ATOMIC STRUCTURE AND INTERACTIONS

Atomic structure and interactions Electrons As noted above, the electron was the first subatomic particle discovered. Its interactions determine atomic structure, the chemical behaviour of atoms in molecules, and the properties of larger aggregations of atoms such as bulk solids. There are four kinds of forces in nature, and the electron is subject to three of themgravity, electromagnetism, and weak interaction. Only the electromagnetic force is significant in determining the properties of atoms and their chemistry. Within the framework of the equations used to describe the motion of the electron, the only two numerical properties that need to be specified are the electron's mass and its charge. These are given in Table 1 along with other basic atomic constants. There are four levels of complexity in the equations used to describe the properties of electrons. At the simplest level, classical equations such as Newton's equation are applied. The motion of the electron beam in a television tube is adequately described by classical physics. The equations of classical physics, however, become invalid when one attempts to describe the motion within distances smaller than the de Broglie wavelength of the electron (see quantum mechanics). Atomic properties fall into this small-distance regime, and quantum equations must be used. The simplest quantum equation is the above-mentioned Schrdinger equation, which is valid when the velocity of the electron is small compared to the speed of light. The electron is described by a function that obeys a wave equation. The function may be visualized as a cloud; where the function is large, the cloud is dense and the electron's presence is strongly felt. Atomic and chemical structure is well described by the Schrdinger wave functions when two additional nonclassical properties of the electron are included. The first property is spin, which is like an internal rotational motion of the electron. The magnitude of an electron's spin is fixed, but its orientation in space can vary. Spin appears in the Schrdinger equation as an attribute of the electron. There are, in fact, two separate wave functions associated with the probability that spin will be pointed in a particular direction or in the opposite direction. Other spin orientations are obtained by suitably combining the two functions to make intermediate directions. The other nonclassical property involved is the Pauli exclusion principle, which states in its general form that the wave function of identical particles must reverse sign when the coordinates of the particles are interchanged. As a consequence of the Pauli principle, two electrons cannot have the same wave function. This principle is extremely important in determining the structure of atoms, molecules, and bulk matter. The third level of complexity in the description of the electron is Dirac's equation (see above), which is a quantum equation postulated to satisfy the requirements of Einstein's relativity. Any particle governed by the Dirac equation is called a fermion; the electron is the most familiar example. The mathematics of the Dirac equation require that its particles have two spin orientations and obey the generalized Pauli principle. Thus, all the properties needed for determining atomic structure are built automatically into the theory. Furthermore, the Dirac equation predicts that for each kind of fermion there is an oppositely charged antiparticle with the same mass. The electron's antiparticle is the positron. The electron can be made to disappear by combining with a positron. When the two particles annihilate each other, their rest energy is converted to gamma rays or some other form of energy. Electrons can also be created from other forms of energy, but always in association with positrons. According to the Dirac equation, charged fermions such as the electron have a magnetic moment pointing along the direction of spin. The magnetic moment of electrons is very close to the value predicted by the Dirac equation. The magnetism of permanent magnets arises from the combined effect of individual electrons having spins aligned in the same direction. The most sophisticated level of complexity in describing electrons is the theory of quantum electrodynamics. Using the Dirac and other equations, this theory builds a wave function not only for observable electrons but also for particles and quanta that may be created from a vacuum. The predictions of quantum electrodynamics deviate only slightly from the Dirac equation. For example, the magnetic moment of electrons is predicted to deviate by 0.1 percent from the Dirac value owing to vacuum modifications. These deviations have been measured accurately and agree perfectly with quantum electrodynamics as far as experiments can determine. Electronic structure of atoms Schrdinger's theory of atoms The theory of the electronic structure of atoms is based for the most part on the Schrdinger equation, which in actuality is not a single equation. Each different physical system is described by its own equation, a partial differential equation that has as many variables as there are coordinates of interest in the application. For example, the Schrdinger equation for the hydrogen atom has three coordinatesthe x, y, and z coordinates of the separation between the electron and the hydrogen nucleus. Similarly, there are six variables in the Schrdinger equation of the helium atom, since the positions of two electrons are required. The Schrdinger equation has many solutions, each of which describes a possible state of the atom. Each state is specified by a function that depends on the coordinates of the particle(s) and by its associated energy. The function, called the wave function, is the mathematical representation of what is descriptively called the cloud. The wave function describes, among other things, the probability of an electron being at any given coordinate position. Electrons are more likely to be found in regions of the atom where the wave function is large. More precisely, the probability is proportional to the square of the wave function. At positions where the wave function goes through zero, the probability of finding electrons vanishes. Normally, an atom is found in its ground statei.e., the state with the least energy or the most bound state. States with higher energy are called excited states. Atomic spectra are the light quanta emitted when an atom makes a transition from an excited state to one of lower energy. The Schrdinger equation can be solved exactly only in special circumstances. Schrdinger himself found the solution for the hydrogen atom. It is important for physicists and chemists to understand the properties of the hydrogen atom obtained from the Schrdinger equation because these same properties appear in the behaviour of electrons in more complex atoms. For atoms with more than one electron, no exact solutions of the Schrdinger equation are known. Nevertheless, it is possible to obtain very accurate numerical solutions using approximation methods. An approximation scheme introduced by the English physicist Douglas R. Hartree is the basis for most calculations and for the prevailing physical understanding of the wave mechanics of atoms. In this method it is assumed that the electrons move independently. The electrons are allowed to interact only through the average electric field made by combining the charge of the nucleus with the charge distribution of the other electrons. A Schrdinger equation for individual electrons moving in an average electric field must then be solved. Each electron has its own wave function, which is called an orbital. A technical difficulty with this method is that the electric field is not known ahead of time because it depends on the charge distribution of the electrons. In turn, the charge distribution can be determined only from the wave functions, which require prior knowledge of the field. The difficulty is overcome by solving the Schrdinger equation in successive approximations. First, one makes a guess for the field, finds the wave functions, and then uses the derived charge distribution to make a better approximation for the field. This iterative process is continued until the final charge and electric field distribution agree with the input to the Schrdinger equation. The Hartree method (sometimes called the HartreeFok method to give credit to V. Fok, a Russian physicist who generalized Hartree's scheme) is widely used to describe electrons in atoms, molecules, and solids.