Zinc and Cadmium

An opportunity to talk about Hume-Rothery rules and order-disorder transitions, as well as zinc


  1. Introduction to Zinc and its Uses
  2. Physical Properties of Zinc
  3. Chemistry of Zinc
  4. Mineralogy and Production of Zinc
  5. Brass and its Secrets
  6. Appendix: Sphere Packing
  7. References

Introduction to Zinc and its Uses

The symbol in the title is an early chemical symbol for zinc, in the spirit of alchemical symbols. It had to be invented by Olof Bergman of Uppsala, in the 18th century, because metallic zinc lacked a traditional symbol.

Zinc was named by the Swiss alchemist Theophrastus Bombastus von Hohenheim (Paracelsus, 1493-1541), who coined the new Latin word zincum from antecedents that are not clear. A name was necessary for the newly-prepared metal, although its alloy, brass, had been known since ancient times. In English and French, this became zinc, in German and Dutch zink, in Spanish cinc, in Welsh sinc (pronounced "shink"), in Greek pseudargyros ("false silver") or tsigkos, pronounced "tsingos." In Russian, it is tsink. The kitchen sink has nothing to do with zinc, unless it happens to be made from it.

Cadmium was only discovered in 1817 as an impurity in zinc by Stromeyer, who studied the deposits in lead and zinc furnaces called cadmia fornacis, and accordingly the metal was called cadmium.

After iron, aluminium and copper, zinc is usually the fourth-most used metal, competing with lead. We probably see it every day, use it nearly as often as a source of electrical power, and handle small parts made mostly of it. This would be enough to make the metal interesting, but it also gives us valuable insights in physics and chemistry, and gives us an excuse to discuss them in this article. Some time ago, when new science courses for liberal-arts majors were being discussed at the University of Denver, a course entitled WATER was proposed, probably because of the role of water in our environment. Professor Edgar Everhart, the comet-chaser, considered this less than attractive to the usual liberal-arts student, and suggested another possibility, ZINC. Well, we didn't follow up either suggestion, probably correctly, but preparing this article recalled the incident to me. So here, then, is ZINC. The metal cadmium will also appear, since it is similar to zinc and closely associated with it, and is also useful, though much less used.

The largest use of zinc is as a protective coating for iron. The process is called galvanizing with reference to the cathodic protection the zinc offers to the iron. The name was coined by Sorel, who patented the process in 1836. Some sources say that galvanizing began at Swansea, at the early zinc smelter, in 1740. It could not have been called "galvanizing" then, of course. It is usually carried out by dipping carefully cleaned iron or steel in molten zinc, which gives the rather thick, robust coating required. Tin, another familiar protective coating, is now usually applied electrolytically, since it is easier to make a reliable thin coating this way. Tin is much more costly than zinc, which is an incentive to thin coats. Galvanized steel cannot be used in cans for food preservation as tin is, however, because the zinc coating is attacked by food acids to which tin is immune. Zinc, however, gives excellent protection against the weather and moisture, so it is preferred where this is important, since it is cheaper. Zinc protects the iron by cathodic protection, since it is higher on the electrochemical scale than iron and will sacrifice itself to protect the iron, reducing it to the metal and eliminating rust. This phenomenon was noted by Faraday in 1829. If it sacrifices too much, however, the iron is exposed to oxidation, as is sometimes seen with old or damaged galvanized iron.

The life of galvanized steel depends on the thickness of the coating and the environment. The life is approximately proportional to the thickness of the coating, however it was applied. A coating of 1 ounce per square foot, giving a film 0.0018" thick, has a life of about 25 years in rural locations, 10-15 years in an urban environment. Note that the coating is on both sides of a sheet, so the total zinc use will be 2 oz./sqft in this case. Zinc will not give cathodic protection if it becomes passivated, or covered by a closely adherent layer of hydroxide, since then the necessary currents cannot flow. However, the layer will protect the zinc from corrosion, also protecting the underlying metal. Zn(OH)2 is insoluble for pH between 6 and 13, and in this range the hydroxide will protect the zinc under water. Aluminium and chromium are protected and passivated by the oxides, lead by lead sulphate.

Iron or steel articles can be mixed with zinc dust and tumbled in a steel drum at about 370°C, below the melting point of zinc. The articles must first be thoroughly cleaned, by pickling in 50% HCl, or a similar process. The zinc alloys with the surface to form a thin but very adherent layer that protects the iron underneath, and will not clog fine details such as threads. About 15 mg/cm2 of zinc is used. This process is called Sherardizing, after Sherard Cowper-Coles, who patented it in 1901. Iron can be coated with chromium or aluminium by similar processes. These procedures are known in general as cementation, in which an alloy is formed without melting.

The next use in tonnage is as wrought zinc, that is, as plate, tubing, and other forms made by rolling or other shaping processes. Unlike tin or aluminium, zinc is insufficiently malleable to form good foils, so we do not see them. Sheet zinc was once used for roofing, where it has a very long life in this application destructive to most metals. In contact with the atmosphere, which contains H2O and CO2, a closely-adhering layer of basic zinc carbonate, Zn(OH)2·ZnCO3, forms and protects the metal. Sheet zinc is now used mainly for battery anodes, in which it is effectively "burned" to produce electrical energy. Batteries are little fuel cells burning zinc, and are a more practical device in their field of application than fuel cells burning alcohol or hydrogen and using the oxygen of the air, at least so far.

Closely following this use, and perhaps exceeding it at times, is the use of zinc to make die castings. The most common process is pressure die casting, in which the molten zinc is forced into steel dies that make the mold. Zinc expands on cooling, so it fills the mold exactly, like type metal, and can make precision castings requiring very little machining. The low melting point of zinc gives long die life. Automobiles are full of die castings, from brake cylinders and fuel pumps to door handles, many of which are plated to give them a shiny finish. Carburetors at one time were made up nearly completely of die castings. In mass production, die castings are much cheaper than machined parts, since the large cost of the dies can be amortized over the numerous products. Die castings required the production of extremely pure zinc, since the usual impurities caused the castings to swell and "crystallize" in a short time. They didn't actually crystallize, of course, but the impurities migrated to the crystal boundaries and caused embrittlement. A typical die-casting alloy is Zn 94.9, Al 4.1, Cu 1.0, called "Zamak." Lead, cadmium and tin impurities must be kept to a very low level in the zinc used for this purpose.

Small amounts of zinc are used for other purposes. Zinc has occasionally been used in coins, such as the Albanian 1/2 Leku and the United States cent. Zinc chloride, ZnCl2 is used as a soldering flux for soldering iron with tin-lead solders in a water solution. It hydrolyzes to give an acid reaction, ZnCl2 + 2HOH → Zn(OH)2 + 2H+ + 2Cl-, which is not as corrosive as pure hydrochloric acid would be. Zinc is used in aluminium solders, such as 75 Zn, 20 Cd, 5 Al, which is Bureau of Standards aluminium solder ZN1. Zinc is also used in silver solders, such as 52 Cu, 38 Zn, 10 Ag, which melts at 820°C. This is actually a silver-bearing brass, which gives the name "brazing" to the process. Zinc oxide, ZnO, is a white pigment, and "blue powder," a colloidal dust of small spheres coated with oxide, is used in paints for protecting ships. Zinc oxide, ZnO, with 0.5% ferric oxide, Fe2O3 is the active ingredient in calamine lotion, which soothes irritated skin. Zinc appears in very small amounts in the mineral supplements favored by those who think eating it will improve the health. Zinc is required in the diet in such small amounts that it is always present in a normal diet without the need for supplementation. Zinc appears to be rather non-poisonous, though in large quantities it is carcinogenic. It is used in several lotions applied topically. If zinc oxide is breathed, the strange nervous malady "oxide shakes" seems to result.

Cadmium is a by-product of zinc production, much rarer and used only in small amounts. Cadmium is prettier than zinc, since it is whiter, like tin or silver. A few percent hardens and strengthens copper without decreasing its conductivity greatly. Cadmium copper is used for electrical contact wires, where durability and low resistance are both desirable. It also is used as a protective coating for iron, in very thin films. These films alloy with the iron and are hard to damage in use, so cadmium-plated screws and bolts are found. It is used as the anode in rechargable nickel-cadmium cells. Its main use is in alloys, where it can replace the more expensive bismuth in fusible alloys. It can also replace expensive tin in some applications, such as solders. However, cadmium is extremely poisonous, more so even than mercury or lead. It attacks the kidneys, among other things. Cadmium may be essential to the rat metabolism, it is believed. Its use has been strongly discouraged where its vapor or dust may be created, or it is disposed of carelessly. However, cadmium is being persecuted, like mercury and lead, although it is a completely negligible hazard in its normal uses.

Physical Properties of Zinc

Zinc is a bluish-grey metal covered by a protective transparent layer of basic carbonate in air. A sheet of zinc looks very much like a sheet of aluminium, but it is more than twice as heavy, and does not bend easily. Zinc is not very ductile or malleable, especially when pure. Its atomic number is 30, atomic weight 65.38. Its naturally occurring isotopes are 64 (49%), 66 (28%), 67 (4%), 68 (19%) and 70 (0.6%). Its density is 7.14 g/cc, electrical resistivity 6.16 μΩ-cm, heat capacity 0.0925 cal/g-K, and heat conductivity 0.268 cal/cm-s-K. Its coefficient of linear expansion is 40.0 x 10-6 per K. Zinc melts at 419.5°C and boils at 907°C. The heat of fusion is 24.09 cal/g. In the cast form, its tensile strength is only 4-12 ksi, but the cold work of rolling gives 28-36 ksi. Hard-drawn zinc has a strength of about 10 ksi. The Young's modulus is 12.4 x 106 psi. Zinc, at Mohs 2.5, is harder than tin or cadmium. Its crystal form is hexagonal close packed, with a = 0.266 nm, c = 0.494 nm. The ionic radius of Zn++ is 0.074 nm. The ionization potentials of zinc are 9.36V and 17.89V.

Cadmium is a white, sivery metal covered by a protective layer of grayish-white hydrated oxide in moist air. It is malleable and ductile, and can be hammered into a thin foil. At Mohs 2, it is harder than tin, but like it emits sound when it is bent. Above 80°C it becomes brittle and can be pulverized. Its atomic number is 48, atomic weight 112.41. Stable isotopes of cadmium exist for all the even numbers between 106 and 116, plus 111 and 113. Its density is 8.65 g/cc, electrical resistivity 7.59 μΩ-cm, heat capacity 0.0552 cal/g-K, and heat conductivity 0.217 cal/cm-s-K. Its coefficient of linear expansion is 29.8 x 10-6 per K. Cadmium melts at 321°C and boils at 765°C. The heat of fusion is 13.17 cal/g. Its tensile strength is 13.5 ksi. Its crystal form is hexagonal close packed, with a = 0.297 nm, c = 0.561 nm. The ionic radius of Cd++ is 0.097 nm. The ionization potentials of cadmium are 8.96V and 16.84V.

Chemistry of Zinc

The electron configuration of zinc is 1s22s22p63s23p63d104s2. Cadmium has a further 18 electrons in the N shell, with the valence electrons 5s2. Both atoms show a valence +2 in their compounds. Pure zinc shows almost no reaction with water or dilute acids, because of the formation of a thin layer of hydrogen gas on its surface, or "polarization." Impure zinc, or zinc in the presence of copper or platinum, reacts readily with the evolution of hydrogen. Arsenical zinc is used for making hydrogen in the chemical laboratory. The reduction potential of zinc is -0.76V, and of cadmium, -0.40V, so both are relatively reactive.

Zinc forms a hydroxide, Zn(OH)2 that can dehydrate to form the anhydrous oxide ZnO, form Zn++ salts in acid solution, or zincates, ZnO2 --, in alkaline solution. In the former case, an acid strips off the OH- to form water, and in the latter case the alkali strips off the H+ with the same end in mind. Therefore, zinc is amphoteric, like aluminium. ZnO will not dissolve in water to form the hydroxide, but will dissolve in acids to form zinc salts. In the presence of ammonium ion, zinc forms a tetrammino complex, Zn(NH3)4--.

Zinc chloride is hydrolyzed in solution, but does not give off HCl on evaporation. Its use as a soldering flux has already been mentioned. It is a drying agent and a catalyst, and is used for preserving wood. It is not as good as creosote for this purpose, and is easily leached out. If zinc chloride solution is swallowed, the whites of egg are an antidote. Zinc salts are poisonous, if not violently so. Zinc sulphate is less toxic, and is used in medicine as an astringent and antiseptic, and also as a mordant, because gelatinous zinc hydroxide is produced when it is hydrolyzed. Zinc sulphide, remarkably, is white, unlike all other sulphides. When it is found in nature as sphalerite, however, impurities darken it and it is even called Black Jack. When pure, it has a large band gap and there are few charge carriers. Impurity centers phosphoresce easily, and ZnS is one of the most useful phosphors, also responding to alpha particle bombardment as well as to electron bombardment and ultraviolet. As with most phosphors, the ZnS is merely the vehicle that holds the fluorescence centers provided by impurities. Zinc cyanide, Zn(CN)2, is only very slightly soluble, a fact used in the cyanide process for the production of gold and silver. The nitrate, carbonate, acetate, borate and stearate also have their uses.

The chemistry of cadmium is very much like that of zinc, except that it is less active and not as acidic in alkaline solution. Cd(OH)2 is produced by alkalis, but the hydroxide does not dissolve in them to form cadmates. CdO is brown, and CdS is bright yellow, the pigment called cadmium yellow. Cadmium also forms a tetrammino complex ion. In a few compounds, cadmium exhibits valence +1, such as cadmous oxide, Cd2O and cadmous chloride Cd2Cl2, which may resemble mercurous chloride. Cadmous ions are strong reducing agents. The bright yellow sulphide is good confirmation of the existence of cadmium in qualitative analysis, as the white sulphide is of zinc.

Mineralogy and Production of Zinc

The source of most zinc is the sulphide, sphalerite, ZnS. Sphalerite is closely associated with galena, PbS, pyrites, FeS2 and other sulphides in hypogene ore deposits, and must be separated from them in smelting. Sphalerite is also called blende, which means "blind" or "deceiving," because although resembling galena and occurring with it, yields no lead. Since many minerals may do this, it may specifically be called Zinc Blende. Sphalerite, indeed, is from the Greek for "treacherous." Another term is Black Jack, from its typical dark appearance, though the pure substance is white. Good crystals can be transparent, with a resinous lustre, and make attractive gems. Transparent sphalerite is often green. Sphalerite is cubic in crystal structure. ZnS crystallized in hexagonal crystals instead is called wurtzite.

In the United States, an outstanding occurrence of sphalerite was in the tri-state area of Missouri, Oklahoma and Kansas, centered on Joplin. Here (and in other Mississippi valley localities) it was associated with lead, but silver was absent. In other areas, silver is usually present, and its recovery makes the ore more valuable. The gravelly slag from the smelting of zinc is called "chats" and was a good railway ballast.

Greenockite, CdS, is associated with sphalerite, but in such dispersion that there are no workable deposits of it. It is yellow or orange, and forms hexagonal crystals, though crystals are rare, and it is mainly a powdery incrustation. It is found in the tri-state ores. Cadmium would be very difficult to obtain if it were not a by-product of zinc smelting.

Franklinite is a complex iron, zinc and manganese oxide found only at Franklin, NJ in company with other zinc minerals. It is slightly magnetic, which can help in separation. Zincite is ZnO. Although the oxide is white, zincite is usually colored red or orange by a manganese impurity. Willemite, the third member of the trio found at Franklin, is ZnSiO4, zinc orthosilicate. It is almost always fluorescent in ultraviolet light, like its companions. These minerals are found in limestone, and seem to be metamorphic products. They were the second most important source of zinc in the United States.

Hemimorphite is a hydrated zinc silicate, Zn2Si2O7(OH)2·H2O, fairly hard but light, and strongly pyroelectric. It is usually found in supergene enriched zones, together with smithsonite, ZnCO3. Each of these minerals has been called calamine at different places and different times, so to avoid confusion the term calamine is no longer recommended, though if it is used it refers to the silicate. In Britain, it always referred to the carbonate. Smithsonite is very easy to reduce to zinc. It is usually brownish, and often contains considerable cadmium. Smithsonite was found in the zinc ores of Leadville, CO.

The usual process for the extraction of zinc from sulphide ores is as follows. First, the ore must be roasted in air to burn off the sulphur as SO2. When this gas is released to the atmosphere, it devastates the vegetation over a considerable area. However, it can be recovered and made into sulphuric acid, which can be used in the smelting process. A good deal of acid is left over for sale to improve the economics. The oxide from the roasting is then leached with the acid produced from the sulphur to produce zinc sulphate, which is then electrolyzed to the metal. This produces a good, pure zinc and is the currently favored process. An important electrolytic zinc plant was operated at Midvale, Utah.

Alternatively, the oxide from roasting can be mixed with carbon, from anthracite coal for example, and sintered if desired. When this is heated in a retort to about 1100°C, the zinc vapor is driven off and condensed in molten zinc. Under certain conditions, "blue powder" can be formed, that is a valuable product in its own right. It consists of colloidal zinc particles covered with ZnO, produced by the reaction of Zn with CO2. It will not melt to zinc metal. The condensed zinc from the retort is cast into ingots, and the material is called spelter. Before metallic zinc was produced in Europe around 1600, it was imported from India (and China, but was produced mainly in India) and called "spiauter" or something like that, which gave us spelter.

Spelter can be refined by liquation, where impurities are skimmed from the top, and the liquids separated into one containing lead and cadmium, and the other purified zinc. Better purification can be obtained by fractional distillation, since zinc vaporizes at a conveniently low temperature. There is also electrolysis, but zinc produced by this method is quite pure anyway. All purification costs money and loses zinc, so where less purity is satisfactory, it is allowed. Lead, copper and tin, for example, do not have a bad effect on brass. For die casting material, on the other hand, it is essential to reduce lead, copper and tin to very small amounts to avoid swelling and embrittlement.

Cadmium is recovered in the purification of spelter, or in the initial treatment of the ore. At Midvale, copper, cadmium and a little zinc and other impurities are precipitated from the zinc sulphate solution that is prepared for electrolysis. This is roasted, to reduce arsenic, leached with sulphuric acid, and then the sulphates of zinc, iron and lead are precipitated by lime, and copper by sodium sulphide. It is then filtered; the filtrate contains about 12% Cd. The cadmium is deposited by electrolysis on sheet aluminium cathodes. The cathodes are stripped, melted under NaOH and cast into ingots. There are various methods for cadmium recovery, depending on the characteristics of the particular zinc smelter.

The history of zinc is peculiar. It was widely used before the metal was discovered. By adding zinc carbonate to copper ores that were being smelted, brass was the product. This, as we shall see, was a very desirable metal that had a color like gold, was easy to work, and corrosion-resistant. Brass coinage was issued in the time of Augustus, and is well attested. Brass was just another variety of aes, like bronze and copper itself. It was just a flavor, as lead and tin were flavors of lead, black and white. If zinc metal had been known then (and it might have been) it would be considered just another flavor of lead, which had so many, as copper did. In fact, brass was called aurichalcum, "gold-copper."

We saw above that the critical process in making zinc was reduction in a closed retort, away from the air, and the condensation of the resulting vapors. This was not a customary process in classical metallurgy, which was a matter of heating things in an open hearth and adding all kinds of magical ingredients. There was no way that zinc would run as a shining silvery stream from any such hearth, as lead, tin, copper, silver and gold did. It would be found as ZnO dust in furnace cracks, as impure blue powder, but never as a recognizable metal. The example of mercury, which was condensed from vapor, might serve as a hint, but zinc, unlike mercury, was chemically active and combined with oxygen. So, it is not impossible, merely unlikely, that anyone would have been inquisitive enough to try to keep the air away and condense zinc metal. If it had been done, the zinc would not even be attractive enough to make acceptable trinkets.

The retort was apparently hit upon in India, and zinc metal produced, which was traded with China, and may have been used in alloys. Although we have credited Paracelsus with the name, he was certainly not the discoverer, merely one of the workers at the frontiers of science. Apparently Agricola had heard of zinc too, and mentions it in his book, which was published before Paracelsus's was. There was, as usual, great confusion in the names of compounds in the absence of certain knowledgy of their composition. Cadmia furnacis was one variety of deposits of furnace dust that contained zinc, as well as the cadmium discovered much later. Cadmia fossilis was, apparently, Smithsonite or something similar, that made brass. The recognition of zinc as a particular metal occurred about 1600 in Europe, but its first use was some two thousand years earlier.

The early centers of zinc smelting in modern Europe were at Swansea (1720) and Bristol (1740) in England. Zinc ores were typically transported large distances for smelting. Today, when the SO2 is recovered, this allows the sulphuric acid to be produced near its markets, instead of at the distant mines, improving the overall profitability of the operation. Roasting of sulphide ores was an environmental disaster, earlier best kept near the mines, as at Anaconda, Montana. Like its reserves of iron ore, bauxite and copper, the United States reserves of zinc and lead are now greatly depleted, and dependence on foreign sources is increasing. The current price of zinc is about $797 per metric ton.

Brass and its Secrets

Brass is an attractive, easily worked alloy of copper and zinc that has been appreciated since antiquity. It will serve us here as an example of crystalline metals and alloys, for it has a lot to teach. Copper melts at 1083°C, while zinc boils at 907°C. Tin does not boil until 2260°C, so when we make bronze we do not lose tin by evaporation. In making brass, however, some loss of zinc is inevitable. We must make brass by first melting the copper at the white heat required, and then add small quantities of zinc as quickly as possible. As we do so, the melting point of the alloy falls, and we can let the ladle cool to get below the boiling point of zinc as soon as possible. Unfortunately, this does not happen until the zinc content is about 30%, close to the alloy we may want. If we keep adding zinc, we eventually get a liquid melting just above 420°C, the melting point of zinc. Quite a few things happen along this road, as we shall see.

The toughness and strength of brass increase steadily as zinc is added up to a composition of 70 Cu, 30 Zn, called cartridge brass for its ability to be formed into cartridge cases and other shapes requiring deep deformation. Its strength is about 38 ksi. Commercial brass, 90 Cu 10 Zn, Red brass, 85 Cu, 15 Zn, and Low brass, 80 Cu, 20 Zn are weak alloys with less zinc. Standard brass is 65 Cu, 35 Zn. The strength has increased a little, to about 42 ksi, but the hardness is rapidly increasing. Muntz metal, at 60 Cu, 40 Zn, is about the highest proportion of zinc used in brass. The tensile strength is the maximum, about 46 ksi, but it must be worked hot because at room temperature it is too hard, since it contains the hard, brittle β-brass. Muntz metal (and similar alloys) are used in condensers for ship's engines, which is rugged duty because of the sea water. 2% to 3% lead makes high-zinc brass easy to machine, since the lead acts as a lubricant. It does not enter the alloy, precipitating in tiny lumps. Adding 1% tin to 70-30 brass or Muntz metal gives increased corrosion resistance. Above 50% Zn, new weak, brittle phases are formed, which make the alloys useless. Brass would seem to be a good metal for coinage, but it has seldom been used for that purpose, though a low-zinc alloy called "bronze," Cu 95, Zn 4.5, Sn 0.5, is quite popular, though not as good-looking or durable. German silver, or Neusilber, 60 Cu, 25 Zn, 15 Ni, has been used for coins, however. It is a high brass with a silvery color, lent by the nickel, that electroplates easily. It was used in the EPNS (electro-plated nickel silver) tableware from Sheffield, which was an excellent and popular product.

Let's now look at the reasons for these properties that depend on the zinc concentration. They all are consequences of the crystalline structure of the alloy. Copper has one valence electron. When it is removed, the Cu+ ion has a diameter of 0.255 nm, and is a fuzzy sphere of electrons with the nucleus deep in the center. The Zn++ ion is a similar fuzzy sphere, of diameter 0.266 nm. When copper or zinc atoms are brought together, their equilibrium state is with the atoms ionized, and surrounded by a sea of almost free electrons, not associated with any particular ions. If the metal is in contact with a thermal reservoir at temperature T (the rest of the universe), then the equilibrium state will be the one for which the free energy F = U - TS is a minimum. Here, U is the internal energy and S is the entropy. If the system were isolated in space, of constant volume, and completely up to its own devices, the condition for equilibrium would be that S is a maximum. If it is isolated, it cannot change its energy U, so energy is of no consequence in looking for equilibrium. Maximum S occurs when the state is such that the largest number of quantum states are randomly available to it.

We are used to thinking that a system is in equilibrium at its minimum energy, but this is not useful in the realm of atoms. In general, the energy and the entropy compete to determine equilibrium. The energy U of a system is a function of its volume and its entropy, U = U(V,S). In a small change, dU = pdV - TdS, which defines the pressure p and the temperature T. In fact, T is only ∂U/∂S, which shows us that if two systems are isolated, but in thermal contact with each other, the condition dS = 0 is the same as T1 = T2. If one of the systems is very much larger than the other, it is a "thermal reservoir" of temperature T, and dS = 0 for the universe becomes dF = 0 for the small system, provided the volume is held constant. Entropy is a more basic concept than temperature, and one cannot exist without the other. The entropy is S = k ln W, where k is Boltzmann's Constant, and W is the number of available states of equal probability that the system could be in. Entropy (like energy) is not a substance, simply a mathematical parameter describing a system. The usefulness of entropy is excellent evidence for the existence of atoms!

In our system of ions and electrons, the electric forces are very strong, and the minimum energy comes when they are snuggling as closely as possible. On the other hand, the entropy is greatest when everything is moving freely, ions as well as electrons. When the ions pack like marbles, U will be the least, since the charges are closest together. However, S will be low, because of the order of the packing. At room temperature, for most metals, U wins out over S and the ions pack like marbles and the electrons fill the space between them as glue. This is the reason crystals exist, because of strong attractive forces that overcome the blandishments of entropy. If we raise the temperature, however, the entropy becomes more and more favored, and at some point the large entropy of randomness becomes more compelling than the low energy of regularity, and the crystal melts. In the case of mercury, entropy has already won out at room temperature.

There are very many electrons, and since the electrons are very light, there are hardly enough quantum states available to accommodate them. These states occupy volume in space, and in momentum space as well. In momentum space, we specify the momentum of the electron states, which is inversely proportional to the wavelengths of the quantum states. The available number of states increases as the square of the kinetic energy. There are so many electrons that they fill up the states to energies far above the thermal energies they would have at room temperature. In fact, ordinary temperatures hardly affect them at all, and they are pretty much as they would be at absolute zero. In that case, they fill up momentum space to an energy called the Fermi energy. This energy is of the order of 8 eV. Note that the thermal energy at room temperature is only about 0.025 eV. All directions in momentum space are equivalent, so this energy surface is a sphere. As we add more electrons, it expands like a balloon.

The ions form a regular crystal lattice. Real lattices are not perfect, and the imperfections are interesting, but like Powdermilk Biscuits, they are perfect, mostly. For every lattice in space, there is a reciprocal lattice in momentum space. To find out more about this somewhat difficult subject, refer to Kittel, Chapter 2. Just as the space lattice is made by infinite repetition of a unit cell, so is the reciprocal lattice. The unit cell in the reciprocal lattice is called the (first) Brillouin zone. The reciprocal lattice for the fcc structure is bcc, and the Brillouin zone is a truncated octahedron. Now imagine the Fermi sphere inflating in the Brillouin zone as electrons are added to the system. For each electron, we simultaneously add an ionic positive charge, so the system remains neutral. For each copper ion, we get one electron, while for each zinc ion, we get two electrons. Therefore, by varying the proportions of copper and zinc, we can get anything between 1 and 2 electrons per ion. Copper and zinc ions are about the same size, so except for the electric charge, they are pretty much the same so far as the crystal is concerned.

It happens, however, that the Fermi sphere soon hits the boundary of the Brillouin zone, and cannot go any farther. A balloon would begin to fill up the corners, but the Fermi sphere cannot do this. Instead, the electrons must go into states of considerably higher energy if they are to fit in the system. This raises U, and destabilizes the system. There is, however, more than one way to stack the ions, and each way will have a different Brillouin zone, whose size may be different even for ions of the same size. Now marbles can be stacked in many ways. If they are as close together as they can be, they are said to be close-packed, and there are two regular ways to do this, one of which gives a face-centered cubic crystal lattice (fcc), and the other a hexagonal close-packed crystal lattice (hcp). There is also a body-centered cubic crystal lattice (bcc) that is not quite close packed, but has certain advantages. In addition to these, there are others that the ingeniuity of the ions can discover in certain cases. Anyway, what is important is that the Brillouin zone for bcc is larger than that for fcc, and the Brillouin zone for hcp is larger than that for bcc.

For small electron density, at one electron per ion, the lowest-energy lattice is the fcc. As we increase the number of electrons per ion, the Fermi sphere collides with the zone boundary. Then, instead of using the high-energy states, the system wiggles into a bcc lattice, where the zone boundary is farther away. Again we can add more electrons, keeping the energy as low as possible, until the Fermi sphere again collides with the boundary. The system wiggles again, this time into a hcp structure, which is a little higher in energy, but has a still more commodious Brillouin zone. The inflation of the Fermi sphere continues, and it will fit in the zone for two electrons per ion. This was first deduced by Hume-Rothery, and the electron densities for the changes are called the Hume-Rothery rules.

If we examine the crystal structures of pure metals, we find that those with only one electron per ion are invariably fcc (except for the alkali metals, that crystallize as bcc). Those with two electrons always crystallize hcp or fcc. There are exceptions, since Pb, Ni and Ca are fcc in spite of having two valence electrons, and Mo is bcc in spite of having only one valence electron. Nevertheless, this is good evidence there is someting in what Hume-Rothery says, and more detailed calculations bear it out. It is remarkable that we can make a pretty good guess as to the crystal structure simply by knowing the number of valence electrons!

In making brass, we can vary the electron ratio from 1 to 2, so something should happen. Copper and zinc ions are pretty much the same size, as we have noted, so the electron concentration n should determine the structure. Copper is fcc, and α-brass, the solid solution that occurs for small zinc concentrations, is also fcc. At n = 1.38, or about 40% Zn, the structure begins to change to bcc. β-brass begins at n = 1.48 and continues up to n = 1.58, where a structure called γ-brass begins. Around n = 1.66 we get δ-brass, but only at elevated temperatures. At n = 1.78, the hcp ε-brass comes in for a while, until n = 1.87, and finally we get hcp η-brass, which is the zinc structure, as n approaches 2. In each case, the alloy is determined to fit the Fermi sphere in its Brillouin zone by being flexible. Only the α and β phases are useful alloys; the others are too hard, except for the final η-brass, which is really just zinc. These different structures are called electron compounds since they are determined by the electron concentration.

For an equilibrium diagram of the Cu-Zn alloys, see Kittel, p. 578, or Leighou, p. 228. This is a temperature-composition plot showing the areas in which the different phases are stable, and is a little too complex to receive justice here. It is well worth studying, as containing examples of most of the things that go on in alloys.

An additional phenomenon is shown by β-brass. Its bcc structure consists of two interpenetrating simple cubic lattices, and it exists where the number of copper and zinc ions is about equal. If all the copper is on one lattice, and all the zinc is on the other, such that a cube of 8 copper ions has a single zinc ion in the center, the alloy is said to be ordered. This is a low-entropy structure, so we expect it would be stable at lower temperatures, which indeed it is. At a higher temperature, the lure of higher entropy would suggest that a state where the copper and zinc ions were located at random on the two lattices would be more stable. After all, there is very little difference in energy in the two cases. At about 460°C, the phase changes abruptly from ordered to disordered as the temperature increases. This is a second-order phase transition, where there is no latent heat (the energy is the same in the two phases) but there is a discontinuity (a "lambda point") in the specific heat. This is an example of an order-disorder transition, driven purely by entropy, not energy.

Appendix: Sphere Packing

The ions in a metal pack together something like marbles, so it is good to know something about this. Get some marbles, or better styrofoam spheres, and investigate their packing for yourself. I will only hit the highlights here.

Two layers of close-packed spheres are shown in the diagram. The upper layer is dotted. Note the locations of the centers of the spheres in each layer, marked by dots. When you have laid down the second layer, there is a set of points between the spheres without dots, whichever way you have chosen. In a third layer, the spheres can go exactly over the spheres in the first layer, which is easy to see. Subsequent layers can just repeat this pattern. You are looking down the c-axis of a hexagonal lattice. It is a kind of body-centered lattice with a unit cell composed of four ions in the first and third layers, together with the ion they enclose. This is the hcp structure.

However, the third layer could go over the empty locations as well. With every subsequent layer, a new choice could be made. If the pattern in the first three layers is repeated, then the spheres form a cubic lattice, and you are looking down a 111-axis, that is along a threefold axis that joins opposite corners of the cube, as shown in the diagram. This is the fcc structure. It is not particularly easy to see the cube looking at the layers of spheres, unless you make a model out of styrofoam spheres and glue them together. Then you can easily rotate the model to find the fourfold axes.

The calculation of the distance between layers in terms of the sphere diameter is not difficult. Refer to the diagram at the right. The centers of three adjacent spheres and the center of the one resting on them form a regular tetrahedron whose sides are of length d. We wish to find the altitude of this tetrahedron, and can easily do so from the triangle shown. The hypotenuse is the diameter d, while the long side is 2/3 of the altitude of one of the sides of the tetrahedron, or d/√3. The Pythagorean theorem then gives h = (√2/3)d. Therefore, for the hcp structure the c axis is of length 2h = (8/3)1/2d = 1.6330 d. The a and b axes are equal, and of length d. Therefore, the ratio c/a = 1.633 for an ideal hcp lattice. Many actual hcp lattices have ratios close to this, but some have larger ratios, up to about 1.8. Nevertheless, all of these are classed as hcp and have the same symmetry. It is left as an exercise for the reader to determine the lattice constant a for the fcc structure in terms of the ion diameter.

In the bcc structure, the lattice constant a = (2/√3)d, as can be deduced from the diagram at the left, which shows a 110-plane. The unit cell in the bcc structure or the hcp structure contains two ions, in the fcc structure, 4 ions. The volume per ion is the volume of the unit cell divided by the number of ions in the unit cell, while the volume of an ion is (π/6)d3. In the hcp or fcc structure, the volume per ion is d3/√2, so the fraction of volume occupied is 74%. In the bcc structure, the volume per ion is (4/3√3)d3, so the fraction of the volume occupied is 68%. Note that in either case there is lots of room for electrons, even with the ions in contact.


C. Kittell, Introduction to Solid State Physics, 3rd ed. (New York: John Wiley & Sons, 1966). Direct and reciprocal lattices, Brillouin zones, energy bands, Fermi levels, first- and second-order transitions, alloys and much else. Alloys: pp 576-586. The best solid-state text.

W. N. Jones, Inorganic Chemistry (Philadelphia: Blakiston, 1949), Chapter 34.

J. L. Bray, Non-Ferrous Production Metallurgy, 2nd ed. (New York: John Wiley & Sons, 1947), Chapter 26.

R. A. Higgins, Engineering Metallurgy, 3rd ed. (London: The English Universities Press, 1971). pp. 315-324, 363f.

R. B. Leighou, Chemistry of Engineering Materials (New York: McGraw-Hill, 1942). pp 164-174.

C. S. Hurlbut, Dana's Manual of Mineralogy, 16th ed. (New York: John Wiley & Sons, 1952). pp. 195, 198, 237, 273, 378, 440f, 454f.

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Composed by J. B. Calvert
Created 18 November 2002
Last revised 20 August 2007