Silicate Structures 

As we discussed in a previous lecture, the relative abundance of elements in the Earth's crust determines what minerals will form and what minerals will be common.  Because Oxygen and Silicon are the most abundant elements, the silicate minerals are the most common.  Thus, we will spend some time here discussing the structure, chemistry, and occurrence of silicate minerals.  Our systematic discussion of the common rock forming minerals will follow in the lectures throughout the remainder of the course.

Element Wt% Atomic% Volume% O 46.60 62.55 ~94 Si 27.72 21.22   ~6 Al   8.13   6.47 Fe   5.00   1.92 Ca   3.63   1.94 Na   2.83   2.34 K   2.59   1.42 Mg   2.09   1.84 Total 98.59 100.00 100

In order to discuss the silicates and their structures it is first necessary to remember that the way atoms are packed together or coordinated by larger anions, like oxygen depends on the radius ratio of the cation to the anion, Rx/Rz.

Rx/Rz C.N. Type 1.0 12 Hexagonal or Cubic Closest Packing 1.0 - 0.732 8 Cubic 0.732 - 0.414 6 Octahedral 0.414 - 0.225 4 Tetrahedral 0.225 - 0.155 3 Triangular <0.155 2 Linear

Since oxygen is the most abundant element in the crust, oxygen will be the major anion that coordinates the other other cations. Thus, for the major ions that occur in the crust, we can make the following table showing the coordination and coordination polyhedra that are expected for each of the common cations.

Ion	C.N.  (with Oxygen)	Coord. Polyhedron	Ionic Radius,  Å
K+	8 - 12	cubic to closest	1.51 (8) - 1.64 (12)
Na+	8 - 6	cubic to octahedral	1.18 (8) - 1.02 (6)
Ca+2	8 - 6		1.12 (8) - 1.00 (6)
Mn+2	6	Octahedral	0.83
Fe+2	6		0.78
Mg+2	6		0.72
Fe+3	6		0.65
Ti+4	6		0.61
Al+3	6		0.54
Al+3	4	Tetrahedral	0.39
Si+4	4		0.26
C+4	3	Triangular	0.08

The radius ratio of Si+4 to O-2 requires that Si+4 be coordinated by 4 O-2 ions in tetrahedral coordination.  

In order to neutralize the +4 charge on the Si cation, one negative charge from each of the Oxygen ions will reach the Si cation. Thus, each Oxygen will be left with a net charge of -1, resulting in a SiO4-4 tetrahedral group that can be bonded to other cations.  It is this SiO4-4 tetrahedron that forms the basis of the silicate minerals.

Since Si+4 is a highly charged cation, Pauling's rules state that it should be separated a far as possible from other Si+4 ions.  Thus, when these SiO4-4 tetrahedrons are linked together, only corner oxygens will be shared with other SiO4-4 groups.  Several possibilities exist and give rise to the different silicate groups

Minerals found in earth crust:

Most minerals are made up of a cation (a positively charged ion) or several cations and an anion (a negatively charged ion (e.g., S2–)) or an anion complex (e.g., SO42–). For example, in the mineral hematite (Fe2O3), the cation is Fe3+ (iron) and the anion is O2– (oxygen). We group minerals into classes on the basis of their predominant anion or anion group. These include oxides, sulphides, carbonates, silicates, and others. Silicates are by far the predominant group in terms of their abundance within the crust and mantle.  Some examples of minerals from the different mineral groups are given in Table

Group	Examples
Oxides	Hematite (iron oxide Fe2O3), corundum (aluminum oxide Al2O3), water ice (H2O)
Sulphides	Galena (lead sulphide PbS), pyrite (iron sulphide FeS2), chalcopyrite (copper-iron sulphide CuFeS2)
Sulphates	Gypsum (calcium sulphate CaSO4·H2O), barite (barium sulphate BaSO4) (Note that sulphates are different from sulphides. Sulphates have the SO4–2 ion while sulphides have the S–2 ion)
Halides	Fluorite (calcium flouride CaF2), halite (sodium chloride NaCl) (Halide minerals have halogen elements as their anion — the minerals in the second last column on the right side of the periodic table, including F, Cl, Br, etc. — see Appendix 1.)
Carbonates	Calcite (calcium carbonate CaCO3), dolomite (calcium-magnesium carbonate (Ca,Mg)CO3)
Phosphates	Apatite (Ca5(PO4)3(OH)), Turquoise (CuAl6(PO4)4(OH)8·5H2O)
Silicates	Quartz (SiO2), feldspar (sodium-aluminum silicate NaAlSi3O8), olivine (iron or magnesium silicate (Mg,Fe)2SiO4)   (Note that in quartz the anion is oxygen, and while it could be argued, therefore, that quartz is an oxide, it is always classed with the silicates.)
Native minerals	Gold (Au), diamond (C), graphite (C), sulphur (S), copper (Cu)

Table:  The main mineral groups and some examples of minerals in each group

Oxide minerals have oxygen (O2–) as their anion, but they exclude those with oxygen complexes such as carbonate (CO32–), sulphate (SO42–), and silicate (SiO44–). The most important oxides are the iron oxides hematite and magnetite (Fe2O3 and Fe3O4, respectively). Both of these are important ores of iron. Corundum (Al2O3) is an abrasive, but can also be a gemstone in its ruby and sapphire varieties. If the oxygen is also combined with hydrogen to form the hydroxyl anion (OH–) the mineral is known as a hydroxide. Some important hydroxides are limonite and bauxite, which are ores of iron and aluminium respectively. Frozen water (H2O) is a mineral (an oxide), but liquid water is not because it doesn’t have a regular lattice.

Sulphides are minerals with the S–2 anion, and they include galena (PbS), sphalerite (ZnS), chalcopyrite (CuFeS2), and molybdenite (MoS2), which are the most important ores of lead, zinc, copper, and molybdenum respectively. Some other sulphide minerals are pyrite (FeS2), bornite (Cu5FeS4), stibnite (Sb2S3), and arsenopyrite (FeAsS).

Sulphates are minerals with the SO4–2 anion, and these include anhydrite (CaSO4) and its cousin gypsum (CaSO4.2H2O) and the sulphates of barium and strontium: barite (BaSO4) and celestite (SrSO4). In all of these minerals, the cation has a +2 charge, which balances the –2 charge on the sulphate ion.

The halides are so named because the anions include the halogen elements chlorine, fluorine, bromine, etc. Examples are halite (NaCl), cryolite (Na3AlF6), and fluorite (CaF2).

The carbonates include minerals in which the anion is the CO3–2 complex. The carbonate combines with +2 cations to form minerals such as calcite (CaCO3), magnesite (MgCO3), dolomite ((Ca,Mg)CO3), and siderite (FeCO3). The copper minerals malachite and azurite are also carbonates.

In phosphate minerals, the anion is the PO4–3 complex. An important phosphate mineral is apatite (Ca5(PO4)3(OH)), which is what your teeth are made of.

The silicate minerals include the elements silicon and oxygen in varying proportions ranging from Si : O2 to Si : O4.

Native minerals are single-element minerals, such as gold, copper, sulphur, and graphite.

Silicate Minerals

The vast majority of the minerals that make up the rocks of Earth’s crust are silicate minerals. These include minerals such as quartz, feldspar, mica, amphibole, pyroxene, olivine, and a great variety of clay minerals. The building block of all of these minerals is the silica tetrahedron, a combination of four oxygen atoms and one silicon atom. These are arranged such that planes drawn through the oxygen atoms form a tetrahedron (Figure 2.6). Since the silicon ion has a charge of +4 and each of the four oxygen ions has a charge of –2, the silica tetrahedron has a net charge of –4.

In silicate minerals, these tetrahedra are arranged and linked together in a variety of ways, from single units to complex frameworks (Figure 2.9). The simplest silicate structure, that of the mineral olivine, is composed of isolated tetrahedra bonded to iron and/or magnesium ions. In olivine, the –4 charge of each silica tetrahedron is balanced by two divalent(i.e., +2) iron or magnesium cations. Olivine can be either Mg2SiO4 or Fe2SiO4, or some combination of the two (Mg,Fe)2SiO4. The divalent cations of magnesium and iron are quite close in radius (0.73 versus 0.62 angstroms[1]). Because of this size similarity, and because they are both divalent cations (both have a charge of +2), iron and magnesium can readily substitute for each other in olivine and in many other minerals.

T

As already noted, the +2 ions of iron and magnesium are similar in size (although not quite the same). This allows them to substitute for each other in some silicate minerals. In fact, the common ions in silicate minerals have a wide range of sizes, as shown in Figure 2.11. All of the ions shown are cations, except for oxygen. Note that iron can exist as both a +2 ion (if it loses two electrons during ionization) or a +3 ion (if it loses three). Fe2+ is known as ferrous iron. Fe3+ is known as ferric iron. Ionic radii are critical to the composition of silicate minerals, so we’ll be referring to this diagram again.

Figure 2.11 The ionic radii (effective sizes) in angstroms, of some of the common ions in silicate minerals

The structure of the single-chain silicate pyroxene is shown on Figures 2.12 and 2.13. In pyroxene, silica tetrahedra are linked together in a single chain, where one oxygen ion from each tetrahedron is shared with the adjacent tetrahedron, hence there are fewer oxygens in the structure. The result is that the oxygen-to-silicon ratio is lower than in olivine (3:1 instead of 4:1), and the net charge per silicon atom is less (–2 instead of –4), since fewer cations are necessary to balance that charge. Pyroxene compositions are of the type MgSiO3, FeSiO3, and CaSiO3, or some combination of these. Pyroxene can also be written as (Mg,Fe,Ca)SiO3, where the elements in the brackets can be present in any proportion. In other words, pyroxene has one cation for each silica tetrahedron (e.g., MgSiO3) while olivine has two (e.g., Mg2SiO4). Because each silicon ion is +4 and each oxygen ion is –2, the three oxygens (–6) and the one silicon (+4) give a net charge of –2 for the single chain of silica tetrahedra. In pyroxene, the one divalent cation (2+) per tetrahedron balances that –2 charge. In olivine, it takes two divalent cations to balance the –4 charge of an isolated tetrahedron.

The structure of pyroxene is more “permissive” than that of olivine — meaning that cations with a wider range of ionic radii can fit into it. That’s why pyroxenes can have iron (radius 0.63 Å) or magnesium (radius 0.72 Å) or calcium (radius 1.00 Å) cations.

Figure 2.12 A depiction of the structure of pyroxene. The tetrahedral chains continue to left and right and each is interspersed with a series of divalent cations. If these are Mg ions, then the formula is MgSiO3.

Figure 2.13 A single silica tetrahedron (left) with  four oxygen ions per silicon ion (SiO4). Part of a single chain of tetrahedra (right), where the oxygen atoms at the adjoining corners are shared between two tetrahedra (arrows). For a very long chain the resulting ratio of silicon to oxygen is 1 to 3 (SiO3).

Exercises

Exercise 2.4 Oxygen Deprivation

The diagram below represents a single chain in a silicate mineral. Count the number of tetrahedra versus the number of oxygen ions (yellow spheres). Each tetrahedron has one silicon ion so this should give the ratio of Si to O in single-chain silicates (e.g., pyroxene).

The diagram below represents a double chain in a silicate mineral. Again, count the number of tetrahedra versus the number of oxygen ions. This should give you the ratio of Si to O in double-chain silicates (e.g., amphibole).

In amphibole structures, the silica tetrahedra are linked in a double chain that has an oxygen-to-silicon ratio lower than that of pyroxene, and hence still fewer cations are necessary to balance the charge. Amphibole is even more permissive than pyroxene and its compositions can be very complex. Hornblende, for example, can include sodium, potassium, calcium, magnesium, iron, aluminum, silicon, oxygen, fluorine, and the hydroxyl ion (OH–).

In mica structures, the silica tetrahedra are arranged in continuous sheets, where each tetrahedron shares three oxygen anions with adjacent tetrahedra. There is even more sharing of oxygens between adjacent tetrahedra and hence fewer charge-balancing cations are needed for sheet silicate minerals. Bonding between sheets is relatively weak, and this accounts for the well-developed one-directional cleavage (Figure 2.14). Biotite mica can have iron and/or magnesium in it and that makes it a ferromagnesian silicate mineral (like olivine, pyroxene, and amphibole). Chlorite is another similar mineral that commonly includes magnesium. In muscovite mica, the only cations present are aluminum and potassium; hence it is a non-ferromagnesian silicate mineral.

Figure 2.14 Biotite mica (left) and muscovite mica (right). Both are sheet silicates and split easily into thin layers along planes parallel to the sheets. Biotite is dark like the other iron- and/or magnesium-bearing silicates (e.g., olivine, pyroxene, and amphibole), while muscovite is light coloured. (Each sample is about 3 cm across.)

Apart from muscovite, biotite, and chlorite, there are many other sheet silicates (or phyllosilicates), which usually exist as clay-sized fragments (i.e., less than 0.004 mm). These include the clay minerals kaolinite, illite, and smectite, and although they are difficult to study because of their very small size, they are extremely important components of rocks and especially of soils.

All of the sheet silicate minerals also have water in their structure.

Silica tetrahedra are bonded in three-dimensional frameworks in both the feldspars and quartz. These are non-ferromagnesian minerals — they don’t contain any iron or magnesium. In addition to silica tetrahedra, feldspars include the cations aluminum, potassium, sodium, and calcium in various combinations. Quartz contains only silica tetrahedra.

The three main feldspar minerals are potassium feldspar, (a.k.a. K-feldspar or K-spar) and two types of plagioclase feldspar: albite (sodium only) and anorthite (calcium only). As is the case for iron and magnesium in olivine, there is a continuous range of compositions (solid solution series) between albite and anorthite in plagioclase. This is because the calcium and sodium ions are almost identical in size (1.00 Å versus 0.99 Å). Any intermediate compositions between CaAl2Si3O8 and NaAlSi3O8 can exist (Figure 2.15). This is a little bit surprising because, although they are very similar in size, calcium and sodium ions don’t have the same charge (Ca2+ versus Na+). This problem is accounted for by corresponding substitution of Al3+ for Si4+. Therefore, albite is NaAlSi3O8 (one Al and three Si) while anorthite is CaAl2Si2O8 (two Al and two Si), and plagioclase feldspars of intermediate composition have intermediate proportions of Al and Si. This is called a “coupled-substitution.”

The intermediate-composition plagioclase feldspars are oligoclase (10% to 30% Ca), andesine (30% to 50% Ca), labradorite (50% to 70% Ca), and bytownite (70% to 90% Ca). K-feldspar (KAlSi3O8) has a slightly different structure than that of plagioclase, owing to the larger size of the potassium ion (1.37 Å) and because of this large size, potassium and sodium do not readily substitute for each other, except at high temperatures. These high-temperature feldspars are likely to be found only in volcanic rocks because intrusive igneous rocks cool slowly enough to low temperatures for the feldspars to change into one of the lower-temperature forms.

In quartz (SiO2), the silica tetrahedra are bonded in a “perfect” three-dimensional framework. Each tetrahedron is bonded to four other tetrahedra (with an oxygen shared at every corner of each tetrahedron), and as a result, the ratio of silicon to oxygen is 1:2. Since the one silicon cation has a +4 charge and the two oxygen anions each have a –2 charge, the charge is balanced. There is no need for aluminum or any of the other cations such as sodium or potassium. The hardness and lack of cleavage in quartz result from the strong covalent/ionic bonds characteristic of the silica tetrahedron.

Exercises

Exercise 2.5 Ferromagnesian Silicates?

Silicate minerals are classified as being either ferromagnesian or non-ferromagnesian depending on whether or not they have iron (Fe) and/or magnesium (Mg) in their formula. A number of minerals and their formulas are listed below. For each one, indicate whether or not it is a ferromagnesian silicate.

Mineral	Formula	Ferromagnesian Silicate?
olivine	(Mg,Fe)2SiO4
pyrite	FeS2
plagioclase	CaAl2Si2O8
pyroxene	MgSiO3
hematite	Fe2O3
orthoclase	KAlSi3O8
quartz	SiO2

Mineral	Formula*	Ferromagnesian Silicate?
amphibole	Fe7Si8O22(OH)2
muscovite	K2Al4 Si6Al2O20(OH)4
magnetite	Fe3O4
biotite	K2Fe4Al2Si6Al4O20(OH)4
dolomite	(Ca,Mg)CO3
garnet	Fe2Al2Si3O12
serpentine	Mg3Si2O5(OH)4

                                    Quantum Theory

Planck’s radiation law

By the end of the 19th century, physicists almost universally accepted the wave theory of light. However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light. All bodies radiate electromagnetic energy as heat; in fact, a body emits radiation at all wavelengths. The energy radiated at different wavelengths is a maximum at a wavelength that depends on the temperature of the body; the hotter the body, the shorter the wavelength for maximum radiation. Attempts to calculate the energy distribution for the radiation from a blackbody using classical ideas were unsuccessful. (A blackbody is a hypothetical ideal body or surface that absorbs and reemits all radiant energy falling on it.) One formula, proposed by Wilhelm Wien of Germany, did not agree with observations at long wavelengths, and another, proposed by Lord Rayleigh (John William Strutt) of England, disagreed with those at short wavelengths.

In 1900 the German theoretical physicist Max Planck made a bold suggestion. He assumed that the radiation energy is emitted, not continuously, but rather in discrete packets called quanta. The energy E of the quantum is related to the frequency ν by E = hν. The quantity h, now known as Planck’s constant, is a universal constant with the approximate value of 6.62607 × 10−34 joule∙second. Planck showed that the calculated energy spectrum then agreed with observation over the entire wavelength range.

Einstein and the photoelectric effect

In 1905 Einstein extended Planck’s hypothesis to explain thephotoelectric effect, which is the emission of electrons by a metal surface when it is irradiated by light or more-energetic photons. The kinetic energy of the emitted electrons depends on the frequency ν of the radiation, not on its intensity; for a given metal, there is a threshold frequency ν0 below which no electrons are emitted. Furthermore, emission takes place as soon as the light shines on the surface; there is no detectable delay. Einstein showed that these results can be explained by two assumptions: (1) that light is composed of corpuscles or photons, the energy of which is given by Planck’s relationship, and (2) that an atom in the metal can absorb either a whole photon or nothing. Part of the energy of the absorbed photon frees an electron, which requires a fixed energy W, known as the work function of the metal; the rest is converted into the kinetic energy meu2/2 of the emitted electron (me is the mass of the electron and u is its velocity). Thus, the energy relation is 

If ν is less than ν0, where hν0 = W, no electrons are emitted. Not all the experimental results mentioned above were known in 1905, but all Einstein’s predictions have been verified since.

Bohr’s theory of the atom

BRITANNICA LISTS & QUIZZES

A major contribution to the subject was made by Niels Bohr of Denmark, who applied the quantum hypothesis to atomic spectra in 1913. The spectra of light emitted by gaseous atoms had been studied extensively since the mid-19th century. It was found that radiation from gaseous atoms at low pressure consists of a set of discrete wavelengths. This is quite unlike the radiation from a solid, which is distributed over a continuous range of wavelengths. The set of discrete wavelengths from gaseous atoms is known as a line spectrum, because the radiation (light) emitted consists of a series of sharp lines. The wavelengths of the lines are characteristic of the element and may form extremely complex patterns. The simplest spectra are those of atomic hydrogen and the alkali atoms (e.g., lithium, sodium, and potassium). For hydrogen, the wavelengths λ are given by the empirical formula 

where m and n are positive integers with n > m and R∞, known as theRydberg constant, has the value 1.097373157 × 107 per metre. For a given value of m, the lines for varying n form a series. The lines for m = 1, the Lyman series, lie in the ultraviolet part of the spectrum; those for m = 2, the Balmer series, lie in the visible spectrum; and those for m = 3, the Paschen series, lie in the infrared.

Bohr started with a model suggested by the New Zealand-born British physicist Ernest Rutherford. The model was based on the experiments ofHans Geiger and Ernest Marsden, who in 1909 bombarded gold atoms with massive, fast-moving alpha particles; when some of these particles were deflected backward, Rutherford concluded that the atom has a massive, charged nucleus. In Rutherford’s model, the atom resembles a miniature solar system with the nucleus acting as the Sun and the electrons as the circulating planets. Bohr made three assumptions. First, he postulated that, in contrast to classical mechanics, where an infinite number of orbits is possible, an electron can be in only one of a discrete set of orbits, which he termed stationary states. Second, he postulated that the only orbits allowed are those for which the angular momentum of the electron is a whole number n times ℏ (ℏ = h/2π). Third, Bohr assumed that Newton’s laws of motion, so successful in calculating the paths of the planets around the Sun, also applied to electrons orbiting the nucleus. The force on the electron (the analogue of the gravitational force between the Sun and a planet) is the electrostatic attraction between the positively charged nucleus and the negatively charged electron. With these simple assumptions, he showed that the energy of the orbit has the form

where E0 is a constant that may be expressed by a combination of the known constants e, me, and ℏ. While in a stationary state, the atom does not give off energy as light; however, when an electron makes a transition from a state with energy En to one with lower energy Em, a quantum of energy is radiated with frequency ν, given by the equation

 Inserting the expression for En into this equation and using the relation λν = c, where c is the speed of light, Bohr derived the formula for the wavelengths of the lines in the hydrogen spectrum, with the correct value of the Rydberg constant.

Bohr’s theory was a brilliant step forward. Its two most important features have survived in present-day quantum mechanics. They are (1) the existence of stationary, nonradiating states and (2) the relationship of radiation frequency to the energy difference between the initial and final states in atransition. Prior to Bohr, physicists had thought that the radiation frequency would be the same as the electron’s frequency of rotation in an orbit.