# Ernest Rutherford (1871 – 1937)

Ernest Rutherford, in the beginning of the twentieth century, made major breakthroughs in understanding the structure of the atom and demonstrating the inadequacy of classical physics. He was awarded the Nobel Prize in Chemistry in 1908.

(A) Radioactivity. Shortly after Röntgen‘s announcement of his discovery of X-rays, the French physicist Henri Becquerel became interested in the subject as a result of a lecture given by Henri Poincare. In an attempt to discover some connection between X-rays and the luminescence exhibited by uranium salts, he wrapped a photographic plate in black paper, placed a thin crystal of the salt (potassium uranyl sulphate) upon the paper, and then exposed the whole to sunlight. When the photographic plate was developed, it was found to be darkened. Becquerel finally showed that the uranium salt emitted rays, even without exposure to sunlight, which were to X-rays. This work was done in the early part of 1896, and the phenomenon was named radioactivity by Marie Curie in 1898.

Ernest Rutherford, in 1899, showed that there were two different types of radiation emitted by a uranium compound: the first, which he called alpha (α) rays, were unable to penetrate more than about 0.002 cm of aluminum. the second, the beta (β) rays, required a much thicker sheet of aluminum to absorb them completely.

A third type of radiation, which could not be deflected in a magnetic field, was detected by P. Villard in 1900. These radiations are now called gamma (γ) rays. It was shown by E. Rutherford and A. N. da C. Andrade in 1914 that they had a wave nature similar to X-rays.

By studying deflection in a magnetic field, it was shown by F. E. Dorn, Marie and Pierre Curie, that beta rays were negatively charged; Rutherford and T. Des Coudes showed in 1903 that, that alpha rays were positively charged. Rutherford believed that alpha rays were helium atoms carrying two unit positive charges, on account of the fact that both radium and actinium salts had been found to liberate helium. Unequivocal proof of this hypothesis was found by Rutherford and Royds in 1909.

The differing electrical properties of the radiations are summarised in the form of diagram below, which Marie Curie included in her doctoral thesis, published in 1903:

(B) Law of Radioactive Decay. In 1902, Rutherford and Soddy suggested that the atoms of radioelements undergo spontaneous disintegration, with the emission of alpha or neta particles and the formation of a new element. From the shape of the curve representing the rate of decay of radioactivity, it appeared to them that the activity was diminishing in an exponential manner. This would mean that the rate of decay is proportional to the number of atoms of the species of element present.

Suppose that a given instant there are present N atoms of a particular radioelement, suppose, further, that in the extremely small subsequent time interval dt, the number of atoms disintegrating is dN. The rate of disintegration is then represented by dN/dt. It was postulated above that the rate of disintegration is proportional to the total number of atoms N; hence,

$\- dN/dt =\lambda N$

(1)

where λ is a constant, which Rutherford and Soddy called the radioactive constant of the element under consideration. Upon rearranging equation (1) into the form

$\frac{dN}{N} = - \lambda dt$

and carrying out the process of integration, the result is

$\ln (N_{i}/N_{0}) = - \lambda t$

(2)

or, in the equivalent exponential form

$\ N_{t} = N_{0}e^{ - \lambda t}$

(3)

where NO is the number of atoms present at any arbitrary time and Nt is the number remaining after the lapse of a further time interval t.

A constant, introduced by Rutherford in 1904, called the half-life, is commonly employed as a characteristic property of a radioelement. The half-life is the time required for the radioactivity of a given amount to decay to half its initial value. Denoting it by T, we have

$\ T = \frac{0.693}{\lambda }$

(4)

If ΔN is the number of atoms disintegrating, i.e., the measured number of alpha or beta particles emitted in a definite time interval Δt, then the ratio ΔN/Δt may be taken as a good approximation to the instantaneous rate of disintegration dN/dt. It follows from equation (1) that

$\lambda = - \frac{\Delta N/\Delta t}{N}$

(5)

It follows that the probability of an atom decaying in time Δt is always λ. This is in stark contradiction to the deterministic view of physics that existed in the nineteenth century.

(C) The Atomic Model. The first definite ideas concerning the interior structure of atoms were put forward by J. J. Thomson in 1898. He suggested that the atom consists of a number of negative corpuscules moving about in a sphere if positive electrification..

It was observed that when alpha particles from a radioactive source fell on a photographic plate, after penetrating a thin sheet of metal, the resulting trace was diffuse, fading off the edges, instead of being sharp. This diffuseness was attributed to the scattering of the alpha particles. The variation of the scattering angle was taken to be due to multiple collisions through small angles. However, Thomson’s model did not account for the observed shape of the scattering curve.

In his classical paper of 1911, Rutherford postulated that, contrary to the Thomson model, the positive charge is concentrated in a small region , which he later (1912) called the nucleus, at the centre of the atom.

Rutherford calculated the distribution of α-particles to be expected as a result of single-scattering processes by atoms of this type. According to the Rutherford scattering formula, derived below, the number Nd of scattered α-particles reaching a small detector of area A is given by

$\ N_{d} = \frac{N_{i}NtA}{4R^{2}}\frac{4Z^{2}e^{4}}{16\pi ^{2}\epsilon _{0}^2m^{2}v_{0}^{2}}\frac{1}{sin^{4}(\phi /2)}$

 where $\ N_{t} =$ number of α-particles incident on a foil of thickness t with n nuclei of atomic number Z per unit volume R = distance of detector from foil m = mass of incident α-particles $\ v_{0} =$ speed of incident α-particles $\phi =$ scattering angle e = charge of a photon $\varepsilon _{0} =$ permittivity of free space

This formula is subject to correction, however, for the motion induced in the scattering nucleus, which may be appreciable for light atoms.

According to Rutherford’s formula, the number of particles striking a small detector with its sensitive face perpendicular to the direction of the motion of the scattered particles should be:

1. Proportional to the square of the atomic number Z of the scattering nuclei
2. Proportional to the thickness t of the foil
3. Inversely proportional to the square of the kinetic energy of the particles, i.e., to vo4
4. Inversely proportional to the fourth power of the sin of half the scattering angle $\phi$

Derivation: Let the charge on the incident particle be ze and its mass m, while the nucleus bears charge Ze and has a mass so much greater than the incident particle that, for a first approximation, it may be assumed to remain at rest. Let the α particle approach at a speed v0 along a line passing at a distance b from the nucleus; its energy is thus mv02/2, and its angular momentum about the nucleus at O is mvb. Let r and θ be polar coordinates for the α particle with origin at O, θ being measured from a line drawn from O in a direction toward the distantly approaching particle (Fig. 1), and let $\dot{r}$ and $\dot{\theta }$ be the time derivatives of r and θ respectively.

Then the particle has perpendicular components of velocity $\dot{r}$ and $\ r\dot{\theta }$, and its angular momentum about O at any time is $\ mr^{2}\dot{\theta }$. It also has potential energy due to repulsion by the nucleus of magnitude zZe2/4πε0r and initially zero. Thus, adding the kinetic energy, we have from the laws of conservation of energy and of angular momentum

$\frac{1}{2}m(\dot{r}^{2} + r^{2}\dot{\theta }^{2}) + \frac{zZe^{2}}{4\pi \epsilon _{0}r} = \frac{1}{2}mv_{0}^{2}$

(1)

$\ mr^{2}\dot{\theta } = mv_{0}b$

(2)

By eliminating dθ/dt it follows that

$\dot{r} = - v_{0}(1 - \frac{2q}{r} - \frac{b^{2}}{r^{2}})^{1/2}$

(3a)

$\ q = \frac{zZe^{2}}{4\pi \epsilon _{0}mv_{0}^2}$

(3b)

in which the negative sign is chosen because, during the approach, dr/dt < 0. Dividing this into Eq. (2) and noting that $\frac{\dot{\theta }}{\dot{r}} = d\theta /dr$, we have

$\frac{d\theta }{dr} = - \frac{b}{r^{2}}(1 - \frac{2q}{r} - \frac{b^{2}}{r^{2}})^{1/2}$

(4)

The integral of this equation that vanishes at r = ∞, as is easily verified by substitution in Eq. (4), is

$\theta = cos^{-1}[\frac{b}{\sqrt{b^{2} +q^{2}}}(1 - \frac{2q}{r} - \frac{b^{2}}{r^{2}})^{1/2}] - cos^{-1} \frac{b}{\sqrt{b^{2} + q^{2}}}$

At the point of closest approach to the nucleus, dr/dt = 0. Hence, by (3), the radical must vanish; then θ = θ0 = $\pi /2 - cos^{-1}(\frac{b}{\sqrt{b^{2} +q^{2}}})$. Hence, the total increase in θ is 2θ0; and the final deflection of the total change in the direction of motion of the particle is

$\phi = \pi - 2\theta _{0} = 2 cos^{-1} \frac{b}{\sqrt{b^{2} + q^{2}}}$

(5a)

therefore

$\ b = q \cot \frac{\phi }{2}$

(5b)

From Eq. (5b) we see that the scattering angle depends on the impact parameter b, with large-angle scattering associated with small b. if the impact parameter is increased to b + db, the scattering angle φ changes by dφ where, by Eq. (5b),

$\ db = - \frac{q}{2}\frac{d\phi }{\sin ^{2}(\phi /2)}$

(6)

To find the statistical distribution of the α particles scattered by a foil of thickness t which contains n nuclei per unit volume, let N 0 α particles per unit area approach the foil along parallel paths normal to the foil. Then the number of incident particles per unit area which are incident along lines passing at a distance between b and b + db of some nucleus is N0(2πb db)nt, provided the foil is thin enough. These particles are scattered through angles between $\phi$ and $\phi + d\phi$. Hence the number of α particles per unit ares scattered into the angular cone between $\phi$ and $\phi + d\phi$; is

$\ dN = N(\phi ) d\phi = N_{0}nt\pi (\frac{zZe^{2}}{4\pi \epsilon _{0}mv_{0}^{2}})^{2} \frac{\cos (\phi /2)}{\sin ^{2} (\phi /2)} d\phi$

(7)

In an experiment one typically measures the number of particles scattered at an angle $\phi$ which strike a detector of area A at a distance R from the small area element of a scattering foil. It is assumed that the useful foil is small enough for all points to be essetially equidistant from the detector. Let the number Ni represent the number of α particles incident on the element of foil; clearly, Ni is N0 times the effective area of the foil. Of the particles scattered between $\phi$ and $\phi + d\phi$ the detector intercepts the fraction $\ A/(2\pi R\sin \phi )(R d\phi )$ (Fig.2). From Eq. (7) and the fact that $\sin \phi = 2 \sin (\phi/2) \cos (\phi/2)$, we find that the number Nd of α particles reaching the detector to be

$\ N_{d} = \frac{N_{i}ntA}{4R^{2}} \frac{z^{2}Z^{2}e^{4}}{16\pi ^{2}\epsilon_{0}^{2}m^{2}v_{0}^{2}} \frac{1}{\sin ^{4} (\phi /2)}$

(8)

as stated.

Size of the nucleus. The size of the nucleus is given by the minimum value of b, which is determined from the maximum scattering angle by Eq. (5b). It is of the order of 10-12 to 10-13 cm.

Some Difficulties. The objections to the nuclear atom were based on questions of stability. No stable arrangement of positive and negative charges at rest can be invented; it can be proved, in fact that such an arrangement is impossible (Earnshaw’s theorem). An electron might be imagined to revolve around a nucleus as the earth revolves around the sun. But according to the laws of classical electromagnetic theory, it would radiate energy. The electron would therefore approach the nucleus along a spiral path and give out radiation of constantly decreasing frequency. This does not agree with the observed emission of spectral lines of fixed frequency.

It was at this point that Bohr introduced his epoch-making theory of the structure of the atom and of the origin of spectra, based on Planck‘s theory of quanta and Rutherford’s model.

The theorem referred to above was proved in 1842 by Samuel Earnshaw (Earnshaw, S., On the nature of the molecular forces which regulate the constitution of the luminiferous ether., 1842, Trans. Camb. Phil. Soc., 7, pp. 97 – 112). It was almost ignored until Roy Harrigan built a levitation device in the late 1970s, which seemingly voilates the theorem (U.S. Patent, 3 May 1983). Maglev trains, which also operate on magnetic levitation, were finally constructed in 2004. Earnshaw’s Theorem rules out stable equilibrium configurations, but unstable equilibrium positions can occur. Such unstable equilibrium, coupled with quantum mechanical uncertainty, once more establishes non-determinism in physics.

(C) Biography. (More details are given at http://www.teara.govt.nz/en/biographies/3r37/1/1)

Image Source: http://www.antiquewireless.org/otb/rutherford.htm

Ernest Rutherford was born at Spring Grove in rural Nelson, New Zealand, on 30 Aug 1871, the 4th child of twelve, born to James Rutherford, a mechanic, and his wife, Martha Thompson, who had been the schoolteacher at Spring Grove. Ernest boarded at Nelson College from 1877 to 1879. In 1889 he was head boy, played rugby and, on the second attempt, won a scholarship at a college of the University of New Zealand. In 1893 he graduated B.A. in Mathematics, Latin, English, French and Physics.

He then won the one Senior Scholarship in Mathematics available in New Zealand and returned for a further (honours) year in both Mathematics and Physics. He obtained an M.A. in Mathematics and Mathematical Physics.

Rutherford left New Zealand in 1895 to work with Professor J. J. Thomson of the University of Cambridge’s Cavendish Laboratory, where he worked on electromagnetic waves. In 1898, he accepted a professorship at McGill University, Montreal, Canada. Here, he worked on radioactivity, later with the help of Frederick Soddy. he returned to New Zealand in 1900 to marry Mary Georgina Newton, the daughter of his landlady in Christchurch. He won the Nobel Prize in Chemistry in 1908, was knighted in 1914, and became the Director of the Cavendish Laboratory in 1919.

Ernest Rutherford died at Cambridge on 19 Oct 1937, the result of delays in operating on his partially strangulated umbilical hernia. His ashes were interred in London’s Westminister Abbey, under an inscribed flagstone near the choir screen in the nave.

(D) References.