Physical crystallography before X-rays

Short description: History of physical crystallography to 1895

Physical crystallography before X-rays describes how physical crystallography developed as a science up to the discovery of X-rays by Wilhelm Conrad Röntgen in 1895. In the period before X-rays, crystallography can be divided into three broad areas: geometrical crystallography culminating in the discovery of the 230 space groups in 1891–1894, chemical crystallography and physical crystallography.

Physical crystallography is concerned with the physical properties of crystals, such as their optical, electrical, magnetic, thermal, and mechanical properties.

* The interaction between crystals and electromagnetic radiation is covered in the following sections: double refraction, rotary polarization, conical refraction, absorption and pleochroism, luminescence, reflection from opaque materials, and infrared optics.
* The effect of electricity and magnetism on crystals is covered in: electrical conduction, magnetic properties, and dielectric properties.
* The effect of temperature change on crystals is covered in: thermal expansion, thermal conduction, thermoelectricity, and pyroelectricity.
* The effect of mechanical force on crystals is covered in: elasticity, photoelasticity, and piezoelectricity.

The study of crystals in the time before X-rays was focused more on their geometry and mathematical analysis than their physical properties.[1] Unlike geometrical crystallography, the history of physical crystallography has no central story, but is a collection of developments in different areas.

## Contents

* 1 Symmetry
* 2 Interaction with electromagnetic radiation
* 2.1 Double refraction
* 2.2 Rotary polarization
* 2.3 Conical refraction
* 2.4 Absorption and pleochroism
* 2.5 Luminescence
* 2.6 Reflection from opaque materials
* 2.7 Infrared optics
* 3 Effect of electricity and magnetism
* 3.1 Electrical conduction
* 3.2 Magnetic properties
* 3.3 Dielectric properties
* 4 Effect of temperature change
* 4.1 Thermal expansion
* 4.2 Thermal conduction
* 4.3 Thermoelectricity
* 4.4 Pyroelectricity
* 5 Effect of mechanical force
* 5.1 Elasticity
* 5.2 Photoelasticity
* 5.3 Piezoelectricity
* 6 Research community
* 7 See also
* 8 Citations
* 9 Works cited

## Symmetry

During the 19th century crystallography was progressively transformed into an empirical and mathematical science by the adoption of symmetry concepts.(Scholz, 1989b) In 1832 Franz Ernst Neumann used symmetry considerations when studying double refraction.[2] Woldemar Voigt, who was a student of Neumann, in 1885 formalized Neumann's principle as "if a crystal is invariant with respect to certain symmetry operations, any of its physical properties must also be invariant with respect to the same symmetry operations".[3](Lalena, 2006) Neumann's principle is sometimes referred to as the Neumann–Minnigerode–Curie principle based on later work by Bernhard Minnigerode(Minnigerode, 1884) (another student of Neumann) and Pierre Curie.[4] Curie's principle "the symmetries of the causes are to be found in the effects" is a generalization of Neumann's principle.(Curie, 1894) At the end of the 19th century Voigt introduced tensor calculus to model the physical properties of anisotropic crystals.(Voigt, 1898)

## Interaction with electromagnetic radiation

### Double refraction

Rasmus Bartholin

Étienne-Louis Malus

Christiaan Huygens

Jean-Baptiste Biot

Researchers in double refraction

Double refraction occurs when a ray of light incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. The double refraction and rhomboidal cleavage of crystals of calcite, or Iceland spar, were first recorded in 1669 by Rasmus Bartholin(Bartholin, 1669) In 1690 Christiaan Huygens analyzed double refraction in his book Traité de la lumière.[5] Huygens reasoned that the cleavage rhombohedron resulted from the stacking of spherical particles[6] and that the peculiarities of the transmission of light can be traced to the particular asymmetry of the crystal.[7]

In 1810 Étienne-Louis Malus determined that natural light, too, when reflected through a certain angle, behaves like one of the rays exiting a double-refracting crystal.[8] Malus called this phenomenon polarization.[9] In 1812 Jean-Baptiste Biot defined optically positive and negative crystals for the first time.(Biot, 1812a) In 1819 David Brewster found that all crystals could be classified as isotropic, uniaxial or biaxial.(Brewster, 1819c) Augustin-Jean Fresnel was a significant researcher in the whole field of crystal optics, and published a detailed paper on double refraction in 1827 in which he described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization.(Fresnel, 1827) Crystal optics was an active research area during the 19th century[10] and comprehensive accounts of the field were published by Lazarus Fletcher (1891),[11] Theodor Liebisch (1891)[12] and Friedrich Pockels (1906).[13]

### Rotary polarization

François Arago

Augustin-Jean Fresnel

George Biddell Airy

Louis Pasteur

Researchers in rotary polarization

In 1811 François Arago, who favoured the corpuscular theory of light, discovered the rotation of the plane of polarization of light travelling through quartz.[14] In 1812 Jean-Baptiste Biot, who favoured the wave theory of light, enunciated the laws of rotary polarization and their application to the analysis of various substances.(Biot, 1812a) Biot discovered that while some crystals rotate the light to the right others rotate it to the left, and determined that the rotation is proportional to the thickness of substance traversed and to the wavelength of the light.[15]

In 1821 John Herschel pointed out the relation between the direction of rotation and the development of faces on quartz crystals.[16] Suspecting that rotatory polarization is an effect of a lack of symmetry, Herschel established that quartz crystals often present faces placed in such a way that those belonging to certain crystals are mirror images of the corresponding faces of other crystals. He explained the connection between this arrangement and the respective rotation of light to the right and to the left.[17] In 1822 Augustin-Jean Fresnel explained the rotation by postulating oppositely circularly polarized beams travelling with different velocities along the optic axis.[18] In 1831 George Biddell Airy gave an explanation of the formation of the spirals which bear his name.(Airy, 1831) In 1846 Michael Faraday discovered that the plane of polarization may also be rotated when light passes through an isotropic medium when it is in a magnetic field.(Faraday, 1846a) The corresponding Kerr effect can be observed on reflecting plane-polarized light from a polished ferromagnetic mirror when in a magnetized state.

In 1848 Louis Pasteur gave the general relation between crystal morphology and rotatory polarization.[19] Pasteur solved the mystery of polarized light acting differently with chemically identical crystals and solutions. Pasteur discovered the phenomenon of molecular asymmetry, that is that molecules could be chiral and exist as a pair of enantiomers. Pasteur's method was to physically separate the crystals of a racemic mixture of sodium ammonium tartrate into right- and left-handed crystals, and then dissolve them to make two separate solutions which rotated polarized light in opposite directions.({{{1}}}, {{{2}}})

In 1855 Christian August Hermann Marbach (de) discovered that crystals of sodium chlorate, sodium bromate, sodium ammonium sulfate and sodium amyl acetate have the property of rotating the polarization plane.(Marbach, 1855) In 1857 Alfred Des Cloizeaux advanced a general theory of rotatory polarization whilst studying cinnabar and strychnine sulphate.(Des Cloizeaux, 1857a) In 1864 Josef Stefan introduced the banded spectrum in the study of rotatory polarization.[20] Theories of magnetic optics in ferromagnetic crystals were published in 1892 by D. A. Goldhammer,[21] and in 1893 by Paul Drude.[22][23]

### Conical refraction

William Rowan Hamilton

Humphrey Lloyd

Researchers in conical refraction

Conical refraction is an optical phenomenon in which a ray of light, passing through a biaxial crystal along certain directions, is refracted into a hollow cone of light. There are two possible conical refractions, one internal and one external.

In 1821–1822 Augustin-Jean Fresnel developed a theory of double refraction in both uniaxial and biaxial crystals.(Fresnel, 1852) Fresnel derived the equation for the wavevector surface in 1823, and André-Marie Ampère rederived it in 1828.[24] Many others investigated the wavevector surface of the biaxial crystal, but they all missed its physical implications.

William Rowan Hamilton, in his work on Hamiltonian optics, discovered the wavevector surface has four conoidal points and four tangent conics.[25] This implies that, under certain conditions, a ray of light could be refracted into a cone of light within the crystal.[26] He termed this phenomenon "conical refraction" and predicted two distinct types: internal and external, corresponding respectively to the conoidal points and tangent conics. Hamilton announced his discovery on 22 October 1832. He then asked Humphrey Lloyd to prove his theory experimentally. Lloyd first observed conical refraction on 14 December 1832 with a specimen of aragonite, and published his results in early 1833.(Lloyd, 1833a) In 1833 James MacCullagh claimed that Hamilton's work was a special case of a theorem he had published in 1830.(MacCullagh, 1830) Hamilton also exchanged letters with George Biddell Airy who was skeptical that conical refraction could be observed experimentally but became convinced after Lloyd's report.(O'Hara, 1982)

Hamilton and Lloyd's discovery was a significant victory for the wave theory of light and solidified Fresnel's theory of double refraction.[27] The discovery of conical refraction is an example of a mathematical prediction being subsequently verified by experiment.(O'Hara, 1982)

Later theoretical work on conical refraction was published in 1860 by Robert Bellamy Clifton[28] and in 1874 by Jules Antoine Lissajous,[29] and experimental work in 1888 by Theodor Liebisch[30] and in 1889 by Albrecht Schrauf.[31][23]

### Absorption and pleochroism

David Brewster

Louis Cordier

Researchers in absorption and pleochroism

In 1809 Louis Cordier discovered the phenomenon of pleochroism while investigating a new mineral that he named dichröıte (cordierite), whereby its crystals showed different colors when viewed along different axes.[32] From 1817 to 1819 David Brewster made a systematic study of light absorption and pleochroism in various minerals and showed that, in uniaxial crystals, the absorption is smallest in the direction of, and greatest at right angles to, the optical axis.(Brewster, 1819a) In 1820 John Herschel studied the absorption of light in biaxial crystals and explained the interference rings first observed by David Brewster.[33] In 1838 Jacques Babinet discovered that the greatest absorption in a crystal generally coincided with the direction of greatest refractive index.[34] In 1845 Wilhelm Haidinger published a general account of pleochroism in crystals.[35] In 1854 Henri Hureau de Sénarmont showed that transparent crystals stained by a dye during crystal growth became pleochroic.[36][37]

In 1877 Paul Glan (de) performed photometric observations on absorption.[38] In 1880 Hugo Laspeyres (de) pointed out the existence of absorption axes (directions of least, intermediate, and greatest absorption). He investigated certain biaxial crystals and found that the absorption axes, although subject to the symmetry of the crystal, did not necessarily coincide with the principal directions of the indicatrix.(Laspeyres, 1880) In 1888 Henri Becquerel made qualitative and quantitative observations.(Becquerel, 1888a) Woldemar Voigt (1885) and Paul Drude (1890) presented theories of the absorption of light in crystals.(Voigt, 1885) In 1906 Friedrich Pockels published his Lehrbuch der Kristalloptik[13] which gave an overview of the subject.[23]

### Luminescence

Francis Bacon

Edmond Becquerel

John Herschel

William Crookes

Researchers in luminescence, fluorescence and phosphorescence

Luminescence is the non-thermal emission of visible light by a substance; an example is the emission of visible light by minerals in response to irradiation by ultraviolet light. The term luminescence was first used by Eilhard Wiedemann in 1888;[39] he stated that luminescence was separate from thermal radiation, and he distinguished six different forms of luminescence according to their excitation.(Wiedemann, 1888)

Fluorescence is luminescence which occurs during the irradiation of a substance by electromagnetic radiation; fluorescent materials stop emitting light nearly immediately after the irradiation is halted,[40] except in the case of certain materials exhibiting delayed fluorescence (e.g., TADF, TTA). The term fluorescence was coined by George Stokes in 1852, and was derived from the behavior of fluorite when exposed to ultraviolet light.(Stokes, 1852)

Phosphorescence is long-lived luminescence; phosphorescent materials continue to emit light for some time after the radiation stops. In 1857 Edmond Becquerel invented the phosphoroscope, and in a detailed study of phosphorescence and fluorescence, showed that the duration of phosphorescence varies by substance, and that phosphorescence in solids is due to the presence of finely dispersed foreign substances. Becquerel suggested that fluorescence is simply phosphorescence of a very short duration.(Becquerel, 1859) The most prominent phosphorescent material for 130 years was ZnS doped with Cu+, or later Co2+, ions. The material was discovered in 1866 by Théodore Sidot who succeeded in growing tiny ZnS crystals by a sublimation method.(Sidot, 1866)

Some additional kinds of luminescence from crystals can arise from energy sources other than electromagnetic radiation.

* Crystalloluminescence is the emission of light during crystal growth from solution, specifically during nucleation.[41] The first observation was that of potassium sulfate which was reported by a number of researchers in the eighteenth century; other substances reported in the early literature which exhibit crystalloluminescence include strontium nitrate, cobalt sulfate, potassium hydrogen sulfate, sodium sulfate, and arsenious acid.[42] In 1918 Harry Weiser summarised the research on crystalloluminescence up to that date.[43] Neither the spectral distribution nor the excitation mechanisms of crystalloluminescence are understood.[44]
* Triboluminescence is the generation of light when certain materials, for example quartz, are rubbed;[45] fractoluminescence is the emission of light from the fracture of a crystal. The first recorded observation is attributed to Francis Bacon when he recorded in his 1620 Novum Organum that sugar sparkles when broken or scraped in the dark.[46] The scientist Robert Boyle also reported on some of his work on triboluminescence in 1664.[47]
* In 1677 Henry Oldenburg described the luminescence of fluorite, CaF2, on heating;(Oldenburg, 1677) this is termed thermoluminescence.
* In 1830 Thomas Pearsall observed that colourless fluorite could be coloured by discharging sparks from a Leyden jar held against it.(Pearsall, 1830a) In 1881 luminescence excited by cathode rays was described by William Crookes,(Crookes, 1881) and in 1885 Edmond Becquerel found that when crystals were bombarded by cathode rays they became coloured and also emitted light;[48] this has been termed cathodoluminescence. In 1894 Eugen Goldstein showed that ultraviolet light has the same effect as cathode rays.[49][50]

### Reflection from opaque materials

Henry Clifton Sorby

Paul Drude

Researchers in reflection from opaque materials

The study of the optical properties of opaque substances has been closely linked with the development of suitable microscopes.[51] The first instrument adapted to reflected light was the Lieberkühn reflector attributed to Johann Nathanael Lieberkühn.[52] The use of polished and etched surfaces for this type of study was introduced by Jöns Jacob Berzelius in 1813.[52] A theory of the light reflected from metals was put forward by Augustin-Louis Cauchy in 1848.[53] In 1858 Henry Clifton Sorby established the technique of cutting minerals and crystals into thin sections for examination under the polarizing microscope.[54] In 1864 Sorby studied the microscopical structure of minerals from meteorites.[55] In 1888 Paul Drude published work on reflection from antimony sulfide.[56]

### Infrared optics

Heinrich Rubens measured the dependence of the refractive index of quartz on wavelength, and found absorption in particular infrared wavelength ranges. By 1896 Rubens saw these bands as a potential filter that would allow him to separate out an almost monochromatic beam from the broad range of infrared radiation that his sources produced.[57] In 1897 Rubens and his student Ernest Fox Nichols studied the reststrahlen (residual rays)[58] obtained when infrared rays of appropriate wavelength are reflected from the surfaces of crystals.(Nichols, 1897)

## Effect of electricity and magnetism

### Electrical conduction

The first observations on the variation of electrical conductivity with direction in a crystal (anisotropy) were made by Henri Hureau de Sénarmont in 1850 on 36 different substances. The results showed a correlation between the axes of symmetry and the directions of maximum or minimum conductivity.[59] In 1855 Carlo Matteucci performed experiments on bismuth.(Matteucci, 1855a) In 1888, Helge Bäckström (sv) performed electrical conduction measurements on hematite, another crystal of rhombohedral symmetry.(Bäckström, 1888)

Electrical conductivity in a crystal is now defined as a second rank symmetric tensor relating two vectors: 𝐉i=σij𝐄j, where 𝐉i is the current density, σij is the electrical conductivity tensor, and 𝐄j is the electric field intensity.[60]

### Magnetic properties

Julius Plücker

John Tyndall

Researchers in magnetic properties

Until the 19th century crystals were regarded either as magnetic or nonmagnetic. Magnetic crystals are now called ferromagnetic to distinguish them from the several other kinds which have since been discovered. Siméon Denis Poisson (1826) put forward a theory of magnetism as applied to crystals and predicted the behaviour of crystals in a magnetic field[61] which was verified by Julius Plücker in 1847. Plücker studied various natural crystals, such as quartz and related the reaction of the crystal to a magnetic field to its symmetry. All these crystals were repelled from a strong field, unlike ferromagnetic crystals. They were therefore called diamagnetic.[62] In 1850 a number of investigations were carried out by Plücker and August Beer using torsion balances to measure the small forces involved in most observations. Not only were some crystals repelled from a strong field but others were slightly attracted.({{{1}}}, {{{2}}}) These were called paramagnetic. Between 1850 and 1856 John Tyndall studied diamagnetism in crystals.({{{1}}}, {{{2}}})

By the end of the 19th century the three types of crystal—ferromagnetic, diamagnetic and paramagnetic—were well established and successful theories had related diamagnetic and paramagnetic crystals to their crystal symmetry. Ferromagnetic properties were dealt with by Pierre Weiss (1896) who explained the hysteresis by assuming that the atoms have permanent magnetic poles which are normally in random positions, but arrange themselves in parallel under the influence of a magnetic field.[63] On removing the field the mutual effect of the parallel dipoles tends to maintain the magnetized state. He further postulated that there were domains within which all the atomic dipoles were similarly orientated and that the N-S axis could be differently orientated in neighbouring domains.(Hoddeson, 1992)

### Dielectric properties

A dielectric is an electrical insulator that can be polarised by an applied electric field. In 1851 the first experiments on the behaviour of crystals in an electric field were carried out by Hermann Knoblauch in a manner similar to that used for the study of magnetic properties.[64] The conductivity of the crystals, both over the surface and through the body of the crystal, made these experiments unreliable.[65] In 1876 Elihu Root avoided some of these difficulties by employing a rapidly alternating field between parallel plates.(Root, 1876a) In 1893 Friedrich Pockels gave an account of the abnormally large piezoelectric constants of Rochelle salt.[66] A brief history on the theories of dielectrics in the 19th century has been written.[67]

## Effect of temperature change

### Thermal expansion

Eilhard Mitscherlich

Hippolyte Fizeau

Researchers in thermal expansion

In 1824 Eilhard Mitscherlich observed that the angle between the cleavage faces of calcite changed with the temperature of the crystal. Mitscherlich concluded that, on heating, calcite contracts (has a negative coefficient of thermal expansion) in a direction perpendicular to the trigonal axis while expanding (positive coefficient) along that axis. This implies that there is a cone of directions along which there is no thermal expansion.(Mitscherlich, 1824) In 1864 Hippolyte Fizeau used an optical interference method to make measurements on many crystals.[68] The measurements of the change of interfacial angle and the expansion of cut plates and bars were applied to crystals of all symmetries.[69]

Crystals with less than cubic symmetry are anisotropic and will generally have different expansion coefficients in different directions. If the crystal symmetry is monoclinic or triclinic, even the angles between the axes are subject to thermal changes. In these cases the coefficient of thermal expansion is a tensor. If the temperature T of a crystal is raised by an amount ΔT, a deformation takes place that is described by the strain tensor uij = αijΔT. The quantities αij are the coefficients of thermal expansion. Since uij is a symmetrical polar tensor of second rank and T is a scalar, αij is a symmetric tensor of second rank.(Authier, 2014) The contemporary usage of the term tensor was introduced by Woldemar Voigt in 1898.[70]

### Thermal conduction

Henri Hureau de Sénarmont

George Stokes

Researchers in thermal conduction

Joseph Fourier was an early researcher in thermal conduction, publishing Théorie analytique de la chaleur in 1822.[71] The first experiments on thermal conduction in crystals were carried out by Jean-Marie Duhamel in 1832.[72]

Henri Hureau de Sénarmont conducted experiments to determine if heat would move through crystals with directional dependence.[73] He found that, for non-cubic crystals, the isothermal envelope surrounding a point source of heat in a crystal plate had an elliptical shape whose exact form depended on the orientation of the crystal.[74] Sénarmont's results qualitatively established that thermal conductivity is directionally dependent (thermal anisotropy), with characteristic directions related to crystallographic axes. In 1848 Duhamel provided an analysis of Sénermont's findings.[75]

George Gabriel Stokes and William Thomson provided mathematical theories to explain Sénarmont's observations.(Stokes, 1851) Stokes acknowledged the connection between the phenomena and the symmetry of the crystal, and showed that the number of constants of heat conductivity reduces from nine to six in the case of two planes of symmetry.[76] The matrix of thermal conductivity components resulting from Stoke's derivation constituted a tensor.[73] Experiments by Franz Stenger (de) in 1884[77] examined the theories put forward by Stokes and Thomson and disproved some of their theoretical speculations.[78]

### Thermoelectricity

Thomas Johann Seebeck

Theodor Liebisch

Researchers in thermoelectricity

Thomas Johann Seebeck discovered the thermoelectric effect in 1821, although it has been claimed that Alessandro Volta should be given the priority.[79] In 1844 Wilhelm Gottlieb Hankel investigated thermoelectricity in cobalt and iron sulfide crystals. Hankel showed that when certain external faces were developed the crystals were thermoelectrically positive relative to copper, whereas with other facial forms they were negative.[80] In 1850 Jöns Svanberg used bismuth and antimony crystals to demonstrate a directional variation of the thermoelectric effect.[81] In 1854 William Thomson put forward a mechanical theory of thermoelectric currents in crystalline solids.(Thomson, 1854a) In 1889 Theodor Liebisch analyzed the dependence of the thermoelectric force on the crystallographic direction in anisotropic crystals.(Liebisch, 1889)

### Pyroelectricity

File:Carl von Linné, 1707-1778, botanist, professor (Alexander Roslin) - Nationalmuseum - 15723.tif

Carl Linnaeus

William Thomson

Researchers in pyroelectricity

Pyroelectricity is the generation of a temporary voltage in a crystal when subjected to a temperature change.[82] The appearance of electrostatic charges upon a change of temperature has been observed since ancient times, in particular with tourmaline and was described, among others, by Steno, Linnaeus, Aepinus and René Just Haüy. Aepinus published an account of his observations in 1756.[83] Haüy made detailed investigations of pyroelectricity;[84] he detected pyroelectricity in calamine and showed that electricity in tourmaline was strongest at the poles of the crystal and became imperceptible at the middle. Haüy published a book on electricity and magnetism in 1787.[85] Haüy later showed that hemihedral crystals are electrified by temperature change while holohedral (symmetric) crystals are not.

Research into pyroelectricity became more quantitative in the 19th century.[86] In 1824 David Brewster gave the effect the name it has today.[87] In 1840 Gabriel Delafosse, Haüy's student, theorized that only molecules which are not symmetrical can be polarized electrically.[88] Both William Thomson in 1878[89] and Woldemar Voigt in 1897[90] helped develop a theory for the processes behind pyroelectricity.

A detailed history of pyroelectricity has been written by Sidney Lang;[91] shorter histories have also been published.(Lang, 2004)

## Effect of mechanical force

### Elasticity

Friedrich Mohs

Woldemar Voigt

Wilhelm Röntgen

Lambros Koromilas

Researchers in elastic properties

Some minerals, for example mica, are highly elastic, springing back to their original shape after being bent. Others, for example talc, may be readily bent but do not return to their original form when released. The initial theory of the elasticity of solid bodies were developed in the 1820s. Augustin-Louis Cauchy and Siméon Denis Poisson published theories of the mutual action of a regular arrangement of particles for a non-cubic body in 1823[92] and 1829 respectively.(Poisson, 1829a) In 1827 Claude-Louis Navier published a theory for an isotropic body.[93] Also during the 1820s Friedrich Mohs introduced his eponymous scale of hardness.[94] In 1834 Franz Ernst Neumann published a paper on the elasticity of homohedral crystals.[95]

In 1828 Cauchy generalised the problem and showed that 36 independent constants were required to describe elasticity in crystals.[96] George Green (1837) introduced the limitation that the force between any two elements of a crystal, however small, must lie along the line joining their centres.[97] This reduced the number of constants from 36 to 21. William Thomson (1857) showed that Green's assumption was unnecessary and that the thermodynamic requirements of a reversible process require only 21 constants, without any special assumptions.[98] In 1874 Woldemar Voigt measured the elasticity of rock salt(Voigt, 1874) and G. Baumgarten measured the elasticity of calcite.[99] In 1887 Wilhelm Röntgen and J. Schneider measured the cubic compressibility of sodium and potassium chlorides.[100] In 1877 Lambros Koromilas measured the elasticity of gypsum and mica by twisting mineral bars.;[101] in 1881 H. Klang carried out similar experiments with fluorites.[102]

In the period 1874–1888 Voigt was the leading researcher on the elasticity of crystals. Voigt showed that the number of elasticity constants reduces as more symmetry is introduced into the crystal. For a triclinic crystal, which is the most general case, 21 elasticity constants are required. For a monoclinic crystal there are 13 elasticity constants, for a rhombic crystal 9, for a hexagonal crystal 7, for a tetragonal crystal 6, and finally for a cubic crystal there are only 3.(Voigt, 1888) A summary of developments in the field was published by W. A. Wooster.[103]

Franz Ernst Neumann

Friedrich Pockels

Researchers in photoelasticity

### Photoelasticity

Photoelasticity describes changes in the optical properties of a material under mechanical deformation. The photoelastic phenomenon in transparent, non-crystalline materials (gels and glasses) was first discovered by David Brewster in 1815.(Brewster, 1815) Brewster then detected the effect in crystals(Brewster, 1818) and showed that uniaxial crystals could be made biaxial.[23] In 1822 Augustin-Jean Fresnel experimentally confirmed that the photoelastic effect was a stress-induced birefringence.(Fresnel, 1822a)

Franz Ernst Neumann investigated double refraction in stressed transparent bodies. In 1841 Neumann published his elastic equations, which describe, in differential form, the changes which polarized light experiences when travelling through a stressed body.[104] The Neumann equations are the basis of all subsequent photoelasticity research.[105]

The photoelastic effect was analyzed by Friedrich Pockels, who also discovered the Pockels electro-optic effect (the production of birefringence of light on the application of an electric field). In 1889–1890 Pockels produced a phenomenological theory for both of these effects for all crystal classes.(Pockels, 1889)

### Piezoelectricity

Pierre Curie

Gabriel Lippmann

Researchers in piezoelectricity

In 1880 Pierre and Jacques Curie discovered piezoelectricity (an electric charge that accumulates in response to applied mechanical stress) in certain crystals, including quartz, tourmaline, cane sugar and sodium chlorate.({{{1}}}, {{{2}}})(Authier, 2013) The Curies, however, did not predict the converse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electric field). The converse effect was deduced by Gabriel Lippmann in 1881.[106] The Curies immediately confirmed the existence of the effect,[107] and went on to obtain quantitative proof of the complete reversibility of electro-elasto-mechanical deformations in piezoelectric crystals.({{{1}}}, {{{2}}})

In 1890 Woldemar Voigt published a phenomenological theory[108] of the piezoelectric effect based on the symmetry of crystals without centrosymmetry.(Voigt, 1890a)

## Research community

Paul Groth

Z. Krist., 1, 1877

Founder of Zeitschrift für Krystallographie
(later spelled Kristallographie)

Before the 20th century crystallography was not a well-established academic discipline. There were no academic positions specifically in crystallography. Workers in the field normally carried out their crystallographic research as an ancillary to other employment(s), or had independent means. The leading workers in the field of physical crystallography were employed as follows:

* Professors
* Mathematics or science: Airy,[109] Arago,[110] E. Becquerel,[111] Biot,[112] Curie,(Curie, 1923) Drude,[113] Hamilton,[114] Linnaeus,[115] Mitscherlich,[116] Pasteur,[117] Pockels,[118] Plücker,[119] Stokes,[120] Tyndall,[121] Thomson,[122] Voigt[123]
* Mineralogy: Groth,[124] Haüy,(Authier, 2013) Liebisch,[125] Mohs,[126] Neumann,(Authier, 2013) Sénarmont[127]
* Other employment: Bartholinus (physician),(Authier, 2013) Brewster (editor),[128] Fresnel (engineer),[129] Hooke (municipal official),(Authier, 2013) Malus (military officer)[130]
* Independently wealthy: Herschel, [131] Huygens(Authier, 2013)

In the nineteenth century there were informal schools of physical crystallography researchers in France (Arago, E. Becquerel, Biot, Fresnel, Haüy, Sénarmont),(Mauskopf, 1976) Germany (Drude, Groth, Liebisch, Mitscherlich, Mohs, Neumann, Pockels, Voigt)(Scholz, 1989c) and the British Isles (Airy, Brewster, Hamilton, Stokes, Thomson).(Deas, 1959)

Until the founding of Zeitschrift für Krystallographie und Mineralogie by Paul Groth in 1877 there was no lead journal for the publication of crystallographic papers. The majority of crystallographic research was published in the journals of national scientific societies, or in mineralogical journals.[132] The inauguration of Groth's journal marked the emergence of crystallography as a mature science independent of geology.[133]

## See also

* Chemical crystallography before X-rays
* Geometrical crystallography before X-rays
* Timeline of crystallography

## Citations

1. ↑ Shinn 2013.
2. ↑ Neumann 1832.
3. ↑ Neumann's principle.
4. ↑ Brandmüller 1986.
5. ↑ Huyghens 1690.
6. ↑ Lalena 2006, p. 131.
7. ↑ Simonyi 2012, pp. 283–288.
8. ↑ Malus 1810.
9. ↑ Simonyi 2012, p. 360.
10. ↑ Wooster 1990, pp. 64–65.
11. ↑ Fletcher 1891.
12. ↑ Liebisch 1891.
13. ↑ 13.0 13.1 Pockels 1906.
14. ↑ Arago 1811.
15. ↑ Buchwald 1989, pp. 86–107.
16. ↑ Herschel 1821.
17. ↑ Lalena 2006, p. 144.
18. ↑ Fresnel 1822b.
19. ↑ Pasteur 1848a.
20. ↑ Stefan 1864.
21. ↑ Goldhammer 1892.
22. ↑ Drude 1893.
23. ↑ 23.0 23.1 23.2 23.3 Wooster 1990, p. 65.
24. ↑ Ampère 1828.
25. ↑ Hamilton 1832.
26. ↑ Hankins 1980, pp. 88–95.
27. ↑ Berry & Jeffrey 2007.
28. ↑ Clifton 1860.
29. ↑ Lissajous 1874.
30. ↑ Liebisch 1888.
31. ↑ Schrauf 1889.
32. ↑ Cordier 1809.
33. ↑ Herschel 1820.
34. ↑ Babinet 1838.
35. ↑ Haidinger 1845.
36. ↑ Sénarmont 1854.
37. ↑ Shtukenberg & Punin 2007, p. 132.
38. ↑ Glan 1877.
39. ↑ Valeur & Berberan-Santos 2011, p. 731.
40. ↑ Harvey 1957, pp. 390–409.
41. ↑ Garten & Head 1966.
42. ↑ Harvey 1957, pp. 387–389.
43. ↑ Weiser 1918.
44. ↑ Zink & Chandra 1982.
45. ↑ Harvey 1957, pp. 378–387.
46. ↑ Bacon 1620.
47. ↑ Boyle 1664.
48. ↑ Becquerel 1885.
49. ↑ Goldstein 1894.
50. ↑ Wooster 1990, pp. 65–66.
51. ↑ Wooster 1990, p. 66.
52. ↑ 52.0 52.1 Orcel 1972, p. 301.
53. ↑ Cauchy 1848.
54. ↑ Sorby 1858.
55. ↑ Sorby 1864.
56. ↑ Drude 1888.
57. ↑ Hoddeson 1992, p. 18.
58. ↑ Rubens & Aschkinass 1898.
59. ↑ Sénarmont 1850, pp. 264–265.
60. ↑ Nye 1990, pp. 204, 289.
61. ↑ Poisson 1826.
62. ↑ Plücker 1847.
63. ↑ Weiss 1896.
64. ↑ Knoblauch 1851.
65. ↑ Wooster 1990, p. 64.
66. ↑ Pockels 1893.
67. ↑ Katzir 2006, pp. 104–107.
68. ↑ Fizeau 1864.
69. ↑ Wooster 1990, pp. 62–63.
70. ↑ Voigt 1898.
71. ↑ Simonyi 2012, p. 370.
72. ↑ Duhamel 1832.
73. ↑ 73.0 73.1 Narasimhan 2010.
74. ↑ Sénarmont 1847.
75. ↑ Duhamel 1848.
76. ↑ Katzir 2006, pp. 85–86.
77. ↑ Stenger 1884.
78. ↑ Wooster 1990, p. 63.
79. ↑ Anatychuk 2024.
80. ↑ Hankel 1844.
81. ↑ Svanberg 1850.
82. ↑ Pyroelectricity.
83. ↑ Aepinus 1756.
84. ↑ Burke 1966, p. 85.
85. ↑ Haüy 1787.
86. ↑ Lang 2004, p. 66.
87. ↑ Brewster 1824.
88. ↑ Katzir 2006, pp. 26–27.
89. ↑ Thomson 1878.
90. ↑ Voigt 1897.
91. ↑ Lang 1974.
92. ↑ Cauchy 1823.
93. ↑ Navier 1827.
94. ↑ Mohs 1822.
95. ↑ Neumann 1834.
96. ↑ Cauchy 1828.
97. ↑ Green 1837.
98. ↑ Thomson 1857b.
99. ↑ Baumgarten 1874.
100. ↑ Röntgen & Schneider 1887.
101. ↑ Coromilas 1877.
102. ↑ Klang 1881.
103. ↑ Wooster 1990, p. 67.
104. ↑ Neumann 1841.
105. ↑ Mönch 1986.
106. ↑ Lippmann 1881.
107. ↑ Curie & Curie 1881d.
108. ↑ Katzir 2003b.
109. ↑ MacTutor, George Biddell Airy.
110. ↑ MacTutor, Dominique François Jean Arago.
111. ↑ Gough 1970.
112. ↑ MacTutor, Jean-Baptiste Biot.
113. ↑ Hoffmann 2006.
114. ↑ Hankins 1980.
115. ↑ Blunt 1971.
116. ↑ Schütt 1992.
117. ↑ Debré 1998.
118. ↑ Shtukenberg & Punin 2007, p. 211, "the great systematizers of crystal physics, Voigt and Pockels".
119. ↑ MacTutor, Julius Plücker.
120. ↑ MacTutor, George Gabriel Stokes.
121. ↑ MacLeod 1981.
122. ↑ MacTutor, William Thomson (Lord Kelvin).
123. ↑ Goldbers 1976.
124. ↑ Fischer 1972.
125. ↑ Schuh 2013.
126. ↑ Authier 2013, p. 350.
127. ↑ Fischer 1975.
128. ↑ Authier 2013, p. 349.
129. ↑ MacTutor, Augustin Jean Fresnel.
130. ↑ MacTutor, Étienne Louis Malus.
131. ↑ MacTutor, John Frederick William Herschel.
132. ↑ Schuh 2007, pp. 371–373.
133. ↑ Kahr & McBride 1992, p. 12.

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* Poisson, S. D. (1829a). "Mémoire sur l'équilibre et le mouvement des corps élastiques" (in French). Mémoires de l'Académie des sciences de l'Institut de France 8: 357–570. https://www.biodiversitylibrary.org/item/55232#page/625/mode/1up. Retrieved 2 April 2025.
* Poisson, S. D. (1829b). "Note sur le problème des ondes" (in French). Mémoires de l'Académie des sciences de l'Institut de France 8: 571–580. https://www.biodiversitylibrary.org/item/55232#page/839/mode/1up. Retrieved 2 April 2025.
* "Pyroelectricity". International Union of Crystallography. 17 November 2017. https://dictionary.iucr.org/Pyroelectricity.
* Robertson, Edmund F.; O'Connor, John J.. "The MacTutor History of Mathematics Archive: Biographies". School of Mathematics and Statistics, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/.
* Röntgen, W. C.; Schneider, J. (January 1887). "Ueber die Compressibilität von verdünnten Salzlösungen und die des festen Chlornatriums" (in German). Annalen der Physik 267 [31] (8): 1000–1005. doi:10.1002/ANDP.18872670828. Bibcode: 1887AnP...267.1000R. https://archive.org/details/sim_annalen-der-physik_1887_31_8/page/1000/mode/2up. Retrieved 4 April 2025.
* Root, Elihu (January 1876a). "Zur Kenntniss der dielektrischen Polarisation" (in German). Annalen der Physik 234 (5): 1–35. doi:10.1002/andp.18762340503. Bibcode: 1876AnP...234....1R. https://archive.org/details/sim_annalen-der-physik_1876_158_5/page/n7/mode/2up. Retrieved 31 March 2025.
* Root, Elihu (January 1876b). "Zur Kenntniss der dielektrischen Polarisation" (in German). Annalen der Physik 234 (7): 425–461. doi:10.1002/andp.18762340705. Bibcode: 1876AnP...234..425R. https://archive.org/details/sim_annalen-der-physik_1876_158_7/page/n87/mode/2up. Retrieved 31 March 2025.
* Rubens, H.; Aschkinass, E. (January 1898). "Die Reststrahlen von Steinsalz und Sylvin" (in German). Annalen der Physik 301 [65] (6): 241–256. doi:10.1002/andp.18983010602. Bibcode: 1898AnP...301..241R. https://archive.org/details/sim_annalen-der-physik_1898_65_2/page/n3/mode/2up. Retrieved 9 April 2025.
* Rubens, H.; Nichols, E. F. (January 1897). "Versuche mit Wärmestrahlen von grosser Wellenlänge" (in German). Annalen der Physik 296 [60] (3): 418–462. doi:10.1002/andp.18972960303. Bibcode: 1897AnP...296..418R. https://archive.org/details/sim_annalen-der-physik_1897_60_3/page/418/mode/2up. Retrieved 9 April 2025.
* Sarton, George (January 1932). "Discovery of Conical Refraction by William Rowan Hamilton and Humphrey Lloyd (1833)". Isis 17 (1): 154–170. doi:10.1086/346641. ISSN 0021-1753. https://www.journals.uchicago.edu/doi/10.1086/346641. "I know of no person who has not seen conical refraction that really believed in it. I have myself converted a score of mathematicians by showing them the cone of light". Hamilton replied, "How different from me! If I had seen it only, I should not have believed it. My eyes have too often deceived me. I believe it, because I have proved it".
* Scholz, Erhard (1989b). "Crystallographic Symmetry Concepts and Group Theory (1850-1860)". in Rowe, David E.; McCleary, John. The history of modern mathematics: proceedings of the Symposium on the history of modern mathematics, Vassar college, Poughkeepsie, New York, June 20-24, 1988. New York London Toronto: Academic Press. pp. 3–27. ISBN 0125996624.
* Scholz, Erhard (1989c). "The rise of Symmetry Concepts in the Atomistic and Dynamistic Schools of Crystallography, 1815-1830". Revue d'histoire des sciences 42 (1–2): 110–112. doi:10.3406/rhs.1989.4136. https://www.persee.fr/doc/rhs_0151-4105_1989_num_42_1_4136. Retrieved 20 January 2025.
* Scholz, Erhard (1994). "Crystallography". in Grattan-Guinness, I.. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. 2. London: Routledge. pp. 1269–1274. ISBN 9780415092395.
* Schrauf, A. (January 1889). "Ueber die Verwendung einer Schwefelkugel zur Demonstration singulärer Schnitte an der Strahlenfläche" (in German). Annalen der Physik 273 [37] (5): 127–144. doi:10.1002/andp.18892730508. Bibcode: 1889AnP...273..127S. https://archive.org/details/sim_annalen-der-physik_1889_37_5/page/126/mode/2up. Retrieved 13 April 2025.
* Schuh, Curtis P. (2007). Mineralogy & Crystallography: On the History of these Sciences through 1919. Curtis Schuh. pp. 371–373. https://archive.org/details/History_Mineralogy_2007/page/n382/mode/1up. Retrieved 20 April 2025.
* Schuh, Curtis (2013). "Liebisch, Theodor". The Mineralogical Record. http://minrec.org/libdetail.asp?id=867.
* Schütt, Hans-Werner (1992) (in German). Eilhard Mitscherlich, Baumeister am Fundament der Chemie. München: In Kommission bei R. Oldenbourg. ISBN 3486262734. https://archive.org/details/eilhardmitscherl0000schu/page/n5/mode/2up. Retrieved 28 February 2025.
* Sénarmont, H. (1847). "Sur la conductibilité des substances cristallisées pour la chaleur" (in French). Annales de chimie et de physique. 3e série 21: 457–470. https://archive.org/details/bub_gb_PJ45AAAAcAAJ/page/n459/mode/2up. Retrieved 28 March 2025.
* Sénarmont, H. (March 1850). "Mémoire sur la conductibilité superficielle des corps cristallisés pour l'électricité de tensions" (in French). Annales de chimie et de physique. 3e série 28: 257–278. https://babel.hathitrust.org/cgi/pt?id=hvd.hx3dy7&seq=267. Retrieved 1 April 2025.
* Sénarmont, H. (1854). "Experiénces sur la production artificielle du polychroïsme dans les substances crystallisées" (in French). Annales de chimie et de physique 3: 319–336. https://archive.org/details/bub_gb_W5o5AAAAcAAJ/page/n321/mode/2up. Retrieved 25 March 2025.
* Shinn, Terry (2013). "The silicon tide". in Buchwald, Jed Z.; Fox, Robert. The Oxford handbook of the history of physics (First ed.). Oxford ; New York, NY: Oxford University Press. p. 868. ISBN 978-0-19-969625-3. "Much pre-twentieth-century study of crystals focused on their geometry, and mathematical description became increasingly central. There was then little reflection on the mechanical, optical, and electrical properties of crystals. This changed early in the twentieth century ..."
* Shtukenberg, A. G.; Punin, Yurii O. (2007). Kahr, Bart. ed. Optically anomalous crystals. Dordrecht: Springer. ISBN 978-1-4020-5287-3.
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* Simonyi, Károly (2012). A cultural history of physics. Boca Raton: CRC press. ISBN 978-1-56881-329-5.
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* Sorby, H. C. (1864). "On the microscopical structure of meteorites". Proceedings of the Royal Society of London 13: 333–334. doi:10.1098/rspl.1863.0075. https://royalsocietypublishing.org/doi/10.1098/rspl.1863.0075. Retrieved 11 April 2025.
* Stefan, J. (January 1864). "Ueber die Dispersion des Lichtes durch Drehung der Polarisationsebene im Quarz" (in German). Annalen der Physik 198 [122] (8): 631–634. doi:10.1002/andp.18641980812. Bibcode: 1864AnP...198..631S. https://gallica.bnf.fr/ark:/12148/bpt6k15207b/f656.item. Retrieved 14 April 2025.
* Stenger, Franz (January 1884). "Zur Wärmeleitungsfähigkeit des Turmalins" (in German). Annalen der Physik 258 (8): 522–528. doi:10.1002/andp.18842580806. Bibcode: 1884AnP...258..522S. https://archive.org/details/sim_annalen-der-physik_1884_22_8/page/522/mode/2up. Retrieved 28 March 2025.
* Stokes, George Gabriel (1851). "On the Conduction of Heat in Crystals". The Cambridge and Dublin Mathematical Journal 6: 215–238. https://archive.org/details/sim_cambridge-and-dublin-mathematical-journal_the-cambridge-and-dublin-mathematical-jour_1851_6/page/n217/mode/2up. Retrieved 28 March 2025.
* Stokes, George Gabriel (1852). "On the Change of Refrangibility of Light". Philosophical Transactions of the Royal Society of London 142: 463–562. doi:10.1098/rstl.1852.0022. https://archive.org/details/philosophicaltra1421roya/page/462/mode/2up. Retrieved 8 April 2025.
* Stokes, George Gabriel (1853). "On the Change of Refrangibility of Light – No. II". Philosophical Transactions of the Royal Society of London 143: 385–396. doi:10.1098/rstl.1853.0016. https://archive.org/details/philosophicaltra1431roya/page/n401/mode/2up. Retrieved 8 April 2025.
* Stokes, George Gabriel (2009). "On the Conduction of Heat in Crystals". Mathematical and Physical Papers. Cambridge University Press. pp. 203–227. doi:10.1017/CBO9780511702266.008. ISBN 978-1-108-00264-6.
* Svanberg, J. (1850). "Expériences sur le pouvoir thermoélectrique du bismuth et de l'antimoine cristallisés" (in French). Comptes rendus hebdomadaires des séances de l'Académie des sciences 31: 250–252. https://www.biodiversitylibrary.org/item/16573#page/260/mode/1up. Retrieved 26 March 2025.
* Thomson, William (1854a). "A mechanical theory of thermo-electric currents in crystalline solids". Mathematical and physical papers. 1. Cambridge University Press. pp. 324–325. https://archive.org/details/mathematicalphys01kelv/page/324/mode/2up. Retrieved 26 March 2025.
* Thomson, William (1854b). "On thermo-electric currents in linear conductors of crystalline sunstances". Mathematical and physical papers. 1. Cambridge University Press. pp. 266–291. https://archive.org/details/mathematicalphys01kelv/page/266/mode/2up. Retrieved 26 March 2025.
* Thomson, William (1854c). "On thermo-electricity in crystalline metals in a state of mechanical strain". Mathematical and physical papers. 1. Cambridge University Press. pp. 467–468. https://archive.org/details/mathematicalphys01kelv/page/466/mode/2up. Retrieved 26 March 2025.
* Thomson, William (1857a). "On the Dynamical Theory of Heat . Part V. Thermo-electric Currents". Transactions of the Royal Society of Edinburgh 21 (1): 123–171. doi:10.1017/S0080456800032014. https://archive.org/details/per_transactions-of-the-royal-society-of-edinburgh_transactions-of-the-royal-society-of_1857_21/page/122/mode/2up. Retrieved 27 March 2025.
* Thomson, William (1857b). "On the thermo-elastic, thermo-magnetic and pyro-electric properties of matter". The Quarterly Journal of Pure and Applied Mathematics 1: 57–77. https://babel.hathitrust.org/cgi/pt?id=uc1.$b417523&seq=75. Retrieved 2 April 2025.
* Thomson, William (1878). "On the thermoelastic, thermomagnetic and pyroelectric properties of matter". Philosophical Magazine. Series 5 5 (28): 4–26. doi:10.1080/14786447808639378. https://books.google.com/books?id=A5YOAAAAIAAJ&pg=PA4. Retrieved 20 March 2025.
* Tutton, A. E. H. (1922). "Elasticity of crystals and its measurement". Crystallography and practical crystal measurement. 2: Physical and Chemical (2nd ed.). London: Macmillan & Co.. p. 1337. https://archive.org/details/crystallographyp02tuttrich/crystallographyp02tuttrich/page/1336/mode/2up. Retrieved 2 April 2025.
* Tyndall, John (September 1851). "On diamagnetism and magnecrystallic action". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2 (10): 165–188. doi:10.1080/14786445108646857. https://archive.org/details/londonedinburghd02lond/page/164/mode/2up. Retrieved 30 March 2025.
* Tyndall, John (September 1855a). "On the nature of the force by which bodies are repelled from the poles of a magnet; to which is prefixed, an account of some experiments on molecular influences". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10 (65): 153–179. doi:10.1080/14786445508641950. https://archive.org/details/londonedinburghp10maga/page/152/mode/2up. Retrieved 30 March 2025.
* Tyndall, John (October 1855b). "On the nature of the force by which bodies are repelled from the poles of a magnet; to which is prefixed, an account of some experiments on molecular influences". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10 (66): 257–290. doi:10.1080/14786445508641969. https://archive.org/details/londonedinburghp10maga/page/256/mode/2up. Retrieved 30 March 2025.
* Tyndall, John (February 1856). "On the relation of diamagnetic polarity to magnecrystallic action". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 11 (70): 125–137. doi:10.1080/14786445608642034. https://archive.org/details/londonedinburghp11maga/page/124/mode/2up. Retrieved 30 March 2025.
* Tyndall, John (1870). Researches on diamagnetism and magne-crystallic action. London: Longmans, Green, and co.. https://archive.org/details/researchesondia02tyndgoog/page/n10/mode/2up. Retrieved 30 March 2025.
* Tyndall, John; Knoblauch, Hermann (March 1850). "On the deportment of crystalline bodies between the poles of a magnet". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36 (242): 178–183. doi:10.1080/14786445008646457. https://archive.org/details/londonedinburghp36lond/page/178/mode/2up. Retrieved 30 March 2025.
* Valeur, Bernard; Berberan-Santos, Mário N. (1 June 2011). "A Brief History of Fluorescence and Phosphorescence before the Emergence of Quantum Theory". Journal of Chemical Education 88 (6): 731–738. doi:10.1021/ed100182h. Bibcode: 2011JChEd..88..731V.
* Voigt, Woldemar (1874). Untersuchung der Elasticitätsverhältnisse des Steinsalzes [Investigation of the elasticity of rock salt] (Doctoral thesis) (in German). Leipzig: E. Pöschel. doi:10.58016/retro_991016492889706476_004099. Retrieved 2 April 2025.`{{cite thesis}}`: CS1 maint: unrecognized language (link)
* Voigt, Woldemar (1876). "Bestimmung der Elasticitätsconstaten des Steinsalzes" (in German). Annalen der Physik und Chemie Erganzung 7: 177–214. https://gallica.bnf.fr/ark:/12148/bpt6k15009g/f186.item. Retrieved 3 April 2025.
* Voigt, W. (1885). "Erklärung der Farbenerscheinungen pleochroitischer Krystalle" (in German). Neues Jahrbuch für Mineralogie, Geologie und Paläontologie: 119–141. https://archive.org/details/bub_gb_P8UKAAAAYAAJ/page/119/mode/2up. Retrieved 25 March 2025.
* Voigt, W. (1888). "Bestimmung der Elasticitätsconstanten von Flusspath, Pyrit, Steinsalz, Sylvin" (in German). Annalen der Physik und Chemie 35 (12): 642–661. https://archive.org/details/sim_annalen-der-physik_1888_35_12/page/642/mode/2up. Retrieved 2 April 2025.
* Voigt, W. (1890a). "Ueber die innere Reibung der festen Körper, inbesondere der Krystalle" (in German). Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 36: 1–47. https://www.biodiversitylibrary.org/item/176239#page/621/mode/1up. Retrieved 16 March 2025.
* Voigt, W. (1890b). "Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen an Krystallen" (in German). Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 36: 47–99. https://www.biodiversitylibrary.org/item/176239#page/667/mode/1up. Retrieved 16 March 2025.
* Voigt, W. (1897). "Versuch zur Bestimmung des wahren specifischen electrischen Momentes eines Turmalins" (in German). Annalen der Physik 60: 368–375. https://books.google.com/books?id=iX0EAAAAYAAJ&pg=PA368. Retrieved 20 March 2025.
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