CarnotCycle is a thermodynamics blog but occasionally its enthusiasm spills over into other subjects, as is the case here.

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When one considers the great achievements in radioactivity research made at the start of the 20th century by Ernest Rutherford and his team at the Victoria University, Manchester it seems surprising how little progress they made in finding an answer to the question posed above.

They knew that radioactivity was unaffected by any agency applied to it (even temperatures as low as 20K), and since the radioactive decay law discovered in 1902 by Rutherford and Soddy was an exponential function associated with probabilistic behavior, it was reasonable to think that radioactivity might be a random process. Egon von Schweidler’s work pointed firmly in this direction, and the Geiger-Nuttall relation, formulated by Hans Geiger and John Nuttall at the Manchester laboratory in 1911 and reformulated in 1912 by Richard Swinne in Germany, laid a mathematical foundation on which to construct ideas. Yet despite these pointers, Rutherford wrote in 1912 that *“it is difficult to offer any explanation of the causes operating which lead to the ultimate disintegration of the atom”*.

The phrase *“causes operating which lead to”* indicates that Rutherford saw the solution in terms of cause and effect. Understandably so, since he came from an age where probability was regarded as a measure of uncertainty about exact cause, rather than something reflecting a naturally indeterministic process. C.P. Snow once said of Rutherford, *“He thought of atoms as though they were tennis balls”*. And therein lay the essence of his problem: he didn’t have the right kind of mind to answer this kind of question.

But someone else did, namely the pioneer who introduced the term radioactivity and gave it a quantifiable meaning – Maria Sklodowska, better known under her married name Marie Curie.

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**Mme. Curie’s idea**

When all the great men of science (and one woman) convened for the second Solvay Conference in 1913, the hot topic of the day was the structure of the atom. Hans Geiger and Ernest Marsden at Rutherford’s Manchester lab had recently conducted their famous particle scattering experiment, enabling Rutherford to construct a model of the atom with a central nucleus where its positive charge and most of its mass were concentrated. Rutherford and his student Thomas Royds had earlier conducted their celebrated experiment which identified the alpha particle as a helium nucleus, so the attention now focused on trying to explain the process of alpha decay.

It was Marie Curie who produced the most fruitful idea, foreshadowing the quantum mechanical interpretation developed in the 1920s. Curie suggested that alpha decay could be likened to a particle bouncing around inside a box with a small hole through which the particle could escape. This would constitute a random event; with a large number of boxes these events would follow the laws of probability, even though the model was conceptually based on simple kinetics.

Now it just so happened that a probability distribution based on exactly this kind of random event had already been described in an academic paper, published in 1837 and rather curiously entitled *Recherches sur la probabilité des jugements en matière criminelle et matière civile* (Research on the probability of judgments in criminal and civil cases). The author was the French mathematician Siméon Denis Poisson (1781-1840).

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**The Poisson distribution**

At the age of 57, just three years before his death, Poisson turned his attention to the subject of court judgements, and in particular to miscarriages of justice. In probabilistic terms, Poisson was considering a large number of trials (excuse the pun) involving just two outcomes – a correct or an incorrect judgement. And with many years of court history on the public record, Poisson had the means to compute a time-averaged figure for the thankfully rare judicial failures.

In his 1837 paper Poisson constructed a model which regarded an incorrect judgement as a random event which did not influence any other subsequent judgement – in other words it was an independent random event. He was thus dealing with a random variable in the context of a binomial experiment with a large number of trials (n) and a small probability (p), whose product (pn) he asserted was finite and equal to µ, the mean number of events occurring in a given number of dimensional units (in this case, time).

In summary, Poisson started with the binomial probability distribution

where p is the probability of success and q is the probability of failure, in which successive terms of the binomial expansion give the probability of the event occurring exactly r times in n trials

Asserting µ = pn, he evaluated P(r) as n goes to infinity and found that

This is the general representation of each term in the Poisson probability distribution

which can be seen from

As indicated above, the mean µ is the product of the mean per unit dimension and the number of dimensional units. In the case of radioactivity, µ = λt where λ is the decay constant and t is the number of time units

If we set t equal to the half-life t_{½} the mean µ will be λt_{½} = ln 2. Mapping probabilities for the first few terms of the distribution yields

Unlike the binomial distribution, the Poisson distribution is not symmetric; the maximum does not correspond to the mean. In the case of µ = ln2 the probability of no decays (r = 0) is exactly a half, as can be seen from

At this point we turn to another concept introduced by Poisson in his paper which was taken further by the Russian mathematician P.L. Chebyshev – namely the law of large numbers. In essence, this law says that if the probability of an event is p, the average number of occurrences of the event approaches p as the number of independent trials increases.

In the case of radioactive decay, the number of independent trials (atoms) is extremely large: a µg sample of Cesium 137 for example will contain around 10^15 nuclei. In the case of µ = ln2 the law of large numbers means that the average number of atoms remaining intact after the half-life period will be half the number of atoms originally present in the sample.

The Poisson distribution correctly accounts for half-life behavior, and has been successfully applied to counting rate experiments and particle scattering. There is thus a body of evidence to support the notion that radioactive decay is a random event to which the law of large numbers applies, and is therefore not a phenomenon that requires explanation in terms of cause and effect.

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**Geiger and Nuttall**

Despite Ernest Rutherford’s protestations that atomic disintegration defied explanation, it was in fact Rutherford who took the first step along the path that would eventually lead to a quantum mechanical explanation of α-decay. In 1911 and again in 1912, Rutherford communicated papers by two of his Manchester co-workers, physics lecturer Hans Geiger (of Geiger counter fame) and John Nuttall, a graduate student.

Rutherford’s team at the Physical Laboratories was well advanced with identifying radioactive decay products, several of which were α-emitters. It had been noticed that α-emitters with more rapid decay rates had greater α-particle ranges. Geiger and Nuttall investigated this phenomenon, and when they plotted the logarithms of the decay constants (they called them transformation constants) against the logarithms of the corresponding α-particle ranges for decay products in the uranium and actinium series they got this result (taken from their 1911 paper):

This implies the existence of a relationship log λ = A + B log R, where A has a characteristic value for each series and B has the same value for both series. Curiously, Geiger and Nuttall did not express the straight lines in mathematical terms in either of their papers; they were more interested in using the lines to calculate the immeasurably short periods of long-range α-emitters. But they made reference in their 1912 paper to somebody who had *“recently shown that the relation between range and transformation constant can be expressed in another form”*.

That somebody was the German physicist Richard Swinne (1885-1939) who sent a paper entitled *Über einige zwischen den radioaktiven Elementen bestehene Beziehungen* (On some relationships between the radioactive elements) to *Physikalische Zeitschrift*, which the journal received on Tuesday 5th December 1911 and published in volume XIII, 1912.

The other form that Swinne had found, which he claimed to represent the experimental data at least as well as the (unstated) formula of Geiger and Nuttall, was log λ = a + bv^n, where a and b are constants and v is the particle velocity.

When it came to n, Swinne was rangefinding: he tried various values of n and found that *“n kann am besten gleich 1 gesetzt werden”*; he was thus edging towards what we now call the Geiger-Nuttall law, namely that the logarithm of the α-emitter’s half-life is inversely proportional to the square root of the α-particle’s kinetic energy

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**Gurney and Condon, and Gamow**

In 1924, the British mathematician Harold Jeffreys developed a general method of approximating solutions to linear, second-order differential equations. This method, rediscovered as the WKB approximation in 1926, was applied to the Schrödinger equation first published in that year and resulted in the discovery of the phenomenon known as quantum tunneling.

It was this strange effect, by which a particle with insufficient energy to surmount a potential barrier can effectively tunnel through it (the dotted line DB) that was seized upon in 1928 by Ronald Gurney and Edward Condon at Princeton – and independently by George Gamow at Göttingen – as a way of explaining alpha decay. Gurney and Condon’s explanation of alpha emission was published in *Nature* in an article entitled *Wave Mechanics and Radioactive Disintegration*, while Gamow’s considerably more academic (and mathematical) paper *Zur Quantentheorie des Atomkernes* was published in *Zeitschrift für Physik*.

In the quantum mechanical treatment, the overall rate of emission (i.e. the decay constant λ) is the product of a frequency factor f – the rate at which an alpha particle appears at the inside wall of the nucleus – multiplied by a transmission coefficient T, which is the (independent) probability that the alpha particle tunnels through the barrier. Thus

At this point it is instructive to recall Marie Curie’s particle-in-a-box idea, a concept which involves the product of two quantities: a large number of escape attempts and a small probability of escape.

The frequency factor f – or escape attempt rate – is estimated as the particle velocity v divided by the distance across the nucleus (2R) where R is the radius

Here, V_{0} is the potential well depth, Q_{α} is the alpha particle kinetic energy and µ is the reduced mass. The escape attempt rate is quite large, usually of the order of 10^{21} per second. By contrast the probability of alpha particle escape is extremely small. In calculating a value for T, Gamow introduced the Gamow factor 2G where

Typically the Gamow factor is very large (2G = 60-120) which makes T very small (T = 10^{-55}-10^{-27}).

Combining the equations

or

which is the Geiger-Nuttall law.

The work of Gurney, Condon and Gamow provided a convincing theoretical explanation of the Geiger-Nuttall law on the basis of quantum mechanics and Marie Curie’s hunch, and put an end to the classical notions of Rutherford’s generation that radioactive decay required explanation in terms of cause and effect.

So to return to the question posed at the head of this post – *What determines the moment at which a radioactive atom decays?* – the answer is chance. And the law of large numbers.

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**An important consequence**

The successful application of quantum tunneling to alpha particle emission had an important consequence, since it suggested to Gamow that the same idea could be applied in reverse i.e. that projectile particles with lower energy might be able to penetrate the nucleus through quantum tunneling. This led Gamow to suggest to John Cockroft, who was conducting atom-smashing experiments with protons, that protons with more moderate speeds could be used. Gamow’s suggestion proved correct, and the success of these trials ushered in a new era of intensive development in nuclear physics.

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**Links to original papers mentioned in this post**

G. Gamow (1928) *Zur Quantentheorie des Atomkernes*, Zeitschrift für Physik; 51: 204-212

https://link.springer.com/article/10.1007/BF01343196

H. Geiger and J.M. Nuttall (1911) *The ranges of the α particles from various radioactive substances and a relation between range and period of transformation*, Phil Mag; 22: 613-621

https://archive.org/stream/londonedinburg6221911lond#page/612/mode/2up

H. Geiger and J.M. Nuttall (1912) *The ranges of α particles from uranium*, Phil Mag; 23: 439-445

https://archive.org/stream/londonedinburg6231912lond#page/438/mode/2up

R.W. Gurney and E.U. Condon (1928) *Wave Mechanics and Radioactive disintegration*, Nature; 122 (Sept. 22): 439

https://www.nature.com/articles/122439a0

R. Swinne (1912) *Über einige zwischen den radioaktiven Elementen bestehene Beziehungen*, Physikalische Zeitschrift; XIII: 14-21

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P Mander August 2017