A Focus on Crystallography
Modern crystallography – creation and development of crystal structure databases
using the Inorganic Crystal Structure Database (ICSD) as an example

The important thing in science is not so muchto obtain new facts as to discover new ways ofthinking about them.
William Henry Bragg

Title Photo: Spinel ©FIZ KarlsruheA Focus on Crystallography

Table of contents

Preface4

Inorganic Crystal Structure Database (ICSD)

6

Extended symbols of coordination polyhedra in ICSD
and their pictorial representation

11

Deposition of crystal structures at FIZ Karlsruhe

18

Analysis of spinel compounds in ICSD

20

Bond lengths statistics as a tool for crystal structure
analysis – a case study with a crystalline silicophosphate

23

Using ICSD to predict new properties, compounds
and modifications

27

Crystals and mathematics

37

Three centuries of crystallography through thelens of zbMATH

44

Facts and figures

51

3

4

A Focus on Crystallography

Preface
About one hundred years ago, Max von Laue discovered the diffraction of X-rays by crystals. This won him the Nobel Prize in Physics
in 1914. At almost the same time William Henry Bragg and his son,
William Lawrence Bragg, recognized that X-rays could be used to
identify crystal structures and were awarded the Nobel Prize inPhysics in 1915.

Sabine Brünger-Weilandt
President & CEO of FIZ Karlsruhe

Today, 100 years later, these ground-breaking discoveries are honored and remembered on the occasion of the International Yearof Crystallography. Over the years, crystallography has steadily
evolved into one of the most basic and most important sciencesof our times, which is reflected, among other things, in more thantwenty Nobel Prizes based on a crystallographic background.
The crystal structure is essential for all solids. It determines the difference in hardness and appearance between graphite and diamond. It serves to detect the effect ofdrugs; important Nobel Prize-winning findings were the identification of the structureof penicillin, insulin, and, last but not least, of the DNA. Today, the Curiosity Rovermeasures the composition of crystalline soil samples on Mars. Crystallography impacts
the development of everyday items such as innovative materials for the automotive or
the aviation industry. It can also help to cope with the challenges of the German “Energiewende“ (i.e. transition from fossile to alternative or renewable energy resources);
Crystallography serves, for example, to develop new insulating materials or to discoversuitable material structures for a new generation of cost-saving and efficient solar cells.
Material properties depend on the crystal structure. Therefore it is important, whendeveloping new materials or products, to know how the atoms are aligned in the compound and which properties result from this. Crystallography helps answer questions
on how certain compounds can be modified in order to obtain specific properties.
FIZ Karlsruhe has been operating and developing the Inorganic Crystal Structure
Database (ICSD) for almost 30 years, its beginnings dating back to 1978. The oldest
structures contained in ICSD are those of common salt and diamond, published by

A Focus on Crystallography

William Henry Bragg and his son William Lawrence Bragg in 1913.
The structures described and recorded today are much more complex, but the measurements are still based on the same assumptionof the periodicity of crystal lattices and on the same measuring
procedures. However, the mathematical methods applied are moresophisticated, and modern technology is used. In particular, the useof computers has significantly shortened the time required to identify
a structure from several weeks to one or two days.
An interesting side phenomenon are the quasicrystals. They had
already been known as a mathematical model in earlier years, but
their existence could only be proven by modern technology. The
discovery of the quasicrystal structures was awarded with the Nobel
Prize in 2011. Their most distinctive property compared to all otherknown solids is their non-periodic crystal lattice. This makes it difficult to record such compounds in traditional structure databases.

FIZ Karlsruhe

Today, ICSD is the most comprehensive database for inorganic crystal structures worldwide. It serves not only as a reference work; the complex search algorithms also allow
for comparative structure determination, for analysis, or for using the data for the development of new materials.
This brochure describes the ICSD’s evolution and shows its potential applications. It
also demonstrates the close relationship between crystallography and mathematics,
and it answers questions, e.g., on how the bibliographic database zbMATH (formerly“Zentralblatt für Mathematik“) can be used to gain an overview on the development ofcrystallography-related mathematical methods.
Crystallography will maintain its high importance in the future. True to its motto”Advancing Science“, FIZ Karlsruhe as the producer of ICSD will live up to the newchallenges of modern crystallography and will continue to provide inorganic chemists
and materials scientists with a highly sophisticated tool enabling them to generatescientific value and innovation.

5

6

A Focus on Crystallography

Ongoing popularity of factual databases even in times of large search engines

Inorganic Crystal Structure Database (ICSD)
Stephan Rühl

Introduction
In spite of manifold possibilities to gain expert information today, the use of factual databases is still essential.
Because of highly specialized requests nowadays many
tailor-made products are available [1].
Crystal structures in general play an important role inunderstanding physical properties of materials. A lot of
special databases cover particular interests like zeolite
structures [2] or mineral structures [3], but for general
purposes overall collections of data are required. Examples to mention here are ICSD [4], CSD [5] or PDB [6].
The huge amount of valuable information stored in crystal
structure databases helps researchers in many ways, forexample to provide input for a Rietveld refinement [7] ordata-mining parameters for structure prediction [8] oroptimization procedures. But crystal structure databases
contain a lot more valuable information than the obvious
unit cells and atomic coordinates. For example, ICSD canbe used to find similar structures by comparing certain
features, like the space group or the ANX formula, thatdefine different structure types [9, 10]. For many more applications of ICSD see article “Using ICSD to predict new
properties, compounds and modifications” by Schön onpage 27.
For a database to be of any use to researchers, it has
to cover several important aspects. The first essential
aspect is that data can be easily compared and hence
the data have to be supplied in some standardized form.
For crystallographic data this kind of standardization ispartly inherited from the principles of crystallography
itself and further enforced by the application of standardization tools to the published crystal structure. Even for
the exchange of crystallographic information a generally
accepted format is defined (Crystallographic Information
File – CIF) [11]. Another important point is the completeness of the information provided. A statistical interpretation based only on a small subset will likely produce
unreliable results. Also it is at least highly inconvenient
having to check several databases and combining results. Unfortunately, this step might become necessary if

the scope of a database only partly includes the field ofinterest. The third and most decisive factor for a database is the quality of the data. Therefore, carefully checking and evaluating new information is fundamental.

History of ICSDThe original idea of ICSD goes back to an initiative ofProf. Günter Bergerhoff in 1978 at the Institute for Inorganic Chemistry of the University of Bonn, Germany [4].
FIZ Karlsruhe started to maintain the database in collaboration with the University of Bonn in 1985. In 1989 a
joint venture between the Gmelin Institute, Germany, and
FIZ Karlsruhe, both non-profit institutions, took over the
responsibility for ICSD. Since 1997 ICSD is produced cooperatively by FIZ Karls-ruhe, Germany, and the National
Institute of Standards and Technology (NIST), U.S.A. The
production, quality control and general software development is done at FIZ Karlsruhe, while the still widely used
PC-Windows based graphical interface FindIt is generated
under the responsibility of NIST. A first WWW interface
was developed by Alan Hewat at the Institute Laue-Langevin, Grenoble, but was replaced in 2009 by a new interface developed by FIZ Karlsruhe. Since many years FIZ
Karlsruhe has been working together with Prof. Rudolf Allmann concerning the quality assurance of the databaseentries and the development of some enhancementsof ICSD, e.g., the Structure Types of the database. Prof.
Allmann was awarded with the Will-Kleber medal of theGerman Association for Crystallography for his extraordinary achievements
for ICSD. FIZ Karlsruhe
is grateful for the tremendous commitment
Prof. Allmann has
shown towards ICSD
during these years.
At about the time thatthe database wasinitiated the Crystal
Structure Deposition
at FIZ Karlsruhe began.
It started as an archiveof crystal structure

Prof. Rudolf Allmann

A Focus on Crystallography

records because the important crystal structure information was missing in many relevant publications. Since
then, many publishers provide footnote references tothe crystal structure depot of FIZ Karlsruhe, where additional structure information can be obtained on request
using a Crystal Structure Deposition number. Since February 1999, there has been an agreement between the
Cambridge Crystallographic Data Centre (CCDC) and FIZ
Karlsruhe that all organic and organometallic compounds
should be deposited at CCDC and all inorganic and intermetallic compounds at FIZ Karlsruhe. The deposited data
are also used as input for the respective databases.

Content of ICSDThe Inorganic Crystal Structure Database (ICSD) contains
information on structures• that have no C-C and no C-H bonds,• whose atomic coordinates have been fully determinedor were derived from a corresponding structure type.
Since 2003, crystal structures of metallic and inter-metallic compounds have been included into ICSD. The number of records in ICSD has roughly doubled in the last
12 years (Fig. 1). Apart from currently more than 6,000
new entries per year FIZ Karlsruhe is working continually

Fig. 1: ICSD growth since 1980

7

on filling in the gaps in older data (Fig. 2). In 2001, ICSD
changed to a relational database system with at that time
25 tables and about 200 database fields. Continual extension of the content results in more than 35 tables andmore than 300 database fields. This allows for queries
with much more criteria. Innovations like the inclusion of
structure types and calculation of standardized data provided new search options.
ICSD records provide the full structural information published, the bibliographic references, and additional value-added content. Especially the structure types, whichwere introduced in 2005 [12] and were subsequently
expanded during the last years, represent valuable information in comparing inorganic crystal structures.
The two defining properties for several crystal structures
to be considered belonging to one structure type arethat the structures are isopointal and isoconfigurational.
These rather unhandy properties are broken down tosome easily checkable properties like the ANX formula,
Pearson symbol, Wyckoff sequence, etc. A complete description of this procedure would be beyond the scope of
this paper and can be found in the article by Allmann and
Hinek [12].

8

A Focus on Crystallography

Fig. 2: Continually filling the gaps in older data – distribution of ICSD records by publication year

ICSD now contains a comprehensive and searchable listof 8,230 structure types according to which approximately 79,8% of the crystal structures have been assigned.
For each structure type one representative is used as aprototype. For all prototypes the atomic environments of
all atomic positions (except for Hydrogen) are given, i.e.
the kind and number of neighboring atoms (coordination
number CN). In questionable cases the neighbors arecalculated by constructing the Dirichlet-domains for all
atomic positions (more details on this can be found in thearticle “Extended symbols of coordination polyhedra in
ICSD and their pictorial representation” by Allmann et al.
on page 11). Additionally, the shape of the coordination
polyhedron is given for all atoms of the prototype using
the nomenclature of Lima-de-Faria [13] and an extension
by Allmann and Hinek [12].

Input and data evaluation
The data acquisition for the input into the databaseutilizes a wide range of procedures like screening of theoriginal papers in crystallographic journals on relevancy,
searching in the Chemical Abstracts File, and including
the information from the Crystal Structure Deposition
depot or direct contributions from authors and crystallographic experts. Each structure determination reported in
the literature leads to a separate record in the database.

Even with evaluating all these sources of information it
seems clear that comprehensiveness is a difficult task sothat some structures are still missing and any feedback
from the scientific community is highly appreciated.
During the production process the data are excerpted byscientists and checked automatically by computer programs and manually by our experts. This evaluation procedure includes, but is not limited to, checks for correct
syntax (in case of CIF files), duplicate records, missing
fields and, most importantly, plausibility. In addition,some details of the new entry can easily be verified likethe space group being compatible with the cell or theassigned oxidation states fulfilling electroneutrality in
combination with the multiplicity and site occupations for
all atoms. Also very important are checks for interatomic
distances which should be within the range of similar
distances in the database. This is just a short overview
of the evaluation process; more details on the checking
routines are described by Behrens [14].
Lastly, additional content not directly given in the article
is generated either automatically or manually. Amongthese data items are the Wyckoff sequence, Pearson
symbol, molecular formula, molecular weight, calculated
density, ANX or AN formula, lists and histograms of interatomic distances, the reduced cell, the standardized cell
(if necessary), mineral names or mineral groups.

A Focus on Crystallography

Interfaces
ICSD is offered as a stand-alone version for local PC installation (e.g. FindIt, produced in cooperation with the
National Institute of Standards and Technology), a local
intranet version for small user groups, and as a Web version hosted by FIZ Karlsruhe.
We have developed a locally installable version based on
the ICSD Web interface (Fig. 3 and 4). This version will replace the FindIt interface which was originally developed
for Windows XP.
Developing a new local version based on an existing interface offers many synergies and will speed up the inclusion of new features for both interfaces. In the near future
the intranet version will also be based on this framework.
The user can then switch from one interface to any otherinterface without having to learn how to use this specificinterface.
All interfaces offer search options for the relevant crystallographic information as well as the added content, visu-

Fig. 3: Search screen of ICSD-Desktop/Web interface

9

alization of the crystal structure, simulation of the powder
pattern, and export as CIF (Crystallographic Information
File).
Apart from FindIt, which is limited to Windows operating
systems, ICSD can be used on all major platforms. Although there is no special ICSD App one can access the
ICSD Web version on mobile devices via the internet. As
mobile operating systems do not support Java applets sofar, the visualization of crystal structures is not yet possible.

Outlook
ICSD offers a wide spectrum of useful information forchemists, physicists, crystallographers, mineralogists orgeologists, and it is widely used from teaching basic inorganic structural chemistry to research on new materials
in academia and industry. As new knowledge is gained,
the needs of the users change and so the content of databases in general is adjusted to reflect these increasing
demands.

10

A Focus on Crystallography

Fig. 4: Results page of ICSD Web interface

The scope of ICSD has changed several times over thelast 35 years. Some smaller improvements are continuously being made, e.g., the inclusion of DOIs for easieraccess to the primary literature. Further extensions withadditional information are planned.
Significant changes are currently made to the ICSD Web
and ICSD-Desktop interfaces, for which some importantnew features and improvements of existing features areincluded. We have some more improvements – basedon the wishes of our users – planned for the near future.
Among these will be more export options for crystal structure data and more customization options on the display.
In addition we are evaluating the option to directly include external programs in our interface to offer better
analysis of the results obtained in the search. This could
also include simple tools to compute information not directly contained in the database like bond valence sums.

References[1]

Böhme, U.; Tesch, S. (2014). Nachr. Chem. 62, p. 43.

[2] Baerlocher, C. ; McCusker, L.B. Database of Zeolite Structures:
http://www.iza-structure.org/databases/ (November 25, 2014).
[3] Downs, R.T. ; Hall-Wallace, M. (2003). Am. Mineral. 88, p. 247.
[4] Bergerhoff, G.; Brown, I.D. (1987). Crystallographic Databases,
International Union of Crystallography, Chester. pp. 77-95; Belsky,
A.; Hellenbrandt, M.; Karen, V.L. ; Luksch, P. (2002). Acta Cryst.
B58, p. 364.
[5] Allen, F. H. (2002). Acta Cryst. B58, p. 380.
[6] Bernstein, F.C.; Koetzle, T.F.; Williams, G.J.; Meyer Jr., E. E.; Brice,
M.D., Rodgers J.R.; Kennard, O.; Shimanouchi, T.; Tasumi, M. (1977).
J. Mol. Biol. 112, p.535.
[7] Rietveld, H.M.; (1967). Acta Cryst. 22, p. 151; Rietveld, H. M. (1969).
J. Appl. Cryst. 2, p. 65.
[8] Hofmann, D.; Kuleshova, L. (2012). Data Mining in Crystallography,
Springer Verlag, Heidelberg; Buchsbaum, C.; Höhler-Schlimm, S.; 		
Rehme, S. (2010) Struct. Bond 134, p. 37.
[9] Kaduk, J.A. (2002). Acta Cryst. B58, p. 370.
[10] Abrahams, S.C. (1994). Acta. Cryst. A50, p. 658.
[11] Hall, S.R.; Allen, F.H.; Brown, I.D. (1991). Acta Cryst. A47, p. 655.
[12] Allmann, R.; Hinek, R. (2007). Acta Cryst. A63, p. 412.
[13] Lima-de-Faria, J.; Hellner, E.; Liebau, F.; Makovicky, E.;
Parthe, E. (1990). Acta Cryst. A46, p.1.
[14] Behrens, H. (1996) J. Res. Natl. Inst. Stand. Technol. 101, p. 365 .

A Focus on Crystallography

11

Classification of related structures in ICSD

Extended symbols of coordination polyhedra
in ICSD and their pictorial representation
Rudolf Allmann, Roland Hinek, Stephan Rühl

In 2014 the Inorganic Crystal Structure Database (ICSD)
contains 173,473 structures in total and about 79% areassigned to one of the 8,230 distinct structure types. The
structure types were introduced in ICSD in 2007 as an approach to describe structural similarity between inorganic
structures. ICSD structure types can best be classified as
isoconfigurational, according to the definition by the IUCr[1]. In this definition isoconfigurational structures must
conform to certain structural parameters (descriptors)
like space group, Pearson symbol, Wyckoff sequence,
c/a ratio, ANX formula. Of these descriptors the Wyckoff
sequence is crucial, because it describes the atomic arrangement in the unit cell. Inorganic structures are often
visualized by coordination polyhedra of single atoms orions. Each structure type is characterized by some or allof the descriptors as has been explained in detail by Allmann and Hinek [2].
For each structure type a “prototype” is selected from all
structures belonging to this structure type. The prototype
and thus the name of the structure type is mostly basedon the first publication of this structure. The prototypes
play an important role in ICSD as the atomic environment(AE) of suitable non-H-atoms in this structure is listed as

Fig. 1: Coordination polyhedra proposed by Parthé

an additional comment. The AE represents the first coordination sphere and includes all first neighbors forming
the coordination polyhedron. For example, in the spinel
structure Al2MgO4 the oxygen atoms are arranged in a cubic close packed lattice while the Al and Mg occupy every
second octahedral void and every eighth tetrahedral void,
respectively. The comment for this entry in ICSD (collection code 31373) includes the AE:
AE: Al: 6o O6; Mg: 4t O4; O: 4t: Al3 Mg; c.c.p. O (O: 12co O12)
In this systematic description, the atomic environment for
Al, Mg and O is specified. The aluminum is octahedrally(6o) coordinated by six oxygen ions (O6); the magnesium
shows a tetrahedral (4t) coordination by four oxygen ions(O4) and the oxygen is also tetrahedrally (4t) coordinatedby three aluminum ions and one magnesium ion (Al3 Mg).
The text in italics describes the second coordination
sphere around the oxygen, but usually only the first
sphere is given in the prototype entries.
At a conference in Kiel (Germany) in 1964, Donnay, Hellner and Niggli introduced the denotation of short symbols
for polyhedra, which at that time were formed exclusively
by symmetry-equivalent atoms in each of the 230 spacegroups [3]. Parthé extended this nomenclature to the firstneighbors (first coordination sphere) of an atom, which inaddition may include non-symmetry-equivalent neighbors[4] (e.g. a tetragonal pyramid “5y” having four equivalent

12

A Focus on Crystallography

Fig. 1: Coordination polyhedra proposed by Parthé

atoms at the base and one non-equivalent atom at the
top of the pyramid). In this book Parthé showed 29 frequently observed coordination polyhedra with their shortsymbols (Fig. 1). This denotation with short symbols was
accepted by the nomenclature commission of the IUCr in
1990 and 1998 [1].

iicosahedrontt

truncated tetrahedronpd

pentagon-dodecahedroncocuboctahedron

In 2010, Allmann and Hinek introduced a more precise
definition of the symbol “c” describing the presence of“capping” atoms. The capping atoms in prisms and antiprisms can be differentiated by being centered on equatorially (c) or axially located faces (c’) [5].

aco

anticuboctahedronrdrhombic dodecahedron

bds

bisdisphenoid

tds

trisdisphenoid

The following letters are used to distinguish between thecoordination polyhedra:

totruncated octahedrontcb

truncated cubelcoplanar or collinear

snubcbsnub cubennon-coplanar or non-collinear

FK

Frank-Kasper polyhedra

p

prism

c

ap

antiprism

capped faces for non-prisms/non-antiprismsypyramidbybipyramidttetrahedronooctahedron

cb

cube

equatorially capped faces for prisms/
antiprisms
c’axially capped faces for prisms/antiprisms(outside):

The central atom is outside of thepolyhedron.

A Focus on Crystallography

The following list shows for each coordination number
the symbols observed in ICSD, and figures 2a and 2bshow the 40 most observed polyhedra in an idealized
geometry:2:2l, 2n3:

3l, 3n4:

4l, 4n, 4t, 4y (central atom at the center of the
base)

5:

5l, 5n, 5y, 5by

6:

6l, 6n, 6o (deformed: 6by, 6ap), 6p, 6y

7:

7y, 7by, 6o1c (= 6ap1c), 6p1c; 6p1c’

8:8cb, 8ap, 8bds, 8by, 6p2c (similar to 8ap),
6p2c’, 4l4t

9:6p3c (similar to 8ap1c’), 8p1c’, 6n3n10:10p, 10ap, 8p2c’, 8ap2c’, 3n6n1c’ (=7y3c’)

11:

6p3c2c’, 10p1c’, 10ap1c’

12:12co, 12aco, 12i, 12p, 10p2c’, 10ap2c’, 12tt,
12tds, 6p6c

13:

12p1c’, 12ap1c’, 13FK(= 10ap3c’)

14:

14FK (= 12ap2c’), 12p2c’, 14rd

15:

15FK(=12ap3c’), 10p5c

16:16FK (=12tt4c)

17:

10p5c2c’

18:

12p6c

20:

12p6c2c’, 20pd

24:

24to, 24tcb, 24snubcb

The coordination numbers in ionic structures usually do
not exceed 12. Very frequent coordination polyhedra are
4t (tetrahedron) and 6o (octahedron). Cations are surrounded by anions and vice versa. The anion-anion contacts are neglected despite the fact that they play an important role in space filling and the unit cell dimension.
In spinels, for example, each oxygen has 12 O-O contacts
as second neighbors resulting in the coordination polyhedron “12co”. Assuming an O-O distance of 2.8 Å thiscorresponds to a cubic lattice constant of about 7.92 Å(=2.8*sqrt(2)*2), which is in good agreement with thecell parameter a=8.08 Å for the ICSD entry with collection
code 31373.
In structures of metals frequent coordination numbers
are 12 to 16 (12i-16FK) due to the closer packing in comparison to ionic structures. A metal structure consists ofcations embedded in an electron gas. The existence of

13energy bands of the electronic states in metal structures
is the reason for characteristic properties like thermaland electrical conductivity. The variety of ion-radii of atoms in metallic structures is smaller than for ionic ones
and the anions are generally greater than the cations.
Usually the first and second coordination spheres of agiven atom are well separated by a significant “distancegap” (Fig. 3).

			Fig. 3: ICSD entry with collection code 31373 (spinel type): distancehistogram around O with a gap between first (3Al +1 Mg) and second
coordination sphere (c.c.p. O (O: 12co O12))

The best way to determine the first neighbors is bymeans of Dirichlet domains [6]. The Dirichlet domain of
an atom is an artificial construction of the part of space
which is nearest to the central atom. It depends on thespecific atomic environment, i.e., of its nearest neighbors. The Dirichlet domain itself is a polyhedron. It is
enclosed by planes located midway between the central
atom and each nearest neighbor and perpendicular tothe distance between them. Figures 4a and 4b illustrate
the creation of the Dirichlet domain for the cuboctahedron as the second coordination sphere around an
oxygen in the spinel as discussed earlier. If only the distances from the central oxygen to the 12 nearest oxygens
forming the cuboctahedron in figure 4a are drawn, theresulting image consists of only the red lines in figure 4b.
Now planes are constructed that are each perpendicular
to one of the red lines and include the center of this line.
The central atom is then enclosed by the Dirichlet domain. This method is used in the program TOPOS [7].
Another method to generate the Dirichlet domain does
not place the planes midway between the central atomand the coordinated atom but takes into account thepossibly different sizes of the atoms. This concept of14

A Focus on Crystallography

6ap

6y

6by

6ap1c‘

trigonal antiprism= deformed 6opentagonal
pyramid

tetrag. bipyramid= deformed 6oaxial cappedhexagonal antiprism

7y

4l4t

8bds

6p2c‘

hexagonal pyramid

deformed 8bds
(1sphenoid→square)bisdisphenoid

axial bicapped
trigonal prism

8p1c‘

8ap1c‘(≈6p3c)

6p3c1c‘

3n6n1c‘ = 7y3c‘

axial capped
tetragonal prism

axial capped
tetragonal antiprism

equat.tricapped axial
capped trig. prism

axial tricappedhexagonal pyramid

8p2c‘

10p

10ap

6p3c2c‘axial bicapped
tetragonal prism

pentagonal
prism

pentagonal
antiprism

equat. tricapped axial
dicappaed trig. prism

10ap1c‘

10p1c‘

6p6c (≠12aco)

8p4c

axial capped
pentagonal antiprism

axial capped
pentagonal prism

equat. hexacapped
trigonal prism

equat. tetracapped
tetrag. prism (≈12co)Fig. 2a: Additional polyhedra observed in ICSDA Focus on Crystallography

15

10ap2c’

10p2c‘

8p4t=12tds

12tt

axial bicapped
pentagonal antiprism= deformed 12i

axial bicapped
pentagonal prism

trisdesphenoid

Truncated tetrahedron

12ap

12ap1c’

12p1c‘

12-1ap2c’=13FK

hexagonal antiprism

axial cappedhexagonal antiprism

axial cappedhexagonal prism

13 Frank-Kasper
polyhedron

8cb6c(≈14rd)

14rd

12ap2c‘

12p2c’

hexacapped cube

rhombic dodecahedronaxial bicapped hexagon.
antiprismaxial bicapped hexagon.
prism

10p5c

10p5c2c’

12p6c

12p6c2c‘equat. pentacapped
pentagonal prism

equat. pentacapped
axial bicapped
pentagonal prism

equat. hexacappedhexagonal prism

equat. hexacapped
axial bicappedhexagonal prism

20pd

24snubc

24tcb

24to

pentagon-dodecahedronsnubcubetruncated cube

truncated octahedron

Fig. 2b: Additional polyhedra observed in ICSD16

A Focus on Crystallography

angles are large for short distances between two atoms
and the corresponding face of the Dirichlet domain isthen also large. The area of the face of the Dirichletdomain can be used instead of the spatial angle as an
estimate for the strength of the bond.
For example, in a body-centered cubic structure each
atom has 8+6 neighbors forming a rhombic dodecahedron (14rd) coordination polyhedron. The spatial angles
for each of the 8 nearest neighbors is 9.84% and for the
6 farer away neighbors 3.54% corresponding to a truncated octahedral Dirichlet domain (which is not dual to
the rhombic dodecahedral coordination polyhedron). The
spatial angles of all 14 neighbors sum up to 100%.
Fig. 4a: spinel-2nd AE: O↔O12

Although there is a strong geometrical correspondence
between the coordination polyhedron and the Dirichletdomain for an atom, there is also a remarkable difference
between both constructions: The Dirichlet domains of all
atoms fill the space completely like, e.g., unit cells, while
the coordination polyhedra do not.
In ICSD, the coordination polyhedra are specified in thecomments and are only given for the structure type prototypes. A coordination found in a prototype will also appear in all members belonging to this structure type.

Fig. 4b: spinel-2nd Dirichlet domain: O↔O12 (TOPOS [7])Here are some examples of how to search for coordination polyhedra using the different retrieval interfaces:radical planes is used in the program DIDO [8]. The radical planes generally are located closer to the smaller atoms. This program is mostly used to calculate the atomic
environments in ICSD.
Synonyms of the term Dirichlet domain are Voronoi polyhedron or Wigner-Seitz cell. A Brillouin zone is a Dirichletdomain constructed in reciprocal space instead of real
space.
Dirichlet domains and coordination polyhedra (AE representation) are often “dual” to each other, i.e., they
depend on each other in the way that the number of vertices and faces are interchanged. Examples of dual pairsof polyhedra are 4t-4t, 6o-8cb, 12i-20pd, 12co-14rd.
AE representation ← “DUAL” → Dirichlet domain (Dc)					
Cuboctahedra “12co” ← “DUAL” → “14rd = 8cb6c”
The “strength” of a bond can be measured by the spatial
angle of the corresponding face of the Dirichlet domain
projected onto a sphere. Spatial angles are usually given
as a percentage of the surface of the sphere. The spatial

Cuboctahedron(12co)

Icosahedron

(12i)

Cube

(8cb)

In FindIt comments can be searched on the Reference tab
using the comments search field available when the radio
button “Free text” is selected (Fig. 5). In FindIt automatic
truncation is applied to the search terms, so the numberof hits tends to be higher (see examples in table 1). In ICSD
Web or ICSD-Desktop this search field is accessible in the
Advanced Search on the Experimental Information mask
(Fig. 6). No automatic truncation takes place. This meansthat the given terms are searched exactly, if no wildcards
have been explicitly set by the user.

Polyhedron

FindItICSD Web/ICSD-Desktop

Hits

12co

AE*12coAE* *12co*

572

12i

AE*12iAE* *12i*

361

8cb

AE*8cb

AE* *8cb*

489

Table 1: Examples of how to search for polyhedra in FindIt and ICSD
Web/ICSD-Desktop

A Focus on Crystallography

17

Fig. 5: Screenshot of a search for a polyhedron in FindIt

Fig. 6: Screenshot of a search for a polyhedron in ICSD Web/ICSD-Desktop

References[1]

Lima-de-Faria, J.; Hellner, E.; Liebau,F.; Makovicky, E.; Parthè, 		E., (1990, 1998). Report of the International Union of Crystallography Commission on Crystallographic Nomenclature Subcommittee on the Nomenclature of Inorganic Structure Types.

[2] Allmann, R.; Hinek, R., (2007). The introduction of structure types
into the Inorganic Crystal Structure Database ICSD, Acta Cryst., 		
A63, pp. 412-417.
[3] Donnay, J.D.H.; Hellner, E.; Niggli, A., (1964). Coordination 		
polyhedra, Z. Kristallogr. 120, pp. 364-374.
[4] Parthé, E. (1964). Elements of inorganic structural chemistry,
2nd edition, K. Sutter Parthé Publisher, Petit-Lancy, Switzerland;
Lima-de-Faria, J. (1994). Structural mineralogy, Kluwer Academic		Publishers, Dordrecht, Netherlands.

[5] Allmann, R.; Hinek, R.,(2010). Extended symbols for coordination 		
polyhedra after Parthé, Acta Cryst., A66, p. 174.
[6] Shannon, R.D. ; Prewitt, C.T. (1969. ) Effective ionic radii in oxides 		
and fluorides, Acta Cryst. B25, pp. 925-946.
Shannon, R.D. (1976). Revised effective ionic radii and systematicstudies of interatomic distances in halides and chalcogenides, 		Acta Cryst., A22, pp. 751-767.
[7] Blatov, V.A. (2004). Voronoi-Dirichlet polyhedra in crystal 		chemistry: theory and applications, Cryst. Rev. 10, pp. 249-318.
[8] Koch, E.; Fischer, W. (1996). DIDO95 and VOID95 - programs for the
calculation of Dirichlet domains and coordination polyhedra,
Z. Kristallogr. 211, pp. 251-253.

18

A Focus on Crystallography

The central address for crystallographers in inorganic chemistry

Deposition of crystal structuresat FIZ Karlsruhe
Helmut Müller

For now 30 years FIZ Karlsruhe has been offering its
service to deposit crystal structures for free. At the beginning, and this means the period when digital networks
were new and storage capacity was expensive, deposition at FIZ Karlsruhe was a valuable aid for researchers
worldwide, providing access to (mostly) inorganic crystal
structure data at a central point so that they could be
used by all interested scientists.
Fig. 1

Nowadays it has become more important than ever tohave FIZ Karlsruhe as an independent manager of research data, so that everyone can request structure information, and regarding the increasing number of incoming
structures it seems that there will be a further demand forthis free service.
Moreover, the storage of research data like structural information in crystallography is nowadays a real hot topic
in many branches. Today it is clear that the availability ofscientific data and the reproducibility of scientific findings are essential for scientific progress. Besides, deposition of structures means securely storing them at a neutral institution with reliable access for the next decades.
When the depot started, all structures were collected, butsince 1999 there has been an agreement between the
Cambridge Crystallographic Data Centre (CCDC) and FIZ
Karlsruhe that all organic and organometallic compounds
should be deposited at CCDC, and all inorganic and intermetallic compounds belong to FIZ Karlsruhe. So, crystal
structures should not contain C-C or C-H bonds if submitted to FIZ Karlsruhe.
During the first years, data were transmitted and filed in
paper form including partially handwritten entries (Fig. 1),
and we were always glad to get clearly legible notes.
From this period a lot of shelves with dusty folders still
exist at the FIZ basement archive (Fig. 2).
Fig. 2

A Focus on Crystallography

As a next step on the route towards electronic data, hardcopies on fan-fold paper (Fig. 3) were sent to FIZ Karlsruhe. Now, in the last decade, the electronic submissionof crystal structure data via CIF file is preferred, whichsignificantly facilitates the daily work. Nevertheless there
is still a demand for really old datasets. Since most ofthese structures are also stored in the ICSD database, FIZ
Karlsruhe can in most cases send these data in electronic
form, too.

19

The CIF format (Crystallographic Information File) was
developed by the International Union of Crystallography(IUCr) and has become a worldwide standard for crystallographic information (http://www.iucr.org/iucr-top/cif/index.html).
During the last years, about 1,300 to 1,500 structures
were deposited on average, and an increasing number ofabout 600 to 1,300 structures were called up by users.
By the way, registered users of the Inorganic Crystal
Structure Database (ICSD) can also find (with a short production-dependent delay) the deposited inorganic crystal
structure data sets in the database, provided that they
have been published.
FIZ Karlsruhe is still interested in depositing as many
structures as possible, irrespective of whether they arealso collected on a publisher’s server or not. Furthermore,authors may send their own data sets not intended for
publication in a journal (e.g. private communications) to
FIZ Karlsruhe for deposition.
In summary, regarding the use of the depot by the crystallographic community it is obvious that the deposition of
structure data by FIZ Karlsruhe is still needed and highly
appreciated. FIZ Karlsruhe, on its part, is pleased to offer
its free service to the scientific community in the future.

Fig. 3

20

A Focus on Crystallography

Searching across all compound classes

Analysis of spinel compounds in ICSD
Andreas BarthThe Inorganic Crystallographic Database ICSD on STN
International contains 169,800 records (with update May2014). Each record consists of bibliographic information,
chemical compound information, and crystallographic
data. The specific combination of information supports
searches which are difficult to perform in other compound databases. For example, it is possible to search
for text in titles in combination with chemical compound
information. This provides an easy possibility to searchfor compound classes which cannot be describedthrough specific compound information such as themolecular formula. An interesting example for such acompound class is provided by the spinel compounds.
The spinel compound class is named after the magnesium aluminum spinel MgAl2O4 and it describes a specificcrystallographic structure. MgAl2O4 is a colorless, almost
transparent crystal, but it can become a very beautiful
colored gemstone resulting from traces of other chemical elements, e.g., in rubies and sapphires. The oxygen
atoms (O2-) build a cubic face-centered crystallographic
system with the Mg2+ atoms occupying 1/8 of the tetrahedral sites and the Al3+ atoms occupying 1/2 of the octahedral sites in the lattice.
The different spinel compounds may differ considerably
in their molecular composition. Hence, it is not possible
to search for combinations of chemical elements. In theliterature, spinels are often associated with the formula
type AB2X4 and with the Pearson symbol [1] cF56 (cubic – all faces centered with 56 atoms in the unit cell). A
search in ICSD will show how these concepts are related
to each other. A simple search for these terms showsthat there is indeed an overlap, but spinel compounds
may have other formula types than AB2X4 and are not
restricted to the Pearson symbol cF56 (s. Table 1). The
total number of spinel compounds (search for spinel*) is
3182 and the overlap with both AB2X4 and cF56 is 1821.
A search is rather complicated since the word “spinel” isan unspecific text term, the molecular formulas may differ
significantly, and neither the Pearson symbol nor the formula type are unique. An illustration of the relationshipbetween spinels and the Pearson symbol cF56 is shownin Figure 1.

Search Term

# HitsSpinel* (*: wildcard)

3182

cF56

3608

AB2X4

6704

Spinel* AND cF56

2465

Spinel* AND AB2X4

1968

Spinel* AND cF56 AND AB2X4

1821

Table 1: Correspondence between spinel compounds, the formula type
AB2X4, and the Pearson symbol (PRS)

3182

Spinel

717

3182

2465

1143

PRS = cF56

4325
Fig. 1: Search results for “spinel*” vs. Pearson symbol PRS = cF56 (seealso Table 1)

In addition to the “standard” spinel compound with molecular formula MgAl2O4 there are many variations of elements and stoichiometries; the most frequent molecular
formulas (in Hill order) are shown in Table 2. MgAl2O4 is
described in 201 documents, MgFe2O4 occurs 145 times,
and NiAl2O4 105 times. In positions 17, 18, and 20 there
are also spinels with Cr atoms and small impurities of Mn
and Zn.
It has been shown that the formula type of spinel compounds is not restricted to the type AB2X4 (s. Table 1).
There are 116 different types in total, and the top 20 formula types are given in Figure 2.
The oxide and sulfide spinels with three elements were
further analyzed in order to see which elements may besubstituted for Mg (an analogous analysis for the substitution of Al can as well be made) and the results areshown in a heat map in Figure 3. Beside Mg, the element

A Focus on Crystallography

TERM #

# DOC

% DOC

MF

1

201

6,32

Al2 Mg1 O4

2

145

4,56

Fe2 Mg1 O4

3

105

3,30

Al2 Ni1 O4

4

96

3,02

Fe3 O4

5

58

1,82

Mg 2 O4 Ti1

6

52

1,63

Al2 O4 Zn1

7

43

1,35

Al2 Fe1 O4

8

40

1,26

Al2 Mg0.99 O4

9

35

1,10

Fe2 O4 Zn1

10

25

0,79

Al2 Co1 O4

11

23

0,72

Al1 Fe1 Mg1 O4

12

23

0,72

Al2 Cu1 O4

13

23

0,72

Ni2 O4 Si1

14

16

0,50

Co3 O4

15

16

0,50

Cr 2 Mg1 O4

16

16

0,50

Cr 2 O4 Zn1

17

15

0,47

Al1.834 Cr0.162 Mg0.995 Mn0.002 O4 Zn0.007

18

15

0,47

Al1.868 Cr0.123 Mg0.999 Mn0.002 O4 Zn0.007

19

15

0,47

Li1 Mn2 O4

20

14

0,44

Al1.928 Cr0.066 Mg1.003 Mn0.001 O4 Zn0.002

20

14

0,44

Al1.928 Cr0.066 Mg1.003 Mn0.001 O4 Zn0.002

Table 2: Distribution of molecular formulas in spinel compounds

Fig. 2: Distribution of Formula Types in Spinel Compounds (logarithmic scale)21

Fe shows the highest occurrence in this table.
It seems that Mg is not well represented in theother spinel compounds.
Obviously, the most important spinel is the“traditional” spinel MgAl2O4; it occurs in 201
documents (s. Table 2, with mostly different
crystallographic data) but there are only 26 different stoichiometries of Mg-Al-O compounds(Mg ranges from 1 to 0.35 while Al ranges from1.998 to 2.44). A further analysis of the stoichiometries of the Al-Mg-O spinels shows a
distinct relationship between the two elements
Al and Mg as shown in Figure 4. The ratio Mg/
Al ranges from 0.5 (1:2) to 0.14 (0.35:2.43) withtwo clusters around 2.05 and 2.4 Al atoms.
This means that Al can partly substitute Mg inspinel compounds.
The other Al-R-O systems (with 4 oxygen atoms, R: any element except Mg) do not show
this linear stoichiometric relationship: the R/Al
ratio is mostly around 1:2 and sometimes 2:1(e.g. for R=Fe both ratios are possible).
A visualization of the unit cell of two different
Mg-Al-O systems shows how Mg is partly substituted by Al (see Figure 5). The first unit cellhas been generated from the crystallographic
data of MgAl2O4 (e.g. see [2] and the second
is based on data for a synthetic defect spinelwith the molecular formula Mg0.388 Al2.408 O4 [3].

22

A Focus on Crystallography

Fig. 3: Heatmap of oxide and sulfide spinels with elements of group 13 and a total number of 3 elements. The number in the cells designates thenumber of compounds, e.g. there are 26 Al-Mg-O compounds with different stoichiometries

Fig. 4: Stoichiometric relationship between Al and Mg in Al-Mg-O
spinels (all systems have four oxygens)

In the latter case the aluminum atoms occupy not only theoctahedral sites but also some of the tetrahedral sites in
the lattice. The Pearson symbol in this case is cF54.
Finally, it should be noted that molecular analysis canalso be performed in the highly comprehensive databases from Chemical Abstracts Service (except for the
crystallographic data and the formula type). The analysis
is a little bit more complicated since the concepts haveto be searched in two separate databases: CAplusSM andRegistrySM [4]. However, a search in these databases may
provide more comprehensive results.

References[1] Hubbard, C. R.; Calvert, L. D. (1981). The Pearson Symbol. Bulletin ofAlloy Phase Diagrams Vol. 2 (2), pp. 153-157.
[2] Zorina, N. G.; Kvitka, S. S. (1968). Refinement of the structure of thespinel Al2 Mg O4. Kristallografiya, 13, pp. 599-600.
[3] Basso, R.; Carbonin, S.; della Giusta, A. (1991). Cation and vacancy
distribution in a synthetic defect spinel. Zeitschrift fuer Kristallographie, 194, pp. 111-119.
[4] Barth, A.; Marx, W. (2012). Stimulation of ideas through compoundbased bibliometrics: Counting and mapping chemical compounds 		
for analyzing research topics in chemistry, physics, and materials
science. ChemistryOpen Vol. 1 (6), pp. 276–283. http://online
library.wiley.com/doi/10.1002/open.201200029/pdf.
(November 25,2014)

Fig. 5: Comparison of unit cells from MgAl2O4 and Mg0.388 Al2.408 O4(Mg: blue, Al: red, oxygen: green)A Focus on Crystallography

23

Using the Cambridge Structural Database to assign atom types

Bond lengths statistics as a tool for crystal
structure analysis – a case study with acrystalline silicophosphate
Uwe Böhme* and Sandra Jähnigen*

Abstract
The assignment of atom types is one of the first steps
after solving a single crystal X-ray structure analysis. It is
often difficult or even impossible to assign correct atom
types if there are atoms with similar atomic number present in the structure solution. The Cambridge Structural
Database (CSD) allows to perform statistical analyses
of bond lengths and delivers an elegant solution for the
problem of ambiguous atom types. Mean bond lengthsof the structural units SiO4, SiO6, PO4, and PO6 were calculated from CSD data. These were compared with bondlengths from the structure solution of a hitherto unknown
silicophosphate. This procedure allowed for an unambiguous assignment of atom types in the structure solution.

The Cambridge Structural Database
The Cambridge Structural Database (CSD) is a complementary database to ICSD. The CSD is compiled anddistributed by the Cambridge Crystallographic Data Centre (CCDC). This is a non-profit company and has been aregistered charity since 1987. The producers of the CSD
describe it as the world’s repository of experimentally
determined small-molecule organic and metal-organic
crystal structures. The database contains the results ofover half a million x-ray and neutron diffraction analyses. This unique database of accurate crystal structures
has become an essential resource to scientists aroundthe world. Each crystal structure undergoes extensive
validation and cross-checking by expert chemists andcrystallographers to ensure that the CSD is maintained
to the highest possible standards. Also, each database
entry is enriched with bibliographic, chemical and physical property information, adding further value to the raw
structural data. These editorial processes are vital for
enabling scientists to interpret structures in a chemically* TU Bergakademie Freiberg, Institut für Anorganische Chemie

meaningful way. The CSD is continually updated with new
structures (>40,000 new structures each year) and withimprovements to existing entries [1].
We use the in-house version of the CSD. Searches are
performed with the software ConQuest. Geometric parameters obtained from a search with ConQuest are exported to the software module VISTA for statistical analysis.
Further analysis of the data is possible with Mercury. This
software offers a comprehensive range of tools for structure visualization, the exploration of crystal packing andfurther statistical analyses.

The problem with the silicophosphate structure
Silicophosphates (aka SiPOs) are promising materials
for the use as lasers, optical fibers and orthopaedic
implants. That is because of their special material andoptical properties such as low melting points and highrefractive indices. Furthermore, they contribute to theunderstanding of geological processes. Until now, manydifferent silicophosphate structures were discovered. Remarkably, therein the silicon atoms are octrahedrally-coordinated to six oxygen atoms. This is in contrast to mostsilicates in nature where silicon occurs in tetrahedral coordination. All known SiPOs are composed of SiO4 andPO4 tetrahedra as well as SiO6 octahedra. These structural building blocks are connected differently to eachother and yield a multitude of various silicophosphate
structures [2]. The classical synthesis of such silicophosphates is performed by melting processes, chemicaltransport reactions or sol-gel-techniques. Recently, we
discovered another way to prepare silicophosphate materials with octahedral silicon structures. The syntheses
were performed at room temperature and normal pressure in diethylether solution. The reaction of tetraethoxysilane (TEOS) and anhydrous H3PO4 gives amorphous
Si-O-P materials with SiO4, PO4 and SiO6 structural units.
On the other hand, chloroethoxysilanes and anhydrous
phosphoric acid yielded oligomeric silicophosphates.

24

A Focus on Crystallography

From the latter reactions one crystalline product was obtained. The X-ray structure analysis of this product leads
to several problems which will be discussed in the following sections.
The structure solution delivered a large complex ion
which consists of silicate and phosphate subunits. Furthermore, tetraethylammonium and solvent molecules
were identified. After several refinement cycles it became

obvious that the unambiguous assignment of the atom
types phosphorous and silicon in the structure would
be no simple task (Fig. 1). Both elements have similar
atomic radii, comparable electron densities, and are able
to create the same coordination geometry with oxygen.
Especially the subunits SiO4 / PO4 and SiO6 / PO6 could
both be present in the structure. It would be most helpful
to discriminate these subunits on the basis of different
bond lengths. Therefore we were in need of reliable bondlengths for these four substructures.

Bond lengths statistics from the CSD

Fig. 1: Structure of the silicophosphate anion after isotropic refinement(dark grey – carbon, red – oxygen, blue – silicon or phosphorous?)

Fig. 2: Graphical user interface of the CSD

A search for the substructures SiO4 / PO4 and SiO6 / PO6
was performed with the CSD in order to obtain statistically reliable bond lengths derived from experimental data.
For that purpose the structural elements were drawn
within the module ConQuest and the number of bonded
atoms was fixed to 4 or 6. The option “ADD 3D” was used
to extract the bond lengths from the structures in the
database (Fig. 2). The search for the SiO4 unit gave 587
hits. These were analysed in the software module VISTA.
The mean bond lengths for this hit list is 1.618 Å withminimal and maximal values of about 1.53 and 1.71 Å, respectively (see Fig. 3). The outliers were analyzed. These
outliers could be eliminated from the statistical analysis
within the VISTA software. But this should be done only
for good reason. The same procedure was applied to thestructural units PO6, PO4, and SiO6. The result of this statistical analysis is summarized in Table 1.

A Focus on Crystallography

25

TRN07

Plot Data
File=/home/aoch/aschv
Test=1
Tot.Obs.=1036
Obs.=1036
Supp.=0((Figure 3.ps oder Figure 3.pdf))

wo ist die GrafiK????????

X-axisMin.=1.527
Max.=1.709
Range=0.181
Mean=1.618
Mean SE=0.001
Sample SD=0.020
Histogram
Median=1.620
Skew=0.190
Quantile=10.000
LQ=1.593
HQ=1.639
Bin Width=0.005
Max. Bin=149.000

TRN07
Fig. 3: Bond lengths statistics for the SiO4 unit

structural unit

mean bond lengths in Å

hitsstandard deviation

lower quartile

upper quartile

PO4

1.535

3217

0.016

1.523

1.546

SiO4

1.618

587

0.020

1.593

1.639

PO6

1.717

58

0.020

1.705

1.719

SiO6

1.782

53

0.039

1.766

1.788

Tab. 1: Mean bond lengths of structural units derived from the CSD

26

A Focus on Crystallography

The assignment of atom types in the silicophosphate structure
The mean bond lengths derived from the CSD were used
to assign the correct atom types in the X-ray structure
under investigation. Three different atom types X, Y, andZ had to be assigned. These are schematically shown in
Figure 4. One hexacoordinate atom X is situated at theheart of the complex. The X-O bond lengths are between
1.757 to 1.770 Å. These bonds are too long for a PO6
unit (see Table 1). That means the central atom X corresponds to a silicon atom. The central SiO6 octahedron issurrounded by six YO4 tetrahedra. These feature bondlengths of 1.513 to 1.574 Å and thus are too short for Si-O
bonds. Furthermore, the atoms Y are supplemented by aterminal oxygen atom, which is typical for a phosphate
unit. Therefore the atoms Y were assigned as phosphorous atoms. The peripheral atoms Z each link to two
phosphate groups and are bound to two diethoxy groups.
The bond lengths Z-O are between 1.579 and 1.633 Å.
This is substantially longer than the previous set of bondlengths and corresponds to silicon-oxygen bond lengthsof tetrahedral silicon atoms. In this way the atoms types
in question were assigned.
The final assignment of atom types is shown in Figure 5.
Obviously, a polynuclear silicophosphate complex wasformed. Within the complex anion a hexacoordinated silicon atom (Si1) is in the middle of the complex. The SiO6
octahedron is bonded to six PO4 tetrahedra (P1 to P6).
These in turn are bridged by six diethoxysilicate groups(Si2 to Si7).

Fig. 5: [Si VI(PO4)6(SiIVO4Et 2)6]-Ion with final assignment of atom types

Summary and Outlook
It was possible to assign the correct atom types in the
X-ray structure of a new silicophosphate with the help ofbond length statistics drawn from the Cambridge Crystallographic Database. The structure has been publishedin Chemical Communications [2] and inspired our further
work [3]. This case study demonstrates the ability of the
CSD as a helpful tool in structure elucidation. Until now
such analyses are not possible with ICSD. Here, a search
can be done by adding names, specific compositions or
structural formula. It would be desirable to perform bondlengths statistics in the future also with ICSD.

References[1] Information taken from the homepage of CSD:
www.ccdc.cam. ac.uk/Solutions/CSDSystem/Pages/CSD.aspx 		
(November 25, 2014)[2] Jähnigen, S.; Brendler, E.; Böhme, U.; Kroke, E. (2012). Synthesisof silico-phosphates containing SiO6-octahedra under ambientconditions – reactions of anhydrous H3PO4 with alkoxysilanes, 		Chem. Commun. 48, pp. 7675–7677.
[3] Jähnigen, S.; Brendler, E.; Böhme, U.; Heide, G.; Kroke, E.(2014).
Silicophosphates containing SiO6 octahedra – anhydrous 		synthesis under ambient conditions, New J. Chem. 38, pp. 744-751.

Fig. 4: Schematic representation of the structural problem

A Focus on Crystallography

27

Predicting new compounds by combining and analyzing large data volumes

Using ICSD to predict new properties,
compounds and modifications
J. C. Schön*

An overview over possible prediction related applications of ICSD as a searchable database is given.

Introduction
ICSD constitutes a large database of crystalline structures found in solid state chemistry and materials science for a multitude of chemical systems. In particular,
one finds an overview over the possible modifications
in chemical systems that have been discovered experimentally (plus some theoretically predicted structures).
Experience has shown that there appear to exist somecorrelations between the structure(s) a chemical compound takes on and the other physical properties weobserve for the compound.
This had led to optimistic suggestions that one can establish perfect structure-property relationships which
will produce easy ways for predicting new materials. In
practice, we note, however, that there exist numerous
examples where the same crystal structure is found fore.g. both insulators and metals, and thus these considerations need to be taken with a large grain of salt. Nevertheless, it is often reasonable to assume that if two
compounds belong to the same general class of chemical systems, then they will share some of their properties, and thus some trends in these properties may truly
be correlated with the various modifications possible
in these systems. This might well allow us to predict theproperties of not-yet-synthesized compounds from thestructure type they crystallize in.
But this requires us to first take the step to figure out,
whether a given structure type will constitute a stable
modification in a particular chemical system. For practical purposes, this is equivalent to predicting, which possible crystalline modifications can exist in a given chemical system. Such modifications can be divided into twogroups: those that agree with structure types that have

* MPI for Solid State Research, Stuttgart

been observed in other chemical systems (which may beeither chemically similar to the system under consideration, or actually quite different and only show the same
composition), and those structure types that have neverbeen observed in any chemical system or only been realized in chemical systems that are so different chemicallyfrom the compound under consideration that ”chemical
intuition” will not suggest these structure types to us.
Clearly, in the case where a completely new structure
type is going to appear during the synthesis of a newcompound, a database can only be of very limited assistance in predicting the structure. In that case, we have toturn to fundamental physical principles that tell us thatevery (meta)stable chemical compound corresponds
to a locally ergodic region on the energy landscape ofa chemical system. At low temperatures such a region
corresponds to a local minimum of the potential energy
in the space of atom configurations, but at elevatedtemperatures such a region can be quite large encompassing several or even many local minima. Over the
past twenty, twenty-five years finding such local minimaof the potential energy via global optimization methods
has been developed to a certain proficiency, with different research groups proposing different global optimization algorithms (see e.g. [1–4] and references therein).
However, the number of such local minima grows exponentially with the size of the system. Even if one
excludes defect minima and amorphous structures byfocussing on only small periodically repeated simulation
cells containing up to, say, ten formula units, the numberof minima can be quite overwhelming, and the danger
that one misses an important modification increases
with system size. Furthermore, we often need to employ
simplified energy functions for the global search, refining the resulting minima afterwards with more accurate
ab initio energy functions using Hartree-Fock- or Density
Functional Theory (DFT)-based computer codes since theab initio energy calculations are too time-consuming forlarge global searches.

28

A Focus on Crystallography

In this case, information drawn from databases like ICSD
can be a valuable resource to heuristically guide us tofind promising candidates for modifications in a given
chemical system.
In this essay, we are going to outline some ongoing andpossible future approaches to structure and property
prediction with the help of database analyses, specifically with the analysis of ICSD. Many of the concepts andapproaches discussed in this essay have already beenpresented in the literature in some fashion, together
with applications. Thus, we are not going to discuss
methods or their applications in detail, but focus on providing a general presentation of the use of databases forstructure prediction and closely related questions thatmight inspire the reader to proceed further along the directions mentioned.

Energy landscape, chemical similarity, andstructure prediction
Energy landscape
The world of all conceivable atom arrangements for N
atoms is called the configuration space (N ≈ NAvogadro ).
A point in the configuration space can be visualized as a➔vector R with 3N coordinates (each atom contributes its➔
position vector r = (x, y, z)). For each such configuration,
we can compute the potential energy, and the 3N-dimensional hypersurface of the energy over the configuration
space is the so-called (potential) energy landscape. As
we know from classical mechanics, the dynamics of thechemical system is given by the forces acting on the atoms, i.e. the gradient of the potential energy.
➔

Clearly, if one picks such a vector R at random, the structure associated with it will be a random arrangement of
atoms as one finds in the gas or liquid phase, and the
crystalline structures listed in ICSD are singular points
that will never be seen by randomly picking points in
configuration space. However, most of these randomly
selected atom arrangements are quite high in energy,
and physics tells us that the chemical system will preferentially occupy those regions of configuration space thatare local minima of the free energy F = E − TS at a given
temperature. Specifically, a metastable modification ofa chemical compound corresponds to a locally ergodic
region of configuration space, i.e. a region containing
many (similar) atom configurations, which is locallyequilibrated with a low free energy and kinetically stableenough such that the system does not leave this region,
on experimental time scales.

Quite generally, the sets of configurations around a minimum or groups of minima on an energy landscape represent the locally ergodic regions at low or intermediary
temperatures, respectively, and the kinetic stability ofthese regions grows with the height of the energetic andentropic barriers surrounding the region. Furthermore,the (crystal) structure that we associate with such ametastable modification is the average over all the configurations in the locally ergodic region; this average is,
at least partly, reflected in the so-called thermal ellipsoids derived for the atoms in a crystal structure fromthe experimental data.
Structure prediction
From a physical point of view, structure prediction istherefore equivalent to finding all the locally ergodic
regions of a chemical system. Typically, one proceeds by
first finding all minima, and then identifying the barriers
surrounding them to estimate their stability.
Usually, the focus is on the energetically low-lying minima, and thus the first step of the search is equivalent toa global optimization on a highly complex multi-minima
energy landscape. Such a procedure is very time-consuming, the computational effort typically growing
exponentially with the size of the system. This remains
true even if one employs as many simplifications as
possible, such as simulation cells containing few formula units with periodic boundary conditions and simple
fast-to-evaluate cost functions instead of ab initio energies. As long as we are only interested in crystalline
structures for systems where the energy landscape iswell-approximated by empirical potentials, this simplification is reasonable, but if one wants to analyze amorphous or defect-controlled compounds much larger
simulation cells are needed, of course.
Much progress has been made in developing efficient
ways to perform such global optimizations for crystalline chemical systems, but as one knows from ICSD,
there exist many crystal structures that contain moreformula units (Fig. 1) units or cannot be well described byempirical potentials, such that even highly refined global
search methods cannot guarantee success within reasonable computation times. Furthermore, as we haveseen, in general it is not sufficient to find only the global
minimum because in many instances it is a metastable
modification, perhaps only realizable as a nano-crystal
or thin film, which exhibits the desirable properties. One
way to address this problem is by exploiting our knowledge about chemical systems collected in databases
such as ICSD.

A Focus on Crystallography29

we subsume all these similar structures under one structure type, we realize that instead of hundreds of thousands of crystalline structures, ”only” tens of thousands
remain, some of which are found to occur in hundredsof different chemical systems. A nice such classification
into isopointal structure types based on symmetry considerations, with a refinement to isoconfigurational
structure types based on additionally using cell parameters and atom coordinates, has recently been introducedin ICSD [8]; by now most of the structures found in ICSD
have been assigned to one of these types.

Fig. 1: Example of a structure with z=10 (CaCO 3 at high pressure)

While the purist might reject such heuristic procedures,
it is obvious that one should not refuse any methodology that can assist us in identifying candidates for metastable compounds in a chemical system, even if it is only
heuristic and clearly limited. In particular, if one’s goal
is not to prove the feasibility of a particular prediction
methodology but to find all possible modifications for a
specific given chemical system, then one should employ
every method available. For more information about the
many global optimization methods that are currently being employed, we refer to the literature (see references
mentioned above), and for the remainder of this essay,
we are only going to discuss approaches that rely on theavailability of such databases.
Similarity of chemical systems
From an abstract point of view, every chemical system isdifferent, and thus there is no a priori reason to assume
that the possible modifications are going to exhibit thesame or even a similar structure as any other system.
And if one defines a ”structure” via the exact cell parameters of a crystalline unit cell and the specific atom
positions within this cell, the numerical values of thesecoordinates will be different from those found in any other chemical system. But the experience of the practicing
crystallographer, chemist and materials scientist has
shown that among the structures observed in the worldof crystalline compounds there are many, which differ
from each other only by slight changes in atom parameters and/or cell parameters, such that they appear tobe the same, at least visually, once one disregards the
specific chemical identity of the atoms involved (Fig. 2).
The crystallographic literature contains many algorithms
that have been suggested to quantify this similarity, e.g.
based on symmetry [5], geometry [6], or topology [7]considerations, each being most useful in specific contexts. But at the moment the decisive issue is that once

Fig. 2a + b: Structures of InPt(a) and AlCu(b) which both belong to thesame structure type

From a physical point of view, the occurrence of the samestructure type T̂ in two different chemical systems meansthat the energy landscapes of these two systems eachpossesses a local minimum that belongs to this structure type T̂ . And if, as not infrequently happens, severalof the modifications present in one of the systems arealso observed in the second system, then it is a reasonable expectation that the two energy landscapes aregoing to be quite similar. Then one can hypothesize thatmany of the other modifications that are only found in
one of the two systems are nevertheless capable of existence in the other one.
Clearly, such an educated guess is very helpful in our
search for new modifications in a particular chemical

30

A Focus on Crystallographysystem since we can right away perform a local minimization to verify whether this structure type is present asa kinetically stable modification in the second system,
which is much faster than the global search necessary
otherwise. We note here that this assumption of landscape similarity with the consequence of a highly desirable transferability of the results found for one landscapeto another one, has also been supported by many global
landscape explorations.
One caveat is, of course, that in most cases the global
search has been performed only with simplified energyfunctions, and thus we are comparing only models of
the true energy landscapes (belonging to similar classes, e.g. hard-sphere two-body interaction models or
all-electron ab initio calculations or density functional
pseudo-potential models) of the two systems, whichmight result in misleading similarities due to the inherent features of the models and interactions. Thus, onemust be aware that the true landscapes of the two systems are usually still going to be different in certain aspects: for example, not all minima found in one system
will be present in the other, or the energy rankings ofthose minima present in both systems may be different.
From a chemical point of view, the existence of modifications with the same structure type in two different chemical systems can often be correlated with the ”chemicalsimilarity” of the two systems. Quantities such as differences in electronegativity, cation-anion-(size) ratios or
total valence electron concentration are commonly employed to categorize chemical compounds, and are used
to postdict, and sometimes even to predict, by analogyor ”chemical intuition” (if the straightforward analogy
is not an obvious one), the structure of a new chemicalcompound. This is a time-honored and very successful
approach in experimental chemistry, as most famously
demonstrated by the success of the periodic table thatstill is one of the most valuable organizational tools of
the practicing chemist.
In this context, we note that from our experience withthe energy landscapes of many chemical systems wehave found that even for relatively small simulation cellswith few atoms the minima that are associated withknown structure types constitute not more than half ofthe local minima found. But among the lowest-energy
minima, this proportion can increase to 90%, at least forsystems with simple AnBm-type compositions (Fig. 3). Of
course, this mainly highlights the fact that, with regard
to synthesis, we have already carefully explored most of
the binary systems with simple composition ratios
n : m. Thus the number of unknown structure types inthis class of systems is rather small (especially if one

Fig. 3a + b: Example of simple structure types: (a) TiO2 (Rutile) and (b)ZnS (Zincblende)

allows for a certain amount of distortions within a structure family). But this also supports the suggestion thatthe energy landscapes of chemically similar systems canshow a relatively high degree of similarity. However, thishigh proportion is much less frequently found for compounds with more complex compositions.

ICSD-based structure prediction for given
chemical systems
One obvious way to exploit the putative similarity ofthe energy landscapes of many chemical systems withthe same composition formula consists in replacing theatoms in a known structure type by their analogues in
the chemical compound of interest, and then to perform
a local minimization, in order to check for the kinetic
stability of this type in the new system. By repeating thisprocedure for all appropriate structure types found in
ICSD, we can gain a certain overview over the minimum
structures of the new system.
A more refined way is to use these minima as starting
points [9] for a global optimization technique such as
the threshold algorithm [10], which explores the configuration space below an energy lid that is accessible froma given energy minimum. By increasing this limiting lid,
one can, in principle, globally explore the whole energy
landscape.
We note that one important preliminary task required
before such an ICSD-based search can be performed is
the identification of all structure types that are present
in ICSD by using one of the comparison algorithms mentioned above. To a certain extent, the classification already provided by ICSD can be employed, but since thisclassification is based primarily on symmetry one needs
to be careful. Hence, one would first classify all ICSD
structures into structure families via a geometry-based
structure comparison, and then define the center of this

A Focus on Crystallography

31

group of structures as the structure father or structure
type representative [11]. Once this has been achieved,
one can take these prototypical structures as starting
structures for the further global search or local minimizations, respectively.
Here, we note that for the use in energy landscape exploration and structure prediction, a geometrical comparison criterion is usually most appropriate: the neighbors
in the real configuration space of atom arrangements aredefined via sums over the Euclidean distances between
corresponding atoms – a geometrical quantity – while
topological similarity is based on the bond network,
which only rather indirectly correlates with the shape ofthe energy landscape, and symmetry-based similarity
cannot really be mapped to the energy landscape at all.
Nevertheless, following the dictum that one should notdiscard any systematic way to generate structure candidates, one should not hesitate to add all those structure
types that have been gained on the basis of topologicaland symmetry arguments to the starting points that are
selected based on geometrical similarity.
In this context, one should keep in mind that there exist
many chemical correspondences between e.g. binary
and ternary or higher phases. Thus, it might well be thatthe unknown structure of the new modification or compound in a binary A/B system might be identical to thestructure of some ternary A’/B’/C’ system, if one only
identifies A’ with A and B’ and C’ with B! (Fig. 4) To explore this, we have in the past analyzed ICSD with regard
to similarity between structures and structure types inbinary, ternary, quaternary, etc. systems.
This procedure will result in a family relationship treethat connects every structure not only with other structures of the same type (forming a structure family), butalso with structures belonging to related families in thesense described above. This procedure can provide us
with additional ICSD-based structure types, even if they
do not exist, strictly speaking, so far in form of a synthesized compound.
Trying to predict structures by chemical analogy has
been the oldest method in the literature [12]. If we consider the statistics mentioned above, it is no surprisethat there have been quite a number of successes, especially in the field of high-pressure structures, where theknowledge that two chemically similar compounds havethe same structure at standard pressure and system one
has a certain high-pressure structure, makes it a relatively easy target to predict that the second system will
also exhibit a high-pressure structure with the samestructure type. And rules such as the pressure-coordi-

Fig. 4a + b: A binary (Ag 2O2 (a)) and a ternary (AgCuO2 (b)) structure
belonging to the same structure type (AgO)

nation rule allow us a pretty well-educated guess about
the similarity of the structures of chemically analogous
systems, e.g. a high-pressure modification of system
two exhibiting a structure identical with the one of thestandard pressure modification of system one.
From the point of view of exploiting ICSD, this combination of chemical intuition and hard structural data inthe database has been a reasonably successful one,leading many people to the conviction that databasemining is all that is needed for successfully predicting
the structures of new compounds. In particular, groups
around Curtarolo and Ceder [13] have been developing
this approach, culminating in the so-called Materials
Genome Project [14]. And molecular crystal structure
prediction and the prediction of secondary and ternary
structures of proteins have also been guided by thiskind of data-mining approach (for a review see e.g.
[15]): In the case of molecular crystals, the statistics has
shown that the symmetry groups for about 90 % of all
structures of molecular crystals belong to only about ten
different space groups, and that many of these structures can be described with only one or two moleculesin the asymmetric unit. As a consequence, many searchmethods for molecular crystal structures exploit thisinformation by e.g. systematically scanning all possible
atom arrangements that can be generated by applying
these space group symmetries to one or two molecules(where the scanning includes the systematic change ofthe cell parameters and the orientation of the moleculeswith respect to the cell axes and to each other).
Similarly, for protein structure prediction, much successhas been achieved by taking the primary sequence ofthe bases of a protein, and then comparing pieces ofthese sequences, or the whole sequence, to corresponding pieces of known protein structures, deriving in thisfashion the secondary structure elements like the a-helices and b-sheets for the unknown protein, and evenestablishing good guesses regarding the tertiary folded32

A Focus on Crystallography

structure. Again, the large number of already solved
protein sequences and structures serve as a confidencebuilding foundation for such ”structure prediction byanalogy”. One even has attempted to use statisticalmethods to estimate, how many ”basic protein sequences” and ”protein structures” are still missing, although
there is clearly a self-reinforcing feedback going on: proteins that are similar can be solved with regard to theirsequence and structure by similar methods, thus giving
too much weight to the ”known” structures, while theset of the unknown, and also the unsolved, structures
is more likely to contain more not-yet-known structure
types than one would expect from extrapolation fromthe set of known and solved proteins.
Nevertheless, even with this caveat, it is clear that employing similarity analyses plus chemical intuition willquite likely continue to contribute to our ability to predict the structures of not-yet-synthesized compounds,
and especially to the structure solution of synthesized
but not-yet fully analyzed compounds. But one should
not forget the fact that the lack of simple but relatively
rigid bonding rules as one finds in covalent compounds
like molecules, makes it much harder to feel fully confident that extrapolation from known cases will resultin the truly new structure also in the case of bulk solid
compounds. Again, the statistics is looking too goodin some way, because those systems that are easy tomake tend to be the ones that are similar to other chemical compounds whose structures are known. Thus themore complex compounds and the seemingly simple
but not-yet-synthesized compounds are statisticallymore likely to exhibit new structure types never seen
so far. These will often even be difficult to post-dict or”explain” after a synthesis, no matter how much one
distorts simple structure motifs such as dense spherepackings or basic coordination polyhedra.

Prediction of appearance of given structure
types
Another complementary task where database information can be put to good use is the issue of predicting theappearance of a given structure type T̂ (for an example,
see e.g. [16]).
Here, the goal is to figure out, which chemical system
might support a particular modification that mighte.g. fit structurally to a technologically useful layered
compound (Fig. 5), giving us more ways to fine-tune
the properties of an electronic device. In some ways,
this task is the dual to the standard structure prediction problem. Up to now, one usually has only relied on

chemical intuition and trial-and-error experience, but
computational capabilities have increased to the point
where they can assist in this task.

Fig. 5: Example of a layered compound (MoS2)

Clearly, one can pursue the brute force approach: Minimize the desired structure type for all chemical systems
of correct composition type that would be compatible
with the T̂ structure type. This is usually quite a computational effort, especially if one keeps in mind that one
needs to a) also verify that the desired modification iskinetically sufficiently stable, and b) perform at least
some global search to check how many competing modifications with lower energy exist in the chemical system.
Furthermore, in this brute force approach, one would
not really exploit all the pre-knowledge one possesses
regarding the structure type, which might be much more
extensive than just the overall composition type. Thus,
one would like to perform some pre-selection in order tofocus on the most likely candidates.
Such a pre-selection should take both structural information about the desired structure type and chemical
information about the systems considered into account.
For example, certain coordination polyhedra might bepresent in T̂, and thus one would want to first look at
those chemical systems which are known to crystallize
in structure types that also contain these polyhedra(Fig. 6). Scanning ICSD for compounds of the right composition with modifications where these polyhedra havebeen observed, would thus be a fruitful data-mining
approach. Of course, this would require us to automate
the search for such coordination polyhedra through thewhole database, but with an efficient use of scripts thiscan be achieved.
In principle, one might even want to include such coordination-polyhedra information for each structure directlyin the database itself.

A Focus on Crystallography

33

Prediction of multinary phases

Fig. 6: Example of a structure (Cr 3Si) with a (slightly distorted) icosahedral polyhedron

Similarly, chemical intuition will be of help by providing
some heuristic rules such as the radius-ratio rule forionic compounds – again, information regarding theionic radii can be taken either from tables compiled in
the past or be generated from a systematic perusal of
the structures in ICSD. Other kinds of chemical information would be the valence-electron concentration or thenumber of covalent bonds expected or involved for the
structure type T̂ – again a systematic pre-scanning of thechemical systems with respect to these attributes could
be very helpful for restricting the search range.
Finally, a third way to attack this problem might be calledthe ”reverse approach”: In this method, we look at thosesystems S(k) where a modification with the type T̂1 is
known to exist, and find all additional structure types
T̂ i ≠ 1 that appear as modifications in these systems. Now,
we argue from a possible similarity of the energy landscapes: If in another chemical system S(j ≠ k) (where wehave not yet observed type T̂1 ) one of the modificationsof structure type Ti≠ 1 exists, then the likelihood is increased that in addition the modification with structure
type T̂1 is also a local minimum, at least compared to arandomly selected system that only exhibits the same
composition type. Thus, we select such a system S(j),and compute/relax all structure types T̂i in these systems, to see whether T̂1 might be stable in such a system,and how it compares with the other competing structure
types Ti ≠ 1.
In most cases, we will find that the already known structures in system S(j) are the thermodynamically stablestructures, but experience has shown that in many cases the desired structure T̂1 is at least kinetically stableand often quite competitive energy-wise. We note that
this approach can e.g. also be used to suggest various
high-pressure phases.

In many applications in materials science one deals
with complex multinary phases. Many of these actuallyconstitute solid-solutions, and while their structure prediction can be addressed by computational means (for areview see e.g. [2]), here we will focus only on the caseof ordered crystalline modifications where ICSD canserve as a resource. In general, we can employ ICSD forstructure prediction of multinary compounds completely
analogously to the case we discussed above in the general structure prediction section. However, there are notreally enough such structures available in ICSD to makethis approach as reliable as it has been for binary andeven ternary compounds (Fig. 7). Thus, one needs to finddifferent ways to tackle the prediction of the structuresof multinary compounds.
The most promising approach consists in reversing the”family” analysis from the previous section where wefound e.g. possible A/B structure candidates from a contraction of known A’/B’/C’ structure types. Instead we
now consider chemical systems A’/B’/C’ where we expect
their structure (based on chemical/structural information, such as e.g. local coordination polyhedra) to berelated to some AB-structure type, and assign the atomsof types A’, B’ and C’ to locations of atoms A and B in the
known AB-structure. Next, we minimize the energy for all
the structure candidates generated in this fashion andcheck their kinetic and thermodynamic stability. Thisapproach for predicting structures of multinary phasesfrom related binary or ternary phases by substitution ofions/atoms has a long tradition in experimental chemistry. Nevertheless, it is not clear to what extent one has,in the past, really systematically exploited all the data
available in ICSD that can be used to predict new compounds in this fashion.
One issue one should keep in mind is that there exists
an enormous number of ways to perform such substitutions – even if one focusses on small unit cells –, andthus chemical intuition, for what it is worth in this case,needs to be appealed to, in order to control the combinatorial explosion of possible substitutional atom arrangements. Sometimes, simple arguments based on e.g.
minimizing the electrostatic repulsion of the ions willlead to a successful restriction in the number of possible
configurations that need to be checked regarding theirkinetic and thermodynamic stability. But in many cases
the number of candidate structures will still be overwhelmingly large. Furthermore, as mentioned earlier, thechances of encountering a truly new structure type will
increase with the complexity of the chemical system. For

34

A Focus on Crystallography

Fig.7: Decline of the number of entries in ICSD for crystal structures with many different elements present

example, due to the slightly different ionic radii of thesubstituting atoms, it may well be that the real structureof the multinary system will contain a different overallarrangement of coordination polyhedra than those in
any of the binary or ternary systems found in ICSD, andthus a deduction of the new type from substituting into
one of the known structure types cannot succeed.

ion by chemical intuition and personal preference, but
by now the computational tools have become efficientenough for a systematic exploration of their energy landscapes once such promising missing compounds havebeen identified via analysis of ICSD.

Search for missing compounds
So far, we have discussed the use of ICSD to aid in theprediction of the existence and structure of specificchemical compounds, or, alternatively, the search of
possible chemical systems where a particular crystalline structure is realized. But we can also use ICSD asa negative-positive screening tool, i.e. we search ICSD
for holes, and try to use theory and/or experiment to fillthese lacunae. What we mean by a hole are ”missing”
chemical compounds, i.e. we want to identify thosechemical systems where one would expect a crystalline
compound to exist but nothing is registered in ICSD.
Every practicing chemist knows of such missing compounds, the most famous being perhaps C3N4 (Fig. 8),
which is one of the first compounds for which a structure
was predicted about twenty-five years ago [12].

Of course, one can again follow a brute force route ande.g. for binary systems pick every possible AnBm composition observed in ICSD for some chemical system, andperform local optimizations for every chemical system
for all A/B-structure types found in ICSD.

But there are many other chemical systems, not only
multinary ones but even among the binary and ternarycompounds, which one might expect to exist, but whichhave not yet been synthesized. So far, such systems
have been identified in a more or less haphazard fash-

In this fashion, we would be able to identify a plethora ofkinetically stable binary compounds that would provide
synthesis targets for the experimentalists. But while
this might just be possible for binary systems, alreadyfor ternary systems, it will be extremely expensive com-

Fig.8: The predicted crystal structure of b-C3N4

A Focus on Crystallography

35

putationally, although the Materials Genome project
mentioned earlier goes some way in this direction.
Furthermore, most of the crystalline modifications predicted will be only marginally kinetically and thermodynamically stable, resulting in a truly momentous heap ofessentially dead data.
Preferably, one would want to preselect promising candidates from among the suggested chemical systems,
their chemical compositions and for each among thestructural modifications, and then perform local ab initio energy minimizations for only these candidates, atleast at the beginning. Such a preselection step should
occur only after the analysis of the database, of course.
In this step, we would employ chemical intuition about
the bonding situations in the various chemical systems
and their compositions (ideally quantified via a statistical analysis of ICSD and not only by ”experience”), andfurthermore treat each of the chemical systems again as
we did for the standard structure prediction for a given
chemical compound of interest: Look in ICSD for modifications that occur in more or less analogous chemicalsystems, and use these as the first candidates for feasible polymorphs in the system under investigation.
Of course, if we proceed like this, the usual caveats
apply: if the thermodynamically stable modification in
one of the chemical systems we study does not have ananalogue among the already synthesized (or perhaps
simulated!) crystalline structures, then we are stumped.
Still, a systematic perusal of ICSD for ”missing chemicalcompounds” should be of great help in our attempts to
explore and understand the world of crystalline chemicalcompounds.

Network structures
So far, we have only talked about the whole structureof a chemical compound and the many similarities withother compounds noted in ICSD. But there is anotheraspect of structural similarity frequently employed whentrying to understand crystal structures: the local coordination polyhedra around cations and/or anions formedby their counter ions, or the presence of complex anions
and cations that can similarly be represented by a ”rigid” polyhedron (Fig. 9). Superficially similar to the local
bonding coordination of atoms in molecular chemistry,
one often finds the same local building blocks in many
compounds that contain the same types of atoms asanions and cations. Frequently, these polyhedra exhibit
slight distortions for different compositions, or whendifferent additional cations/anions are present in the
structure but are not part of the polyhedra. Once we

Fig.9: Example of a structure with polyhedra formed by anions aroundtheir counter ionshave investigated all structures in ICSD with respect tothe presence of such ”typical” building blocks, we canthen, guided by the analogy to molecules, select all,or only the most common, building blocks to generate
structure models for new not-yet-synthesized compounds. There exist several such ”coordination graph”
based approaches to structure prediction in the literature [17, 18].
While this type of method seems to avoid the global
search we talked about above by yielding many apparently feasible structure candidates for a given chemicalsystem, we quickly realize that there are very many ways
one can combine these building blocks, even if one uses
certain heuristic rules to restrict the exponentially large
number of such polyhedra networks. Furthermore, it is
not obvious that every candidate structure can be reduced to a network of known building blocks, and the
addition of hypothetical coordination polyhedra might
be required to globally scan the landscape, increasing
the number of candidates enormously. Thus one canquickly get overwhelmed by the number of local minimizations required - after all there is no a priori information
available, which of the network models are high or low
in energy.

Prediction of structure-property relationships
Finally, we return shortly to the issue of using ICSD topredict materials with specific properties. With the availability of truly gigantic computers and computer farms,
one can provide for each entry in the ICSD database not
only structural but also many other physical properties
by direct computation. By proposing to employ massive
computations of this type, the materials genome typeprojects have set themselves the goal to provide the materials scientist with a large smorgasbord of compounds
that might be useful at some point.

36

A Focus on Crystallography

But going beyond just accumulating large amounts ofdata, the grand aim here would be to not only compute
various physical properties of interest for all compounds
in ICSD, but also to use this information to automaticallyset up structure-property relationships.
In particular, after computing the properties of many
thousands of structures, the database should be large
enough that we can go beyond chemical intuition whenchoosing the structural parameters in these structureproperty plots. Instead we would use statistical analyses
on a large scale to establish significant relationships
among structural, chemical and physical properties ofvarious groups of chemical compounds.
In the end, this type of information will then guide us in
the choice of which chemical systems will be the most
likely ones that exhibit certain physical and chemical
properties. In particular, this will be true even for instances when the compound has not yet been synthesized but has only been predicted to exist as a kinetically stable modification.

Conclusion
In philosophy and pedagogy one sometimes considers
phase transitions in our understanding: The abstract
transition from quantity to quality, as Hegel and his successors have discussed, or the deeper understanding ofa mathematical theorem gained once we have studied
enough examples of its applications, or even in the
study of chemistry the profound intuition acquired by
the experimental chemist due to the many syntheses heor she has performed. Also in computer learning, suchtransitions are observed during the teaching of a neuralnetwork whose parameters are optimized by feeding ita multitude of teaching examples until the network isready to correctly analyze and classify instances of input
never presented before. Similarly, one often speaks inphysics of the new paradigm of emergent properties incomplex systems where there is no longer a simple direct relation between the e.g. macroscopic and/or longtime features of the system and the microscopic and/or shorttime aspects of the system that are ruled by e.g.
the laws of quantum mechanics. Perhaps the most exciting such dynamical transition is suspected to lie behindthe emergence of life and consciousness in biologicalsystems that are governed on the microscopic level byphysical and chemical laws.

One can propose that this kind of transition will haveits counterpart in the field of database applications:
Once the amount of information inside the database hasgrown large enough, then there exists, at least in potentia, the foundation for the emergence of higher order
structures among the wealth of information containedin the database. In particular, with regard to chemical
databases like ICSD, we can hope and anticipate that the
combination of large amounts of data and sophisticatedstatistical analysis tools will result in deeper insights
into the relationships among chemical systems and theirstructural, chemical and physical properties. The final
aim would be that with the help of the database we cananswer questions about chemical compounds that are
not included in the database or not even known to bepossible to exist.

Acknowledgements
My thanks go to W. Marx for suggesting the topic of thisessay, and to H. Schier for valuable comments, and to
Stephan Rühl for providing the figures for this essay.
Most of the ideas presented in this essay are the outcome of discussions with colleagues and friends, too
numerous to list, and I would like to thank them for their
exciting conversations and contributions.
Note: This essay is based on an article recently published in Zeitschrift für anorganische und allgemeine
Chemie (DOI: 10.1002/zaac201400374)

References[1]

Woodley, S. M.; Catlow, C. R. A. (2008). Nature Mater. 7, p. 937.

[2] Schön, J. C.; Jansen, M. (2009). Int. J. Mat. Res. 100, p. 135.
[3] Oganov, A. R., ed. (2011). Modern methods of crystal structure 		prediction (Wiley VCh, Weinheim).
[4] Wang, Y.; Ma, Y. (2014). J. Chem. Phys. 140, p. 040901.
[5] Burzlaff, H.; Malinovsky,Y. (1997). Acta Cryst. A53, p.217.
[6] Hundt, R.; Schön, J. C. ; Jansen, M. (2006). J. Appl. Cryst. 39, p. 6.
[7] Blatov, V. A. (2006). IUCr CompComm Newsletter 7, p. 4.
[8] Allmann, R; Hinek,R. (2007). Acta Cryst. A63, p. 412.
[9] Schön, J. C. (2004). Z. Anorg. Allg. Chem. 630, p. 2354.
[10] Schön, J. C. (1996). Ber. Bunsenges. 100, p. 1388.
[11] Sultania, M.; Schön, J. C.; Fischer, D.; Jansen, M. (2012). Struct. 		
Chem. 23, p. 1121.
[12] Liu, A. Y.; Cohen, M. L. (1990). Phys. Rev. B 41, p. 10727.
[13] Curtarolo, S.; Morgan, D.; Persson, K.; Rodgers, J.; Ceder, G. (2003).
Phys. Rev. Lett. 91, p. 135503.
[14] Kalil T.; Wadia, C. (2011). Materials genome initiative for global 		
competitiveness, www.whitehouse.gov (November 25, 2014)[15] Schön, J. C.; Jansen, M. (2001) Z. Krist. 216, p. 307.
[16] Zagorac, D.; Schön, J. C.; Jansen, M. (2013). Process. Appl. Ceram.
7, p. 111.
[17] Delgado-Friedrichs, O.; Dress, A. W. M.; Huson, D. H.; Klinowski, J.;
Mackay, A. L. (1999). Nature 400, p. 644.
[18] Winkler, B.; Pickard, C. J.; Milman, V.; Klee, W. E.; Thimm, G. (1999).
Chem. Phys. Lett. 312, p. 536.

A Focus on Crystallography

37

A historical outline of the interaction between two disciplines

Crystals and mathematics
Gert-Martin Greuel

People have always been fascinated by crystals and surprised and delighted by their regular geometry, their special symmetry, and their mysterious color refraction. Many
natural minerals are crystals. They are parts of stones,
but due to the irregular grain boundaries their crystalline
nature is often hidden. In cavities, however, minerals canfreely grow into recognizable crystals. The crystal shapeis also obvious in snowflakes, the growth of which is not
limited. Crystals are often found on display in museums
or private collections, or as polished gems. Less known
is that the largest part (about 98%) of the solid ground is
crystalline. This means that crystals are a stable state of
the condensed matter.
The term “crystal“ is derived from the Greek “krystallos“(meaning “ice“) and was used for the first time in connection with rock crystals. People probably thought thatcrystals formed in extreme cold, which is not true. Most
crystalline minerals form at high temperatures and under
extreme pressure during the cooling process of magmafrom the interior of the earth.

crystallography, a crystal is defined and its geometricalstructure described by means of diffraction patterns
obtained through exposure to x-rays or other sources ofradiation.
This article will briefly outline the history of crystallography and the mathematical-geometrical description of
crystal structures.

1. The beginnings of research on crystals
The first known systematic treatise dedicated to minerals
is that of natural scientist Theophrastos of Eresos (371287 B.C.), a student of Plato and Aristotle. It is containedin his publication “On Stones“ (see [1]).
In the encyclopedia “Naturalis Historia“ by Pliny the Elder(23-79 AD), a comprehensive scientific work consistingof 37 volumes, there is also a treatise on rare stones andminerals (cf. [2]), which was the basis of mineralogicalknowledge until the Middle Ages. During the Middle Ages,
crystals and stones were also thought to have healing
properties – a belief still shared by some New Agers today.

Crystals are solids whose components (atoms, ions,molecules) constitute a regular crystal structure – mathematically described as a lattice with a basis. In modern

As neither the chemical composition, nor the opticalcharacteristics of crystals could be determined before thebeginning of the 19th century, these old studies are onlyof historical interest to scientists.

Fig. 1: Fluorite octahedron on calcite

Fig. 2: Fluorite crystal – Grube Clara38

A Focus on Crystallography

2. Crystal shapes and polyhedra
The regular shapes of crystals immediately remind mathematicians of regular, convex polyhedra, especially the
Platonic solids with congruent faces and the same number of faces meeting at each vertex. There are exactly five
Platonic solids (tetrahedron, hexahedron or cube, octahedron, dodecahedron, and icosahedron, see Fig. 3); theirfaces are equilateral triangles, squares, or pentagons.
The classification of regular polyhedra (claiming that
there are only these five regular polyhedra) by the Greek
mathematicians, in particular Plato and Euclid, may beconsidered the first strict approach to classification inmathematics.
In fact, there are minerals whose crystals are almost
proper, regular tetrahedra, hexahedra (see Fig. 2), or
octahedra (see Fig. 1). Dodecahedra and icosahedra,
however, can never be crystal shapes (for reasons oftranslational symmetry, see chapters 8 and 10), but they
can be found in living organisms, e.g., viruses.
Non-regular dodecahedra and icosahedra, on the other
hand, exist as crystals. Pyrite, for example, can have theshape of a cubic pentagondodecahedron which by its appearance can be easily mistaken for a regular dodecahedron. Each of its faces is a pentagon, but with four shorterand one longer side. It has cubic symmetry.
An important role beside the Platonic solids play theArchimedean solids. They consist of various types ofregular polygons and are symmetric in such a way thatevery vertex can be taken to any other vertex by rotation
or reflection. Archimedean solids can be constructed as
truncated Platonic solids, i.e. by cutting away a corner of
the Platonic solid so that a regular face is formed (e.g., aregular triangle in place of the corner of a cube).

Fig. 3: Platonic solids

Even more important for crystallography are the Catalansolids or Archimedean duals that consist of only one typeof face. They have, however, different types of vertices,because duality means that the face of a Catalan solid isformed at the vertex of an Archimedean solid. For example, the rhombic dodecahedron is a Catalan solid whichis dual to the cuboctahedron. The rhombic dodecahedron is a typical crystal form and is present in garnets.
Other frequent crystal structures are prisms, which consist of two parallel, congruent, regular polygons and thelateral surfaces that connect them.

3. Crystals and Greek mathematics
The Greek mathematicians certainly knew the crystal
shapes and their symmetries. However, their studies onregular polyhedra were neither motivated by, nor related
to crystals. At least, it seems that there is not any proof
for this ([3], p. 342). On the contrary – platonic philosophers considered the total symmetry of the regular polyhedra an expression of perfect, everlasting beauty, independent of short-lived, earthly manifestations (cf. [3],p. 340).
In this, Greek philosophy shows some similarities toparts of modern, in particular to pure, mathematics, theresearch of which is directed towards mathematics-inherent structures and does not serve any particular purpose. It is only committed to its own axioms and logicaldeductions, without any need or regard for practicability
or experience. Still we note that mathematics, also puremathematics, is – today more than ever – indispensable
for understanding scientific processes, and that it has
become the driving force of industrial and economic innovation.

A Focus on Crystallography

39

However, there is most certainly a connection between
the crystal shape, which is considered beautiful by man,and the fact that Greek mathematicians were dealing withregular polyhedra. After all, they could have examinedother things instead. But according to Plato, the searchfor everlasting perfection, for the Good, is related tobeauty and regularity. „The good is always beautiful, andthe beautiful never lacks proportion“. (Platon, Timaios,
87c, 4-5)

4. Kepler and the snowflake
It seems that the connection between examinations of
the crystal shapes and the ancient studies on mathematical structures of polyhedra was only made in the Renaissance ([3], p. 342). Johannes Kepler (1571-1630) used theharmony of the Platonic solids as a means to describe theharmony of the celestial spheres. However, his attempt toprove that the distances of the planets can be defined bya model of nested, regular polyhedra encased by spheres
failed. He therefore abandoned this model and developed for the first time the model of elliptical planetary
orbits around the sun that is still in use today.
Kepler’s interest for crystals is also reflected in his studies on the symmetry of snowflakes ([4], see Fig. 4). He
discovered that the unique geometry and six-fold symmetry of the snowflake is caused by natural forces. The exact
physical reason behind this, the fact that the components
of matter – atoms and molecules – are always aligned insuch a way that they are in a state of minimal energy, wasstill unknown at his times. This results in beautiful symmetrical structures in snowflakes and other crystals.
Kepler‘s attempt to describe the hexagonal structureof snowflakes as a structure of miniscule particles withminimal distance between them, caused him to study
the maximum density of circle and sphere arrangements,
published in the same work. He assumed that the mostclosely packed arrangement of spheres is the cubic close
packing (as seen, for example, with oranges stacked inform of a pyramid at a market stall). Only some 400 years
later, in 2003, this Kepler conjecture was proven by theAmerican mathematician Thomas Hales, partly by complex computer calculations.

5. The beginnings of crystallography
As early as around 1669, Nicolaus Steno (1638-1686)
discovered the law of constancy of the interfacial angle
when examining quartz. This means that the angles
between two identical faces of a crystal are always thesame, independent of its size, shape, and the conditions
under which it was formed. Steno assumed this law ofFig. 4: Snowflake

constancy of the interfacial angle for all crystals, and his
examinations marked the beginnings of crystallography,
i.e. the science dealing with crystals.
The general validity of the law of constancy of the interfacial angle was empirically proven in later years, around
1783, by Jean Baptist Romé de L’Isle through systematic
examination and detailed description of about 500 different crystals.
One step further goes the “Traité de minéralogie etc.“, byRené-Just Haüy, published in Paris in 1801. Based on theobservation of fragments of a piece of calcite shatteredon the ground, Haüy described the crystals as consistingof smaller units (which he called “integrating molecules“)that always have the same structure as the crystal itself.
That the external shape of crystals can be derived froma periodical alignment of elementary components is afundamental finding for crystallography. For example, it
makes it easy to explain the law of constancy of the interfacial angle. Therefore, Haüy is nowadays also referred toas the “Father of Crystallography”.

6. Modern crystallography
The proof that crystals really consist of regularly aligned
units, however, was only furnished by Max von Laue
(1879-1960) and his colleagues in 1912. They discovered
the diffraction of x-rays by crystals and the crystal-dependent, regular pattern of points resulting from it. This
discovery and its theoretical substantiation won von Laue
the Nobel Prize in Physics in 1914.
Already in 1913, William Henry and his son William Lawrence Bragg had shown that x-rays could even be used toexactly define the position of atoms within a crystal, thus
identifying its three-dimensional structure. They bothwere awarded the Nobel Prize in Physics for this discovery
in 1915.

40

A Focus on Crystallography

While up to then geometrical optics had been the mostpopular analytical method in crystallography, x-ray
crystallography or other sources of radiation are almost
exclusively used in modern crystallography to determine
diffraction patterns. So, today some hundreds of thousands of such structures, ranging from small inorganicand organic compounds to large biomolecules, have been
stored in databases, with their number steadily growing.
The methods of x-ray crystallography were continuously
refined, thus making it possible to determine also thestructure of important biological molecules (e.g. cholesterine, penicillin, insulin) between 1920 und 1970. This
had a great impact on healthcare. The most importantdiscovery in this connection certainly was that of thestructure of the DNA by James Watson and Francis Crickby analyzing diffraction experiments, for which they wereawarded the Nobel Prize in 1962 in Physiology or Medicine together with Maurice Wilkens.

that the whole crystal is formed by the translations of one
basic unit.
Of course, ideal crystals do not exist. A real crystal isalways finite and has defects, i.e. deviations from regularity. Still it suffices for many purposes, especially to classify crystals, to look at ideal crystals.
Each translation within the three-dimensional Euclidian space is defined by a translation vector that can be
described as a combination of the multiples of three(once chosen) independent basic vectors. The total of
all translations that transform the basic unit of a crystal
into other basic units, and transform the (ideal) crystal
into itself, have three linearly independent basic vectors
so that each of these translations can be described as a
combination of integer multiples of the basic vectors. Alltranslation vectors (or their end points) that transform thecrystal into itself form a three-dimensional lattice within
the space, the crystal lattice.

In recent times, two discoveries are particularly noteworthy: Graphene as the first example of a new class of
two-dimensional crystalline materials with unique electronic and mechanical properties (Nobel Prize in Physics
awarded to Andre Geim and Konstantin Novoselov in2010) and the quasicrystals (Nobel Prize in Chemistryawarded to Dan Shechtman in 2011). Especially the discovery of the quasicrystals in 1982 was a total surprise forcrystallography. These were materials whose diffraction
patterns showed a clear pattern of points just like those ofordinary periodic crystals (unlike amorphous substances,which produce blurred diffraction patterns), but with fivefold symmetry – although this should not be possible,
due to the periodicity of a crystal (see sections 9 and 10).
Fig. 5: Lattice of a diamond

It is worth mentioning that nowadays, besides the mathematical crystallography (i.e., the classification of crystals
by symmetry), which will be discussed in detail below,
there are many more important aspects of crystallography.
These include the mineralogical and biological crystallography, but also crystal physics, in particular diffraction
physics and crystal growth, with manifold fields of applications, e.g. medicine and materials research.

7. Crystal lattices
The discovery made by Steno, Romé und Haüy, that crystals consist of periodically repeating basic units having
the same shape as the crystal itself, leads us to the concept of the crystal lattice. However, for this purpose weare looking at ideal crystals. These are infinite and havetranslational symmetry, i.e., their basic units, or, fromtoday’s point of view, the atom positions, are merged into
one another by a translational movement within space, so

The points of this lattice do not represent atoms but only
depict the periodicity of the crystal structure. Of course,
the crystal lattice depends on the outward shape of thecrystal or the basic unit, but since it translates the basic
units within the crystal (within the range of atom radii)
it is not possible to deduce the crystal lattice from theshape of the crystal. The parallelepiped defined by thethree basic vectors is called unit cell or primitive cell.
The choice of the basis of a lattice is not unambiguous.
Therefore one tries to find a basis where the unit cellreflects the shape of the crystal as exactly as possible.
Since this is not always possible, but, on the other hand,
it is important to be able to easily view the crystal’ssymmetry, one waives the requirement that the vectors
forming the unit cell be a lattice basis and uses threevectors forming larger (non-primitive) unit cells instead.
In this case, the translations of the crystal lattice can beA Focus on Crystallography

41

described as a combination of rational multiples of these
vectors. These vectors are also called a non-primitive basis of the lattice.
Around the year 1843, Auguste Bravais (1811-1863) classified the different possible crystal lattices by indicating
both primitive and non-primitive unit cells. They are
named Bravais lattices after him. In three dimensions
there are exactly 14 Bravais lattices, i.e. exactly 14 translation groups of all possible ideal crystals.

8. Crystallographic groups
Crystals are classified according to their symmetric properties, i.e., we examine the isometries (distance- andangle-preserving mapping) of the three-dimensional Euclidian space that transform the crystal or crystal lattice(which is actually the same) into itself.
The inverse operation of such an isometry and subsequent random operations again result in isometries of
the crystal lattice; mathematicians call this a group. The
isometry group of a crystal lattice is called crystallographic group or (crystallographic) space group.
Of course, the translations of the lattice belong to the
crystallographic group. They describe the “long-rangeorder“ of the crystal. However, there are also isometriesof the crystal lattice with (at least) one fixed point, e.g. rotations around an axis, point reflections or reflection in aplane, or combinations thereof. This group describes thesymmetry of the basic unit and thus of the crystal itself.
It is called the crystal’s point group. There are exactly 32
of such crystallographic point groups, also called crystal
classes. Being abstract groups, these groups are of mathematical interest, but for our purposes it is important tonote that they are applied to the isometry of the crystals
and serve to define the different crystal classes.
Each element of a crystal’s point group operates on a
crystal with a finite order, i.e. after a finite number of subsequent operations the crystal is back in its initial position (because there is only a finite number of positions ofa vertex, an edge or a face of a crystal). For example, thereflection in a plane has order 2, the rotation of a cube
around an axis through opposite plane centers (or edge
centers or corners) has order four (or two or three).
So there are 32 crystallographic point groups (crystal
classes) besides the 14 translation groups (crystal lattices).
There are 230 different crystallographic groups in total, i.e.,
isometry groups of crystal lattices (in three dimensions).
They were identified in 1891 by Arthur Moritz Schoenflies
and Jewgraf Stepanowitsch Fjodorow independently.

Fig. 6: Example of a point group

Different point groups are combined in one crystal
system. Today, we differentiate between seven crystal
systems: triclinic, monoclinic, orthorhombic, tetragonal,
trigonal, hexagonal, and cubic. The crystal systems wereintroduced by Christian Samuel Weiss (1780-1856) when
he translated the textbooks by Haüys. They were basedon the analysis of the alignment of particularly striking
directions of the crystals, the axes “around which everything was equally distributed“. Thus, the crystal systems
constitute a symmetry-related classification of crystals
by means of crystallographic axes of coordinates. Weiss
used the axes to clearly denote for the first time the position of all crystal faces or planes within the lattice by
the ratios between their intercepts on the axes, the Weiss
indices. Nowadays we mainly use the lowest integral
common multiples of the reciprocal intercepts, the Miller
indices.

9. Mathematical groups
We can have a look at the isometries of space that transform any body into itself. These are called the isometrygroup or symmetry group of the body. Most bodies innature are irregular or asymmetrical. In this case, there
is no isometry besides the identity, which transforms the
body into itself. A group that only consists of the identity
is called trivial. Bodies with a trivial isometry group arethus asymmetrical, and bodies with a non-trivial isometrygroup are called symmetrical. The larger the isometrygroup of a body, the more symmetrical its appearance.
For example, the sphere as a symbol of a symmetricalbody is transformed into itself through any isometry, andits symmetry group contains an uncountably infinite number of elements, i.e. as many as there are real numbers(e.g., every rotation around any angle and any axis andevery reflection in any plane through the central point).
The isometry group of a crystal is not trivial, but always
countable, with the point group itself only containing a42

A Focus on Crystallography

finite number of elements. In any case, we have to notethat the colloquial terms “symmetric” and “regular” aremathematically described by the concept of the group.
A (abstract) mathematical group consists of a set and an
operation which assigns, to any two elements of the set,
one element of the set. Several subsequently performed
operations can be arbitrarily grouped into operations of
two (put into brackets). In addition, each operation has
an inverse operation, so that performing subsequent
operations results in the identity, which is always one
element of each group.
From a historical point of view, the abstract group concept defined in this manner (something taught today rightat the beginning of mathematical studies) is still quite
young and an abstraction of the concept of the symmetrygroup which has been in existence much longer. This is agood example of how mathematical concepts are derived
from real experience (symmetrical, regular) by abstraction, and how these mathematical concepts are then exclusively defined by abstract axioms, independent of theoriginal observation.

10. The impossibility of the icosahedron
Why can crystals not take the shape of regular icosahedra or dodecahedra? We will learn that this is due to thetranslational symmetry, which strongly limits the point
group of the crystal.
Five regular triangles meet at each vertex of the icosahedron. If we look at a rotation around an axis through two
opposite vertices, which transforms the icosahedron into
itself and each triangle into the neighboring triangle, thisrotation has order 5. The same applies to the dodecahedron if we are looking at the rotation axis through thecenters of two opposite faces.
Now we are going to show that a crystal cannot havesuch a rotation of order 5. Let us look at a crystal rotatingaround any given axis. This axis has a finite order n, andtransforms a unit cell of the crystal into a translated one.
The case n=2, i.e. a rotation of 180°, is of course possible
and we can therefore assume in the following that n islarger than or equal to 3. Now we imagine a plane locatedperpendicularly to the rotation axis and intersecting avertex of the unit cell which is not located on the rotation
axis. With a full rotation, the vertex describes a regularpolygon with n edges within the plane. All vertices ofthis polygon are vertices of unit cells, i.e., they are transformed into each other by translation of the crystal grid.
Each translation shifts the rotation axis so that therotation around the shifted axis generates an adjacentn-edged polygon within the same plane. This process canbe repeated for an infinite number of times until the wholeplane is completely (without any gaps or overlapping) tiled
with regular, n-edged polygons.
By examining the angles we will now show that this is only
possible if n= 3, 4 or 6. Let us assume that r regular polygons with n edges meet at a point within the plane, andthat the vicinity of this point is completely tiled with thesepolygons.
The angle between two adjacent edges of a polygon with nedges is 180° - 360°/n, and r(180° - 360°/n) = 360° applies accordingly. If we divide by 360°, then r/2 - r/n = 1.
Multiplying by 2n and rearranging the equation givesn(r-2) = 2r. If we set r-2 = s, then n = 2 + 4/s. As n is a positive integer, s can only take the values of 1, 2 or 4, whichmeans that n can only be 6, 4 or 3.
Therefore, the point group of a crystal can only containrotations of orders 2, 3, 4 or 6. Thus, crystals cannot havethe shape of an icosahedron or of a dodecahedron.

11. Quasicrystals
We have learned that a crystal’s point group can only contain rotations of orders 2, 3, 4 or 6. The diffraction of x-rays
in crystals shows a sharp pattern of points that allows todetermine the crystal’s point group. For a long time, scientist thought that purely point-shaped diffraction reflexes
were only present in crystals and used this assumption todefine a crystal. Accordingly, the translational symmetryor the crystal lattice were considered an equivalent for the
existence of point-shaped diffraction patterns.
Therefore it came as a shock to crystallography when, in
1982, Dan Shechtman discovered clear, point-shaped diffraction patterns in materials having the same rotational
symmetry as the icosahedron, i.e. of order 5. Since these
new materials certainly could not be considered crystals ina traditional sense, they were named quasicrystals.
The discovery of quasicrystals led to a new crystallographic definition of (traditional) crystals: now, translational
symmetry was required as an independent property. Quasicrystals have clear diffraction points, but no periodictranslational-symmetric structure. A review article aboutquasicrystals published on the occasion of the awardingof the Nobel Prize to Shechtman is provided in [8].
If a quasicrystal is cut in an appropriate manner, the slice
plane, e.g. in Shechtman’s aluminum-manganese alloy,
shows a local five-fold rotational symmetry and a tiling of
the plane that is not periodic, but “quasi-periodic”. This

A Focus on Crystallography

quasi-periodic tiling of the plane had already been discovered in the 1970s by the British mathematician Roger
Penrose and is named Penrose tiling after him.
The Penrose tiling has many interesting properties. For
example, any patch from the tiling (independent of itssize) can occur for an infinite number of times, but cannot
periodically repeat itself. An easily understandable description of many mathematically interesting characteristics of quasicrystals and Penrose tilings is provided by themathematicians Baake, Grimm, and Moody in [9].

43Acknowledgements:
I would like to thank Prof. Stephan Klaus at the Mathematical Research Institute Oberwolfach for his useful advice
and also Mr. Werner Günter, Chairman of Verein der Freunde von Mineralien und Bergbau Oberwolfach for his advice
and the permission to publish his photos of crystals. Many
thanks also to Regina Karl from FIZ Karlsruhe for her excellent English translation of the original German text.
The following references are only a part of the sources
I used. I think they contain particularly useful further information on the topic.
References[1] Caley, E. R.; Richards, J. F. C. (1956). Theophrastus: On Stones.
Annotated edition with Greek text and English translation, Ohio
State University. www.farlang.com/gemstones/theophrastuson-stones/page_001 (November 25, 2014)[2] Nies, A. (1884).Zur Mineralogie des Plinius. Mainz. Buchdruckerei
von H. Prickarts. Full text on Archive.org. https://archive.org/details/
zurmineralogied01niesgoog (November 25, 2014)[3] Brieskorn, E. Lineare Algebra, III. 13.7, Reguläre Polyeder, Einleitung.
Manuskript (soon to be found at imaginary.org). The page numbers
refer to the unpublished typoscript.
[4] Kepler, J. (1611). Strena seu de nive sexangula (about the hexagonalsnowflake). Keplers Gesammelte Werke, (Hrsg. Caspar, M.;
Hammer, F.) Bd. IV (1941). German Translation: Goetz, D. (1987).
Ostwalds Klassiker Bd. 273, Leipzig. Text in Latin and English by
Hardie, C. (1966). Oxford, Clarendon P.

Fig. 7: Quasi-periodic rhombic penrose tiling

Quasicrystals are becoming increasingly important fortechnical applications because they can give special
properties to materials when used as additives.
The discovery of quasicrystals has gained the Penrose
tiling a lot of attention. One found out that similar quasiperiodic patterns had already been existing in Islamic
ornaments of mosques and palaces from the Middle Ages.
How these beautiful but complicated patterns were constructed, was only discovered a short time ago (cf. [10]).

[5] Articles on Wikipedia: http://de.wikipedia.org/wiki/Kristall (November 25, 2014) http://de.wikipedia.org/wiki/Kristallmorphologie(November 25, 2014)[6] Kristallographie, na klar! German translation of “Crystallographymatters!” by UNESCO 2013. http://dgk-home.de/veroeffentlichun		gen/kristallographie-na-klar (November 25, 2014)[7] Kristallographie in Deutschland, Deutsche Gesellschaft für Kristallographie e.V. www.kristall.ethz.ch/DGK/ und www.dgkristall.de 		(November 25, 2014)[8] Trageser, G. Kristalle mit unmöglicher Symmetrie, Spektrum.de. 		
www.spektrum.de/alias/nobelpreis-fuer-chemie/kristalle-mitunmoeglicher-symmetrie/1129068 (November 25, 2014)[9] Baake, M.; Grimm, U.; Moody, R. V. (2002). Die verborgene Ordnung
der Quasikristalle, Spektrum der Wissenschaft 2/2002, p. 64.
[10] Schönfelder, V. Das Geheimnis der Girih, Spektrum.dewww.spektrum.de/alias/angewandte-mathematik/das-geheimnisder-girih/866284, (November 25, 2014).

Images:Fig. 1: Fluoride octahecron on calcite, Photo: Werner Günter, Oberwolfach
Fig. 2: Fluorite crystal – Grube Clara, Photo: Werner Günter, Oberwolfach
Fig. 3: Platonic solids, Photo: Zumthie at de.wikipedia, source:
http://commons.wikimedia.org/wiki/File:Platonische_
Koerper_im_Bagno.jpg
Fig. 4: Snowflake, Photo: Jochen Burger, source:
www.natur-portrait.de/foto-26022-frau-holles-erster-gruss.html
Fig. 5: Lattice of a diamond; screenshot from the Crystal Flight software,
source: http://imaginary.org/program/crystal-flight
Fig. 6: Example of a point group; Author: Jean Constant; source: http://
imaginary.org/gallery/jean-constant-crystallographic-points- 		groupsFig. 7: Quasi-periodic rhombic penrose tiling; Author: Uli Gaenshirt 		source: http://imaginary.org/gallery/quasicrystalline-wickerwork
Fig. 8: Jigsaw puzzle of a penrose tiling; Mineralien- und Mathematikmuseum Oberwolfach (MiMa) source: http://mima.museum/		
mathematik-penrose.php
Fig. 8: Jigsaw puzzle of a penrose tiling

44

A Focus on Crystallography

Practical importance and implementation of abstract mathematical concepts

Three centuries of crystallography throughthe lens of zbMATH
Olaf Teschke

As already outlined in the article “Crystals and mathematics”, crystallography has a natural connection to mathematics: its objects are defined by inherent symmetries
which can be directly translated into abstract mathematical formulations. This is quite specific, since most natural phenomena are so complex that only approximate
mathematical models exist, which usually do not allow
for exhaustive explicit solutions. Contrary to typical cases
like problems of celestial mechanics for which closed
solutions cannot be expected, chaotic systems of climate
or the intricate geometries of elementary particles – therapid advances in group theory provided already in the
19th century a complete classification of the underlying
mathematical structures of crystallography. However,even these seemingly “final” results gave rise to further
research which proved to be of great interest even from apurely mathematical viewpoint, but also of subsequentvalue in material sciences, therefore once more affirming
the “Unreasonable effectiveness of mathematics” as postulated by Eugene Wigner).
In this contribution, we will outline the development ofsome mathematical problems arising from crystallography since the 19th century, as reflected in the database
zbMATH produced by FIZ Karlsruhe, which indexes, classifies and reviews all relevant mathematical research
publications since 1868. Especially the post-publication
reviews will turn out to be very useful in understanding
the historical developments: The special tradition of having the essential results and context of a mathematical
publication reviewed by an independent expert – which
has been introduced in the 19th century to handle theoverwhelming complexity and density of papers andprevails in the core areas of mathematics till today – provides a highly valuable resource of information on the
progress of the subject.

The notion of crystal in mathematics: direct,
indirect, and apparent connections to minerals
Searching in zbMATH for “crystal*” or “kristall*” returns
from the corpus of about 3.4 million documents a comparatively small, but still considerable set of ca. 14,000
publications. A first rough overview of the distinct areas
where the notion arises is given by the Mathematical
Subject Classification: The main areas pertaining to
the notion of crystals are statistical mechanics (section
82, 3900 documents), solid mechanics (74, 3139), fluid
mechanics (76, 1578) – the last two often in connection with partial differential equations (35, 1388) – andgroup theory (20, 1203). This coincides basically withthe areas where the notion of crystal exists explicitly inthe classification, namely, 74E15 (crystalline structure),
74N05 (crystals in solid mechanics), 76A15 (liquid crystals), 82D25 (crystals in statistical mechanics), 82D30(random media and disordered materials including liquid
crystals and spin glasses) and 20H15 (other geometricgroups, including crystallographic groups), where the firstsections basically arise from mathematical physics, i.e.,mathematical descriptions of real mineral structures, andthe last one from abstract structure properties of the corresponding symmetries. One should, however, be awarethat this is not exhaustive: the mathematical language
knows, e.g., also the notion of crystalline cohomology(with associated definitions of crystals and isocrystals,
corresponding to the mathematical subject area 14F30) [1],or crystal bases of certain modules on quantum groups(mathematical subject area 17B37) [2] which notions bear
only some conceptual resemblance with the classicalcrystal objects defined by space symmetries.
While a description of the latter mathematical concepts
would be beyond the scope of this note, we will try tooutline in the following some aspects of crystallography’s
impact on mathematics structure theory.

A Focus on Crystallography

Crystallographic groups in the 19th century:
structure theories and classification theorems
Reading 19th century scientific texts is often difficult, however, the review of Arthur Moritz Schönflies’ “Krystallsysteme und Krystallstructur” [3] (which was, unusually,
done by himself, due to the lack of experts), written in the
Jahrbuch’s spirit of making results broadly accessible,
gives today’s readers still a very lively impression of his
breakthrough in mathematical modelling of crystal symmetries, and its historical context (see Fig. 1). It starts with
Die krystallisirte Materie unterscheidet sich bekanntlich
dadurch von den übrigen festen Körpern, dass ihr physikalisches Verhalten längs verschiedener Richtungen imallgemeinen verschieden ist. Nennt man alle Richtungen,in denen sich ein Krystall in jeder Beziehung gleichartig
verhält, gleichwertige Richtungen, und denkt man sichvon irgend einem Punkte O des Krystalles aus eine Gerade
g und alle mit ihr gleichwertigen Geraden g1 ,g2 ,... gezogen, so ist die Lage dieser N Geraden, wie die Erfahrung
lehrt, durch bestimmte Symmetrieeigenschaften, wie
Symmetrieaxen, Symmetrieebenen u. s. w., ausgezeichnet. Die Symmetrieeigenschaften der N Geraden g,
g1, g2,... sind davon unabhängig, wie die Ausgangsrichtung g innerhalb der Krystallmasse angenommen wird;
sie erhalten sich überdies während der wechselnden
physikalichen Zustände, in denen sich der Krystall befinden kann. Diese Thatsache kann als das “definirende
Grundgesetz der krystallisirten Materie” betrachtet
werden und wird vom Verfasser zweckmässig als “Symmetriegesetz” bezeichnet.

45(As is generally known, crystal matter is distinguishedamong other solids by its well-defined physical behaviour
along distinct directions. If we denote all directions inwhich the crystal behaves homogeneously as equivalent,
and assume that from an arbitrary point 0 of the crystal aline g and all its equivalent lines g1 ,g2 , … are drawn, experience shows then that the arrangement of this N lines
is characterized by certain properties like axes and planesof symmetry etc. The symmetric properties of these lines
are independent on assumptions of the direction of g in
the crystal, and are stable in the various physical statesof the crystal. This fact can be considered the defining
basic law of crystal matter, and is denoted by the author
as “Law of Symmetry”.)
In the following, Schönflies outlines why only a finite
number of possible structures exists, and reviews the
known results on the 32 classes of point symmetries
and the 14 lattices reflecting the translation invariance.
Both finite numbers derive from particular symmetrylaws explaining different properties of the matter, and alarge part of the review (as well as, naturally, the book)
explains how physical structure theories are reflected intheir mathematical models. The result of how both translation and point symmetries can be joined into a generalsymmetry law, and the explicit derivation of the 230 possible space groups, is the core of the work and defines
the notion of crystallographic groups since then. Schönflies’ results are also a striking proof of the conceptual
power of the Erlangen program envisioned by his teacher
Felix Klein, which aimed to reflect geometric properties by
the classification of its transformation groups.

Fig. 1: The beginning of the zbMATH review of Schönflies’ “Krystallsysteme und Krystallstructur”

46

A Focus on Crystallography

Though from a classification viewpoint, the mathematical problems of crystallography have been “solved” bySchönflies’ list, the end of his review points toward what
has not been fully understood:
Die Möglichkeit, zwischen die beiden vorstehend genannten Structurtheorien noch eine Reihe anderer Structurauffassungen einzuordnen, beruht darauf, dass sich die
Gesamtsymmetrie eines Krystalles im allgemeinen so
in zwei Teile zerlegen lässt, dass einer von beiden der
Molekel aufgeprägt wird, während sich der andere in derAnordnung, d. h. in der Art des Aufbaues darstellt. Jederderartigen Zweiteilung der Krystallsymmetrie entsprichteine andere Structurvorstellung; je höher die Symmetrieeines Krystalles ist, um so mannigfaltiger ist daher im allgemeinen die Art, auf welche die Zweiteilung ausgeführt
werden kann. Welche Structurauffassung in jedem speciellen Fall am zweckmässigsten zu Grunde gelegt wird,
ist eine Frage, deren Entscheidung dem Krystallographenüberlassen bleiben muss. Von mathematischer Seite
könne es sich, wie der Verfasser bemerkt, nur darum handeln, die Gesamtheit aller überhaupt möglichen Structurauffassungen anzugeben und zu kennzeichnen.
(The possibility of having further intermediate structure
theories between this two notions [Bravais lattices andpoint groups], relies on the fact that the total symmetryof the crystal can be usually split into two parts of which
one is stamped onto a molecule while the second reflects
its composition. Every such dichotomy of the symmetry
corresponds to a different structure theory, and a larger
intrinsic symmetry of a crystal allows for an increased
possibility of such splits. Which structure theory is mostsuitable for a special crystal is a question which must beleft to the mineralogist; a mathematician can only label
the number of the distinct possible structure theories.)
As we will see later, the description of these possibilities
has led to extended powerful mathematical theories;
at this point, we will only note that the Jahrbuch also
reflects the obstacles of scientific communication backthen. Though Fedorov’s “Basic Law of Crystallography”
is mentioned as a title [4] it has not been reviewed due
to lack of accessibility, and only in 1905 a review of Fedorov’s “Lessons on Syngony” [5] appeared, ensuring
the dissemination of his results within the mathematicalcommunity.
From a historical viewpoint, it is interesting to note thatmuch information on the context of Schönflies’ book canalso be found through zbMATH, though in a much later
review of the 1984 reprint by Johann Jakob Burckhardt [6]:Wir bemerken, daß der Autor übersehen hatte, daß nicht
Hessel (1830/31), sondern Frankenheim die 32 Klassen

erstmals herleitete. Wohl zitiert der Autor zwei Arbeiten
dieses Forschers und weist darauf hin, daß dieser erstmals die sog. Bravais-Gitter aufstellte. Leider übersah
er aber den 1826 in der Zeitschrift ISIS von Oken erschienenen Artikel ”Crystallonomische Aufsätze”, worin
erstmals diese Klassen hergeleitet werden. […] Zu Recht
wird daher sein Werk als einer der beiden Klassiker dermathematischen Kristallographie bezeichnet. Der andere
Klassiker wäre zweifelsohne sein Zeitgenosse E. S. vonFedorov, falls dieser seine Entdeckungen in lesbarer Formveröffentlicht hätte. Allein im Ural arbeitend, fehlte ihm
die Schulung im geläufigen mathematischen Ausdruck.
Dies hat wohl auch Chebychev veranlaßt, die Veröffentlichung der Arbeiten abzulehnen. Glücklicherweise wissen
wir aus dem Briefwechsel von Schoenflies und Fedorov,
daß nur eine enge Zusammenarbeit der beiden Forscher
zur endgültigen und bereinigten Anzahl der 230 Gruppen
führte. Allerdings scheint der Autor nicht bemerkt zu haben, daß der Begriff der symmorphen (= arithmetischen)
Klasse von Fedorov tiefer in die Struktur der Raumgruppen
eingreift als derjenige der 32 Klassen, obschon er diese
73 Systeme auf Seite 598 erwähnt.
Der Nachdruck ist von hervorragender Qualität und es
ist zu hoffen, daß er heute, wo das Gebiet neue Aktualität
genießt, viele Leser finden wird. Sie werden reich belohnt,
insbesondere auch durch die Nennung von etwa 40
früheren Autoren und deren Werken, von Hauy, Delafosse,Frankenheim, und Hessel bis zu Fedorov und Barlow, wasbezeugt, daß der Autor ein ausgezeichneter Kenner derGeschichte der Kristallographie war.
(We note that the author has overlooked that not Hessel(1830/31) but Frankenheim first derived the 32 classes.
Though the author quotes two of his articles and points
out that he first compiled the so-called Bravais lattices,
but he overlooked that these classes were derived for the
first time in his article ”Crystallonomische Aufsätze”, published in 1826 in Oken’s journal ISIS.
[…] This work is rightfully considered one of the two classics of mathematical crystallography. The other classic
would have been his contemporary E. S. von Fedorov if
he had been able to publish his results in a readableform. Working isolated in the Ural mountains, he lacked
the knowledge of common mathematical notation, whichhas probably led Chebychev to the rejection of his work.
Fortunately, we know from the correspondence of Schoenflies und Fedorov that only a close collaboration led
both teachers to the final and simplified number of 230
groups. However, it seems that the author didn’t noticethat Fedorov’s concept of symmorphic (=arithmetic) classcatches more of the structure of space groups than the 32
classes, though he mentions these 73 systems at p. 598.

A Focus on Crystallography

The reprint is of extraordinary quality and it remains tohope that, given the renewed interest in the subject today, it will find many readers. They will be amply rewarded also by the references to 40 earlier authors and theirwork, from Hauy, Delafosse, Frankenheim, and Hessel
to Fedorov and Barlow, which show that the author has
been an intimate expert of the history of crystallography.)
While we will explain the “renewed interest” in the next
section, we would like to note that some of the historicalfacts pointed out by Burckhardt have been frequently
ignored in earlier and later historical surveys.

Some waves of 20th century mathematicalcrystallography
Looking at the presence of mathematics in crystallographic journals (like the Zeitschrift für Kristallographie,which has been indexed in zbMATH for about 100 years)
reveals some peaks of mathematical articles separated
by longer gaps of stagnancy (see Fig. 2). After decadesof silence following the breakthroughs of Schönflies andFedorov, the interest of mathematicians in crystallographic questions rose again in the 1920s, followed bya very active period between 1930 und 1940. A second
wave of interest took shape at the end of the 70s, andhas continued ever since.

47

What developments in mathematical understanding havebeen behind this? Two main driving forces in mathematics played a major role also here: the urge to understandthe underlying concepts, and the search for generalizations. A first step into both directions was taken by Pólya
und Niggli in 1924 by deriving the classification of thetwo-dimensional analogue of crystal structures by strictly algebraic methods [7]. This has been in line with thecomplete reshaping of abstract algebra which took placein the 1920s in the course of Emmy Noether’s ideas andconcepts. Consequently, Johann Jakob Burckhardt (whomwe already met as a reviewer above – in total, he contributed about 700 reviews from 1939 until 2004, at theage of 101!), who has been strongly influenced by thesedevelopments, undertook the task of adapting the newformal concepts to crystallographic questions. By introducing the notion of arithmetic crystal class (which gavea precise meaning to some concepts used by Fedorov 40
years earlier), he derived the crystallographic groups by
purely algebraic means [8] – an approach which he could
generalize to higher dimensions. His research on crystallographic questions cumulated in the standard book “Die
Bewegungsgruppen der Kristallographie” [9] which, in thewords of the reviewer György Hajós, made “for the firsttime the mathematical theory of crystal classes and spacegroups accessible also to non-mathematicians” [10].
While these results were again conclusive to a certain point, Burckhardt’s algorithmic approach anticipated a recent driving force in the
study of the subject, namely, the useof computer algebra in the study ofsuch group structures (which will beconsidered in the next section).

Fig. 2: Exploring mathematical crystallography through a zbMATH journal profile

At about the same time, a second
development in the formation of
abstract algebra also maintained
strong connections to the origins ofcrystallographic groups. An essential
part of its understanding was how
the possible structures of point symmetries and translation symmetries
fit together. From a mathematical
viewpoint, this meant to consider
the possible group extensions. Thatthe underlying structure is quite
involved is somewhat obvious fromthe fact that the 32 point groups andthe 14 possible lattices make up toprecisely 230 different space groups
structures. Since similar questions

48

A Focus on Crystallographyof how simpler structures (so-called factors) can beextended to a larger one arose also in different areas of
19th century mathematics (for instance, the question ofsolvability of polynomial equations in radicals led to theconsider extensions of Galois groups). The successful
discovery of the abstract mechanism guiding those extensions relied heavily on the concepts developed in the
1920s, leading to the notion of group cohomology which
was finally discovered by Heinz Hopf and Beno Eckmann
at the ETH Zürich (where also Burckhardt worked at thattime), Samuel Eilenberg and Saunders Mac Lane at Columbia University, and by Hans Freudenthal at the University of Amsterdam, and which has evolved into an indispensable tool in studying such structures ever since [11].
The third parallel development does not seem to fit into
the pattern of increased understanding through abstract
concepts: Harold Coxeter revived the interest in theseemingly old-fashioned area of classical geometry. However, a key ingredient of his studies turned out to be theconsideration of reflection groups, which are a generalization of symmetry groups arising in crystallography [12].
The Coxeter diagrams he developed for the descriptionof these groups were not only a large conceptual step
forward, they also allowed for a surprising application
to the seemingly distant problem of classifying simple
Lie groups of any dimension. A Lie group, contrary to
the groups considered so far, is an infinite group whichcomes along with the geometric structure of a manifold,while simple just means to be a basic building block in
the factor sense described above in the context of groupextensions. Lie groups correspond to Lie algebras (a nonassociative structure fulfilling a certain identity), and it
turned out that they can be described by a finite “rootsystem” of vectors encoding its geometry. The symmetries of the root system give rise to the so-called Weylgroup, which is a special Coxeter group whose diagrams
fulfils precisely the condition that only 2, 3, 4, and 6 are
permitted as its edge labels. This corresponds surprisingly to the space-filling condition we know from crystallography (analogously to the proof of the impossibility of the
icosahedron in the article “Crystals and mathematics”)
and has become known as the “crystallographic restriction theorem”. The importance of these structures isunderlined by the fact that such Dynkin diagrams (i.e.,
Coxeter diagrams fulfilling the crystallographic conditions) have appeared since then in many different areasof mathematics, as representation theory, singularity theory, quantum mechanics or category theory.

The appearance of computers and quasicrystals: some recent developments
With the theoretical foundations laid in the 1930s, acomplete classification of the (as proven) finitely many
crystallographic structures in higher dimensions would
have been possible in principle. However, it turned outthat the concrete handling of the vast amount of structures was extremely difficult. Not much progress could
be made even in dimension four until the availabilityof sufficiently powerful computers changed the game.
A major milestone was the work of Harold Brown, Rolf
Bülow, Joachim Neubüser, Hans Wondratschek and Hans
Zassenhaus [13]. While Zassenhaus – also a descendantof the Noether school – had already found an effective algorithm in 1948 [14], it took about 30 years until sufficientcomputing power was available to implement it effectively. The complete list of 4895 four-dimensional crystallographic groups was one of the first major results in mathematics solely verified by computers (besides the better
known case of the four-colouring of graphs in 1976), as
pointed out by Burckhardt in his zbMATH review. This wasa starting point of computational approaches in grouptheory which bore many fruits during the next decades.
Since questions concerning finite groups often imply thenecessity to handle extensive classification lists (manyof them only for intermediate purposes) with the inherent danger of error propagation, it is an incredible help
to have a formalization available which is not affectedby human calculation errors. If one keeps in mind theimmense effort which had been necessary for the classification of all finite simple groups (which is distributed into
hundreds of research articles between 1955 and 2004,
by about hundred authors, assembling to about 25,000
pages), which can hardly be surveyed by a single humananymore, the advantage of computer support becomes
obvious. The arguably most powerful tool developed so
far is the GAP system [15]. As can be seen from swMATH,
the software facet of zbMATH, more than 800 mathematical research publications have employed this software
so far [16], including a number of formal verificationsof earlier human-generated lists. Included in this large
project are also packages with specific crystallographic
features like Cryst [17] and CARAT [18]. A typical example of ongoing developments may look like the paper“Computations with almost-crystallographic groups”
of Karel Dekimpe and Bettina Eick [19] which explores
the opportunities of the GAP system when dealing withalmost-crystallographic groups, a natural generalizationof crystallographic groups which are finite extensions offinitely generated, torsion-free nilpotent groups (insteadof free abelian groups, i.e. lattices, in the case of crystallographic groups).

A Focus on Crystallography

49

this discovery could also be duly attributed to the creators of the Islamicgirih ornaments) [22]. Today, quasicrystals are established mathematicalobjects which even made their way into
the Mathematical Subject Classification
as a subfield of convex and discretegeometry (52C23 - Quasicrystals, aperiodic tilings). Interesting enough, one of
the most powerful tools for describing
quasicrystalline symmetries are againroot systems and their diagrams (thistime, naturally, violating the crystalline
restrictions) which were constructedlong before the mathematicians became aware of the existence of suchobjects.

Fig. 3: The GAP system in swMATH

Another, probably more famous, generalization of crystalline structures are quasicrystals, which have been
already introduced in other contributions of this volume.
From a mathematical viewpoint, it is fascinating that theunderlying abstract structures were derived before theirdiscovery by Shechtman in the “real world” – namely, as
non-periodic tilings of the plane first discovered by Robert Berger [20] and the especially aesthetic example ofRoger Penrose which needed only two tiles [21] (though

It would be beyond the scope of thiscontribution to explore the recent ramifications of this subject which includes
many different aspects of combinatorics, geometry, algebra or computational
mathematics, and we recommend to
explore them by browsing through thezbMATH database. We would only liketo conclude with a nice example of how
the aesthetics of quasicrystalline structures can be used even for the popularization of mathematics at school level:
The mathematician John Horton Conway (also famous for, e.g., the “Gameof Life”) constructed a Penrose typetiling which uses only a pair of simple
equilateral triangles as building blocks(which project to the more complicated“kite and dart”-components of Penrose)
which fit together as a surface in the three-dimensional
space which can be constructed by a simple and beautiful
origami folding [23].
So, if you have a sheet of paper at hand (of course, apartfrom this brochure!) – start to explore the beauty of extended symmetry which has been long hidden both inminerals and in mathematical formalism!

50

A Focus on Crystallography

Fig. 4: Quasiperiodic origami surface (source: R. J. Lang and B. Hayes, Paper Pentasia: an aperiodic surface in modular origami. Math. Intell. 35, No. 4,6174 (2013))

References[1] introduced by Grothendieck, A. (1968).[Crystals and the De Rham
cohomology of schemes, in: Dix Exposes Cohomologie Schemas,
Advanced Studies Pure Math. 3, pp. 306-358. ( Zbl 0215.37102)](https://zbmath.org/?q=an:0215.37102v) in the study of the Weyl
conjectures)[2] introduced by Kashiwara, M. (1990).[Bases Crystallines, C.
R. Acad. Sci., Paris, Ser. I 311, No. 6, pp. 277-280 (Zbl 0724.17008)](https://zbmath.org/?q=an:0739.17005)[3] Leipzig. Teubner. XII+638 S. (1891; JFM 23.0554.02)(https://zbmath.org/?q=an:23.0554.02)[4] St. Petersb. Miner. Ges. (2) XXXI. pp. 171-190 (1894; JFM 25.0879.02)(https://zbmath.org/?q=an:25.0879.02)[5] Münch. Abh. 23, pp. 1-88 (1906; JFM 37.0504.01)(https://zbmath.org/?q=an:37.0504.01)[6] Krystallsysteme und Krystallstructur. Reprint des Originals von1891. Berlin etc.: Springer Verlag (1984; Zbl 0536.20001)(https://zbmath.org/?q=an:0536.20001)[7] Pólya, G. Über die Analogie der Kristallsymmetrie in der Ebene,
Z. Kristallogr. 60, No. 1, pp. 278-283 (1924; Zbl 06355163)(https://zbmath.org/?q=an:06355163, November 25, 2014);
Die Flächensymmetrien homogener Kontinuen. Z. Kristallogr. 60,
No. 1, 283-298 (1924; Zbl 06355164);(https://zbmath.org/?q=an:06355164)[8] Comment. Math. Helv. 2, pp. 91-98 (1930; JFM 56.0867.01)(https://zbmath.org/?q=an:56.0867.01)[9] Basel: Birkhäuser, 186 p. (1947; Zbl 0037.14903)(https://zbmath.org/?q=an:0037.14903)[10] Many historical details can be found in Burckhardt’s survey “Die
Symmetrie der Kristalle. Von René-Just Haüy zur kristallographischen Schule in Zürich. Mit einem Beitrag von Erhard Scholz“[Basel etc.: Birkhäuser Verlag. 195 S. (1988; Zbl 0644.01004) 		
(https://zbmath.org/?q=an:0644.01004).

[11] For a detailed history of the subject, see [Weibel, C. A. An introduction to homological algebra. Cambridge Studies in Advanced
Mathematics. 38. Cambridge: Cambridge University Press. (1994; 		
Zbl 0797.18001)] (https://zbmath.org/?q=an:0797.18001)[12] Coxeter, H. S. M. (1935) J. Lond. Math. Soc. 10, pp. 21-25( Zbl 0010.34202) (https://zbmath.org/?q=an:0010.34202)[13] Crystallographic groups of four-dimensional space. Wiley
Monographs in Crystallography. 443 p. (1978; Zbl 0381.20002) 		
(https://zbmath.org/?q=an:0381.20002)[14] Über einen Algorithmus zur Bestimmung der Raumgruppen.
Comment. Math. Helv. 21, pp. 117-141 (1948; Zbl 0030.00902),(https://zbmath.org/?q=an:0030.00902); both works were again 		
reviewed by Burckhardt in zbMATH[15] Groups, Algorithms, Programming (www.gap-system.org ).
(November 25, 2014)[16] www.swmath.org/software/320 (November 25, 2014)[17] www.swmath.org/software/9514 (November 25, 2014)[18] www.swmath.org/software/4293 (November 25, 2014)[19] Lond. Math. Soc. Lect. Note Ser. 304, pp. 110-119(2003; Zbl 1054.20031) (https://zbmath.org/?q=an:1054.20031)[20] The undecidability of the domino problem. Mem. Am. Math. Soc. 66,
72 p. (1966; Zbl 0199.30802)(https://zbmath.org/?q=an:0199.30802)[21] Pentaplexity. A class of non-periodic tilings of the plane.
Math. Intell. 2, pp. 32-37 (1979; Zbl 0426.52005)(https://zbmath.org/?q=an:0426.52005)[22] Lu, P. J.; Steinhardt, P. J. (2007).Decagonal and quasi-crystalline 		tilings in medieval Islamic architecture. Science 315, No. 5815, pp.
1106-1110 (Zbl 1226.00028) (https://zbmath.org/?q=an: 1226. 		
00028)[23] Lang, R. J.; B. Hayes, B. (2013). Paper Pentasia: an aperiodic surface
in modular origami. Math. Intell. 35, No. 4, pp. 61-74 ( Zbl 06249373)(https://zbmath.org/?q=an:06249373)

Facts and figures
ICSD contains

Minerals		

Metals and alloys

		

Inorganic compounds

Available interfaces forInternet

		Intranet		

Local installation

Updates		

twice per year		

Around 3,000 new entries per update

Content (release 2014.2)		173,473
elements		2,019
binary compounds		 33,809
ternary compounds		 65,126
quaternary compounds		 41,665
quintenary compounds		 19,004

Structure types		8,230# of structures assigned to a structure type		138,503 (~79.8 %)Bibliography
Journals		 1,671			
Authors		65,026
Papers		79,336

FIZ Karlsruhe – Leibniz Institute forInformation Infrastructure
Hermann-von-Helmholtz-Platz 176344 Eggenstein-Leopoldshafen
Germany
Phone +49 7247 808-0
contact@fiz-karlsruhe.de
www.fiz-karlsruhe.de
BerlinFIZ Karlsruhe’s information services in mathematics and informatics are produced at the
Department «Mathematics and Computer
Science» in Berlin.
FIZ KarlsruheFranklinstr. 1110587 Berlin
Germany
Phone +49 30 399934 0
editor@zblmath.fiz-karlsruhe.de
www.zentralblatt-math.org/zmath/en

A Focus on Crystallography
Editor: Helmut Müller
Staff of Department Mathematics and Computer Science
Staff of Department Science and Technology
Staff of Department FIZ Products and Services
Staff of Marketing Communications
Translation: Regina Karl
(Preface; “Crystals and Mathematics”)© FIZ Karlsruhe 2015