Plum pudding model

The now obsolete plum pudding model was the first scientific model of the atom with internal structure. It was first proposed by J. J. Thomson in 1904 following his discovery of the electron in 1897 and subsequently rendered obsolete by Ernest Rutherford's discovery of the atomic nucleus in 1911. The model tried to account for two properties of atoms then known: that there are electrons and that atoms have no net electric charge. Logically there had to be a commensurate quantity of positive charge to balance out the negative charge of the electrons, but having no clue as to the source of this positive charge, Thomson tentatively proposed it was everywhere in the atom, the atom being in the shape of a sphere for the sake of mathematical simplicity. Following from this, Thomson imagined that the balance of electrostatic forces in the atom would distribute the electrons in a more or less even manner throughout this hypothetical sphere.

Thomson attempted without success to develop a complete model that could predict any other known properties of the atom such as emission spectra or valencies. Based on experimental studies of alpha particle scattering, Ernest Rutherford developed an alternative model for the atom featuring a compact nuclear center. This model was taken up by Niels Bohr as the basis of the first quantum atom model.

Thomson's model is popularly referred to as the "plum pudding model" with the notion that the electrons are distributed with similar density as raisins in a plum pudding. Neither Thomson nor his colleagues ever used this analogy. It seems to have been conceived by popular science writers to make the model accessible to the layman. The analogy is perhaps misleading because Thomson likened the sphere to a liquid rather than a solid since he thought the electrons moved around in it.

Significance
Thomson's Plum-pudding model of the atom is one of a series of atomic models running back to the philosophical models of the ancient Greeks through John Dalton's chemistry based atom through to the modern quantum atom of atomic physics. Among these models, Thomson's can be consider the first modern model. Thomson's model is distinguished by being the first with internal structure; it was the best available model between 1904 and 1910. Thomson's model introduced the idea that successive elements in the periodic chart would be characterized by additions of single electrons. His concentric rings of electrons introduced an idea that later became "core" and "valence" electrons. And his model was based on a model of mechanical stability. Being based on experimentally studied subatomic "corpuscles", now known as electrons, Thomson's model was the first model to be subject to direct experimental tests. By 1909 these tests began to reveal new ideas and in 1911 Ernest Rutherford used experimental scattering data to propose a new atomic model.

Background
Throughout the 19th century evidence from chemistry and statistical mechanics accumulated that matter was composed of atoms. The structure of the atom was discussed, and by the end of the century the leading model was vortex theory of the atom, proposed by William Thomson (later Lord Kelvin) in 1867. By 1890, J.J. Thomson had his own version called the "nebular atom" hypothesis, in which atoms were composed of immaterial vortices and suggested there were similarities between the arrangement of vortices and periodic regularity found among the chemical elements.

Thomson's discovery of the electron in 1897 changed his views. While Thomson called them "corpuscles" (particles), but they were more commonly called "electrons", the name G. J. Stoney had coined for the "fundamental unit quantity of electricity" in 1891. However even late in 1899, few scientists believed in subatomic particles.

Another emerging scientific theme of the 19th century was the discovery and study of radioactivity. Thomson discovered the electron by studying cathode rays and in 1900 Henri Becquerel determined that the highly penetrating radiation from uranium, now called beta particles, had the same charge/mass ratio as cathode rays. These beta particles were believed to be electrons traveling at much high speeds. These beta particles would be used by Thomson to probe atoms to find evidence for his atomic theory. The other form of radiation critical to this era of atomic models was alpha particles. Heavier and slower than beta particles, these would be the key tool used by Rutherford to find evidence against Thomson's model.

In addition to the emerging atomic theory, the electron, and radiation, the last element of history was the many studies of atomic spectra published near the end of the 19th century. Part of the attraction of the vortex model was its possible role in describing the spectral data as vibrational responses to electromagnetic radiation. Neither Thomson's model or its successor, Rutherford's model made progress towards understanding atomic spectra. That would have to wait until Niels Bohr built the first quantum-based atom model.

Development
Thomson's model was the first to assign a specific inner structure to an atom, though his earliest descriptions did not include mathematical formulas. From 1897 through 1913, Thomson proposed a series of increasingly detailed polyelectron models for the atom. His first versions were qualitative culminating in his 1906 paper and follow on summaries. Thomson's model changed over the course of its initial publication, finally becoming a model with much more mobility containing electrons revolving in the dense field of positive charge rather than a static structure. Thomson attempted unsuccessfully to reshape his model to account for some of the major spectral lines experimentally known for several elements.

1897 Corpuscles inside atoms
In a paper titled Cathode Rays, Thomson demonstrated that cathode rays are not light but made of negatively charged particles which he called corpuscles. He observed that cathode rays can be deflected by electric and magnetic fields, which does not happen with light rays. In a few paragraphs near the end of this long paper Thomson discusses the possibility that atoms were made of these corpuscles, calling them primordial atoms. Thomson believed that the intense electric field around the cathode caused the surrounding gas molecules to split up into their component corpuscles, thereby generating cathode rays. Thomson thus showed evidence that atoms were in fact divisible, though he did not attempt to describe their structure at this point.

Thomson notes that he was not the first scientist to propose that atoms are actually divisible, making reference to William Prout who in 1815 noted that the atomic weights of various elements were multiples of hydrogen's atomic weight and hypothesized that all atoms were hydrogen atoms fused together. While Prout's hypothesis was dismissed by chemists when it was found by the 1830s that some elements seemed to have a non-integer atomic weight—e.g. chlorine has an atomic weight of about 35.45—the concept continued to have influence. Eventually the discrepancies would be explained with the discovery of isotopes and nuclear structure in the early 20th century.

A few months after Thomson's paper appeared, George FitzGerald suggested that the corpuscle identified by Thomson from cathode rays and proposed as parts of an atom was a "free electron", as described by physicist Joseph Larmor and Hendrik Lorentz. While Thomson did not adopt the terminology, the connection convinced other scientists that cathode rays were particles, an important step in their eventual acceptance of an atomic model based on sub-atomic particles.

In 1899, reiterated his atomic model in a paper that showed that negative electricity created by ultraviolet light landing on a metal (known now as the photoelectric effect) has the same mass-to-charge ratio as cathode rays; then he applied his previous method for determining the charge on ions to the negative electric particles created by ultraviolet light. By this combination he estimated that the electron's mass was 0.0014 times that of the hydrogen ion (as a fraction: $1⁄714$). In the conclusion of this paper he writes: "I regard the atom as containing a large number of smaller bodies which I shall call corpuscles; these corpuscles are equal to each other; the mass of acorpuscle is the mass of the negative ion in a gas at low pressure, i.e. about 3 × 10-26 of a gramme. In the normal atom, this assemblage of corpuscles forms a system which is electrically neutral. The negative effect is balanced by something which causes the space through which the corpuscles are spread to act as if it had a charge of positive electricity equal in amount to the sum of the negative charges on the corpuscles."

1904 Mechanical model of the atom
Thomson provided his first detailed description of the atom in his 1904 paper On the Structure of the Atom. Thomson starts with a short description of his model "... the atoms of the elements consist of a number of negatively electrified corpuscles enclosed in a sphere of uniform positive electrification, ..." Primarily focused on the corpuscles, Thomson adopted the positive sphere from Kelvin's atom model proposed a year earlier. He then gives a detailed mechanical analysis of such a system, distributing the corpuscles uniformly around a ring. The attraction of the positive electrification is balanced by the mutual repulsion of the corpuscles. His analysis focuses on stability, looking for cases were small changes in position are countered by restoring forces.

After discussing his many formulae for stability he turned to analyzing patterns in the number of electrons in various concentric rings of stable configurations. These regular patterns Thomson argued are analogous to the periodic law of chemistry behind the struture of the periodic table. This concept, that a model based subatomic particles could account for chemical trends, encouraged interest in Thomson's model and influenced future work even if the details Thomson's electron assignments turned out to be incorrect.

Thomson believed that all the mass of the atom was carried by the electrons. This would mean that even a small atom would have to contain thousands of electrons, and the positive electrification the encapsulated them was without mass.

In a lecture delivered to the Royal Institution of Great Britain in 1905, Thomson reviewed his 1904 paper and demonstrated some of its concepts with a practical experiment invented by Alfred M. Mayer in 1878. The demonstration involved magnetized pins pushed into cork disks and set afloat in a basin of water. The magnetized pins were oriented such that they repelled each other. Above the center of the basin was suspended an electromagnet that attracted the pins towards the center. The equilibrium arrangement the pins took informed Thomson on what arrangements the electrons in an atom might take and he provided a brief table.

For instance, he observed that while five pins would arrange themselves in a stable pentagon around the center, six pins could not form a stable hexagon. Instead, one pin would move to the center and the other five would form a pentagon around the center pin, and this arrangement was stable. As he added more pins, they would arrange themselves in concentric rings around the center.

From this, Thomson believed the electrons arranged themselves in concentric shells, and the electrons could move about within these shells but did not move out of them unless electrons were added or subtracted from the atom.

1906 Estimating electrons per atom
Before 1906 Thomson considered the atomic weight to be due to the mass of the electrons (which he continued to call "corpuscles"). Based on his own estimates of the electron mass, an atom would need tens of thousands electrons to account for the mass. In 1906 he used three different methods, X-ray scattering, beta ray absorption, or optical properties of gases, to estimate that "number of corpuscles is not greatly different from the atomic weight". This reduced the number of electrons to tens or at most a couple of hundred and it required that the positive sphere in Thomson's atom model contain most of the mass of the atom. This in turn meant that Thomson's mechanical stability work from 1904 and the comparison to the periodic table were no longer valid. Moreover the alpha particle, so important to the next advance in atomic theory by Rutherford, would no longer be viewed as an atom containing thousands of electrons.

In 1907, Thomson published The Corpuscular Theory of Matter which reviewed his ideas on the atom's structure and proposed further avenues of research.

In Chapter 6, he further elaborates his experiment using magnetized pins in water, providing an expanded table. For instance, if 59 pins were placed in the pool, they would arrange themselves in concentric rings of the order 20-16-13-8-2 (from outermost to innermost).

In Chapter 7, Thomson summarized his 1906 results on the number of electrons in an atom. He included one important correction: he replaced the beta-particle analysis with one based on the cathode ray experiments of August Becker, giving a result in better agreement with other approaches to the problem. Experiments by other scientists in this field had shown that atoms contain far fewer electrons than Thomson previously thought. Thomson now believed the number of electrons in an atom was a small multiple of its atomic weight: "the number of corpuscles in an atom of any element is proportional to the atomic weight of the element — it is a multiple, and not a large one, of the atomic weight of the element."

This would mean that almost all of the atom's mass was carried by the positive sphere. In this book he now estimates that a hydrogen atom is 1,700 times heavier than an electron (the current measurement is 1,837). Thomson still did not know what substance constituted the positive electrification, though he noted that no scientist had yet found a positively-charged particle smaller than a hydrogen ion.

1910 Multiple scattering
Thomson's difficulty with beta scattering in 1906 lead him to renewed interest in the topic. He encouraged J. Arnold Crowther to experiment with beta scattering through thin foils and, in 1910, Thomson produced a new theory of beta scattering. The two innovations in this paper was the introduction of scattering from the positive sphere of the atom and analysis that multiple or compound scattering was critical to the final results. This theory and Crowther's experimental results would be confronted by Rutherford's theory and Geiger and Mardsen new experiments with alpha particles.

Rutherford's new evidence
Between 1908 and 1913, Ernest Rutherford, Hans Geiger, and Ernest Marsden collaborated on a series of experiments in which they bombarded metal foils with a beam of alpha particles and measured the intensity versus scattering angle of the particles. Gold was their preferred material because gold is very malleable and can therefore be made into an especially thin foil. They found that the gold foil could scatter alpha particles by more than 90 degrees. This should not have been possible according to the Thomson model: the scattering into large angles should have been negligible. The positive charge in the Thomson model is too diffuse to produce an electric field of sufficient strength to cause such scattering and the electrons are too light to alter the course of the alpha particle. Rutherford deduced that the positive charge of the atom, along with most of the atom's mass, was concentrated in a tiny nucleus at the center of the atom. Only such an intense concentration of charge and mass, could have scattered the alpha particle beam so dramatically.

Alpha particle scattering from Thomson's atom
Historically, Thomson and Crowther studied the scattering of beta particles from metal foils while Rutherford and Geiger used alpha particles. Alpha particles had an advantage unknown at the time: they had enough momentum to ignore interactions with atomic electrons but not enough energy to penetrated the nucleus. Prior to Rutherford's 1911 paper, there was no alpha particle scattering model.

Beiser has analyzed alpha scattering from a plum-pudding-like atomic model. The analysis is divided into three parts: scattering from electrons, scattering from the positive sphere, and the consequences of multiple scattering.

Scattering from atomic electrons
Beiser notes that the electron is 7000 times lighter than the alpha particle. In any collision, the electron will be forcefully pushed aside. The largest deflection of the alpha particle would occur when the electron is pushed away at 90 degrees. For an upper limit on the sideways push, Beiser calculates the largest possible push for any angle: a direct head-on collision. That is, he uses the geometry of 90 degree scattering as shown in the diagram, but gets the size of $$\Delta p$$ from head-on collision as follows.

Using $$M, V'$$ for the alpha particle and $$m, v'$$ for the atomic electron, with unprimed velocities measured before collision, conservation of momentum means the electron momentum change equals the alpha particle momentum change: $$\Delta p = MV - MV' = mv'$$ Similarly, all of the energy lost by the alpha particle is gained by the electron: $$(1/2)MV^2 - (1/2)MV'^2 = (1/2)mv'^2$$ Separating the mass and velocity terms and using $$V^2 - V'^2 = (V+V')(V-V')$$ allows the energy formula to be divided by the momentum formula giving $$ V + V' = v'$$ Next Beiser eliminates the alpha particle's final velocity $$V'$$ using the momentum conservation equation again: $$v' = \frac{2M}{M+m} V$$ Since the mass of the alpha particle is much greater than the mass of the electron, $$v' \approx 2V$$ and the head-on momentum change is $$\Delta p = mv' = 2mV$$.

From the geometry in the diagram $$\sin\theta = \Delta p / p'$$ but the angle is so small that we can use $$\theta \approx \Delta p / p$$, giving the angle for alpha scattering by the atomic electrons as $$\theta_{e-\alpha} = \frac{2mV}{MV} = \frac{2}{7000} \approx 3\times 10^{-4} \text{radians} = 0.02^{\circ}.$$ Thus alpha particles scatter very little from the very light electrons.

Scattering from the positive sphere
Next Beiser considers the effect of scattering from the positive sphere as Thomson had used for his model of beta scattering from atoms. The alpha particle will experience maximum deflection if it just grazes the edge of the positive sphere because that is where the electric field is at its strongest. If the alpha particle were to pass through the sphere, not all of its positive charge would be pushing it out.



In the impulse model the average repelling force is applied for a time where that force is large. The alpha particle's lateral change in momentum Δpy can be approximated using the Coulomb force over the distance equal to the radius of the atom, $$\frac{kq_a q_g}{r^2}$$, applied just for the time the alpha particle passes, which is equal to $2r⁄v$:

$$\Delta p_\text{y} = \bar{F} t \approx \frac{kq_a q_g}{r^2} \cdot \frac{2r}{v}$$

where


 * qg = positive charge of the gold atom = $79 e$ = $1.26 C$
 * qa = charge of the alpha particle = $2 e$ = $3.2 C$
 * r = radius of the gold atom = $1.44 m$
 * v = speed of the alpha particle = $1.53 m/s$
 * m = mass of the alpha particle = $6.64 kg$
 * k = Coulomb constant = $8.987 N·m^{2}/C^{2}$

Geometrically, the tangent of the deflection angle θ is the ratio of the lateral and forward momentum but for small angles $$\tan \theta \approx \theta$$ giving: $$\theta_{+-\alpha} = \frac{\Delta p_\text{y}}{p_\text{x}} \approx \frac{kq_a q_g}{r^2} \cdot \frac{2r}{v} \cdot \frac{1}{mv} = 0.0003~\text{radians}~(\text{or}~0.02^\circ)$$ Alpha particle scatter very little from the relatively large, diffuse positive sphere.

Scattering by multiple collisions
Despite the extremely small scattering angles from either atomic electrons or a positive sphere, a gold foil like the one Rutherford and his colleagues used would be around 10,000 atoms thick. Beiser concludes his analysis of alpha particle scattering by considering the combination of many collisions.

Each collision may increase or decrease the total scattering angle. Only very rarely would a series of collisions all line up in the same direction. The result is similar to the standard statistical problem called a random walk. If the average deflection angle of the alpha particle in a single collision with an atom is $$\bar{\theta}$$, then the average deflection after n collisions is $$\bar\theta_n = \bar{\theta}\sqrt{n}$$ The probability that an alpha particle will be deflected by a total of more than 90° after n deflections is given by: $$e^{-(90 / \bar\theta_n)^2}$$ Where e is Napier's constant. If we assume an average deflection per collision of 0.01°, and therefore an average deflection of 1° after 10,000 collisions, then the probability of an alpha particle being deflected by more than 90° will be $$e^{-(90 / 1)^2} = e^{-8100} \approx 10^{-3518}$$ While in Thomson's "plum pudding" model it is possible that an alpha particle could be deflected by more than 90° after 10,000 collisions, the probability of such an event is so low as to be undetectable. This extremely small number shows that Thomson's model of 1906 cannot explain the Geiger-Mardsen results of 1909.

Contemporary reactions
Rutherford's 1911 paper on alpha particle scattering contained largely the same points as described above and yet in the years immediate following its publication few scientists took note. The scattering model predictions were not considered definitive evidence against Thomson's plum pudding model. Thomson and Rutherford had pioneered scattering as a technique to probe atoms, its reliability and value were unproven. Before Rutherford's paper the alpha particle was considered an atom, not a compact mass. It was not clear why it should be a good probe. Rutherford's paper did not discuss the atomic electrons vital to practical problems like chemistry or atomic spectroscopy. Rutherford's nuclear model would only become widely accepted after the work of Niels Bohr.

Mathematical Thomson problem
The Thomson problem in mathematics seeks the optimal distribution of equal point charges on the surface of a sphere; it is a generalization of the plum pudding model in the absence of its uniform positive background charge.

Origin of the nickname
The first known writer to compare Thomson's model to a plum pudding, a British dessert with whole raisins, was an anonymous reporter who wrote an article for the British pharmaceutical magazine The Chemist and Druggist in August 1906.

"While the negative electricity is concentrated on the extremely small corpuscle, the positive electricity is distributed throughout a considerable volume. An atom would thus consist of minute specks, the negative corpuscles, swimming about in a sphere of positive electrification, like raisins in a parsimonious plum-pudding, units of negative electricity being attracted toward the centre, while at the same time repelling each other."

The analogy was never used by Thomson nor his colleagues. It seems to have been a conceit of popular science writers to make the model easier to understand for the layman.