Lawrence Bragg

How to use this course

In 1912 a German group showed that crystals diffract X-rays, but the photographs were hard to interpret. Within months Lawrence Bragg, a 22-year-old research student at Cambridge, found the simple picture: X-rays are reflected by planes of atoms, and the reflections reinforce at angles given by \(n\lambda=2d\sin\theta\). With his father, William Henry Bragg, he then solved the first crystal structures. Over the next fifty years he led the field from simple salts to minerals, metals and finally proteins.

This course follows that path. The technical lessons follow the order of his work: the law and the first structures (1912–1914), the measurement of intensities and Fourier methods (1920s), minerals and alloys at Manchester (1920s–1930s), models of metals and the protein programme at Cambridge (1940s–1950s).

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lengths are in ångströms (1 Å = \(10^{-10}\) m), the unit crystallographers have used since Bragg’s time.

Timeline

Adelaide, 1890–1909

William Lawrence Bragg was born in Adelaide on 31 March 1890. His father, William Henry Bragg, had come from England in 1886 to be Elder Professor of Mathematics and Physics at the University of Adelaide. His mother, Gwendoline Todd, was the daughter of Charles Todd, South Australia’s government astronomer and the builder of the overland telegraph line across Australia. He had a younger brother, Robert, and later a sister.

The boy was shy and bookish, with a naturalist’s eye. He collected shells on the beaches of South Australia, eventually about 500 species, and one new cuttlefish was named Sepia braggi after him. He attended Queen’s School in North Adelaide and then St Peter’s College. He was well ahead of his age group in mathematics, and he entered the University of Adelaide at fifteen.

His father’s research career began late. In his forties W. H. Bragg turned to the new science of radioactivity and gained an international reputation for his study of the ranges of alpha particles. In 1908 Lawrence graduated with first-class honours in mathematics, and in 1909 his father accepted the Cavendish chair of physics at Leeds. The family sailed for England.

Cambridge and the autumn of 1912

Bragg entered Trinity College, Cambridge, in the autumn of 1909 with a scholarship in mathematics. He changed to physics and in 1912 took first-class honours in the Natural Sciences Tripos. He then began research at the Cavendish Laboratory under J. J. Thomson.

That summer, news reached the Braggs of the Munich photographs of Walter Friedrich, Paul Knipping and Max von Laue, showing spots produced by X-rays passing through a crystal of zinc blende. W. H. Bragg held that X-rays were particle-like, a view based on how they ionize gases, and father and son first discussed whether the spots could be produced by particles channelled along open avenues in the crystal. Lawrence soon convinced himself that the spots were a wave effect, but that Laue’s analysis was wrong in its details.

Laue had assumed a simple cubic lattice and had needed several discrete wavelengths in the beam to match the spots. Bragg, thinking about the problem on walks by the river in Cambridge, saw two things. First, the diffraction could be treated as reflection of a continuous spectrum of X-rays by planes of atoms. Each family of planes picks out the wavelengths that satisfy a simple condition. Second, the spots that appeared and those that were missing matched a face-centred cubic arrangement. He later credited C. T. R. Wilson’s lectures, which treated X-rays as short pulses, with suggesting that a pulse would be reflected by atomic planes as by a mirror.

On 11 November 1912 Thomson presented Bragg’s paper, The diffraction of short electromagnetic waves by a crystal, to the Cambridge Philosophical Society. In December Bragg tested the reflection idea directly. He sent X-rays onto a sheet of mica at a glancing angle and recorded a reflected spot on a photographic plate, which moved with the mica as the reflection law requires. The result appeared in Nature that month.

The spectrometer, the prize and the war, 1913–1919

At Leeds, W. H. Bragg built an X-ray spectrometer. A crystal on a turntable reflected a narrow X-ray beam into an ionization chamber, which measured the intensity at each angle. It showed at once that the X-rays from a tube contained sharp characteristic lines on top of the continuous spectrum. With the spectrometer the Braggs could measure angles precisely and compare intensities, which the photographic method could not do well. The division of labour between them followed their strengths: the father built and measured, the son interpreted.

In 1913 Lawrence Bragg determined the structures of sodium chloride, potassium chloride and related salts, and with his father that of diamond (Lesson 05). In 1914–15 the work extended to fluorite, pyrite, calcite and other minerals. The two wrote X Rays and Crystal Structure (1915). In public the work was often credited to the father, and the son felt this keenly for years.

When the war began in 1914 Bragg was commissioned in the Royal Horse Artillery. In 1915 he was sent to France as technical adviser on sound ranging, the location of enemy guns from the arrival times of their reports at a line of microphones. The low-frequency boom of heavy guns was hard to detect, and his section eventually built a hot-wire microphone that responded to the pressure wave. The method became a standard tool of the British artillery. On 2 September 1915 his brother Robert was killed at Gallipoli. That November, while serving in France, he learned that he and his father had been awarded the Nobel Prize in Physics for 1915, for their services in the analysis of crystal structure by means of X-rays. He was 25, and remains the youngest laureate in physics. He received the Military Cross and the OBE in 1918.

Bragg’s law and the reflection picture

For the general reader. Imagine a crystal as a stack of transparent mirrors, each reflecting a tiny fraction of an X-ray beam. Reflections from successive mirrors add up only when the extra distance travelled by each is a whole number of wavelengths. That single condition tells you, from the angle of a reflected beam, either the spacing of the mirrors or the wavelength of the X-rays.

Derivation

Consider two adjacent planes. A ray reflected from the lower plane travels farther than one reflected from the upper plane by the two segments marked in the hero figure, each of length \(d\sin\theta\). The total path difference is \(2d\sin\theta\). Within one plane, all atoms scatter in phase in the mirror direction, because the incoming and outgoing paths to any two atoms of the plane differ by equal and opposite amounts. So the whole stack reflects in phase exactly when \(2d\sin\theta\) is a multiple of \(\lambda\). Since each plane reflects only about \(10^{-3}\) to \(10^{-5}\) of the amplitude, thousands of planes contribute, and the condition is very sharp.

It is often convenient to absorb \(n\) into the spacing. The \(n\)-th order reflection from planes \((hkl)\) is the first-order reflection from the fictitious planes \((nh\,nk\,nl)\) of spacing \(d/n\). Then the law reads \(\lambda=2d_{hkl}\sin\theta\) with \(h,k,l\) allowed common factors. In vector form, with \(\mathbf k_0\) and \(\mathbf k\) the incident and reflected wavevectors and \(\mathbf G_{hkl}\) the reciprocal lattice vector normal to the planes with \(|\mathbf G_{hkl}|=2\pi/d_{hkl}\), the law is Laue’s condition \(\mathbf k-\mathbf k_0=\mathbf G_{hkl}\).

The zinc blende photographs

In a Laue photograph the crystal is fixed and the beam contains all wavelengths between some \(\lambda_{\min}\) and \(\lambda_{\max}\). Each family of planes has a fixed \(\theta\), so it reflects the wavelengths \(\lambda=2d\sin\theta/n\) that lie in the band. A plane appears as a spot whenever at least one of these falls in the band and the planes reflect strongly enough. Bragg showed that the observed spots of zinc blende correspond to planes of the face-centred cubic lattice of the zinc atoms: for a face-centred cube, the planes whose indices are all odd or all even. In a simple cubic lattice, many planes with mixed indices would have given spots that were not there.

Rock salt, diamond and the first structures

For the general reader. Before 1913 nobody knew how atoms were arranged in any solid. Bragg’s first structure, common salt, showed that there are no molecules of NaCl in the crystal: each sodium sits among six chlorines and each chlorine among six sodiums. Months later, diamond showed carbon bonded to four neighbours at the corners of a tetrahedron, as chemists had guessed for molecules.

Rock salt and sylvine

Bragg compared photographs and spectrometer data for KCl, NaCl, KBr and KI. The KCl pattern looked like that of a simple cubic lattice with one atom per cell. NaCl showed extra reflections, at positions for which a simple cubic cell of half the edge would give none. He explained both with one structure: two interpenetrating fcc lattices of the two ions, displaced by half a cube edge. In KCl the two ions have the same number of electrons and scatter almost equally, so the pattern is that of a simple cubic lattice of edge \(a/2\). In NaCl, sodium has 10 electrons as an ion and chlorine 18, so the reflections with all-odd indices, which depend on the difference, survive.

Absolute wavelengths

Once the structure is known, the edge of the cube follows from the density \(\rho\), molar mass \(M\) and Avogadro’s number, since the cell contains four formula units: \(a^3=4M/(\rho N_A)\). Then \(\lambda=2d\sin\theta\) gives absolute X-ray wavelengths. This is how the Braggs measured the characteristic lines of X-ray tubes, and how Henry Moseley, using a crystal spectrometer in 1913–14, measured the lines of the elements and found that their frequencies follow the atomic number.

Diamond

In diamond the carbon atoms occupy an fcc lattice and a second fcc lattice displaced by \((\tfrac14,\tfrac14,\tfrac14)\). Each atom has four neighbours at the corners of a regular tetrahedron, at distance \(a\sqrt3/4\). The Braggs found that the (222) reflection, which an fcc lattice of single atoms would give strongly, is almost absent.

Intensities and Fourier series

For the general reader. The angles of reflected beams tell you the shape of the repeating box. The brightness of each beam tells you what is inside the box. At Manchester Bragg and his colleagues learned to measure brightness on an absolute scale, and then to turn the measured numbers into a map of the electrons by summing waves, exactly as a musical tone is built from harmonics.

Absolute intensities

Charles Galton Darwin showed in 1914 that the integrated reflection from an ideally imperfect (mosaic) crystal is proportional to \(|F|^2\), and that a perfect crystal reflects far less strongly in proportion because of extinction. In 1921–22 Bragg, Reginald James and C. H. Bosanquet measured the absolute integrated reflection of rock salt and confirmed Darwin’s formula. For a small crystal of volume \(\delta V\), rotated through the reflecting position at angular velocity \(\omega\), the integrated energy reflected is \(E\omega/I_0=Q\,\delta V\) with

where \(N_c\) is the number of unit cells per unit volume, \(r_e=2.82\times10^{-15}\) m, the fraction is the polarization factor for an unpolarized beam, and \(e^{-2M}\) is the Debye–Waller factor for thermal motion. Measured intensities thus give \(|F|\) on an absolute scale, the number of electrons that scatter.

Fourier synthesis

W. H. Bragg suggested in 1915 that the electron density of a crystal, being periodic, should be written as a Fourier series whose coefficients are fixed by the reflections. With fractional coordinates \(\mathbf x=(x,y,z)\) and \(\mathbf h=(h,k,l)\),

In 1929 Lawrence Bragg applied the method to diopside, \(\mathrm{CaMgSi_2O_6}\), whose structure Bertram Warren and he had found the year before. He used a two-dimensional series, a projection of the density onto one face of the cell, with signs taken from the approximate structure. The map showed every atom as a peak and refined their positions. From then on, Fourier synthesis became the standard last step of a structure analysis.

The Patterson function

In 1934 Arthur Lindo Patterson showed that the Fourier series with coefficients \(|F|^2\), which need no phases, gives

Its peaks sit at the vectors between pairs of atoms, with heights proportional to the products of their electron numbers. Heavy atoms stand out, which made the Patterson function the starting point of the protein work of Lesson 10.

Atomic sizes and the silicates

For the general reader. In 1920 Bragg noticed that atoms in crystals behave roughly like hard spheres of definite sizes. That observation turned crystal chemistry into a game of packing. The silicates, which make up most of the Earth’s crust, had baffled chemists with their formulas. Bragg’s school showed they are built from one unit, a silicon atom inside four oxygens, linked in chains, sheets and frameworks.

Atomic radii

In 1920 Bragg compared interatomic distances in many crystals and found that they are close to sums of radii assigned to each element. His radii treated atoms as neutral; Victor Goldschmidt and Linus Pauling later produced tables of ionic radii, in which large anions such as O²⁻ (about 1.40 Å) dominate the volume and small cations fill the gaps. In this picture many oxides and silicates are close-packed arrays of oxygen with metal ions in the holes.

The silicates

For Si⁴⁺ and O²⁻ the ratio is about 0.29, so silicon takes four oxygens at the corners of a tetrahedron. Bragg and his students, among them W. H. Taylor, J. West, B. E. Warren and W. H. Zachariasen, solved one silicate after another between 1926 and 1930, beginning with beryl (Bragg and West, 1926). The structures fall into classes according to how many oxygens each \(\mathrm{SiO_4}\) tetrahedron shares with its neighbours: none (olivine), two (pyroxene chains), three (mica sheets) or four (quartz, feldspar frameworks). Aluminium can replace silicon in a tetrahedron, with a charge compensated by an extra cation. Bragg summarized the work in Atomic Structure of Minerals (1937).

Order and disorder in alloys

For the general reader. In some alloys, such as brass with equal numbers of copper and zinc atoms, the two kinds of atom take alternating sites at low temperature, like a chessboard. Heating mixes them up. In 1934 Bragg and Evan James Williams wrote a simple theory of this process. It predicts that order vanishes suddenly at a critical temperature, and it became the prototype of the mean-field theories used everywhere in physics.

The model

Take an AB alloy on a lattice with two equal sublattices, \(\alpha\) and \(\beta\), such as β-brass on a body-centred cubic lattice. Let \(r\) be the fraction of \(\alpha\) sites occupied by A atoms. The long-range order parameter is \(S=2r-1\): \(S=1\) for perfect order, \(S=0\) for a random alloy. Bragg and Williams assumed that the energy needed to move an A atom from a right site to a wrong one, with a B atom moving the other way, is proportional to the existing order, \(V=V_0S\). The order is reinforced by itself: this is the mean-field approximation.

Derivation. On the \(\alpha\) sublattice the fraction of wrong atoms is \(1-r\); on \(\beta\) it is also \(1-r\). An exchange that creates a pair of wrong atoms requires a right A on \(\alpha\) and a right B on \(\beta\), probability \(\propto r^2\), and costs \(V\). The reverse requires two wrong atoms, probability \(\propto(1-r)^2\). Detailed balance gives the equation above. With \(r=(1+S)/2\), \((1-S)/(1+S)=e^{-V_0S/2kT}\), which is \(S=\tanh(V_0S/4kT)\). The slope of the right side at \(S=0\) is \(V_0/4kT\); a non-zero crossing needs slope above 1.

The same equation describes a ferromagnet in Pierre Weiss’s molecular-field theory and the Ising model in the mean-field approximation, with \(S\) playing the role of magnetization. Mean-field theory gets the existence of the transition right and the critical exponents wrong (\(S\propto(T_c-T)^{1/2}\), whereas the three-dimensional Ising value is about 0.33). Hans Bethe improved the treatment in 1935 by including short-range order.

Seeing order with X-rays

Ordering makes the two sublattices different, so reflections that were forbidden by the body-centring become allowed. These superlattice lines have intensity proportional to \(S^2\).

Bubble rafts and dislocations

For the general reader. Metals bend far more easily than a perfect crystal should. The reason is defects called dislocations, rows where the atomic planes do not match. They were proposed in 1934 but nobody could watch one move. In the 1940s Bragg blew thousands of identical soap bubbles onto a liquid surface. They packed like atoms, and the raft showed dislocations, grain boundaries and vacancies moving under stress.

Why metals are soft

To slide one plane of atoms over another in a perfect crystal, every bond across the plane must be stretched at once. Yakov Frenkel estimated the stress for this in 1926.

For copper, with \(G\approx45\) GPa, this is about 7 GPa, while pure single crystals yield at around 1 MPa, thousands of times less. In 1934 Egon Orowan, Michael Polanyi and G. I. Taylor independently explained the gap: slip proceeds by the motion of dislocations, which break only a line of bonds at a time.

The bubble raft

Bragg first tried bubble models with George Crowe in 1940–42. In 1947 Bragg and John Nye described the technique in detail. Bubbles of a millimetre or so in diameter, all the same size, were blown at constant pressure from a fine nozzle under the surface of a soap solution. They rose and packed into a two-dimensional hexagonal raft held together by surface tension. The raft showed grain boundaries, vacancies and dislocations. Compressing or shearing the raft made the dislocations glide. The forces between neighbouring bubbles resemble interatomic forces in a metal, repulsive at short range and attractive beyond, which is why the raft behaves like a crystal.

Proteins: helices and phases

For the general reader. A protein crystal contains thousands of atoms per molecule, and its X-ray pattern has tens of thousands of spots. In 1947 Bragg put his authority behind a long gamble on solving such structures at Cambridge. The first attempt to guess the shape of the protein chain went wrong, and Linus Pauling got there first. The second idea, attaching heavy atoms to the protein to recover the missing phases, worked, and it gave the first protein structures.

Chain models, 1950–51

Max Perutz had begun X-ray work on haemoglobin in 1937, and Bragg supported him from 1938. In 1950 Bragg, John Kendrew and Perutz published a survey of possible helical configurations of the polypeptide chain, each held by hydrogen bonds between the C=O and N–H groups along the chain. They assumed an integral number of residues per turn and did not insist that the peptide group be planar. None of their models was right. In 1951 Pauling, Robert Corey and Herman Branson proposed the α-helix, with a planar peptide group and a non-integral 3.6 residues per turn. Perutz quickly saw that the α-helix predicts a reflection at 1.5 Å spacing, from the rise per residue along the axis. He found it in horse hair, in synthetic polypeptides and in haemoglobin within days.

Isomorphous replacement

In 1953 Perutz showed that mercury atoms could be attached to specific sites of haemoglobin without changing the crystal structure. Then for each reflection

where \(\mathbf F_P\) and \(\mathbf F_{PH}\) are the complex structure factors of the native and derivative crystals and \(\mathbf F_H\) that of the heavy atoms alone. The heavy atoms can be located from a difference Patterson map, so \(\mathbf F_H\) is known. The measured magnitudes \(|F_P|\) and \(|F_{PH}|\) then fix the phase of \(\mathbf F_P\) up to a twofold ambiguity, which a second derivative resolves. With this method Kendrew solved myoglobin (low resolution 1957, high resolution 1959) and Perutz haemoglobin. They shared the Nobel Prize in Chemistry for 1962. At the Royal Institution, David Phillips’s group, which Bragg had built, solved the first enzyme, lysozyme, in 1965.

Manchester and the Cavendish, 1919–1953

In 1919, aged 29, Bragg succeeded Ernest Rutherford as Langworthy Professor of Physics at Manchester. He found large classes of returning soldiers and a department to rebuild. He recruited former members of his sound-ranging section, among them Reginald James, and built the leading school of crystal-structure analysis of the interwar years. Collaborators and visitors included Douglas Hartree, A. J. Bradley, W. H. Taylor, Bertram Warren and Willie Zachariasen. In 1921 he married Alice Hopkinson, and was elected a Fellow of the Royal Society; they had two sons and two daughters. He was also an early supporter of Alfred Wegener’s continental drift, and is credited with arranging the English translation of Wegener’s book in 1922.

Bragg was a careful, generous director but sometimes doubted himself as a lecturer and administrator. In 1937 he became director of the National Physical Laboratory. After Rutherford’s death the same year, he was chosen as Cavendish Professor of Experimental Physics at Cambridge and took up the post in 1938. He was knighted in 1941. During the Second World War he advised on sonar and sound ranging and served on scientific committees, while the Cavendish turned to war work.

After 1945 he reorganized the laboratory. Nuclear physics, Rutherford’s domain, required large machines that Cambridge could not easily afford, and Bragg favoured smaller groups: radio astronomy under Martin Ryle, low-temperature physics, metal physics, and crystallography. In 1947 he persuaded the Medical Research Council to fund Perutz and Kendrew as a unit for the study of the molecular structure of biological systems, the seed of the Laboratory of Molecular Biology. In 1951 Francis Crick and James Watson joined it. Bragg tried to stop their unofficial work on DNA after an early failed model, and later encouraged it once they had the right structure. On 8 April 1953 he announced their double helix at the Solvay conference in Brussels, and he wrote the foreword to Watson’s The Double Helix (1968).

The Royal Institution and legacy, 1954–1971

In 1954 Bragg moved to the Royal Institution in London as Fullerian Professor of Chemistry and director of its Davy–Faraday Research Laboratory, the posts his father had held from 1923 until his death in 1942. He set out to restore the institution’s research and its public role. In the laboratory he built a protein group under David Phillips, which collaborated with Perutz and Kendrew and solved the structure of lysozyme in 1965. For the public he started lectures for schoolchildren, attended by tens of thousands each year, and gave television broadcasts. He had already given the Royal Institution Christmas Lectures in 1934 and gave them again in 1961.

In December 1965 he gave the first Nobel Guest Lecture in Stockholm, fifty years after his prize. He retired in 1966. His later honours included the Copley Medal (1966) and appointment as a Companion of Honour (1967). He died on 1 July 1971 in hospital at Ipswich, near his home at Waldringfield in Suffolk, aged 81.

Legacy

• Method: Bragg’s law is used every day in X-ray, electron and neutron diffraction, in materials science, mineralogy, chemistry and biology. Every one of the hundreds of thousands of structures in today’s databases rests on it.
• Crystal chemistry: atomic radii, the silicate classification and the packing view of minerals.
• Physics of solids: the Bragg–Williams theory as the first mean-field model of an order–disorder transition, and the bubble raft as a visual model of crystal defects.
• Molecular biology: his support for Perutz, Kendrew, Phillips, and for Watson and Crick, made Cambridge and London the centres of structural biology.
• Honours: Nobel Prize in Physics (1915), Fellow of the Royal Society (1921), Hughes (1931), Royal (1946) and Copley (1966) Medals, knighthood (1941), Companion of Honour (1967). The Institute of Physics awards the Lawrence Bragg Medal and Prize for physics education, and the Australian Institute of Physics the Bragg Gold Medal.

Connections

People linked to Lawrence Bragg across this collection. A name without a link has no page here yet. See Lawrence Bragg in the network of all 337 lives →

Teachers and advisors

• J. J. Thomson 1912

Students

• Max Perutz
• William Zachariasen

Influenced by

• C. T. R. Wilson
• Max von Laue

Influenced

• Henry Moseley

Colleagues and collaborators

• Ernest Rutherford
• Henry Moseley
• Bertram Warren 1927-1929
• Charles Galton Darwin
• David Phillips 1955-1966
• Douglas Hartree
• Egon Orowan
• Evan James Williams 1934-1935
• Francis Crick
• James Watson
• John Kendrew
• John Nye 1947
• Reginald James 1919-1937

Rivals

• Linus Pauling

Family

• Charles Todd
• William Henry Bragg

Also mentioned in this life

• Amedeo Avogadro
• Hans Bethe
• Ludwig Boltzmann
• Henry Cavendish
• Humphry Davy
• Peter Debye
• Michael Faraday
• Joseph Fourier
• G. I. Taylor
• Robert Hooke

Fields

• solid-state physics 6 lives
• mineralogy 3 lives

Sources and further reading

Biography

1. Lawrence Bragg: Biographical, NobelPrize.org.
2. Lawrence Bragg, Wikipedia (with extensive references).
3. Memorial of Sir Lawrence Bragg, American Mineralogist 59 (1974) 408–411.
4. D. C. Phillips, William Lawrence Bragg, Biographical Memoirs of Fellows of the Royal Society 25 (1979).
5. G. K. Hunter, Light Is a Messenger: The Life and Science of William Lawrence Bragg, Oxford University Press, 2004.
6. J. Jenkin, William and Lawrence Bragg, Father and Son, Oxford University Press, 2008.
7. D. C. Phillips and J. M. Thomas (eds.), Selections and Reflections: The Legacy of Sir Lawrence Bragg, Science Reviews, 1990.

Physics

1. W. L. Bragg, The diffraction of short electromagnetic waves by a crystal, Proc. Cambridge Philos. Soc. 17 (1913) 43–57 (read 11 November 1912).
2. W. H. Bragg and W. L. Bragg, The reflection of X-rays by crystals, Proc. R. Soc. A 88 (1913) 428–438.
3. W. L. Bragg, The structure of some crystals as indicated by their diffraction of X-rays, Proc. R. Soc. A 89 (1913).
4. W. H. Bragg and W. L. Bragg, The structure of the diamond, Proc. R. Soc. A 89 (1913).
5. W. L. Bragg, The diffraction of X-rays by crystals, Nobel lecture, 6 September 1922.
6. W. L. Bragg, The determination of parameters in crystal structures by means of Fourier series, Proc. R. Soc. A 123 (1929).
7. W. L. Bragg and E. J. Williams, The effect of thermal agitation on atomic arrangement in alloys, Proc. R. Soc. A 145 (1934).
8. W. L. Bragg and J. F. Nye, A dynamical model of a crystal structure, Proc. R. Soc. A 190 (1947) 474–481.
9. W. L. Bragg, J. C. Kendrew and M. F. Perutz, Polypeptide chain configurations in crystalline proteins, Proc. R. Soc. A 203 (1950) 321–357.
10. L. Pauling, R. B. Corey and H. R. Branson, The structure of proteins: two hydrogen-bonded helical configurations of the polypeptide chain, PNAS 37 (1951) 205–211.
11. W. L. Bragg, Atomic Structure of Minerals, Cornell University Press, 1937.
12. Bragg centennial, Acta Crystallographica A 69 (2013): on Bragg’s work on metals at the Cavendish.