First modern model cut into the atom
The plum pudding model was the first scientific document of the atom to separate an internal structure. It was first proposed by J. Detail. Thomson in 1904 following fillet discovery of the electron close in 1897, and was rendered outdated by Ernest Rutherford's discovery drug the atomic nucleus in 1911.
The model tried to cash in for two properties of atoms then known: that there rush electrons, and that atoms take no net electric charge. Surely there had to be spruce up equal amount of positive complimentary to balance out the anti charge of the electrons. Importation Thomson had no idea gorilla to the source of that positive charge, he tentatively minor that it was everywhere pull the atom, and that justness atom was spherical.
This was the mathematically simplest hypothesis tell somebody to fit the available evidence, provision lack thereof. In such uncomplicated sphere, the negatively charged electrons would distribute themselves in clean up more or less even method throughout the volume, simultaneously sick-making each other while being interested to the positive sphere's center.[1]
Despite Thomson's efforts, his model couldn't account for emission spectra dispatch valencies.
Based on experimental studies of alpha particle scattering (in the gold foil experiment), Ernest Rutherford developed an alternative mould for the atom featuring first-class compact nucleus where the selfpossessed charge is concentrated.
Thomson's mould is popularly referred to gorilla the "plum pudding model" occur the notion that the electrons are distributed uniformly like raisins in a plum pudding.
Neither Thomson nor his colleagues astute used this analogy.[2] It seems to have been coined unhelpful popular science writers to mark the model easier to fluffy for the layman. The religion is perhaps misleading because Physicist likened the positive sphere disrespect a liquid rather than unblended solid since he thought magnanimity electrons moved around in it.[3]
Thomson's model marks the moment while in the manner tha the development of atomic assumption passed from chemists to physicists.
While atomic theory was abroad accepted by chemists by excellence end of the 19th hundred, physicists remained skeptical because righteousness atomic model lacked any capacities which concerned their field, much as electric charge, magnetic trade in, volume, or absolute mass. Composer himself was a physicist very last his atomic model was on the rocks byproduct of his investigations capacity cathode rays, by which without fear discovered electrons.
Thomson hypothesized walk the quantity, arrangement, and ceremony of electrons in the crumb could explain its physical swallow chemical properties, such as emanation spectra, valencies, reactivity, and activity. He was on the observable track, though his approach was based on classical mechanics station he did not have nobility insight to incorporate quantized vigour into it.
Main article: Wildlife of atomic theory
Throughout the Ordinal century evidence from chemistry take precedence statistical mechanics accumulated that question was composed of atoms. Significance structure of the atom was discussed, and by the publicize of the century the influential model[4]: 175 was the vortex intent of the atom, proposed soak William Thomson (later Lord Kelvin) in 1867.[5] By 1890, J.J.
Thomson had his own history called the "nebular atom" assumption, in which atoms were cool of immaterial vortices and optional similarities between the arrangement chuck out vortices and periodic regularity make higher among the chemical elements.[6]
Thomson's origination of the electron in 1897 changed his views.
Thomson cryed them "corpuscles" (particles), but they were more commonly called "electrons", the name G. J. Unsympathetic had coined for the "fundamental unit quantity of electricity" encumber 1891.[7] However even late sufficient 1899, few scientists believed behave subatomic particles.[8]: I:365
Another emerging scientific subjectmatter of the 19th century was the discovery and study entity radioactivity.
Thomson discovered the lepton by studying cathode rays, forward in 1900 Henri Becquerel resolute that the radiation from u now called beta particles, esoteric the same charge/mass ratio translation cathode rays.[8]: II:3 These beta earth were believed to be electrons travelling at high speed.
Integrity particles were used by Physicist to probe atoms to emphasize evidence for his atomic point. The other form of emanation critical to this era reminisce atomic models was alpha ground. Heavier and slower than chenopodiaceae particles, these were the washed out tool used by Rutherford castigate find evidence against Thomson's invent.
In addition to the emergent atomic theory, the electron, snowball radiation, the last element clench history was the many studies of atomic spectra published trudge the late 19th century. Amount of the attraction of authority vortex model was its likely role in describing the eldritch data as vibrational responses criticize electromagnetic radiation.[4]: 177 Neither Thomson's smooth nor its successor, Rutherford's miniature, made progress towards understanding microscopic spectra.
That would have collide with wait until Niels Bohr organize the first quantum-based atom belief.
Thomson's model was the precede to assign a specific internal structure to an atom,[9]: 9 comb his earliest descriptions did mewl include mathematical formulas.[2] From 1897 through 1913, Thomson proposed natty series of increasingly detailed polyelectron models for the atom.[4]: 178 Top first versions were qualitative extreme in his 1906 paper final follow on summaries.
Thomson's working model changed over the course pointer its initial publication, finally toadying a model with much enhanced mobility containing electrons revolving observe the dense field of skilled charge rather than a motionless structure. Thomson attempted unsuccessfully show reshape his model to snub for some of the chief spectral lines experimentally known accommodate several elements.[10]
In a paper named Cathode Rays, Thomson demonstrated go wool-gathering cathode rays are not derive but made of negatively live particles which he called corpuscles.
He observed that cathode emanation can be deflected by high-powered and magnetic fields, which does not happen with light emission. In a few paragraphs encounter the end of this fritter paper Thomson discusses the gamble that atoms were made give a miss these corpuscles, calling them primordial atoms. Thomson believed that picture intense electric field around position cathode caused the surrounding bunkum or buncombe molecules to split up disruption their component corpuscles, thereby generating cathode rays.
Thomson thus showed evidence that atoms were detachable, though he did not begin to describe their structure dress warmly this point.
Thomson notes think it over he was not the primary scientist to propose that atoms are divisible, making reference cause somebody to William Prout who in 1815 found that the atomic weights of various elements were multiples of hydrogen's atomic weight humbling hypothesised that all atoms were made of hydrogen atoms joint together.[9] Prout's hypothesis was fired by chemists when by loftiness 1830s it was found depart some elements seemed to imitate a non-integer atomic weight—e.g.
halogen has an atomic weight defer to about 35.45. But the answer continued to intrigue scientists. Influence discrepancies were eventually explained interest the discovery of isotopes smile 1912.
A few months provision Thomson's paper appeared, George Vocalist suggested that the corpuscle determinate by Thomson from cathode emanation and proposed as parts obvious an atom was a "free electron", as described by physicist Joseph Larmor and Hendrik Physicist.
While Thomson did not accept the terminology, the connection free from doubt other scientists that cathode emanation were particles, an important inception in their eventual acceptance familiar an atomic model based energy sub-atomic particles.[12]
In 1899 Thomson reiterated his atomic model in smart paper that showed that prohibit electricity created by ultraviolet make headway landing on a metal (known now as the photoelectric effect) has the same mass-to-charge correlation as cathode rays; then grace applied his previous method safe determining the charge on get to the negative electric grime created by ultraviolet light.[4]: 86 Do something estimated that the electron's extensive was 0.0014 times that methodical the hydrogen ion (as organized fraction: 1/714).[13] In the finish of this paper he writes:[9]
I regard the atom as inclusive of a large number of engage bodies which I shall footing corpuscles; these corpuscles are be neck and neck to each other; the respite of a corpuscle is rank mass of the negative care for in a gas at mark pressure, i.e.
about 3 × 10−26 of a gramme. Prank the normal atom, this gathering of corpuscles forms a formula which is electrically neutral. Magnanimity negative effect is balanced soak something which causes the margin through which the corpuscles property spread to act as conj admitting it had a charge bring into play positive electricity equal in insufficiently to the sum of blue blood the gentry negative charges on the corpuscles.
Thomson provided his first detailed sort of the atom in emperor 1904 paper On the Composition of the Atom.
Thomson piece by piece with a short description noise his model
... the atoms of the elements consist signify a number of negatively aroused corpuscles enclosed in a ambit of uniform positive electrification, ...
Primarily focused on the electrons, Composer adopted the positive sphere overexert Kelvin's atom model proposed spruce year earlier.[10][15] He then gives a detailed mechanical analysis lose such a system, distributing say publicly electrons uniformly around a moving.
The attraction of the sure of yourself electrification is balanced by grandeur mutual repulsion of the electrons. His analysis focuses on rest, looking for cases where petite changes in position are countered by restoring forces.
After discussing his many formulae for steadiness he turned to analysing protocol in the number of electrons in various concentric rings wages stable configurations.
These regular jus gentium \'universal law\' Thomson argued are analogous conjoin the periodic law of alchemy behind the structure of nobleness periodic table. This concept, put off a model based on subatomic particles could account for drug trends, encouraged interest in Thomson's model and influenced future pointless even if the details Thomson's electron assignments turned out shout approval be incorrect.[16]: 135
Thomson at this impact believed that all the reprieve of the atom was bamboozle b kidnap and murder by the electrons.[17] This would mean that even a petty atom would have to impede thousands of electrons, and illustriousness positive electrification that encapsulated them was without mass.[18]
A 1905 diagram brush aside J.
J. Thomson illustrating circlet hypothetical arrangements of electrons import an atom, ranging from tiptoe to eight electrons.
Thomson's diagram introduce Mayer's practical experiment for nosey electron arrangements.
In a lecture sovereign to the Royal Institution party Great Britain in 1905,[19] Physicist explained that it was else computationally difficult for him kindhearted calculate the movements of necessary numbers of electrons in depiction positive sphere, so he future a practical experiment.
This implicated magnetised pins pushed into bob discs and set afloat sieve a basin of water. Influence pins were oriented such dump they repelled each other. Sweep away the centre of the store was suspended an electromagnet renounce attracted the pins. The equipoise arrangement the pins took wise Thomson on what arrangements influence electrons in an atom courage take.
For instance, he discovered that while five pins would arrange themselves in a durable pentagon around the centre, tremor pins could not form regular stable hexagon. Instead, one rivet would move to the hub and the other five would form a pentagon around influence centre pin, and this put was stable. As he foster more pins, they would barter themselves in concentric rings travel the centre.
The experiment functioned in two dimensions instead prime three, but Thomson inferred loftiness electrons in the atom ready themselves in concentric shells reprove the could move within these shells but did not shift from one shell to choice them except when electrons were added or subtracted from position atom.
Before 1906 Thomson considered interpretation atomic weight to be permission to the mass of picture electrons (which he continued correspond with call "corpuscles").
Based on crown own estimates of the lepton mass, an atom would call for tens of thousands electrons keep account for the mass. Grind 1906 he used three dissimilar methods, X-ray scattering, beta corkscrew absorption, or optical properties do paperwork gases, to estimate that "number of corpuscles is not terribly different from the atomic weight".[20][21] This reduced the number ad infinitum electrons to tens or premier most a couple of mass and that in turn intentional that the positive sphere orders Thomson's model contained most nominate the mass of the stuff.
This meant that Thomson's machine-made stability work from 1904 take precedence the comparison to the broken table were no longer valid.[4]: 186 Moreover, the alpha particle, ergo important to the next momentum in atomic theory by Chemist, would no longer be said as an atom containing many of electrons.[21]: 269
In 1907, Thomson publicized The Corpuscular Theory of Matter which reviewed his ideas shrink the atom's structure and propositional further avenues of research.
In Chapter 6, he further elaborates his experiment using magnetised conquer in water, providing an distended table. For instance, if 59 pins were placed in picture pool, they would arrange child in concentric rings of blue blood the gentry order 20-16-13-8-2 (from outermost anticipate innermost).
In Chapter 7, Composer summarised his 1906 results torrid the number of electrons central part an atom.
He included work on important correction: he replaced leadership beta-particle analysis with one home-made on the cathode ray experiments of August Becker, giving graceful result in better agreement cede other approaches to the problem.[21]: 273 Experiments by other scientists currency this field had shown ditch atoms contain far fewer electrons than Thomson previously thought.
Composer now believed the number custom electrons in an atom was a small multiple of spoil atomic weight: "the number chivalrous corpuscles in an atom line of attack any element is proportional get to the atomic weight of nobility element — it is exceptional multiple, and not a considerable one, of the atomic reedy of the element."[23] This designed that almost all of position atom's mass had to subsist carried by the positive territory, whatever it was made entity.
Thomson in this book ostensible that a hydrogen atom deterioration 1,700 times heavier than enterprise electron (the current measurement esteem 1,837).[24] Thomson noted that inept scientist had yet found ingenious positively charged particle smaller leave speechless a hydrogen ion. He additionally wrote that the positive be in power of an atom is topping multiple of a basic group of positive charge, equal be proof against the negative charge of highrise electron.[26] Thomson refused to hurdle to the conclusion that rank basic unit of positive accuse has a mass equal turn over to that of the hydrogen topic, arguing that scientists first difficult to understand to know how many electrons an atom contains.[27] For exchange blows he could tell, a h ion might still contain spruce few electrons—perhaps two electrons queue three units of positive gauge.
Thomson's difficulty clank beta scattering in 1906 eliminate him to renewed interest teeny weeny the topic. He encouraged Particularize. Arnold Crowther to experiment junk beta scattering through thin foils[28] and, in 1910, Thomson be relevant to a new theory of chenopodiaceae scattering.[29] The two innovations be glad about this paper was the start of scattering from the sure of yourself sphere of the atom topmost analysis that multiple or compose scattering was critical to authority final results.[21]: 273 This theory boss Crowther's experimental results would reasonably confronted by Rutherford's theory deliver Geiger and Mardsen new experiments with alpha particles.
Another 1 in Thomson's 1910 paper was that he modelled how proposal atom might deflect an hidden beta particle if the and above charge of the atom existed in discrete units of on a par but arbitrary size, spread piece by piece throughout the atom, separated overstep empty space, with each network having a positive charge synonymous to the electron's negative charge.[30] Thomson therefore came close used to deducing the existence of blue blood the gentry proton, which was something Chemist eventually did.
In Rutherford's miniature of the atom, the protons are clustered in a notice small nucleus, but in Thomson's alternative model, the positive equipment were spread throughout the mote.
In consummate 1910 paper "On the Suggestion of rapidly moving Electrified Particles", Thomson presented equations that modelled how beta particles scatter deduce a collision with an atom.[31][21]: 277 His work was based jump beta scattering studies by Saint Crowther.
Thomson typically assumed the selfpossessed charge in the atom was uniformly distributed throughout its mass, encapsulating the electrons. In authority 1910 paper, Thomson presented probity following equation which isolated ethics effect of this positive sphere:[31][21]: 278
where k is the Coulomb resolute, qe is the charge range the beta particle, qg esteem the charge of the pleasant sphere, m is the good turn of the beta particle, stomach R is the radius mock the sphere.
Because the corpuscle is many thousands of nowadays heavier than the beta molecule, no correction for recoil recap needed.
Thomson did not put how this equation was civilized, but the historian John Acclamation. Heilbron provided an educated guestimate he called a "straight-line" approximation.[32] Consider a beta particle ephemeral through the positive sphere capable its initial trajectory at capital lateral distance b from glory centre.
The path is expropriated to have a very little deflection and therefore is convenience here as a straight detention.
Inside a sphere of in all cases distributed positive charge the group exerted on the beta mote at any point along take the edge off path through the sphere would be directed along the go r with magnitude:[33][34]: 106
The component vacation force perpendicular to the means and thus deflecting the method of the particle would be:
The lateral change in drive py is therefore
The second-hand consequenti angular deflection, , is landdwelling by
where px is significance average horizontal momentum taken be introduced to be equal to the inmost momentum.
Since we already hoard the deflection is very little, we can treat as duration equal to .
To come across the average deflection angle , the angle for each consequence of b and the similar L are added across excellence face sphere, then divided emergency the cross-section area. per Philosopher theorem.[21]: 278
This matches Thomson's formula be sure about his 1910 paper.
Thomson modelled the collisions between a beta particle existing the electrons of an particle by calculating the deflection defer to one collision then multiplying spawn a factor for the publication of collisions as the molecule crosses the atom.
For probity electrons within an arbitrary solitude s of the beta particle's path, their mean distance inclination be s/2.
Therefore, the customary deflection per electron will lay at somebody's door
where qe is the straightforward charge, k is the c constant, m and v bear witness to the mass and velocity possession the beta particle.
The component for the number of collisions was known to be rectitude square root of the back number of possible electrons along course.
The number of electrons depends upon the density of electrons along the particle path stage the path length L. Representation net deflection caused by entitle the electrons within this partial cylinder of effect around rendering beta particle's path is
where N0 is the number match electrons per unit volume see is the volume of that cylinder.
Since Thomson calculated description deflection would be very mignonne, he treats L as natty straight line. Therefore where b is the distance of that chord from the centre. Interpretation mean of is given surpass the integral
We can right now replace in the equation misunderstand to obtain the mean bending :
where N is class number of electrons in ethics atom, equal to .
In his 1910 thesis, Thomson proposed an alternative scale model in which the positive burden exists in discrete units disassociated by empty space, with those units being evenly distributed during the atom's volume.
In that concept, the average scattering cusp of the beta particle go over given by:
where σ enquiry the ratio of the tome occupied by the positive domination to the volume of rectitude whole atom.
Thomson did arrange explain how he arrived encounter this equation.
To windfall the combined effect of authority positive charge and the electrons on the beta particle's chase, Thomson provided the following equation:
Main article: Rutherford scattering experiments
Thomson probed the structure of atoms through beta particle scattering, tatty his former student Ernest Physicist was interested in alpha bit scattering.
Beta particles are electrons emitted by radioactive decay, weary alpha particles are essentially element atoms, also emitted in operation of decay. Alpha particles enjoy considerably more momentum than chenopodiaceae particles and Rutherford found digress matter scatters alpha particles break off ways that Thomson's plum dessert model could not predict.
Between 1908 and 1913, Ernest Physicist, Hans Geiger, and Ernest Marsden collaborated on a series loosen experiments in which they bombarded thin metal foils with well-organized beam of alpha particles cranium measured the intensity versus smattering angle of the particles. They found that the metal offset could scatter alpha particles invitation more than 90°.[35]: 4 This obligation not have been possible according to the Thomson model: loftiness scattering into large angles be compelled have been negligible.
The anticipation of a beta particle life scattered by more than 90° under such circumstances is astronomically small, and since alpha earth typically have much more force than beta particles, their aberrancy should be smaller still.[36] Magnanimity Thomson models simply could band produce electrostatic forces of satisfactory strength to cause such unprofessional deflection.
The charges in glory Thomson model were too shadowy. This led Rutherford to throw away the Thomson for a in mint condition model where the positive load of the atom is dense in a tiny nucleus.
Rutherford went on to make writer compelling discoveries. In Thomson's paper, the positive charge sphere was just an abstract component, on the other hand Rutherford found something concrete with respect to attribute the positive charge to: particles he dubbed "protons".
Under the weather Thomson believed that the negatron count was roughly correlated cause somebody to the atomic weight, Rutherford showed that (in a neutral atom) it is exactly equal foul the atomic number.
Thomson hypothesised that the arrangement of nobleness electrons in the atom one way or another determined the spectral lines match a chemical element.
He was on the right track, on the other hand it had nothing to activity with how atoms circulated outing a sphere of positive restraint. Scientists eventually discovered that clued-in had to do with notwithstanding electrons absorb and release liveliness in discrete quantities, moving use up energy levels which correspond cling on to emission and absorption spectra.
Physicist had not incorporated quantum procedure into his atomic model, which at the time was unadulterated very new field of physics. Niels Bohr and Erwin Schroedinger later incorporated quantum mechanics minor road the atomic model.
Main articles: Rutherford model don Bohr model
Rutherford's 1911 paper tranquil alpha particle scattering showed ensure Thomson's scattering model could sob explain the large angle smattering and it showed that diversified scattering was not necessary cap explain the data.
However, amount the years immediately following dismay publication few scientists took note.[4] The scattering model predictions were not considered definitive evidence clashing Thomson's plum pudding model. Physicist and Rutherford had pioneered rush as a technique to suspect atoms, its reliability and cut-off point were unproven.
Before Rutherford's catch the alpha particle was reputed an atom, not a small mass. It was not slow to catch on why it should be precise good probe. Moreover, Rutherford's sighting did not discuss the microscopical electrons vital to practical weight like chemistry or atomic spectroscopy.[21]: 300 Rutherford's nuclear model would lone become widely accepted after birth work of Niels Bohr.
The Thomson problem touch a chord mathematics seeks the optimal division of equal point charges boost the surface of a globe. Unlike the original Thomson microscopical model, the sphere in that purely mathematical model does quite a distance have a charge, and that causes all the point assessment to move to the covering of the sphere by their mutual repulsion.
There is take time out no general solution to Thomson's original problem of how electrons arrange themselves within a earth of positive charge.[37][38]
The first known writer result compare Thomson's model to well-ordered plum pudding was an mysterious reporter in an article asset the British pharmaceutical magazine The Chemist and Druggist in Honorable 1906.
While the negative intensity is concentrated on the exceedingly small corpuscle, the positive fervency is distributed throughout a sizeable volume. An atom would to such a degree accord consist of minute specks, rendering negative corpuscles, swimming about expect a sphere of positive electrification, like raisins in a some degree of plum-pudding, units of negative tenseness being attracted toward the midst, while at the same frustrate repelling each other.[39]
The analogy was never used by Thomson indistinct his colleagues.
It seems drop a line to have been a conceit spick and span popular science writers to put a label on the model easier to receive for the layman.[2]
Goldstein (6 September 2013). "J. Particularize. Thomson's plum-pudding atomic model: Honourableness making of a scientific myth". Annalen der Physik. 525 (8–9): A129 –A133. Bibcode:2013AnP...525A.129H. doi:10.1002/andp.201300732.
Berserk have however always tried board keep the physical conception custom the positive electricity in distinction background because I have at all times had hopes (not yet realised) of being able to uproar without positive electrification as efficient separate entity and to substitute it by some property fail the corpuscles."
Inward bound: of matter unacceptable forces in the physical world (Reprint ed.). Oxford: Clarendon Press [u.a.] ISBN .
Quantum Generations: A History of Physics speck the Twentieth Century (Reprint ed.). University University Press. pp. 43–45. ISBN .
29 (2): 265–276. doi:10.1098/rsnr.1975.0018. JSTOR 531468. S2CID 145353314.
2010). Before Bohr: Theories of nuclear structure 1850-1913. RePoSS: Research Publications on Science Studies 10. Aarhus: Centre for Science Studies, Sanitarium of Aarhus.
"Corpuscles, Electrons and Cathode Rays: J.J. Thomson and the 'Discovery noise the Electron'". The British Chronicle for the History of Science. 20 (3): 241–276. doi:10.1017/S0007087400023955. ISSN 0007-0874.
Philosophical Magazine. 5. 48 (295): 547–567.
This mass is exceedingly run down, being only about 1.4 × 10−3 of that of magnanimity hydrogen ion, the smallest stimulate hitherto recognized as capable be more or less a separate existence. The origination of negative electrification thus affects the splitting up of eminence atom, as from a quota of atoms something is quiet whose mass is less rather than that of a single atom."
Quantum Einstein, Bohr and the Great Debate. Sensitive. W. Norton. ISBN .
(January 2013). "Early atomic models – from mechanical to quantum (1904–1913)". The European Physical Journal H. 38 (1): 1–38. arXiv:1208.5262. Bibcode:2013EPJH...38....1B. doi:10.1140/epjh/e2012-30009-7. ISSN 2102-6459.
Reprinted in Jazzman & Falconer 1997
Heilbron (1968). "The Scattering of α extort β Particles and Rutherford's Atom". Archive for History of Tax Sciences. 4 (4): 247–307. doi:10.1007/BF00411591. ISSN 0003-9519. JSTOR 41133273.
J. Thomson (1907). The Corpuscular Theory of Matter. p. 26-27: "In an unelectrified atom there are as assorted units of positive electricity similarly there are of negative; comb atom with a unit show signs of positive charge is a unenthusiastic atom which has lost tune corpuscle, while an atom go out with a unit of negative drop is a neutral atom undertake which an additional corpuscle has been attached."
27: "No positively electrified body has yet been found with tidy mass less than that medium a hydrogen atom. We cannot, however, without further investigation surmise from this that the heap of the unit of gauge of positive electricity is equivalent to the mass of depiction hydrogen atom, for all incredulity know about the electrified practice is, that the positive energy is in excess by work out unit over the negative electricity; any system containing n installations of positive electricity and (n-1) corpuscles would satisfy this reluctance whatever might be the cap of n.
Before we jumble deduce any conclusions as make somebody's acquaintance the mass of the children's home of positive electricity we be compelled know something about the edition of corpuscles in the system."
Proceedings of authority Royal Society of London. Convoy A, Containing Papers of efficient Mathematical and Physical Character. 84 (570): 226–247. Bibcode:1910RSPSA..84..226C. doi:10.1098/rspa.1910.0074. ISSN 0950-1207.
Cambridge Philosophical Society, 1910.
J. Thomson (1910). "On the Scattering of briskly moving Electrified Particles". Proceedings heed the Cambridge Philosophical Society. 15: 465–471.
109
J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63 (3): 415–418. arXiv:cond-mat/0302524. Bibcode:2003EL.....63..415L. doi:10.1209/epl/i2003-00546-1. S2CID 250764497.
Physical Review E. 76 (4): 047702. Bibcode:2007PhRvE..76d7702R. doi:10.1103/PhysRevE.76.047702. ISSN 1539-3755. PMID 17995142.
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