Editing Van der Waals radius (section)

=== Crystallographic measurements ===

The molecules in a [[molecular crystal]] are held together by [[van der Waals force]]s rather than [[chemical bond]]s. 
In principle, the closest that two atoms belonging to ''different'' molecules can approach one another is given by the sum of their van der Waals radii. 
By examining a large number of structures of molecular crystals, it is possible to find a minimum radius for each type of atom such that other non-bonded atoms do not encroach any closer. 
This approach was first used by [[Linus Pauling]] in his seminal work ''The Nature of the Chemical Bond''.<ref name="Pauling1945">{{cite book | first = Linus | last = Pauling | author-link = Linus Pauling | year = 1945 | title = The Nature of the Chemical Bond | location = Ithaca, NY | publisher = Cornell University Press | isbn = 978-0-8014-0333-0| title-link = The Nature of the Chemical Bond }}</ref> 
Arnold Bondi also conducted a study of this type, published in 1964,<ref name="Bondi1964"/> although he also considered other methods of determining the van der Waals radius in coming to his final estimates. 
Some of Bondi's figures are given in the table at the top of this article, and they remain the most widely used "consensus" values for the van der Waals radii of the elements. 
Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09&nbsp;Å for the van der Waals radius of [[hydrogen]] as opposed to Bondi's 1.20&nbsp;Å.<ref name="RowlandRS1996"/> A more recent analysis of the [[Cambridge Structural Database]], carried out by Santiago Alvarez, provided a new set of values for 93 naturally occurring elements.<ref name="Alvarez2013">{{cite journal |last=Alvarez |first=Santiago |year=2013 |title=A cartography of the van der Waals territories |journal=[[Dalton Trans.]] |volume=42 |issue=24 |pages=8617–36 |doi=10.1039/C3DT50599E |pmid=23632803 |doi-access=free |hdl-access=free |hdl=2445/48823}}</ref> The values of different authors are sometimes very different, so that one has to chose the ones which are closest in their physical meaning to those one wants to compare with. Here is a table with entries of four different authors. The valuse of Bondi from 1966 are those mostly used in crystallography:
{| class="wikitable"
|
|
|''r''<sub>vdW</sub> / Å
|''r''<sub>vdW</sub>  / Å
|''r''<sub>vdW</sub>  / Å
|''r''<sub>vdW</sub>  / Å
|-
|'''Element'''
|'''Atomic'''
'''number'''
|'''Bondi'''<ref name="Bondi1964" />    
1966
|'''Batsanov'''<ref>{{Cite journal |last=Batsanov |first=S. S. |date=2001 |title=[No title found] |url=https://link.springer.com/10.1023/A:1011625728803 |journal=Inorganic Materials |volume=37 |issue=9 |pages=871–885 |doi=10.1023/A:1011625728803}}</ref> 
2001
|'''Hu'''<ref>{{Cite journal |last=S. Z. Hu, Z. H. Zhou, B. E. Robertson |date=2009 |title=Consistent approaches to van der Waals radii for the metallic elements |url= |journal=Z. Kristallogr. |volume=224 |issue=8 |pages=375–383|doi=10.1524/zkri.2009.1158 |bibcode=2009ZK....224..375H }}</ref>           
2009
|'''Alvarez'''<ref name="Alvarez2013" /> 
2014
|-
|H
|1
|1.2
|1.0
|1.08
|1.2
|-
|Li
|3
|1.82
|2.15
|2.14
|2.12
|-
|Be
|4
|
|1.85
|1.69
|1.98
|-
|B
|5
|
|1.75
|1.68
|1.91
|-
|C
|6
|1.7
|1.7
|1.53
|1.77
|-
|N
|7
|1.55
|1.6
|1.51
|1.66
|-
|O
|8
|1.52
|1.55
|1.49
|1.5
|-
|F
|9
|1.47
|1.45
|1.48
|1.46
|-
|Na
|11
|2.27
|2.45
|2.38
|2.5
|-
|Mg
|12
|1.73
|2.15
|2.00
|2.51
|-
|Al
|13
|
|2.05
|1.92
|2.25
|-
|Si
|14
|2.1
|2.05
|1.93
|2.19
|-
|P
|15
|1.8
|1.95
|1.88
|1.9
|-
|S
|16
|1.8
|1.8
|1.81
|1.89
|-
|Cl
|17
|1.75
|1.8
|1.75
|1.82
|-
|Se
|17
|1.90
|1.9
|1.92
|1.82
|-
|K
|19
|2.75
|2.85
|2.52
|2.73
|-
|Ca
|20
|
|2.45
|2.27
|2.62
|-
|Sc
|21
|
|2.25
|2.15
|2.58
|-
|Ti
|22
|
|2.10
|2.11
|2.45
|-
|V
|23
|
|2.05
|2.07
|2.42
|-
|Cr
|24
|
|2.0
|2.06
|2.45
|-
|Mn
|25
|
|2.0
|2.05
|2.45
|-
|Fe
|26
|
|2.0
|2.04
|2.44
|-
|Ni
|28
|1.63
|1.95
|1.97
|2.4
|-
|Cu
|29
|1.40
|1.9
|1.96
|2.38
|-
|Zn
|30
|1.39
|2.0
|2.01
|2.39
|-
|Ga
|31
|1.87
|2.05
|2.03
|2.32
|-
|Ge
|32
|
|2.05
|2.05
|2.29
|-
|As
|33
|1.85
|2.05
|2.08
|1.88
|-
|Br
|35
|1.85
|1.9
|1.9
|1.86
|-
|Rb
|37
|
|3.0
|2.61
|3.21
|-
|Sr
|38
|
|2.6
|2.42
|2.84
|-
|Y
|39
|
|2.4
|2.32
|2.75
|-
|Zr
|39
|
|2.3
|2.23
|2.52
|-
|Nb
|41
|
|2.15
|2.18
|2.56
|-
|Mo
|42
|
|2.1
|2.17
|2.45
|-
|Tc
|43
|
|2.1
|2.16
|2.44
|-
|Ru
|44
|
|2.05
|2.13
|2.46
|-
|Rh
|45
|
|2.0
|2.1
|2.44
|-
|Pd
|46
|1.63
|2.05
|2.1
|2.15
|-
|Ag
|47
|1.72
|2.05
|2.11
|2.53
|-
|Co
|47
|
|1.95
|2
|2.4
|-
|Cd
|48
|1.62
|2.2
|2.18
|2.43
|-
|In
|49
|1.93
|2.25
|2.21
|2.43
|-
|Sn
|50
|2.17
|2.2
|2.23
|2.42
|-
|Sb
|51
|
|2.25
|2.24
|2.47
|-
|Te
|52
|2.06
|2.15
|2.11
|1.99
|-
|I
|53
|1.98
|2.1
|2.09
|2.04
|-
|Cs
|55
|
|3.15
|2.75
|3.48
|-
|Ba
|56
|
|2.7
|2.59
|3.03
|-
|La
|57
|
|2.5
|2.43
|2.98
|-
|Hf
|72
|
|2.25
|2.23
|2.63
|-
|Ta
|73
|
|2.2
|2.22
|2.53
|-
|W
|74
|
|2.15
|2.18
|2.57
|-
|Re
|75
|
|0.21
|2.16
|2.49
|-
|Os
|76
|
|2.0
|2.16
|2.48
|-
|Ir
|77
|
|2.0
|2.13
|2.41
|-
|Pt
|78
|1.72
|2.05
|2.13
|2.32
|-
|Au
|79
|1.66
|2.0
|2.14
|2.32
|-
|Hg
|80
|1.70
|2.1
|2.23
|2.45
|-
|Tl
|81
|1.96
|2.25
|2.27
|2.47
|-
|Pb
|82
|2.02
|2.3
|2.37
|2.6
|-
|Bi
|83
|
|2.35
|2.38
|2.54
|-
|Th
|90
|
|2.45
|2.45
|2.93
|-
|U
|91
|1.86
|2.4
|2.41
|2.71
|}
A simple example of the use of crystallographic data (here [[neutron diffraction]]) is to consider the case of solid helium, where the atoms are held together only by van der Waals forces (rather than by [[Covalent bond|covalent]] or [[metallic bond]]s) and so the distance between the nuclei can be considered to be equal to twice the van der Waals radius. 
The density of solid helium at 1.1&nbsp;K and 66&nbsp;[[Atmosphere (unit)|atm]] is {{val|0.214|(6)|u=g/cm3}},<ref name="Henshaw1958">{{cite journal | first = D.G. | last = Henshaw | year = 1958 | title = Structure of Solid Helium by Neutron Diffraction | journal = [[Physical Review]] | volume = 109 | issue = 2 | pages = 328–330 | doi = 10.1103/PhysRev.109.328|bibcode = 1958PhRv..109..328H }}</ref> corresponding to a [[molar volume]] ''V''{{sub|m}}&nbsp;= {{val|18.7|e=-6|u=m3/mol}}. 
The van der Waals volume is given by
<math display="block">V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}}</math>
where the factor of π/√18 arises from the [[Sphere packing|packing of spheres]]: ''V''{{sub|w}}&nbsp;= {{val|2.30|e=-29|u=m3}}&nbsp;= 23.0&nbsp;Å{{sup|3}}, corresponding to a van der Waals radius ''r''{{sub|w}}&nbsp;= 1.76&nbsp;Å.