The Properties of Waves

Waves have quantifiable properties. In our physical world of time and space, the maximum value that a wave can attain is its amplitude. The distance it travels before it repeats is its wavelength, the time required for it to travel one wavelength, or to complete one full oscillation is its period. The number of periods it completes per unit time is its frequency. Period and frequency are inverses of one another. We normally say that the period of a wave corresponds to 360° (or 2 r), and that any point within the period corresponds to some angle between zero and 360°. 

Introduction to Macromolecular Crystallography, Second Edition By Alexander McPherson Copyright 2009 John Wiley Sons, Inc. 

When we have a large number of individual waves, like those produced by the scattering of X-rays from families of planes, or from all of the unit cells in a crystal, or from all of the atoms within a unit cell, we are ultimately interested in knowing how all of the waves add together to yield a resultant wave that we can observe, characterize, and use. Waves are more complicated to sum than simple quantities like mass or temperature because they have not only an amplitude, a scaler, but also a phase angle 0 with respect to one another. This must be taken into account when waves are combined. As will be seen below, waves share identical mathematical properties with vectors (and with complex numbers, which are really nothing but vectors in two dimensions). 

Any sinusoidal wave such as sound, light, electrons, or X rays traveling through space and varying with time may be described by an expression of the form 

X rays are electromagnetic waves and like all such waves, they travel at the speed of light and they contain energy (we will not demonstrate all of that here, but we refer the reader to Maxwell, 1878). The energy of an electromagnetic wave is hco, where co is the frequency and h is Planck s constant. In X-ray crystallography we are usually not consciously concerned with energies, but in a sense, we do have to pay attention to how it is distributed in the wave. 

---

This section reviews some basic ideas on the properties of waves, and is an introduction to the wave properties of radiation discussed later. 

You should recall from your general chemistry course that electrons have some of the properties of waves. Chemists use the equations of wave mechanics to describe these electron waves. Solving these wave equations for an electron moving around the nucleus of an atom gives solutions that lead to a series of atomic... 

Although the detailed solution of the Schrodinger equation for the hydrogen atom is not appropriate in this text, we will illustrate some of the properties of wave mechanics and wave functions by using the wave equation to describe a very simple, hypothetical system commonly called the particle in a box, a situation in which a particle is trapped in a one-dimensional box that has infinitely high sides. It is important to recognize that this situation... 

The idea of electrons existing in definite energy states was fine, but another way had to be devised to describe the location of the electron about the nucleus. The solution to this problem produced the modern model of the atom, often called the quantum mechanical model. In this new model of the hydrogen atom, electrons do not travel in circular orbits but exist in orbitals with three-dimensional shapes that are inconsistent with circular paths. The modern model of the atom treats the electron not as a particle with a definite mass and velocity, but as a wave with the properties of waves. The mathematics of the quantum mechanical model are much more complex, but the results are a great improvement over the Bohr model and are in better agreement with what we know about nature. In the quantum mechanical model of the atom, the location of an electron about the nucleus is described in terms of probability, not paths, and these volumes where the probability of finding the electron is high are called orbitals. 

It is found that the symbols f, g, h, etc. representing such arrays can be manipulated by an algebra closely related to ordinary algebra, differing from it mainly in the process of multiplication. The rules of this algebra can be easily derived from the properties of wave functions, which we already know. 

Theoretically, Vineyard described GIXD with a distorted-wave approximation in the kinematical theory of x-ray diffraction [4]. In terms of the ordinary dynamical theory of Ewald [5] and Lane [6], Afanas ev and Melkonyan [7] worked out a formulation for the dynamical diffraction of x-rays under specular reflection conditions and Aleksandrov, Afanas ev, and Stepanov [8] extended this formalism to include the diffraction geometry of thin surface layers. Subsequently, the properties of wave Adds constructed during specularly diffracted reflections have been discussed in more detail by Cowan [9] and Sakata and Hashizume [10]. Meanwhile, a geometrical interpretation of GIXD based on a three-dimensional dispersion surface has been proposed by Hoche, Briimmer, and Nieber [11]. 

Electromagnetic waves have all the properties of waves in general reflection, refraction, interference, diffraction. However, through the magic of quantum theory ( wave-particle duality ), they may also behave like quantized particles or photons . The energy ( ) of any photon is related to its frequency ... 

In 1903, Marie Curie, her husband and Henri Becquerel received the Nobel Prize in physics Marie won another Nobel prize (chemistry) in 1911. In 1900, Max Planck had postulated that light energy must be emitted and absorbed in discrete particles, called quanta. In Paris in 1924, Victor de Broglie concluded that if light could act as if it were a stream of particles, particles could have the properties of waves. Both quanta and waves are central to quantum physics. Quantum theory states that energy comes in discrete packets, called quanta, which travel in waves. The principle of wave-particle duality states that all subatomic particles can be considered as either waves or particles. Light is a stream of photon particles that travel in waves. 

The nature of electromagnetic radiation baffled scientists for many years. At times light appears to behave like a wave at other times it behaves as though it were composed of small particles. While we now understand the wave-particle duality of all matter, including electromagnetic radiation, in terms of quantum mechanics, it is still convenient to consider electromagnetic radiation as having the properties of waves in many cases. 

Finally, there is remarkable that the current density probability current has as one of its major consequences the property of wave functions to be square integrable, i.e., with finite constant value for the integral... 

As a last point in this chapter we will address the coherence properties of hght The concept of coherence is related to the stabihty, or predictability, of the phase of an electromagnetic wave. Therefore, in the broadest sense, coherence is defined as the property of waves (or wave-like states like wave packets) that... 

Remember that the quantum mechanical approach treats atomic orbitals as wave functions[ H Section 6.5], and that one of the properties of waves is their capacity for both constructive combination and destructive combination r W Section 6.1]. 

Chapter 10 (Modern Atomic Theory) In Chapter 10 we have expanded our treatment of the atom by adding a section on the historical development of the structure of the atom. This material on Rutherford s work will give students a better perspective on how the current model of the atom was conceived. We have also greatly expanded our treatment of light with new emphasis on the properties of waves and the dual nature of light. 

---