Hammond C. The Basics of Crystallography and Diffraction

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344 Appendix 2

(D) Coordination and space-ﬁlling polyhedra (Voronoi polyhedra or

Dirichlet domains)

Coordination polyhedra, which represent the pattern of atoms or lattice points surround-

ing an atom or lattice point, should be clearly distinguished from the domains (Voronoi

polyhedra or Dirichlet domains) which deﬁne the environment around an atom or lattice

point and which are always space-ﬁlling (see Section 3.4). A vertex of a coordination

polyhedron corresponds to the face of the corresponding domain, but the polyhedra are

not necessarily duals.

They are duals, for example, in the cubic F lattice in which the coordination poly-

hedron is a cubeoctahedron (Fig. A.2.1(b) and (e)) and the space-ﬁlling polyhedron is

a rhombic dodecahedron (Fig. A.2.4(a)) but not in the case of the cubic I lattice in

which the coordination polyhedron is a rhombic dodecahedron

and the space-ﬁlling

polyhedron is a truncated octahedron.

The procedure for constructing the space-ﬁlling Voronoi polyhedra for lattices is

described in Section 3.4. For crystal structures consisting of only one kind of atom

—

e.g. the simple cubic, bcc, fcc and hcp metal structures the procedure is the same. For

structures consisting of more than one kind of atom the procedure is to construct the

space-ﬁlling polyhedron around one kind of atom and its nearest neighbours of the same

kind.

The atoms of other kinds are then shown as occupying the appropriate vertices of

the space-ﬁlling polyhedron.

Afew examples will make this clear. IntheNaCl structure the space-ﬁlling polyhedron

around a Cl atom is a rhombic dodecahedron and the Na atoms are situated at the six

vertices of the polyhedron where four edges (or faces) meet, as shown in Fig. A.2.6(a)

(the same pattern occurs of course if the space-ﬁlling polyhedron is constructed around

an Na atom).

In the NiAs or α-Al

structures in which the As or O atoms are (closely) hexagonal

close packed the space-ﬁlling polyhedron is a trapezorhombic dodecahedron. In NiAs the

Ni atoms are situated at all six vertices where four edges (or faces) meet (Fig. A.2.6(b))

and in α-Al

the Al atoms are situated at four such vertices (Fig. A.2.6(c)).

Coordination polyhedra also determine the radius ratios of interstitial sites, e.g.

between the atoms in metals as described in Section 1.6 or between close-packed oxy-

gen anions. The data are summarized in Table A.2.2 which includes the interstitial sites

delineated by squares and triangles.

Table A.2.3 lists the coordination and space-ﬁlling polyhedra for some simple

structures.

(E) Brillouin zones

Brillouin zones are ‘nests’of polyhedra in k-space (reciprocal space with the scale factor

2π). Their surfaces represent the directions and wavelengths (or wavenumbers) of free

The eight ‘obtuse’ vertices correspond to the lattice points at the corners of the cell and the six ‘acute’

vertices correspond to the lattice points at the centres of neighbouring cells as indicated in Fig. A.2.4(a)

The term ‘atom’ encompasses charged species, i.e. both cations and anions.

E.W. Gorter. Classiﬁcation representation and prediction of crystal structures of ionic compounds. J. of

Solid State Chem. 1, 279 (1970).

Appendix 2 345

(a)

(b)

(c)

Fig. A2.6. Space-ﬁllingpolyhedra for (a)the NaCl structure, (b)the NiAs structureand (c)the α-Al

structure.

346 Appendix 2

Table A2.2 Coordination polyhedra for interstitial sites and radius

ratios

Structure type Co-ordination polyhedron

around interstitial site

Radius ratio

Simple cubic Cubic 0.732

Square 0.414

Cubic close packed Octahedral 0.414

Square 0.414

Tetrahedral 0.225

Triangular 0.155

Hexagonal close packed Octahedral 0.414

Tetrahedral 0.225

Triangular 0.155

Body centered cubic ‘Distorted’ octahedral 0.154

‘Distorted’ tetrahedral 0.291

Simple hexagonal Triangular prism 0.528

Triangular 0.155

Table A2.3 Coordination and space-ﬁlling polyhedra

Structure/lattice type Co-ordination polyhedron Space-ﬁlling polyhedron

cubic P (simple cubic) octahedron (square dipyramid) cube (dual)

cubic F (fcc) cubeoctahedron rhombic dodecahedron (dual)

cubic I (bcc) rhombic dodecahedron truncated octahedron

hexagonal P (simple

hexagonal)

hexagonal dipyramid hexagonal prism (dual)

hexagonal close-packed (hcp) hexagonal cubeoctahedron trapezorhombic dodecahedron (dual)

NaCl (structure type) octahedron, Na around Cl or Cl

around Na

rhombic dodecahedron, with Na or Cl

at the 6 vertices where 4 edges meet

CsCl (structure type) cube, Cs around Cl or Cl

around Cs

cube, with Cs or Cl at the 8 vertices

NiAs (structure type) trigonal prism, Ni around As trapezorhombic dodecahedron, with

Ni atoms at the 6 vertices where 4

edges meet

α−Al

tetrahedron (distorted), Al

around O

trapezorhombic dodecahedron, with

Al atoms at the 4 of the 6 vertices

where 4 edges meet

CaF

and Li

O (ﬂuorite and

anti-ﬂuorite structure types)

cube, F or Li around Ca or O rhombic dodecahedron, with F or Li

at the 8 vertices where 3 edges meet

tetrahedron, Ca or O around F

or Li

cube, with Ca or O situated at 4 of

the 8 vertices forming a tetrahedron

electrons in crystals which are Bragg-reﬂected by the lattice planes. The ﬁrst, innermost

Brillouin zone corresponds to the reﬂections from the planes of largest d-spacing, the

Appendix 2 347

next Brillouin zone corresponds to reﬂections from the planes of next largest spacing,

and so on.

The ﬁrst Brillouin zone is equivalent to the Voronoi polyhedron or Wigner–Seitz cell

for the corresponding reciprocal lattice. For example, for the cubic F lattice, for which

the corresponding reciprocal lattice is cubic I, it is a truncated octahedron.

The constructional procedure for Brillouin zones may be illustrated for a cubic P

(simple cubic) lattice with a cell edge length a and for which the corresponding reciprocal

(k-space) lattice is also cubic P with a cell edge length

2π



(Fig. A.2.7).

Consider electrons travelling in the x

∗

-y

∗

plane reﬂected from lattice planes parallel

to the z

∗

axis. In order of decreasing d

hkl

-spacing (or increasing d

∗

hkl

spacing) these are

100, 110, 200, 210, … Consider an electron with wavevector k as indicated:

k =

2π

For reﬂection from the 100 planes, the corresponding value of k is found by substituting

for λ in Bragg’s law

i.e. k =

2π

2a sin θ

2π

2 sin θ

110

––

010

–

110

100

–

000

010

110

100

–

110

–

Fig. A2.7. Construction for the ﬁrst Brillouin zone for a cubic P lattice. θ is shown for reﬂection from

the 100 planes.

348 Appendix 2

which value is indicated by point P. The locus of P, for all values of θ, and for reﬂections

from the 010, etc. planes, gives rise to the outline of the ﬁrst Brillouin zone, as indicated

by the dashed lines. In three dimensions the Brillouin zone is a cube.

The second Brillouin zone arises from reﬂections from the 110 planes and, proceeding

as before, the corresponding values of k are indicated by the dotted lines. In three

dimensions this second Brillouin zone is a rhombic dodecahedron (like the ﬁrst Brillouin

zone for the cubic I reciprocal lattice).

Appendix 3

Biographical notes on

crystallographers and scientists

mentioned in the text

Ernst Abbe 1840–1905

Abbe’s father was a mill worker in Eisenach, Germany, and it was his employers who

provided the scholarships which enabled Abbe to proceed from High School to study

physics at the Universities of Jena and Göttingen. In 1863 Abbe achieved his ambition

of becoming a lecturer at Jena, but his career may be said to commence in 1866 when he

began his long and fruitful collaboration with Carl Zeiss. As a theoretician he established

the relation between the aperture and the limit of resolution of a lens (the equation is

engraved on his memorial at Jena); as an optical designer he invented the apochromatic

objective lens (made possible with the new glasses developed by the Schott glassworks);

and as an industrialist he and Zeiss developed systematic microscope production on a

large scale.

In 1876 he was made a partner in the Carl Zeiss ﬁrm and became the sole owner

following the death of Zeiss in 1888. In 1891 he created the Carl Zeiss Foundation

whose charter was used in Prussia as a model for progressive social legislation.

George Biddell Airy 1801–92

Airy’s life was characterized, to a remarkable degree, by a determination to succeed and

meticulous organization: he made a record of everything that he wrote, with dates, and

discarded not the merest note or receipt. His entry to Cambridge in 1819, which was

largely facilitated by a wealthy uncle in Colchester, provided him with the opportunity to

break away from a modest parental background and thence his career developed rapidly.

He was appointed Lucasian Professor of Mathematics in 1826, Plumian Professor of

Astronomy in 1828 andAstronomer Royal in 1835, a post he ﬁlled for 46 years and which

entirely ﬁtted his temperament. He largely re-equipped the Greenwich Observatory and

transformed it into a highly efﬁcient Institution—but at a price: it provided no centre for

the training of innovative scientiﬁc minds and the discoveries of John Herschel and John

Couch Adams were made elsewhere. Airy was, in effect, the prototype of the modern

government scientist and played a major role in the development of institutional science

in Victorian England. He designed (with the lawyer, Edmund Beckett Denison) the

clock for the new Palace of Westminister—it had to be redesigned in 1852 when it was

discovered that the interior walls had been made too small—but its continued functioning

almost without interruption for over 140 years is a measure of their achievement.

350 Appendix 3

Airy calculated the magnitude of the image formed by a telescope objective of a

star—a point source of light—and thereby laid the foundations for the theory of the

resolving power of optical instruments. He was knighted in 1872, having refused that

honour three times previously on the grounds that he couldn’t afford the fees.

William Thomas Astbury 1898–1961

Astbury was born in Longton in Staffordshire, England, one of the ‘ﬁve towns’ of the

pottery manufacturing industry in what was then called the ‘Black Country’. His family

was poor, but Astbury’s success in winning scholarships provided him with an excellent

early education and entry into Jesus College, Cambridge, which would otherwise have

been beyond his reach. His initial studies in mathematics and physics were interrupted by

the First World War, during which time he served with the X-ray unit of the RoyalArmy

Medical Corps. On his return to Cambridge, through the inﬂuence ofArthur Hutchinson,

Demonstrator in Mineralogy, he studied in addition chemistry, mineralogy and the newly

emerging science of X-ray crystallography pioneered by the Braggs. Arthur Hutchinson

indeed was the ‘source’ of many of the crystals ﬁrst studied by the Braggs which he

‘borrowed’ (without permission) from the Mineralogical Museum.

In 1921Astbury joined W.H. Bragg’s newly formed research group at University Col-

lege, London and in 1923 moved with the group to the Royal Institution where (together

with Kathleen Yardley) he developed the ﬁrst space group tables to help determine,

from X-ray evidence, to which of the 230 space groups a crystal belongs. W. H. Bragg

thought them unnecessary—all that was needed was common sense. Astbury replied

that not everyone has common sense. However, the major direction of his life’s work,

in macromolecular structures, came about as a result of a request from Bragg in 1926

that he take some X-ray photographs of ﬁbres which Bragg wished, if possible, to use in

one of his Royal Institution lectures. This was typical of Bragg’s subtle way of directing

research—to simply ask for help on some particular topic. Astbury was so successful

in applying X-ray techniques to the study of cellulose, silk and wool ﬁbres that in 1928

Bragg recommended him for the post of Lecturer in Textile Physics and Director of the

Textile Physics Research Laboratory at Leeds University which, particularly through the

work of J. B. Speakman, was pre-eminent in the study of the physical and chemical prop-

erties of ﬁbres. Here he made rapid progress in distinguishing the fully-extended chain

structures characteristic of cellulose and silk (the simplest polypeptide protein structure)

and the much more complex protein structure, keratin—the basic material not only of

wool and hair ﬁbres but also of nails horn and quills. He showed, in wool, that the ker-

atin polypeptide chains occurred in two forms, a folded form, which he called α-keratin

and an extended form, β-keratin, a discovery of enormous scientiﬁc and technological

importance in understanding the extensibility of wool ﬁbres and in the processing of

woollen fabrics. He disseminated this information in a series of University Extension

lectures in 1932 and which formed the basis of his book Fundamentals of Fibre Structure

(Oxford University Press, 1933).

Astbury proposed a model for α-keratin in which the amino acid peptide links were

folded in the form of hexagons, parallel to the chain axis and perpendicular to the cross-

linking side arms. This model was superseded in 1951 by Pauling’s α-helix model but

Appendix 3 351

it did nevertheless represent the ﬁrst attempt to model a protein chain structure in which

speciﬁc linkages held the polypeptide chains in a characteristic conformation.

In 1935, together with a research student, Florence Ogilvy Bell, Astbury began to

study the structure of nucleic acids and in 1938 together they published the ﬁrst X-ray

diffraction photograph (or ﬁbre diagram) of a nucleic acid and attempted, unsuccessfully,

to interpret its structure.

In 1945 Astbury was appointed to the new Chair of Biomolecular Structure at Leeds

which he occupied until his early death in 1961.

As a pioneer it was perhaps inevitable that Astbury should see his own work super-

seded. His great legacy is his early stimulation to so many others in the ﬁeld of molecular

biology.

Johannes Martin Bijvoet 1892–1980

Bijvoet was born and brought up in an old waterside house in the centre of Amsterdam.

He recalls that his wish to become a chemist arose from the excellent teaching at his

secondary school, where the emphasis was on understanding rather than rôte learning.

As a result (and also having passed the then rigorous examinations in Latin and Greek

required for university entrance) he enteredAmsterdam University in the years preceding

the First World War. Here the emphasis was on routine analytical chemistry, which gave

him much less satisfaction.

During the war, Bijvoet, like all young men of his age, was conscripted for the military

service, but since the Netherlands was a non-combatant nation, it was simply a period of

enforced idleness which Bijvoet used ‘to study quietly thermodynamics and statistical

mechanics’—an indication perhaps of his serious attitude to life.

Bijvoet’s move to the area of crystallography arose from the excitement aroused by

Bragg’s model of NaCl, a model which was by no means accepted by the professorial

staff at the University, including his own supervisor, as an adequate representation of the

structure. However, these objections had a happy consequence in that it was decided that

X-ray investigations should be undertaken in which Bijvoet would have a leading role.

However, despite sound progress made in the solution of crystal structures, the

‘chemists’ remained unconvinced. Bijvoet recalls a ‘painful memory’ of a visit by

W. H. Bragg in which in despair at the persistent hackneyed arguments against the

results of X-ray analysis, Bragg ‘raised his arms up to heaven’. Even after Bijvoet’s

appointment in 1928 as lecturer in crystallography and thermodynamics he records that

he ‘passed through a time of struggle for the raison d’etre of X-ray analysis’.

In 1939 Bijvoet was appointed to the Chair of Chemistry at the van’t Hoff Laboratory

of the University of Utrecht, a post which the previous holder, Ernst Cohen, was forced,

as a Jew, to vacate and who, despite his support for Germany following the First World

War, was later to be killed by the Nazis.

It was at Utrecht, in the years 1949–1951, that Bijvoet and his colleagues, made

their most outstanding contribution to X-ray crystallography: the use of anomalous

scattering to determine absolutely (and not relatively) the right and left handed forms

of enantiomorphic crystals—a fulﬁlment, in a sense, of van’t Hoff’s work of 1874 on

tetrahedrally bound carbon in organic crystals. The experimental work, carried out with

352 Appendix 3

crystals of sodium rubidium tartrate, was of great difﬁculty. Exposures of over 100 hours

were needed, using an improvised X-ray tube with a zirconium target and a ‘freakish’

pump which had to be watched continuously.

Bijvoet was closely involved in the establishment of the International Union of Crys-

tallography in 1946 and became President in 1951. He retired to Winterswijk, a rural

district of the Netherlands, in 1962.

William Barlow 1845–1934

Following an inheritance from his father, Barlow, one of the last great English amateurs

in science, took up the study of crystallography in his early thirties and attempted to

relate the properties of crystals to the packing arrangements of their constituent atoms.

He collaborated with W. J. Pope at Cambridge and guessed (correctly) the structure of

the alkali halides. His mathematical ability and developing interest in symmetry enabled

him to extend the methods of Bravais and Sohncke to ﬁnd the ‘symmetrical groupings

to ﬁt the forms and compositions of a variety of different substances’ but he did not

publish his derivation of the 230 space groups until 1894, three years after Fedorov and

Schoenﬂies, of whose work he almost certainly had no prior knowledge.

William Henry Bragg 1862–1942

W.H. Bragg was born in Westward, Cumberland. At the age of seven, following his

mother’s death, he went to live ﬁrst with an Uncle in Market Harborough, Leicestershire

and then to a boarding school in the Isle of Man. In 1881, he won a scholarship at

Trinity College, Cambridge to read mathematics; he graduated as a ‘Third Wrangler’ in

Part I of the mathematical tripos in 1884 and obtained a ﬁrst class in Part II in 1885.

On the basis of this remarkable mathematical achievement in the same year he was

appointed Professor of Mathematics and Physics at the University of Adelaide. The

story goes that some friends pointed out that he did not know any physics. ‘Ah’, said

Bragg ‘but the boat takes ten weeks to reach Australia, plenty of time to read up on the

subject.’

For nearly twenty years, Bragg immersed himself in the affairs of the University,

developed his clear teaching style but did little serious research. It was only in 1903,

at the age of 41, that he identiﬁed a research area suited to his particular gifts—the

problems of radioactive emission, the ionization of gases and the nature of X-rays which

he considered to be ‘corpuscular’ in nature. His rapidly growing reputation resulted in

election to the Royal Society in 1907 and, in the same year, Bragg was offered the

Cavendish Chair of Physics in the University of Leeds, a post which he took up in 1909.

The contrast between Leeds and Adelaide could not have been more great: his Aus-

tralian wife, Gwen, was particularly appalled by Leeds, the dirt and smoke, the rows

of back-to-back houses and the poor rickety children. It was no better in the Univer-

sity; Bragg was restricted by the syllabus, found the students inattentive and also found

himself in a losing battle with his corpuscular theory of X-rays and γ -rays. However,

he derived support from Arthur Smithells, Professor of Chemistry, from Ernest Ruther-

ford at Manchester and not least from the emerging scientiﬁc abilities of his elder son,

Appendix 3 353

Lawrence. Gwen immersed herself in social work for poor children and for the family

there were weekends spent in the beautiful surrounding Dales countryside.

In June and July 1912, Laue’s two papers announcing the diffraction of X-rays were

presented at meetings of the Bavarian Academy of Sciences. They were published in

late August, but even before then word ‘got around’ it did so to the Braggs during a

family holiday at Cloughton, a village on the Yorkshire coast, by way of a letter from

Lars Vegard, a former colleague, who enclosed a photograph which he had obtained

from Laue. Laue’s photograph clearly showed the wave nature of X-rays but both father

and son were reluctant to abandon the corpuscular theory and on their return to Leeds,

attempted to interpret the patterns in terms of corpuscles travelling along ‘avenues’

between the atoms. This work resulted in a letter to Nature in October 1912, authored

solely by W.H. Bragg. It represents the last attempt to ‘save’ the corpuscular theory.

Thereafter, the wave theory was unquestioned.

Bragg’s signiﬁcant achievement at this time was the design and construction in the

Leeds physics workshops in the winter of 1912–1913 of the X-ray spectrometer and it

was this instrument that enabled him and Lawrence to make the ﬁrst determinations of

crystal structures, for which work they were jointly awarded the Nobel Prize in 1915.

Bragg records how he and Lawrence worked together in the physics laboratory far

into the night and it is near impossible to disentangle the relative contributions of father

and son. Certainly Lawrence’s deep physical insight and his exploitation of the law of

reﬂection that bears his name was of primary importance. But set against this is Bragg’s

foresight, expressed in the Bakerian lecture to the Royal Society in 1915, that the series

of spectra (reﬂections) from a given set of crystal planes represented a series of harmonic

terms which could be analysed by Fourier methods—a notion of profound importance

in crystallography which awaited some ten years before it could be exploited in crystal

structure determination.

In 1915 Bragg accepted the Quain Chair of Physics and University College London

but only took up the post in 1918, following secondment for research at the Admiralty

during the course of the War. At University College he assembled and encouraged a

remarkable team of researchers—W.T. Asbury, G. Shearer, A. Muller, K. Yardley—and

made the ﬁrst attacks on the structure of organic crystals (by tacit agreement Lawrence,

then at Manchester, worked on inorganic, mainly mineral and silicate, crystals).

In 1919 Bragg was invited to give the Christmas lectures ‘to a Juvenile Auditory’ at

the Royal Institution, continuing a long-standing tradition that had begun with Michael

Faraday. The lectures, subsequently published as ‘TheWorld of Sound’, show his powers

of simple exposition and his affection for young people (he dedicated the book ‘to Peggy,

Gwendy and Phyllis who discussed with me so many things in this book as we walked

to school in the mornings’).

It was perhaps as a result of these lectures that Bragg was elected in 1923 to the head-

ship of the Royal Institution and the title, inter alia of Fullerian Professor of Chemistry.

Bragg was by now the distinguished icon of British science, he was elected to the Order

of Merit in 1931, yet in spite of his heavy administrative duties continued to attract to

the Institution younger scientists such as J.D. Bernal who maintained Britain’s leading

role in X-ray crystallography. His book The Universe of Light (1933) was based on his

1931 Royal Institution Christmas lectures.