Crystalline nature of metals
In the solid state, atoms in a metal are bonded together by metallic bond. Outer (valence) electrons are given up and form an “electron sea” that glues the positive ion cores (nclei and inner electrons) together. The metallic bond is non-directional, so atoms pack around each other densely. The metallic bond is a strong bond and the bond strength accounts for the melting point and the Young’s modulus relating to elastic deformation.
Under the metal bonding, atoms in a solid metal are arranged in an orderly pattern with a constant equilibrium distance between adjacent atoms. The so-called “crystalline lattice” is three dimensional inspace and can have different arrangements.
Three commonest “lattices” found in most metals are:
1. BCC- body centre cubic
2. HCP-hexagonal close-packed
3. FCC-face centred cubic
They give the shape and arrangement of a “unit cell” which is the repeating basic element in the lattice.
Those 3 lattices can be built by stacking two-dimensional layers of close-packed atoms one over the other
1. Stacking square packing over one another in ABAB… manner –BCC
2. Stacking diamond packing over one another in ABAB… manner-HCP
3. Stacking diamond packing over one another in ABCABC… manner-FCC
Slip in a single crystal
Solid metals have always been formed from solidification of molten metals. As such, there are a large number of small crystals in a piece of metals. Metals are poly-crystalline in nature. The crystals are called grains and have different orientations.
Consider a piece of single metal crystal (which can only be formed under highly controllable condition) under loading. When the load is low, it just stretches the distances between adjacent atoms to produce some elastic deformation. Upon release of the load, the original metallic bond will pull the atoms back to the same equilibrium distance as before.
When the load is high enough, it will stretch the distances between the atoms to the extent that the metal bond is broken. The atoms then separate and the crystal break. Another possibility arises if the “slip” planes of the crystal lie at an oblique direction to the applied load. In that case, slip occurs leading to plastic deformation.
In a crystal, there exist planes on which the atoms are closely packed (the original stacking planes in BCC, FCC and HCP). They are called slip planes. On a slip plane, the directions along which the atoms are arranged in the highest density is called slip direction.
When the slip planes are inclined to the applied load the component of the load along the slip planes and along the slip direction will have a shear effect on the layer of close-packed atoms. When the load component is high enough, it will drag one layer of atoms (and the part of crystal above it) ober the lower layer. The metal bonds among the original neighboring atoms are broken and then now bonds are formed among the new neighbors. This mechanism is called “slip” which leads to plastic defomation.
In a large real metal crystal, large groups of atoms do not slide over other in a simultaneous manner (“block slip”). This is because the process requires too high a load to break many bonds simultaneously. Another mechanism requiring less energy is possible and this is due to the presence of imperfections in metal crystals.
In the Simple Cubic (SC) unit cell there is one lattice point at each of the eight corners of a cube. Unit cells in which there are lattice points only at the eight corners are called primitive. In general, the number of lattice points is denoted by the letter "Z"; thus, for SC, Z = 1.
Let a host atom of radius r occupy each lattice point, and assume that each atom touches as many adjacent atoms as possible (in this case, there are six such contacts). Then each of the three unit cell edges is equal to the sum of two atomic radii: a = b = c = 2r. The volume of the cell is thus
Vc = 8r3
In a simple cubic cell, there is one host atom wholly inside the cube, because each of the eight corner atoms contributes one eighth of an atom to the cell interior. In general, the total volume of the cell which is occupied by the host atoms is
Vs = 4/3r3.Z.
The packing efficiency of a lattice is defined as the ratio Vs:Vc. Thus, for SC, the packing efficiency is about 52%.
In the Body Centered Cubic (BCC) unit cell there is one host atom (lattice point) at each corner of the cube and one host atom in the center of the cube: Z = 2. Each corner atom touches the central atom along the body diagonal of the cube, and it is easy to show by that the unit cell edge, an irrational number, is about 2.3r. Thus, the corner atoms do not touch one another.
The packing efficiency in this lattice can be shown to be about 68%, much higher than the packing efficiency in a simple cubic lattice.
In the Face Centered Cubic (FCC) unit cell there is one host atom at each corner and one host atom in each face. Since each corner atom contributes one eighth of its volume to the cell interior, and each face atom contributes one half of its volume to the cell interior (and there are six faces), then Z = 1/8.8 + 1/2.6 = 4.
The corner and face atoms touch along the face diagonal, and it is easy to show that the cube edge (a) is about 2.8r. Thus, the corner atoms do not touch one another.
The packing efficiency is about 74%. This is the maximum packing efficiency for spheres of equal radius and is call closest packing. Thus a face centered lattice of atoms is also called Cubic Closest Packing (CCP).
In any lattice it is always possible to choose a primitive (Z = 1) unit cell. In fact, there is an infinite number of such choices, and it can be shown that all of the primitive unit cells on a lattice have the same volume. However, only one of these primitive unit cells has the three shortest cell edges (a,b,c), and this unit cell is called the standard reduced cell.
In a FCC lattice, the standard reduced cell is a rhombohedron, with a = b = c = 2r. The three interior angles formed between unit cell edges are called:
(alpha, between edges b & c)
(beta, between edges a & c)
(gamma, between edges a & b)
In the FCC rhombohedral standard reduced cell, it can be shown that = = = 60o. Note that a cube is a just a special rhombohedron, with = = = 90o.
In a plane, spheres of equal size are most densely packed (with the least amount of empty space) when each sphere touches six other spheres arranged in the form of a regular hexagon.
When two such hexagonally closest packed planes are stacked directly on top of one another, a primitive hexagonal array results.
The primitive unit cell, outlined in black in the crossed stereo pair above, has cell edges
a = b = 2r and c = 2r. Thus, the ratio c:a = 1, and it can be shown that the packing efficiency of this three dimensional lattice is only about 60% (compared to 74% for closest packing), even though the atoms are closest packed in two dimensions.
Instead of stacking hexagonal closest packed planes directly above one another, they can be stacked such that atoms in successive planes nestle in the triangular "grooves" of the adjacent plane. (note that there are six of these "grooves" surrounding each atom in the hexagonal plane, but only three of them can be covered by atoms in the adjacent plane).
Let the first plane (at the bottom) be labeled "A" and the next plane above it be labeled "B". If a third hexagonal closest packed plane is stacked above B but in the "A" orientation, and succeeding planes are stacked in the repeating pattern ABABA... = (AB), a hexagonal unit cell can be chosen (using the nine atoms labeled "h"), with Z = 2.
For the stack of hexagonally closest packed spheres of equal radius (r) described above, the interplanar spacing between adjacent planes is proportional to r. The proportionality constant is an irrational number called the Closest Packed Interlayer Spacing, CPIS, and its value is about 1.6. Thus, the interlayer spacing is about 1.6.r (compared to 2.r for simple hexagonal stacking).
Thus, the "c" unit cell edge (in the stacking direction) has a length c = 2.CPIS.r, the ratio
c:a = CPIS, and it can be shown that this lattice has a packing efficiency which is identical to the FCC lattice. Thus, the name Hexagonal Closest Packing (HCP) for this array is justified.
Each host atom in an HCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r:
• There are six atoms in the planar hexagonal array (the central A layer);
• There are three atoms in the B layer above the A layer;
• There are three atoms in the B layer below the A layer.
The six atoms in the two B layers form a trigonal prism around the central atom in the A layer; the length of this prism is 2.CPIS.r.
Suppose the first two layers of hexagonal closest packed planes are stacked in "AB" fashion but the third layer is positioned so that its atoms lie over the three grooves in the A layer which were not covered by the atoms in the B layer. Then the third layer is in a different orientation from either A or B and is labeled "C". If a fourth layer then repeats the A layer orientation, and succeeding layers repeat the pattern ABCABCA... = (ABC), the resulting unit cell is hexagonal with three host atoms (Z = 3), unit cell edge c = 3.CPIS.r and c:a = 1.5.CPIS. Note that for identical atoms in all layers, (ACB) is identical to (ABC).
It can be shown that this is a closest packed structure because the three host atoms occupy 74% of the total hexagonal unit cell volume. Furthermore, the standard reduced cell in this array is as follows: choose the two central atoms in the top and bottom "A" layers, and connect them to the six atoms shown in the "B" and "C" layers . This unit cell is identical to the standard reduced cell chosen for the face centered cubic lattice. Thus, the (ABC) repeat structure is identical to the face centered cubic lattice (CCP = FCC), with the stacking direction along the body diagonal of the cubic unit cell.
In order for the (ABC) layered lattice to be closest packed, the interlayer spacing must be exactly equal to CPIS.r with c:a = 1.5.CPIS. Thus, the standard reduced cell is then the special rhombohedron found in the face centered cubic lattice.
If the c:a ratio differs from the closest packed value, then the standard reduced unit cell is still a rhombohedron (a = b = c and = = , but the cell edges need not be of length 2r and/or the inter edge angles need not be 60o. In these quasi-closest packed structures, either the hexagonal
(Z = 3) cell or the primitive (Z = 1) unit cell may be used to describe this lattice, which is called, by convention, rhombohedral.
Note that quasi-closest packed (AB) lattices are also possible if c:a differs from CPIS.
In closest and quasi-closest packings, the only stipulations are
• two adjacent layers of hexagonal closest packed planes must be different (A, B or C);
• the first and last named layers in the repeat unit must be different (because they are adjacent in the whole pattern);
• different letters arranged in the same pattern represent the same lattice.
Thus, while,there are eighteen possible permutations of three letters in four-layers, as illustrated in the cascade diagrams shown here, not all of these patterns are unique. For example, in each cascade one of the three permutations is actually a two-layer (HCP) pattern. Furthermore, many of the remaining twelve patterns are equivalent. For example, pattern (ABAC) is equivalent to pattern (ACAB):
= AC)(ABAC)(AB =
After all coincident patterns are eliminated (using, for example, a spreadsheet string-search function), there are aparently three unique four-layer closest packing patterns: (ABAC), (ABCB), and (ACBC). However, (ABCB) and (ACBC) represent the same lattice with different choices of the unit cell. Thus, there are only two unique four-layer packings shown here.
In the same way, it can be shown that of the 30 possible five-letter permutations, four are apparently unique: (ABABC), (ABACB), (ABCAC) and (ABCBC). However, by rechoosing axes and turning the stack end-for-end (reading the stack backwards), it is seen that (ABABC) and (ABCAC) are equivalent. Thus, only three five-layer closest packed patterns are unique: (ABABC), (ABACB), and (ABCBC).
Thursday, 15 January 2009
Crystalline nature of metals