Atomphysik
Klassische Atommodelle
Originalarbeit von RUTHERFORD
 Welche Vorstellungen hatten die alten Griechen von Atomen?
 Was versteht man unter dem „PlumpuddingModell“?
 Welche Vorhersagen macht das BOHRsche Atommodell?
 Mit welchen Atommodellen arbeitet die moderne Physik?
Originalarbeit von RUTHERFORD
Für historisch und in der Theorie interessierte Lehrer und Schüler bieten wir die Originalarbeit von Rutherford zum lesen an. Dabei sollen Sie einen Eindruck bekommen, wie eine solche Originalarbeit, die 1911 in einer entsprechenden naturwissenschaftlichen Zeitschrift abgedruckt wurde, abgefasst ist.
Auch wenn mancher theoretische Schritt in der Arbeit für Sie nicht nachvollziehbar ist, können Sie an geeigneten Textstellen diejenigen Aussagen finden, die heute noch in den Schulbüchern (in einfacherer Form) wieder zu finden sind.
The Scattering of α and β Particles by Matter and the Structure of the Atom
E. Rutherford, F.R.S.*Philosophical Magazine
Series 6, vol. 21
May 1911, p. 669688
669 § 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 Marsden** 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 goldfoil about 0.00004 cm. thick, which was equivalent in stoppingpower of the α particle to 1.6 millimetres of air. Geiger*** 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.
* Communicated by the Author. A brief account of this paper was communicated to the Manchester Literary and Philosophical Society in February, 1911. 
670 Recently Sir J. J. Thomson**** 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 Crowther* 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.
* Crowther, Proc. Roy. Soc. lxxxiv. p. 226 (1910)


671 § 2. We shall first examine theoretically the single encounters** 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 \[\begin{array}{l}X = Ne\left( {\frac{1}{{{r^2}}}  \frac{r}{{{R^3}}}} \right)\\V = Ne\left( {\frac{1}{r}  \frac{3}{{2R}}  \frac{{{r^2}}}{{2{R^3}}}} \right)\end{array}\] 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 \[{\textstyle{1 \over 2}}m{u^2} = NeE\left( {\frac{1}{b}  \frac{3}{{2R}} + \frac{{{b^2}}}{{2{R^3}}}} \right)\] 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 2,09·10^{9} 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 
672 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 OP'.
OP and OP' 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. 

673
Let angle POA = θ.
Let V = velocity of particle on entering the atom, v its velocity at A, then from consideration of angular momentum pV = SA·v. From conservation of energy \[\begin{array}{l}{\textstyle{1 \over 2}}m{V^2} = {\textstyle{1 \over 2}}m{v^2}  \frac{{NeE}}{{SA}},\\{v^2} = {V^2}\left( {1  \frac{b}{{SA}}} \right)\end{array}\]
Since the eccentricity is sec q, SA = SO + OA = p cosecθ·(1 + cosθ) = p cotθ/2 p^{2} = SA(SA  b) = p cot θ/2(p cot θ/2  b), therefore b = 2·p·cot θ. The angle of deviation \(\phi \)of the particles is π  2θ and cot\(\phi \)/2 = (2p/b) * . . . . (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 \(\phi \) 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·R^{2}·n·t in the thickness t. * A simple consideration shows that the deflexion is unaltered if the forces are attractive instead of repulsive. 
674
The probability m of entering an atom within a distance p of its center is given by m = π·p^{2}·n·t. Chance dm of striking within radii p and p + dp is given by dm = 2π·p·n·t . dp = (π/ 4)·n·t·b^{2} cot \(\phi \)/2 cosec^{2} \(\phi \)/2 d\(\phi \) . . . . (2) since cot \(\phi \)/2 = 2p/b The value of dm gives the fraction of the total number of particles which are deviated between the angles \(\phi \) and \(\phi \) + d\(\phi \). The fraction ρ of the total number of particles which are deflected through an angle greater than\(\phi \) is given by ρ = (π/ 4)·n·t·b^{2}·cot^{2} \(\phi \)/2 . . . . . . (3) The fraction ρ which is deflected between the angles \(\phi \)_{1} and \(\phi \)_{2} is given by ρ = (π/ 4)·n·t·b^{2} ·(cot^{2} \(\phi \)_{1}/2  cot^{2} \(\phi \)_{2}/2) . . . . . . . . . . . . . (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 a 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\(\phi \) is given by \[y = \frac{{Qdm}}{{2\pi {r^2}\sin \phi \cdot d\phi }} = \frac{{nt{b^2} \cdot Q\cos e{c^4}\phi /2}}{{16{r^2}}}.\,\quad .\quad .\quad (5)\] Since \(b = \frac{{2 \cdot N \cdot e \cdot E}}{{m \cdot {u^2}}}\), 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


675 Incidence of the rays is proportional to (1) cosec^{4} \(\phi \)/2 or 1/\(\phi \)^{4} if \(\phi \) be small; 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\(\phi \) in passing through a thickness t is 1/1000, the probability of two successive deflexions each of value \(\phi \) is 1/10^{6} , 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. Geiger,* who found that the distribution for particles deflected between 30° and 150° from a thin goldfoil 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 ß 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. * Manch. Lit. & Phil. Soc. 1910.

676
Let m be mass of the particle, v_{1} = velocity of approach,v_{2} = 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 mv_{1} of the entering particle, and OB the momentum of the receding particle which has been turned through an angle AOB =\(\phi \). Then BA represents in magnitude and direction the momentum MV of the recoiling atom. (MV)^{2} = (mv_{1})^{2} + (mv_{2})^{2}  2m^{2}v_{1}v_{2} cos f . . . (1) By conservation of energy MV^{2} = mv_{1}^{2}  mv_{2}^{2} . . . . .(2) Suppose M/m = K and v_{2} = ρv_{1}, where ρ < 1. \[\left( {K + 1} \right) \cdot {\rho ^2}  2 \cdot \rho \cdot \cos \phi = K  1\] \[\rho = \frac{{\cos \phi }}{{K + 1}} + \frac{1}{{K + 1}}\sqrt {{K^2}  {{\sin }^2}\phi } \] 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, \[\rho = \sqrt {\frac{{K  1}}{{K + 1}}} = 0,979\] 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 \(\phi \) = 90° ρ = 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 

677 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 \(\phi \)_{1} due to the field of the sphere of positive electricity of radius R and quantity Ne was found by him to be \[{\phi _1} = \frac{\pi }{4} \cdot \frac{{NeE}}{{m{u^2}}}\frac{1}{R}\] The average deflexion \(\phi \)_{2} due to the N negative corpuscles supposed distributed uniformly throughout the sphere was found to be \[{\phi _2} = \frac{{16}}{5}\frac{{eE}}{{m{u^2}}}\frac{1}{R}\sqrt {\frac{{3N}}{2}} \] The mean deflexion due to both positive and negative electricity was taken as (\(\phi \)_{1}^{2} + \(\phi \)_{2}^{2})^{1/2} 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 
678 centre is given by \[X = Ne\left( {\frac{1}{{{r^2}}}  \frac{r}{{{R^3}}}} \right)\] it is not difficult to show that the deflexion (supposed small) of an electrified particle due to this field is given by \[\theta = \frac{b}{p} \cdot {\left( {1  \frac{{{p^2}}}{{{R^2}}}} \right)^{3/2}}\] 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 \(\phi \). 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 ß 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 \[{\phi _1} = \frac{{3\pi }}{8} \cdot \frac{b}{R}\] This value of \(\phi \)_{1} 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 \[{\left( {\phi _1^2 + \phi _2^2} \right)^2}\quad or\quad \frac{b}{{2R}}{\left( {5 \cdot 54 + \frac{{15 \cdot 4}}{N}} \right)^{1/2}}\] 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. 

679 Neglecting the second term, the average deflexion per atom is \(\frac{{3 \cdot \pi \cdot b}}{{8 \cdot R}}\). 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_{t} after passing through a thickness t of matter is proportional to the square root of the number of encounters and is given by \[{\Theta _t} = \frac{{3\pi b}}{{8R}}\sqrt {\pi {R^2}.n.t} = \frac{{3\pi b}}{8}\sqrt {\pi nt} \] where n as before is equal to the number of atoms per unit volume. The probability p_{1} for compound scattering that the deflexion of the particle is greater than \(\phi \) is equal to \({e^{  {\phi ^2}/\Theta _t^2}}\). Consequently \[{\phi ^2} =  \frac{{9{\pi ^3}}}{{64}}{b^2} \cdot n \cdot t \cdot \log {p_1}\] Next suppose that single scattering alone is operative. We have seen (§3) that the probability p_{2} of a deflexion greater than\(\phi \) is given by \[{p_2} = \frac{\pi }{4} \cdot {b^2} \cdot n \cdot t \cdot {\cot ^2}\phi /2\] By comparing these two equations p_{2} log p_{1}=  0.181\(\phi \)^{2} cot^{2} \(\phi \)/2 , \(\phi \) is sufficiently small that tan \(\phi \)/2 = \(\phi \)/2, p_{2} log p_{1}= 0.72 If we suppose that p_{2} = 0.5, then p_{1} = 0.24 If p_{2} = 0.1, then p_{1} = 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. 
680 §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 \(\phi \) and \(\phi \) + d\(\phi \) where \(\phi \) 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 ß particles, The following points will be discussed:  (a) The 'diffuse reflexion' of a 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 ß 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. (b) 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.


681
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 . A^{2}t , 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 \(1/\sqrt A \) . 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 A^{3/2} or z / A^{3/2} 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.* 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 a 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 * The effect of change of velocity in an atomic encounter is neglected in this calculation. 
682 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 p_{1} and p_{2} for compound and single scattering respectively is given by p_{2} log p_{1}= 0.721. The probability q of the combined effects may as a first approximation be taken as q = (p_{1}^{2} +p_{2}^{2})^{1/2}. If q = 0.5, it follows that p_{1} = 0.2 and p_{2} = 0.46 We have seen that the probability p_{2} of a single deflexion greater than\(\phi \)is given by p_{2} = (p / 4)n . t . b^{2} (cot^{2} \(\phi \)/2) . Since in the experiments considered f is comparatively small \[\frac{{\phi \sqrt {{p_2}} }}{{\sqrt {\pi nt} }} = b = \frac{{NeE}}{{m{u^2}}}\]
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 \(\phi \) through which half the a particles are tuned thus corresponds to 2° nearly. t = 0.00017 cm.; n = 6.07 x 10^{22}; u (average value) = 1.8 x 10^{9}. E/m = 1.5 x 10^{14} E.S. units; e = 4.65 x 10^{10}, 

683 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, \(\phi \) = 4°.4, and average u =1.7 x 10^{9}, 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 goldfoil 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 ß rays.  p = (p / 4)n . t . b^{2} (cot^{2} \(\phi \)/2) . In most of Crowther's experiments f is sufficiently small that tan f/2 may be put equal to f/2 without much error. Consequently \(\phi \)^{2} = 2πn . t . b^{2} if p =1/2 On the theory of compound scattering, we have already seen that the chance p_{1} that the deflexion of the particles is greater than \(\phi \) is given by \[{\phi ^2}/\log {p_1} =  \frac{{9{\pi ^3}}}{{64}}n.t.{b^2}\] Since in the experiments of Crowther the thickness t of matter was determined for which p_{1} = 1/2, \(\phi \)^{2} = 0.96π n t b^{2}. For the probability of 1/2, the theories of single and compound 
684 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 t_{m} be the thickness for which half the particles are scattered through an angle f, Crowther showed that \(\phi /\sqrt {{t_m}} \) and also \(\frac{{m \cdot {u^2}}}{E} \cdot \sqrt {{t_m}} \) were constants for a given material when \(\phi \) 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 \(\phi \) 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 mu^{2} / 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 \(\phi \) is proportional to 1  e^{μ / t} where m is a constant for any given value of \(\phi \). The correctness of this latter formula was tested by Crowther by measuring electrically the fraction I / I_{o} of the scattered ß particles which passed through a circular opening subtending an angle of 36° with the scattering material. If \[I/{I_0} = 1  {e^{  \mu /t}}\] the value of I should decrease very slowly at first with


685 increase of t. Crowther, using aluminium as scattering material, states that the variation of I/I_{o} 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/I_{o} and t should be nearly linear in the initial stages. The experiments of Marsden* 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 \(\phi /\sqrt {{t_m}} \) for different elements for ß rays of velocity 2.68 x 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 \(\phi /\sqrt {{t_m}} \) the fraction of the ß particles deflected by single scattering through an angle greater than \(\phi \) 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 ß 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 * Phil. Mag. xviii. p. 909 (1909) 
686 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.* 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 b 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 nA^{2} 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 ß 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 a 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 Annal. d. Phys. iv. 23. p. 671 (1907)


687 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 Ne^{2} instead of N^{2}e^{2}. 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 (§ bc). 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 ß particles; for the α particle must approach much closer to the center of the atom than the ß particle of average speed to suffer the same large deflexion. The general data available indicate that the value of this central charge for different atoms is approximately proportional to their atomic weights, at any rate of atoms heavier than aluminium. It will be of great interest to examine 
688 experimentally whether such a simple relation holds also for the lighter atoms. In cases where the mass of the deflecting atom (for example, hydrogen, helium, lithium) is not very different from that of the α particle, the general theory of single scattering will require modification, for it is necessary to take into account the movements of the atom itself (see § 4). It is of interest to note that Nagaoka* has mathematically considered the properties of the Saturnian atom which he supposed to consist of a central attracting mass surrounded by rings of rotating electrons. He showed that such a system was stable if the attracting force was large. From the point of view considered in his paper, the chance of large deflexion would practically be unaltered, whether the atom is considered to be disk or a sphere. It may be remarked that the approximate value found for the central charge of the atom of gold (100 e) is about that to be expected if the atom of gold consisted of 49 atoms of helium, each carrying a charge of 2 e. This may be only a coincidence, but it is certainly suggestive in view of the expulsion of helium atoms carrying two unit charges from radioactive matter. The deductions from the theory so far considered are independent of the sign of the central charge, and it has not so far been found possible to obtain definite evidence to determine whether it be positive or negative. It may be possible to settle the question of sign by consideration of the difference of the laws of absorption of the ß particles to be expected on the two hypothesis, for the effect of radiation in reducing the velocity of the ß particle should be far more marked with a positive than with a negative center. If the central charge be positive, it is easily seen that a positively charged mass if released from the center of a heavy atom, would acquire a great velocity in moving through the electric field. It may be possible in this way to account for the high velocity of expulsion of α particles without supposing that they are initially in rapid motion within the atom. Further consideration of the application of this theory to these and other questions will be reserved for a later paper, when the main deductions of the theory have been tested experimentally. Experiments in this direction are already in progress by Geiger and Marsden. University of Manchester Nagaoka, Phil. Mag. vii. p. 445 (1904). 