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2. mathematics
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ELECTRON ARRANGEMENTS IN MULTI-ELECTRON ATOMS
AND THE ORIGIN OF ATOMIC SPECTRA
F. B. CARLETON
London N16
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©FBC/London/Lisk/24thFeb2013
ABSTRACT
The arrangement of multi-electron atomic systems in the rotating nuclear model of the
atom is examined. It is shown that variations in potential energy, alone, are sufficient to
account for the arrangement of well defined numbers of electrons into shells and that
variation of the kinetic energy by displacement from the equatorial axis of rotation
accommodates the repulsive forces within these shells. The spectra of helium, lithium
and beryllium are consistent with the proposed structural arrangements of electrons into
well defined shells and indicate that the position of these additional electrons are
associated with a reduction in the shielding of the nuclear charge when the electron is in
its respective ground state configuration. The shielding provided by the electrons in orbit
around the nucleus can be predicted from classical considerations.
INTRODUCTION
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©FBC/London/Lisk/24thFeb2013
The Mendeleevian periodicity of the elements has long been established. In 1869 his
arrangement of the elements in order of increasing atomic weights and their assignment
to groups of similar properties confirmed a long suspected 'natural' order of the elements.
Gaps were left to accommodate new elements and predictions of their properties made.
Relative atomic mass
Eka Silicon
(1871 prediction)
72
Germanium
(discovered 1886)
72.32
Specific gravity
5.5
5.47
Specific heat
0.073
0.076
Atomic volume
13
13.22
Colour
Dark grey
Greyish white
Specific gravity of Ge Cl 4
1.9
1.88
Specific gravity of Ge O
4.7
4.703
100
86
160
160
Property
2
Boiling point of Ge Cl 4
Boiling point of Ge ( C 2 H 5 )
4
With the advent of the Bohr Periodic Table certain anomalies of Mendleev's Table were
explained, the elements were now classified according to atomic number, electronic
configuration - today. The ionisation potentials of the electrons of all the elements of the
periodic table display a similar periodicity both within a given atomic system and
between atomic systems. For example, the ionisation potentials of the electrons in the
calcium atom, Z = 20, display the following pattern of behaviour. Electron number refers
to the order in which the electrons are added to the atomic system to form the atom.
Electrons 1 and 2 have ionisation potentials markedly greater than all the other electrons
in this atom. There is a sharp transition to the ionisation potential of the third electron
accompanied by a gradual successive diminution in the potentials of the following seven
electrons. After electron number ten there is again a sharp transition, though reduced in
magnitude, and electrons 11 to 18 again display a gradually decreasing ionisation
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©FBC/London/Lisk/24thFeb2013
potential pattern. Similar patterns, gradual transitions and sharp discontinuities, may be
discerned in the ionisation potentials of all atoms and the discontinuities are observed at
the same recurring intervals. When we consider the
Ionisation potential (eV)
6000
5000
4000
3000
2000
1000
0
0
10
20
30
Electron number
Figure 1. Ionisation potentials of the electrons in the calcium atom.
ionisation potentials of the last electrons, i.e. the electron which renders the charge of
each atom electrically neutral, an equally clear pattern exists. As we proceed across the
periodic table ionisation potentials change gradually until we reach the potential
associated with the inert gases. There is then a sharp discontinuity and the following
element's ionisation potential is considerably reduced. In every case this element is
metallic and readily loses it's last electron to form a positive ion - the alkali metals.
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©FBC/London/Lisk/24thFeb2013
30
He
Ionisation potential (eV)
Ne
20
Ar
Kr
10
Li
Na
K
Rb
0
0
10
20
30
40
50
Atomic number (Z)
Figure 2. Ionisation potentials of the electron which renders an atom
electrically neutral versus atomic number (Z).
Discussion
The rotating nuclear atomic model conforms to all the constraints of classical physics.
Stability of the lone ground state electron can be achieved only if there is no relative
motion between the electron and the nucleus. Simultaneous changes of potential and
kinetic energy conserve the total energy of the system. From the ground state, the lone
electron must change its potential energy, alone, in order to absorb energy, the ground
state kinetic energy of the lone electron is unique. A decrease in the kinetic energy of a
lone ground state electron is prohibited by the properties of the simple harmonic
oscillator model.
The geometric position of the new permissible orbits are simple
multiple integers of the ground state orbit and the associated potential energy change
(kinetic energy is constant) produces electrical effects which vary as the square of these
multiple integer changes of orbit. When a second electron is added to the lone electron
system, the first electron becomes displaced from the equatorial plane of orbit and both
electrons eventually assume identical orbits which are equally displaced north and south
of the equatorial plane of orbit associated with the lone electron system.
This
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©FBC/London/Lisk/24thFeb2013
displacement of orbit reduces the centrifugal forces acting on the electron. Stability of
the two electron system is achieved by a change of kinetic energy of the electrons at
constant potential energy. A change in the effective nuclear charge for a lone ground
state electron is affected by the introduction of a second electron into the atomic system.
Zeff = Z - σ
(σ = shielding factor)
(1)
0≤ σ ≤ 1
The second electron will experience a force of attraction, by the nucleus, this force of
attraction will be less than that of the nucleus itself due to the presence of the first
electron. Therefore, the value of, σ, is greater than zero. A value of zero would belie the
existence of the charge, itself. If the value of the shielding factor were unity, the
effective nuclear charge presented by an atomic system of known, Z, would be reduced to
that of the element preceding it in the periodic table and the associated spectral properties
of the two electron system would duplicate those of each preceding element with one
electron. This is not the case, nor is it to be expected, since the electron is not an integral
part of the nucleus. It has a negative charge with respect to the nucleus and its presence
only partly reduces the overall effect of the atomic number, Z. The charge of any nuclear
system minus some other constant charge (an electron) placed at a constant distance (r1)
from the nucleus will present a total effective charge which is equal to the charge of the
system less a constant amount of charge. All electrons are identical. This reasoning
successfully accounts for the structure of all two electron helium like atomic systems,
where the electrons are confined at the ground state radius of orbit. However, in general
the arrangement and the effect of electron interactions must be different. Due to the
presence of other electrons in an atom, any one electron does not experience the full
effects of the nuclear charge, Z. At the electrons given location it will experience a
shielded nuclear charge and the true charge, Z, is reduced to Zeff (the effective nuclear
charge) by an amount called the shielding constant, σT .
Zeff = n Z = Z - σT
(2)
In the rotating nuclear model of the atom, the charge Z, experienced by a lone electron is
dependent on the position (i.e. the radial distance from the nucleus, as denoted by the
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©FBC/London/Lisk/24thFeb2013
value of, N) of the electron from the nucleus. The maximum charge that an electron in
the Nth orbit may experience is Z/N. Thus, in the complete absence of other electrons, an
electron in the Nth orbit is shielded from the full nuclear charge by virtue of its position
Z-
Z
= σP
N
(3)
An electrical shielding factor is defined as the difference between the total shielding
factor, σΤ, and the positional shielding factor σP, so that,
σΤ − σP = σE
∴
Ζ − n N Z = N σE
(5)
Z
- σ
E
N
(6)
nZ =
F.R. = (
1 - nN
N
since
(4)
1
N
2
n = (
2
-
) =
σ
Ε
NZ
(6a)
n
= F.R.
N
Z - N σE
NZ
)
(7)
When an electron is removed to some considerable distance from an atomic system of
nuclear charge, Z, with x electrons in orbit around the nucleus, the effective charge in the
far electric field exerted by such an atomic system on the distant electron must accurately
approach, Z - x, by the laws of classical physics. When a second electron is in the far
electric field, approaching a lone electron hydrogen like atomic system (NσΕ = x = 1),
n = (
Z - 1
)
NZ
(8)
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©FBC/London/Lisk/24thFeb2013
For the helium atom, (Z = 2), the values of, n, derived from empirical spectra (Atomic
Energy Levels) are given in Table 1 and Figure 3 where they are plotted against the
theoretical values derived, from equation (8), for each corresponding value of, N.
Equation (8) accurately predicts the spectra of not only the helium atom, it has been fitted
with equal success to the two electron helium like spectra of lithium and beryllium. It
should be noted that the approach to the equilibrium position for a particular electron
leads to increasing deviation from the simple model proposed, compare numerical values
in Table 1. The explanation of this apparent deviation from the model has its origins in
the measurement of the electrons associated electrical energy and the measurement scale
employed. The treatment of this aberration is addressed in another paper.
0.3
(Z-1)/NZ
0.2
0.1
0.0
0.0
0.1
0.2
0.3
n
Figure 3. Theoretical spectrum of the second electron in a helium atom compared
with empirical spectrum, expressed in terms of displacement, n.
The addition of a third electron to the two electron system will change, yet again, the
energy of the atomic system. Since simultaneous change of potential and kinetic energy
is an energy conservative process for the basic rotating nuclear atomic model, either the
kinetic energy alone or the potential energy alone of the two electron system must
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©FBC/London/Lisk/24thFeb2013
undergo change, initially. The values of, N, derived for isoelectronic series containing
three to ten electrons are shown in Figure 4, see equation 6.
12
10
nZ(3)
nZ(4)
nZ
8
nZ(5)
nZ(6)
6
nZ(7)
4
nZ(8)
nZ(9)
2
0
nZ(10)
0
10
20
30
40
Z
Figure 4. The relationship between electron numbers 3 to 10 (bracketed
number) and the parameter N.
8
6
nZ(11)
nZ(12)
nZ
nZ(13)
nZ(14)
4
nZ(15)
nZ(16)
nZ(17)
2
0
nZ(18)
10
20
30
40
Z
Figure 5. The relationship between electron numbers 11 to 18 (bracketed number)
and the parameter N.
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©FBC/London/Lisk/24thFeb2013
For nuclear systems containing three to ten electrons the value of N is 2. When more
than ten electrons are present about the atomic nucleus there is an abrupt change in the
gradient (1/N) of the relationship and the assignment of electrons 11 to 18, to a distance
N = 3, from the nucleus is indicated. For this reason the two electron groupings are
shown separately. In other words there are eight electrons accommodated in the second
orbit (N = 2) about the atomic nucleus. The transitions between the various possible
radii, denoted by the value of N, occur by alteration of the potential energy at constant
kinetic energy. Whereas, at a constant radius the linear velocity of the electrons can
change to accommodate repulsive forces acting on the electron or to accommodate the
addition of more electrons to the atomic system. The values of the parameter, σΕ,
determined from the Figures 4 and 5, for electron numbers 3 to 18 are given in Figure 6.
6
Electron shielding factor
5
4
3
2
1
0
0
10
20
Electron number
Figure 6. Shielding factor, σΕ, as a function of electron number.
The spectrum of the third electron in the lithium atom (Z = 3, x = 2) is given in Table 2
and Figure 7, for all values of N > 2. In this example stability of the electron which
renders the atom electrically neutral is obtained when this electron resides in the second
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©FBC/London/Lisk/24thFeb2013
orbit about the nucleus, N =2, in accordance with the foregoing analysis, see Figure 4.
0.12
(Z-2)/NZ
0.10
0.08
0.06
0.04
0.02
0.02
0.04
0.06
0.08
0.10
0.12
n
Figure 7. Theoretical spectrum of the third electron in a lithium atom compared with
empirical spectrum, expressed in terms of displacement, n.
Conclusions
In the introduction, the calcium atom was employed to illustrate the regularities and
patterns displayed by the ionisation potentials of the electrons in a given atomic system.
Comparisons between atomic systems were made by considering the patterns displayed
by the electron, which renders the atom electrically neutral. In this paper, periodicity has
been demonstrated by considering the same numbered electrons (isoelectronic series) in
various atoms from Z = 3 to Z = 18, where "same numbered electron" refers to the
number in the order in which electrons are added about the nucleus to construct the atom.
This procedure establishes the existence of shells at clearly defined distances (integer
multiples, N, of the ground state orbit) with characteristic numbers of electrons allocated
to each of the shells. The origin of spectra has been examined by making comparison
with the spectra of helium, lithium and beryllium. It has been shown that these spectra
are in accordance with the predictions of classical physics when the effective electric
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©FBC/London/Lisk/24thFeb2013
field is considered to be the resultant of the difference between the number of positive
charges and negative charges, as experienced by a single electron, in the far electric field.
A full explanation of the electrons stability as it approaches its position of maximum
stability is discussed elsewhere. The reasons for the pronounced increase in stability on
the final transition, N + 1 to N, is related to the treatment of the associated electrical
energy that contributes to a particular ground state configuration and is beyond the scope
of this paper.
REFERENCES
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©FBC/London/Lisk/24thFeb2013
1. CRC Handbook of Chemistry and Physics, 59th Edition, CRC Press Inc., (1978-79).
2. Atomic Energy Levels, As Derived From The Analyses of Optical Spectra., C. E.
Moore, Circular of the National Bureau of Standards 467,Washington, D. C., 1947
3. F. B. Carleton, Ph. D. thesis, QUB, 1972.
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©FBC/London/Lisk/24thFeb2013
Table 1
I.P. 1= 54.40
-1
1/λ (cm. )
Designation
L
2P
3P
4P
5P
6P
7P
8P
9P
10
11P
12P
13P
14P
15P
16P
17P
18P
19P
20P
198305.00
171129.14
186203.62
191486.95
193936.75
195269.17
196073.41
196595.56
196953.95
197210.41
197400.18
197544.56
197656.95
197746.15
197818.12
197877.04
197925.87
197966.80
198001.44
198031.02
Table 2
1/λ (cm. -1)
43487.19
30925.38
36469.55
39015.56
40390.84
41217.35
41751.63
42118.27
42379.16
42569.10
42719.14
L = Limit
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
1po
I.P. 1= 122.419
N
L
3P
4P
5P
6P
7P
8P
9P
10P
11P
12P
2
3
4
5
6
7
8
9
10
11
12
I.P. (eV)
x
5.390
1.556
.8695
.5540
.3835
.2811
.2148
.1694
.1371
.1135
.0949
2
1/2 m0 vx
2
=
1/2 m0 v1
vx
v1
=
2
n
(Z-1)/NZ
.67216
.24883
.16604
.12463
.09976
.08316
.07130
.06240
.05548
.04993
.04540
.04162
.03842
.03568
.03330
.03122
.02939
.02775
.02629
.02498
.5000
.2500
.1667
.1250
.1000
.0833
.0714
.0625
.0555
.0500
.0454
.0416
.0384
.0357
.0333
.0312
.0294
.0277
.0263
.0250
L = Limit
Designation
2p o
2p o
2p o
2p o
2p o
2p o
2p o
2p o
2p o
2p o
2p o
I.P. (eV)
x
24.585
3.3690
1.5001
.84514
.54142
.37624
.27653
.21179
.16736
.13557
.11204
.09414
.08021
.06915
.06023
.05292
.04687
.04179
.03750
.03383
I.P. x
I.P. 1
=
I. P. x
v0 Z1
(Z-2)/NZ
.2098
.1127
.0842
.0672
.0559
.0479
.0418
.0372
.0334
.0304
.0278
.1666
.1111
.0833
.0666
.0555
.0476
.0416
.0370
.0333
.0303
.0277
(4 )
I.P. 1
v0 Z x
n
= n
(5)
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