Close-packed metals

Close packing in two dimensions

For many elements, the overriding principle that governs the crystal structure is the need to have the highest possible density. How this can be achieved when chemical bonding does not throw up any constraints can be seen by looking at the packing of atoms in two dimensions, Fig. 2.1. Each atom has six neighbours,

Fig. 2.1 Two-dimensional close-packed stru-cture, with atoms in contact with six neigh-bours to give the maximum density.

and there are groups of three atoms in close contact.

Using the two-dimensional close-packed structure as an example, we can identify a number of important components involved in the description of any crystal structure. First we consider how the atoms are packed into a periodically repeating arrangement. The periodicity is obvious from Fig. 2.1, and to formalize this may seem to be a distraction at this point, but it will quickly become apparent that the formalism will help to make the description of crystal structures much easier. Our formalism involves separating the operation of the periodicity from the objects that are repeated. The process in two dimensions is illustrated in Fig. 2.2. We start with an infinite periodic array of points, which we will call thelattice.

Each lattice point represents a small region of space, which can be represented as a polygon bounded by four lattice points. Three examples are marked in Fig. 2.2. The whole of the space can be tiled by these parallelograms. One of the parallelograms (a) has edges of equal lengths and internal angles of 120^◦ and 60^◦. This shape reflects the basic hexagonal symmetry of the close-packed structure, and is therefore the most natural parallelogram to consider as the basic tile. The region of space associated with each lattice point is called the unit cell.

In principle a unit cell can contain one or more atoms. The crystal is constructed by replacing each lattice point by the contents of the unit cell. The unit cell (a) of the two-dimensional structure in Fig. 2.1 is the trivial case of a unit cell containing just one atom, so in this case the process of replacing each lattice point by the contents of the unit cell is simply to place an atom on each lattice point. However, when the unit cell may contain several atoms, each lattice point will be replaced by a small arrangement of atoms. For future reference, the process of constructing an infinite crystal by replacing each lattice point

Fig. 2.2 Underlying lattice of the two-dimensional close-packed structure. The actual crystal structure is constructed by replacing the points on the lattice by atoms.

The figure shows three possibleunit cells. The unit cell marked a is a rhombus (internal angles 60^◦and 120^◦), whose shape reflects the underlying rotational symmetry. The shape of the unit cell marked b bears no obvious relationship to the symmetry and would not be a logical choice. The rectangular unit cell, c, has sides with lengths in the ratio 1:√

3. The orthogonal axes reflect some of the symmetry of the structure. This cell encompasses two lattice points. All three unit cells represent areas that can be repeated periodically and fill all the space.

a

b

c

by a small group of atoms is an example of the mathematical operation called convolution – this operation will be very important when we discuss diffraction methods in Chapter 6.

As we have noted, the natural unit cell of Figs 2.1 and 2.2 has edges of equal lengths, and angles of 120^◦and 60^◦, reflecting the hexagonal symmetry of the close-packed structure. We could also have defined a unit cell with orthogonal axes but with twice the area, as shown by c in Fig. 2.2. In this case, one cell edge would be√

3 times as long as the other. This unit cell is not as perverse as the parallelogram marked b, because orthogonal axes are particularly easy to handle. We will see below that there are three-dimensional arrangements of atoms for which the most natural unit cell to reflect the symmetry has orthogonal axes, even though it may have a volume that is twice or four times larger than the volume associated with a single lattice point.

Before we move away from our two-dimensional example, we note that we can also formalize thesymmetry of the structure. First, we note that from a point at the centre of each atom we could rotate the crystal structure by any multiple of 60^◦ and get an exact copy of the starting structure. This is an example of rotational symmetry. In this case we can rotate the structure by multiples of

1

6-th of a complete revolution, so the particular symmetry is called a 6-fold rotation symmetry. The structure also has points with 3-fold rotational symmetry (located in the middle of the small triangles described by three touching atoms) and2-fold rotation symmetry (located at the points where pairs of atoms touch).

The axes associated with these rotational symmetries are shown in Fig. 2.3.

Fig. 2.3 Another view of the close-packed layer, showing some of the axes of the rotational symmetry (hexagons for 6-fold rotational symmetry, triangles for 3-fold sym-metry, and ellipses for 2-fold symmetry), and the lines of the mirror planes.

The structure also has a type of symmetry associated with reflection rather than rotation. Consider a line through the centres of atoms in a horizontal row in Fig. 2.1. If we reflect the crystal through this line we get an exact copy of the structure. This line is called amirror plane (because in three dimensions it is a plane, and one can imagine that the mirror is three-dimensional lying normal to the plane of the diagram). Because of the 6-fold rotation symmetry, there are identical mirror planes passing through the centre of each atom rotated by±60^◦. There is another set of mirror planes passing through the centres of each atoms lying at 30^◦ to the first set, and, again, these are reproduced by rotations of±60^◦. Both sets of mirror planes are shown in Fig. 2.3.

To formally list all the rotation and mirror symmetries may seem to be a little gratuitous at this point, but later (Chapter 3) we will see that there is great power in being able to describe the complete symmetry of a crystal structure.

For the moment, the important point to note is that we have introduced and defined both rotational symmetry and mirror planes.

Three-dimensional close-packing: hexagonal close-packed structure The close-packed layers of Fig. 2.1 can be stacked on top of each other in two ways that continue the close packing, as illustrated in Fig. 2.4. The atoms in

B A

Fig. 2.4 Another view of the close-packed layer, showing the sites for the stacking of another close-packed layer. TheA sites are occupied by the atoms in the layer, and the B and C sites fit into dimples on the top and bottom surfaces.

adjacent layers are displaced with respect to each other in order to continue the close packing. If the first layer is denotedA in Fig. 2.4, the atoms in the next layer will be in either theB or C positions. With either of these stackings, an atom in one layer is able to touch three atoms in the layers above or below.

When added to the six touching neighbours in the same layer, this means that each atom has 12 touching neighbours in a three-dimensional close-packed arrangement.

Because the layer on top ofA can be in either B or C positions (Fig. 2.4), it is clear that there is not one unique way of building a three-dimensional crystal layer by layer. Given the tendency of nature to like to produce order, there are two main types of stacking favoured by the elements. The first is the case where the atoms in the third layer have the same relative positions as in the first layer, producing a repeat every other layer. In terms of the labels we have given the layers, the double-repeat can be described by the sequence –ABABAB–. This structure is calledhexagonal close-packed, and is abbreviated as hcp.

We can define a three-dimensional unit cell. The base of the unit cell will be the parallelogram with equal sides and angles of 120^◦and 60^◦defined for the two-dimensional packed layer. The third axis is normal to the close-packed layer, and extends to the next-but-one layer up – recall that the stacking of layers in hcp repeats every other layer. The hcp unit cell is shown in Fig. 2.5a.

We use the vectors a and b to represent the unit cell edges within the close-packed layer, and the vector c to represent the unit cell edge normal to the close-packed plane. The moduli of these three vectors are written as a, b and c respectively, noting that a = b in this case.

A straightforward geometric analysis (see problem 2.2) can show that if the dimension of the unit cell in the horizontal plane, a, is equal to 2r, where r is the radius of the atoms, the dimension of the unit cell in the third direction, c, would ideally be 4√

2/3r, so that c/a =√

8/3= 1.633. In practice, some of the elements with hcp structure have values of a and c that come close to satisfying this ratio, such as Ce (1.63), La (1.62) and Mg (1.62), but because this ratio is not fixed by symmetry it will not be satisfied exactly. In some cases the ratio is broken to a much larger extent, being either larger or smaller than the ideal value. Examples are Cd (1.89), Zn (1.86), Be (1.56) and Ho (1.57). These examples are evidence that the bonding between atoms is rather more complex than the simple stacking of spheres, as will be discussed in Chapter 5.

The three-dimensional unit cell of the hcp structure contains two atoms, as shown in Fig. 2.5a. One of the atoms is at the origin of the unit cell, and the other is associated with the alternative close-packed layer. We can describe the

Fig. 2.5 The hexagonal (left) and cubic (right) close-packed structures showing the

nearest-neighbour contacts. (a) (b)

position of the second atom using a convention calledfractional coordinates.

This provides a more general method than using absolute positions. We can describe the absolute position of an atom within a unit cell in terms of a combination of vectors parallel to each of the three lattice vectors. If the atom is at position r, it is described this way by

r= xa + yb + zc (2.1)

where x, y, and z are numbers with values between±1. It may seem strange to want to use negative numbers, but in practice the use of negative numbers can sometimes help to highlight the coordination around the origin. The set of numbers x, y, z are thefractional coordinates. The lattice symmetry means that addition or subtraction of any integer to any one of the fractional coordinates simply gives an equivalent site in another unit cell. The use of fractional coordinates is valuable since it permits a description of the positions of atoms in the unit cell that is independent of the values of the lattice parameters. As a result, the set of fractional coordinates for all atoms in the unit cell can show the symmetry of a crystal structure. For hcp, the fractional coordinates of the two atoms are at 0, 0, 0 (the atom at the origin), and ¹₃,²₃,¹₂.

The important point to note is that the vector from the origin to the second atom cannot be reduced to a lattice vector; doubling the length of the vector takes us to the point with fractional coordinates ²₃,⁴₃,1 ≡ ²₃,¹₃,0, which is not another lattice point. In addition, the environments of the two atoms are z=0 layer

z=1/2 layer

Fig. 2.6 Projection of the hcp structure show-ing the two close-packed layers, highlightshow-ing the different layers by using different size spheres. The point to note is that the atoms in the two layers have different environments, which are related to each other by a rotation of 180^◦.

not equivalent, but are related by 180^◦or a plane of reflection. This is seen in Fig. 2.6.

Three-dimensional close-packing: cubic close-packed structure

A second common three-dimensional arrangement of close-packed layers is where the repeat occurs every third layer. From Fig. 2.4, this arrangement of layers can be described by the sequence –ABCABC–. However, this structure can be described more easily than as just another version of hcp. In Fig. 2.5 we show a small part of this three-layer repeat structure, and draw in a new unit cell. By geometry it can be shown that the angles of this unit cell are 90^◦, and that the three lengths of the unit cell are equal (see problem 2.3 at the

end of the Chapter). The new unit cell is shown in a different orientation in Fig. 2.5. It contains four atoms, and is described as a cube with an atom on each corner and one at the centre of each face. This structure is called cubic packed, abbreviated as ccp, because it is the cubic version of the close-packed arrangement. This is the simplest version of a type of structure called face-centred cubic (fcc).

The ccp structure now introduces another aspect of the formalism of the description of the crystal structure. Each atom has an equivalent environment, unlike the case of hcp. Thus each atom can be said to be occupying a lattice point. The fcc unit cell, containing four lattice points, is therefore not the most fundamental unit cell that can be drawn. Quite how one then describes the fundamental unit cell is arbitrary, but the most sensible unit cell would be similar to that shown in Fig. 2.7. Denoting the three vectors describing the

Fig. 2.7 One possible primitive unit cell of the ccp structure, shown as the large outline, with the outline of the conventional cubic unit cell.

cubic unit cell by a, b and c, as shown in Fig. 2.7, the vectors describing the reduced unit cell can be written as

a^=1 2(a+ b) b^=1

2(b+ c) (2.2)

c^=1 2(c+ a)

If the edge of the fcc unit cell has length a, the new unit cell has all edges of length a/√

2, and angles equal to 60^◦. This type of transformation will be discussed in more detail in Section 4.3.

This discussion concerning the two unit cells of the fcc structure illustrates the issue involved in deciding how to best describe a crystal structure. The unit cell with angles of 60^◦is the more fundamental since there is one lattice point per unit cell. It is called theprimitive unit cell. However, it is much more complicated to think about than the fcc unit cell, and usually people stick to this one. As a result, it is called theconventional unit cell. The issue of having a conventional unit cell that differs from the primitive unit cell, and which contains two or four lattice points, is actually quite common.

It is worth comparing the symmetry of the ccp structure with that of the hcp structure. Two views of the ccp structure are shown in Fig. 2.8. The way that the close-packed layers stack together means that the 6-fold rotation symmetry is broken, preserving only the 3-fold symmetry. But we now also have 4-fold rotation symmetry down the cube axes, and 2-fold rotation symmetry axes lying between two orthogonal 4-fold axes. These rotation axes are shown in Fig. 2.8.

Inspection of the ccp structure in Fig. 2.8 also shows the presence of several different types of mirror planes. There is one other important symmetry operation that we should highlight. If you could view the environment from the centre of an atom in the ccp structure, you would find that the view along any direction is exactly the inverse of that seen from the opposite direction. Put more formally, anything at the position x, y, z would be identical to anything at the position−x, −y, −z. This is a very important symmetry operation that has profound implication for the physical properties of materials. The origin is called thecentre of symmetry, and a crystal with a centre of symmetry is called centrosymmetric. The centre of symmetry is given the symbol 1.

Fig. 2.8 Two views of the ccp structure, highlighting the symmetry operations active in the two directions. Left shows the view down one of the cube axes, labelled[001], showing the 4-fold rotation symmetry.Right shows the view down the cube diagonal and normal to the close-packed planes, labelled [111]. This view shows the 3-fold rotation symmetry. In both views atoms in the same horizontal layer are connected by the sticks drawn as guides.

[001] [111]

Packing efficiency

To close the discussion of the two close-packed three-dimensional structures, we address the question of the efficiency of the packing of atoms in the close-packed structures. Clearly both the ccp and ideal hcp structures have the same efficiency of packing, so we will consider only the ccp structure because the geometry is a little easier. Let us consider each atom to be approximated by a hard sphere of radius R. In the ccp arrangement, the diagonal of the face of the cubic unit cell is equal to√

2a= 4R. The total volume of the cubic unit cell is equal to a³. There are four atoms in the cubic unit cell, each of volume 4π R³/3. Thus the fraction of space filled by the atoms is equal to 4× (4πR³/3)/a³ = π/3√

2= 0.74.

This value will be compared with the packing efficiencies of other structures we will introduce later in this chapter.

Close-packed crystal structures

The number of elements that have either the hcp or ccp crystal structures can be seen by looking at the representation of the periodic table in Fig. 2.9. Many of the metals exist in one of either of these structures. There are around 30 elements that have the hcp structure, and a similar number with the ccp structure.

Some elements can be found in either structure, sometimes because of a phase transition (as in Ca), sample preparation (Co), or by sample treatment (Ni).

Because both structures are close-packed, it is to be expected that they will have similar energies. Moreover, since the hcp and ccp structures differ in how the close-packed layers are stacked on top of each other, it is possible to imagine that structures with other stacking sequences are possible. Some elements (e.g. Am, La) can be prepared with a hexagonal structure that has twice the repeat length and four atoms in the unit cell. From Fig. 2.4, this arrangement of layers can be described by the sequence –ABACABAC–, with the repetition occuring on the fourth layer. It is worth noting that we will briefly discuss other types of layer structures later in this chapter, and that layer structures can adopt quite complex periodic stacking sequences, such as –ABCACBABCACB– (sixth-neighbour repeat). Phases with long-period repeats are calledpolytypes. Some of these long-period repeats, particular those extending over tens of layers, can arise as non-equilibrium states due to crystal growth conditions, but there is evidence that other polytypes exist as equilibrium states, arising from the

H

Fig. 2.9 The periodic table of the elements,

Fig. 2.9 The periodic table of the elements,

In document OXFORD MASTER SERIES IN CONDENSED MATTER PHYSICS (pagina 41-47)