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# Periodic table

The periodic table organizes the chemical elements of matter (hydrogen, helium, lithium, etc.) according to properties shared with other elements. The table was first compiled by Dmitri Mendeleev in 1869.[1] He left a few gaps in the table and predicted the discovery of elements sharing properties with other elements depending on their placement on the table; his predictions were later shown to be correct. Mendeleev's achievement is all the more remarkable when we consider that he was working with atomic masses (generally expressed as a real number to three or four decimal places according to the weight of the most stable isotope) and was unaware of atomic numbers, those integers that enumerate how many protons and electrons an atom has. Nevertheless, Mendeleev was smart enough to place tellurium before iodine[2] even though tellurium is about 0.7 heavier than iodine.

Chemists generally refer to atomic elements by their atomic symbols, which are one or two letters that abbreviate the English or Latin name of the element. For example, gold is Au and lead is Pb.[3] (Elements that have been discovered but not officially named get a 3-letter symbol based on their atomic number). Mendeleev's original table was of course in black and white. Nowadays the table is often color-coded but neither the colors nor the key are standard.

## Mendeleev's periodic table

Mendeleev periodic table
Group 1 2   3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Period Noble gases, A018227
1 1
H
Alkali metals
A099955
Nonmetals
A101647
Halogens
A097478
2
He
2 3
Li
4
Be
Alkaline earth metals
A099956
Metalloids
A101648
5
B
6
C
7
N
8
O
9
F
10
Ne
3 11
Na
12
Mg
Inner transition metals
A??????
Transition metals
A??????
Poor metals
A101649
13
Al
14
Si
15
P
16
S
17
Cl
18
Ar
4 19
K
20
Ca
21
Sc
22
Ti
23
V
24
Cr
25
Mn
26
Fe
27
Co
28
Ni
29
Cu
30
Zn
31
Ga
32
Ge
33
As
34
Se
35
Br
36
Kr
5 37
Rb
38
Sr
39
Y
40
Zr
41
Nb
42
Mo
43
Tc
44
Ru
45
Rh
46
Pd
47
Ag
48
Cd
49
In
50
Sn
51
Sb
52
Te
53
I
54
Xe
6 55
Cs
56
Ba
Lanthanides
[Ln] 57–71
72
Hf
73
Ta
74
W
75
Re
76
Os
77
Ir
78
Pt
79
Au
80
Hg
81
Tl
82
Pb
83
Bi
84
Po
85
At
86
Rn
7 87
Fr
88
Ra
Actinides
[An] 89–103
104
Rf
105
Db
106
Sg
107
Bh
108
Hs
109
Mt
110
Ds
111
Rg
112
Cn
113
Uut
114
Uuq
115
Uup
116
Uuh
117
Uus
118
Uuo
Elements with placeholder name pending approval of official name by IUPAC.

Halogens and noble gases are also nonmetals, so perhaps we could call the other nonmetals "the plainer nonmetals." The inner transition metals, i.e. the lanthanides and actinides, are generally placed on rows by themselves below the main table.

 [Ln] LanthanidesA000027 (subset) 57La 58Ce 59Pr 60Nd 61Pm 62Sm 63Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 71Lu [An] ActinidesA000027 (subset) 89Ac 90Th 91Pa 92U 93Np 94Pu 95Am 96Cm 97Bk 98Cf 99Es 100Fm 101Md 102No 103Lr

## Conway's periodic table

********** AND HERE SHOULD GO SOMETHING ABOUT CONWAY'S PERIODIC TABLE, I THINK THERE'S A CHAPTER ON IT AT THE END OF BOOK OF NUMBERS **********

Conway's periodic table of elements is associated with his look and say audioactive sequence.[4][5]

## Orbitals and suborbitals

Matter particles (e.g. protons, neutrons, electrons, ...) are fermions, thus abiding by the Pauli exclusion principle (no two fermions may occupy the same quantum state, two fermions with opposite spins being in a different quantum state.)

Orbitals and suborbitals
$n \,$ Orbitals

$l \,$

(0 to $\scriptstyle n \,$-1)

Suborbitals

$m \,$

($\scriptstyle -l \,$ to $\scriptstyle +l \,$)

Spin

$s \,$

($\scriptstyle \frac{-1}{2} \,$ and $\scriptstyle \frac{+1}{2} \,$)

1 (0)

(1s)

(0) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
2 (0, 1)

(2s, 2p)

(-1, 0, 1) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
3 (0, 1, 2)

(3s, 3p, 3d)

(-2, -1, 0, 1, 2) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
4 (0, 1, 2, 3)

(4s, 4p, 4d, 4f)

(-3, -2, -1, 0, 1, 2, 3) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
5 (0, 1, 2, 3, 4)

(5s, 5p, 5d, 5f, 5g)

(-4, -3, -2, -1, 0, 1, 2, 3, 4) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
6 (0, 1, 2, 3, 4, 5)

(6s, 6p, 6d, 6f, 6g, 6h)

(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)
7 (0, 1, 2, 3, 4, 5, 6)

(7s, 7p, 7d, 7f, 7g, 7h, 7i)

(-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6) ($\scriptstyle \frac{-1}{2},\, \frac{+1}{2} \,$)

where the quantum numbers [6] are

• $\scriptstyle n \,$ is the principal quantum number, or energy level quantum number (both $\scriptstyle n \,$ and $\scriptstyle l \,$, but $\scriptstyle n \,$ more so, affect the distance of the orbitals from the atom's nucleus,)
• $\scriptstyle l \,$ is the azimuthal quantum number, or angular quantum number, (giving the shape of the electronic orbital,)
• $\scriptstyle m \,$, or $\scriptstyle m_l \,$, is the magnetic quantum number (giving the orientation of the electronic suborbital,)
• $\scriptstyle s \,$, or $\scriptstyle m_s \,$, is the spin quantum number of the electron.

and

• s orbitals ($\scriptstyle l \,=\, 0 \,$) have 1 possible value of $\scriptstyle m \,$ to hold 2 electrons
• p orbitals ($\scriptstyle l \,=\, 1 \,$) have 3 possible values of $\scriptstyle m \,$ to hold 6 electrons
• d orbitals ($\scriptstyle l \,=\, 2 \,$) have 5 possible values of $\scriptstyle m \,$ to hold 10 electrons
• f orbitals ($\scriptstyle l \,=\, 3 \,$) have 7 possible values of $\scriptstyle m \,$ to hold 14 electrons
• g orbitals ($\scriptstyle l \,=\, 4 \,$) have 9 possible values of $\scriptstyle m \,$ to hold 18 electrons
• h orbitals ($\scriptstyle l \,=\, 5 \,$) have 11 possible values of $\scriptstyle m \,$ to hold 22 electrons
• ...

the s orbitals being spherical (hence no distinct orientations), all other orbitals having increasingly complex shapes, e.g. more and more lobes, offering more and more distinct orientations, each of them maximizing the distances between them (the electrons repelling each other.)

### Number of electrons per filled orbital

The sequence of number of electrons per filled orbital (Cf. A016825) is

{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, ...}

with formula

$o(l) = 2 (2 l + 1),\quad l \ge 0, \,$
$G_{\{o(l)\}}(x) = \frac{2 (1 + x)}{(1 - x)^2} \,$
$\coth(\tfrac{1}{2}) = \frac{e + 1}{e - 1} = a_0 ~+~ \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cfrac{1}{a_5 + \cfrac{1}{a_6 + \cfrac{1}{a_7 + \cfrac{1}{\ddots}}}}}}}} = 2 ~+~ \cfrac{1}{6 + \cfrac{1}{10 + \cfrac{1}{14 + \cfrac{1}{18 + \cfrac{1}{22 + \cfrac{1}{26 + \cfrac{1}{30 + \cfrac{1}{\ddots}}}}}}}} \,$

## Shells and subshells

### Bohr's aufbau principle (building up principle)

Bohr's aufbau principle (building up principle) tells in which order each shell's subshells fill up with electrons, a shell being a cluster of subshells.

Bohr's aufbau principle (building up principle)
Shell

(period)

$k \,$

Subshells

$\scriptstyle ((n \,=\, k,\, 0);\, (n \,=\, k+1-l,\, l \,=\, \lfloor\frac{k}{2}\rfloor),\, ...,\, (n \,=\, k+1-l,\, l \,=\, 1)) \,$

Subshells'

numbers of

electrons

Shell's

number of

electrons

1 (1s)

((1, 0))

(2) 2
2 (2s; 2p)

((2, 0); (2, 1))

(2; 6) 8
3 (3s; 3p)

((3, 0); (3, 1))

(2; 6) 8
4 (4s; 3d, 4p)

((4, 0); (3, 2), (4, 1))

(2; 10, 6) 18
5 (5s; 4d, 5p)

((5, 0); (4, 2), (5, 1))

(2; 10, 6) 18
6 (6s; 4f, 5d, 6p)

((6, 0); (4, 3), (5, 2), (6, 1))

(2; 14, 10, 6) 32
7 (7s; 5f, 6d, 7p)

((7, 0); (5, 3), (6, 2), (7, 1))

(2; 14, 10, 6) 32
8 (8s; 5g, 6f, 7d, 8p)

((8, 0); (5, 4), (6, 3), (7, 2), (8, 1))

(2; 18, 14, 10, 6) 50
9 (9s; 6g, 7f, 8d, 9p)

((9, 0); (6, 4), (7, 3), (8, 2), (9, 1))

(2; 18, 14, 10, 6) 50

Madelung's rule says that the orbitals fill by increasing $\scriptstyle n + l \,$, then with $\scriptstyle l \,=\, \big\lceil\frac{n+l}{2}\big\rceil - 1 \,$ down to 1. This gives the sequence of orbitals in the filling order.

$n+l \,$ Orbital Number of

electrons

$2 ~ (2l+1) \,$

$(n = 1) + (l = 0) = 1 \,$ 1s 2

$(n = 2) + (l = 0) = 2 \,$ 2s 2

$(n = 2) + (l = 1) = 3 \,$ 2p 6
$(n = 3) + (l = 0) = 3 \,$ 3s 2

$(n = 3) + (l = 1) = 4 \,$ 3p 6
$(n = 4) + (l = 0) = 4 \,$ 4s 2

$(n = 3) + (l = 2) = 5 \,$ 3d 10
$(n = 4) + (l = 1) = 5 \,$ 4p 6
$(n = 5) + (l = 0) = 5 \,$ 5s 2

$(n = 4) + (l = 2) = 6 \,$ 4d 10
$(n = 5) + (l = 1) = 6 \,$ 5p 6
$(n = 6) + (l = 0) = 6 \,$ 6s 2

$(n = 4) + (l = 3) = 7 \,$ 4f 14
$(n = 5) + (l = 2) = 7 \,$ 5d 10
$(n = 6) + (l = 1) = 7 \,$ 6p 6
$(n = 7) + (l = 0) = 7 \,$ 7s 2

$(n = 5) + (l = 3) = 8 \,$ 5f 14
$(n = 6) + (l = 2) = 8 \,$ 6d 10
$(n = 7) + (l = 1) = 8 \,$ 7p 6
$(n = 8) + (l = 0) = 8 \,$ 8s 2

... ... ...

### Janet's periodic table

Janet started from the fact that the series of chemical elements is a continuous sequence, which he represented as a helix traced on the surfaces of four nested cylinders. By various geometrical transformations he derived several striking designs, one of which is his "left-step periodic table," in which hydrogen and helium are placed above lithium and beryllium. It was only later that he realized that his arrangement concorded perfectly with quantum theory and the electronic structure of the atom. He placed the actinides under the lanthanides twenty years before Glenn Seaborg, and he continued the series to element 120.

Janet's table differs from the standard table in placing the s block elements on the right, so that the blocks of the periodic table are arranged in the natural order s, p, d, f from right to left. There is then no need to divide the table or move the f block into a 'footnote'. He believed that no elements with atomic number $\scriptstyle Z \,$ higher than 120 would be found, so he did not envisage a g block. In terms of atomic quantum numbers, each row corresponds to one value of the sum $\scriptstyle n+l \,$ where $\scriptstyle n \,$ is the principal quantum number and $\scriptstyle l \,$ the azimuthal quantum number. The table therefore corresponds to Madelung's rule, which states that atomic subshells are filled in order of increasing values of $\scriptstyle n+l \,$.

 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 p1 p2 p3 p4 p5 p6 s1 s2 H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt Ds Rg Cn Uut Uuq Uup Uuh Uus Uuo Uue Ubn Janet's Periodic Table (with current element symbols)[7]

Janet also envisaged an 'element zero' – whose 'atom' would consist of two neutrons, and he speculated that this would be the link to a mirror-image table of elements with negative atomic numbers – in effect anti-matter. He also conceived heavy hydrogen. He died just before the discovery of the neutron, the positron and heavy hydrogen.[8]

### Maximum number of electrons of subshells of nth shell

The maximum number of electrons of subshells of $\scriptstyle n \,$th shell gives the infinite sequence of finite sequences

{{2}, {2, 6}, {2, 6}, {2, 10, 6}, {2, 10, 6}, {2, 14, 10, 6}, {2, 14, 10, 6}, {2, 18, 14, 10, 6}, {2, 18, 14, 10, 6}, {2, 22, 18, 14, 10, 6}, {2, 22, 18, 14, 10, 6}, {2, 26, 22, 18, 14, 10, 6}, {2, 26, 22, 18, 14, 10, 6}, ...}

whose concatenation gives Janet's sequence (subshells with 2 electrons being the first subshell of a shell) (Cf. A167268$\scriptstyle (n),\, n \,\ge\, 1 \,$)

{2, 2, 6, 2, 6, 2, 10, 6, 2, 10, 6, 2, 14, 10, 6, 2, 14, 10, 6, 2, 18, 14, 10, 6, 2, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, ...}

### Maximum number of electrons of nth shell

By summing the subsequences in

{{2}, {2, 6}, {2, 6}, {2, 10, 6}, {2, 10, 6}, {2, 14, 10, 6}, {2, 14, 10, 6}}

we get the number of elements for period $\scriptstyle k,\, 1 \,\le\, k \,\le\, 7, \,$ which is the number of electrons in the $\scriptstyle k \,$th shell

{2, 8, 8, 18, 18, 32, 32}

By summing the subsequences in

{{2}, {2, 6}, {2, 6}, {2, 10, 6}, {2, 10, 6}, {2, 14, 10, 6}, {2, 14, 10, 6}, {2, 18, 14, 10, 6}, {2, 18, 14, 10, 6}, {2, 22, 18, 14, 10, 6}, {2, 22, 18, 14, 10, 6}, {2, 26, 22, 18, 14, 10, 6}, {2, 26, 22, 18, 14, 10, 6}, ...}

we get the number of possible elements for periods $\scriptstyle k,\, k \,\ge\, 1, \,$ giving the sequence (Cf. A093907)

{2, 8, 8, 18, 18, 32, 32, 50, 50, 72, 72, 98, 98, 128, 128, 162, 162, 200, 200, 242, 242, 288, 288, 338, 338, 392, 392, 450, 450, 512, 512, 578, 578, 648, 648, 722, 722, 800, 800, 882, 882, 968, 968, ...}

A period corresponds to the filling of an electronic shell within the atom. Electronic shells are said to occur when the separation between energy levels is significantly greater than the local mean separation between subshells.

The number of elements for period $\scriptstyle k \,$ is the number of electrons which may occupy the $\scriptstyle k \,$th shell, given by the formulae

$s(k) = 2 + \sum_{l=1}^{\lfloor\frac{k}{2}\rfloor} o(l) = 2 + \sum_{l=1}^{\lfloor\frac{k}{2}\rfloor} 2 (2 l + 1) = 2 \Bigg(1 + \sum_{l=1}^{\lfloor\frac{k}{2}\rfloor} (2 l + 1)\Bigg) = 2 \sum_{l=0}^{\lfloor\frac{k}{2}\rfloor} (2 l + 1) = 2 {\Bigg({\Bigg\lfloor \frac{k}{2} \Bigg\rfloor} + 1\Bigg)}^2,\quad k \ge 1, \,$

where $\scriptstyle \lfloor n \rfloor \,$ is the floor function, and where $\scriptstyle o(l) \,$ is the number of electrons which may occupy the subshell with angular quantum number $\scriptstyle l \,$, and the shell's subshells actually fill up in the reverse order that the summation indicates, i.e. from $\scriptstyle l \,=\, \lfloor\frac{n}{2}\rfloor \,$ down to 1, $\scriptstyle l \,=\, 0 \,$ being always first. Also, note that for $\scriptstyle l \,=\, 0 \,$ we have $\scriptstyle n+l \,=\, k \,$, while for $\scriptstyle l \,>\, 0 \,$ we have $\scriptstyle n+l \,=\, k+1 \,$. This is given by the Aufbau principle.[9]

The generating function is

$G_{\{s(k)\}}(x) = \frac{2x (1 + 3x - 2 x^2 - x^3 + x^4)}{(1+x)^2 (1-x)^3} \,$

### Periods in Mendeleev's periodic table

#### Period 1

Elements of period 1 fills the following orbitals (belonging to shell 1) in the given order (per Aufbau principle)[9]

• 1s, $\scriptstyle n+l \,=\, 1+0 \,=\, 1 \,$, allowing for 2 electrons.

The first period contains 2 elements with the following atomic numbers (Cf. A??????)

{1, 2}

#### Period 2

Elements of period 2 fills the following orbitals (belonging to shell 2) in the given order (per Aufbau principle)

• 2s, $\scriptstyle n+l \,=\, 2+0 \,=\, 2 \,$, allowing for 2 electrons;
• 2p, $\scriptstyle n+l \,=\, 2+1 \,=\, 3 \,$, allowing for 6 electrons.

The second period contains 8 elements with the following atomic numbers (Cf. A??????)

{3, 4, 5, 6, 7, 8, 9, 10}

#### Period 3

Elements of period 3 fills the following orbitals (belonging to shell 3) in the given order (per Aufbau principle)

• 3s, $\scriptstyle n+l \,=\, 3+0 \,=\, 3 \,$, allowing for 2 electrons;
• 3p, $\scriptstyle n+l \,=\, 3+1 \,=\, 4 \,$, allowing for 6 electrons.

The third period contains 8 elements with the following atomic numbers (Cf. A??????)

{11, 12, 13, 14, 15, 16, 17, 18}

#### Period 4

Elements of period 4 fills the following orbitals (belonging to shell 4) in the given order (per Aufbau principle)

• 4s, $\scriptstyle n+l \,=\, 4+0 \,=\, 4 \,$, allowing for 2 electrons;
• 3d, $\scriptstyle n+l \,=\, 3+2 \,=\, 5 \,$, allowing for 10 electrons;
• 4p, $\scriptstyle n+l \,=\, 4+1 \,=\, 5 \,$, allowing for 6 electrons.

The fourth period contains 18 elements with the following atomic numbers (Cf. A??????)

{19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}

#### Period 5

Elements of period 5 fills the following orbitals (belonging to shell 5) in the given order (per Aufbau principle)

• 5s, $\scriptstyle n+l \,=\, 5+0 \,=\, 5 \,$, allowing for 2 electrons;
• 4d, $\scriptstyle n+l \,=\, 4+2 \,=\, 6 \,$, allowing for 10 electrons;
• 5p, $\scriptstyle n+l \,=\, 5+1 \,=\, 6 \,$, allowing for 6 electrons.

The fifth period contains 18 elements with the following atomic numbers (Cf. A??????)

{37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54}

#### Period 6

Elements of period 6 fills the following orbitals (belonging to shell 6) in the given order (per Aufbau principle)

• 6s, $\scriptstyle n+l \,=\, 6+0 \,=\, 6 \,$, allowing for 2 electrons;
• 4f, $\scriptstyle n+l \,=\, 4+3 \,=\, 7 \,$, allowing for 14 electrons;
• 5d, $\scriptstyle n+l \,=\, 5+2 \,=\, 7 \,$, allowing for 10 electrons;
• 6p, $\scriptstyle n+l \,=\, 6+1 \,=\, 7 \,$, allowing for 6 electrons.

The sixth period contains 32 elements with the following atomic numbers (Cf. A??????)

{55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86}

#### Period 7

Elements of period 3 fills the following orbitals (belonging to shell 7) in the given order (per Aufbau principle)

• 7s, $\scriptstyle n+l \,=\, 7+0 \,=\, 7 \,$, allowing for 2 electrons;
• 5f, $\scriptstyle n+l \,=\, 5+3 \,=\, 8 \,$, allowing for 14 electrons;
• 6d, $\scriptstyle n+l \,=\, 6+2 \,=\, 8 \,$, allowing for 10 electrons;
• 7p, $\scriptstyle n+l \,=\, 7+1 \,=\, 8 \,$, allowing for 6 electrons.

The seventh period contains 32 elements with the following atomic numbers (Cf. A??????)

{87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118}

#### Periods 8 and higher

If there are ever some elements in those higher periods with a somewhat stable nucleus... which is rather unlikely!

## Blocks and groups

### Main blocks

The main blocks are the s block and the p block. Chemical and physical properties of elements of the main blocks are very different depending on which group they belong.

#### s block

The s block includes groups filling up the outermost shell's s orbital with one then two electrons, allowing for 2 elements per period. The elements of period 1 (i.e. hydrogen and helium) and elements of group 1 (i.e. hydrogen and the alkali metals) and group 2 (i.e. alkaline earth metals) belong to the s block. Elements of the s block have atomic numbers $\scriptstyle Z \,$ (Cf. A??????)

{1, 2, 3, 4, 11, 12, 19, 20, 37, 38, 55, 56, 87, 88}

#### p block

The p block includes groups progressively filling up the outermost shell's p orbital with electrons, allowing for 6 elements per period. Groups 13 to 18 belong to the p block. Elements of the p block have atomic numbers $\scriptstyle Z \,$ (Cf. A??????)

{5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 35, 36, 49, 50, 51, 52, 53, 54, 81, 82, 83, 84, 85, 86, 113, 114, 115, 116, 117, 118}

### Transition blocks

The transition blocks are the d block and the inner transition blocks. Chemical and physical properties of elements of the transition blocks are much more similar to each other than elements of the main blocks.

#### d block

The d block includes groups progressively filling up the outermost shell's d orbital with electrons, allowing for 10 elements per period. Groups 3 to 12 belong to the d block. The transition metals and the last 15th elements of either lanthanides and actinides series, e.g. Lu (71) and Lr (103) respectively, belong to the d block. Elements of the d block have atomic numbers $\scriptstyle Z \,$ (Cf. A??????)

{21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112}

#### Inner transition blocks

The inner transition blocks are the f block and possibly, if the nucleus of any were ever stable against nuclear decay, even more inner blocks, e.g. g block, h block, i block, ... Chemical and physical properties of elements of the inner transition blocks are even much more similar to each other, the inner you could get the more similar..., than elements of the outer transition block, namely the d block.

##### f block

The f block includes groups progressively filling up the outermost shell's f orbital with electrons, allowing for 14 elements per period. The inner transition metals, which include the lanthanides and actinides series, belong to the f block, with the exception of the last 15th elements of either series, e.g. Lu (71) and Lr (103) respectively, which belong to the d block. Elements of the f block have atomic numbers $\scriptstyle Z \,$ (Cf. A??????)

{57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102}
##### g, h, i, ... blocks

If elements with ever higher atomic numbers $\scriptstyle Z \,$ had stable nucleus, then you would have even more inner (within inner...) blocks filling the outermost shell's

• g orbital with electrons, allowing for 18 elements per period;
• h orbital with electrons, allowing for 22 elements per period;
• i orbital with electrons, allowing for 26 elements per period;
• ...

## Atomic numbers

The atomic number $\scriptstyle Z \,$ of an element is the number of protons in the atom's nucleus, and thus the number of electrons of the neutral (non-ionized) atom.

Note that hydrogen (atomic number 1) has been included in both nonmetals and alkali metals, although at normal pressures and temperatures it behaves as a nonmetal. Only at extremely high pressures may hydrogen become a metallic solid or liquid.

### Metals

#### Non-transition metals

##### Pre-transition metals

The pre-transition metals are strongly electropositive metals, since they only have a only 1 or 2 electrons, except aluminum which has 3, in their outermost shell, which they are eager to offer to attain the stable configuration of the preceding noble gas. The pre-transition metals include metals from periods above, i.e. aluminum, or groups to the left, i.e. the alkali metals and the alkaline earth metals, of the transition metals. Pre-transition metals have atomic number $\scriptstyle Z \,$ (Cf. A??????)

{3, 4, 11, 12, 13, 19, 20, 37, 38, 55, 56, 87, 88}
###### Alkali metals

We can compute this sequence mathematically by taking the formula for electronic magic numbers, which happen to be the atomic numbers of the most stable elements, the noble gases, to which you add 1 (ignoring elements 117 and heavier as these have not been studied enough to properly classify). The outermost shell of a noble gas atom has as many electrons as it can possibly have. Alkali metal atoms, on the other hand, have only one electron on their outer shell, and need only loose that one electron to have a full outer shell.

Atomic numbers of the alkali metals (Cf. A099955)

{1, 3, 11, 19, 37, 55, 87}
###### Alkaline earth metals

We can compute this sequence mathematically by taking the formula for electronic magic numbers, which happen to be the atomic numbers of the most stable elements, the noble gases, to which you add 2 (ignoring elements 117 and heavier as these have not been studied enough to properly classify). The outermost shell of a noble gas atom has as many electrons as it can possibly have. Alkaline earth metal atoms, on the other hand, have only two electrons on their outer shell, and need only loose those two electrons to have a full outer shell.

Atomic numbers of the alkaline earth metals (Cf. A099956)

{4, 12, 20, 38, 56, 88}
##### Post-transition metals

The post-transition metals are weakly electropositive metals, since they have more than 2 electrons in their outermost shell, which they are less eager to offer to attain the stable configuration of the preceding noble gas. The post-transition metals include the poor metals, with the exception of aluminum, which is the only pre-transition metal with more than 2 electrons in its outer shell, and thus is more akin to a post-transition metal. Post-transition metals have atomic number $\scriptstyle Z \,$ (Cf. A??????)

{31, 49, 50, 81, 82, 83}
###### Poor metals

Poor metals have more than 2 electrons in their outermost shell.

{13, 31, 49, 50, 81, 82, 83}

#### Transition metals

Atomic numbers of the transition metals (Cf. A??????)

{21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 72, 73, 74, 75, 76, 77, 78, 79, 80, 104, 105, 106, 107, 108, 109, 110, 111, 112}

#### Inner transition metals

Atomic numbers of the inner transition metals (Cf. A??????)

{57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103}
##### Lanthanides

Atomic numbers of the lanthanides (Cf. A000027, from a(57) to a(71))

{57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71}
##### Actinides

Atomic numbers of the actinides (Cf. A000027, from a(89) to a(103))

{89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103}

### Non-metals

#### Metalloids

Atomic numbers of the metalloids or semimetals (Cf. A101648)

{5, 14, 32, 33, 51, 52, 84}

#### Nonmetals

Atomic numbers of nonmetals (Cf. A101647)

{1, 6, 7, 8, 15, 16, 34}

#### Halogens

We can compute this sequence mathematically by taking the formula for magic numbers, which happen to be the atomic numbers of the noble gases, from which you subtract 1 (ignoring hydrogen, which is not a halogen, and elements 117 and heavier as these have not been studied enough to properly classify). The outermost shell of a noble gas atom has as many electrons as it can possibly have. Halogen atoms, on the other hand, can admit precisely one more electron to their outer shell.

Atomic numbers of halogens (Cf. A097478)

{9, 17, 35, 53, 85, 117}

#### Noble gases

The outermost shell of a noble gas (or inert gas) atom has as many electrons as it can possibly have.

Atomic numbers of noble gases, i.e. magic numbers: atoms with full shells containing any of these numbers of electrons are considered electronically stable (Cf. A018227)

{2, 10, 18, 36, 54, 86, 118}

This sequence demonstrates that there is a mathematical structure to physical matter. We can compute this sequence mathematically by taking the formula for magic numbers, which happen to be the atomic numbers of the noble gases.

The magic numbers are given by

$a(n) = \sum_{k=1}^{n} s(k) = 2 \sum_{k=1}^{n} {\Bigg({\Bigg\lfloor \frac{k}{2} \Bigg\rfloor} + 1\Bigg)}^2 = \frac{n ~ ((n+3)^2 + 2)}{6} + \frac{(n+2)(1+(-1)^n)}{4} - 1,\quad n \ge 1, \,$
$a(n) = \frac{(n+1)(n+2)(n+3)}{6} + \frac{(n+2)(1+(-1)^n)}{4} - 2 = {n+3 \choose 3} + \frac{(n+2)(1+(-1)^n)}{4} - 2 = T_{n+1} + \frac{(n+2)(1+(-1)^n)}{4} - 2,\quad n \ge 1, \,$

where $\scriptstyle T_n \,$ is the $\scriptstyle n \,$th tetrahedral number, and where $\scriptstyle s(k) \,$ is the number of elements of period $\scriptstyle k \,$, and ignoring elements 117 and heavier as these have not been studied enough to properly classify.

The magic numbers give the extended sequence

{2, 10, 18, 36, 54, 86, 118, 168, 218, 290, 362, 460, 558, 686, 814, 976, 1138, 1338, 1538, 1780, 2022, 2310, 2598, 2936, 3274, 3666, 4058, 4508, 4958, 5470, 5982, 6560, 7138, 7786, 8434, 9156, ...}

with generating function

$G_{\{a(n)\}}(x) = \frac{2x (1 + 3 x - 2 x^2 - x^3 + x^4)}{(1 + x)^2 (1 - x)^4} \,$

## Notes

1. Technically, Mendeleev's table was not the first periodic table, but such details are beyond our scope here. See p. xxii in the Halka & Nordstrom book.
2. Halka & Nordstrom, p. xxiii
3. David A. Katz, NAMES AND SYMBOLS OF COMMON ELEMENTS, ©2002, 1992, 1990 by David A. Katz.
4. Eric W. Weisstein, Cosmological Theorem, from MathWorld.
5. Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements, © Copyright July 1999 Henry Bottomley.
6. About.com, Quantum Numbers and Electron Orbitals.
7. WebElements: The Janet Periodic Table.
8. Stewart, Philip (April 2010). "Charles Janet:unrecognized genius of the Periodic System". Foundations of Chemistry. doi:10.1007/s10698-008-9062-5.
9. 9.0 9.1 About.com, The Aufbau Principle - Electronic Structure and the Aufbau Principle.

## References

• Halka, Monica; Nordstrom, Brian (2010). Halogens and Noble Gases. Facts On File.