The Dreams That Stuff is Made of

by Stephen Hawking

  it is not difficult to show that the deflexion (supposed small) of an electrified particle due to this field is given by

  Where p is the perpendicular from the center on the path of the particles and b has the same value as before. It is seen that the value of θ increases with diminution of p and becomes great for small value of φ.

  Since we have already seen that the deflexions become very large for a particle passing near the center of the atom, it is obviously not correct to find the average value by assuming θ is small.

  Taking R of the order 10−8 cm., the value of p for a large deflexions is for α and β particles of the order 10−11 cm. Since the chance of an encounter involving a large deflexion is small compared with the chance of small deflexions, a simple consideration shows that the average small deflexion is practically unaltered if the large deflexions are omitted. This is equivalent to integrating over that part of the cross section of the atom where the deflexions are small and neglecting the small central area. It can in this way be simply shown that the average small deflexion is given by

  This value of φ1 for the atom with a concentrated central charge is three times the magnitude of the average deflexion for the same value of Ne in the type of atom examined by Sir J. J. Thomson. Combining the deflexions due to the electric field and to the corpuscles, the average deflexion is

  It will be seen later that the value of N is nearly proportional to the atomic weight, and is about 100 for gold. The effect due to scattering of the individual corpuscles expressed by the second term of the equation is consequently small for heavy atoms compared with that due to the distributed electric field.

  Neglecting the second term, the average deflexion per atom is 3πb/8R. We are now in a position to consider the relative effects on the distribution of particles due to single and to compound scattering. Following J. J. Thomson’s argument, the average deflexion θ after passing through a thickness t of matter is proportional to the square root of the number of encounters and is given by

  where n as before is equal to the number of atoms per unit volume.

  The probability p1 for compound scattering that the deflexion of the particle is greater than φ is equal to

  Consequently

  Next suppose that single scattering alone is operative. We have seen (§ 3) that the probability p2 of a deflexion greater than φ is given by

  By comparing these two equations

  φ is sufficiently small that

  If we suppose thatp2 = 0.5, then p1 = 0.24

  Ifp2 = 0.1, then p1 = 0.0004

  It is evident from this comparison, that the probability for any given deflexion is always greater for single than for compound scattering. The difference is especially marked when only a small fraction of the particles are scattered through any given angle. It follows from this result that the distribution of particles due to encounters with the atoms is for small thicknesses mainly governed by single scattering. No doubt compound scattering produces some effect in equalizing the distribution of the scattered particles; but its effect becomes relatively smaller, the smaller the fraction of the particles scattered through a given angle.

  §6. COMPARISON OF THEORY WITH EXPERIMENTS

  On the present theory, the value of the central charge Ne is an important constant, and it is desirable to determine its value for different atoms. This can be most simply done by determining the small fraction of α or β particles of known velocity falling on a thin metal screen, which are scattered between φ and φ + dφ where φ is the angle of deflexion, The influence of compound scattering should be small when this fraction is small.

  Experiments in these directions are in progress, but it is desirable at this stage to discuss in the light of the present theory the data already published on scattering of α and β particles,

  The following points will be discussed:—(a) The ‘diffuse reflexion’ of α particles, i.e. the scattering of α particles through large angles (Geiger and Marsden.)

  (b) The variation of diffuse reflexion with atomic weight of the radiator (Geiger and Marsden.)

  (c) The average scattering of a pencil of α rays transmitted through a thin metal plate (Geiger.)

  (d) The experiments of Crowther on the scattering of β rays of different velocities by various metals.

  (a) In the paper of Geiger and Marsden (loc. cit.) on the diffuse reflexion of α particles falling on various substances it was shown that about 1/8000 of the α particles from radium C falling on a thick plate of platinum are scattered back in the direction of the incidence. This fraction is deduced on the assumption that the α particles are uniformly scattered in all directions, the observation being made for a deflexion of about 90°. The form of experiment is not very suited for accurate calculation, but from the data available it can be shown that the scattering observed is about that to be expected on the theory if the atom of platinum has a central charge of about 100 e.

  In their experiments on this subject, Geiger and Marsden gave the relative number of α particles diffusely reflected from thick layers of different metals, under similar conditions. The numbers obtained by them are given in the table below, where z represents the relative number of scattered particles, measured by the of scintillations per minute on a zinc sulphide screen.

  On the theory of single scattering, the fraction of the total number of α particles scattered through any given angle in passing through a thickness t is proportional to n · A2t, assuming that the central charge is proportional to the atomic weight A. In the present case, the thickness of matter from which the scattered α particles are able to emerge and affect the zinc sulphide screen depends on the metal. Since Bragg has shown that the stopping power of an atom for an α particle is proportional to the square root of its atomic weight, the value of nt for different elements is proportional to 1/[square root of] A. In this case t represents the greatest depth from which the scattered α particles emerge. The number z of α particles scattered back from a thick layer is consequently proportional to A3/2 or z/A3/2 should be a constant.

  To compare this deduction with experiment, the relative values of the latter quotient are given in the last column. Considering the difficulty of the experiments, the agreement between theory and experiment is reasonably good.am

  The single large scattering of α particles will obviously affect to some extent the shape of the Bragg ionization curve for a pencil of α rays. This effect of large scattering should be marked when the α rays have traversed screens of metals of high atomic weight, but should be small for atoms of light atomic weight.

  (c) Geiger made a careful determination of the scattering of α particles passing through thin metal foils, by the scintillation method, and deduced the most probable angle through which the α particles are deflected in passing through known thickness of different kinds of matter.

  A narrow pencil of homogeneous α rays was used as a source. After passing through the scattering foil, the total number of α particles are deflected through different angles was directly measured. The angle for which the number of scattered particles was a maximum was taken as the most probable angle. The variation of the most probable angle with thickness of matter was determined, but calculation from these data is somewhat complicated by the variation of velocity of the α particles in their passage through the scattering material. A consideration of the curve of distribution of the α particles given in the paper (loc.cit. p. 498) shows that the angle through which half the particles are scattered is about 20 per cent greater than the most probable angle.

  We have already seen that compound scattering may become important when about half the particles are scattered through a given angle, and it is difficult to disentangle in such cases the relative effects due to the two kinds of scattering. An approximate estimate can be made in the following ways:—From (§ 5) the relation between the probabilities p1 and p2 for compound and single scattering respectively is given byp2 log p1 = −0.721.

  The probability q of the combined eff
ects may as a first approximation be taken as

  If q = 0.5, it follows that p1 = 0.2 and p2 = 0.46p1 = 0.2 and P2 = 0.46

  We have seen that the probability P2 of a single deflexion greater than φ is given by

  Since in the experiments considered φ is comparatively small

  Geiger found that the most probable angle of scattering of the α rays in passing through a thickness of gold equivalent in stopping power to about 0.76 cm. of air was 1° 40’. The angle φ through which half the α particles are tuned thus corresponds to 2° nearly.

  Taking the probability of single scattering = 0.46 and substituting the above value in the formula, the value of N for gold comes out to be 97.

  For a thickness of gold equivalent in stopping power to 2.12 cms, of air, Geiger found the most probable angle to be 3° 40’. In this case, t = 0.00047, φ = 4°.4, and average u = 1.7 × 109, and N comes out to be 114.

  Geiger showed that the most probable angle of deflexion for an atom was nearly proportional to its atomic weight. It consequently follows that the value for N for different atoms should be nearly proportional to their atomic weights, at any rate for atomic weights between gold and aluminum.

  Since the atomic weight of platinum is nearly equal to that of gold, it follows from these considerations that the magnitude of the diffuse reflexion of α particles through more than 90° from gold and the magnitude of the average small angle scattering of a pencil of rays in passing through gold-foil are both explained on the hypothesis of single scattering by supposing the atom of gold has a central charge of about 100 e.

  (d) Experiments of a Crowther on scattering of α rays.—We shall now consider how far the experimental results of Crowther on scattering of β particles of different velocities by various materials can be explained on the general theory of single scattering. On this theory, the fraction of β particles p turned through an angel greater than φ is given byp = (π/4) n · t · b2 (cot2 φ/2).

  In most of Crowther’s experiments φ is sufficiently small that tan φ/2 may be put equal to φ/2 without much error. Consequently

  On the theory of compound scattering, we have already seen that the chance p1 that the deflexion of the particles is greater than φ is given by

  Since in the experiments of Crowther the thickness t of matter was determined for which p1 = 1/2,

  For the probability of 1/2, the theories of single and compound scattering are thus identical in general form, but differ by a numerical constant. It is thus clear that the main relations on the theory of compound scattering of Sir J. J. Thomson, which were verified experimentally by Crowther, hold equally well on the theory of single scattering.

  For example, it tm be the thickness for which half the particles are scattered through an angle φ, Crowther showed that φ/[square root of] tm and also mu2/E times [square root of] tm were constants for a given material when φ was fixed. These relations hold also on the theory of single scattering. Notwithstanding this apparent similarity in form, the two theories are fundamentally different. In one case, the effects observed are due to cumulative effects of small deflexion, while in the other the large deflexions are supposed to result from a single encounter. The distribution of scattered particles is entirely different on the two theories when the probability of deflexion greater than φ is small.

  We have already seen that the distribution of scattered α particles at various angles has been found by Geiger to be in substantial agreement with the theory of single scattering, but can not be explained on the theory of compound scattering alone. Since there is every reason to believe that the laws of scattering of α and β particles are very similar, the law of distribution of scattered β particles should be the same as for α particles for small thicknesses of matter. Since the value of mu2/E for β particles is in most cases much smaller than the corresponding value for the α particles, the chance of large single deflexions for β particles in passing through a given thickness of matter is much greater than for α particles. Since on the theory of single scattering the fraction of the number of particles which are undeflected through this angle is proportional to kt, where t is the thickness supposed small and k a constant, the number of particles which are undeflected through this angle is proportional to 1 − kt. From considerations based on the theory of compound scattering, Sir J.J. Thomson deduced that the probability of deflexion less than Φ is proportional to 1 − eµ/t is where µ is a constant for any given value of φ.

  The correctness of this latter formula was tested by Crowther by measuring electrically the fraction I/Io of the scattered β particles which passed through a circular opening subtending an angle of 36° with the scattering material. IfI/I0 = 1 − 1 − eµ/t,

  the value of I should decrease very slowly at first with increase of t. Crowther, using aluminium as scattering material, states that the variation of I/Io was in good accord with this theory for small values of t. On the other hand, if single scattering be present, as it undoubtedly is for α rays, the curve showing the relation between I/Io and t should be nearly linear in the initial stages. The experiments of Marsdenan on scattering of β rays, although not made with quite so small a thickness of aluminium as that used by Crowther, certainly support such a conclusion. Considering the importance of the point at issue, further experiments on this question are desirable.

  From the table given by Crowther of the value φ/[square root of] tm for different elements for β rays of velocity 2.68 × 10−10 cms. per second, the value of the central charge Ne can be calculated on the theory of single scattering. It is supposed, as in the case of the α rays, that for given value of φ/[square root of] tm the fraction of the β particles deflected by single scattering through an angle greater than φ is 0.46 instead of 0.5

  The value of N calculated from Crowther’s data are given below.

  It will be remembered that the values of N for gold deduced from scattering of the α rays were in two calculations 97 and 114. These numbers are somewhat smaller than the values given above for platinum (viz. 138), whose atomic weight is not very different from gold. Taking into account the uncertainties involved in the calculation from the experimental data, the agreement is sufficiently close to indicate that the same general laws of scattering hold for the α and β particles, notwithstanding the wide differences in the relative velocity and mass of these particles.

  As in case of the α rays, the value of N should be most simply determined for any given element by measuring the small fraction of the incident β particles scattered through a large angle. In this way, possible errors due to small scattering will be avoided.

  The scattering data for the β rays, as well as for the α rays indicate that the central charge in an atom is approximately proportional to its atomic weight. This falls in with the experimental deductions of Schmidt.ao In his theory of absorption of β rays, he supposed that in traversing a thin sheet of matter, a small fraction α of the particles are stopped, and a small fraction β are reflected or scattered back in the direction of incidence. From comparison of the absorption curves of different elements, he deduced that the value of the constant β for different elements is proportional to nA2 where n is the number of atoms per unit volume and A the atomic weight of the element. This is exactly the relation to be expected on the theory of single scattering if the central charge on an atom is proportional to its atomic weight.

  §7. GENERAL CONSIDERATIONS

  In comparing the theory outlined in this paper with the experimental results, it has been supposed that the atom consists of a central charge supposed concentrated at a point, and that the large single deflexions of the α and β particles are mainly due to their passage through the strong central field. The effect of the equal and opposite compensation charge supposed distributed uniformly throughout a sphere has been neglected. Some of the evidence in support of these assumptions will now be briefly considered. For concreteness, consider the passage of a high speed α particle through an atom having a positive central charge Ne, and surrounded
by a compensating charge of N electrons. Remembering that the mass, momentum, and kinetic energy of the α particle are very large compared with the corresponding values of an electron in rapid motion, it does not seem possible from dynamic considerations that an α particle can be deflected through a large angle by a close approach to an electron, even if the latter be in rapid motion and constrained by strong electrical forces. It seems reasonable to suppose that the chance of single deflexions through a large angle due to this cause, if not zero, must be exceedingly small compared with that due to the central charge.

  It is of interest to examine how far the experimental evidence throws light on the question of extent of the distribution of central charge. Suppose, for example, the central charge to be composed of N unit charges distributed over such a volume that the large single deflexions are mainly due to the constituent charges and not to the external field produced by the distribution. It has been shown (§ 3) that the fraction of the α particles scattered through a large angle is proportional to (NeE)2, where Ne is the central charge concentrated at a point and E the charge on the deflected particles, If, however, this charge is distributed in single units, the fraction of the α particles scattered through a given angle is proportional of Ne2 instead of N2e2. In this calculation, the influence of mass of the constituent particle has been neglected, and account has only been taken of its electric field. Since it has been shown that the value of the central point charge for gold must be about 100, the value of the distributed charge required to produce the same proportion of single deflexions through a large angle should be at least 10,000. Under these conditions the mass of the constituent particle would be small compared with that of the α particle, and the difficulty arises of the production of large single deflexions at all. In addition, with such a large distributed charge, the effect of compound scattering is relatively more important than that of single scattering. For example, the probable small angle of deflexion of pencil of α particles passing through a thin gold foil would be much greater than that experimentally observed by Geiger (§ b–c). The large and small angle scattering could not then be explained by the assumption of a central charge of the same value. Considering the evidence as a whole, it seems simplest to suppose that the atom contains a central charge distributed through a very small volume, and that the large single deflexions are due to the central charge as a whole, and not to its constituents. At the same time, the experimental evidence is not precise enough to negative the possibility that a small fraction of the positive charge may be carried by satellites extending some distance from the centre. Evidence on this point could be obtained by examining whether the same central charge is required to explain the large single deflexions of α and β particles; for the α particle must approach much closer to the center of the atom than the β particle of average speed to suffer the same large deflexion.