Max Born

Zur Gittertheorie der Ionenkristalle

ZusammenfassungDie Gittertheorie der Ionenkristalle wird durch drei Änderungen des Energieansatzes verschärft: Das Abstoßungspotential wird nicht als Potenz des Gitterabstandes, sondern als Exponentialfunktion angenommen. Dabei wird das Gesetz der Additivität der Ionenradien berücksichtigt. Endlich werden die van der Waalsschen Kohäsionskräfte mit in Rechnung gesetzt. Es wird gezeigt, daß hierdurch die verschiedene Stabilität der Gittertypen NaCl und CsCl verständlich gemacht werden kann.

Thermodynamics of Crystals and Melting

The Helmholtz free energy, A, of a rigid body is a function of temperature, and of the six homogeneous strain components. If the crystal is to be rigid, three inequalities must be satisfied for the derivatives of A with respect to the six strain components, for a regular (cubic) lattice. This enables one to limit the pressure‐temperature range for which the crystal is stable. The violation of the condition c44>0, that the crystal resist shearing, is interpreted as leading to melting. From a knowledge of the forces between the molecules the phase integral, and therefore the free energy, may be calculated as a function of T, V, and the six strain components. The numerical calculations are carried out for a body‐centered cubic lattice. The product of all the frequencies is calculated directly, so that the assumption that the Debye equation for the frequency distribution holds, is not necessary. The melting curve, pressure against temperature, is then determined.

Beweis des Adiabatensatzes

ZusammenfassungDer Adiabatensatz in der neuen Quantenmechanik wird für den Fall des Punktspektrums in mathematisch strenger Weise bewiesen, wobei er sich auch bei einer vorübergehenden Entartung des mechanischen Systems als gültig erweist.

On the stability of crystal lattices. I

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.

A General Kinetic Theory of Liquids. I. The Molecular Distribution Functions

This paper outlines a general theory whose object is to provide a basis from which all the equilibrium and dynamical properties of liquids can be investigated. A set of multiform distribution functions is defined, and the generalized continuity equations satisfied by these functions are derived. By introducing the equations of motion, a set of relations is obtained from which the distribution functions may be determined. It is shown that Boltzmann’s equation in the kinetic theory of gases follows as a particular case, and that, in equilibrium conditions, the theory gives results consistent with statistical mechanics. An integral equation for the radial distribution function is obtained which is the natural generalization of one obtained by Kirkwood for ‘rigid spherical molecules’. Finally, it is indicated how the theory may be applied to solve both equilibrium and dynamical problems of the liquid state.

Zur Quantenmechanik. II

Die aus Heisenbergs Ansatzen in Teil I dieser Arbeit entwickelte Quantenmechanik wird auf Systeme von beliebig vielen Freiheitsgraden ausgedehnt. Die Storungstheorie wird fur nicht entartete und eine grose Klasse entarteter Systeme durchgefuhrt und ihr Zusammenhang mit der Eigenwerttheorie Hermite scher Formen nachgewiesen. Die gewonnenen Resultate werden zur Ableitung der Satze uber Impuls und Drehimpuls und zur Ableitung von Auswahlregeln und Intensitatsformeln benutzt. Schlieslich werden die Ansatze der Theorie auf die Statistik der Eigenschwingungen eines Hohlraumes angewendet.

Optik : ein Lehrbuch der elektromagnetischen Lichttheorie

Dieses Buch unterscheidet sich von alteren Darstellungen der Optik durch die Grenzziehung gegen andere Gebiete der Physik. Die liberkommene Einteilung (Mechanik, Elektrizitat und Magnetismus, Optik, Thermodynamik, erganzt durch kinetische Theorie der Materie und Atomphysik) ist wohl vorlaufig flir den Unterricht noch unentbehrlich, so wenig sie auch der Einheit des Lehr gebaudes Rechnung tragt. Die Optik ist seit langem als elektromagnetische Lichttheorie ein Sonderkapitel der allgemeinen Lehre yom elektromagnetischen Felde. Man kann sich dabei natlirlich nicht auf das sichtbare Licht beschranken, sondern muB den Frequenzbereich nach oben und unten erweitern. Die HERTZ schen Wellen pflegt man aber nicht hinzuzunehmen; nach kurzen Wellen zu scheint es geboten, die Rontgen-und y-Strahlen auszuschlieBen oder wenigstens nur andeutungsweise zu behandeln. Auch hier wird diesem Brauche gefolgt. Die Optik bewegter Korper durfte frliher in einem Lehrbuche der Lichttheorie nicht fehlen. lch halte das nicht fUr zeitgemaB; diese Dinge gehoren zur Rela tivitatstheorie, die sich zu einem besonderen Kapitel der Physik entwickelt hat.

A Suggestion for Unifying Quantum Theory and Relativity

There seems to be a general conviction that the difficulties of our present theory of ultimate particles and nuclear phenomena (the infinite values of the self energy, the zero energy and other quantities) are connected with the problem of merging quantum theory and relativity into a consistent unit. Eddington’s book, “Relativity of the Proton and the Electron”, is an expression of this tendency; but his attempt to link the properties of the smallest particles to those of the whole universe contradicts strongly my physical intuition. Therefore I have considered the question whether there may exist other possibilities of unifying quantum theory and the principle of general invariance, which seems to me the essential thing, as gravitation by its order of magnitude is a molar effect and applies only to masses in bulk, not to the ultimate particles. I present here an idea which seems to be attractive by its simplicity and may lead to a satisfactory theory. 1. Reciprocity The Motion of a free particle in quantum theory is represented by a plane wave A exp[i/ℏ pkxk], where x1, x2, x3, x4 are the co-ordinates of space-time x, y, z, ct, and p1, p2, p3, p4 the components of momentum-energy px, py, pz, E. The expression is completely symmetric in the two 4-vectors x and p. The transformation theory of quantum mechanics extends this “reciprocity” systematically. In a representation of the operators xk, pk in the Hilbert space for which the xk are diagonal (δ-funcions), the pk are given by ℏ/i ∂/∂xk; and vice versa, if the pk are diagonal the xk are given by —ℏ/i ∂/∂pk. Any wave equation in the x-space can be transformed into another equation in the p-space, by help of the transformation φ(p) = ∫ψ(x) exp [i/ℏ pkxk]dx.

CHAPTER VIII – ELEMENTS OF THE THEORY OF DIFFRACTION

Introduction IN carrying out the transition from the general electromagnetic field to the optical field, which is characterized by very high frequencies (short wavelengths), we found that in certain regions the simple geometrical model of energy propagation was inadequate. In particular, we saw that deviations from this model must be expected in the immediate neighbourhood of the boundaries of shadows and in regions where a large number of rays meet. These deviations are manifested by the appearance of dark and bright bands, the diffraction fringes. Diffraction theory is mainly concerned with the field in these special regions; such regions are of great practical interest as they include the part of the image space in which the optical image is situated (region of focus). The first reference to diffraction phenomena appears in the work of Leonardo da Vinci (1452–1519). Such phenomena were, however, first accurately described by Grimaldi in a book, published in 1665, two years after his death. The corpuscular theory, which, at the time, was widely believed to describe correctly the propagation of light, could not explain diffraction. Huygens, the first proponent of the wave theory, seems to have been unaware of Grimaldis discoveries; otherwise he would have undoubtedly quoted them in support of his views. The possibility of explaining diffraction effects on the basis of a wave theory was not noticed until about 1818. In that year there appeared the celebrated memoir of Fresnel (see Historical introduction) in which he showed that diffraction can be explained by the application of Huygens’ construction (see §3.3.3) together with the principle of interference. Fresnels analysis was later put on a sound mathematical basis by Kirchhoff (1882), and the subject has since then been extensively discussed by many writers.

Das Adiabatenprinzip in der Quantenmechanik

ZusammenfassungAuf Grund der statistischen Auffassung der Quantenmechanik, die kürzlich an Hand der Stoßvorgänge entwickelt wurde, läßt sich der dem Ehrenfestschen Adiabatenprinzip analoge Satz formulieren und beweisen.