Hammond C. The Basics of Crystallography and Diffraction

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374 Appendix 3

In 1932, armed with the ﬁrst of a series of large grants from the Rockefeller Foun-

dation, Pauling abandoned the analysis of inorganic compounds and entered the ﬁeld of

biology and medicine—a major phase in his life’s work that was ultimately to lead to the

‘α-helix’ model for the structure of proteins. Pauling’s initial work was on haemoglobin

and led, from 1936, to his long collaboration with Robert Corey.

During the war years 1940–1945, Pauling virtually abandoned scientiﬁc work and

became involvedwith political and socialissues which initially stemmedfrom his outrage

at the treatment of Japanese-American citizens and the viliﬁcation of his doctor—an

outspoken communist whose treatment in 1941 for a serious illness certainly saved

Pauling’s life. After the war, Pauling became a strong supporter of the Peace Movement,

an advocate of rapprochement with the Soviet Union and an opposer of nuclear weapons,

activities which during the McCarthy era stigmatized him as a security risk and which

led to the withdrawal of his passport in 1952.

The story of Pauling’s discovery of the α-helix protein structure is a curious one.

He recalls that during a visit to Oxford as a visiting Professor in 1948 he was laid up

in bed with a bad cold. Sick of reading detective stories he drew pictures of amino-

acid molecules on pieces of paper, folding the papers in such a way as to achieve the

correct bond angles and then ﬁtting them together to form a helix—a helix with the

unexpected property that the successive turns of the helix did not correspond with an

integral number of amino acid residues. On his return to Caltech he set a senior fellow,

Herman Branson, with the problem of ﬁnding possible helical structures and Branson,

largely independently, came up with two solutions—the α and γ helices. The discovery

of the α-helix was formally announced at a congress in Stockholm in 1951. It did not

go down well at ﬁrst with the English scientists—Astbury, Perutz and Bragg—because

it contravened their innate feeling that the turns of the helix should correspond with an

integral number of amino acid residues.

In 1952, W. T. Astbury organized a Discussion Meeting on the Structure of Proteins

at the Royal Society, to which of course Pauling was invited as a principal speaker. But

without a passport, he was unable to attend and a colleague, Edward Hughes, had to

deputize on his behalf.

The great opus of Pauling’s scientiﬁc work was recognized with the award of his ﬁrst

Nobel Prize for Chemistry in 1954. His second Nobel Prize, for Peace, was announced

on 10th October 1963, the same day as the announcement of the (partial) Test Ban Treaty

for which Pauling had campaigned so hard.

William Jackson Pope 1870–1939

Pope’s major scientiﬁc work was in the ﬁeld of organic chemistry. He was educated at

Finsbury Technical College and the City and Guilds of London Institute in Kensington;

in 1908 he was appointed Professor of Chemistry at Cambridge at the early age of 38.

His crystallographic work extends over the period 1906–10 when, in collaboration with

W. Barlow, he developed models of crystal structures based on the close-packing of ions

or atoms of different sizes—models which were to prove so valuable to the Braggs in

their analyses of crystal structures.

Appendix 3 375

Jean Baptiste Louis Rome de L’Isle 1736–90

Rome de L’Isle, the son of a cavalry ofﬁcer, was born in Gray in France. In 1756 he

joined the Royal Corps of Artillery and Engineering and travelled to Pondicherry in the

French East Indies. When Pondicherry was seized by the English in 1761 he was taken

prisoner and transported to China where he remained until 1764 when he returned to

France. He ﬁrst studied mineralogy and in 1767 he was asked to prepare a catalogue of a

collection of mineral ‘curiosities’. In doing so he followed the method of Linnaeus (the

great Swedish botanist) in emphasising the importance of crystalline form in minera-

logical description. In 1772 he published Essai de cristallographie and in 1783 his major

work Cristallographie. Here he identiﬁed 450 ‘crystalline forms’ and characterized the

cube, regular octahedron, parallelipiped, rhomboidal octahedron and dodecahedron. In

the course of measuring the angles between the faces of terracotta (clay) crystal models

using a contact goniometer, he recognized the constancy of interfacial angles.

Rome de L’ Isles’great contribution, through his experimental work, was to establish

crystallography as the ﬁrm basis for the study of minerals although he did not follow

(and was critical of) Hauy’s notion of ‘molecules integrantes’.

Wilhelm Conrad Röntgen 1845–1923

Röntgen was born in Lennep, a small town near Remscheid in the Rhine province of

Germany. His father was a prosperous cloth merchant and both parents came from estab-

lished Lutheran Rhineland families. When Röntgen was aged three, the family moved to

Appeldoorn in Holland. During his boyhood Röntgen showed no special aptitudes and

his education was interrupted as a result of being expelled from school for refusing to

name a fellow pupil who had been involved in some misdemeanour. He entered Utrecht

Technical School (rather than University) in 1862 and then went on to study mechanical

engineering at Zurich Polytechnic, gaining the PhD degree in 1869.

By this time Röntgen’s academic ability, and moreover his outstanding practical

experiment skills, were evident. In 1871 he became assistant to Professor August Kundt

in the University of Würtzburg and moved with him to Strasbourg before being himself

appointed in 1879 to the Chair of Physics at the University of Geissen—a post he

held until 1888 despite the offer of Chairs at the prestigious Universities of Jena and

Utrecht. However, in 1889 he returned to Würtzburg as Professor of Physics: perhaps the

atmosphere of that old Bavarian town suited his own reticent personality—for Röntgen

shunned all public engagements. He worked almost entirely alone in the laboratory such

that there should be no disturbance to his concentration and developed acute powers of

observation which undoubtedly led to his discovery of X-rays on 8th November 1895.

Röntgen was experimenting with discharge (cathode ray) tubes and the action of the

rays in generating ﬂuorescence in crystals and luminescence in gases—hardly a ‘hot

topic’ since such tubes had been developed by William Crookes some twenty years ear-

lier. Röntgen had enclosed the tube in a light-tight cardboard box and noticed that every

time the induction coil discharged through the tube, a crystal of barium platinocyanide,

lying on a nearby table, gave a ﬂash of ﬂuorescence. Röntgen, ever cautious, did not

376 Appendix 3

immediately announce his discovery, but in a feverish bout of work found that the rays

(which, not knowing their nature, he called X-rays), originated from where the cathode

rays fell on the glass walls of the tube; that they were neither refracted, reﬂected nor

diffracted (from optical gratings), were more heavily absorbed by denser elements and

blackened photographic plates. On 22nd December he took the famous radiograph of

his wife’s hand and on 28th December communicated his observations to the Physical

and Medical Society of Würtzburg which he followed up by sending friends and col-

leagues reprints of his paper as New Year gifts. On 5th January 1896 the news of these

amazing rays which could reveal the living skeleton broke around the World and on

15th January, Röntgen received a Royal Summons from the young Kaiser Wilhelm II to

demonstrate his discovery at the Royal Palace in Berlin (a summons which, of course,

he could not refuse), following which he was immediately awarded the Prussian Order

of the Crown, Second Class. Further honours followed. In 1896, jointly with Lenard,

he was awarded the Rumford Medal of the Royal Society and in 1901 the ﬁrst Nobel

Prize for Physics; at the presentation ceremony Röntgen declined to give the expected

lecture.

Röntgen’s discovery is an example of serendipity in science as well as a testament to

his observational and experimental skills. For X-rays were being generated long before

1895. As early as 1879 Crookes complained of fogged photographic plates and Good-

speed and Jennings in Philadelphia in 1890 noted a ‘peculiar’blackening of photographic

plates after a demonstration with a Crookes discharge tube. Finally, we should record that

Röntgen’s other great achievement was to detect and measure the (very weak) magnetic

effects, predicted by Maxwell’s theory, when a dielectric is moved between charged

condenser plates.

Alexei Vasilievich Shubnikov 1887–1970

Shubnikov’s childhood in Moscow was one of great poverty, a family of six children

being raised by his mother following the early death of his father. In 1906 Shubnikov

attended a ‘popular’ course in crystallography given by G.V. Wulff and following his

enrolment in 1908 at the State University, became Wulff’s devoted pupil and assistant.

At this time the seeds of the Russian Revolution were already being sown: in 1911 Wulff,

Shubnikov and many of the academic staff resigned in protest against the reactionary

policy of theMinister(of Education) LevA. Kasso and transferred to theShaniavskiiPeo-

ple’s University from which Shubnikov graduated in 1912. He was seriously wounded

in 1914 and spent the remaining war years on non-combat duty as a chemist, working

during his free time on geometrical crystallography and corresponding with Fedorov. In

1918 he returned to the People’s University as Wulff’s assistant but the political circum-

stances of the time—the isolation from Western Europe and the almost complete lack of

equipment—made it impossible to do any serious research work. However, by 1927 the

situation had eased and Shubnikov was ‘sent’ to Germany and Norway to gain ﬁrst hand

experience of X-ray analysis and where he met many of the leading ﬁgures of German

science. He recalls asking Laue to show him the laboratory where he (Laue) investi-

gated crystal structures and being amazed to ﬁnd that Laue did not have a laboratory

at all!

Appendix 3 377

In 1925 Wulff died and Shubnikov took up a position in the Mineralogical Museum of

the SovietAcademy of Science in Leningrad where he set up an X-ray diffraction facility

and also studied piezoelectricity in quartz, work which entailed the full use of his great

experimental skills. This work was severely interrupted in 1934 when the government

transferred the Academy to Moscow and again in 1941 when the crystallography labo-

ratory was evacuated to the Urals. It was during this time that Shubnikov developed the

idea of ‘antisymmetry’ (surely an unsatisfactory and potentially misleading word) which

rather means the uniﬁcation of opposite concepts – plus and minus, black and white,

etc., initially an abstract ﬁeld of crystallography which has later found wide application

in the study of magnetic and other properties of crystals.

John William Strutt, third Baron Rayleigh 1842–1919

Like his near contemporary the geologist and microscopist Henry Clifton Sorby, Lord

Rayleigh (as he became when he succeeded to his father’s title in 1873) had the leisure

and ﬁnancial resources to devote his life to science. His early life was dogged by ill

health; in 1865 he graduated in mathematics at Cambridge and remained there until

1872 when he went to Egypt for a period of convalescence. It was not, however, a period

of idleness; he wrote his classic book The Theory of Sound on a houseboat on the Nile.

On his return to England he built his own private laboratory and carried out most of

his work there except for a short period (1879–1884) when he was Cavendish Professor

of Experimental Physics at Cambridge.

Rayleigh’s main contributions to science were in optics and acoustics; his work on

the scattering of light by small particles led to the explanation of the blue colour of the

sky and his work on the density of gases led to the discovery of argon.

Paul Hermann Scherrer 1890–1969

Scherrer wasbornat St. Gall, Switzerland. During his educationhisinterests moved away

from commercial topics towards natural history, ﬁrst to botany and then to mathematics

and physics. In 1913 he entered the University of Göttingen and became a member of

Debye’s research group. The stimulus for the investigation of polycrystalline specimens

appears to have come from Debye although the construction of the cylindrical camera

was due to Scherrer. With it, Debye and Scherrer took the ﬁrst X-ray powder photographs

(of lithium ﬂuoride) in 1915. A very similar method was devised in 1917 by Albert W.

Hull at the General Electric Laboratories, USA.

In later years Scherrer was active in the foundation of the European Centre for Nuclear

Research (CERN). He was also proliﬁc teacher and brought to his lectures a considerable

element of showmanship.

Arthur Schoenﬂies 1853–1928

Schoenﬂies was born in Landsberg an der Warte, now in Poland. He studied mathe-

matics at Berlin and became successively high school teacher, Associate Professor of

378 Appendix 3

Mathematics (at Göttingen), Professor (at Königsberg) and Rector at the University at

Frankfurt. He extended the work of Sohncke on periodic discrete groups by taking into

consideration rotation–reﬂection and inversion axes of symmetry, adding another 165 to

Sohncke’s sixty-ﬁve groups. The work was completed in his book Kristall-systeme und

Kristallstruktur, published in Leipsig in 1891, a few months after Fedorov’s paper.

William Bradford Shockley 1910–89

William Shockley was born in London to American parents and moved to California

(with them) in 1913. He was educated at the California Institute of Technology and

Massachusetts Institute of Technology and, after war service with the U.S. Navy’s Sub-

marine Warfare Operations Research Group, became a supervisor in the semiconductor

research group at Bell Telephone Laboratories in New York. There, in 1947, together

with John Bardeen and Walter Brattain, he discovered the principle of the point contact

transistor and then went on to develop the junction transistor. This may be said to be the

starting point of the electronic transformation of the latter part of the twentieth century,

which was recognized by the award of the Nobel prize for Physics, jointly with Bardeen

and Brattain, in 1956.

Shockley then left ‘Bell Labs’ to set up the Shockley Semiconductor Corporation in

California and from 1963 became Professor of Engineering at Stanford University.

The latter partof his life was clouded bycontroversybecause of his uncompromisingly

held views on dysgenics, i.e. retrogressive evolution through the excessive reproduc-

tion of the genetically disadvantaged, and for which, as one example of his increasing

isolation, the University of Leeds refused to conﬁrm his nomination for an Honorary

Doctorate.

Leonhard Sohncke 1842–97

Sohncke was born in Halle, Germany, where his father was Professor of Mathematics

at the University. Like his father he followed an academic career, being appointed to a

succession of professorships in physics, ﬁrstly at Halle and at the end of his career at

the Technische Hochschule in Munich. Sohncke considered the possible arrays of points

which have identical environments but not necessarily in the same orientation (as in the

deﬁnition of Bravais lattices), and arrived at sixty-ﬁve of the possible 230 space groups.

He published his ﬁndings in 1867, while he was professor of Physics at the Technische

Hochschule in Karlsruhe, and used cigar-box models by way of illustrations.

Nicolaus Steno (Niels Stensen) 1638–86

Stensen (Steno being the Latinized form of his name) was born in Copenhagen where his

father, a goldsmith, kept a shop close to the famous Round Tower. He studied medicine

at the University of Copenhagen and carried out anatomical researches in Florence,

ultimately being appointed Royal Anatomist in Denmark in 1672.

Stensen combined his medical and anatomical career with an interest in minerals,

metals, the refraction of light, telescopes and microscopes which presumably stemmed

Appendix 3 379

both from boyhood associations in his father’s shop and also from his later travels in

Italy where he became acquainted with the works of Kepler and Galileo. He published

in Florence in 1669 the Prodromum (i.e. an advance notice) to a treatise on solid bodies

naturally enclosed in other solids—a work in which he outlined the principles of geology.

This was translated into English by Henry Oldenberg, Secretary of the Royal Society,

and re-published in London in 1671. In this work Stensen shows that crystals grow by

external deposition on faces already laid down and not (as was then supposed by analogy

with plants) by the absorption of ﬂuids and the deposition of material from within. The

Law of the Constancy of Interfacial Angles, for which he is best known, is illustrated in

just one ﬁgure and stated in two sentences. In the latter part of his life Stensen’s strongly

held religious beliefs led him to renounce scientiﬁc research.

Gustav Heinrich Johann Apollon Tammann 1861–1938

Tammann can be regarded as a pioneer physical metallurgist in the light of his work in

metallography and the study of solid-state chemical reactions. He was born in Yamburg

(now Kingisepp) in Russia and, after a career as a lecturer at Charlottenburg and Leipzig,

was chosen to head the newly formed Institute at Göttingen in 1903. It was here that he

began hisstudies on metalliccompounds, thecrystalstructures and mechanicalproperties

of metals and alloys, and the general problems of the mechanisms of plastic deformation

and work hardening in terms of crystallographic slip and crystalline rearrangements. He

continued to work on these problems until shortly before his death, applying his ideas

at the same time to the ﬂow of ice.

Georgy Fedoseevich Voronoi 1868–1908

Voronoi was born in Zhuravka, Russia, into an academic family—his father was Super-

intendent (a sort of Government Inspector) of the Gymnasiums in towns in Southern

Ukraine. His mathematical ability was apparent from an early age; he attended the

Gymnasium at Priluka, graduated in mathematics in the University of St. Petersburg in

1889 and was appointed Professor of Pure Mathematics in the University of Warsaw

at the early age of 26. Voronoi’s main interests were in the Theory of Numbers, for

which he planned a series of memoirs—a series cut short by his early death. It was in

the second of these that he established the possible methods of ﬁlling n-dimensional

Euclidean space with identical non-intersecting convex polyhedra (Voronoi polyhedra

or parallelipipeds), all of which possessed contiguous boundaries—an analysis which

extended the work of his near contemporary, Fedorov.

Wilhelm Eduard Weber 1804–91

Weber was Professor of Physics at Göttingen from 1831 for most of his professional life

and worked in collaboration with Gauss on magnetic phenomena. His main achieve-

ments were the introduction of a logical system of units for electricity, related to the

fundamental units of mass, length and time and also the connection between electro-

magnetic phenomena and the velocity of light—a connection which was later thoroughly

380 Appendix 3

worked out by Maxwell. Weber was also one of the ‘Göttingen Seven’who, in 1837, lost

their university posts because of their opposition to the autocracy of King Ernst August

of Hanover.

Christian Samuel Weiss 1780–1856

After studying medicine at Leipzig, Weiss switched to chemistry and physics, becoming

Professor of Physics there in 1808. In 1810 he was appointed Professor of Physics at the

newly established and prestigious University of Berlin, a post which he held until his

death. Weiss developed the idea of crystal axes, from which he was able to distinguish

crystal systems by the ways in which crystal faces were related to such axes. He also

formulated the concept of a zone: originally conceived as a direction of prominent crystal

growth, the term was deﬁned as a collection of crystal faces parallel to a line—the zone

axis; from this concept the zone law was derived.

Georg (Yuri Viktorovich) Wulff 1863–1925

GeorgWulff (the name he used in his German-language publications) was born in Nezhin

in the Ukraine. He published his ﬁrst papers (on the morphology and physical properties

of crystals) in 1883 and 1884 whilst still a student at Warsaw University. In 1907 he was

appointed Professor of Crystallography at Moscow University, a post which he held until

his death. He proposed what has become known as the Wulff net (the stereographic pro-

jection of a sphere orientated with the polar ‘north-south’ axis in the plane of projection

rather than perpendicular to it) in 1909.

Wulff’s contribution to X-ray crystallography was his demonstration of the geometri-

cal equivalence of Laue’s diffraction and W. L. Bragg’s reﬂection interpretation of X-ray

diffraction. The equation which he derived in 1913

is equivalent to that which W. L.

Bragg presented in the previous year;

in both cases the equation is expressed in the

form nλ = 2d cos  where  = (90 −θ ). On this basis some writers link Wulff’s name

with Bragg’s—the Bragg-Wulff equation. However, the present form of the equation

was ﬁrst given in a later joint paper by W. H. and W. L. Bragg

and it was of course the

Braggs who made the ﬁrst great leap forward in crystal structure analysis.

Thomas Young 1773–1829

Young was an infant prodigy who could read at the age of 2 and at the age of 14 knew

Latin, Greek, French and Italian, and had begun to study several ancient languages—

Hebrew, Chaldean, Syriac and the like—linguistic knowledge that he put to use in his

contribution to the translation of the text of the Rosetta stone. Young intended to take up

a medical career and commenced his education in 1793 at St. Bartholomew’s Hospital,

moving from there to Edinburgh, Göttingen and Cambridge. He was elected a Fellow of

Phys. Zeitschrift (15 Mar. 1913) Vol. 14, p. 217.

Proc. Cam. Phil. Soc. (11 Nov. 1912) Vol. 17, pp. 43–57.

Proc. Roy. Soc. A (17 April 1913) Vol. 88, pp. 428–438.

Appendix 3 381

the Royal Society as early as 1794 for his explanation on how the ciliary muscles of the

eye change the shape (and focal length) of the lens.

Young’s great contribution to science was his revival in 1800–04 of Huygens’ wave

theory of light, to which he added the concept of interference to explain rectilinear

propagation and the diffraction phenomena at single or double slits or pinholes and in

which he was closely shadowed by the work of Fresnel in France. In England Young

was strongly attacked because he questioned the hegemony of Newton, but his work

quickly won acceptance in the USA and Europe. Young was also ﬁrst to suggest (in a

letter to François Arago in 1816) that light might be propagated as a transverse wave,

thus accounting for polarization and meeting the main objection of Newton. Regrettably,

Young did not live to witness Foucault’s demonstration, in 1850, that the speed of light

was less in water than in air—the crucial experiment which distinguished between the

corpuscular and wave theories.

Young was also active in many other ﬁelds. His name comes down to us through his

work on the elastic behaviour of solids as the constant of proportionality (E) in Hooke’s

law, called Young’s (or latterly Young) modulus.

Appendix 4

Some useful crystallographic

relationships

In Sections A4.l–A4.3 the rather lengthy expressions for the triclinic system are

omitted.

A4.1 Interplanar spacings, d

hul

(see Section 6.5.4)

Orthorhombic:

hkl

(Tetragonal: a = b; cubic: a = b = c.)

Hexagonal:

hkl



+ hk + k



Rhombohedral:

hkl

+ k

+ l

) sin

α + 2(hk + kl +hl)(cos

α − cos α)

(1 − 3 cos

α + 2 cos

α)

Monoclinic:

hkl

sin



sin

−

2hl cos β



A4.2 Interplanar angles, ρ, between planes (h

)of

spacing d

and (h

) of spacing d

(see Section 6.5.5)

Orthorhombic:

cos ρ =

) + (k

) + (l

)



[(h

) + (k

) + (l

)][(h

) + (k

) + (l

)]

(Tetragonal: a = b; cubic: a = b = c.)

Appendix 4 383

Hexagonal:

cos ρ =

+ k

+ h

) + (3a

/4c



[(h

+ k

+ h

+ (3a

/4c

][h

+ k

+ h

+ (3a

/4c

]

Rhombohedral:

cosρ = (a

)[sin

α(h

+ k

+ l

)

+ (cos

α − cos α)(k

+ k

+ l

+ h

)].

Monoclinic:

cos ρ =

sin

−

+ l

) cos β

A4.3 Volumes of unit cells (see Section 6.5.6)

Orthorhombic:

V = abc.

(Tetragonal: α = b; cubic: α = b = c.)

Hexagonal:

V = (

√

3/2)a

c = 0.866a

Rhombohedral:

V = a



(1 − 3 cos

α + 2 cos

α).

Monoclinic:

V = abc sin β.

A4.4 Angles between planes in cubic crystals (variants

between 100 and 221) (see Section 6.5.5)

100 100 0.00 90.00

110 45.00 90.00

111 54.74

210 26.56 63.43 90.00

211 35.26 65.90

221 48.19 70.53

110 110 0.00 60.00 90.00

111 35.26 90.00

210 18.43 50.77 71.56

211 30.00 54.74 73.22 90.00

221 19.47 45.00 76.37 90.00