A model which I wish to show you was constructed to illustrate the motion of the electrons around the nucleus of the Rutherford atom, and the changing dimensions and varying plane of their orbit. More violent disturbance produces a series of secondary waves in each orbit, which divide the orbit in the manner suggested by the wave and quantum theories. The model is a very simple one, and my only excuse for bringing it to your notice is the pleasure which it has given to myself, and the fact that it has enabled me to realise as I could not previously do, Rutherford's theory of atomic structure. Rutherford's atom, which at present holds the field, consists essentially of a nucleus around which rotate a number of electrons. This number is the atomic number of the atom, so that with hydrogen there is one circulating electron, with helium there are two, and the number of electrons steadily rises with the atomic number until we reach uranium, which has ninety-two circulating electrons. Each of these electrons rotates independently, and its orbit is subject to very great variation. Indeed the motion of the electron is so complex that it is exceedingly difficult to form any concrete idea of what a variable orbit really means, or what perturbations in the orbit signify. It is with the idea of assisting the imagination that I have constructed this model, consisting of two prisms so placed that a horizontal beam of light is reflected downwards by the first prism on to the second, by which it is again reflected horizontally.
A model which I wish to show you was constructed to illustrate the motion of the electrons around the nucleus of the Rutherford atom, and the changing dimensions and varying plane of their orbit. More violent disturbance produces a series of secondary waves in each orbit, which divide the orbit in the manner suggested by the wave and quantum theories. The model is a very simple one, and my only excuse for bringing it to your notice is the pleasure which it has given to myself, and the fact that it has enabled me to realise as I could not previously do, Rutherford's theory of atomic structure. Rutherford's atom, which at present holds the field, consists essentially of a nucleus around which rotate a number of electrons. This number is the atomic number of the atom, so that with hydrogen there is one circulating electron, with helium there are two, and the number of electrons steadily rises with the atomic number until we reach uranium, which has ninety-two circulating electrons. Each of these electrons rotates independently, and its orbit is subject to very great variation. Indeed the motion of the electron is so complex that it is exceedingly difficult to form any concrete idea of what a variable orbit really means, or what perturbations in the orbit signify. It is with the idea of assisting the imagination that I have constructed this model, consisting of two prisms so placed that a horizontal beam of light is reflected downwards by the first prism on to the second, by which it is again reflected horizontally.
Abstract A theory of the helium atom was developed by Heisenberg, following his famous principle of “resonance.” His work is based upon Schrodinger’s equation, and in order to allow for the spin of the electrons he introduces extra energy terms, expressing the classical energy of two appropriate magnets. It is the perturbation by this spin energy which produces the triplet separations. Heisenberg’s calculation of these separations is open to three criticisms: (1) He assumes that certain simple forms are correct first approximations to the wave-functions; whereas, owing to degeneracy, it is necessary to take linear combinations of these forms. (2) He estimates the mean values of the spin energy by means of a model built up of precessing vectors. (3) He neglects the radius of the inner orbit in comparison with that of the outer orbit, even when the principal quantum number of the latter is only 2; this is the least satisfactory of several necessary simplifications. Schrödinger’s equation is likely to be supplanted as the foundation of wave-mechanics by the equation recently put forward by Dirae. This fits the spin of the electron neatly into the theory of relativity, and produces the doublets of the hydrogen-like atom in a beautiful manner. Dirac’s q-number theory has been translated into wave-mechanics by Darwin. Apart altogether from the above criticisms of Heisenberg, it seemed expedient to proceed to the theory of an atom with two electrons on the basis of the new equation. Calculations which are independent of the spin, such as the approximate energylevels, and the separation between ortho- and para-terms, are the same on either theory, and are not the concern of this paper. We deal here with spin effects, such as the fine structure of the triplets and intercombinations between ortho- and para-states.
The description of the physical and chemical properties of matter is a central issue that has occupied the minds of scientists since the age of the ancient Greeks. In their route to dissect matter down to what cannot be divided any further, they coined the term atom , the indivisible . Matter became then a collection of atoms. More than twenty centuries had to pass until the development of a more precise concept of atom, thanks, amongst others, to the systematic studies of Mendeleyev and the establishment in 1869 of the periodic table of the elements (Mendeleyev, 1869). The discovery of the electron in 1897 and the first modern model of the atomic structure by Sir Joseph Thomson were soon refined by his student, Sir Ernest Rutherford, who in 1910 showed that an atom was made of a positively charged small nucleus and a number of negatively charged electrons that neutralize the nuclear charge. Much in the spirit of planetary systems, and drawing from the analogy between gravitational and electrostatic interactions, scientists in the beginning of the twentieth century built an image of the atom that consisted of a number Z of electrons – of elementary charge – e – orbiting around the nucleus of charge Ze . A number of experimental observations, though, were incompatible with this idea of orbiting electrons. In particular, according to the successful electromagnetic theory, charged electrons in orbital (radially accelerated) motion should radiate energy, thus decelerating and eventually collapsing onto the nucleus. Clearly, such a picture would imply that matter is essentially unstable, in flagrant contradiction with our everyday experience of the very existence of matter.
In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons. Rutherford subsequently found in 1911 that the electrons orbited at some distance from a central positively charged nucleus. Rutherford proposed that electrostatic attraction between the nucleus and the electron was balanced by the centrifugal force arising from the orbital motion. However, if this were the case, then the electrons (being accelerated inwards towards the center of rotation) would continuously radiate all their energy as electromagnetic waves and very quickly fall into the nucleus. In 1913, Bohr postulated two important additions to Rutherford’s theory of atomic structure: 1. Electrons can orbit the nucleus in what are called stationary states in which no emission of radiation occurs and in which the angular momentum L is constrained to have values: 2. Electrons can make transitions from one state to another accompanied by the emission or absorption of a single photon of energy E = hf thus leading to absorption and emission spectra.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The original intention of the author’s preoccupation with the quantum-mechanical behaviour of simple atoms and molecules such as Hydrogen and Helium was, on the one hand, the elegant simplicity of Niels Bohr’s atom model for Hydrogen, describing the metastable states of the excited electrons by planar concentric electron orbits, and, on the other hand, the hardly intelligible wave mechanical approach of Heisenberg, Schrödinger and others, describing mainly atoms with multiple electrons by three-dimensional orbitals which were characterized by probabilities of presence. Thereby the question arose whether it would be possible to find alternative atom models with well-defined electron trajectories. Therein, Louis de Broglie’s thesis of the wavy nature of electron motion implicating standing waves would have to be implemented. Nevertheless, as reviewed in the introduction, the orthodox three-dimensional concept influenced the own thinking in such a way that three-dimensional constellations for the electronic excited states were conceived. The break-through was achieved for the electronic ground state in the form of the spin-orbit coupling where the spin acts as a perpetuum mobile, inducing the orbital angular momentum h/2π. Furthermore, the insight was gained that a circularly rotating electron intrinsically corresponds to a harmonic oscillator, thus fulfilling the condition of a standing wave. Based on this concept, a double planar model was established for the H2-molecule which could be empirically verified by X-ray data from literature. However, for the two electrons containing Helium a 2D-array seemed impossible since the Pauli-principle seemed to be violated. After a long stepwise succession of 3D-attempts which turned out to be impossible—not least since eccentric forces are not possible in such a system—the here presented 2D-version for Helium was found, composed by two imaginary orthogonal electron orbits. It will enable in a subsequent publication the quantum mechanical interpretation of the thermal-radiative behaviour of Helium which was reported in the author’s publication nine years ago.