electrons they are called orbitals. The word comes from orbit, which describes the path that a planet follows when it moves about the sun. An orbit, however, consists of a specific path, typically a circle or an ellipse. In contrast, an orbital is a three-dimensional volume for example, a sphere or an hourglass. The shape of a particular orbital shows how an atomic or a molecular electron fills three-dimensional space. Just as energy is quantized, orbitals have specific shapes and orientations. We describe the details of orbitals in Section 7-1. 

Gray and Wozny [101, 102] later disclosed the role of quantum interference in the vibrational predissociation of He Cl2(B, v, n = 0) and Ne Cl2(B, v, = 0) using three-dimensional wave packet calculations. Their results revealed that the high / tail for the VP product distribution of Ne Cl2(B, v ) was consistent with the final-state interactions during predissociation of the complex, while the node at in the He Cl2(B, v )Av = — 1 rotational distribution could only be accounted for through interference effects. They also implemented this model in calculations of the VP from the T-shaped He I C1(B, v = 3, n = 0) intermolecular level forming He+ I C1(B, v = 2) products [101]. The calculated I C1(B, v = 2,/) product state distribution remarkably resembles the distribution obtained by our group, open circles in Fig. 12(b). 

First, the orbital or quantum theory of matter assumes that the electron is not a particle, as we normally think of particles. Orbital theory considers the electron as a three-dimensional wave that can exist at several energy levels (orbitals), but not at the same time. 

The Three-Dimensional Wave System of Spinning Detonation , Ibid, pp 839-50 890 R. Cheret J. Brossard, "Cylindrical and Spherical Detonations in Gases , Ibid, Paper 84, p 149 (Abstracts only) 89g)... 

Macpherson (Ref 31) discussed the three-dimensional wave system of spinning detonation... 

The three dimensional wave system in spinning detonation was examined by Mac Pherson (Ref 105)... 

Under this term ate known waves which ate generated by condensed expls developing such high pressures (10s to l05atm) iQ the detonation reaction that the flow behind the front has a component radially outward Evans Ablow (Ref 66) described three-dimensional waves under the titles "Three-Dimensional, Axially Symmetric, Steady-State Detonation Waves With Finite Reaction Rate (pp 157-67), and Three-Dimensional, Transient Detonation Waves (pp 173-75)... 

The theories of transient processes leading to steady detonation waves have been concerned on the one hand with the prediction of the shape of pressure waves which will initiate, described in Section VI, A of Ref 66, and on the other hand with the pressure leading to the formation of such.an initiating pulse, described in Section VI, B. In Section V it was shown that the time-independent side boundary conditions are important in determining the characteristics of steady, three-dimensional waves. It now becomes necessary to take into consideration time-dependent rear boundary conditions. For one-dimensional waves, the side boundary conditions are not involved... 

A probability cloud is therefore a close approximation of the actual shape of an electrons three-dimensional wave. 

J. P. Vigier, Explicit mathematical construction of relativistic non-linear de Broglie waves described by three-dimensional (wave and electromagnetic) solitons piloted (controlled) by corresponding solutions of associated linear Klein-Gordon and Schrodinger equations, Found. Phys. 21(2) (1991). 

I hope the foregoing helps you to imagine three-dimensional waves. What do the equations of such waves look like A three-dimensional wave has three frequencies, one along each of the x-, y-, and z-axes. So three variables h, k, and l are needed to specify the frequency in each of the three directions. 

In words, Eq. (5.9) says that the complicated three-dimensional wave f(x,y,z) can be represented by a Fourier series. Each term in the series is a simple three-dimensional wave whose frequency is h in the x-direction, k in the y-direction, and 1 in the z-direction. For each possible set of values h, k, and the associated wave has amplitude Fhkl and, implicitly, phase oLhkl. The triple sum simply means to add up terms for all possible sets of integers h, k, and 1. The range of values for h, k, and 1 depends on how many terms are required to represent the complicated wavef(x,y,z) to the desired precision. 

This equation gives the desired electron density as a function of the known amplitudes IFI and the unknown phases ot hkl of each reflection. Recall that this equation represents p(x,y,z) in a now-familiar form, as a Fourier series, but this time with the phase of each structure factor expressed explicitly. Each term in the series is a three-dimensional wave of amplitude IF I, phase (x hkl, and frequencies h along the x-axis, k along the y-axis, and 1 along the z-axis. 

Although this equation is rather forbidding, it is actually a familiar equation (5.15) with the new parameters included. Equation (7.8) says that structure factor Fhk[ can be calculated (Fc) as a Fourier series containing one term for each atom j in the current model. G is an overall scale factor to put all Fcs on a convenient numerical scale. In the /th term, which describes the diffractive contribution of atom j to this particular structure factor, n- is the occupancy of atom j f- is its scattering factor, just as in Eq. (5.16) Xj,yjt and zf are its coordinates and Bj is its temperature factor. The first exponential term is the familiar Fourier description of a simple three-dimensional wave with frequencies h, k, and / in the directions x, y, and 7. The second exponential shows that the effect of Bj on the structure factor depends on the angle of the reflection [(sin 0)/X]. 

Manthe, U. and Koppel, H. (1991). Three-dimensional wave-packet dynamics on vibron-ically coupled dissociative potential energy surfaces, Chem. Phys. Lett. 178, 36-42. 

Manthe, U., Koppel, H., and Cederbaum, L.S. (1991). Dissociation and predissociation on coupled electronic potential energy surfaces A three-dimensional wave packet dynamical study, J. Chem. Phys. 95, 1709-1720. 

The divergent contributions from the soft (21) and hard (22) scales cancel each other, so that in the sum of all contributions we can put d = 3. We thus get the final expression for the 0(ma6) correction to a singlet iS-state energy of the helium atom (all average values below are over the three-dimensional wave function) ... 

In 1926, Erwin Schrodinger made use of the wave character of the electron and adapted a previously known equation for three-dimensional waves to the hydrogen atom problem. The result is known as the Schrodinger wave equation for the hydrogen atom, which can be written as... 

The wave mechanical treatment of the hydrogen atom does not provide more accurate values than the Bohr model did for the energy states of the hydrogen atom. It does, however, provide the basis for describing the probability of finding electrons in certain regions, which is more compatible with the Heisenberg uncertainty principle. Note that the solution of this three-dimensional wave equation resulted in the introduction of three quantum numbers (n, /, and mi). A principle of quantum mechanics predicts that there will be one quantum number for... 

The amplification rates are different in different directions and of course, different d are themselves function of ip. For three- dimensional waves the spatial theory comes with lot more complications as compared to temporal theory. In addition to the wave orientation angle ip, the amplification direction ip must also be specified before any calculation can be made. Once again, if... 

In the absence of charges and currents, Maxwell equations can be transformed into three-dimensional wave equations... 

Untch A., Weide, K., and Schinke, R. (1985) The direct photodissociation of ClNO(Si) An exact three-dimensional wave packet analysis, J. Chem. Phys. 95, 6496-6507. 

Gray, S.K. and Wozny, C.E. (1991) Fragmentation mechanisms from three-dimensional wave packet studies I ibrational predissociation of NcCb, HcCb, Nc-ICl and HelCl, J. Chem. Phys. 94, 2817-2832. 

From Figure 3.2, a crystal emerges as a virtually infinite array of identical unit cells that repeat in three-dimensional space in a completely periodic manner. Like a simple sine wave in one dimension, it repeats itself identically after a period of a, b, or c along each of the three axes. A crystal is in fact a three-dimensional periodic function in space, a three-dimensional wave. The period of the wave in each direction is one unit cell translation, and the value of the function at any point xj, yj, Zj within the cell, or period, is the density of electrons at that point, which we designate p(xj, yj, Zj). 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

Mathematicians1 have studied the conditions under which the wave equation is separable, obtaining the result that the three-dimensional wave equation can be separated only in a limited number of coordinate systems (listed in Appendix IV) and then only if the potential energy is of the form... 

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