1 Cubic Close-Packed (Fm3 – m)
The noble metals (Cu, Ag, Au), the metals of higher valence (Al and Pb), the later transition metals (Ni, Rh, Pd, Ir, Pt) and the inert gases (Ne, Ar, Kr, Xe) when in the solid state all possess this structure (see Appendix 7), as does Fe between 912 and 1394 °C and Co above 420 °C. The lattice is face-centred cubic and there is one atom associated with each lattice point. A con ventional unit cell of the structure with lattice parameter a is shown in Figure 3.1a. The conventional unit cell has four lattice points, and therefore four atoms, per unit cell. The relationship of this unit cell to the primitive rhombohedral unit cell for this crystal structure is shown in Figure 3.1b.
The coordinates of the atoms in the conventional unit cell are therefore (0, 0, 0), ( ) 1 1, , 2 2 0 , ( ) 1 1 , , 2 2 0 and ( ) 1 1 , , 2 2 0 . Each atom possesses 12 nearest neighbours at a distance of a/ 2 . The coordination number is thus 12. This structure is the one obtained if equal spheres are placed in contact at the lattice points of a face-centred cubic lattice. A diagram illustrating this is shown in Figure 3.2a. An alternative packing arrangement of equal spheres which produces the hexagonal close-packed structure is shown in Figure 3.2b (Section 3.2.2). If the radius of the spheres is R then R a = / (2 2). The proportion of space filled by the spheres in the c.c.p. crystal structure is π / (3 2) or 74%. This is described by saying that the packing fraction is 0.74. A packing fraction of 0.74 is the closest packing of equal spheres that can be achieved [2–8], and is the closest packing that can be achieved with equal spheres at the lattice points of a Bravais lattice (see Section 3.6).
Rows of spheres are in contact along the line joining their centres. These lines are 〈1 1 0〉 directions in the lattice. Such directions are the closest packed and are termed close-packed directions. There are 12 such directions in all, if account is taken of change in sign. Taking the atomic centres to be at the lattice points, it is apparent from Figure 1.27 that in the {1 1 1} planes, which lie normal to triad axes, the atom centres form a triequiangular net of points. If the atoms are spheres of radius a/2 2 , the appearance of a {1 1 1} plane is shown by the full circles in Figure 3.3. Each sphere is in contact with six equidistant spheres with centres in the plane. Since the arrangement shown in Figure 3.3 is the closest packing of circles in a plane, the {111} planes are spoken of as being closest packed, or close-packed. There are eight such planes in the lattice if distinction is drawn between parallel normals of opposite sense (so that, for example, (11– 1) and (1– 11– ) are counted separately) and each contains six close-packed directions. The spacing of the {111} planes is a R / 3 2 2/3 = and so is equal to 2/3 of the atomic spacing in the {111} planes.
The centres of the atoms in a particular (111) plane occupy points such as A in Figure 3.3. If the positions of the centres of the atoms in adjacent (111) planes are projected on to this (111) plane, they occupy positions such as those marked B or C in Figure 3.3. The projections of the spheres centred on points B are shown dotted in the figure. A given crystal can be considered to be made up by stacking, one above the other, successive planar rafts of closest-packed spheres so that proceeding in the [111] direction the centres of the atoms in adjacent rafts follow the sequence ABCABCABC… (Figure 3.4a). The same crystal structure, but in a different orientation, would be described by the sequence of rafts ACBACBACB… Any atom in a c.c.p. crystal has 12 nearest neighbours along < 110 > directions at a distance of a/ 2 (= 2R), 6 second-nearest along < 100 > directions at a ( 22) = R , 24 third-nearest along < 211 > directions at 3/2 ( 2 3 ) a R = and 12 fourth-nearest along < 110 > directions at 2( 2 4 ) a R = .
When the c.c.p. crystal structure is regarded as being made up of spheres in contact, the size of the interstices between spheres is important, because many other crystal structures contain at least one set of atoms in either a c.c.p. arrangement or a very close approximation to it. The largest interstice occurs at positions in the unit cell with coordinates ( ) 111 , , 222 and equivalent positions ( ) 1 11 2 22 i.e. 0, , 0; 0, 0, and 0, 0 . There are four of these interstice s per unit cell and, hence, one per lattice point; one is illustrated in Figure 3.5. The largest sphere which can be placed in this position without disturbing the arrangement of spheres at the lattice points has radius r RR = −= ( 2 1) 0.414 . This sphere would have octahedral coordination with six nearest neighbours (Figure 3.5); for this reason these interstices are commonly referred to as octahedral interstices. The sites of these octahedral interstices by themselves form a face-centred cubic lattice.
The second largest interstice occurs at points with coordinates ( ) 111 , , 444 and equivalent positions (Figure 3.6). The largest sphere which can be placed here has radius r RR = −= ( 3/2 1) 0.225 and possesses tetrahedral coordination; for this reason these interstices are commonly referred to as tetrahedral interstices. There are eight such points in the unit cell and hence two per lattice point. If the atoms in the c.c.p. crystal structure are ignored, the sites of the tetrahedral interstices can be seen to lie on a primitive cubic lattice.
This structure is exhibited by the early transition metals Sc, Ti, Y and Zr, by the divalent metals Be, Mg, Zn and Cd, and by a number of the rare earth metals (see Appendix 7). The hexagonal primitive unit cell contains two atoms with coordinates (0, 0, 0) and ( ) 211 , , 332 (Figure 3.7). Hence, there are two atoms associated with each lattice point. There is no pure rotational hexagonal axis, but instead 63 axes located at 00z; that is, at the origin of the unit cell (and equivalent sites) running parallel to the z-axis. This structure can be produced by packing together equal spheres, as shown in Figure 3.2b. If each sphere has 12 nearest neighbours equally far away from it, the axial ratio c/a must equal 8/3 1.633 = . The packing fraction is then 0.74, as in the c.c.p. structure.
Octahedral interstices in the h.c.p. crystal structure have coordinates ( ) 121 , , 334 and ( ) 1 2 3 , , 334 (Figure 3.8a). There are two such interstices per unit cell and hence one per atom. If c/a = 8/3 then the largest sphere that can be inserted without disturbing the spheres of radius R packed in contact has radius r = 0.414R, just as in the c.c.p. crystal structure. The second-largest interstices, the tetrahedral interstices, lie at ( ) 3 8 0, 0, , ( ) 5 8 0, 0, , ( ) 211 , , 338 and ( ) 2 1 7 , , 338 (Figure 3.8b). There are four such interstices per cell and hence two per atom. If c/a = 8/3 these have r/R = 0.225, again, just as in the c.c.p. crystal structure.
3 Double Hexagonal Close-Packed (P63 /mmc)
In the eight rare earth metals La, Pr, Nd, Pm, Am, Cm, Bk and Cf, the hexagonal close packing is a four-layer repeat, ABAC. This crystal structure is often referred to as the double h.c.p. structure. The axial ratios characteristic of these metals vary between 3.192 for Pm and 3.262 for Cf, so that in each case they are just below the ideal value of 2 8/3 = 3.266. There are four atoms per unit cell, one octahedral interstice per atom and two tetrahedral interstices per atom.
4 Body-Centred Cubic (Im3 – m)
This structure is shown by the alkali metals Li, Na, K, Rb and Cs, and by the transition metals V, Cr, Fe, Nb, Mo, Ta and W. It is also shown by Ti and Zr at high temperature. The lattice is b.c.c. with one atom at each lattice point, so the atomic coordinates are (0, 0, 0) and ( ) 111 , , 222 (Figure 3.9). Each atom has eight nearest neighbours along 〈111〉 directions at a separation of 3 / a 2, where a is the lattice parameter. These eight 〈111〉 directions are close-packed directions.
Unlike in the c.c.p. and h.c.p. structures, none of the nearest neighbours of one particular atom are nearest neighbours of each other. If the b.c.c structure is presumed to be made up with spheres of equal size, these each have radius R = 3 / a 4. The packing fraction is π 3/8 = 0.68. The second-nearest neighbours of an atom are closer than in the c.c.p. structure. There are six second-nearest neighbours along <100> directions at a distance 4/3 R = 2.309R (in c.c.p., second-nearest neighbours are at 2 2R = 2.828R) and 12 third-nearest along <110> directions at 42 / R 3. There are no closest-packed planes of atoms in this structure.
This structure has its largest interstices at coordinates ( ) 1 1, 2 4 , 0 and equivalent positions (Figure 3.10). There are 12 such positions per cell: six per lattice point, and therefore six per atom. The largest sphere fitting in such an interstice has radius r = ( 5/3 1 − ) R = 0.291R. This interstice is therefore significantly smaller than the largest one in c.c.p. It has four nearest neighbours equidistant from it, and so it is a tetrahedral interstice, but the tetrahedron that these neighbours form is not regular. The second-largest interstice is at ( ) 1 1, 2 2 , 0 and equivalent positions (Figure 3.10). There are six such sites per cell ( 1 2 6 × at the centres of cube faces and 1 4 12 × at the midpoints of cube edges) and so three per lattice point. Each can accommodate a sphere of radius r = (2/ 3 1) − R = 0.155R; such a sphere is at the centre of a distorted octahedron, and so these interstices are distorted octahedral interstices.
Reference: https://crystal-algerien1970.blogspot.com/2020/03/crystallography-and-crystal-defects.html
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