Stacking Spheres. The dimensions of a tetrahedron. The basic structure of a stack of spheres is a tetrahedron.

Transcription

1 Stacking Sphees The dimensions of a tetahedon The basic stuctue of a stack of sphees is a tetahedon. The dimensions of a tetahedon of unit side ae as follows: Now the volume of a pyamid is 1/6 th of the volume of the enclosing ectangula bo so the volume of a tetahedon is 1/6 1 3/ (/3) = 1/6 In a close-packed lattice, evey sphee is associated with one and only one tetahedon and the lattice consists of an infinite aay of these tetaheda placed vete to vete. If we conside one laye of tetaheda viewed fom above, it looks something like this:

2 The bases of thee of the tetaheda in the net laye ae also shown. Now thee of the tetaheda in the base laye fom the bottom thee cones of a lage tetahedon whose empty middle is, in fact, an octahedon. The whole of the base plane can theefoe be filled with an aay of homboheda (made up of two tetaheda and one octahedon) whose volume is equal to that of 6 tetaheda i.e. 1/. It follows fom this that the atio of space filled by a close-packed aay of sphees is equal to the volume of a unit sphee divided by the volume of this hombohedon. This comes to 0.74 Cubic and heagonal close-packed lattices In both the face-cented cubic and the heagonal close packed lattices, each sphee is suounded by 1 neighbouing sphees. If we conside the sphee whose cente is at the ape of the cental tetahedon in the diagam above, it is clea that 6 of these neighbous fom a heagon aound it. 3 othes lie below and 3 moe above. The only diffeence between the ccp and hcp lattices is in the elative oientation of the thee sphees above and below. If they ae vetically above one anothe, the lattice is hcp. If they ae diametically opposite one anothe, the lattice is ccp. In the diagam above, if the shaded tiangles ae the bases of the tetaheda on the net laye, each vete will lie vetically above a vete in the base laye and hence will geneate a hcp lattice. If, on the othe hand, the unshaded tiangles ae used (like the wie-fame tetahedon) a ccp lattice will esult. If the sphees have a unit adius, the coodinates of the 6 hoizontal junctions ae: (1, 0, 0) (1/, 3/, 0) (-1/, 3/, 0) (-1, 0, 0) (-1/, - 3/, 0) (1/, - 3/, 1/) Refeing back to the diagam showing the dimensions of a tetahedon, the coodinates of the 3 junctions below ae: (1/, -1/ 3, - (/3)) (0, 1/ 3, - (/3)) (-1/, -1/ 3, - (/3)) In the hcp lattice, the thee in the laye above ae vetically above those in the base laye so thei coodinates ae: (1/, -1/ 3, (/3)) (0, 1/ 3, (/3)) (-1/, -1/ 3, (/3)) In the ccp lattice, they ae diametically opposed, hence: (-1/, 1/ 3, (/3)) (0, -1/ 3, (/3)) (1/, 1/ 3, (/3))

3 The Wigne-Seitz cell The question now aises, what is the eact shape of the solid that suounds each sphee in the two diffeent aangements? (This solid is called a Wigne-Seitz cell and will, of couse, fill space completely. It also shaes the same symmety as that of the lattice which geneates it.) This solid may be defined as follows: at the junction between the taget sphee and each of its 1 neighbous, constuct a plane paallel to the suface of the sphee. The solid we ae afte is enclosed by these 1 planes. Suppose a plane intesects the thee aes at the points a, b and c. It is easy to see that it is epesented by the following equation: a y b z c = 1 (1) We need to find the equation of the plane which passes though a point (p, q, ) and which is nomal to the line joining this point to the oigin. This situation is shown below. fom which we see that d = d c hence c = d Applying the same logic to the othe two aes we obtain: so the equation we equie is; a = d p, b = d q, c = d whee d = p q The equations fo the 6 vetical planes ae theefoe: = 1 = 1 p qy z = d () 3 y 3 y = 1 3 y = 1 and 3 y Fist of all, it is clea that the 6 planes defined by the 6 hoizontal junctions will intesect in 6 vetical lines whose equations ae: = 1 = 1

4 = 1, y = 1 / 3 = 0, y = / 3 = 1, y = 1/ 3 = 1, y = 1/ 3 = 0, y = / 3 = 1, y = 1/ 3 The equations fo the 3 planes below the oigin ae: (A) (B) (C) (D) (E) y 3 z 3 = 1 (I) (F) y 3 z 3 = 1 (J) y 3 z 3 = 1 (K) By symmety, we know that these thee planes will intesect the Z ais at the same point and it is easy to see that this point will be (0, 0, - (3/)) In ode to wok out whee the emaining vetices ae, we must see whee the vetical lines listed above intesect with the thee planes. The solutions ae as follows A is intesects I at (1, 1/ 3, - (/3)) A is intesects J at (1, 1/ 3, - (/3)) A is intesects K at (1, 1/ 3, -5/ 6) We can obviously eject the thid solution as being outside the solid leaving just the single point (1, 1/ 3, - (/3)) (line A) A simila calculation eveals that the othe five points below the oigin ae: (0, / 3, -1/ 6) (line B) (-1, 1/ 3, - (/3)) (line C) (-1, -1/ 3, -1/ 6) (line D) (0, -/ 3, - (/3)) (line E) (1, -1/ 3, -1/ 6) (line F) Fo a hcp lattice, the 6 vetices above the oigin will be the same with a positive Z coodinate; and fo a ccp lattice, the two diffeent Z coodinates will be intechanged. hcp ccp

5 Both solids have 1 quadilateal faces but in the case of the Wigne-Seitz cell fo the cubic closepacked lattice, all the faces ae identical hombuses with the smalle angle being equal to This solid is a hombic dodecahedon. It is immediately obvious that the ccp lattice has a much highe degee of symmety that the hcp lattice. Both have 14 vetices of which 6 ae of ode 4 and 8 of ode 3 but in the ccp case, the 6 ode 4 vetices ae mutually at ight angles. Stacking Sphees If you stack sphees on a tiangula base, you will build up a tetahedal stuctue with evey thid laye being diectly ove the one beneath. This is obviously a ccp lattice. What is not so clea is that if you stat with a squae base and build up a ectangula pyamid, you still end up with a ccp lattice. The diagam below shows a cube with the mid-points of the faces maked. Hidden inside the cube is a tetahedon. Now populate the cones of the cube and the cente points of the faces with atoms and you will see that the two coloued layes and the neaest cone fom the ABC sequence of tiangula lattice planes which ae chaacteistic of the ccp lattice. It is fo this eason that the cubic close-packed lattice is moe commonly called the face-cented cubic lattice. Now since the sphees in the tiangula planes ae in contact, it is clea that the diagonal of the cube has length sphee diametes. The size of the unit cube fo the face-cented lattice is theefoe diametes in length.