Hammond C. The Basics of Crystallography and Diffraction

Подождите немного. Документ загружается.

34 Crystals and crystal structures

.... Stacking of the Zn (or S) atoms in zinc blende and the ∇∇ ... stack-

ing of the Zn (or S) atoms in wurtzite. Many carbides, including silicon carbide,

SiC, also possess such close-packed structures with the carbon atoms occupying the

tetrahedral sites and the metal or metalloid atoms stacked in various combinations of

... and ∇∇ .... Figure 1.26 shows ﬁve structures or polytypes of silicon

carbide (B3–B7) in which the silicon atoms are represented as solid spheres and the

carbon atoms as small open spheres. The low-temperature form, β-SiC (B3) has the fcc

structure, isomorphous with zinc blende. There are a number of high-temperature poly-

types known collectively as α-SiC, the simplest structure of which, B4, is isomorphous

with wurtzite. The other polytypes of α-SiC, three of which (B5, B6, B7) are shown in

Fig. 1.26, have more complex stacking sequences, resulting in longer unit cell repeat

distances.

For example, the stacking sequence in B5 (Carborundum III) is (reading up) ABA-

CABAC giving a four-layer repeat distance. In the Frank notation this is represented

as ∇∇∇∇... i.e. by inversions in the stacking sequence every two layers, rather

than every layer as in B4, the wurtzite structure. Similarly, for B6 (Carborundum II)

the stacking sequence is ABCACBA… giving a six-layer repeat distance which in the

Frank notation is ∇∇∇...i.e. an inversion every three layers.

Figure 1.27 shows an alternative representation of the polytypes of SiC showing (for

clarity) only the close-packed layers of the metalloid (Si)atoms. The number below each

polytype refers to the number of layers in the unit cell repeat distance and the letter

refers to the type of unit cell (C—cubic; H—hexagonal; R—rhombohedral). All these

polytypes of silicon carbide should not be thought of as distinct ‘species’, rather they

should be regarded as interrelated as a result of different sequences of stacking faults and

the transformations β  α SiC appear, from electron microscopy evidence, to occur

as the result of the passage of partial dislocations across the close-packed planes in the

same way as for the generation of a twinned crystal, as shown in Fig. 1.20. For example

6H, a common form of α-SiC, may be regarded as a ‘microtwinned’ form of 3C, the

three-layer thick twins being generated by the passage of three partial dislocations on

successive close-packed planes as shown in Fig. 1.20(d).

In aluminium oxide, A1

, the large oxygen anions occur in close-packed or nearly

close packed layers with the aluminium cations occupying two-thirds of the octahedral

interstitial sites. We now have an added complexity; not only may the oxygen anions

be packed in different sequences but also the aluminium cations may be distributed

differently throughout the interstitial sites—i.e. there may be different distributions

of the one-third ‘empty’ sites. In the well-characterized form, α-Al

(corundum-

isomorphous with α-Fe

), the oxygen anions are stacked in the hcpABAB … stacking

sequence and the aluminium cations between them are stacked in a rhombohedral

sequence in exactly the same pattern as the carbon atoms in the rhombohedral form

of graphite (see Fig. 1.37(b)). Hence the structure of α-Al

is rhombohedral with a

six-layer unit cell repeat distance of oxygen anions. The other polytypes of alumina,

called the transition aluminas, are not so well characterized, particularly with respect

to the distribution of the aluminium cations. A common form, γ -Al

is based on an

ABCABC … (fcc) stacking of oxygen anions with a distribution of aluminium cations

which gives rise to a maghemite structure, described in Section 1.11.3.

1.11 Some more complex crystal structures 35

(a) (b) (c)

(d) (e) (f)

6H 8H

15R

Fig. 1.27. The crystal structures of six SiC polytypes. 3C is β-SiC, the fcc low temperature form and

the others with 2, 4, 6 and 8-layer repeat hexagonal cells or a 15-layer repeat rhombohedral cell are

the α-SiC high temperature forms (from Ceramic Microstructures by W. E. Lee and W. M. Rainforth,

Chapman & Hall, 1994).

1.11.3 The oxides and oxy-hydroxides of iron

Iron is remarkable for the range of oxides and hydroxides which can be formed and is

one of the few elements which form compounds intermediate between these two—the

oxy-hydroxides, the crystal structures of which are also of interest, and we will now

describe them (as far as we are able) and trace their interrelationships.

We will begin with the simplest oxide, wustite, ferrous oxide, FeO and consider

the structural changes which take place in the progressive oxidation of FeO to Fe

(ferroso-ferric oxide) and Fe

(ferric oxide). FeO has the NaCl structure—the Fe

ions are situated in the octahedral interstitial sites between the oxygen anions

. How-

ever, FeO is rarely stoichiometric but has vacant Fe

sites in the face-centred cubic

The terms ‘anion’and ‘cation’are used when we want to draw speciﬁc attention to the charge on the ionic

species, otherwise the more general term ‘atom’ is used—it being understood that the term encompasses both

neutral and charged species.

36 Crystals and crystal structures

structure, electrical neutrality being preserved by the presence of equal numbers of Fe

ions (some of which are situated in the tetrahedral interstitial sites). The ‘oxidation’ of

FeO to Fe

proceeds, not by the ‘addition’of oxygen atoms to the structure, but by the

migration of Fe atoms to the surface (to combine with atmospheric oxygen there)—to a

ﬁrst approximation the close-packed oxygen atoms in the original FeO structure remain

undisturbed. Within the structure the Fe

ions are progressively replaced by Fe

ions,

half of which are situated in the tetrahedral interstitial sites. The structure of Fe

(mag-

netite) thus formed is that of an inverse spinel with the general formula AB

in which

the ‘A’ tetrahedral sites are occupied by Fe

ions and the ‘B’ octahedral sites by equal

numbers of Fe

and Fe

ions. Fe

is therefore better represented by the formula

(Fe

. However, the occupancy of the interstitial sites is not random, but

ordered such that the unit cell of Fe

has twice the side-edge (eight times the volume)

of the original FeO unit cell and contains 32 O

2−

ions, 8 Fe

ions and 16 Fe

ions.

The distinction betweenan‘inverse’and a ‘normal’spinel (bothbadnames!) is simple:

spinel is the mineral MgAl

. The Mg

ions occupy the ‘A’ tetrahedral sites and the

ions occupy the ‘B’ octahedral sites—with regard to the ‘A’ sites only the inverse

of Fe

. Again, we see that the occupancies of the interstitial sites are determined not

simply by the valencies of the ions but also their sizes. Figure 1.28 shows a projection,

or crystal plan of the spinel structure (see Section 1.8), split into two halves to make the

atom/ion positions more clear.

As oxidation proceeds further the remaining eight Fe

ions in the unit cell are

replaced by two-thirds of their number by Fe

ions (i.e. to maintain electrical neutrality)

giving a total of 16 +5

= 21

ions, 32 O

2−

ions and the overall composition

. Except for a small reduction in volume (on account of the increased number

of vacant lattice sites) the large cubic unit cell is unchanged. The structure is called

maghemite, γ -Fe

and is isostructural with the γ -Al

structure (see Section 1.11.2).

040

400

: B;: A;: X

Fig. 1.28. Plan view of the unit cell of the cubic spinel structure AB

projected on to a plane

perpendicular to a cube axis. The heights of the atoms are indicated in units of one-eighth the cubic

cell edge length. For clarity, the upper and lower halves of the cell are shown separately and only the

tetrahedral coordination of the A ions is indicated. (From Crystal Chemistry, 2nd Edn, by R. C. Evans,

Cambridge University Press, 1966.)

1.11 Some more complex crystal structures 37

However, γ -Fe

(like γ -Al

) is not the stable structural form and its conversion to

α-Fe

(haematite) occurs essentially by the rearrangement of the oxygen atoms from

the fcc to the hcp stacking sequence.

Similar relationships exist in the oxy-hydroxides in which iron is in the Fe

(ferric)

form. The formulae are best written FeO·OH (rather than Fe

·H

O) to emphasize the

replacement of oxygen ions by hydroxyl, (OH)

−

groups rather than the presence in the

structure of discrete ‘molecules’ of water. In γ -FeO·OH (lepidocrocite), the oxygen and

hydroxyl ions are approximately cubic close-packed and in α-FeO·OH (goethite), they

are approximately hexagonal close-packed. The deviation from perfect close-packing

arises from the formation of directed hydrogen bonds between the hydroxyl groups

in different layers; the structures are not cubic or hexagonal but have orthorhombic

symmetry (see Chapter 3). On heating, these oxy-hydroxide structures dehydrate to

γ -Fe

and α-Fe

, respectively.

Lepidocrocite and goethite are not the only oxy-hydroxides which occur. Of perhaps

more biological or technological importance is ferrihydrite, a poorly crystallized mineral

which is ubiquitous across the Earth’s surface. It is a common product of weathering of

iron-bearing minerals and of the microbial oxidation of ferrous ions and doubtless of the

rusting of iron itself.

The determination of its crystal structure and composition is a matter of considerable

difﬁculty because of the very small crystallite size, typically in the range 2–10 nm. How-

ever, it appears that the oxygen-hydroxyl groups are arranged in an hcp (2H, 4H or 6H)

stacking sequence (see Fig. 1.27), i.e. closer to goethite than lepidocrocite and possibly

isostructural with a natural alumina hydrate phase, akdaleite. The composition is com-

monly given as Fe

(OH)

.4H

O but the water content appears to depend on particle size.

Ferrihydrite is an example of a naturally occuring nanocrystalline material—upon

which a whole new science and technology is now being built. Their importance consists

in the simple fact that at these sizes a large proportion of the atoms or ions are at, or

near, the surface and the conditions for the stability of the structures are found to be (as

might be expected) quite different at the centres of the crystals and at their surfaces.

Such nano-sizedcrystals of ferrihydrite alsoappear to constitute theinorganic ‘core’of

the ferritin molecule—the iron storage molecule that occurs principally within the liver.

The core is enclosed within a protein ‘shell’ (Fig. 1.29) which has a cubic symmetry not

found in any inorganic crystals (see Chapter 4).

The mechanisms by which iron leaves and enters the ferritin molecule, and how

it is taken up, and released, from the ferrihydrite, are matters of much research. But

undoubtedly the crystallography—the valency and distribution of iron ions across the

interstitial sites both at and below the crystal surfaces—will emerge as factors of great

importance.

1.11.4 Silicate structures

Silicates—which constitute by far the most important minerals in the earth’s crust—

are based on the different ways in which SiO

tetrahedra

may be joined together, each

The SiO

tetrahedra may be referred to more generally as Si–O tetrahedra since, as described in sub-

sections (a)–(g), the silicon–oxygen ratio depends on how they are linked.

38 Crystals and crystal structures

Fig. 1.29. The cubic (point group 432) structure of the ferritin protein shell viewed along a cube axis.

(From Mineralization in Ferritin: An Efﬁcient Means of Iron Storage by N Dennis Chasteen and Pauline

M Harrison, Journal of Structural Biology 126, 182–, 1999.)

Fig. 1.30. A perspective of the SiO

tetrahedron. The oxygen anions at the corners of the tetrahedron

are nearly close packed, as shown in the models, e.g. in Fig. 1.7 (the radius of the oxygen anion,

= 0.132 nm and that of the silicon cation, r

= 0.039 nm, give a radius ratio, r

= 0.296,

slightly larger than the ideal value r

= 0.225, Fig. 1.10(a) and (b)).

1.11 Some more complex crystal structures 39

tetrahedron being made up of four oxygen anions with the silicon cation in the tetrahedral

interstice in the centre (Fig. 1.30). Silicate chemistry is based on the linking of the SiO

tetrahedra, i.e. whether they occur separately, or whether they are linked by common

oxygen anions to form chains, rings, sheets or complete frameworks. This provides the

initial basis for the classiﬁcation of silicate structures. If the Si–O bond is considered

to be purely ionic, there are four positive charges associated with each silicon cation

and two negative charges associated with each oxygen anion; hence there are four net

negative charges associated with each SiO

tetrahedron. The ‘charge balance’in silicates

may be achieved in seven possible ways (see Fig. 1.31):

(a) Separate SiO

tetrahedra (nesosilicates): the charge balance (four net negative

charges) is achieved with metal cations, e.g. Mg

,Fe

, which also link the tetra-

hedra together. Typical minerals are forsterite, Mg

(SiO

), or fayalite, Fe

(SiO

the end-members of the olivine group, MgFe(SiO

(b) Two tetrahedra linked together sharing one oxygen anion (sorosilicates): the Si:O

ratio is now Si

giving six net negative charges which are balanced with

metal cations. Typical minerals are melilite, Ca

Mg(Si

O7), or hemimorphite,

(OH)H

O(Si

two oxygen anions (cyclosilicates): the Si:O ratio is Si

, where n is the number

of tetrahedra in the ring. A typical mineral is beryl, with a ring of six tetrahedra,

(Si

(d) Many tetrahedralinked together toform single chains(inosilicates): each tetrahedron

shares two oxygen anions, as in the ring structures above, and which therefore give

rise to the same Si:O ratio. This is the basis of the group of minerals called the

pyroxenes. Typical examples are enstatite, Mg

(Si

), or diopside, CaMg(Si

(e) Tetrahedra linked together to form double chains (inosilicates), each tetrahedron

sharing alternately two and three oxygen anions, giving the Si:O ratio Si

. This

is the basis of the group of minerals called the amphiboles. Typical examples are

anthophyllite, Mg

(OH)

(Si

)

, or tremolite, Ca

(OH)

(Si

)

(f) Tetrahedra linked together to form sheets (phyllosilicates), each tetrahedron sharing

three oxygen anions giving the Si: O ratio Si

. This is the basis of the micas,

chlorites and the clay minerals.

(g) Tetrahedra linked together such that all the oxygen anions are shared giving a three-

dimensional framework (tectosilicates). The Si:O ratio is now SiO

and there is an

overall charge balance without the necessity of any linking cations.

Figure 1.31 shows the arrangements of SiO

tetrahedra in these seven silicate

structures. However, having established the basic pattern there are very important com-

plications both in the chemistry and the arrangements of the tetrahedra which must not

be overlooked. In all the silicates, and in the chain, sheet and framework silicates in par-

ticular, the silicon in the centre of the tetrahedron can be substituted by aluminium—a

trivalent rather than a tetravalent ion. For each such substitution an additional positive

charge by way of a ‘linking cation’is required. All sheet silicates or phyllosilicates show

40 Crystals and crystal structures

Class

(a)

Nesosilicates

Oxygen

(SiO

)

4–

Olivine,

(Mg, Fe)

SiO

Hemimorphite

(OH)·H

Beryl,

Pyroxene

e.g. Enstatite,

MgSiO

(Si

)

6–

(Si

)

12–

(Si

)

4–

(b)

Sorosilicates

(c)

Cyclosilicates

(d)

Inosilicates

(single chain)

Arrangement of SiO

tetrahedra

(central Si

not shown)

Unit

composition

Mineral example

1.11 Some more complex crystal structures 41

(e)

Inosilicates

(double chain)

Amphibole

e.g. Anthophyllite,

SiO

(OH)

(Si

)

2–

(SiO

)

Mica

e.g. Phologopite,

KMg

(Alsi

)(OH)

High cristobalite,

SiO

(f)

Phyllosilicates

(g)

Tectosilicates

(Si

)

6–

Fig. 1.31. The arrangements of the SiO

tetrahedra and Si:O ratios in the seven main types of silicate

structures, with examples of typical minerals (from Manual of Mineralogy, 21st edn, by C. Klein and

C. S. Hurlbut Jr., John Wiley & Sons Inc., 1993).

42 Crystals and crystal structures

(a)

(f)

(b)

(c)

(d)

(e)

(g)

(h) (i)

Fig. 1.32. The arrangements of the tetrahedra in the inosilicates. They are described (in German)

according to the number of the tetrahedra in the repeat distance: a (zweier); b (dreier); c (vierer);

d (fünfer); e (sechser); f (siebener); g (neuner); h (zwölfer); i (24er) (from Structural Chemistry of

Silicates by F. Liebau, Springer-Verlag, 1985).

such substitution to a greater or lesser extent, for example phlogopite in which one sil-

icon is replaced by one aluminium cation giving the formula KMg

(OH)

(Si

AlO

In framework silicates (tectosilicates), substitution gives rise to an important new class

of minerals—the feldspars, the most abundant minerals in the earth’s crust, e.g. albite,

Na(Si

A1O

), or orthoclase, K(Si

A1O

) (one silicon cation substituted), or anorthite,

Ca(Si

) (two silicon cations substituted).

1.11 Some more complex crystal structures 43

With respect to the arrangements of the tetrahedra, one example—that of the

inosilicates—will sufﬁce to show the principles involved. Figure 1.32 shows nine pos-

sible patterns (a)–(i) or conformations of the (unbranched) single chains, giving rise

to different repeat distances as indicated: (a) (the simplest—known as zweier single

chains because there are two tetrahedra in the repeat distance) is that for diopside and

enstatite; (b) (drier single chains with three tetrahedra in the repeat distance) is that for

wollastonite, Ca

(Si

), and so on.

Clearly, there are also many possible arrangements of the tetrahedra in the cyclo-

silicates, phyllosilicates and tectosilicates, and it is these which give rise (in part) to

the many structural differences in silicate minerals. For example, in the tectosilicates

the three different crystallographic forms of silica—quartz, tridymite and cristobalite—

simply correspond to different ways in which the SiO

tetrahedra are linked together.

1.11.5 The structures of silica, ice and water

Of the three structural forms of silica—quartz, tridymite and cristobalite (not counting

the high-pressure forms, coesite and stishovite)—quartz is by far the most common and

is structurally stable at ambient temperatures, whereas tridymite is stable between 857

and 1470

◦

C and cristobalite is stable from 1470

◦

C to the melting point. At ambient

temperatures, these latter two forms of silica are therefore metastable but they do not

transform to quartz because in order to do so, a rearrangement of the linking of the

SiO

tetrahedra needs to take place—in short, a reconstructive phase transformation

must occur in contrast to a displacive transformation in which atomic bonds are not

broken. However, displacive transformations occur in all three forms of silica by small

rotations of the SiO

tetrahedra, giving rise to the ‘more open, high temperature’ β

forms and the ‘more closed, low temperature’ α forms. This is illustrated, for quartz, in

Fig. 1.33. Figure 1.33(b) is a plan view or projection of the hexagonal β-quartz structure

perpendicular to the c-axis. For simplicity, only the Si atoms are indicated and their

relative heights along the c-axis: white (0), grey (1/3) and black (2/3). Figure 1.33(a)

is the corresponding projection for α-quartz; the structure is ‘twisted’ but no bonds are

broken. The symmetry also changes from hexagonal to trigonal, as will be described in

Section 3.3.

However, it is worth noticing at this stage one very important structural feature of

quartz: the silicon atoms (and hence the SiO

tetrahedra) are arranged in a helical pattern

along the c-axis. If we imagine a spiral staircase in the centre of a hexagon then the steps

go: 0, 1/3, 2/3 … in a clockwise fashion giving rise to a left-handed screw or helix. Now,

we can interchange the positions of the grey and black atoms such that the steps go 0,

1/3, 2/3 … in an anticlockwise fashion giving rise to a right-handed screw or helix. In

short, quartz (both α and β) has two forms, one ‘left-handed’ and one ‘right-handed’and

is an example of an enantiomorphous crystal structure (see Table 3.1 and Section 4.5).

Quartz is the densest structural form (not counting the high-pressure form, coesite),

(∼2.66 g/cm

); tridymite and cristobalite have much more ‘open’ structures (∼2.33

g/cm

). In their β forms, these two structures are similar to those of wurtzite and zinc

blende (the two structural forms of zinc sulphide), respectively, the SiO

tetrahedra

being in the positions of the Zn and S atoms. More speciﬁcally, they correspond to the