Solid Edge Units

[Nicodemo Aidara]

Thestructure of solids can be described as if they werethree-dimensional analogs of a piece of wallpaper. Wallpaper hasa regular repeating design that extends from one edge to theother. Crystals have a similar repeating design, but in this casethe design extends in three dimensions from one edge of the solidto the other.

We can unambiguously describe a piece of wallpaper byspecifying the size, shape, and contents of the simplestrepeating unit in the design. We can describe a three-dimensionalcrystal by specifying the size, shape, and contents of thesimplest repeating unit and the way these repeating units stackto form the crystal.

The simplest repeating unit in a crystal is called a unitcell. Each unit cell is defined in terms of lattice pointsthepoints in space about which the particles are free to vibrate ina crystal.

These unit cells are important for two reasons. First, anumber of metals, ionic solids, and intermetallic compoundscrystallize in cubic unit cells. Second, it is relatively easy todo calculations with these unit cells because the cell-edgelengths are all the same and the cell angles are all 90.

The simple cubic unit cell is the simplest repeatingunit in a simple cubic structure. Each corner of the unit cell isdefined by a lattice point at which an atom, ion, or molecule canbe found in the crystal. By convention, the edge of a unit cellalways connects equivalent points. Each of the eight corners ofthe unit cell therefore must contain an identical particle. Otherparticles can be present on the edges or faces of the unit cell,or within the body of the unit cell. But the minimum that must bepresent for the unit cell to be classified as simple cubic iseight equivalent particles on the eight corners.

The body-centered cubic unit cell is the simplestrepeating unit in a body-centered cubic structure. Once again,there are eight identical particles on the eight corners of theunit cell. However, this time there is a ninth identical particlein the center of the body of the unit cell.

The face-centered cubic unit cell also starts withidentical particles on the eight corners of the cube. But thisstructure also contains the same particles in the centers of thesix faces of the unit cell, for a total of 14 identical latticepoints.

The face-centered cubic unit cell is the simplest repeatingunit in a cubic closest-packed structure. In fact, the presenceof face-centered cubic unit cells in this structure explains whythe structure is known as cubic closest-packed.

The lattice points in a cubic unit cell can be described interms of a three-dimensional graph. Because all three cell-edgelengths are the same in a cubic unit cell, it doesn't matter whatorientation is used for the a, b, and caxes. For the sake of argument, we'll define the a axis asthe vertical axis of our coordinate system, as shown in thefigure below.

Thinking about the unit cell as a three-dimensional graphallows us to describe the structure of a crystal with aremarkably small amount of information. We can specify thestructure of cesium chloride, for example, with only four piecesof information.

Because the cell edge must connect equivalent lattice points,the presence of a Cl- ion at one corner of the unitcell (0,0,0) implies the presence of a Cl- ion atevery corner of the cell. The coordinates 1/2,1/2,1/2 describe alattice point at the center of the cell. Because there is noother point in the unit cell that is one cell-edge length awayfrom these coordinates, this is the only Cs+ ion inthe cell. CsCl is therefore a simple cubic unit cell of Cl-ions with a Cs+ in the center of the body of the cell.

NaCl should crystallize in a cubic closest-packed array of Cl-ions with Na+ ions in the octahedral holes betweenplanes of Cl- ions. We can translate this informationinto a unit-cell model for NaCl by remembering that theface-centered cubic unit cell is the simplest repeating unit in acubic closest-packed structure.

The figure below shows that there is an octahedral hole in thecenter of a face-centered cubic unit cell, at the coordinates1/2,1/2,1/2. Any particle at this point touches the particles inthe centers of the six faces of the unit cell.

ZnS crystallizes as cubic closest-packed array of S2-ions with Zn2+ ions in tetrahedral holes. The S2-ions in this crystal occupy the same positions as the Cl-ions in NaCl. The only difference between these crystals is thelocation of the positive ions. The figure below shows that thetetrahedral holes in a face-centered cubic unit cell are in thecorners of the unit cell, at coordinates such as 1/4,1/4,1/4. Anatom with these coordinates would touch the atom at this corneras well as the atoms in the centers of the three faces that formthis corner. Although it is difficult to see without athree-dimensional model, the four atoms that surround this holeare arranged toward the corners of a tetrahedron.

Because the corners of a cubic unit cell are identical, theremust be a tetrahedral hole in each of the eight corners of theface-centered cubic unit cell. If S2- ions occupy thelattice points of a face-centered cubic unit cell and Zn2+ions are packed into every other tetrahedral hole, we get theunit cell of ZnS shown in the figure below.

Nickel is one of the metals that crystallize in a cubicclosest-packed structure. When you consider that a nickel atomhas a mass of only 9.75 x 10-23 g and an ionic radiusof only 1.24 x 10-10 m, it is a remarkable achievementto be able to describe the structure of this metal. The obviousquestion is: How do we know that nickel packs in a cubicclosest-packed structure?

In 1912, Max van Laue found that x-rays that struck thesurface of a crystal were diffracted into patterns that resembledthe patterns produced when light passes through a very narrowslit. Shortly thereafter, William Lawrence Bragg, who was justcompleting his undergraduate degree in physics at Cambridge,explained van Laue's resultswith an equation known as the Braggequation, which allows us to calculate the distance betweenplanes of atoms in a crystal from the pattern of diffraction ofx-rays of known wavelength.

The pattern by which x-rays are diffracted by nickel metalsuggests that this metal packs in a cubic unit cell with adistance between planes of atoms of 0.3524 nm. Thus, thecell-edge length in this crystal must be 0.3524 nm. Knowing thatnickel crystallizes in a cubic unit cell is not enough. We stillhave to decide whether it is a simple cubic, body-centered cubic,or face-centered cubic unit cell. This can be done by measuringthe density of the metal.

Atoms on the corners, edges, and faces of a unit cell areshared by more than one unit cell, as shown in the figure below.An atom on a face is shared by two unit cells, so only half ofthe atom belongs to each of these cells. An atom on an edge isshared by four unit cells, and an atom on a corner is shared byeight unit cells. Thus, only one-quarter of an atom on an edgeand one-eighth of an atom on a corner can be assigned to each ofthe unit cells that share these atoms.

If nickel crystallized in a simple cubic unit cell, therewould be a nickel atom on each of the eight corners of the cell.Because only one-eighth of these atoms can be assigned to a givenunit cell, each unit cell in a simple cubic structure would haveone net nickel atom.

Because they have different numbers of atoms in a unit cell,each of these structures would have a different density. Let'stherefore calculate the density for nickel based on each of thesestructures and the unit cell edge length for nickel given in theprevious section: 0.3524 nm. In order to do this, we need to knowthe volume of the unit cell in cubic centimeters and the mass ofa single nickel atom.

The Pythagorean theorem states that the diagonal across aright triangle is equal to the sum of the squares of the othersides. The diagonal across the face of the unit cell is thereforerelated to the unit-cell edge length by the following equation.

A similar approach can be taken to estimating the size of anion. Let's start by using the fact that the cell-edge length incesium chloride is 0.4123 nm to calculate the distance betweenthe centers of the Cs+ and Cl- ions inCsCl.

Before we can calculate the distance between the centers ofthe Cs+ and Cl- ions in this crystal,however, we have to recognize the validity of one of the simplestassumptions about ionic solids: The positive and negative ionsthat form these crystals touch.

If we had an estimate of the size of either the Cs+or Cl- ion, we could use the results to calculate theradius of the other ion. The ionic radius of the Cl-ion is 0.181 nm. Substituting this value into the last equationgives a value of 0.176 nm for the radius of the Cs+ion.

The results of this calculation are in reasonable agreementwith the value of 0.169 nm known for the radius of the Cs+ion. The discrepancy between these values reflects the fact thationic radii vary from one crystal to another. The tabulatedvalues are averages of the results of a number of calculations ofthis type.

I'm having a hard time getting my prints to be the correct dimensions. I model using Solid Edge and export as .STL. However, when i drop the stl file into repetier, it's tiny. I can always scale it up to about the size i want, but I don't know the dimensions that it starts out, so i cannot get a scale factor to bring it to the exact dimensions I want. This happens for everything I create using Solid Edge. I don't have any other CAD software to try, so I am unsure if this is specific to Solid Edge or if it's something else entirely, but every STL that i load from other sources shows up in repetier the correct size the first time.

Most modelling, particularly in the 3D world, is done in metric (it divides by 10, and has far less 0's for the same given accuracy). Both of you I'm guessing are displaying/viewing in CAD in Inches or part there of, and then exporting it? The STL export will be invariably in Metric.

Scale up your object in Slic3r by 25.4 , this will convert mm's back to inches.

If you experience the reverse (happens occasionally, particularly importing STLs into CAD), then scale down .0393 .

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