The Kelvin-Thomson
atom
Part 2: The manyelectron atoms
ALAN J WALTON
Physics Department, Open University,
Milton Keynes
In part 1 of this article (Phys. Educ. 1977 12 326-8)
we examined the structure of atoms containing up to
six electrons. We did so on the assumption that the
electrons were at rest withina sphere of uniform
charge. In this article we shall sec how it is possibleto
have a mechanically stable ring of more than three
electrons, provided the ring is rotating. We shall also
examine some of properties predicted for elements in
the Kelvin-Thomson model.
The manyelectron atom
Having discussed the electronic arrangements within
the one- to sixelectron atoms, Kelvin (1902) remarks,
apropos of the problems of dealing with more than
seven electrons, ‘Allof these cases, with questions of
stability or instability, and of the different amounts of
work required to pluck all the electrions [sic] out of
the atom and remove them to infimite distances,
present most interesting subjects for not difficult
mathematical work; and I regret not being able to
pursue them at present’. To quote Thomson (1907),
‘The general problem of findinghow n corpuscles
[electrons] will distribute themselves inside the sphere
isverycomplicated,
and I have not succeededin
solving it.’
Thomson (1904) washowever able to solve the
special case where the electrons wereconfiined to a
plane passing through the centre of the sphere. Figure
1 shows a ring, radius b, of, say, n electrons. Since the
total positive charge on the sphere of radius a is ne it
follows (from Gauss’s theorem) that the resultant
force acting on each electron, arising from the
positively charged sphere, is equal to that produced by
a charge of ne(b3/a3)located at the centre of the
sphere. This inwardly-directed force of magnitude
n $b/4ne& acting on an electron, such as P in figure
1, must of course be balanced by an equal and
opposite resultant arising from the repulsive forces
exerted by the other n- 1 electrons. The repulsive
370
force on P, resolved along lineOP, due to electron Q is
givenby (e2/4neoPQ2)cosOPQ or, since PQ = 2b
sin[t(2n/n)l and msOPQ = sin[+(2n/n)l, by
(e2/16neob9cosec (nln). Writing down the corresponding contributions due to electrons R, S, T, etc,
and equating theirsum to the resultant inwardlydirected force gives
Equation ( 1 ) predicts, for example, that when n = 2,
b/a = 0.5 (a result we derived in part l ) , when n = 4,
b/a = 0.621 and when n = 6, b/a= 0,673.
In view of the instability of a ring of static electrons
with n > 3 Thomson (1904) next assumed that the
ring of electrons, rather than being at rest,in fact
rotated with an angular velocity W . Equating the
centripetal force acting on each electron of mass m to
mbw2 gives
n e2b
4ne,a3
16ne0b2
”
-
[,,,,g72 + cosec 2n
+ . . . + cosec (n 1
n
= mbo2
where b wil, of course, be different to that predicted
by the static model. Thomson nowexamined the
stability of an electron within the rotating ring by
calculating the forces acting on the electron as it was
displaced slightly both within and out of the plane of
its orbit. What he found was that up to n = 3 the orbit
is stable evenwhen W = 0,and that for n = 4 and
n = 5 the orbit is stable provided W exceeds a certain
(calculated) value.Howeverwhen n > 5 stability is
onlypossible if there are other electrons inside the
ring. Thus, for example,
a
rotating ring of six
electrons placed at the comers of a regular hexagon is
unstable by itself; but it becomes stable when another
electron is placed at the centre of the hexagon. Indeed
byexamining
the conditions for stability of the
rotating ring systems Thomson was able to deduce the
lowest number nl of electrons which must be placed
inside a ring of n electrons for that ring to be in stable
equilibrium. The valueshe obtained are set out in
table 1. For large values of n the values of nI are proportional to n3.
In an earlier paper Thomson (1903) had examined
the problem of the radiant energyemittedbya
rotating ringofelectrons.
He found that the total
radiation emittedbyaring
of six electrons each
moving at one-hundredth of the speed of light was
only 1 6 X
of that ofsingle
a
electron
describing the same orbit with the same speed. Had he
considered rather more than six electrons the radiation
losses could have been reduced to almost insignificant
levels. Indeed it isjust possible that he did not wish to
reduce the losses further! As he points out in his 1904
Physics Education September 1 9 7 7
Table 1 Number of electrons nl required inside a ring
of n electrons
1 2 3 4 5 6 7 8 9 1 0 1 2 1 3 1 5 2 0 30 40
nl
00000 1 11 2
3 8 10 15 101
39
232
/ ”\
a
\
n
,,
l
l
,
The solution of this equation-presently to be solved
graphically-gives us n,. To findn,, the number of
electrons in the outermost but one ring, we write down
the stability conditions for the electrons in this ring,
namely
N - n,
- n,
=f(nJ.
(3)
Since n, is known from the solution of equation (2)
this equation wil, when solved, give us n2. Likewise
the number of electrons n3 in the next ringis given by
N - n,
Figure 1 Ring of n electrons at equilibrium under
mutual repulsions and attraction of a positive-charge
cloud
paper, ‘In consequence of the radiation from the
moving corpuscles, their velocities wil slowly-very
slowly-diminish;when,aRera
long interval, the
velocity reaches the critical velocity [the minimum
consistent with orbital stability] there will be what is
equivalent to an explosion of the corpuscles . and
we should have, as in the case of radium, a part of the
atom shot off’. Thomson in fact believed at this time
that all atoms shouldbe Bemitters (though some
would have a very long halflife).
..
- n, - n3 =An3
(4)
and so on.
The family of equations (2), (3), (4), etc, c m be
solved very rapidly by agraphical mcthcd. Using table
1 draw the graph nl = f(n), plottingn along the
ordinate and nl = f(n) along the ablicissa. This graph
is shown in figure 2. To find out how the N electrons
arrange themselves measure off along the abscissa a
distance OP equal to N. Through P draw the straight
line PQ inclined at an angle QPO = 45’. The integral
part of QM gives the value of n,, the number of
electrons in the outermost ring. To prove this we note
that since angles MQP and QPM are both 4 5 O , QM
= MP, and so
OM=OP-MP=OP-QM=N-QM.
But OM = AQM) and therefore
Electronic structureof the N-electronatom
The next problem to consider is how N electrons w
li
arrange themselves inside a sphere (of charge Ne). A
few distributions can be solved directly from table 1.
We see, for example, that if there are 12 electrons in
the outermost ring there must be 8 electrons in toto in
the interior rings(givinga grand total of N = 20
electrons in the atom). But how are these 8 electrons
arranged? Table 1 tells us the answer: a ring of seven
electrons and asingle electron at the centre of the
atom. So in the case of the 20-electron atom the ring
populations, starting at the centre, are (1, 7, 12).
To pose the problem in a more general vein: given
the relationship between the total number of electrons
nl inside the ring and number n of electrons around
that ring-a
relationsbip we shall write as n, =
f(n)-how
are N electrons arranged in an atom?
Clearly, if there are n, electrons in the outer ring there
will be a total of N-n, electrons in all the interior
rings, and if these are sufficient to keep the outer ring
in stable equilibrium, then
nl = N
- n, =f(n,).
Physics Educatlon September 1977
(2)
N - QM =f(QM).
Comparing this equation with equation (2) we see that
QM = n,. In the particular case illustrated in figure 2,
N=70andQM=21.4;son1=21.
To find the value of n,, mark off along the abscissa
the distance OP, = N - n,. (Only if QM happens to
be an integer will P, and M coincide.) From P, draw
line Q,P, parallel to QP. The integral part of Q,M, is
the value of n,. (In the example illustrated in f g w e 2
Q,M, = 17.3 and so n, = 17.) To find n3 mark off
OP, = N - n, - n2 and draw Q2P2parallel to QP;
the integral part of QzM, will be the value of n3 (here
15). By repeating this procedure one readily discovers
the values of n,,n,, etc, for the given total electron
number N . To find the arrangement of electrons for a
different value of N you simply set OP equal to the
new value of N and repeat the entire procedure.
Table 2 givesthe populations of the various electron
rings for atoms with N = 1 to N = 90. Here ring
number 0 corresponds to the centre of the sphere, ring
number 1 to the first ring out, number 2 to the second
ring out from the centre, and so on.
371
- 30
-20
n
-1 0
0
10
20
30
LO
ni = f ( n )
50
60
70
80
90
100
Figure 2 Number of electrons ni which are sufficient to stabilize an orbit containing n electrons
The Kelvin-Thomson periodic table
Although itwouldbesilly
to spend too muchtime
attempting to correlate the electronic properties
predicted for the elements from table 2 with the actual
(1907) makes
properties of the elements-Thomson
such an attempt-certain features are stillworth
noting. Firstly, we look at the family of atoms with N
= 1, 6, 17, 32,49, 70,etc. As can be seen from table 2
in passing N = 1 to N = 6 wesimply add an
Table 2 Electronic arrangement for atoms of N
electrons
Ring number
N O 1 2 3 4
1 1
2 2
33
4 4
55
61 5
71 6
81 7
91 8
102 8
11 3 8
123 9
13 3 10
14 4 10
15 5 10
16 5 1 1
17 1 511
18 1 6 1 1
19 1 7 1 1
20 1 7 12
21 1 8 12
222 812
232 813
243 813
25 3 9 13
26 3 1013
27 4 10 13
28 4 1014
29 5 10 14
30 5 l015
372
King number
N O 1 2 3 4
31 5 l 1 15
32 1 5 1 1 15
33 1 6 1 1 15
34 1 7 1 1 15
35 1 7 1 1 16
36 1 7 12 16
37 1 8 12 16
38 2 8 12 16
39 2 8 13 16
40 3 8 13 16
41 3 9 13 16
42 3 9 13 17
43 3 10 13 17
44 4 10 13 17
45 4 10 14 17
46 5 10 14 17
47 5 10 15 17
48 5 1 1 15 17
49 1 5 1115 17
50 1 5 1 1 15 18
51 1 6 1 1 15 18
52 1 71115 18
53 1 7111618
54 1 7121618
55 1 7121619
56 1 8 12 1619
57 2 812 16 19
58 2 813 1619
59 2 8 13 16 20
60 3 8 13 1620
-
Ring number
N O 1 2 3 4 5
61 3 913 1620
62 3 9 13 1720
63 3 l0131720
64 4 1013 1720
65 4 10 14 1720
66 5 10 14 1720
67 5 10 l5 1720
68 5 1015 1721
69 5 1 1 15 1721
70 1 5 1115 1721
71 1 5 l115 1821
72 1 611151821
73 1 7 1 1 I5 18 21
74 1 7 1 1 16 18 21
75 1 712161821
76 1 712161921
77 1 81216 1921
78 1 8 12161922
79 2 8 12 161922
80 2 8 13161922
81 2 8 13 16 2022
82 3 8 13 162022
83 3 913 162022
84 3 9 13 1720 22
85 3 10 13 1720 22
86 3 1013 172023
87 4 1013 172023
88 4 1014172023
89 5 l014172023
90 5 10 15 17 20
additional ring of 5 electrons; in passing from N = 6
to N = 17 we simply add a new ring of 1 1 electrons;
in passing from N = 17 to N = 32 we add a ring of
15 electrons. Similar remarks apply to many other
series, such as N = 5 , 16, 31, 48, 69, etc. Thomson
was quick to capitalize on such periodic properties but
(perhaps because of the then uncertain relationship
between N and atomic weights) scarcely attempted to
relate
the
periodicity
tothat
observed in the
Mendeleev classification.
Thesecond
feature of interest is that certain
elements are, inThornson’s (1907) words, ‘on the
verge of instability and therefore very liable to lose a
negative corpuscle’. As an example, the atom with N
= 59 has 20 electrons in its outer ring and therefore a
total of 39 electrons in its inner rings. However these
39 electrons are barely adequate to stabilize the outer
ring. So an electron is readily lost, leaving 39 electrons
to stabilize 19 electrons. To quote Thomson, ‘This ring
is exceedingly
stable’.
Likewise,
instability again
occurs at N = 68 (where the 21 outer electrons are
barely stabilizedby the inner 47 electrons); an electron
can be readily lost to give the stable element N = 67.
So atoms with N = 59 and N = 68 (to name but
two) are electron donors. Thomson argued that atoms
lying between N = 6 0 and N = 67 would, to varying
degrees, act as electron acceptors since the added
electrons all go to innerrings, thereby further stabilizing the outer ring.With the notion of electron
donors, electron acceptors and filled orbits it is
obvious
(from
our viewpoint) that Thomson’s
attempts at explaining chemical valencies wouldbe at
least partially successful.
Perhaps the singlemost damning failure of the
Kelvin-Thomson atom was its inability to explain the
complexity of atomic spectra. Assuming that the
emission spectra is due to the vibrations of the
electrons, an atom of N electrons can at most have 3 N
spectral lines(thisbeing
the number of degrees of
freedom). Thomson (1 907) did however point out that
the gas is ionized in a discharge tube or flame(the
23
sources which produce atomic spectra). Under these
Physxs Education September 1977
circumstances, says Thomson, ‘Apositivelyelectrifiedion
and a corpuscle might form a system
analogous to the solar system, in which the positively
electrified ion, with its large mass, takes the part of the
sun while the corpuscles circulate around it as planets
. . . In order that the corpuscle outside the ionmay
give a definite line it must revolve in a closed orbit; if
orbits having all possible periods within certhin limits
were possible, then the system of ions and corpuscles
would give a continuous and not a tine spectrum. Now
if the forces between the positive ionand the corpuscle
weresimply a central force vaiying inversely as the
square of the distance there would be an infinite
number of elliptic orbits for the corpuscle with continuously varying periods, and the spectrum would be
a continuous one.When,however
. . the force
between the ion and the corpuscle ismuchmore
complex, the number of possibleperiodic
orbits
becomes more limited’.
.
An experimental simulation
It is possible to simulate (partially) the
Kelvin-Thomson atom using the apparatus shown in
figure 3. (This technique-was developed by Mayer, an
American physicist.) Short bar magnets (or magnetized needles) pushed through corks and floating on
watersimulate
the electrons. These bar magnets,
which must have their poles all pointing in the same
way, repel each other like electrons. The attractive
force is produced by an iron-cored solenoidplaced
above the surface of the water, the lower pole of this
magnet having the opposite sign to that of the upper
poles of the floating magnets. The component of the
force due to this magnet along the surface of the water
is directed to a point where the axis of the solenoid
meets the water, and is approximately proportional to
the distance from this point. The forces acting on the
floating magnets therefore simulate those acting on the
electrons in the Kelvin-Thomson atom. (Actually the
experiment, being static, really only simulates an atom
inwhich other forces constrain the electrons to one
Figure 3 Layout for experiment simulatingelectronic
arrangements in a Thomson-Kelvin atom in which
electrons are constrained to tie on a plane
U
Physics Educatlon September 1977
plane. The water-and gravity-provide these constraining forces. Without them, stability is
only
possible in an orbiting arrangement of electrons.)
If one throws bar magnet after bar magnet into the
water one finds that they w
li arrange themselves in
definite p a t t e r n s a e e magnets at the comer of an
equilateral triangle, four at the comers of a square,
five at the corners of a pentagon. When, however,one
throws in a sixth magnet the sequence is broken; five
go to the corners of a pentagon and one goes to the
middle.When one throws in a seventhmagnet one
gets a ring of six withone magnet at the centre. By this
means one can simulate at least some of the lowpopulation atoms whose (dynamic) structure is summarized in table 2.
summary
Perhaps any summary of the key features of the
Kelvin-Thomson atom should best be leR to one of its
authors. To quote Thomson (1904), ‘We have thus in
the first place a sphere of uniformpositive electrification, and inside this sphere a number of corpuscles
arranged in a series of parallel rings, the number of
corpuscles in a ring varying from ring to ring: each
corpuscle is travelling at a high speed round the circumference of the ring in which it is situated, and the
rings are so arranged that those which contain a large
number of corpuscles are near the surface of the
sphere while those in which there are a smaller number
of corpuscles are more in the inside. If the corpuscles,
like the poles of the tittle magnets in Mayer’s experiments with the floatingmagnets, are constrained to
move in one plane they would, even if not in rotation,
bein equilibrium when arranged in a series of rings
just described. The rotation is required to make the
arrangement stable when the corpuscles can move at
right anglesto the plane of the ring’.
A useful reference forreaders wishing to explore the
historical context of the Thomson-Kelviq atom is
Schonland (1968). A (perhaps not disinterested)
account of the development of J J Thomson’s ideas
may be found in G P Thomson’s (1964) biography of
his father.
REFERENCES
Kelvin Lord 1902 Phil. Mag. Series 6 3 257-83 (This article
is, according to a footnote, fromthe Jubilee Volume
presented to Professor Bosscha in November 1 9 0 1 )
Schonland B 1968 The Atomists (Oxford:Clarcndon Press)
Thomson G P 1964 J J Thornson and the Cavendish
Laboratory in his Day (London:Thomas Nelson)
Thomson J J 1903 Phil. Mag.Series 6 6 673-93
Thomson J J 1904 Phil. Mag.Series 6 7 237-65
Thomson J J 1907 The Corpuscular Theory of Matter
(London: Constable) (This book is based on a course of
lecturesgiven by Thomson at the Royal Institution in
1906)
373