The emission and absorption spectra of the elements depend on the electronic structure of the atom. An atom consists of a number of negatively charged electrons bound to a nucleus containing an equal number of positively charged protons. The nucleus contains a certain number (Z) of protons and a generally different number (N) of neutrons. The diameter of a nucleus depends on the number of protons and neutrons and is typically 10−14 to 10−15 metre (3.9 × 10−13 to 3.9 × 10−14 inch). The distribution of electrons around the nuclear core is described by quantum mechanics.
The chemical and spectroscopic properties of atoms and ions are primarily determined by their electronic structure—i.e., by the number and arrangement of electrons surrounding their nucleus. Typical energies of electrons within an atom range from a few electron volts to a few thousand electron volts. Chemical reactions and other processes occurring in spectroscopic sources usually involve energy exchanges on this order of magnitude. Processes that occur within nuclei (e.g., electromagnetic transitions between energy states of the nucleus, beta decay, alpha decay, and electron capture) typically involve energies ranging from thousands to millions of electron volts; hence the internal state of nuclei are nearly unaffected by the usual processes occurring in chemical reactions, light absorption, and light sources. On the other hand, nuclear magnetic moments can be oriented by light through their coupling to the atom’s electrons. A process known as optical pumping, in which the atom is excited with circularly polarized light, is used to orient the spin of the nucleus.
The forces holding an atom together are primarily the electrostatic attractive forces between the positive charges in the nucleus and the negative charge of each electron. Because like charges repel one another, there is a significant amount of electrical repulsion of each electron by the others. Calculation of the properties of the atom first require the determination of the total internal energy of the atom consisting of the kinetic energy of the electrons and the electrostatic and magnetic energies between the electrons and between the electrons and the nucleus.
The size scale of the atom is determined by the combination of the fact that the atom prefers to be in a state of minimum energy and the Heisenberg uncertainty principle. The Heisenberg uncertainty principle states that the uncertainty in the simultaneous determination of the position and the momentum (mass times velocity) of a particle along any direction must be greater than Planck’s constant. If an electron is bound close to the nucleus, the electrostatic energy decreases inversely with the average distance between the electron and the proton. Lower electrostatic energy corresponds to a more compact atom and, hence, smaller uncertainty in the position of the electron. On the other hand, if the electron is to have low kinetic energy, its momentum and its uncertainty in momentum must be small. According to the Heisenberg principle, if the uncertainty in momentum is small, its uncertainty in position must be large, thus increasing the electrostatic energy. The actual structure of the atom provides a compromise of moderate kinetic and electrostatic energies in which the average distance between the electron and the nucleus is the distance that minimizes the total energy of the atom.
Going beyond this qualitative argument, the quantitative properties of atoms are calculated by solving the Schrödinger wave equation, which provides the quantum mechanical description of an atom. The solution of this equation for a specified number of electrons and protons is called a wave function and yields a set of corresponding eigenstates. These eigenstates are analogous to the frequency modes of a vibrating violin string (e.g., the fundamental note and the overtones), and they form the set of allowed energy states of the atom. These states of the electronic structure of an atom will be described here in terms of the simplest atom, the hydrogen atom.
The hydrogen atom is composed of a single proton and a single electron. The solutions to the Schrödinger equation are catalogued in terms of certain quantum numbers of the particular electron state. The principal quantum number is an integer n that corresponds to the gross energy states of the atom. For the hydrogen atom, the energy state En is equal to −(me4)/(2ℏ2n2) = −hcR∞/n2, where m is the mass of the electron, e is the charge of the electron, c is the speed of light, h is Planck’s constant, ℏ = h/2π, and R∞ is the Rydberg constant. The energy scale of the atom, hcR∞, is equal to 13.6 electron volts. The energy is negative, indicating that the electron is bound to the nucleus where zero energy is equal to the infinite separation of the electron and proton. When an atom makes a transition between an eigenstate of energy Em to an eigenstate of lower energy En, where m and n are two integers, the transition is accompanied by the emission of a quantum of light whose frequency is given by ν =|Em − En|/h = hcR∞(1/n2 − 1/m2). Alternatively, the atom can absorb a photon of the same frequency ν and be promoted from the quantum state of energy En to a higher energy state with energy Em. The Balmer series, discovered in 1885, was the first series of lines whose mathematical pattern was found empirically. The series corresponds to the set of spectral lines where the transitions are from excited states with m = 3, 4, 5,… to the specific state with n = 2. In 1890 Rydberg found that the alkali atoms had a hydrogen-like spectrum that could be fitted by series formulas that are a slight modification of Balmer’s formula: E = hν = hcR∞[1/(n − a)2 − 1/(m − b)2], where a and b are nearly constant numbers called quantum defects.
There are a set of angular momentum quantum numbers associated with the energy states of the atom. In terms of classical physics, angular momentum is a property of a body that is in orbit or is rotating about its own axis. It depends on the angular velocity and distribution of mass around the axis of revolution or rotation and is a vector quantity with the direction of the angular momentum along the rotation axis. In contrast to classical physics, where an electron’s orbit can assume a continuous set of values, the quantum mechanical angular momentum is quantized. Furthermore, it cannot be specified exactly along all three axes simultaneously. Usually, the angular momentum is specified along an axis known as the quantization axis, and the magnitude of the angular momentum is limited to the quantum values Square root of√l(l + 1) (ℏ), in which l is an integer. The number l, called the orbital quantum number, must be less than the principal quantum number n, which corresponds to a “shell” of electrons. Thus, l divides each shell into n subshells consisting of all electrons of the same principal and orbital quantum numbers.
There is a magnetic quantum number also associated with the angular momentum of the quantum state. For a given orbital momentum quantum number l, there are 2l + 1 integral magnetic quantum numbers ml ranging from −l to l, which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values mlℏ. This phenomenon is known as space quantization and was first demonstrated by two German physicists, Otto Stern and Walther Gerlach.
Elementary particles such as the electron and the proton also have a constant, intrinsic angular momentum in addition to the orbital angular momentum. The electron behaves like a spinning top, with its own intrinsic angular momentum of magnitude s = Square root of√(1/2)(1/2 + 1) (ℏ), with permissible values along the quantization axis of msh = ±(1/2)ℏ. There is no classical-physics analogue for this so-called spin-angular momentum: the intrinsic angular momentum of an electron does not require a finite (nonzero) radius, whereas classical physics demands that a particle with a nonzero angular momentum must have a nonzero radius. Electron-collision studies with high-energy accelerators show that the electron acts like a point particle down to a size of 10−15 centimetre, one hundredth of the radius of a proton.
The four quantum numbers n, l, ml, and ms specify the state of a single electron in an atom completely and uniquely; each set of numbers designates a specific wave function (i.e., quantum state) of the hydrogen atom. Quantum mechanics specifies how the total angular momentum is constructed from the component angular momenta. The component angular momenta add as vectors to give the total angular momentum of the atom. Another quantum number, j, representing a combination of the orbital angular momentum quantum number l, and the spin angular momentum quantum number s can have only discrete values within an atom: j can take on positive values only between l + s and |l − s| in integer steps. Because s is 1/2 for the single electron, j is 1/2 for l = 0 states, j = 1/2 or 3/2 for l = 1 states, j = 3/2 or 5/2 for l = 2 states, and so on. The magnitude of the total angular momentum of the atom can be expressed in the same form as for the orbital and spin momenta: Square root of√j( j + 1) (ℏ) gives the magnitude of the total angular momentum; the component of angular momentum along the quantization axis is mjℏ, where mj can have any value between +j and −j in integer steps. An alternative description of the quantum state can be given in terms of the quantum numbers n, l, j, and mj.
The electron distribution of the atom is described as the square of the absolute value of the wave function. The probability of finding an electron at a given point in space for several of the lower energy states of the hydrogen atom is shown in Figure 5 . It is important to note that the electron density plots should not be thought of as the time-averaged locations of a well-localized (point) particle orbiting about the nucleus. Rather, quantum mechanics describes the electron with a continuous wave function in which the location of the electron should be considered as spread out in space in a quantum “fuzz ball” as depicted in Figure 5.