The Dreams That Stuff is Made of

by Stephen Hawking

  Another problem with the planetary model of the atom was that the electrons’ orbit around the nucleus should be unstable. We know from the theory of electrodynamics that an orbiting charged particle will radiate electromagnetic waves, causing it to lose energy and spiral into the nucleus. But atoms are stable—the electrons in an atom do not do this. Why not? In 1913 Niels Bohr tackled this question in his paper, “On the Constitution of Atoms and Molecules.” His answer was the next important step in the development of quantum mechanics. He simply postulated that only discrete distances from the nucleus were allowed for an electron orbiting a nucleus. In other words, he assumed the radius of orbit (and equivalently, the energy) of electrons in atoms to be quantized. It is as if the allowed orbits are steps in a staircase. The electron can be on the third step or the fourth step, for example, but it cannot be in between. Electrons can jump between levels but cannot spiral inward because that would involve being in an intermediate space between levels. With this assumption, Bohr’s model could also explain the atomic spectrum of hydrogen. When an electron jumps from a higher to lower energy level it emits electromagnetic radiation with an energy amount given by the difference in energies of the two levels. Since the energy levels are discrete transitions, atomic spectra have distinct, sharp lines.

  Bohr’s model was immediately recognized as revolutionary, and it won him the 1922 Nobel prize. However, the Bohr model has a number of shortcomings. It only works for atoms with a single electron, and even for single-electron atoms it cannot explain the fine structure of atomic spectra, nor does it provide an explanation of why the electron energy levels should be quantized. It was not until the 1920s, when a more complete theory of quantum mechanics was developed, that we understood why we needed to assume energy level quantization.

  THE SCATTERING OF α AND β PARTICLES BY MATTER AND THE STRUCTURE OF THE ATOM

  BY

  ERNEST RUTHERFORDae

  First appeared in Philosophical Magazine, Series 6, vol. 21, May 1911, p. 669–688

  § 1. It is well known that the α and the β particles suffer deflexions from their rectilinear paths by encounters with atoms of matter. This scattering is far more marked for the β than for the α particle on account of the much smaller momentum and energy of the former particle. There seems to be no doubt that such swiftly moving particles pass through the atoms in their path, and that the deflexions observed are due to the strong electric field traversed within the atomic system. It has generally been supposed that the scattering of a pencil of α or β rays in passing through a thin plate of matter is the result of a multitude of small scatterings by the atoms of matter traversed. The observations, however, of Geiger and Marsdenaf on the scattering of α rays indicate that some of the α particles, about 1 in 20,000 were turned through an average angle of 90 degrees in passing though a layer of gold-foil about 0.00004 cm. thick, which was equivalent in stopping-power of the α particle to 1.6 millimetres of air. Geigerag showed later that the most probable angle of deflexion for a pencil of α particles being deflected through 90 degrees is vanishingly small. In addition, it will be seen later that the distribution of the α particles for various angles of large deflexion does not follow the probability law to be expected if such large deflexion are made up of a large number of small deviations. It seems reasonable to suppose that the deflexion through a large angle is due to a single atomic encounter, for the chance of a second encounter of a kind to produce a large deflexion must in most cases be exceedingly small. A simple calculation shows that the atom must be a seat of an intense electric field in order to produce such a large deflexion at a single encounter.

  Recently Sir J.J. Thomsonah has put forward a theory to explain the scattering of electrified particles in passing through small thicknesses of matter. The atom is supposed to consist of a number N of negatively charged corpuscles, accompanied by an equal quantity of positive electricity uniformly distributed throughout a sphere. The deflexion of a negatively electrified particle in passing through the atom is ascribed to two causes–(1) the repulsion of the corpuscles distributed through the atom, and (2) the attraction of the positive electricity in the atom. The deflexion of the particle in passing through the atom is supposed to be small, while the average deflexion after a large number m of encounters was taken as [the square root of] m · θ, where θ is the average deflexion due to a single atom. It was shown that the number N of the electrons within the atom could be deduced from observations of the scattering was examined experimentally by Crowtherai in a later paper. His results apparently confirmed the main conclusions of the theory, and he deduced, on the assumption that the positive electricity was continuous, that the number of electrons in an atom was about three times its atomic weight.

  The theory of Sir J. J. Thomson is based on the assumption that the scattering due to a single atomic encounter is small, and the particular structure assumed for the atom does not admit of a very large deflexion of diameter of the sphere of positive electricity is minute compared with the diameter of the sphere of influence of the atom.

  Since the α and β particles traverse the atom, it should be possible from a close study of the nature of the deflexion to form some idea of the constitution of the atom to produce the effects observed. In fact, the scattering of high-speed charged particles by the atoms of matter is one of the most promising methods of attack of this problem. The development of the scintillation method of counting single α particles affords unusual advantages of investigation, and the researches of H. Geiger by this method have already added much to our knowledge of the scattering of α rays by matter.

  § 2. We shall first examine theoretically the single encountersaj with an atom of simple structure, which is able to produce large deflections of an α particle, and then compare the deductions from the theory with the experimental data available.

  Consider an atom which contains a charge ±Ne at its centre surrounded by a sphere of electrification containing a charge ±Ne [N.B. in the original publication, the second plus/minus sign is inverted to be a minus/plus sign] supposed uniformly distributed throughout a sphere of radius R. e is the fundamental unit of charge, which in this paper is taken as 4.65 × 10−10 E.S. unit. We shall suppose that for distances less than 10−12 cm. the central charge and also the charge on the alpha particle may be supposed to be concentrated at a point. It will be shown that the main deductions from the theory are independent of whether the central charge is supposed to be positive or negative. For convenience, the sign will be assumed to be positive. The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged parts.

  In order to form some idea of the forces required to deflect an alpha particle through a large angle, consider an atom containing a positive charge Ne at its centre, and surrounded by a distribution of negative electricity Ne uniformly distributed within a sphere of radius R. The electric force X and the potential V at a distance r from the centre of an atom for a point inside the atom, are given by

  Suppose an α particle of mass m and velocity u and charge E shot directly towards the centre of the atom. It will be brought to rest at a distance b from the centre given by

  It will be seen that b is an important quantity in later calculations. Assuming that the central charge is 100 e, it can be calculated that the value of b for an α particle of velocity 2.09 × 109 cms. per second is about 3.4 × 10−12 cm. In this calculation b is supposed to be very small compared with R. Since R is supposed to be of the order of the radius of the atom, viz. 10−8 cm., it is obvious that the α particle before being turned back penetrates so close to the central charge, that the field due to the uniform distribution of negative electricity may be neglected. In general, a simple calculation shows that for all deflexions greater than a degree, we may without sensible error suppose the deflexion due to the field of the central charge alone. Possible single deviations due to the
negative electricity, if distributed in the form of corpuscles, are not taken into account at this stage of the theory. It will be shown later that its effect is in general small compared with that due to the central field.

  Consider the passage of a positive electrified particle close to the centre of an atom. Supposing that the velocity of the particle is not appreciably changed by its passage through the atom, the path of the particle under the influence of a repulsive force varying inversely as the square of the distance will be an hyperbola with the centre of the atom S as the external focus. Suppose the particle to enter the atom in the direction PO (Fig. 1), and that the direction of motion on escaping the atom is OPʹ. OP and OPʹ make equal angles with the line SA, where A is the apse of the hyperbola. p = SN = perpendicular distance from centre on direction of initial motion of particle.

  FIG. 1

  Let angle POA = θ .

  Let V = velocity of particle on entering the atom, v its velocity at A, then from consideration of angular momentumpV = SA · ν.

  From conservation of energy

  Since the eccentricity is sec θ ,

  therefore b = 2 p cot θ.

  The angle of deviation θ of the particles in π − 2θ and

  ak(1)

  This gives the angle of deviation of the particle in terms of b, and the perpendicular distance of the direction of projection from the centre of the atom.

  For illustration, the angle of deviation φ for different values of p/b are shown in the following table:—

  § 3. PROBABILITY OF SINGLE DEFLEXION THROUGH ANY ANGLE

  Suppose a pencil of electrified particles to fall normally on a thin screen of matter of thickness t. With the exception of the few particles which are scattered through a large angle, the particles are supposed to pass nearly normally through the plate with only a small change of velocity. Let n = number of atoms in unit volume of material. Then the number of collisions of the particle with the atom of radius R is π R2nt in the thickness t.

  The probability m of entering an atom within a distance p of its center is given by

  Chance dm of striking within radii p and p + dp is given by

  (2)

  since

  The value of dm gives the fraction of the total number of particles which are deviated between the angles φ and φ + dφ.

  The fraction p of the total number of particles which are deflected through an angle greater than φ is given by

  (3)

  The fraction p which is deflected between the angles φ1 and φ2 is given by

  (4)

  It is convenient to express the equation (2) in another form for comparison with experiment. In the case of the α rays, the number of scintillations appearing on the constant area of the zinc sulphide screen are counted for different angles with the direction of incidence of the particles. Let r = distance from point of incidence of α rays on scattering material, then if Q be the total number of particles falling on the scattering material, the number y of α particles falling on unit area which are deflected through an angle φ is given by

  (5)

  Since b = 2NeE/mu2, we see from this equation that the number of α particles (scintillations) per unit area of zinc sulphide screen at a given distance r from the point of Incidence of the rays is proportional to(1) cosec4 φ/2 or 1/φ4 if φ be small;

  (2) thickness of scattering material t provided this is small;

  (3) magnitude of central charge Ne;

  (4) and is inversely proportional to (mu2)2, or to the fourth power of the velocity if m be constant.

  In these calculations, it is assumed that the α particles scattered through a large angle suffer only one large deflexion. For this to hold, it is essential that the thickness of the scattering material should be so small that the chance of a second encounter involving another large deflexion is very small. If, for example, the probability of a single deflexion φ in passing through a thickness t is 1/1000, the probability of two successive deflexions each of value φ is 1/106, and is negligibly small.

  The angular distribution of the α particles scattered from a thin metal sheet affords one of the simplest methods of testing the general correctness of this theory of single scattering. This has been done recently for α rays by Dr. Geigeral, who found that the distribution for particles deflected between 30° and 150° from a thin gold-foil was in substantial agreement with the theory. A more detailed account of these and other experiments to test the validity of the theory will be published later.

  § 4. ALTERATION OF VELOCITY IN AN ATOMIC ENCOUNTER

  It has so far been assumed that an α or β particle does not suffer an appreciable change of velocity as the result of a single atomic encounter resulting in a large deflexion of the particle. The effect of such an encounter in altering the velocity of the particle can be calculated on certain assumptions. It is supposed that only two systems are involved, viz., the swiftly moving particle and the atom which it traverses supposed initially at rest. It is supposed that the principle of conservation of momentum and of energy applies, and that there is no appreciable loss of energy or momentum by radiation.

  Let m be mass of the particle,

  ν1 = velocity of approach,

  ν2 = velocity of recession,

  M = mass of atom,

  V = velocity communicated to atom as result of encounter.

  Let OA (Fig. 2) represent in magnitude and direction the momentum mν1 of the entering particle, and OB the momentum of the receding particle which has been turned through an angle AOB = φ. Then BA represents in magnitude and direction the momentum MV of the recoiling atom.

  (1)

  By conservation of energy

  (2)

  FIG. 2

  Suppose M/m = K and ν2 = pν1, where p < 1. From (1) and (2).(K + 1) ρ2 − 2ρ cosφ = k − 1,

  or

  Consider the case of an α particle of atomic weight 4, deflected through an angle of 90° by an encounter with an atom of gold of atomic weight 197.

  Since K = 49 nearly,

  or the velocity of the particle is reduced only about 2 per cent. by the encounter.

  In the case of aluminium K = 27/4 and for φ = 90° p = 0.86.

  It is seen that the reduction of velocity of the α particle becomes marked on this theory for encounters with the lighter atoms. Since the range of an α particle in air or other matter is approximately proportional to the cube of the velocity, it follows that an α particle of range 7 cms. has its range reduced to 4.5 cms. after incurring a single deviation of 90° in traversing an aluminium atom. This is of a magnitude to be easily detected experimentally. Since the value of K is very large for an encounter of a β particle with an atom, the reduction of velocity on this formula is very small.

  Some very interesting cases of the theory arise in considering the changes of velocity and the distribution of scattered particles when the α particle encounters a light atom, for example a hydrogen or helium atom. A discussion of these and similar cases is reserved until the question has been examined experimentally.

  § 5. COMPARISON OF SINGLE AND COMPOUND SCATTERING

  Before comparing the results of theory with experiment, it is desirable to consider the relative importance of single and compound scattering in determining the distribution of the scattered particles. Since the atom is supposed to consist of a central charge surrounded by a uniform distribution of the opposite sign through a sphere of radius R, the chance of encounters with the atom involving small deflexions is very great compared with the change of a single large deflexion.

  This question of compound scattering has been examined by Sir J. J. Thomson in the paper previously discussed (§ 1). In the notation of this paper, the average deflexion φ1 due to the field of the sphere of positive electricity of radius R and quantity Ne was found by him to be

  The average deflexion φ2 due to the N negative corpuscles supposed distributed uniformly throughout the sphere was found to be

  The mean deflexion due to both po
sitive and negative electricity was taken as

  In a similar way, it is not difficult to calculate the average deflexion due to the atom with a central charge discussed in this paper.

  Since the radial electric field X at any distance r from the centre is given by