Minerals, Lattices and Gemstones

Crystallography and the Structure of Solids

Dana Ashkenazi and Noam Eliaz

Ever since the dawn of mankind, human beings took a lively interest in crystals, because of their magical beauty, impressive shapes and seemingly endless variety of colors and shades. Precious stones such as diamonds, sapphires, rubies and emeralds are crystals, characterized by long-range periodic configurations of atoms, which gives them their unique properties. In ancient times, people believed that the various gemstones have magical powers able to bestow upon their owners virtues such as health, wisdom and wealth – and many believe it to this day. In Pharaonic Egypt, lapis lazuli – a blue stone sometimes flecked with gold – was considered among the most precious of gemstones. Believed to be the gemstone of the gods, it was often set in expensive jewelry. Lapis is made of a rock composed of several minerals. Ancient Egyptians also used to grind it to powder, mix it with water and use it as makeup. The Old Testament tells us (in Exodus, Ch. 39) that the High Priest used to don a holy garment with a breastplate in which a dozen precious stones were set, representing the Twelve Tribes. These ritual vestments were worn by the High Priest when serving God at the Temple. For thousands of years and to this day, people have been wearing jewels set with crystalline and shiny gemstones. Men and women have given their loved ones crystals as a symbol for their eternal love, and many fights and even wars have broken over these fascinating crystals.

Figure 1: The beauty of crystals and their fascinating shapes have attracted mankind ever since the dawn of history. Graphics: Dana Ashkenazi.

"Fair is what we see,

Fairer what we have perceived,

Fairest what is still in veil"

Nicholas Steno (1638-1688)

From Ancient Greece to Modern Crystallography

The mysterious structure of crystals has obsessed humankind ever since their discovery. The ancient Greeks believed that all matter is made up of four elements: earth, wind, fire, and water. They thought that crystals are fundamentally made of earth. A 13th-century theologian named John Duns Scotus believed crystals live and grow much like plants, and that their structure results from the aspiration of all living things to arrange themselves in an "ideal" shape. In the early 17th century, German astronomer and mathematician Johannes Kepler – best known for formulating the laws of planetary motion – published a short treatise, "The Six-Cornered Snowflake", in which he claimed that snowflakes are composed of tiny, spherical elementary particles.

Later in that century, Danish Geologist Nicolas Steno argued that the growth of any body – be it animate or inanimate – stems from the accumulation of particles secreted from the liquid. Steno believed that an inner force is responsible for the growth of plants, while an outer force is responsible for the growth of crystals. Although he couldn't identify that force, he believed a tiny "seed" in the liquid begins the process of crystalline growth by attracting particles from within the liquid. Robert Hooke, one of the greatest scientists of the 17th century (Hooke's Law models the relationship between stress and strain on an elastic body; he also coined the all-important biological term "cell"), followed Kepler and showed that compressing spheres results in multiple crystalline shapes.

Since the early pioneers of crystallography worked with impure materials, the classification of crystals was a daunting task. Thus, for many years crystallography was shrouded in a veil, until 18th-century German geologist Abraham Werner suggested a systematic classification of minerals. Minerals are natural, inorganic solids, mostly crystalline, with definite physical and chemical properties. As a practical man, Werner suggested grouping crystals according to their visual characteristics. Using this straightforward approach, Werner classified a broad range of crystalline minerals according to color, unevenness, specific weight, odor, transparency/translucency, brittleness, hardness, etc. Werner cataloged each mineral he could find based on these properties, creating a mineral identification "field manual."

Figure 2: The founders and pioneers of crystallography. Graphics: Dana Ashkenazi.

Figure 3: Typical unit cell structure. A unit cell is the smallest structural unit representing the symmetry of the atomic organization in the three-dimensional crystal structure. Seven different geometric systems exist, one of them is cubic, which differs in the basic symmetric arrangements allowing the entire space to be filled. Figure: Dana Ashkenazi.

One of the most important breakthroughs in the study of crystalline structures occurred in 1845, when French physicist August Bravai's succeeded in predicting 14 possible basic combinations of geometric atomic structures of the various crystals, called Bravais Lattices (see image below).

Crystallography, or the science of crystals, studies and empirically identifies crystalline structures, as well as the way crystals are formed and their properties. The most important breakthrough in this area occurred in the early 20th century, with the discovery that crystal structures may be revealed using X-rays. Much of what we know about the structural ordering of crystals come from studies on X-ray diffraction. X rays are electromagnetic waves 0.02-100 Å long [1Å (Angström) = 10^-10 m]. Diffraction occurs when a wave passes through a series of evenly-spaced obstacles such that (1) they are able to scatter the wave, and (2) the space between them is of the same scale as the incident wave's length. The wavelength of visible light is between 3,800 and 7,800Å, much longer than required to characterize crystalline structures. Conversely, in X-ray diffraction a beam with a typical energy of 1-120keV is used, producing a wavelength of 0.1Å to several Ångströms (for the lowest and highest energy limits, respectively). These wavelengths are of the same scale as lattice parameters, thus making X-rays particularly useful for revealing crystalline structures.

In 1912, the German physicist Max von Laue X-radiated a copper sulfate crystal. In a series of experiments, he found that X-rays refract when passing through the crystal and that X-radiation is an electromagnetic, short-wavelength type of radiation, of the same scale as the space between planes of atoms. Irradiating the crystal produced a series of dots (or spots) resulting from the way X-rays had refracted and reflected off the periodically organized crystal atoms. Von Laue deduced the atoms' orientation from the intensity of the radiation reflected from the crystal, using X-rays to produce "portraits" of crystalline materials based on the way the X-rays passing through them were interfered, proving that crystal atoms are periodically arranged. In 1914, this discovery won him the Noble Prize in Physics.

Interference is a phenomenon characterizing wave-like behavior: when several waves pass through a single point in space, the wave's amplitude at that point will be an algebraic sum, taking direction into account, of the amplitudes of all the waves at that point. This wave scheme is called superposition. Constructive interference occurs when the waves' maximum (and minimum) points occur together, that is when the waves' frequencies and phases are equal. Destructive interference, on the other hand, occurs when two waves have identical frequencies, but a half-cycle phase differential.

Von Laue encountered many difficulties when deciphering his images. These difficulties ended in 1914, when two British scientists, William Henry and William Lawrence Bragg, found that the crystal atoms' ordering forms evenly-spaced parallel planes. The father and son concluded that for a given distance between atoms and a given wavelength, there is an angle that will produce maximum reflection (constructive interference). Therefore, X-radiating the crystal and measuring the maximum reflection angle enables us to calculate distances between crystal atoms. This discovery won the two researchers the 1915 Noble Prize in Physics.

Figure 4: Bravais lattices. The three three-dimensional patterns in the image, based on two different geometric structures (cubic and hexagonal), represent the keystones of some of the crystals. The red balls stand for atoms. Filling the space using each of these patterns will produce a crystalline structure. The cubic structure is a relatively simple example since all its axes are equal and all its angles are straight. Most pure solid crystal elements have a body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close-packed (HCP). Figure: Dana Ashkenazi.

Figure 5: Constructive and destructive interference. Figure: Dana Ashkenazi.

The Riddle of Crystals

Everywhere we look around us, we see a wide variety of materials. What all these materials have in common is that they're made up of elementary atoms (or ions). A chunk of iron is made up of the atoms of the element iron; a chunk of titanium is made up of titanium atoms, while water is composed of hydrogen and oxygen atoms. The elements of which matter is made, the atomic bonding, the atoms' arrangement in space, and the type of imperfections in the matter – all these are responsible for the materials' properties.

There are several types of bonds between atoms, including ionic bonding, covalent bonding (in which atoms share electron pairs,) and metallic bonding. In the metallic bond, the atom "gives up" electrons in its outer shell, which it is then "contributes" to a cloud of free electrons inside the material. This structure gives metals properties such as brilliance and heat and electricity conduction.

In addition to the ionic, covalent, and metallic bonds, there are also secondary, weaker bonds, called Van der Waals bonds and hydrogen bonds. Van der Waals bonds result from asymmetry in the motion of electrons surrounding the nucleus. Due to this asymmetry, a certain area in the molecule temporarily becomes slightly negatively charged, while another becomes positively charged. A weak electric attraction is created between those temporarily charged areas, called the Van der Waals bond. For example, weak, easily broken bonds exist between the layers of atoms in graphite, enabling the pencil to leave its trace on a piece of paper.

Hydrogen bonds result from the fact that in certain molecules, such as water molecules, there are (partly) positively charged hydrogen atoms and (partly) negatively charged other atoms. In such adjacent molecules, the negatively charged area of one molecule will be attracted to the positively charged area of another.

Material may be characterized either by a long-range periodic configuration of atoms – or a crystalline structure – or by a short-range ordering, as in liquids. In the case of short-range ordering, we would say the structure is amorphous or glassy. The glassy structure may be likened to a liquid frozen due to rapid cooling with no time to crystallize. The image above shows the difference between amorphous and crystalline structures.

The crystalline ordering of atoms is reminiscent of honeycombs, made up as they are of a huge number of hexagons or the way the famous Dutch painter Escher painted various periodically repetitive motifs (such as fish, butterflies, and lizards). In a crystalline structure, each point in the lattice may be assigned at least one atom (motif), with a periodic arrangement of all the atoms in the material volume. When depicting crystalline structures, it is customary to represent atoms or ions by solid spheres. As you may recall, the basic crystal cell is called a unit cell. The lattice is made up of a series of unit cells whose spatial positions recur periodically, such that a single motif or multiple motifs are located at constant points in the lattice. The basic unit cell geometry may be, for example, a cube. If the particles making up the crystal are metal atoms, with metallic bonds among the atoms, it will be a metallic crystal. If the particles are ions, with ionic bonds among them, it will be an ionic crystal. Many ceramics have such a structure. Table salt is one example of an ionic crystal, as seen in the image below.

Figure 6: The structural difference between amorphous and crystalline matter. To the left, a crystalline structure, with a periodic, long-range atomic ordering. To the right, an amorphous structure, with no long-range atomic ordering. Figure: Dana Ashkenazi.

Figure 7: The periodic ordering of sodium and chloride atoms in the crystalline structure of table salt (NaCl). The blue spheres represent negative chloride ions, while the purple spheres represent positive sodium ions. The crystalline structure, in this case, must ensure both overall electric neutrality and efficient packaging of two types of differently sized ions. The image on the left represents the way atoms are actually arranged in three-dimensional space. The image on the right depicts a package of unit cells, with notional lines stretched between them. Figure: Dana Ashkenazi.

In addition to metallic and ionic crystals, there are also molecular crystals, in which the motifs in the lattice points are molecules, rather than single atoms. The bonds within these molecules are strong, covalent bonds, while the molecules are weakly bonded (Van der Waals and hydrogen bonds).

Materials made up of just one crystal are called single crystal or monocrystals. In the microelectronics industry, a huge silicone (Si) single crystal is sliced into thin wafers. At the end of the production process, each wafer contains thousands of components used to manufacture electronic devices. Most crystalline materials, however, are not single but made up of a large number of crystals growing from various solidification centers and arranged in various crystalline orientations. Such materials are called polycrystals.

The process in which crystals are created is called crystallization. When a polycrystal solidifies, tiny crystals called embryos begin to form in several locations in the liquid. These crystals grow by collecting other atoms from the liquid environment. The interface between two such embryos, or grains, is called grain boundary. At grain boundaries, the crystalline ordering is imperfect, and they contain a high ratio of impurities (that is, imperfections in the spatial ordering of atoms), such as foreign atoms (impurities) concentrated at the boundaries and moving through the quickly, relative to their motion through the grain volume. The ordering of polycrystalline atoms in grains and grain boundaries is depicted in the image below.

Material properties are dictated by the way atoms are arranged inside grains and in their boundaries. For example, the smaller the grains of a polycrystal, the more grain boundaries there are, and consequently, the matter becomes stronger and harder (so long as the temperature is not too high). Furthermore, single crystals, polycrystals, and amorphous materials differ in their properties. Ceramic materials and polymers in their amorphous or single crystalline state tend to be translucent, while the same materials in the polycrystalline state tend to be opaque or sometimes shiny and reflective like a mirror. There are also ceramic materials and polymers with short-range atomic order. These glassy or amorphous materials have very different properties than those of crystalline solids.

Polymorphism

Polymorphism (Greek for "having multiple forms") means that materials have a variety of different crystalline configurations under different pressure and/or temperature conditions, without changing their chemical composition. In pure elements, this phenomenon is also called allotropism.

Many ceramics, such as silica (SiO₂) are polymorphous. In iron – the main component of steels (iron- and carbon-based alloys) – allotropism enables the production of steels with various crystalline structures and properties using various thermal treatments. This is one of the reasons why various steel alloys have become such common construction materials. When pure iron is heated, it goes through two allotropic transformations before melting and liquefying at a 1,538°C temperature. At room temperature, iron has a body-centered cubic structure (called ferrite), it is magnetic and may contain a small amount of carbon atoms (solute) within it. At 912°C the iron goes through an allotropic transformation and changes its unit cell to a face-centered cubic structure (Austenite), which is not magnetic and may contain a large amount of carbon atoms in it. This structure is stable up to 1,394°C, where a second allotropic occurs, ending in the creation of a magnetic, body-centered cubic structure called delta iron. Ferrite and delta iron have similar crystalline structures, but ferrite is softer and capable of containing a lower concentration of carbon.

Another important example is the element carbon. In its solid state, it may exist as graphite (hexagonal structure), diamond (cubic), a C60 sphere containing 60 carbon atoms (fullerene or buckyball), as well as nanotubes. In each of these materials, all made of pure carbon, different bonds exist among the atoms, giving them their different look and properties.

Figure 8: Solidification of material from a liquid to a polycrystalline solid state. Usually, the crystal ordering is maintained in a certain material volume, next to which is an additional volume with a crystalline structure ordered in a different orientation (yellow areas in the image). Between these orderly groupings, called grains, there is a disorderly buffer zone, called grain boundary. The red spheres represent pure atoms, while the blue spheres represent impurities, which move to grain boundaries in the process of solidification. Figure: Dana Ashkenazi.

Figure 9: Diamond. On the left – photograph of a diamond crystal, considered a precious stone. The various facets and differential light refraction indicate a crystalline ordering with different orientations relative to the incoming light ray. On the right – a diamond structure unit cell, with the blue spheres representing carbon atoms. Figure: Dana Ashkenazi.

The transparent diamond is a valuable crystalline mineral made of pure carbon. This is a true natural treasure, first and foremost among precious stones. Diamonds are created naturally deep underground (150-450 km beneath sea level or maybe even deeper), under conditions of very high pressure (about 40,000 atmospheres) and temperatures (1,000-1,200°C). Presumably, diamonds are made of carbonate-rich magmas. After their creation, diamonds may remain buried in the Earth's crust for a long time until they rise to the surface following eruptions of magma rocks called kimberlites (after the town of Kimberley, South Africa). When the kimberlite erodes, the diamonds get washed down rivers, sometimes forming secondary deposits inside the sediment. Although diamonds are energetically unstable under standard environmental conditions, luckily for us their decomposition process is extremely slow, so that we are able to enjoy their brilliance for a long time. Artificial diamonds may be produced in a laboratory using high-pressure and high-temperature plasma waves. Today, this process is used to process tiny fragments of natural diamonds, which act as growth (nucleation) centers. Nevertheless, we haven't yet been able to manufacture artificial diamonds that are as clean, as transparent, and as large as the real natural thing.

The diamond crystal, which is one of carbon's polymorphs, is made up of a cubic unit cell, in which each carbon atom has covalent bonds with four additional carbon atoms. This creates carbon atom tetrahedrons. This structure, whose unit cell is depicted below, is called diamond structure and is also typical of other elements such as silicon (Si) and germanium (Ge). It is a highly stable (low-energetic) structure since the electron pairs involved in the covalent bonds are maximally distant. This structural stability gives diamonds their typical hardness and toughness. On a scale of 1-10 (1 being the softest) suggested by 19th-century mineralogist Friedrich Mohs for classifying the hardness of various minerals, diamonds are rated 10, higher than all other naturally occurring materials. This means diamonds may be used for processing such as cutting, grinding, and polishing other materials, including other diamonds.

Graphite is another carbon polymorph. However, it is made up of layers containing hexagons of carbon atoms, as depicted in the image below. Inside each such layer, the carbon atoms are bonded in strong covalent bonds, such that each carbon atom in the plane is bonded to three adjacent atoms. The layers are connected by weak Van der Waals bonds. These weak bonds make graphite soft (1-2 on the Mohs hardness scale) and allow it to be used (mixed with other materials, such as clay) to produce a pencil. Additional uses: steel manufacturing, nuclear reactor rods, and reinforcing components in composite materials.

In 1985, an additional carbon configuration – fullerene – was discovered. This is a carbon molecule (based on covalent bonds) with a hollow spherical shape (geodesic structure). The discovery of this configuration, which exists in soot (tiny amounts of fullerene may even be found in wax candle soot) and in interstellar space won Robert Curl, Harold Kroto, and Richard Smalley the 1996 Nobel Prize in Chemistry. Buckminsterfullerene, or carbon 60 (after the 60 carbon atoms of which this molecule is composed) is the first fullerene discovered in the fullerene family of molecular crystalline structured materials. In time, additional members of this family were discovered. All fullerenes are made up of a combination of hexagonal and pentagonal rings. Buckminsterfullerene is named after the famous American architect and mathematician Richard Buckminster ("Bucky") Fuller, who designed the American geodesic pavilion at the Montreal Expo of 1967. Since the C60 molecule is shaped like a soccer ball, it won the nickname Buckyball.

Figure 10: Graphite is made up of carbon atom hexagons, bonded in the plane in strong covalent bonds. Each carbon atom (blue spheres) is bonded to three other carbon atoms in the plane. The planar layers are bonded in weak Van der Waals bonds, making graphite soft and suitable to be used in pencils. When we use a pencil, the hexagonal carbon planes slide one over the other, become detached and remain on the paper.

Figure 11: C60 (buckminsterfullerene) molecule made up of sixty carbon atoms forming a geodesic structure. The blue spheres represent carbon atoms. The molecule is made up of 20 hexagons and 12 pentagons – a configuration identical to the classic configuration of leather patches in a soccer ball.

Figure 12: Armchair-type carbon nanotubes. This molecule is reminiscent of a bent graphite surface. The blue spheres represent carbon atoms. Nanotubes are a kind of elongated fullerene, which seems like a graphite plane bent over and closed like a ring, to which a fullerene hemisphere has been added on each side. Figure: Dana Ashkenazi.

There is a molecular crystal in which the motif located in a facet-centered cubic cell's lattice points is made of fullerene molecules, rather than single atoms. Fullerene crystals may be produced in a lab, for instance by conducting a powerful electric current between two graphite electrodes within an atmosphere of inert gases. Recently, research and development in the fullerene area has been expanding and progressing rapidly, a trend which is expected to continue well into the future.

Carbon Nanotubes

Carbon Nanotubes are a type of elongated fullerene combined by bending a single graphite plane and adding a fullerene hemisphere on each side of the cylindrical structure, creating a closed, hollow structure. These nanotubes are made up of a hexagonal carbon layer, much like graphite. However, unlike graphite, nanotubes also contain pentagons, and sometimes even heptagons, preventing the surface from being planar.

This additional carbon structure, first discovered in 1991, does not exist naturally and may only be produced artificially. Carbon nanotubes have distinct properties resulting from their unique structure. They have high electric conductivity, and also offer a rare combination of high strength and excellent flexibility. Potential future uses of nanotubes include precise electronic circuits, high-resolution TV monitors, medical uses (such as focused destruction of cancer cells), bridges, particularly light and strong chassis, and airplane parts.

Bio-Mineralization

Crystals and crystallization processes also occur in the living body. The main inorganic component of bone is a mineral called apatite, a member of the calcium phosphate family. In their synthetic form, ceramic apatite minerals promote the growth and attachment of hard tissues such as bone. Therefore, for the past thirty years or so, many orthopedic and dental implant producers coat their products – made basically out of metals such as titanium and its alloys, with synthetic apatite. These coatings improve the implant's affixation by creating osseous tissue around it, preventing the creation of weak fibrous tissue in the implant/bone interface. In the biomaterial and corrosion lab at Tel Aviv University, to which the authors belong, various processes affecting the interaction of bone-building cells with the surface have been studied over the past few years. Below microscopic images are shown, illustrating the surface morphology of electrochemically produced synthetic hydroxyapatite coatings.

Figure 13: (A) Scanning electron microscope (SEM) photograph illustrating the plated morphology and surface pores of synthetic hydroxyapatite coating, electrochemically sedimented over pure titanium. (B) Atomic force microscope (AFM) photograph taken after 60 minutes of electrochemical coating process over pure titanium.

Figure 14: Characterization of Zr69.5Cu12Ni11Al7.5 quasicrystals (the numbers represent atomic percentages) using transmission electron microscopy (TEM). (A) Bright- field image (created when the main electron beam passes through a shutter, while diffraction electrons do not) shows spherical quasicrystals surrounded by an amorphous matrix (the matrix is the continuous phase). (B) Electron diffraction off the spherical phase, indicating the presence of a quasicrystalline structure.

Israeli Innovativeness

In 1984, Professor Dan Schechtman of the Material Engineering Faculty of the Technion astonished the global crystallography community when he and his colleagues published a revolutionary paper discussing the discovery of crystals with so-called "forbidden" symmetry. They had made this discovery already in 1982, but due to the doubts raised by various researchers in the scientific community, it took no less than two years for the paper to be published in a reputable journal.

The crystals discovered by Dan Schechtman, called quasicrystals, are not arranged according to the symmetry rules of classic crystallography, according to which a "crystal" is defined as a three-dimensional configuration of atoms with translational periodicity along its three main axes. The two-dimensional space may be fully "paved" by square, rectangular, triangular or hexagonal "tiles". This is because these polygons have point angles equal to whole quotients of 2π. Combining 2-, 3-, 4- and 6-order rotations (that is, rotations at a rate of 2π/2, 2π/3, 2π/4, and 2π/6, respectively) with the 14 Bravais Lattices produces 230 space groups.

Quasicrystals, however, embody a new kind of order, somewhere between crystalline and amorphous. They are defined as material with perfect long-range ordering, lacking three-dimensional translational periodicity. The first part of this definition refers to the occurrence of sharp diffraction points, and the second part – to the occurrence of rotational symmetry which is not governed by the rules of classic crystallography (that is, rotations of the fifth or higher-than-sixth order). Such rotations do not allow for periodic space tiling. In pentagonal tiles, for example, the point angles equal 108° or 2π/3.333. This means that around a given lattice point, only three pentagons may be located, with a remaining angle of 36°. A common unidimensional illustration of the quasi-periodic configuration is the Fibonacci Chain.

The discovery made by Schechtman and his colleagues shattered crystallographic conventions and led to the discovery of hundreds of new crystalline configurations, which until then had been considered impossible. It was also found that the quasicrystalline configuration affects material properties. For example, quasicrystals are bad electricity and heat conductors, with a low friction factor, high hardness, and high corrosion resistance.

Summary

Ever since the dawn of humanity, people were curious about crystals and their structure. However, only in modern times did researchers such as Bravais, von Laue and Bragg lead to scientific breakthroughs in crystallography. Crystals are characterized by a long-range periodic three-dimensional ordering of atoms, ions, or molecules.

This configuration affects the properties of the material, whether it synthetic (such as steel) or natural (such as bio-minerals or diamonds). Sometimes, the crystalline structure of a given material changes as a result of temperature or pressure changes, a phenomenon known as polymorphism, affecting the material's properties in turn.

Israeli scientists made an important contribution to crystallography – the discovery of quasicrystals with "forbidden symmetry" by Professor Dan Schechtman and his colleagues. We may expect more breakthroughs in the area of crystallography leading to the mapping of new materials and to further technological developments.

Glossary

Crystal - A material with a long-range three-dimensional periodic ordering of atoms, ions, or molecules.

Crystallography – The science of crystals devoted to the empirical study and mapping of crystalline structures, including the way they are created and their properties.

Interference - A phenomenon characteristic of wavelike behavior. Wave interference is the result of combining two waves (of the same kind). The result of this combination depends not only on their amplitude but also on their phase difference. Whenever two waves of equal amplitude but with one half (or any odd number of halves) of a wavelength's phase difference, the result is destructive interference, as the apex of one wave offsets the other's trough, and vice versa, so that there is no oscillation (zero amplitude).

Minerals - Naturally occurring, inorganic solids, mostly crystalline, with specific physical and chemical properties.

Polymorphism - A phenomenon typical of certain types of materials, in which a material with a constant chemical composition takes on various crystalline structures under certain pressure and/or temperature conditions.