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Taking a break from studying thermodynamics: Mme. Curie measuring picoamp leakage currents

Having read biographies of the Curies, Marie Curie’s doctoral thesis, and a number of scholarly articles about the radium phenomenon, I have come to the conclusion that the Marie Curie legend in popular culture tends to sideline her scientific achievements, focusing more on her imagined saintliness and perceived role as bringer of medical marvels than on her pioneering work as a physical chemist, in which her husband Pierre (above left) played an important facilitating role.

To my mind, the popular press of her day was largely responsible for the misconstruction of the Marie Curie legend, filling the public’s mind with the discovery of a miracle cure for cancer brought about by what it portrayed as an angelic young foreign-born mother slaving away in a dark shed in Paris for no wages.

The media frenzy around radium had consequences. Once radium production was established on a commercial scale, ignorant and unscrupulous marketers quickly morphed the Curies’ discovery of an element that glowed in the dark into revitalizing radium baths, radium drinking water, radium chocolate, radium toothpaste, radium cigarettes and even radium suppositories for restoring male potency while eradicating hemorrhoids:

Also splendid for piles and rectal sores. Try them and see what good results you get!

The dreadful damage these products must have caused doesn’t bear thinking about. Sadly, the Curies themselves seemed carried along similarly radium-dazzled tracks. They failed to connect Pierre’s rapidly deteriorating health with exposure to radioactive emissions, while stoically accepting the painful damage to Marie’s hands as a price worth paying for the greater good they somehow imagined radium to represent.

Irène Curie points to her mother’s radiation-damaged hands

The sensationalist aspects of the Curie legend, while an education in themselves, are however not the subject of this post. Physics and chemistry are the subjects here. When you look at Marie Curie as a physical chemist, and examine her contributions to the science of natural radioactivity, it is clear how crucial a role was played by the miracle machine designed and developed by Pierre Curie.

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The Curie quartz electrometer apparatus

The above diagram is taken from Méthodes de Mesure employées en Radioactivité published in Paris in 1911 by Albert Laborde, a graduate engineer who became Pierre Curie’s assistant in 1902. It shows the quartz electrometer apparatus developed by Pierre Curie and his brother Jacques for the precise measurement of very weak currents (of the order of tenths of picoamperes) following their discovery of piezoelectricity in 1880. It was this discovery that prompted the brothers Curie to build a calibrated electrostatic charge generator using a thin quartz lamella (center) to compensate and thus measure the leakage current from a charged capacitor (left) using a quadrant electrometer (right).

This apparatus was later adapted by Pierre Curie to allow accurate quantification of the tiny leakage currents produced in an ionization chamber by samples of radioactive material.

This is the experimental set-up that Marie Curie can be seen using in the header photograph, which dates from 1898.

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Amazing coincidences

When you consider the train of coincidences that led to Marie Curie’s choice of subject for her doctorate (Recherches sur les substances radioactives) it is nothing short of amazing. At the time she was looking around for suitable topic, five years after having journeyed from Poland to Paris to enroll as a Sorbonne student, Henri Becquerel had just accidentally discovered mysterious rays emanating from uranium which had the property of weakly ionizing air. This was in 1896. Just a year previously, Marie had married Pierre Curie who happened to possess the one instrument capable of accurately measuring small ionization currents, following his discovery of piezoelectricity sixteen years earlier.

Because uranic rays were a new phenomenon, Marie was saved the task of first researching the topic which otherwise would have entailed reading a lot of academic papers in unfamiliar French. This saving of time and effort attracted her to choose to study Becquerel’s uranic rays, something she admitted in later life. Furthermore she had no competition since Becquerel had shown little interest in pursuing his original finding – the big news at the time was X rays, discovered by Wilhelm Röntgen in 1895. No fewer than 1044 papers on X rays were published in 1896, when Becquerel first announced his discovery. Not surprisingly, nobody took any notice. Marie Curie had the field to herself.

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The weight, the watch and the light spot

This magnified portion of the header image shows Marie Curie as she sits at the quartz electrometer apparatus. Her right hand can be seen holding an analytical balance weight in a controlled manner while in her left hand the edge of a stopwatch can be seen. Her eyes are looking fixedly at a horizontal measuring scale above a light source (square hole) mounted on a wooden pedestal.

The light source is shining a beam onto the mirror of a quadrant electrometer out of view to the left. The light spot is reflected onto the horizontal scale (cf. diagram above) and Marie is endeavoring to keep the light spot stationary. She does this by gradually releasing the weight which is attached to the quartz lamella, thereby generating charge to compensate the ionization current produced by the radioactive sample in the ionization chamber also out of view to the left. The entire process of weight release is timed by a stopwatch. Once the weight is fully released the watch is stopped. The weight generates a specific amount of charge Q* on the quartz lamella during the measured time T. Hence Q/T is equal to the ionization current, which is directly proportional to the intensity of the ionizing radiation emitted by the sample, or to use the term Marie Curie coined, its radioactivity.

*The amount of charge Q is calculated from Q = W × K × L/B where W is the applied weight, K is the quartz specific constant, L is the lamella length and B is the lamella thickness.

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Thesis

On Thursday 25th June 1903, at La Faculté des Sciences de Paris, Marie Curie presented her doctoral thesis to the examination committee, two of whose members were later to become Nobel laureates. The committee was impressed; in fact it expressed the view that her findings represented the greatest scientific contribution ever made in a doctoral thesis.

At the outset, Curie coined a new term – radioactivity – to describe the ionizing radiation emitted by the uranium compounds studied by Henri Becquerel. She announced her discovery that the element thorium also displays radioactivity. And she presented a method, using the quartz electrometer apparatus developed by Jacques and Pierre Curie, by which the intensity of radioactive emissions could be precisely quantified and expressed as ionization currents. This was a game-changing advance on the essentially qualitative methods that had been used hitherto e.g. electroscopes and photographic plates.

As one would expect, Curie began her experimental work with a systematic study of uranium and its compounds, measuring and tabulating their ionization currents. There was a considerable range from the largest to the smallest currents, and within the limits of experimental error it was evident that the ionization currents were proportional to the amounts of uranium present in the sample. The same was true for thorium.

From the chemist’s perspective this was a puzzling result. The properties of chemical compounds of the same element generally depend on what it is compounded with and the arrangement of atoms in the molecule. Yet here was a very different finding – the radioactivity Curie measured was independent of compounding or molecular structure.

Curie drew the conclusion that radioactivity was a property of the atom – une propriété atomique she called it. She wasn’t referring to the uranium atom or the thorium atom, but to the atom as a generalized material unit with an implied interior from which radioactive emissions issued. That is a profound conception, with which Marie Curie made a significant contribution to the advancement of physics.

And at this point in her thesis she hadn’t even mentioned radium.

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New elements

For the next part of her thesis, Marie Curie turned her attention to the study of uranium-containing minerals, one of which was the mineral pictured above. Today we call it uraninite but in Curie’s day it was called pitchblende. The sample she obtained was from a uranium mine near the town of Joachimsthal in Austria, now Jáchymov in the Czech Republic. She measured its ionization current and found it to be considerably higher than its uranium content warranted. If her radioactivity hypothesis was correct, there was only one explanation: pitchblende contained atoms that emitted much more intense radiation than uranium atoms, which meant that another radioactive substance must be present in the ore. Curie now had the task of finding it, and was joined in this quest by her husband Pierre and the chemist Gustave Bémont.

The quartz electrometer demonstrated its value yet again, since the various fractions derived from the pitchblende sample during chemical analysis could be tested for radioactivity. In this way, the radioactivity was followed to two fractions: one containing the post-transition metal bismuth* and another containing the alkaline earth metal element barium. The Curies announced their findings in July 1898, stating their belief that these fractions contained two previously unknown metal elements, and suggesting that if the existence of these metals were confirmed, the bismuth-like element should be called polonium and the barium-like element radium.

*unknown to the Curies, the uranium decay series actually produces two radioisotopes of bismuth along with the isotopes of polonium, so the presence of radioactivity in this fraction did not solely indicate the presence of a new element.

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The shed

The fabled shed where Marie Curie labored for four years, chemically processing tons of pitchblende to produce a tiny spoonful of radium.

The heroic work which made Marie Curie a legend took place in a shed at the back of a Grande École* in Rue Vauquelin. In order to produce sufficient quantities to isolate the new elements and determine their atomic weights, tons of pitchblende were needed. This was because the maximum amounts of radium and polonium that can coexist in secular equilibrium with uranium are in the ratios of their respective half lives. This fact and the limited human resources available rendered any attempt to isolate polonium impossible, and the situation with radium was not much better. At best, a quantity of uranium ore containing 3 metric tons of elemental uranium is needed to extract 1 gram of radium at a yield of 100%. In the primitive conditions of the shed, obtaining a gram of radium meant processing 8 or 9 tons of uranium ore. One can only wonder at how Marie Curie found the physical and mental strength for such an arduous task.

*At the time, it was called École supérieure de physique et de chimie industrielles de la ville de Paris. Today it is called ESCPI Paris.

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Did she deserve two Nobel Prizes?

Marie Curie was awarded a quarter of the Nobel Prize for Physics in 1903 for her work on radioactivity. In 1911 she was the sole recipient of the Nobel Prize for Chemistry, awarded for the discovery of radium and polonium.

There can be no doubt about her credentials for the 1903 award, but some biographers have questioned whether the 1911 Prize was deserved, claiming that the discoveries of radium and polonium were part of the reason for the first prize.

As described in this post, the experimental evidence which Marie Curie set forth to reason that radioactivity is an atomic property was based solely on her experiments with uranium and thorium. Neither radium nor polonium had anything to do with it. On these grounds the claims of those biographers can be rejected.

Which leaves the question of under what circumstances the discovery of a new element qualifies for a Nobel Prize in chemistry. Clearly the discovery of a naturally radioactive element is not sufficient, otherwise Marguerite Perey – who worked as Marie Curie’s lab assistant and discovered francium in 1939 – would have qualified. Other aspects of the discovery need to be taken into account, and in 1911 there were many such aspects to Marie Curie’s discovery of radium and polonium, and the isolation of radium.

Reading the award citation, what comes across to me – albeit between the lines – is a recognition of the monumental personal effort and dedication involved in the discovery and characterization of these remarkable elements that led to the modern science of nuclear physics.

The miracle machine

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P Mander June 2017

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A thermodynamic system doesn’t have to be big. Although thermodynamics was originally concerned with very large objects like steam engines for pumping out coal mines, thermodynamic thinking can equally well be applied to very small systems consisting of say, just a few atoms.

Of course, we know that very small systems play by different rules – namely quantum rules – but that’s ok. The rules are known and can be applied. So let’s imagine that our thermodynamic system is an idealized solid consisting of three atoms, each distinguishable from the others by its unique position in space, and each able to perform simple harmonic oscillations independently of the others. At the absolute zero of temperature, the system will have no thermal energy, one microstate and zero entropy, with each atom in its vibrational ground state.

Harmonic motion is quantized, such that if the energy of the ground state is taken as zero and the energy of the first excited state as ε, then 2ε is the energy of the second excited state, 3ε is the energy of the third excited state, and so on. Suppose that from its thermal surroundings our 3-atom system absorbs one unit of energy ε, sufficient to set one of the atoms oscillating. Clearly, one unit of energy can be distributed among three atoms in three different ways – 100, 010, 001 – or in more compact notation [100|3].

Now let’s consider 2ε of absorbed energy. Our system can do this in two ways, either by promoting one oscillator to its second excited state, or two oscillators to their first excited state. Each of these energy distributions can be achieved in three ways, which we can write [200|3], [110|3]. For 3ε of absorbed energy, there are three distributions: [300|3], [210|6], [111|1].

Summarizing the above information

Energy E (in units of ε) Total microstates W Ratio of successive W’s
0 1
1 3 3
2 6 2
3 10 1⅔

 

The summary shows that as E increases, so does W. This is to be expected, since as W increases, the entropy S (= k log W) increases. In other words E and S increase or decrease together; the ratio ∂E/∂S is always positive. Since ∂E/∂S = T, the finding that E and S increase or decrease together is equivalent to saying that the absolute temperature of the system is always positive.

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Adding an extra particle

It is instructive to compare the distribution of energy among three oscillators (N =3)*

E = 0: [000|1]
E = 1: [100|3]
E = 2: [200|3], [110|3]
E = 3: [300|3], [210|6], [111|1]

with the distribution among four oscillators (N = 4)*

E = 0: [0000|1]
E = 1: [1000|4]
E = 2: [2000|4], [1100|6]
E = 3: [3000|4], [2100|12], [1110|4]

*For any single distribution among N oscillators where n0, n1,n2 … represent the number of oscillators in the ground state, first excited state, second excited state etc, the number of microstates is given by

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It is understood that 0! = 1. Derivation of the formula is given in Appendix I.

For both the 3-oscillator and 4-oscillator systems, the first excited state is never less populated than the second, and the second excited state is never less populated than the third. Population is graded downward and the ratios n1/n0 > n2/n1 > n3/n2 are less than unity.

Example calculations for N = 4, E = 3:

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Comparisons can also be made of a single ratio across distributions and between systems. For example the values of n1/n0 for E = 0, 1, 2, 3 are

(N = 4) : 0, ⅓, ½, ⅗
(N = 3) : 0, ½, ⅔, ¾

Since for a macroscopic system

tp06

this implies that for a given value of E the 4-oscillator system is colder than the 3-oscillator system. The same conclusion can be reached by looking at the ratio of successive W’s for the 4-oscillator system sharing 0 to 3 units of thermal energy

Energy E (in units of ε) Total microstates W Ratio of successive W’s
0 1
1 4 4
2 10
3 20 2

 

For the 4-oscillator system the ratios of successive W’s are larger than the corresponding ratios for the 3-oscillator system. The logarithms of these ratios are inversely proportional to the absolute temperature, so the larger the ratio the lower the temperature.

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Finite differences

The differences between successive W’s for a 4-oscillator system are the values for a 3-oscillator system

W for (N =4) : 1, 4, 10, 20
Differences : 3, 6, 10

Likewise the differences between successive W’s for a 3-oscillator system and a 2-oscillator system

W for (N =3) : 1, 3, 6, 10
Differences : 2, 3, 4

Likewise for the differences between successive W’s for a 2-oscillator system and a 1-oscillator system

W for (N =2) : 1, 2, 3, 4
Differences : 1, 1, 1

This implies that W for the 4-particle system can be expressed as a cubic in n, and that W for the 3-particle system can be expressed as a quadratic in n etc. Evaluation of coefficients leads to the following formula progression

For N = 1

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For N = 2

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For N = 3

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For N = 4

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It appears that in general

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Since n = E/ε and ε = hν, the above equation can be written

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For a system of oscillators this formula describes the functional dependence of W microstates on the size of the particle ensemble (N), its energy (E), the mechanical frequency of its oscillators (ν) and Planck’s constant (h).

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Appendix I

Formula to be derived

For any single distribution among N oscillators where n0, n1,n2 … represent the number of oscillators in the ground state, first excited state, second excited state etc, the number of microstates is given by

tp02

Derivation

In combinatorial analysis, the above comes into the category of permutations of sets with the possible occurrence of indistinguishable elements.

Consider the distribution of 3 units of energy across 4 oscillators such that one oscillator has two units, another has the remaining one unit, and the other two oscillators are in the ground state: {2100}

If each of the four numbers was distinct, there would be 4! possible ways to arrange them. But the two zeros are indistinguishable, so the number of ways is reduced by a factor of 2! The number of ways to arrange {2100} is therefore 4!/2! = 12.

1 and 2 occur only once in the above set, and the occurrence of 3 is zero. This does not result in a reduction in the number of possible ways to arrange {2100} since 1! = 1 and 0! = 1. Their presence in the denominator will have no effect, but for completeness we can write

4!/2!1!1!0!

to compute the number of microstates for the single distribution E = 3, N = 4, {2100} where n0 = 2, n1 = 1, n2 = 1 and n3 = 0.

In the general case, the formula for the number of microstates for a single energy distribution of E among N oscillators is

tp02

where the terms in the denominator are as defined above.

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P Mander April 2016

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The Arrhenius equation explains why chemical reactions generally go much faster when you heat them up. The equation was actually first given by the Dutch physical chemist JH van ‘t Hoff in 1884, but it was the Swedish physical chemist Svante Arrhenius (pictured above) who in 1889 interpreted the equation in terms of activation energy, thereby opening up an important new dimension to the study of reaction rates.

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Temperature and reaction rate

The systematic study of chemical kinetics can be said to have begun in 1850 with Ludwig Wilhelmy’s pioneering work on the kinetics of sucrose inversion. Right from the start, it was realized that reaction rates showed an appreciable dependence on temperature, but it took four decades before real progress was made towards quantitative understanding of the phenomenon.

In 1889, Arrhenius penned a classic paper in which he considered eight sets of published data on the effect of temperature on reaction rates. In each case he showed that the rate constant could be represented as an explicit function of the absolute temperature:

arr01

where both A and C are constants for the particular reaction taking place at temperature T. In his paper, Arrhenius listed the eight sets of published data together with the equations put forward by their respective authors to express the temperature dependence of the rate constant. In one case, the equation – stated in logarithmic form – was identical to that proposed by Arrhenius

arr02

where T is the absolute temperature and a and b are constants. This equation was published five years before Arrhenius’ paper in a book entitled Études de Dynamique Chimique. The author was J. H. van ‘t Hoff.

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Dynamic equilibrium

In the Études of 1884, van ‘t Hoff compiled a contemporary encyclopædia of chemical kinetics. It is an extraordinary work, containing all that was previously known as well as a great deal that was entirely new. At the start of the section on chemical equilibrium he states (without proof) the thermodynamic equation, sometimes called the van ‘t Hoff isochore, which quantifies the displacement of equilibrium with temperature. In modern notation it reads:

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where Kc is the equilibrium constant expressed in terms of concentrations, ΔH is the heat of reaction and T is the absolute temperature. In a footnote to this famous and thermodynamically exact equation, van ‘t Hoff builds a bridge from thermodynamics to kinetics by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions

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Substituting this quotient in the original equation leads immediately to

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van ‘t Hoff then argues that the rate constants will be influenced by two different energy terms E1 and E-1, and splits the above into two equations

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where the two energies are such that E1 – E-1 = ΔH

In the Études, van ‘t Hoff recognized that ΔH might or might not be temperature independent, and considered both possibilities. In the former case, he could integrate the equation to give the solution

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From a starting point in thermodynamics, van ‘t Hoff engineered this kinetic equation through a characteristically self-assured thought process. And it was this equation that the equally self-assured Svante Arrhenius seized upon for his own purposes, expanding its application to explain the results of other researchers, and enriching it with his own idea for how the equation should be interpreted.

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Activation energy

It is a well-known result of the kinetic theory of gases that the average kinetic energy per mole of gas (EK) is given by

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Since the only variable on the RHS is the absolute temperature T, we can conclude that doubling the temperature will double the average kinetic energy of the molecules. This set Arrhenius thinking, because the eight sets of published data in his 1889 paper showed that the effect of temperature on the rates of chemical processes was generally much too large to be explained on the basis of how temperature affects the average kinetic energy of the molecules.

The clue to solving this mystery was provided by James Clerk Maxwell, who in 1860 had worked out the distribution of molecular velocities from the laws of probability. Maxwell’s distribution law enables the fraction of molecules possessing a kinetic energy exceeding some arbitrary value E to be calculated.

It is convenient to consider the distribution of molecular velocities in two dimensions instead of three, since the distribution law so obtained gives very similar results and is much simpler to apply. At absolute temperature T, the proportion of molecules for which the kinetic energy exceeds E is given by

arr08

where n is the number of molecules with kinetic energy greater than E, and N is the total number of molecules. This is exactly the exponential expression which occurs in the velocity constant equation derived by van ‘t Hoff from thermodynamic principles, which Arrhenius showed could be fitted to temperature dependence data from several published sources.

Compared with the average kinetic energy calculation, this exponential expression yields very different results. At 1000K, the fraction of molecules having a greater energy than, say, 80 KJ is 0.0000662, while at 2000K the fraction is 0.00814. So the temperature change which doubles the number of molecules with the average energy will increase the number of molecules with E > 80 KJ by a factor of more than a hundred.

Here was the clue Arrhenius was seeking to explain why increased temperature had such a marked effect on reaction rate. He reasoned it was because molecules needed sufficiently more energy than the average – the activation energy E – to undergo reaction, and that the fraction of these molecules in the reaction mix was an exponential function of temperature.

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The meaning of A

But back to the Arrhenius equation

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I have always thought that calling the constant A the ‘pre-exponential factor’ is a singularly pointless label. One could equally write the equation as

arr10

and call it the ‘post-exponential factor’. The position of A in relation to the exponential factor has no relevance.

A clue to the proper meaning of A is to note that e^(–E/RT) is dimensionless. The units of A are therefore the same as the units of k. But what are the units of k?

The answer depends on whether one’s interest area is kinetics or thermodynamics. In kinetics, the concentration of chemical species present at equilibrium is generally expressed as molar concentration, giving rise to a range of possibilities for the units of the velocity constant k.

In thermodynamics however, the dimensions of k are uniform. This is because the chemical potential of reactants and products in any arbitrarily chosen state is expressed in terms of activity a, which is defined as a ratio in relation to a standard state and is therefore dimensionless.

When the arbitrarily chosen conditions represent those for equilibrium, the equilibrium constant K is expressed in terms of reactant (aA + bB + …) and product (mM + nN + …) activities

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where the subscript e indicates that the activities are those for the system at equilibrium.

As students we often substitute molar concentrations for activities, since in many situations the activity of a chemical species is approximately proportional to its concentration. But if an equation is arrived at from consideration of the thermodynamic equilibrium constant K – as the Arrhenius equation was – it is important to remember that the associated concentration terms are strictly dimensionless and so the reaction rate, and therefore the velocity constant k, and therefore A, has the units of frequency (t^-1).

OK, so back again to the Arrhenius equation

arr09

We have determined the dimensions of A; now let us turn our attention to the role of the dimensionless exponential factor. The values this term may take range between 0 and 1, and specifically when E = 0, e^(–E/RT) = 1. This allows us to assign a physical meaning to A since when E = 0, A = k. We can think of A as the velocity constant when the activation energy is zero – in other words when each collision between reactant molecules results in a reaction taking place.

Since there are zillions of molecular collisions taking place every second just at room temperature, any reaction in these circumstances would be uber-explosive. So the exponential term can be seen as a modifier of A whose value reflects the range of reaction velocity from extremely slow at one end of the scale (high E/low T) to extremely fast at the other (low E/high T).

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P Mander September 2016

CarnotCycle is honored to announce that Rutgers, The State University of New Jersey, has included this thermodynamics blog on the reading list for students of physical chemistry. Founded in 1766, Rutgers is the eighth oldest college in the United States and is the largest institution for higher education in New Jersey.

CarnotCycle is committed to making topics in this area of science accessible to students worldwide. Thermodynamics has played – and continues to play – a major role in shaping our world. It can be a difficult subject, but time spent learning about thermodynamics is never wasted. It enriches knowledge and empowers the mind.

Link: http://andromeda.rutgers.edu/~huskey/345f17_lec.html

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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.

– – – –

Mme. Curie’s idea

The 2nd Solvay Conference (1913) La structure de la matière (The structure of matter)

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 t½ the mean µ will be λt½ = 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

The potential well diagram in Gurney and Condon’s article

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, V0 is the potential well depth, Qα is the alpha particle kinetic energy and µ is the reduced mass. The escape attempt rate is quite large, usually of the order of 1021 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

George Gamow and John Cockroft

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
http://www.nature.com/physics/looking-back/gurney/index.html

R. Swinne (1912) Über einige zwischen den radioaktiven Elementen bestehene Beziehungen, Physikalische Zeitschrift; XIII: 14-21
https://babel.hathitrust.org/cgi/pt?id=mdp.39015023176806;view=1up;seq=52;size=125

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P Mander August 2017