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Electron shell
Electron shell
Electron shell
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In chemistry and atomic physics, an electron shell may be thought of as an orbit that electrons follow around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on further and further from the nucleus. The shells correspond to the principal quantum numbers (n = 1, 2, 3, 4 ...) or are labeled alphabetically with the letters used in X-ray notation (K, L, M, ...). Each period on the conventional periodic table of elements represents an electron shell.

Each shell can contain only a fixed number of electrons: the first shell can hold up to two electrons, the second shell can hold up to eight electrons, the third shell can hold up to 18, continuing as the general formula of the nth shell being able to hold up to 2(n2) electrons.[1] For an explanation of why electrons exist in these shells, see electron configuration.[2]

Each shell consists of one or more subshells, and each subshell consists of one or more atomic orbitals.

History

[edit]

In 1913, Niels Bohr proposed a model of the atom, giving the arrangement of electrons in their sequential orbits. At that time, Bohr allowed the capacity of the inner orbit of the atom to increase to eight electrons as the atoms got larger, and "in the scheme given below the number of electrons in this [outer] ring is arbitrary put equal to the normal valency of the corresponding element". Using these and other constraints, he proposed configurations that are in accord with those now known only for the first six elements. "From the above we are led to the following possible scheme for the arrangement of the electrons in light atoms:"[3][4]

Bohr's 1913 proposed configurations
Element Electrons per shell Element Electrons per shell Element Electrons per shell
1 1 9 4, 4, 1 17 8, 4, 4, 1
2 2 10 8, 2 18 8, 8, 2
3 2, 1 11 8, 2, 1 19 8, 8, 2, 1
4 2, 2 12 8, 2, 2 20 8, 8, 2, 2
5 2, 3 13 8, 2, 3 21 8, 8, 2, 3
6 2, 4 14 8, 2, 4 22 8, 8, 2, 4
7 4, 3 15 8, 4, 3 23 8, 8, 4, 3
8 4, 2, 2 16 8, 4, 2, 2 24 8, 8, 4, 2, 2

The shell terminology comes from Arnold Sommerfeld's modification of the 1913 Bohr model. During this period Bohr was working with Walther Kossel, whose papers in 1914 and in 1916 called the orbits "shells".[5][6] Sommerfeld retained Bohr's planetary model, but added mildly elliptical orbits (characterized by additional quantum numbers ℓ and m) to explain the fine spectroscopic structure of some elements.[7] The multiple electrons with the same principal quantum number (n) had close orbits that formed a "shell" of positive thickness instead of the circular orbit of Bohr's model which orbits called "rings" were described by a plane.[8]

The existence of electron shells was first observed experimentally in Charles Barkla's and Henry Moseley's X-ray absorption studies. Moseley's work did not directly concern the study of electron shells, because he was trying to prove that the periodic table was not arranged by weight, but by the charge of the protons in the nucleus.[9] However, because the number of electrons in an electrically neutral atom equals the number of protons, this work was extremely important to Niels Bohr who mentioned Moseley's work several times in his 1962 interview.[10] Moseley was part of Rutherford's group, as was Niels Bohr. Moseley measured the frequencies of X-rays emitted by every element between calcium and zinc and found that the frequencies became greater as the elements got heavier. This led to the theory that electrons were emitting X-rays when they were shifted to lower shells.[11] This led to the conclusion that the electrons were in Kossel's shells with a definite limit per shell, labeling them with the letters K, L, M, N, O, P, and Q.[4][12] The origin of this terminology was alphabetic. Barkla, who worked independently from Moseley as an X-ray spectrometry experimentalist, first noticed two distinct types of scattering from shooting X-rays at elements in 1909 and named them "A" and "B". Barkla described these two types of X-ray diffraction: the first was unconnected with the type of material used in the experiment and could be polarized. The second diffraction beam he called "fluorescent" because it depended on the irradiated material.[13] It was not known what these lines meant at the time, but in 1911 Barkla decided there might be scattering lines previous to "A", so he began at "K".[14] However, later experiments indicated that the K absorption lines are produced by the innermost electrons. These letters were later found to correspond to the n values 1, 2, 3, etc. that were used in the Bohr model. They are used in the spectroscopic Siegbahn notation.

The work of assigning electrons to shells was continued from 1913 to 1925 by many chemists and a few physicists. Niels Bohr was one of the few physicists who followed the chemist's work[15] of defining the periodic table, while Arnold Sommerfeld worked more on trying to make a relativistic working model of the atom that would explain the fine structure of the spectra from a classical orbital physics standpoint through the Atombau approach.[4] Einstein and Rutherford, who did not follow chemistry, were unaware of the chemists who were developing electron shell theories of the periodic table from a chemistry point of view, such as Irving Langmuir, Charles Bury, J.J. Thomson, and Gilbert Lewis, who all introduced corrections to Bohr's model such as a maximum of two electrons in the first shell, eight in the next and so on, and were responsible for explaining valency in the outer electron shells, and the building up of atoms by adding electrons to the outer shells.[16][4] So when Bohr outlined his electron shell atomic theory in 1922, there was no mathematical formula for the theory. So Rutherford said he was hard put "to form an idea of how you arrive at your conclusions".[17][18] Einstein said of Bohr's 1922 paper that his "electron-shells of the atoms together with their significance for chemistry appeared to me like a miracle – and appears to me as a miracle even today".[19] Arnold Sommerfeld, who had followed the Atombau structure of electrons instead of Bohr who was familiar with the chemists' views of electron structure, spoke of Bohr's 1921 lecture and 1922 article on the shell model as "the greatest advance in atomic structure since 1913".[4][20][17] However, the electron shell development of Niels Bohr was basically the same theory as that of the chemist Charles Rugeley Bury in his 1921 paper.[21][4][22]

As work continued on the electron shell structure of the Sommerfeld-Bohr Model, Sommerfeld had introduced three "quantum numbers n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing."[23] Because we use k for the Boltzmann constant, the azimuthal quantum number was changed to ℓ. When the modern quantum mechanics theory was put forward based on Heisenberg's matrix mechanics and Schrödinger's wave equation, these quantum numbers were kept in the current quantum theory but were changed to n being the principal quantum number, and m being the magnetic quantum number.

However, the final form of the electron shell model still in use today for the number of electrons in shells was discovered in 1923 by Edmund Stoner, who introduced the principle that the nth shell was described by 2(n2). Seeing this in 1925, Wolfgang Pauli added a fourth quantum number, "spin", during the old quantum theory period of the Sommerfeld-Bohr Solar System atom to complete the modern electron shell theory.[4]

Subshells

[edit]
3D views of some hydrogen-like atomic orbitals showing probability density and phase (g orbitals and higher are not shown).

Each shell is composed of one or more subshells, which are themselves composed of atomic orbitals. For example, the first (K) shell has one subshell, called 1s; the second (L) shell has two subshells, called 2s and 2p; the third shell has 3s, 3p, and 3d; the fourth shell has 4s, 4p, 4d and 4f; the fifth shell has 5s, 5p, 5d, and 5f and can theoretically hold more in the 5g subshell that is not occupied in the ground-state electron configuration of any known element.[2] The various possible subshells are shown in the following table:

Subshell label ℓ Max electrons Shells containing it Historical name
s 0 2 Every shell  sharp
p 1 6 2nd shell and higher  principal
d 2 10 3rd shell and higher  diffuse
f 3 14 4th shell and higher  fundamental
g 4 18 5th shell and higher (theoretically) (next in alphabet after f)[24]
  • The first column is the "subshell label", a lowercase-letter label for the type of subshell. For example, the "4s subshell" is a subshell of the fourth (N) shell, with the type (s) described in the first row.
  • The second column is the azimuthal quantum number (ℓ) of the subshell. The precise definition involves quantum mechanics, but it is a number that characterizes the subshell.
  • The third column is the maximum number of electrons that can be put into a subshell of that type. For example, the top row says that each s-type subshell (1s, 2s, etc.) can have at most two electrons in it. Each of the following subshells (p, d, f, g) can have 4 more electrons than the one preceding it.
  • The fourth column says which shells have a subshell of that type. For example, looking at the top two rows, every shell has an s subshell, while only the second shell and higher have a p subshell (i.e., there is no "1p" subshell).
  • The final column gives the historical origin of the labels s, p, d, and f. They come from early studies of atomic spectral lines. The other labels, namely g, h, and i, are an alphabetic continuation following the last historically originated label of f.

Number of electrons in each shell

[edit]

Each subshell is constrained to hold 4ℓ + 2 electrons at most, namely:

  • Each s subshell holds at most 2 electrons
  • Each p subshell holds at most 6 electrons
  • Each d subshell holds at most 10 electrons
  • Each f subshell holds at most 14 electrons
  • Each g subshell holds at most 18 electrons

Therefore, the K shell, which contains only an s subshell, can hold up to 2 electrons; the L shell, which contains an s and a p, can hold up to 2 + 6 = 8 electrons, and so forth; in general, the nth shell can hold up to 2n2 electrons.[1]

Shell
name		 Subshell
name		 Subshell
max
electrons		 Shell
max
electrons
K 1s 2 2
L 2s 2 2 + 6 = 8
2p 6
M 3s 2 2 + 6 + 10
= 18
3p 6
3d 10
N 4s 2 2 + 6 +
10 + 14
= 32
4p 6
4d 10
4f 14
O 5s 2 2 + 6 +
10 + 14 +
18 = 50
5p 6
5d 10
5f 14
5g 18

Although that formula gives the maximum in principle, that maximum is only achieved (in known elements) for the first four shells (K, L, M, N). No known element has more than 32 electrons in any one shell.[25][26] This is because the subshells are filled according to the Aufbau principle. The first elements to have more than 32 electrons in one shell would belong to the g-block of period 8 of the periodic table. These elements would have some electrons in their 5g subshell and thus have more than 32 electrons in the O shell (fifth principal shell).

Subshell energies and filling order

[edit]
Further information: Aufbau principle
For multielectron atoms n is a poor indicator of electron's energy. Energy spectra of some shells interleave.
The states crossed by same red arrow have same value. The direction of the red arrow indicates the order of state filling.

Although it is sometimes stated that all the electrons in a shell have the same energy, this is an approximation. However, the electrons in one subshell do have exactly the same level of energy, with later subshells having more energy per electron than earlier ones. This effect is great enough that the energy ranges associated with shells can overlap.

The filling of the shells and subshells with electrons proceeds from subshells of lower energy to subshells of higher energy. This follows the n + ℓ rule which is also commonly known as the Madelung rule. Subshells with a lower n + ℓ value are filled before those with higher n + ℓ values. In the case of equal n + ℓ values, the subshell with a lower n value is filled first.

Because of this, the later shells are filled over vast sections of the periodic table. The K shell fills in the first period (hydrogen and helium), while the L shell fills in the second (lithium to neon). However, the M shell starts filling at sodium (element 11) but does not finish filling till copper (element 29), and the N shell is even slower: it starts filling at potassium (element 19) but does not finish filling till ytterbium (element 70). The O, P, and Q shells begin filling in the known elements (respectively at rubidium, caesium, and francium), but they are not complete even at the heaviest known element, oganesson (element 118).

List of elements with electrons per shell

[edit]

The list below gives the elements arranged by increasing atomic number and shows the number of electrons per shell. At a glance, the subsets of the list show obvious patterns. In particular, every set of five elements (  electric blue) before each noble gas (group 18,   yellow) heavier than helium have successive numbers of electrons in the outermost shell, namely three to seven.

Sorting the table by chemical group shows additional patterns, especially with respect to the last two outermost shells. (Elements 57 to 71 belong to the lanthanides, while 89 to 103 are the actinides.)

The list below is primarily consistent with the Aufbau principle. However, there are a number of exceptions to the rule; for example palladium (atomic number 46) has no electrons in the fifth shell, unlike other atoms with lower atomic number. The elements past 108 have such short half-lives that their electron configurations have not yet been measured, and so predictions have been inserted instead.

Z Element No. of electrons/shell Group
1 Hydrogen 1 1
2 Helium 2 18
3 Lithium 2, 1 1
4 Beryllium 2, 2 2
5 Boron 2, 3 13
6 Carbon 2, 4 14
7 Nitrogen 2, 5 15
8 Oxygen 2, 6 16
9 Fluorine 2, 7 17
10 Neon 2, 8 18
11 Sodium 2, 8, 1 1
12 Magnesium 2, 8, 2 2
13 Aluminium 2, 8, 3 13
14 Silicon 2, 8, 4 14
15 Phosphorus 2, 8, 5 15
16 Sulfur 2, 8, 6 16
17 Chlorine 2, 8, 7 17
18 Argon 2, 8, 8 18
19 Potassium 2, 8, 8, 1 1
20 Calcium 2, 8, 8, 2 2
21 Scandium 2, 8, 9, 2 3
22 Titanium 2, 8, 10, 2 4
23 Vanadium 2, 8, 11, 2 5
24 Chromium 2, 8, 13, 1 6
25 Manganese 2, 8, 13, 2 7
26 Iron 2, 8, 14, 2 8
27 Cobalt 2, 8, 15, 2 9
28 Nickel 2, 8, 16, 2 10
29 Copper 2, 8, 18, 1 11
30 Zinc 2, 8, 18, 2 12
31 Gallium 2, 8, 18, 3 13
32 Germanium 2, 8, 18, 4 14
33 Arsenic 2, 8, 18, 5 15
34 Selenium 2, 8, 18, 6 16
35 Bromine 2, 8, 18, 7 17
36 Krypton 2, 8, 18, 8 18
37 Rubidium 2, 8, 18, 8, 1 1
38 Strontium 2, 8, 18, 8, 2 2
39 Yttrium 2, 8, 18, 9, 2 3
40 Zirconium 2, 8, 18, 10, 2 4
41 Niobium 2, 8, 18, 12, 1 5
42 Molybdenum 2, 8, 18, 13, 1 6
43 Technetium 2, 8, 18, 13, 2 7
44 Ruthenium 2, 8, 18, 15, 1 8
45 Rhodium 2, 8, 18, 16, 1 9
46 Palladium 2, 8, 18, 18 10
47 Silver 2, 8, 18, 18, 1 11
48 Cadmium 2, 8, 18, 18, 2 12
49 Indium 2, 8, 18, 18, 3 13
50 Tin 2, 8, 18, 18, 4 14
51 Antimony 2, 8, 18, 18, 5 15
52 Tellurium 2, 8, 18, 18, 6 16
53 Iodine 2, 8, 18, 18, 7 17
54 Xenon 2, 8, 18, 18, 8 18
55 Caesium 2, 8, 18, 18, 8, 1 1
56 Barium 2, 8, 18, 18, 8, 2 2
57 Lanthanum 2, 8, 18, 18, 9, 2
58 Cerium 2, 8, 18, 19, 9, 2
59 Praseodymium 2, 8, 18, 21, 8, 2
60 Neodymium 2, 8, 18, 22, 8, 2
61 Promethium 2, 8, 18, 23, 8, 2
62 Samarium 2, 8, 18, 24, 8, 2
63 Europium 2, 8, 18, 25, 8, 2
64 Gadolinium 2, 8, 18, 25, 9, 2
65 Terbium 2, 8, 18, 27, 8, 2
66 Dysprosium 2, 8, 18, 28, 8, 2
67 Holmium 2, 8, 18, 29, 8, 2
68 Erbium 2, 8, 18, 30, 8, 2
69 Thulium 2, 8, 18, 31, 8, 2
70 Ytterbium 2, 8, 18, 32, 8, 2
71 Lutetium 2, 8, 18, 32, 9, 2 3
72 Hafnium 2, 8, 18, 32, 10, 2 4
73 Tantalum 2, 8, 18, 32, 11, 2 5
74 Tungsten 2, 8, 18, 32, 12, 2 6
75 Rhenium 2, 8, 18, 32, 13, 2 7
76 Osmium 2, 8, 18, 32, 14, 2 8
77 Iridium 2, 8, 18, 32, 15, 2 9
78 Platinum 2, 8, 18, 32, 17, 1 10
79 Gold 2, 8, 18, 32, 18, 1 11
80 Mercury 2, 8, 18, 32, 18, 2 12
81 Thallium 2, 8, 18, 32, 18, 3 13
82 Lead 2, 8, 18, 32, 18, 4 14
83 Bismuth 2, 8, 18, 32, 18, 5 15
84 Polonium 2, 8, 18, 32, 18, 6 16
85 Astatine 2, 8, 18, 32, 18, 7 17
86 Radon 2, 8, 18, 32, 18, 8 18
87 Francium 2, 8, 18, 32, 18, 8, 1 1
88 Radium 2, 8, 18, 32, 18, 8, 2 2
89 Actinium 2, 8, 18, 32, 18, 9, 2
90 Thorium 2, 8, 18, 32, 18, 10, 2
91 Protactinium 2, 8, 18, 32, 20, 9, 2
92 Uranium 2, 8, 18, 32, 21, 9, 2
93 Neptunium 2, 8, 18, 32, 22, 9, 2
94 Plutonium 2, 8, 18, 32, 24, 8, 2
95 Americium 2, 8, 18, 32, 25, 8, 2
96 Curium 2, 8, 18, 32, 25, 9, 2
97 Berkelium 2, 8, 18, 32, 27, 8, 2
98 Californium 2, 8, 18, 32, 28, 8, 2
99 Einsteinium 2, 8, 18, 32, 29, 8, 2
100 Fermium 2, 8, 18, 32, 30, 8, 2
101 Mendelevium 2, 8, 18, 32, 31, 8, 2
102 Nobelium 2, 8, 18, 32, 32, 8, 2
103 Lawrencium 2, 8, 18, 32, 32, 8, 3 3
104 Rutherfordium 2, 8, 18, 32, 32, 10, 2 4
105 Dubnium 2, 8, 18, 32, 32, 11, 2 5
106 Seaborgium 2, 8, 18, 32, 32, 12, 2 6
107 Bohrium 2, 8, 18, 32, 32, 13, 2 7
108 Hassium 2, 8, 18, 32, 32, 14, 2 8
109 Meitnerium 2, 8, 18, 32, 32, 15, 2 (?) 9
110 Darmstadtium 2, 8, 18, 32, 32, 16, 2 (?) 10
111 Roentgenium 2, 8, 18, 32, 32, 17, 2 (?) 11
112 Copernicium 2, 8, 18, 32, 32, 18, 2 (?) 12
113 Nihonium 2, 8, 18, 32, 32, 18, 3 (?) 13
114 Flerovium 2, 8, 18, 32, 32, 18, 4 (?) 14
115 Moscovium 2, 8, 18, 32, 32, 18, 5 (?) 15
116 Livermorium 2, 8, 18, 32, 32, 18, 6 (?) 16
117 Tennessine 2, 8, 18, 32, 32, 18, 7 (?) 17
118 Oganesson 2, 8, 18, 32, 32, 18, 8 (?) 18

See also

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Wikimedia Commons has media related to Electron shell diagrams.

References

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Electron shell

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In and chemistry, an electron shell refers to a group of atomic orbitals that share the same , n, which defines the and average distance of electrons from the nucleus in an atom. These shells organize the electrons surrounding the positively charged nucleus, with electrons filling lower-energy shells first according to the . The concept originated in the of the atom but is now understood through , where shells correspond to discrete energy states rather than fixed circular orbits. Electron shells are traditionally labeled with letters starting from the innermost: K (n=1), L (n=2), M (n=3), and so on, up to the seventh shell for known elements. Each shell can hold a maximum of 2n² electrons, allowing the first shell to accommodate 2 electrons, the second up to 8, the third up to 18, and higher shells even more, though outer shells in most elements are not fully occupied. This capacity arises from the subshells (s, p, d, f) within each shell, each with specific orbital quantum numbers that determine their electron-holding limits. The arrangement of electrons in shells governs an atom's chemical behavior, particularly through the valence shell—the outermost shell—which contains the valence electrons responsible for bonding and reactivity. Full valence shells, as in , confer stability, while incomplete ones drive elements to form ions or compounds to achieve a stable octet configuration in the valence shell. This shell structure underpins the periodic table's organization, where periods correspond to the filling of successive shells, explaining trends in atomic size, , and across elements.

Fundamentals

Definition and Basic Properties

In atomic physics, electron shells represent discrete energy levels within an atom, conceptualized as concentric spherical regions surrounding the nucleus where electrons are most likely to be found according to quantum mechanical probability distributions. These shells are defined by the principal quantum number nn, a positive integer that specifies the shell's designation, with n=1n = 1 for the innermost shell and increasing sequentially outward as n=2,3,4,…n = 2, 3, 4, \dots. The value of nn indicates the shell's average distance from the nucleus and its associated energy, with higher nn corresponding to larger, higher-energy shells. Electron shells are conventionally labeled using : the K shell for n=1n=1, L shell for n=2n=2, M shell for n=3n=3, N for n=4n=4, and so forth. A key property is that shell radius and binding both increase with nn, meaning in outer shells are less tightly bound to the nucleus and require less energy to remove. Consequently, the occupying the outermost shell—termed valence electrons—predominantly govern an atom's chemical behavior, including its reactivity and bonding tendencies with other atoms. For conceptual clarity, electron shells can be intuitively compared to the discrete orbits in a simplified planetary model of the atom, akin to planets revolving around the sun at fixed distances; this analogy, however, oversimplifies the quantum reality, where electrons do not follow definite paths but exist in probabilistic clouds. Each shell is subdivided into subshells based on the .

Role in Atomic and Molecular Structure

Electron shells play a crucial role in determining the stability of atoms by organizing electrons into energy levels that minimize overall energy. Atoms with completely filled inner shells, such as the noble gases, exhibit exceptional stability due to their closed-shell configurations, where the valence shell is fully occupied, leading to low reactivity and high ionization energies. For instance, helium achieves this stability with its two electrons filling the 1s shell completely, resulting in a configuration that resists chemical interactions under normal conditions. In chemical bonding, electron shells govern how atoms interact to achieve more stable configurations, primarily through the sharing or transfer of from the outermost shell. The describes the tendency of atoms to gain, lose, or share electrons to fill their outer shell with eight electrons, mimicking the stable arrangement and lowering . This leads to , where electrons are transferred—as in sodium, which loses its single 3s to form Na⁺ with a neon-like configuration—creating oppositely charged ions attracted by electrostatic forces. In covalent bonding, atoms share pairs of to satisfy the , forming molecules where the shared electrons are counted toward both atoms' outer shells. The arrangement of electrons in outer shells also influences molecular structure by dictating bond angles and geometries through interactions of valence electrons. In molecular formation, these valence electrons participate in hybridization, where atomic orbitals from the outer shell combine to form hybrid orbitals that overlap effectively, enabling specific molecular shapes like the tetrahedral geometry in . This shell-driven hybridization and repulsion among electron pairs in the valence shell determine overall molecular architecture, affecting properties such as polarity and reactivity.

Historical Context

Pre-Quantum Models

In the early , J.J. Thomson proposed the of the atom in 1904, envisioning it as a uniform sphere of positive charge with negatively charged embedded throughout like plums in a pudding, ensuring overall electrical neutrality without any discrete shell structure. This model accounted for the discovery of the but lacked a concept of organized electron layers, treating electrons as distributed within a diffuse positive medium. Ernest Rutherford's nuclear model, introduced in 1911 based on the gold foil scattering experiments conducted by and , depicted the atom as a tiny, dense, positively charged nucleus surrounded by electrons orbiting in planetary-like paths, much like planets around the sun. However, this classical picture was inherently unstable, as accelerating electrons in circular orbits would, according to Maxwell's electromagnetic theory, continuously radiate energy and spiral inward toward the nucleus, leading to atomic collapse. Niels Bohr addressed these issues in his 1913 model, postulating that occupy discrete, quantized orbits around the nucleus—serving as precursors to modern electron shells—where they do not radiate while in these stable "stationary states." Bohr introduced quantization, with the electron's orbital given by L=nℏL = n \hbar, where nn is a positive and ℏ=h/2π\hbar = h / 2\pi (with hh as Planck's constant), preventing continuous loss. This framework successfully explained the discrete spectral lines of , attributing them to electrons transitioning between quantized levels and emitting photons of specific wavelengths corresponding to the energy differences. Despite its successes, Bohr's model had significant limitations, particularly its inability to accurately describe multi-electron atoms, where electron-electron interactions were not accounted for, leading to incorrect predictions of energy levels and spectra for elements beyond . This shortfall highlighted the need for a more comprehensive quantum mechanical treatment to incorporate wave-like behavior and full atomic complexity.

Development of Quantum Theory

The development of quantum theory marked a pivotal shift in understanding electron behavior within atoms, building upon the limitations of earlier atomic models such as Niels Bohr's 1913 planetary model, which posited quantized electron orbits but failed to fully explain spectral complexities. A key milestone in this evolution came from in 1916, who extended Bohr's model by incorporating relativistic effects and allowing for elliptical orbits rather than strictly circular ones. This introduction of an additional quantum condition for the radial motion, alongside angular quantization, provided a precursor to the concept of subshells by accounting for splittings in atomic spectra, such as the hydrogen fine structure lines observed experimentally. Sommerfeld's relativistic treatment predicted energy level shifts dependent on orbital eccentricity, laying groundwork for more nuanced electron groupings in multi-electron atoms. Further advancements arose from Louis de Broglie's 1924 hypothesis, which proposed wave-particle duality for electrons, asserting that particles like electrons exhibit wave-like properties with a wavelength given by λ=h/p\lambda = h / p, where hh is Planck's constant and pp is the electron's momentum. This idea suggested that electrons in atoms could be standing waves confined to discrete paths, providing a conceptual bridge from classical orbits to wave mechanics and motivating the quantization of electron shells. Building on this, Erwin Schrödinger formulated the time-independent Schrödinger equation in 1926 for the hydrogen atom, expressed as H^ψ=Eψ\hat{H} \psi = E \psi, where H^\hat{H} is the Hamiltonian operator, ψ\psi is the wave function, and EE is the energy eigenvalue. Solving this equation yielded quantized energy levels En=−13.6 eV/n2E_n = -13.6 \, \text{eV} / n^2, where nn is a positive , confirming discrete shells and resolving inconsistencies in Bohr's non-relativistic energies. Complementing these developments, Werner Heisenberg's uncertainty principle, articulated in 1927, introduced fundamental indeterminacy in quantum measurements, stating that the product of uncertainties in position Δx\Delta x and momentum Δp\Delta p satisfies ΔxΔp≥ℏ/2\Delta x \Delta p \geq \hbar / 2, where ℏ=h/2π\hbar = h / 2\pi. This principle prohibited the precise definition of fixed electron orbits, as simultaneous knowledge of position and momentum becomes impossible, thereby necessitating a probabilistic wave description for electron shells rather than deterministic paths. Finally, Wolfgang Pauli's exclusion principle, proposed in 1925, stipulated that no two electrons in an atom can occupy the same quantum state, meaning they cannot share all quantum numbers simultaneously. This rule was essential for establishing the shell structure, as it enforced distinct occupancy limits within energy levels, preventing collapse of electrons into the lowest state and enabling the layered configuration observed in atomic spectra.

Quantum Mechanical Framework

Principal Quantum Number and Shells

In , the nn is a (n=1,2,3,…,∞n = 1, 2, 3, \dots, \infty) that primarily determines the and size of an electron shell in an atom. It labels the discrete energy shells, with n=1n = 1 corresponding to the innermost K shell, n=2n = 2 to the L shell, n=3n = 3 to the M shell, and so on, following established in early . The energy of an electron in a shell depends on nn. For hydrogen-like atoms (single-electron systems such as H or He⁺), the energy levels are given by En=−13.6 Z2n2 eV,E_n = -\frac{13.6 \, Z^2}{n^2} \, \text{eV}, where ZZ is the atomic number; this formula arises from solving the Schrödinger equation for the Coulomb potential. In multi-electron atoms, inner electrons screen the nuclear charge, reducing the effective nuclear charge ZeffZ_{\text{eff}} experienced by outer electrons, which modifies the energy to approximately En≈−13.6 Zeff2n2 eVE_n \approx -\frac{13.6 \, Z_{\text{eff}}^2}{n^2} \, \text{eV} and results in less negative (higher) energies compared to hydrogen-like cases. Spatially, the electron shell's extent is characterized by the radial , where the probability for finding the peaks at an average distance from the nucleus scaling as ∼n2a0\sim n^2 a_0, with a0=0.529 A˚a_0 = 0.529 \, \text{Å} being the (the most probable radius for the of ). This quadratic dependence on nn reflects the increasing orbital size with higher shells, as derived from the radial wavefunctions in the solution. Electron shells are denoted using spectroscopic notation, such as 1s or 2p, where the numeral specifies nn and the letter indicates the subshell (an angular subdivision within the shell); this ties the overall shell structure directly to the principal quantum number.

Subshells and Angular Momentum

Within each electron shell defined by the principal quantum number nn, subshells arise as subdivisions characterized by the azimuthal quantum number ll, which quantifies the orbital angular momentum of the electron. The value of ll ranges from 0 to n−1n-1 in integer steps, with this range constrained by nn. Subshells are conventionally labeled using letters: s for l=0l = 0, p for l=1l = 1, d for l=2l = 2, and f for l=3l = 3, with higher values of ll assigned subsequent letters in alphabetical order (g, h, etc.). The orbital associated with a subshell has a magnitude given by l(l+1)ℏ\sqrt{l(l+1)} \hbar
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